Eur. Phys. J. C (2008) 57: 13–182DOI 10.1140/epjc/s10052-008-0715-2

Review

Flavor physics of leptons and dipole moments*

M. Raidal1,b, A. van der Schaaf2,b, I. Bigi3,b, M.L. Mangano4,a,b, Y. Semertzidis5,b, S. Abel6, S. Albino7, S. Antusch8,E. Arganda9, B. Bajc10, S. Banerjee11, C. Biggio8, M. Blanke8,12, W. Bonivento13, G.C. Branco14,4, D. Bryman15,A.J. Buras12, L. Calibbi16,17,18, A. Ceccucci4, P.H. Chankowski19, S. Davidson20, A. Deandrea20, D.P. DeMille21,F. Deppisch22, M.A. Diaz23, B. Duling12, M. Felcini4, W. Fetscher24, F. Forti25, D.K. Ghosh26, M. Giffels27,M.A. Giorgi25, G. Giudice4, E. Goudzovskij28, T. Han29, P.G. Harris30, M.J. Herrero9, J. Hisano31, R.J. Holt32,K. Huitu33, A. Ibarra34, O. Igonkina35,36, A. Ilakovac37, J. Imazato38, G. Isidori28,39, F.R. Joaquim9, M. Kadastik1,Y. Kajiyama1, S.F. King40, K. Kirch41, M.G. Kozlov42, M. Krawczyk19,4, T. Kress27, O. Lebedev4, A. Lusiani25,E. Ma43, G. Marchiori25, A. Masiero18, I. Masina4, G. Moreau44, T. Mori45, M. Muntel1, N. Neri25, F. Nesti46,C.J.G. Onderwater47, P. Paradisi48, S.T. Petcov16,49, M. Picariello50, V. Porretti17, A. Poschenrieder12,M. Pospelov51, L. Rebane1, M.N. Rebelo14,4, A. Ritz51, L. Roberts52, A. Romanino16, J.M. Roney11, A. Rossi18,R. Rückl53, G. Senjanovic54, N. Serra13, T. Shindou34, Y. Takanishi16, C. Tarantino12, A.M. Teixeira44,E. Torrente-Lujan55, K.J. Turzynski56,19, T.E.J. Underwood6, S.K. Vempati57, O. Vives17

1National Institute for Chemical Physics and Biophysics, 10143 Tallinn, Estonia2Physik-Institut der Universität Zürich, 8057 Zürich, Switzerland3Physics Department, University of Notre Dame du Lac, Notre Dame, IN 46556, USA4Physics Department, CERN, 1211 Geneva, Switzerland5Brookhaven National Laboratory, Upton, NY 11973-5000, USA6Institute for Particle Physics Phenomenology, Durham University, Durham DH1 3LE, UK7II. Institute for Theoretical Physics, University of Hamburg, 22761 Hamburg, Germany8Max-Planck-Institut für Physik, 80805 München, Germany9Departamento de Fisica Teorica and IFT/CSIC-UAM, Universidad Autonoma de Madrid, 28049 Madrid, Spain

10J. Stefan Institute, 1000 Ljubljana, Slovenia11Department of Physics, University of Victoria, Victoria, BC V8W 3P6, Canada12Physics Department, TU Munich, 85748 Garching, Germany13Università degli Studi di Cagliari and INFN Cagliari, 09042 Monserrato (CA), Italy14Departamento de Física and Centro de Física Teórica de Partículas (CFTP), Instituto Superior Técnico (IST), 1049-001 Lisboa, Portugal15Department of Physics and Astronomy, University of British Columbia, TRIUMF, Vancouver, BC, V6T 2A3, Canada16SISSA and INFN, Sezione di Trieste, 34013 Trieste, Italy17Departament de Física Teòrica, Universitat de València-CSIC, 46100 Burjassot, Spain18Dipartimento di Fisica ‘G. Galilei’ and INFN, 35131 Padova, Italy19Institute of Theoretical Physics, University of Warsaw, 00-681 Warsaw, Poland20IPNL, CNRS, Université Lyon-1, 69622 Villeurbanne Cedex, France21Physics Department, Yale University, New Haven, CO 06520, USA22School of Physics and Astronomy, University of Manchester, Manchester M13 9PL, UK23Facultad de Fisica, Pontificia Universita Catolica de Chile, Santiago 22, Chile24Department of Physics, ETH Honggerberg, 8093 Zurich, Switzerland25INFN and Dipartimento di Fisica, Universita di Pisa, 56127 Pisa, Italy26Theoretical Physics Division, Physical Research Lab., Navrangpura, Ahmedabad 380 009, India27III. Physikalisches Institut B, RWTH Aachen, 52056 Aachen, Germany28Scuola Normale Superiore, 56100 Pisa, Italy29Department of Physics, High Energy Physics, University of Wisconsin, Madison, WI 53706, USA30Department of Physics and Astronomy, University of Sussex, Falmer, Brighton BN1 9QH, UK31ICRR, University of Tokyo, Tokyo, Japan32Physics Division, Argonne National Laboratory, Argonne, IL 60439-4843, USA33Department of Physics, University of Helsinki, and Helsinki Institute of Physics, 00014 Helsinki, Finland34Theory Group, DESY, 22603 Hamburg, Germany35Physics Department, University of Oregon, Eugene, OR, USA36NIKHEF, 1098 SJ Amsterdam, The Netherlands37Department of Physics, University of Zagreb, 10002 Zagreb, Croatia38IPNS, KEK, Ibaraki 305-0801, Japan39INFN, Laboratori Nazionali di Frascati, 00044 Frascati, Italy40School of Physics and Astronomy, University of Southampton, SO17 1BJ Southampton, UK41Paul Scherrer Institut, 5232 Villigen, Switzerland42Petersburg Nuclear Physics Institute, Gatchina 188300, Russia43Department of Physics and Astronomy, University of California, Riverside, CA 92521, USA

14 Eur. Phys. J. C (2008) 57: 13–182

44Laboratoire de Physique Théorique, UMR 8627 Université de Paris-Sud XI, 91405 Orsay Cedex, France45ICEPP, University of Tokyo, Tokyo 113-0033, Japan46Università dell’Aquila and INFN LNGS, 67010 L’Aquila, Italy47Kernfysisch Versneller Instituut (KVI), 9747 AA Groningen, The Netherlands48University of Rome “Tor Vergata” and INFN Sezione Roma II, 00133 Roma, Italy49Institute of Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, 1784 Sofia, Bulgaria50INFN and Dipartimento di Fisica, Università del Salento, 73100 Lecce, Italy51Department of Physics and Astronomy, University of Victoria, Victoria, BC V8P 5C2, Canada52Department of Physics, Boston University, Boston, MA 02215, USA53Institute for Theoretical Physics and Astrophysics, University of Würzburg, 97074 Würzburg, Germany54International Centre for Theoretical Physics, Trieste, Italy55Department of Physics, University of Murcia, 30100 Murcia, Spain56Physics Department, University of Michigan, Ann Arbor, MI 48109, USA57Centre for High Energy Physics, Indian Institute of Science, Bangalore 560012, India

Received: 19 February 2008 / Revised: 12 August 2008 / Published online: 12 November 2008© Springer-Verlag / Società Italiana di Fisica 2008

Abstract This chapter of the report of the “Flavor in the eraof the LHC” Workshop discusses the theoretical, phenom-enological and experimental issues related to flavor phenom-ena in the charged lepton sector and in flavor conserving CP-violating processes. We review the current experimental lim-its and the main theoretical models for the flavor structureof fundamental particles. We analyze the phenomenologi-cal consequences of the available data, setting constraintson explicit models beyond the standard model, presentingbenchmarks for the discovery potential of forthcoming mea-surements both at the LHC and at low energy, and exploringoptions for possible future experiments.

Contents

1 Charged leptons and fundamental dipole moments:alternative probes of the origin of flavor and CPviolation . . . . . . . . . . . . . . . . . . . . . . . 15

2 Theoretical framework and flavor symmetries . . . 182.1 The flavor puzzle . . . . . . . . . . . . . . . . 182.2 Flavor symmetries . . . . . . . . . . . . . . . 19

3 Observables and their parameterization . . . . . . 293.1 Effective operators and low scale observables 293.2 Phenomenological parameterizations of quark

and lepton Yukawa couplings . . . . . . . . . 353.3 Leptogenesis and cosmological observables . 44

4 Organizing principles for flavor physics . . . . . . 464.1 Grand unified theories . . . . . . . . . . . . . 464.2 Higher dimensional approaches . . . . . . . . 504.3 Minimal flavor violation in the lepton sector . 53

*Report of Working Group 3 of the CERN Workshop “Flavor in theera of the LHC”, Geneva, Switzerland, November 2005–March 2007.a e-mail: michelangelo.mangano@cern.chbConvenors

5 Phenomenology of theories beyond the standardmodel . . . . . . . . . . . . . . . . . . . . . . . . . 575.1 Flavor violation in non-SUSY models

directly testable at LHC . . . . . . . . . . . . 575.2 Flavor and CP violation in SUSY extensions

of the SM . . . . . . . . . . . . . . . . . . . . 665.3 SUSY GUTs . . . . . . . . . . . . . . . . . . 865.4 R-parity violation . . . . . . . . . . . . . . . . 975.5 Higgs-mediated lepton flavor violation

in supersymmetry . . . . . . . . . . . . . . . . 1015.6 Tests of unitarity and universality

in the lepton sector . . . . . . . . . . . . . . . 1095.7 EDMs from RGE effects in theories

with low energy supersymmetry . . . . . . . . 1166 Experimental tests of charged lepton universality . 122

6.1 π decay . . . . . . . . . . . . . . . . . . . . . 1246.2 K decay . . . . . . . . . . . . . . . . . . . . 1256.3 τ decay . . . . . . . . . . . . . . . . . . . . . 127

7 CP violation with charged leptons . . . . . . . . . 1287.1 μ decays . . . . . . . . . . . . . . . . . . . . 1297.2 CP violation in τ decays . . . . . . . . . . . . 1307.3 Search for T violation in K+ → π0μ+ν decay 1317.4 Measurement of CP violation

in ortho-positronium decay . . . . . . . . . . 1338 LFV experiments . . . . . . . . . . . . . . . . . . 135

8.1 Rare μ decays . . . . . . . . . . . . . . . . . . 1358.2 Searches for lepton flavor violation in τ decays 1418.3 B0

d,s → e±μ∓ . . . . . . . . . . . . . . . . . . 1468.4 In flight conversions . . . . . . . . . . . . . . 147

9 Experimental studies of electric and magneticdipole moments . . . . . . . . . . . . . . . . . . . 1489.1 Electric dipole moments . . . . . . . . . . . . 1489.2 Neutron EDM . . . . . . . . . . . . . . . . . . 1519.3 Deuteron EDM . . . . . . . . . . . . . . . . . 1539.4 EDM of deformed nuclei: 225Ra . . . . . . . . 1569.5 Electrons bound in atoms and molecules . . . 1589.6 Muon EDM . . . . . . . . . . . . . . . . . . . 164

Eur. Phys. J. C (2008) 57: 13–182 15

9.7 Muon g − 2 . . . . . . . . . . . . . . . . . . . 165Acknowledgements . . . . . . . . . . . . . . . . . . 167References . . . . . . . . . . . . . . . . . . . . . . . 167

1 Charged leptons and fundamental dipole moments:alternative probes of the origin of flavor and CPviolation

The understanding of the flavor structure and CP violation(CPV) of fundamental interactions has so far been domi-nated by the phenomenology of the quark sector of the stan-dard model (SM). More recently, the observation of neutrinomasses and mixing has begun extending this phenomenol-ogy to the lepton sector. While no experimental data avail-able today link flavor and CP violation in the quark andin the neutrino sectors, theoretical prejudice strongly sup-ports the expectation that a complete understanding shouldultimately expose their common origin. Most attempts toidentify the common origin, whether through grand unified(GUT) scenarios, supersymmetry (SUSY), or more exoticelectroweak symmetry breaking mechanisms predict in ad-dition testable correlations between the flavor and CP vi-olation observables in the quark and neutrino sector on theone side, and new phenomena involving charged leptons andflavor conserving CP-odd effects on the other. This chapterof the “Flavor in the era of the LHC” report focuses pre-cisely on the phenomenology arising from these ideas, dis-cussing flavor phenomena in the charged lepton sector andflavor conserving CP-violating processes.

Several theoretical arguments make the studies discussedin this chapter particularly interesting.

– The charged lepton sector provides unique opportunitiesto test scenarios tailored to explain flavor in the quark andneutrino sectors, for example by testing correlations be-tween neutrino mixing and the rate for μ→ eγ decays, aspredicted by specific SUSY/GUT scenarios. Charged lep-tons are therefore an indispensable element of the flavorpuzzle, without which its clarification could be impossi-ble.

– The only observed source of CP violation is so far theCabibbo–Kobayashi–Maskawa (CKM) mixing matrix.On the other hand, it is by now well established that this isnot enough to explain the observed baryon asymmetry ofthe universe (BAU). The existence of other sources of CPviolation is therefore required. CP-odd phases in neutrinomixing, directly generating the BAU through leptogene-sis, are a possibility, directly affecting the charged leptonsector via, e.g., the appearance of electric dipole moments(EDMs). Likewise, EDMs could arise via CP violation inflavor conserving couplings, like phases of the gaugino

fields or in extended Higgs sectors. In all cases, the ob-servables discussed in this chapter provide essential ex-perimental input for the understanding of the origin of CPviolation.

– The excellent agreement of all flavor observables in thequark sector with the CKM picture of flavor and CP vio-lation has recently led to the concept of minimal flavor vi-olation (MFV). In scenarios beyond the SM (BSM) withMFV, the smallness of possible deviations from the SMis naturally built into the theory. While these schemesprovide a natural setting for the observed lack of newphysics (NP) signals, their consequence is often a re-duced sensitivity to the underlying flavor dynamics ofmost observables accessible by the next generation offlavor experiments. Lepton flavor violation (LFV) andEDMs could therefore provide our only probe into thisdynamics.

– Last but not least, with the exception of the magnetic di-pole moments, where the SM predicts non-zero valuesand deviations due to new physics compete with the ef-fect of higher order SM corrections, the observation ofa non-zero value for any of the observables discussedin this chapter would be unequivocal indication of newphysics. In fact, while neutrino masses and mixing canmediate lepton flavor violating transitions, as well as in-duce CP-odd effects, their size is such that all these effectsare by many orders of magnitude smaller than anythingmeasurable in the foreseeable future. This implies that,contrary to many of the observables considered in otherchapters of this report, and although the signal interpre-tation may be plagued by theoretical ambiguities or sys-tematics, there is nevertheless no theoretical systematicuncertainty to claim a discovery once a positive signal isdetected.

The observables discussed here are also very interestingfrom the experimental point of view. They call for a verybroad approach, based not only on the most visible tools ofhigh energy physics, namely the high energy colliders, butalso on a large set of smaller-scale experiments that drawfrom a wide variety of techniques. The emphasis of theseexperiments is by and large on high rates and high preci-sion, a crucial role being played by the control of very largebackgrounds and subtle systematics. A new generation ofsuch experiments is ready to start or will start during the firstpart of the LHC operations. More experiments have been onthe drawing board for some time, and could become realityduring the LHC era if the necessary resources were madeavailable. The synergy between the techniques and potentialresults provided by both the large- and small-scale exper-iments makes this field of research very rich and excitingand gives it a strong potential to play a key role in exploringthe physics landscape in the era of the LHC.

16 Eur. Phys. J. C (2008) 57: 13–182

The purpose of this document is to provide a compre-hensive overview of the field, from both the theoretical andthe experimental perspective. While we cover many modelbuilding aspects of neutrino physics that are directly relatedto the phenomenology of the quark and charged lepton sec-tors, for the status of the determinations of the mixing pa-rameters and for the review of the future prospects we referthe reader to the vast existing literature, as documented forexample in [1–4].

Several of the results presented are already well known,but they are nevertheless documented here to provide a self-contained review, accessible to physicists whose expertisecovers only some of the many diverse aspects of this sub-ject. Many results emerged during the workshop, includingideas on possible new experiments, further enrich this re-port. We present here a short outline and some highlights ofthe contents.

Section 2 provides the general theoretical framework thatallows us to discuss flavor from a symmetry point of view.It outlines the origin of the flavor puzzles and lists the math-ematical settings that have been advocated to justify or pre-dict the hierarchies of the mixing angles in both the quarkand neutrino sectors. Section 3 introduces the observablesthat are sensitive to flavor in the charged lepton sector andto flavor conserving CP violation, providing a unified de-scription in terms of effective operators and effective scalesfor the new physics that should be responsible for them. Theexisting data already provide rather stringent limits on thesize of these operators, as shown in several tables. We col-lect here in Table 1 some of the most significant benchmarkresults (for details, we refer to the discussion in Sect. 3.1.2).We constrain the dimensionless coefficients εi of effectiveoperators Oi describing flavor or CP-violating interactions.Examples of these effective operators include

�iσμνγ5�iF

emμν , �iσ

μν�jFemμν , (1.1)

which describe a CP-violating electric dipole moment(EDM) of lepton �i or the flavor violating decay �i → �jγ ,or the four-fermion operators:

�iΓa�j qkΓaql, �iΓ

a�j �kΓa�l, (1.2)

where the Γa represent the various possible Lorentz struc-tures. The overall normalization of the operators is cho-sen to reproduce the strength of transitions mediated byweak gauge bosons, assuming flavor mixing angles and CP-violating phases of order unity. The smallness of the con-straints on ε therefore reflects either the large mass scale offlavor phenomena, or the weakness of the relative interac-tions.

It is clear from this table that current data are already sen-sitive to mass scales much larger than the electroweak scale,or to very small couplings. On the other hand, many of these

constraints leave room for interesting signals coupled to thenew physics at the TeV scale that can be directly discov-ered at the LHC. For example, a mixing of order 1 betweenthe supersymmetric scalar partners of the charged leptonsand a mass splitting among them of the order of the leptonmasses is consistent with the current limits if the scalar lep-ton masses are just above 100 GeV, and it could lead both totheir discovery at the LHC, and to observable signals at thenext generation of �→ �′γ experiments.

Most of this report will be devoted to the discussion ofthe phenomenological consequences of limits such as thosein Table 1, setting constraints on explicit BSM models, pre-senting benchmarks for the discovery potential of forthcom-ing measurements both at the LHC and at low energy, andexploring options for future experiments aimed at increasingthe reach even further.

Section 3 also introduces the phenomenological parame-terizations of the quark and lepton mixing matrices that arefound in the literature, emphasizing with concrete exam-ples the correlations among the neutrino and charged lep-ton sectors that arise in various proposed models of neutrinomasses. The section is completed by a discussion of the pos-sible role played by leptogenesis and cosmological observ-ables in constraining the neutrino sector.

Section 4 reviews the organizing principles for flavorphysics. With a favorite dynamical theory of flavor stillmissing, the extended symmetries of BSM theories can pro-vide some insight in the nature of the flavor structures ofquarks and leptons, and give phenomenologically relevantconstraints on low energy correlations between them. InGUT theories, for example, leptons and quarks belong tothe same irreducible representations of the gauge group,and their mass matrices and mixing angles are consequentlytightly related. Extra dimensional theories provide a possibledynamical origin for flavor, linking flavor to the geometry ofthe extra dimensions. This section also discusses the impli-cations of models adopting for the lepton sector the sameconcept of MFV already explored in the case of quarks.

Section 5 discusses at length the phenomenological con-sequences of the many existing models, and represents themain body of this document. We cover models based on

Table 1 Bounds on CP- or flavor violating effective operators, ex-pressed as upper limits on their dimensionless coefficients ε, scaledto the strength of weak interactions. For more details, in particularthe overall normalization convention for the effective operators, seeSect. 3.1.2

Observable Operator Limit on ε

eEDM eLσμνγ5eRFμν ≤2.1 × 10−12

B(μ→ eγ ) μσμνeFμν ≤3.4 × 10−12

B(τ→ μγ ) τσμνμFμν ≤8.4 × 10−8

B(K0L→ μ±e∓) (μγ μPLe)(sγ

μPLd) ≤2.9 × 10−7

Eur. Phys. J. C (2008) 57: 13–182 17

SUSY, as well as on alternative descriptions of electroweaksymmetry breaking, such as little Higgs or extended Higgssectors. In this section we discuss the predictions and the de-tection prospects of standard observables, such as �→ �′γdecays or EDMs, and connect the discovery potential forthese observables with the prospects for direct detection ofthe new massive particles at the LHC or at a future LinearCollider.

This section underlines, as is well known that the ex-ploration of these processes has great discovery potential,since most BSM models anticipate rates that are within thereach of the forthcoming experiments. From the point ofview of the synergy with collider physics, the remarkableoutcome of these studies is that the sensitivities reachedin the searches for rare lepton decays and dipole momentsare often quite similar to those reached in direct searchesat high energy. We give here some explicit examples. InSO(10) SUSY GUT models, where the charged lepton mix-ing is induced via renormalization-group evolution of theheavy neutrinos of different generations, the observation ofB(μ→ eγ ) at the level of 10−13, within the range of thejust-starting MEG experiment, is suggestive of the existenceof squarks and gluinos with a mass of about 1 TeV, wellwithin the discovery reach of the LHC. Squarks and gluinosin the range of 2–2.5 TeV, at the limit of detectability for theLHC, would push B(μ→ eγ ) down to the level of 10−16.While this is well beyond the MEG sensitivity, it wouldwell fit the ambitious goals of the next-generation μ→ e

conversion experiments, strongly endorsing their plans. Thedecay μ→ eγ induced by the mixing of the scalar part-ners of muon and electron, and with a B(μ→ eγ ) at thelevel of 10−13, could give a χ0

2 → χ01μ

±e∓ signal at theLHC, with up to 100 events after 300 fb−1. Higher statisticsand a cleaner signal would arise at a Linear Collider. Mod-els where neutrino masses arise not from a see-saw mech-anism at the GUT scale but from triplet Higgs fields at theTeV scale can be tested at the LHC, where processes likepp→H++H−− can be detected for mH++ up to 700 GeV,using the remarkable signatures due to B(H++ → τ+τ+)≈B(H++ → μ+μ+)≈ B(H++ → μ+τ+)≈ 1/3.

Should signals of new physics be observed, alternativeinterpretations can be tested by exploiting different pat-terns of correlations that they predict among the various ob-servables. For example, while typical SUSY scenarios pre-dict B(μ→ 3e)∼ 10−2B(μ→ eγ ), these branching ratiosare of the same order in the case of little Higgs modelswith T parity. Important correlations also exist in see-sawSUSY GUT models between B(μ→ eγ ) and B(τ → μγ )

or B(τ → eγ ). Furthermore, SUSY models with CP vio-lation in the Higgs or gaugino mass matrix, be they super-gravity (SUGRA) inspired or of the split-SUSY type, predictthe ratio of electron and neutron EDM to be in the range of

10−2–10−1. Furthermore, in SUSY GUT models with see-saw mechanism correlations exist between the values of theneutron and deuteron EDMs and the heavy neutrino masses.

Section 6 discusses studies of lepton universality. Thebranching ratios Γ (π → μν)/Γ (π → eν) and Γ (K →μν)/Γ (K → eν), for example, are very well known theo-retically within the SM. Ongoing experiments (at PSI andTRIUMF for the pion, and at CERN and Frascati for thekaon) test the existence of flavor-dependent charged Higgscouplings, by improving the existing accuracies by factorsof order 10.

In Sect. 7 we consider CP-violating charged lepton de-cays, which offer interesting prospects as alternative probesof BSM phenomena. SM-allowed τ decays, such as τ →νKπ , can be sensitive to new CP-violating effects. Thedecays being allowed by the SM, the CP-odd asymme-tries are proportional to the interference of a SM amplitudewith the BSM, CP-violating one. As a result, the small CP-violating amplitude contributes linearly to the rate, ratherthan quadratically, enhancing the sensitivity. In the specificcase of τ→ νKπ , and for some models, a CP asymmetry atthe level of 10−3 would correspond to B(τ → μγ ) around10−8. Another example is the CP-odd transverse polariza-tion of the muon, PT , in K → πμν decays. The currentsensitivity of the KEK experiment E246, which resulted inPT < 5× 10−3 at 90% C.L., can be improved to the level of10−4, by TREK proposed at J-PARC, probing models suchas multi-Higgs or R-parity-violating SUSY.

Section 8 discusses experimental searches for chargedLFV processes. Transitions between e, μ, and τ might befound in the decay of almost any weakly decaying parti-cle and searches have been performed in μ, τ , π , K , B ,D, W and Z decay. Whereas the highest experimental sen-sitivities were reached in dedicated μ and K experiments,τ decay starts to become competitive as well. In Sect. 8 theexperimental limitations to the sensitivities for the variousdecay modes are discussed in some detail, in particular for μand τ decays, and some key experiments are presented. Thesensitivities reached in searches for μ+ → e+γ are limitedby accidental e+γ coincidences and muon beam intensitieshave to be reduced now already. Searches for μ–e conver-sion, on the other hand, are limited by the available beam in-tensities, and large improvements in sensitivity may still beachieved. Similarly, in rare τ decays some decay modes arealready background limited at the present B-factories andfuture sensitivities may not scale with the accumulated lu-minosities. Prospects of LFV decays at the LHC are limitedto final states with charged leptons, such as τ → 3μ andB0d,s → e±μ∓, which are discussed in detail. This section

finishes with the preliminary results of a feasibility studyfor in-flight μ→ τ conversions using a wide beam of highmomentum muons. No working scheme emerged yet.

18 Eur. Phys. J. C (2008) 57: 13–182

Section 9 covers electric and magnetic dipole moments.The muon magnetic moment has been much discussed re-cently, so we limit ourselves to a short review of the theo-retical background and of the current and foreseeable ex-perimental developments. In the case of EDMs, we pro-vide an extensive description of the various theoretical ap-proaches and experimental techniques applied to test elec-tron and quark moments, as well as other possible sourcesof flavor diagonal CP-violating effects, such as the gluonicθFF coupling, or CP-odd four-fermion interactions. Whilethe experimental technique may differ considerably, the var-ious systems provide independent and complementary in-formation. EDMs of paramagnetic atoms such as Tl are sen-sitive to a combination of the fundamental electron EDMand CP-odd four-fermion interactions between nucleons andelectrons. EDMs of diamagnetic atoms such as Hg are sen-sitive, in addition, to the intrinsic EDM of quarks, as wellas to a non-zero QCD θ coupling. The neutron EDM moredirectly probes intrinsic quark EDMs, θ , and possible higherdimension CP-odd quark couplings. EDMs of the electron,without contamination from hadronic EDM contributions,can be tested with heavy diatomic molecules with unpairedelectrons, such as YbF. In case of a positive signal the com-bination of measurements would help to disentangle the var-ious contributions.

The experimental situation looks particularly promising,with several new experiments about to start or under con-struction. For example, new ultracold-neutron setups at ILL,PSI and Oak Ridge will increase the sensitivity to a neutronEDM by more than two orders of magnitude, to a level ofabout 10−28 e cm in 5–10 years. This sensitivity probes e.g.CP-violating SUSY phases of the order of 10−4 or smaller.Similar improvements are expected for the electron EDM.One of the main new ideas developed in the course of theworkshop is the use of a storage ring to measure the deuteronEDM. The technical issues related to the design and con-struction of such an experiment, which could have a statis-tical sensitivity of about 10−29 e cm, are discussed here insome detail.

All the results presented in this document prove the greatpotential of this area of particle physics to shed light on oneof the main puzzles of the standard model, namely the originand properties of flavor. Low energy experiments are sensi-tive to scales of new physics that in several cases extend be-yond several TeV. The similarity with the scales directly ac-cessible at the LHC supports the expectation of an importantsynergy with the LHC collider programme, a synergy thatclearly extends to future studies of the neutrino and quarksectors. The room for improvement, shown by the projec-tions suggested by the proposed experiments, finally under-scores the importance of keeping these lines of research atthe forefront of the experimental high energy physics pro-gramme, providing the appropriate infrastructure, supportand funding.

2 Theoretical framework and flavor symmetries

2.1 The flavor puzzle

The presence of three fermion families with identical gaugequantum numbers is a puzzle. The very origin of this repli-cation of families constitutes the first element of the SM fla-vor puzzle. The second element has to do with the Yukawainteractions of those three families of fermions. While thegauge principle allows us to determine all SM gauge inter-actions in terms of three gauge couplings only (once the SMgauge group and the matter gauge quantum numbers havebeen specified), we do not have clear evidence of a guidingprinciple underlying the form of the 3 × 3 matrices describ-ing the SM Yukawa interactions. Finally, a third element ofthe puzzle is represented by the peculiar pattern of fermionmasses and mixing originating from those couplings.

The replication of SM fermion families can be rephrasedin terms of the symmetries of the gauge part of the SM La-grangian. The latter is in fact symmetric under a U(3)5 sym-metry acting on the family indexes of each of the five in-equivalent SM representations forming a single SM family(q,uc, dc, l, ec in Weyl notation). In other words, the gaugecouplings and interactions do not depend on the (canonical)basis we choose in the flavor space of each of the five sets offields qi, uci , d

ci , l, e

ci , i = 1,2,3.

This U(3)5 symmetry is explicitly broken in the Yukawasector by the fermion Yukawa matrices. It is because ofthis breaking that the degeneracy of the three families isbroken and the fields corresponding to the physical masseigenstates, as well as their mixing, are defined. An addi-tional source of breaking is provided by neutrino masses.The smallness of neutrino masses is presumably due to thebreaking of the accidental lepton symmetry of the SM at ascale much larger than the electroweak, in which case neu-trino masses and mixing can be accounted for in the SMeffective Lagrangian in terms of a dimension five operatorbreaking the U(3)5 symmetry in the lepton doublet sector.

As mentioned, the special pattern of masses and mixingoriginating from the U(3)5 breaking is an important elementof the flavor puzzle. This pattern is quite peculiar. It sufficesto mention the smallness of neutrino masses; the hierarchyof charged fermion masses relative to that of the two heavierneutrinos; the smallness of Cabibbo–Kobayashi–Maskawamixing in the quark sector and the two large mixing anglesin Pontecorvo–Maki–Nakagawa–Sakata (PMNS) matrix inthe lepton sector; the mass hierarchy in the up quark sector,more pronounced than in the down quark and charged lep-ton sectors; the presence of a large CP-violating phase in thequark sector and the need of additional CP violation to ac-count for baryogenesis; the approximate equality of bottomand tau masses at the scale at which the gauge couplings

Eur. Phys. J. C (2008) 57: 13–182 19

unify1 and the approximate factor of 3 between the strangeand muon masses, both pointing at a grand unified picture athigh energy.

The origin of family replication and of the peculiar pat-tern of fermion masses and mixing are among the most in-teresting open questions in the SM, which a theory of flavor,discussed in Sect. 2, should address. As seen in Sect. 3, ex-periment is ahead of theory in this field. All the physicalparameters describing the SM flavor structure in the quarksector have been measured with good accuracy. In the lep-ton sector crucial information on lepton mixing and neutrinomasses is being gathered and a rich experimental program isunder way to complete the picture.

Several tools are used to attack the flavor problem. Grandunified theories allow one to relate quark and lepton massesat the GUT scale and provide an appealing framework tostudy neutrino masses, leptogenesis, flavor models, etc. Notethat in a grand unified context the U(3)5 symmetry of thegauge sector is reduced (to U(3) in the case in which allfermions in a family are unified in a single representation,as in SO(10)). Extra dimensions introduce new ways to ac-count for the hierarchy of charged fermion masses (and insome cases for the smallness of neutrino masses) through themechanism of localization in extra dimensions and by pro-viding a new framework for the study of flavor symmetries.The concept of minimal flavor violation may also provide aframework for addressing flavor. The impact of those orga-nizing principles on flavor physics is discussed in detail inSect. 4.

From experimental point of view, however, additionalhandles are needed to gain more insight in the origin of fla-vor. Essentially this requires a discovery of new physics be-yond the SM. New physics at the TeV scale may in fact beassociated with an additional flavor structure, whose originmight well be related to the origin of the Yukawa couplings.Some of the present attempts to understand the pattern offermion masses and mixing do link the flavor structure ofthe SM and that of the new physics sectors. In which casethe search for indirect effects at low energy and for directeffects at colliders may play a primary role in clarifying ourunderstanding of flavor. And conversely, the attempts to un-derstand the pattern of fermion masses and mixing mightlead to the prediction of new flavor physics effects. Thoseissues are addressed in Sect. 5.

Finally, lepton flavor physics is not just related to the lep-ton flavor violation or CP violation in the lepton sector butalso to understanding the unitarity and universality in thelepton sector. Possible deviations from those are discussedin Sect. 5.6.

1Needless to say, precise unification requires an extension of the SM,with supersymmetry doing best from this point of view.

2.2 Flavor symmetries

The SM Lagrangian is U(3)5 invariant in the limit in whichthe Yukawa couplings vanish. This might suggest that theYukawa couplings, or at least some of them, arise fromthe spontaneous breaking of a subgroup of U(3)5. Need-less to say, the use of (spontaneously broken) symmetriesas organizing principles to understand physical phenomenahas been largely demonstrated in the past (chiral symmetrybreaking, electroweak, etc.). In the following, we discuss thepossibility of using such an approach to address the originof the pattern of fermion masses and mixing, the constraintson the flavor structure of new physics, and to put forwardexpectations for flavor observables.

The spontaneously broken “flavour” or “family” sym-metry can be local or global. Many (most) of the conse-quences of flavor symmetries are independent of this. Theflavor breaking scale must be sufficiently high in such a wayto suppress potentially dangerous effects associated with thenew fields and interactions, in particular with the new gaugeinteractions (in the local case) or the unavoidable pseudo-Goldstone bosons (in the global case). In the context of ananalysis in terms of effective operators of higher dimen-sions, a generic bound of about 103 TeV on the flavor scalefrom flavor changing neutral currents (FCNC) processeswould be obtained. Nevertheless, a certain evidence for b–τunification and the appeal of the see-saw mechanism forneutrino masses seem to suggest that these Yukawa cou-plings are already present near the GUT scale. This is indeedwhat most flavor models assume, and we shall also assumein the following.

The SM matter fields belong to specific representations ofthe flavor group, such that in the unbroken limit the Yukawacouplings have a particularly simple form. Typically someor all Yukawa couplings (with the possible exception ofthird generation ones) are not allowed. The spontaneoussymmetry breaking of the flavor symmetry is provided bythe vacuum expectation value (VEV) of fields often called“flavons”. As the breaking presumably arises at a scale muchhigher than the electroweak scale, such flavons are SM sin-glets (or contain a SM singlet in the case of SM extensions)and typically they are only charged under the flavor sym-metry. Flavor breaking is communicated dynamically to theSM fields by some interactions (possibly renormalizable,often not specified) living at a scale Λf not smaller thanthe scale of the flavor symmetry breaking. A typical exam-ple for these interactions that communicate the breaking isthe exchange of heavy fermions whose mass terms respectthe flavor symmetry. In that case the scale Λf would cor-respond to this fermion mass Mf . Many consequences ofthe flavor symmetry are actually independent of the media-tion mechanism. It is therefore useful to consider an effec-tive field theory approach below the scale Λf in which the

20 Eur. Phys. J. C (2008) 57: 13–182

Table 2 Transformation of the matter superfields under the family symmetries. The ith generation SM fermion fields are grouped into the repre-sentation 5i = (Dc,L)i , 10i = (Q,Uc,Ec)i , 1i = (Nc)iField 103 102 101 53 52 51 13 12 11 θ

U(1) 0 2 3 0 0 1 nc3 nc2 nc1 −1

flavor messengers have been integrated out. Once the flavonfields have acquired their VEVs, the structure of the Yukawamatrices (and other flavor parameters) can be obtained froman expansion in non-renormalizable operators involving theflavon fields and respecting the different symmetries (flavorand other symmetries) of the theory.

There are several possibilities for the flavor symmetry,local, global, accidental, continuous or discrete, Abelian ornon-Abelian. Many examples are available in the literaturefor each of those possibilities. Some of them will be dis-cussed in next subsections in relation to the implicationsconsidered in this study.

2.2.1 Continuous flavor symmetries

In order to provide an explicit example, we shortly discusshere one of the simplest possibilities, which goes back tothe pioneering work of Froggatt–Nielsen [5]. In this modelwe have a U(1) flavor symmetry under which the three gen-eration of SM fields have different charges. In the simplestversion we assign positive integer charges to the SM fermi-onic fields, the Higgs field is neutral, and we have a singleflavon field θ of charge −1. The VEV of the flavon fieldis somewhat smaller than the mass of the heavy mediatorfields Mf , so that the ratio ε = v/Mf � 1. In this way thevarious entries in the Yukawa matrices are determined by ep-silon to the power of the sum of the fermion charges with anundetermined order 1 coefficient. This mechanism explainsnicely the hierarchy of fermion masses and mixing angles.

This idea is the basis for most flavor symmetries. It canbe implemented in a great variety of different models. Forthe sake of definiteness, we show here how it works usingas a concrete example a supersymmetric GUT model. Its su-perpotential is of the form

WYukawa = cdij εqi+dcj QiD

cjH1 + cuij εqi+u

cjQiU

cj H2

+ ceij εli+ecj LiE

cjH1 + cνij εli+lj LiLj

H2H2

M,

(2.1)

where the c’s are O(1) coefficients and M is the scale asso-ciated to B − L breaking. The last term in this equation isan effective operator, giving Majorana masses to neutrinos,which can be generated, e.g., through a see-saw mechanism.Notice that the power of ε in each Yukawa coupling is pro-portional to the sum of the fermion charges: Yuij = cuij εqi+u

cj ,

Ydij = cdij εqi+dcj , etc. Hence, this mechanism explains the hi-

erarchy of fermion masses and mixing angles through a con-venient choice of charges. The value of these charges andthe expansion parameter ε are constrained by the observedmasses and angles. A convenient set of charges for exampleis given in Table 2. It turns out that this set of charges is theonly one compatible with minimal SU(5) unification. By in-troducing three right handed neutrinos with positive chargesit is also possible to successfully realize the see-saw mecha-nism.

These charges give rise to the following Dirac Yukawacouplings for charged fermions at the GUT scale

Yu =⎛⎝ε6 ε5 ε3

ε5 ε4 ε2

ε3 ε2 1

⎞⎠ ,

⎛⎝ε4 ε3 ε3

ε3 ε2 ε2

ε 1 1

⎞⎠ , (2.2)

where O(1) coefficients in each entry are understood hereand in the following. With ε =O(λc) (the Cabibbo angle),the observed features of charged fermion masses and mixingare qualitatively well reproduced. It is known that the highenergy relation YTe = Yd is not satisfactory for the lighterfamilies and should be relaxed by means of some mecha-nism [6–8]. The Dirac neutrino Yukawa couplings and theMajorana mass matrix of right handed neutrinos are

Yν =⎛⎝εnc1+1 εn

c2+1 εn

c3+1

εnc1 εn

c2 εn

c3

εnc1 εn

c2 εn

c3

⎞⎠ ,

MR =⎛⎝ε2nc1 εn

c1+nc2 εn

c1+nc3

εnc1+nc2 ε2nc2 εn

c2+nc3

εnc1+nc3 εn

c2+nc3 ε2nc3

⎞⎠ M.

(2.3)

Applying the see-saw mechanism to obtain the effectivelight neutrino mass matrix Mν in the basis of diagonalcharged lepton Yukawa couplings,2 it is well known [9, 10]that if all right handed neutrino masses are positive the de-pendence on the right handed charges disappears:

U∗PMNSm

diagν U

†PMNS =mν =

⎛⎝ε2 ε ε

ε 1 1ε 1 1

⎞⎠ v2

2

M. (2.4)

2Notice that going to the basis of diagonal charged leptons will onlychange the O(1) coefficients, but not the power in ε of the differententries.

Eur. Phys. J. C (2008) 57: 13–182 21

Experiments require M ∼ 5 × 1014 GeV. The featuresof neutrino masses and mixing are quite satisfactorilyreproduced—the weak point being the tuning in the 23-determinant [9, 10] that has to be imposed. For later ap-plication, it is useful to introduce the unitary matrices whichdiagonalize Yν in the basis where both Ye and MR are di-agonal: VLYνVR = Y diag

ν ≈ diag(εnc1 , εn

c2 , εn

c3). Notice that,

as a consequence of the equal charges of the lepton doubletsL2 and L3, the model predicts that VL has a large mixing,although not necessarily maximal, in the 2–3 sector as ob-served in UPMNS.

The literature is very rich of models based on flavorsymmetries. Some references are [5, 9–40]; for more re-cent attempts the interested reader is referred for instanceto [41–64].

2.2.2 Discrete flavor symmetries

2.2.2.1 Finite groups Discrete flavor symmetries havegained popularity because they seem to be appropriate toaddress the large mixing angles observed in neutrino oscil-lations. To obtain a non-Abelian discrete symmetry, a simpleheuristic way is to choose two specific non-commuting ma-trices and form all possible products. As a first example,consider the two 2 × 2 matrices

A=(

0 11 0

), B =

(ω 00 ω−1

), (2.5)

where ωn = 1, i.e. ω = exp(2πi/n). Since A2 = 1 andBn = 1, this group contains Z2 and Zn. For n = 1,2, weobtain Z2 and Z2 ×Z2 respectively, which are Abelian. Forn= 3, the group generated has six elements and is in fact thesmallest non-Abelian finite group S3, the permutation groupof three objects. This particular representation is not the onefound in text books, but it is related to it by a unitary trans-formation [65], and was first used in 1990 for a model ofquark mass matrices [66, 67]. For n= 4, the group generatedhas eight elements which are in fact ±1, ±iσ1,2,3, whereσ1,2,3 are the usual Pauli spin matrices. This is the group ofquaternionsQ, which has also been used [68] for quark andlepton mass matrices. In general, the groups generated by(2.5) have 2n elements and may be denoted as Δ(2n).

Consider next the two 3 × 3 matrices:

A=⎛⎝

0 1 00 0 11 0 0

⎞⎠ , B =

⎛⎝ω 0 00 ω2 00 0 ω−3

⎞⎠ . (2.6)

Since A3 = 1 and Bn = 1, this group contains Z3 and Zn.For n = 1, we obtain Z3. For n = 2, the group generatedhas 12 elements and is A4, the even permutation group of 4

objects, which was first used in 2001 in a model of leptonmass matrices [36, 41]. It is also the symmetry group of thetetrahedron, one of five perfect geometric solids, identifiedby Plato with the element “fire” [69]. In general, the groupsgenerated by (2.6) have 3n2 elements and may be denotedasΔ(3n2) [70]. They are in fact subgroups of SU(3). In par-ticular, Δ(27) has also been used [57, 71]. Generalizing tok × k matrices, we then have the series Δ(knk−1). How-ever, since there are presumably only three families, k > 3is probably not of much interest.

Going back to k = 2, but using instead the following twomatrices:

A=(

0 11 0

), B =

(ω 00 1

). (2.7)

Now again A2 = 1 and Bn = 1, but the group generated willhave 2n2 elements. Call it Σ(2n2). For n= 1, it is just Z2.For n = 2, it is D4, i.e. the symmetry group of the square,which was first used in 2003 [47, 72]. For k = 3, consider

A=⎛⎝

0 1 00 0 11 0 0

⎞⎠ , B =

⎛⎝ω 0 00 1 00 0 1

⎞⎠ , (2.8)

then the groups generated have 3n3 elements and may bedenoted as Σ(3n3). They are in fact subgroups of U(3). Forn= 1, it is just Z3. For n= 2, it is A4 ×Z2. For n= 3, thegroupΣ(81) has been used [73] to understand the Koide for-mula [74] as well as lepton mass matrices [75]. In general,we have the series Σ(knk).

2.2.2.2 Model recipe

1. Choose a group, e.g. S3 or A4, and write down its possi-ble representations. For example S3 has 1, 1′, 2;A4 has 1,1′, 1′′, 3. Work out all product decompositions. For exam-ple 2×2 = 1+1′+2 in S3, and 3×3 = 1+1′+1′′+3+3in A4.

2. Assign (ν, l)1,2,3 and lc1,2,3 to the representations ofchoice. To have only renormalizable interactions, itis necessary to add Higgs doublets (and perhaps alsotriplets and singlets) and, if so desired, neutrino singlets.

3. The Yukawa structure of the model is restricted by thechoice of particle content and their representations. Asthe Higgs bosons acquire vacuum expectation values(which may be related by some extra or residual symme-try), the lepton mass matrices will have certain particularforms, consistent with the known values of me, mμ, mτ ,etc. If the number of parameters involved is less than thenumber of observables, there will be one or more predic-tions.

4. In models with more than one Higgs doublet, flavor non-conservation will appear at some level. Its phenomeno-logical consequences need to be worked out, to ensure

22 Eur. Phys. J. C (2008) 57: 13–182

the consistency with present experimental constraints.The implications for phenomena at the TeV scale canthen be explored.

5. Insisting on using only the single SM Higgs doublet re-quires effective non-renormalizable interactions to sup-port the discrete flavor symmetry. In such models, thereare no predictions beyond the forms of the mass matricesthemselves.

6. Quarks can be considered in the same way. The twoquark mass matrices mu and md must be nearly alignedso that their mixing matrix involves only small angles.In contrast, the mass matrices mν and me should havedifferent structures so that large angles can be obtained.

Some explicit examples will now be outlined.

2.2.2.3 S3 Being the simplest, the non-Abelian discretesymmetry S3 was used already [76] in the early days ofstrong interactions. There are many recent applications[55, 77–86], some of which are discussed in [87]. Typically,such models often require extra symmetries beyond S3 toreduce the number of parameters, or assumptions of howS3 is spontaneously and softly broken. For illustration, con-sider the model of Kubo et al. [77] which has recently beenupdated by Felix et al. [88]. The symmetry used is actuallyS3 ×Z2, with the assignments

(ν, l), lc,N,(φ+, φ0)∼ 1 + 2, (2.9)

and equal vacuum expectation values for the two Higgs dou-blets transforming as 2 under S3. The Z2 symmetry servesto eliminate four Yukawa couplings, otherwise allowed byS3, resulting in an inverted ordering of neutrino masses with

θ23 π/4, θ13 0.0034, mee 0.05 eV, (2.10)

wheremee is the effective Majorana neutrino mass measuredin neutrinoless double beta decay. This model relates θ13 tothe ratio me/mμ.

2.2.2.4 A4 To understand why quarks and leptons havevery different mixing matrices, A4 turns out to be very use-ful. It allows the two different quark mass matrices to bediagonalized by the same unitary transformations, implyingthus no mixing as a first approximation, but because of theassumed Majorana nature of the neutrinos, a large mismatchmay occur in the lepton sector, thus offering the possibil-ity of obtaining the so-called tri-bi-maximal mixing matrix[89, 90], which is a good approximation to the present data.One way of doing this is to consider the decomposition

UPMNS =⎛⎝

√2/3 1/

√3 0

−1/√

6 1/√

3 −1/√

2−1/

√6 1/

√3 1/

√2

⎞⎠

= 1√3

⎛⎝

1 1 11 ω ω2

1 ω2 ω

⎞⎠⎛⎝

0 1 01/√

2 0 −i/√21/√

2 0 i/√

2

⎞⎠ ,

(2.11)

where UPMNS is the observed neutrino mixing matrix andω = exp(2πi/3)=−1/2 + i√3/2. The matrix involving ωhas equal moduli for all its entries and was conjectured al-ready in 1978 [91, 92] to be a possible candidate for the 3×3neutrino mixing matrix.

Since UPMNS = V †e Vν , where Ve , Vν diagonalize the ma-

trices mem†e , mνm†

ν respectively, (2.11) may be obtained ifwe have

V †e = 1√

3

⎛⎝

1 1 11 ω ω2

1 ω2 ω

⎞⎠ (2.12)

and

mν =⎛⎝a + 2b 0 0

0 a − b d

0 d a − b

⎞⎠

=⎛⎝

0 1 01/√

2 0 −i/√21/√

2 0 i/√

2

⎞⎠

×⎛⎝a − b+ d 0 0

0 a + 2b 00 0 −a + b+ d

⎞⎠

×⎛⎝

0 1/√

2 1/√

21 0 00 −i/√2 i/

√2

⎞⎠ . (2.13)

It was discovered in Ref. [36] that (2.12) is naturally ob-tained with A4 if

(ν, l)1,2,3 ∼ 3, lc1,2,3 ∼ 1 + 1′ + 1′′,(φ+, φ0

)1,2,3 ∼ 3

(2.14)

for 〈φ01〉 = 〈φ0

2〉 = 〈φ03〉. This assignment also allows me ,

mμ, mτ to take on arbitrary values, because there are hereexactly three independent Yukawa couplings invariant un-der A4. If we use this also for quarks [41], then V †

u and V †d

are also given by (2.12), resulting in UCKM = 1, i.e. no mix-ing. This should be considered as a good first approximationbecause the observed mixing angles are all small. In the gen-eral case without any symmetry, we would have expected Vuand Vd to be very different.

It was later discovered in Ref. [93] that (2.13) may also beobtained with A4, using two further assumptions. Consider

Eur. Phys. J. C (2008) 57: 13–182 23

the most general 3 × 3 Majorana mass matrix in the form

mν =⎛⎝a + b+ c f e

f a + bω+ cω2 d

e d a + bω2 + cω

⎞⎠ ,

(2.15)

where a comes from 1, b from 1′, c from 1′′, and (d, e, f )from 3 of A4. To get (2.13), we need e = f = 0, i.e.the effective scalar A4 triplet responsible for neutrinomasses should have its vacuum expectation value along the(1,0,0) direction, whereas that responsible for charged lep-ton masses should be (1,1,1) as remarked earlier. This mis-alignment is a technical challenge to all such models [50,94–104]. The other requirement is that b = c. Since theycome from different representations of A4, this is rather adhoc. A very clever solution [50, 94] is to eliminate both,i.e. b = c = 0. This results in a normal ordering of neutrinomasses with the prediction [96]

|mνe |2 |mee|2 +�m2atm/9. (2.16)

Other applications [60, 105–120] of A4 have also been con-sidered. A natural (spinorial) extension of A4 is the binarytetrahedral group [30, 34] which is under active current dis-cussion [64, 121–123].

Other recent applications of non-Abelian discrete flavorsymmetries include those of D4 [47, 72, 124], Q4 [68], D5

[125, 126],D6 [127],Q6 [128–130],D7 [131], S4 [61, 132–135], Δ(27) [57, 71], Δ(75) [15, 136], Σ(81) [73, 75], andB3 ×Z3

2 [137, 138] which has 384 elements.

2.2.3 Accidental flavor symmetries

While flavor symmetries certainly represent one of the lead-ing approaches to understanding the pattern of fermionmasses and mixing, it was recently found that the hierarchi-cal structure of charged fermion masses and many other pe-culiar features of the fermion spectrum in the SM (neutrinosincluded) do not require a flavor symmetry to be understood,nor any other special “horizontal” dynamics involving thefamily indices of the SM fermions [63, 139]. Surprisinglyenough, those features can in fact be recovered in a modelin which the couplings of the three SM families not only arenot governed by any symmetry, but are essentially anarchi-cal (uncorrelated O(1) numbers) at a very high scale.

The idea is based on the hypothesis that the SM Yukawacouplings all arise from the exchange of heavy degrees offreedom (messengers) at a scale not far from the unifica-tion scale. Examples of diagrams contributing to the up anddown quark Yukawa matrices are shown below, where φ isa SM singlet field getting a VEV. As discussed in Sects. 2.2and 2.2.1, the same exchange mechanism is often assumed

Fig. 1 Contributions to the up- and down-type quark Yukawa massmatrices, from the exchange of heavy messengers

to be at work in models with flavor symmetries. Here, how-ever, the couplings of the heavy messengers to the SM fieldsare not constrained by any symmetry.3 An hierarchy amongYukawa couplings still arises because a single set of lefthanded messenger fields (heavy quark doublets Q + Q inthe quark sector and heavy lepton doublets L+ L in the lep-ton sector) dominates the exchange at the heavy scale. Forexample, the diagrams below represents the dominant con-tribution to the quark Yukawa matrices. As only one fieldis exchanged, the Yukawa matrices have rank one. There-fore, whatever are the O(1) couplings in the diagram, thetop and bottom Yukawa couplings are generated (at the O(1)level, giving large tanβ), but the first two families’ onesare not, which is a good starting point to obtain a hierar-chy of quark masses. This mechanism is similar to a thesingle right handed neutrino dominance mechanism, usedin neutrino model building to obtain a hierarchical spectrumof light neutrinos [140–143]. Note that the diagonalizationof the quark Yukawa matrices involves large rotations, asall the couplings are supposed to be O(1). However, therotations of the up and down left handed quarks turn outto be the same (because they have same couplings to theleft handed doublet messenger). Therefore, the two rotationscancel when combined in the CKM matrix, which ends upvanishing at this level.

The Yukawa couplings of the second family, and a non-vanishing Vcb angle, are generated by the subdominant ex-change of heavier right handed messengers Dc , Uc , Ec,Nc. Altogether, the messengers form a heavy (vector-like)replica of a SM family, with the left handed fields lighterthan the right handed ones. The (inter-family) hierarchy be-tween the masses of the second and the third SM familymasses arises from the (intra-family) hierarchy between leftand right handed fields in the single family of messengers.In turn, in a Pati–Salam or SO(10) unified model, the hi-erarchy between right handed and left handed fields can beeasily obtained by giving mass to the messengers through abreaking of the gauge group along the T3R direction. Thisway, the hierarchy among different families is explained interms of the breaking of a gauge group acting on single fam-ilies, with no need of flavor symmetries or other dynamicsacting on the family indexes of the SM fermions.

3A discrete Z2 symmetry, under which all the three SM families (andthe field φ) are odd, is used for the sole purpose of distinguishing thelight SM fields from the heavy messengers.

24 Eur. Phys. J. C (2008) 57: 13–182

It is also possible to describe the mechanism outlinedabove in terms of accidental flavor symmetries. In the effec-tive theory below the scale of the right handed messengers,in fact, the Yukawa couplings of the two lighter families are“protected” by an accidental U(2) symmetry. One can alsoconsider the effective theory below the cut-off of the model,which is supposed to lie one or two orders of magnitudeabove the mass of the right handed messengers. In the effec-tive theory below the cut-off, the second family gets a non-vanishing Yukawa coupling, but the Yukawa of the lightestfamily is still “protected” by an accidental U(1) symmetry.

Surprisingly enough, a number of important features ofthe fermion spectrum can be obtained in this simple andeconomical model. The relation |Vcb| ∼ ms/mb is a directconsequence of the principles of this approach. The strongermass hierarchy observed in the up quark sector is accountedfor without introducing a new scale (besides the left handedand right handed messenger ones) or making the up quarksector somehow different. In spite of the absence of smallcoefficients, the CKM mixing angles turn out to be small. Atthe same time, a large atmospheric mixing can be generatedin a natural way in the neutrino sector, together with normalhierarchical neutrino masses. In fact, a see-saw mechanismdominated by the single right handed (messenger) neutrinoNc is at work. The bottom and tau mass unify at the highscale, while a B − L factor 3 enters the ratios of the muonand strange masses. For a detailed illustration of the model,we refer the reader to [63].

The study of FCNC and CPV effects in a supersymmet-ric context is still under way. Such effects might representthe distinctive signature of the model, due to the sizable ra-diative effects one obtains in the (23) block of the “righthanded” sfermion mass matrices in both the squark and slep-ton sector.

2.2.4 Flavor/CP symmetries and their violation fromsupersymmetry breaking

While the vast literature on flavor symmetries covers a num-ber of interesting aspects of the theory and phenomenologyof flavor, we are interested here in a (non-exhaustive) reviewof only those aspects relevant to new physics. The relevanceof flavor symmetries to new physics follows from the factthat SM extensions often contain new flavor dependent in-teractions. In the following we shall consider the case of su-persymmetry, in which new flavor violating gaugino or hig-gsino interactions can be induced by possible new sources ofSU(5)5 breaking in the soft supersymmetry breaking terms.

While in the SM the Yukawa matrices provide the onlysource of flavor (U(3)5) breaking, the supersymmetric ex-tensions of the SM are characterized by a potentially muchricher flavor structure associated to the soft supersymmetrybreaking Lagrangian. Unfortunately, a generic flavor struc-ture leads to FCNC and CPV processes that can exceed the

experimental bounds by up to two orders of magnitude—the so-called supersymmetric flavor and CP problem. Thesolution of the latter problem can lie in the supersymmetrybreaking and mediation mechanism (this is the case for ex-ample of gauge mediated supersymmetry breaking) or in theconstraints on the soft terms provided by flavor symmetries.

In turn, the implications of flavor symmetries on thestructure of the soft terms depends on the interplay betweenflavor and supersymmetry breaking. Without entering thedetails of specific models, we can distinguish two oppositesituations.

– The soft terms are flavor universal, or at least symmetricunder the flavor symmetry, at the tree level, and

– flavor symmetry breaking enters the soft terms (as forthe Yukawa interactions) already at the tree level, throughnon-renormalizable couplings to the flavon fields.

Let us consider them in greater detail.The first possibility is that the supersymmetry breaking

mechanism takes care of the FCNC and CPV problems. Inthe simplest case, the new sfermion masses and A-terms donot introduce new flavor structure at all. This is the case if

m2ij =m2

0δij , Aij =A0 δij ,

where i, j are family indexes and the universal values m20,

A0 can be different in the different sfermion sectors.4 Thebreaking of the flavor symmetry is felt at the tree level onlyby the Yukawa matrices. Needless to say, the tree level uni-versality of the soft terms will be spoiled by renormaliza-tion effects associated to interactions sensitive to Yukawacouplings [144, 145]. These effects can be enhanced bylarge logarithms if the scale at which the soft terms and theYukawa interactions appear in the observable sector is suf-ficiently high. The radiative contributions of Yukawa cou-plings associated with neutrino masses (or Yukawa cou-plings occurring in the context of grand unification) are par-ticularly interesting in this context, because they offer newpossibilities to test flavor physics by opening a window forphysics at very large scales. For example, in the minimalSUSY see-saw model only the off-diagonal elements forleft-slepton soft supersymmetry breaking mass terms aregenerated, while in supersymmetric GUTs also the righthanded slepton masses get renormalization induced flavornon-diagonal contributions. In any case, all the flavor effectsinduced by the soft terms can be traced back to the Yukawacouplings, which remain the only source of flavor breaking.

4This is the case for example of gauge mediation. In supergravity, su-persymmetry breaking can be fully flavor blind in the case of dila-ton domination. In this case, we expect the diagonal elements of thesoft mass matrices to be exactly universal. However, this is not alwaysthe case. Moduli domination is often encountered, in which case fieldswith different modular weights receive different soft masses.

Eur. Phys. J. C (2008) 57: 13–182 25

Such unavoidable effects of flavor breaking on the soft termswill be discussed in Sects. 5.2 and 5.3.

As we have just seen, the radiative contributions to softmasses represent an unavoidable but indirect effect of thephysics at the origin of fermion masses and mixing. On theother hand, the mechanism generating the soft terms mightnot be blind to flavor symmetry breaking, in which case wemight also expect flavor breaking to enter the soft terms ina more direct way. If this is the case, the soft term providea new independent source of flavor violation. Such model-dependent “tree level” effects of flavor breaking on the softterms add to the radiative effects and will be discussed inSect. 2.2.4.1. The actual presence in the soft terms of flavorviolating effects directly induced by the physics account-ing for Yukawa couplings depends on the interplay of thesupersymmetry breaking and the flavor generation mecha-nisms.

Theoretical and phenomenological [146–151] constraintson supersymmetry breaking parameters essentially force su-persymmetry breaking to take place in a hidden sector withno renormalizable coupling to observable fields.5 The softterms are therefore often characterized by the scale ΛSUSY

at which supersymmetry breaking is communicated to theobservable sector by some mediation mechanism. The softterms arise in fact from non-renormalizable operators in theeffective theory below ΛSUSY obtained by integrating outthe supersymmetry breaking messenger fields. Analogously,in the context of a theory addressing the origin of flavor, wecan define a scale Λf at which the flavor structure arises.Let us consider for definiteness the case of flavor symme-tries. The analogy with supersymmetry breaking is in thiscase even more pronounced. Above Λf , the theory is fla-vor symmetric. By this we mean that we can at least defineconserved family numbers, perhaps part of a larger flavorsymmetry. The family numbers are then spontaneously bro-ken by the VEV of flavons that couple to observable fieldsthrough non-renormalizable interactions suppressed by thescale Λf .

We are now in the position to discuss the presenceof “tree-level” flavor violating effects in the soft terms.A first possibility is to have Λf � ΛSUSY, as for in-stance in the case of gravity mediation, in which we ex-pect Λf �MPlanck = ΛSUSY. The soft breaking terms arealready present below MPlanck. However, the flavor sym-metry is still exact at scales larger than Λf . Therefore, thesoft terms must respect the family symmetries. At the lowerscale Λf the effective Yukawa couplings are generated asfunctions of the flavon VEVs, 〈θ〉/Λf , and analogously thesoft breaking terms will also be functions of 〈θ〉/Λf . Inthe Λf �ΛSUSY case, we therefore expect new “tree-level”

5The fields of the minimal supersymmetric standard model (MSSM)or its relevant extension.

sources of flavor breaking in the soft terms on top of theeffects radiatively induced by the Yukawa couplings.

On the other hand, ifΛSUSY �Λf , the soft terms are notpresent at the scale of flavor breaking. The prototypical ex-ample in this case is gauge mediated supersymmetry break-ing (GMSB) (see [152] and references therein). At Λf theflavor interactions are integrated and supersymmetry is stillunbroken. The only renormalizable remnant of the flavorphysics below Λf are the Yukawa couplings. At the scaleΛSUSY soft breaking terms feel flavor breaking only throughthe Yukawa couplings. Strictly speaking, there could alsobe non-renormalizable operators involving flavon fields sup-pressed by the heavier Λf . The contributions of these termsto soft masses would be proportional to ΛSUSY/Λf andtherefore negligible [152]. We are then only left with theradiatively induced effects of Yukawa couplings. The quali-tative arguments above show that flavor physics can providerelevant information on the interplay between the origin ofsupersymmetry and flavor breaking in the observable sector.

As we just saw, the family symmetry that accounts forthe structure of the Yukawa couplings also constrains thestructure of sfermion masses. In the limit of exact flavorsymmetry, this implies family universal, or at least diago-nal, sfermion mass matrices. After the breaking of the flavorsymmetry giving rise to the Yukawa couplings, we can havetwo cases.

– The SUSY breaking mediation mechanism takes place ata scale higher or equal to the flavor symmetry breakingscale and is usually sensitive to flavor. The flavor symme-try breaking accounts for both the structure of the Yukawacouplings and the deviations of the soft breaking termsfrom universality. This is the general expectation in grav-ity mediation of the supersymmetry breaking from thehidden sector.

– The supersymmetry breaking mediation mechanism takesplace at a scale much smaller than the flavor symmetrybreaking scale. In this case the flavor mediation mech-anism, which is flavor-blind, guarantees the universalityof the soft breaking terms. The flavor symmetry breakinggenerates the Yukawa couplings but flavor breaking cor-rections in the soft mass matrices are suppressed by theratio of the two scales. This is the case of gauge-mediationmodels of supersymmetry breaking [152].

We begin discussing the first case.

2.2.4.1 “Tree level” effects of flavor symmetries in super-symmetry breaking terms After the breaking of the fla-vor symmetry responsible for the structure of the Yukawacouplings, we can expect to have non-universal contribu-tions to the soft breaking terms at tree level. Under cer-tain conditions, mainly related to the SUSY-breaking me-diation mechanism, these tree-level contributions can be

26 Eur. Phys. J. C (2008) 57: 13–182

sizable and have important phenomenological effects. Themain example among these models where the tree level non-universality in the soft breaking terms is relevant is providedby models of supergravity mediation [153–157] (for a niceintroduction see the appendix in [158]).

The structure of the scalar mass matrices when SUSYbreaking is mediated by supergravity interactions is deter-mined by the Kähler potential. We are not going to dis-cuss here the supergravity Lagrangian; we refer the inter-ested reader to Refs. [153–156, 158]. For our purposes,we only need to know that the Kähler potential is a non-renormalizable, real, and obviously gauge-invariant, func-tion of the chiral superfields with dimensions of masssquared. This non-renormalizable function includes cou-plings with the hidden sector fields suppressed by differentpowers of MPlanck, φφ∗(1 + XX∗/M2

Planck + · · · ) with φvisible sector fields and X hidden sector fields. This Kählerpotential gives rise to SUSY breaking scalar masses oncea certain field of the hidden sector gets a non-vanishing F-term. The important point here is that these couplings withhidden sector fields that will eventually give rise to the softmasses are present in the theory at any scale belowMPlanck.Below this scale, we can basically consider the hidden sectoras frozen and renormalize these couplings only with visiblesector interactions.

Therefore, in the following, to simplify the discussion,we concentrate only on the soft masses and treat them ascouplings present at all energies below MPlanck. The struc-ture of the soft mass matrices is easily understood in termsof the present symmetries. At high energies, our flavor sym-metry is still an exact symmetry of the Lagrangian and there-fore the soft breaking terms have to respect this symme-try [46]. At some stage, this symmetry is broken generat-ing the Yukawa couplings in the superpotential. In the sameway, the scalar masses will also receive new contributionsafter flavor symmetry breaking from the flavon field VEVssuppressed by mediator masses.

First we must notice that a mass term φ†i φi is clearly in-

variant under gauge, flavor and global symmetries and hencegives rise to a flavor diagonal contribution to the soft masseseven before the family symmetry breaking.6 Then, after fla-vor symmetry breaking, any invariant combination of flavonfields (VEVs) with a pair of sfermion fields, φ†

i φj , can alsocontribute to the sfermion mass matrix and will break theuniversality of the soft masses.

An explicit example with a continuous Abelian U(1) fla-vor symmetry [5, 11, 13, 16, 19, 21, 44, 48, 54] was givenabove in Sect. 2.2.1.

6As we shall discuss in the following, these allowed contributions maybe universal, the same for the different generations, as in the case ofnon-Abelian flavor symmetries, or they can be different for the threegenerations in some cases with Abelian flavor symmetries.

We turn now to the structure of the scalar mass ma-trices concentrating mainly on the slepton mass matrix[13, 14, 16, 43]. In this case, even before the breaking of theflavor symmetry, we have three different fields with differ-ent charges corresponding to each of the three generations.As we have seen, diagonal scalar masses are allowed by thesymmetry, but being different fields, there is no reason a pri-ori for these diagonal masses to be the same, and in generalwe have

Lsymmm2 =m2

1φ∗1φ1 +m2

2φ∗2φ2 +m2

3φ∗3φ3. (2.17)

Notice, however, that this situation is very dangerous, es-pecially in the case of squarks, given that the rotation to thebasis of diagonal Yukawa couplings from (2.2) will generatetoo large off-diagonal entries [43]. In some cases, like dila-ton domination, these allowed masses can be equal avoid-ing this problem. In the following we assume m2

1 = m22 =

m23 = m2

0. However, even in this case, after the breaking ofthe flavor symmetry we obtain new contributions propor-tional to the flavon VEVs that break this universality. Allwe have to do is to write all possible combinations of twoMSSM scalar fields φi and an arbitrary number of flavonVEVs invariant under the symmetry:

Lm2 = m20

(φ∗1φ1 + φ∗2φ2 + φ∗3φ3 +

( 〈θ〉Mfl

)q2−q1

φ∗1φ2

+( 〈θ〉Mfl

)q3−q1

φ∗1φ3 +( 〈θ〉Mfl

)q3−q2

φ∗2φ3 + h.c.

).

(2.18)

Therefore, the structure of the charged slepton mass matrixwe would have in this model at the scale of flavor symmetrybreaking would be (suppressing O(1) coefficients):

m2L⎛⎝

1 ε ε

ε 1 1ε 1 1

⎞⎠m2

0. (2.19)

This structure has serious problems with the phenomeno-logical bounds coming from μ→ eγ , etc. There are otherU(1) examples that manage to alleviate, in part, these prob-lems [43]. However, large LFV effects are a generic prob-lem of these models due to the required charge assignmentsto reproduce the observed masses and mixing angles.

These FCNC problems in the sfermion mass matrices ofAbelian symmetries were one of the main reasons for theintroduction of non-Abelian flavor symmetries [18, 20]. Themechanism used in non-Abelian flavor models to generatethe Yukawa couplings is again a variation of the Froggatt–Nielsen mechanism, very similar to the mechanism we havejust seen for Abelian symmetries. The main difference isthat in this case the left handed fermions are grouped inlarger representations of the symmetry group. For instance,

Eur. Phys. J. C (2008) 57: 13–182 27

in a SU(3) symmetry all three generations are unified in atriplet. In a SO(3) flavor symmetry we can assign the threegenerations to a triplet or to three singlets. In a U(2) flavorsymmetry the third generation is a singlet and the two lightgenerations are grouped in a doublet. Then we do not haveto assign different charges to the various generations, but inexchange, we need several stages of symmetry breaking bydifferent flavon fields with specially aligned VEVs.

We begin analyzing a non-Abelian U(2) flavor sym-metry. As stressed above, if the sfermions mass matricesare only constrained by a U(1) flavor symmetry there isno reason why m2

1 should be close to m22 in (2.17). Un-

less an alignment mechanism between fermions and sfermi-ons is available, the family symmetry should then suppress(m2

1 − m22)/m

2. At the same time, in the fermion sector,the family symmetry must suppress the Yukawa couplingof the first two families, m1,m2 � m3. If the small break-ing of a flavor symmetry is responsible for the smallness of(m2

1 − m22)/m

2 on one hand and of m1/m3,m2/m3 on theother, the symmetric limit should correspond to m2

1 = m22

and to m1 =m2 = 0. Interestingly enough, the largest fam-ily symmetry compatible with SO(10) unification that forcesm1 = m2 = 0 automatically also forces m2

1 = m22. This is a

U(2) symmetry under which the first two families transformas a doublet and the third one, as well as the Higgs, as a sin-glet [16, 18, 20, 24, 26].

ψ =ψa ⊕ψ3.

The same conclusion can be obtained by using discrete sub-groups [30, 64]. In the limit of unbrokenU(2), only the thirdgeneration of fermions can acquire a mass, whereas the firsttwo generations of scalars are exactly degenerate. While thefirst property is not a bad approximation of the fermion spec-trum, the second one is what is needed to keep FCNC andCP-violating effects under control. This observation can ac-tually be considered as a hint that the flavor structure of themass matrices of the fermions and of the scalars are relatedto each other by a symmetry principle. The same physicsresponsible for the peculiar pattern of fermion masses alsoaccounts for the structure of sfermion masses.

The rank 2 ofU(2) allows for a two step breaking pattern:

U(2)ε→U(1)

ε′→ 0, (2.20)

controlled by two small parameters ε and ε′ < ε, to be at theorigin of the generation mass hierarchies m3 �m2 �m1 inthe fermion spectrum. Although it is natural to view U(2)as a subgroup of U(3), the maximal flavor group in the caseof full intra-family gauge unification, U(3) will be anyhowstrongly broken to U(2) by the large top Yukawa coupling.

A nice aspect of the U(2) setting is that there is littlearbitrariness in the way the symmetry breaking fields cou-ple to the SM fermions. This is unlike what happens e.g.

with the choice of fermion charges in the cases of U(1)symmetries. The Yukawa interactions transform as (ψ3ψ3),(ψ3ψa), (ψaψb) (a, b, c, . . . = 1,2). Hence the only rele-vant U(2) representations for the fermion mass matrices are1, φa , Sab and Aab , where S and A are symmetric and an-tisymmetric tensors, and the upper indices denote a U(1)charge opposite to that of ψa . While φa and Aab are bothnecessary, models with [20, 26] or without [24] Sab are bothpossible.

Let us first consider the case with Sab . At leading order,the flavons couple to SM fermions through D = 5 opera-tors suppressed by a flavor scale Λ. Normalizing the flavonsto Λ, it is convenient to choose a basis in which φ2 = O(ε)and φ1 = 0, while A12 = −A21 = O(ε′). If S is present, itturns out to be automatically aligned with φ [27], in such away that in the limit ε′ → 0 a U(1) subgroup is unbroken.More precisely, S22 = O(ε) and all other components es-sentially vanish. We are then led to Yukawa matrices of theform⎛⎝

0 ε′ 0−ε′ ε ε

0 ε 1

⎞⎠ . (2.21)

All non-vanishing entries have unknown coefficients of or-der unity, while still keeping λ12 =−λ21. In the context ofSU(5) or SO(10) unification, the mass relations mτ ≈ mb ,mμ ≈ 3ms , 3me ≈ md are accounted for by the choice ofthe transformations ofAab , Sab under the unified group. Thestronger mass hierarchy in the up quark sector, a peculiarfeature of the fermion spectrum, is then predicted, due tothe interplay of the U(2) and the unified gauge symmetry.

The texture in (2.21) leads to the predictions

∣∣∣∣Vtd

Vts

∣∣∣∣=√md

ms,

∣∣∣∣Vub

Vcb

∣∣∣∣=√mu

mc. (2.22)

While the experimental determination of |Vtd/Vts | based onone loop observables might be affected by new physics, thetree-level determination of |Vub/Vcb| is less likely to be af-fected and at present is significantly away from the predic-tion in (2.22) [29, 39]. A better agreement can be obtainedby (i) relaxing the condition λ12 = −λ21, (ii) allowing forsmall contributions to the 11, 13, 31 entries in (2.21) or by(iii) allowing for asymmetric textures [39]. The latter pos-sibility is realized in models in which the Sab flavon is notpresent [20].

While the model building degrees of freedom in the quarkand charged lepton sector are limited, a virtue of the U(2)symmetry, the neutrino sector is less constrained. This isdue, in the see-saw context, to the several possible choicesinvolved in the modelization of the singlet neutrino massmatrix. This is reflected for example in the possibility to getboth small and large mixing angles [25, 28, 31, 34, 35].

28 Eur. Phys. J. C (2008) 57: 13–182

In the case of an SU(3) flavor symmetry, all three gener-ations are grouped in a single triplet representation, ψi . Inaddition we have several new scalar fields (flavons) whichare either triplets, θ3, θ23 and θ2, or antitriplets, θ3 and θ23.SU(3)fl is broken in two steps: the first step occurs whenθ3 and θ3 get a large VEV breaking SU(3) to SU(2), anddefining the direction of the third generation. Subsequentlya smaller VEV of θ23 and θ23 breaks the remaining sym-metry and defines the second generation direction. To repro-duce the Yukawa textures the large third generation Yukawacouplings require a θ3 (and θ3) VEV of the order of the me-diator scale, Mfl, while θ23/Mfl (and θ23/Mfl) have smallVEVs7 of order ε. After this breaking chain we obtainthe effective Yukawa couplings at low energies throughthe Froggatt–Nielsen mechanism [5] integrating out heavyfields. The resulting superpotential invariant under SU(3)would be

WY = Hψiψcj[θi3θ

j

3 + θi23θj

23 + εiklθ23,kθ3,lθj

23(θ23θ3)

+ εijkθ23,k(θ23θ3)2 + εijkθ3,k(θ23θ3)(θ23θ23)

+ · · · ]. (2.23)

In this equation we can see that each of the SU(3) indicesof the external MSSM particles (triplets) are either satu-rated individually with an antitriplet flavon index (a “me-son” in QCD notation) or in an antisymmetric couplingswith other two triplet indices (a “baryon”). The presenceof other singlets in the different term is due to the pres-ence of additional global symmetries necessaries to en-sure the correct hierarchy in the different Yukawa elements[37, 45, 46]. This structure is quite general for the dif-ferent SU(3) models we can build. Here we are not spe-cially concerned with additional details and we refer to[37, 45, 46] for more complete examples. The Yukawatexture we obtain with this superpotential is the follow-ing:

Yf =⎛⎜⎝

0 αε3 βε3

αε3 ε2

a2 γ ε2

a2

βε3 γ ε2

a2 1

⎞⎟⎠a2, (2.24)

with a = 〈θ3〉M

, and α,β, γ unknown coefficients of or-der O(1).

Let us now analyze the structure of scalar soft masses. Inanalogy with the Abelian case, in the unbroken limit diago-nal soft masses are allowed. However, the three generationsbelong to the same representation of the flavor symmetryand now this implies the mass is the same for the whole

7In fact, in realistic models reproducing the CKM mixing matrix, thereare two different mediator scales and expansion parameters, ε in the upquark and ε in the down quark sector [37, 45, 46].

triplet. After the breaking of SU(3) symmetry the scalar softmasses deviate from exact universality [46, 160–162]. Anyinvariant combination of flavon fields can also contribute tothe sfermion masses, although flavor symmetry indices canbe contracted with fermion fields. Including these correc-tions the leading contributions to the sfermion mass matricesare given by

(m2f

)ij = m20

(δij + 1

M2f

[θi†3 θ

j

3 + θi†23θj

23

]

+ 1

M4f

(εiklθ3,kθ23,l

)†(εjmnθ3,mθ23,n

)). (2.25)

Notice that each term inside the parentheses is trivially neu-tral under the symmetry because it contains always a fieldtogether with its own complex conjugate field. However, asthe flavor indices of the flavon fields are contracted with theexternal matter fields this gives a non-trivial contribution tothe sfermion mass matrices. Therefore in this model, sup-pressing factors of order 1 we have,

m2f⎛⎝

11

1

⎞⎠m2

0 +

⎛⎜⎜⎝ε2 0 0

0 ε2

a2ε2

a2

0 ε2

a2 1

⎞⎟⎟⎠a2m2

0, (2.26)

with a = 〈θ3〉/Mfl which is still O(1). In the model [37, 45,46], the expansion parameter for right handed down quarksand charged leptons is ε = 0.15. Using (2.24) and (2.26) wecan obtain the slepton mass matrix in the basis of diagonalcharged lepton Yukawa couplings:

m2eR

⎛⎝

1 + ε2 −ε3 −ε3

−ε3 1 + ε2 ε2

−ε3 ε2 1

⎞⎠m2

0, (2.27)

where we have used a3 O(Mfl). Therefore that gener-ates the order ε3 entry in the (1,2) element. The moduloof this entry is order 3 × 10−3 at MGUT. These estimates atMGUT are slightly reduced through renormalization groupevolution to the electroweak scale and is order 1 × 10−3 atMW. This value implies that supersymmetric contribution toμ→ eγ is very big and can even exceed the present boundsfor light slepton masses and large tanβ if we are not in thecancellation region[163–165]. This makes this process per-haps the most promising one to find deviations from uni-versality in flavor models. The presence of the SU(3) flavorsymmetry controls the structure of the sfermion mass ma-trices and the supersymmetric flavor problem can be nicelysolved. However, interesting signals of the supersymmetricflavor structure can be found in the near future LFV experi-ments.

Eur. Phys. J. C (2008) 57: 13–182 29

3 Observables and their parameterization

3.1 Effective operators and low scale observables

In spite of the clear success of the SM in reproducing allthe known phenomenology up to energies of the order ofthe electroweak scale, nobody would doubt the need of amore complete theory beyond it. There remain many funda-mental problems such as the experimental evidence for darkmatter (DM) and neutrino masses, as well as the theoreticalpuzzles posed by the origin of flavor, the three generations,etc., that a complete theory should address. Therefore, wecan consider the SM as the low energy effective theory ofsome more complete model that explains all these puzzles.Furthermore, we have strong reasons (gauge hierarchy prob-lem, unification of couplings, dark matter candidate, etc.)to expect the appearance of new physics close to the elec-troweak scale. Suppose that these new particles from themore complete theory are to be found at the LHC. Exper-iments at lower energies E < mNP are also sensitive [166]to this new physics (NP). Indeed the exchange of new parti-cles can induce:

– corrections to the SM observables (such as S, T and U),and

– the appearance of new observables or new (d > 4) opera-tors, (e.g. the flavor violating dipole operators).

Note that both effects can be parameterized by SU(3) ×SU(2)×U(1)-invariant operators of mass dimension d > 4.We refer to these non-renormalizable operators as effectiveoperators. Any NP proposed to explain new phenomena atthe LHC must satisfy the experimental constraints on the ef-fective operators it generates.

3.1.1 Effective Lagrangian approach: Leff

Considering the SM as an effective theory below the scaleof NP, mNP, where the heavy fields have been integratedout, we can describe the physics through an effective La-grangian, Leff. This effective Lagrangian contains all pos-sible terms invariant under the SM gauge group and builtwith the SM fields. Besides the usual SM fields, we couldintroduce new light singlet fermions with renormalizableYukawa couplings to the lepton doublets (and possibly smallMajorana masses) to accommodate the observed neutrinomasses. In this case we would have more operators allowedin the effective Lagrangian of the SM + extra light sterilestates. On the assumption that the light sterile particles areweakly interacting, if present, and therefore not relevant tothe LHC, we focus on the effective Lagrangian that can beconstructed only from the known SM fields. Then, the effec-tive Lagrangian at energies E � mNP can be written as an

expansion in 1/mNP as,

LSMeff = L0 + 1

mNPL1 + 1

m2NP

L2 + 1

m3NP

L3 + · · · , (3.1)

where L0 is the renormalizable SM Lagrangian contain-ing the kinetic terms of the U(1), SU(2) and SU(3) gaugebosons Aμ, the gauge interactions and kinetic terms of theSM fermions, {f }, and Higgs, and the Yukawa couplings ofthe Higgs and SM fermions. In order to fix the notation, welist the SM fermions as

qi =(uLidLi

), �i =

(νLieLi

),

uRi, dRi, eRi,

(3.2)

where i is a flavor/family/generation index. Note that in thefollowing we use always four-component Dirac spinors inthe different Lagrangians. Explicit expressions, for L0 insimilar notation, can be found in [167].

The different Ln are Lagrangians of dimension d = 4+ninvariant under SU(3)× SU(2)×U(1) and can be schemat-ically written

Ln =∑a

Ca · Oa

(H, {f }, {Aμ}

)+ h.c. (3.3)

The local operators Oa are gauge invariant combinations ofSM fields of dimension 4 + n. Their coefficient, which inthe full Lagrangian has mass dimension −n, is unknown inbottom-up effective field theory, but calculable in NP mod-els. We write this coefficient as a dimensionless Ca dividedby the nth power of the mass scale of the NP mediator,mnNP,which for new physics relevant at LHC energies would bemNP ∼√

sLHC. We shall later normalize to GF (see (3.21)).We are mainly interested in dimension five and di-

mension six operators. We assume that any particles cre-ated at the LHC could generate dimension six operators,and then we can neglect higher dimension operators con-tributing to the same physical processes. Operators of di-mension 7 include the lepton number violating opera-tor εabεcdHa�b[iσμνHc�

dj ]Fμν which gives neutrino transi-

tion moments (flavor-changing dipole moments) after elec-troweak symmetry breaking (EWSB). At dimension 8 aretwo-Higgs-four-fermion operators, which can give four-fermion operators after EWSB, with a different flavor struc-ture from the dimension six terms. We shall not analyzethese operators here, but they are studied in the context ofnon-standard neutrino interactions [168]. Therefore, in thefollowing, we restrict our analysis to L1 and L2.

The unique operator allowed with the standard modelfields and symmetries at dimension five is Oij

�� =εabεmnH

a�cb

i Hm�nj (a, b,n,m are SU(2) indices). Thus we

have,

L1 = 1

4κij

ν�� · εabεmnHa�cb

i Hm�nj + h.c., (3.4)

30 Eur. Phys. J. C (2008) 57: 13–182

where �c is the charge conjugate of the lepton doublet. Af-ter electroweak symmetry breaking, this gives rise to a Ma-jorana mass matrix 1

4κij��〈H 0〉2νciνj + h.c. In the neutrino

mass eigenstate basis, the masses are κii��〈H 0〉2/2. The co-

efficient κij�� = 2YkiM−1k Ykj is generated for instance after

integrating out heavy right handed neutrinos of mass Mk ina see-saw mechanism with Yukawa coupling Y .

L2 is constructed with dimension-six operators built outof SM fields. An exhaustive list is given in [167], includ-ing operators with Higgs, W± and Z0 external legs. Herewe list operators which give interactions among leptons andphotons, and leptons and quarks. We can classify the possi-ble operators according to the external legs as follows:

– operators with a pair of leptons and an (on-shell) photon:

OijeB = �iσμνeRjHBμν,

(3.5)OijeW = �iσμντ I eRjHWIμν,

– four-lepton operators, with Lorenz structure LLLL,RRRR or LRRL, singlet or triplet SU(2) gauge contrac-tions (described in the operator subscript), and all possibleinequivalent flavor index combinations (see Sect. 3.1.2).The SU(2)×U(1) invariant operators, with flavor indicesin the superscript, are

Oijkl(1)�� =

(�iγ

μ�j)(�kγμ�l),

Oijkl(3)�� =

(�iτ

I γ μ�j)(�kτ

I γμ�l),

Oijklee = (eiγ μPRej

)(ekγμPRel),

Oijkl�e = (�iej )(ek�l),

(3.6)

– two lepton two-quark operators, with Lorentz structureLLLL, RRRR or LRRL, singlet or triplet SU(2) gaugecontractions (described in the operator subscript), andall possible inequivalent flavor index combinations (seeSect. 3.1.2). The S(3) × SU(2) × U(1) invariant opera-tors, with color indices implicit and flavor indices in thesubscript, are

Oijkl(1)�q =

(�iγ

μ�j)(qkγμql),

Oijkl(3)�q =

(�iτ

I γ μ�j)(qkτ

I γμql),

Oijkled = (eiγ μPRej

)(dkγμPRdl),

Oijkleu = (eiγ μPRej

)(ukγμPRul),

Oijkl�u = (�iul)(uk�j ), Oijkl

�d = (�idl)(dk�j ),Oijkl�qS = (�iej )(qkul), Oijkl

qde = (�iej )(dkql).

(3.7)

Therefore the Lagrangian L2 for leptons only is

L2 = CijeB · OijeB +CijeW · Oij

eW

+ 1

1 + δ(C

ijkl(1)�� · Oijkl

(1)�� +Cijkl(3)�� · Oijkl

(3)��

+Cijklee · Oijkl

ee + 2Cijkl�e · Oijkl

�e .)+ h.c., (3.8)

where we introduce the parameter δ to cancel possible fac-tors of 2 that can arise from the +h.c.: it is 1 for Oij ...

... =[Oij ...... ]†; otherwise it is 0. The sums over i, j, k, l run over

inequivalent operators, taking an operator to be inequivalentif neither it, nor its h.c., are already in the list. The factor of2 in the definition of O�e is included to compensate the 1/2in the Fierz rearrangement below (second line of (3.13)).8

The effective operators whose coefficients we constrain inthe next section are related to those of (3.8) through an ex-pansion in terms of the SU(2) components of the fields andtaking into account the electroweak symmetry breaking. Forexample, for the lepton operators:

OijeB = �iσμνeRjHBμν = cos θW 〈H 〉eiσμνPRejF em

μν , (3.9)

OijeW = �iσμντ I eRjHWIμν

=− sin θW 〈H 〉eiσμνPRejF emμν , (3.10)

Oijkl(1)�� =

(�iγ

μ�j)(�kγμ�l)

= (νiγ μPLνj + eiγ μPLej)

× (νkγμPLνl + ekγμPLel), (3.11)

Oijkl(3)�� =

(�iτ

I γ μ�j)(�kτ

I γμ�l)

= 2(νiγ

μPLej)(ekγμPLνl)

+ 2(eiγ

μPLνj)(νkγμPLel)

+ [(νiγ μPLνj)(νkγμPLνl)

+ (eiγ μPLej)(ekγμPLel)

− (νiγ μPLνj)(ekγμPLel)

− (eiγ μPLej)(νkγμPLνl)

], (3.12)

Oijkl�e = 2(�iej )(ek�l)

= 2[(νiPRej )(ekPLνl)+ (eiPRej )(ekPLel)

]

=−[(νiγ μPLνl)(ekγμPRej )

+ (eiγ μPLel)(ekγμPRej )

]. (3.13)

All these operators, together with O ijklee , induce dipole

moments and four-charged-lepton (4CL) vertices, as appear

8Note there will sometimes be other 2 s for identical fermions.

Eur. Phys. J. C (2008) 57: 13–182 31

to the right-hand side (RHS) in the above equations. Con-straints on the coefficients of the 4CL operators

OijklPP = 1

1 + δ(eiγ

μP ej)(ekγμPel),

OijklRL = 1

1 + δ(eiγ

μPRej)(ekγμPLel),

(3.14)

where P = PR or PL, are listed in Tables 4, 5, 6 and 7.After electroweak symmetry breaking, the operators Oij

eB

and OijeW become the chirality-flipping dipole moments as

written in (3.9), (3.10) (where we did not include the Z–lepton–lepton operators [169]). These dipoles can be flavorconserving or transition dipole moments. The flavor diag-onal operators are specially interesting because they corre-spond to the anomalous magnetic moments and the electricdipole moments of the different fermions. TakingCijeγ (q2)=CijeB(q

2) cos θW −CijeW (q2) sin θW as the Wilson coefficientwith momentum transfer equal to q2, we have for q2 = 0,

Ciieγ (q2 = 0)

m2NP

〈H 〉eiσμνPReiF emμν + h.c.

= Re{Ciieγ (q2 = 0)}m2

NP

〈H 〉eiσμνeiF emμν

+ Im{Cijeγ (q2 = 0)}m2

NP

〈H 〉ieiσμνγ5eiFemμν

= e aei4mei

eiσμνeiF

emμν + i

2dei eiσ

μνγ5eiFemμν , (3.15)

with aei = (gei −2)/2 the anomalous magnetic moment anddei the electric dipole moment of the lepton ei that can befound in [170].

In a given model, the coefficients of the effective op-erators can be obtained by matching the effective theoryof (3.1) onto the model, at some matching scale (for in-stance, the mass scale of new particles). However, in partic-ular models there can appear various pitfalls in constrainingthe generic coefficients Cijkl

... . This is illustrated, for example,in the model of [171] which corresponds to adding a singletslepton Ec of flavor k, in R-parity violating (RPV) SUSY. Inthis case, after integrating out the heavy slepton we obtainthe following effective operator:

λk[ij ]λ∗k[mn]

M2

((νL)cieLj

)((eL)n(νL)

cm

)

= λk[ij ]λ

∗k[mn]2M2

(enγ

μPLej)(νmγμPLνi), (3.16)

where λk[ij ] is antisymmetric in i, j because the SU(2) con-traction of �i�j is antisymmetric. This is an example of op-

erator O��(1), but since it is induced by singlet scalar ex-change, there is no four-charged-lepton operator (compareto (3.11)). This illustrates that the bounds obtained here, byassuming that Cijkl

... �= 0 for one choice of ijkl at a time, arenot generic. Each process receives contributions from a sumof operators, and that sum could contain cancellations in aparticular model.

Many models of new physics introduce new TeV-scaleparticles carrying a conserved quantum number (e.g. R-parity, T-parity. . . ). Such particles appear in pairs at ver-tices, so they contribute via boxes and penguins to thefour-fermion and dipole moment operators considered here.Generic formulae for the one loop contribution to a dipolemoment can be found in [172], and for boxes in [173]. ExtraHiggses [174, 175] would contribute to the same operatorsconstructed from SM fields, so they are constrained by theexperimental limits on the coefficients of such operators.

3.1.2 Constraints on low scale observables

In this section we present the low energy constraints onthe different Wilson coefficients introduced before. Any NPfound at LHC will necessarily respect the bounds presentedhere.

3.1.2.1 Dipole transitions After electroweak symmetrybreaking, the operators of (3.9), (3.10) generate magneticand electric dipole moments for the charged leptons. Flavor-diagonal operators give rise to anomalous magnetic mo-ments and electric dipole moments as shown in (3.15).The anomalous magnetic moment of the electron ae =(g − 2)e/2 is used to determine αem. The current measure-ment of the muon anomalous moment aμ = (g − 2)μ/2 de-viates from the (uncertain) SM expectation by 3.2σ usinge+e−-data [176], and can be taken as a constraint, or indica-tion on the presence of new physics. Currently there is onlyan upper bound on the magnetic moment of the τ from theanalysis of e+e− → τ+τ− [170, 177]. Electric dipole mo-ments have not yet been observed, although we have veryconstraining bounds specially on the electron dipole mo-ment. In Table 3 we present the bounds of flavor diagonal di-pole moments. The EDMs are discussed in detail in Sect. 5.

The bounds on off-diagonal dipole transitions are pre-sented in Table 3. It is convenient to normalize these coef-ficients, Cijeγ = CijeB cos θW − CijeW sin θW , to the Fermi in-teractions given our ignorance on the scale of new physicsmNP:

Cijeγ

m2NP

= 4GF√2εijeγ . (3.17)

In the literature, it is customary to use the left and right formfactors for lepton flavor violating transitions defined by

32 Eur. Phys. J. C (2008) 57: 13–182

Table 3 Bounds on the different dipole coefficients. Flavor diagonaldipole coefficients are given in terms of the corresponding anomalousmagnetic moment, aei , and the dipole moment, dei . Bounds on transi-

tion moments are given in terms of the dimensionless coefficients |εijeγ |

(defined in (3.17)) from the bounds on the branching ratios given inthe last column. These bounds apply also both to |εijeγ | and |εjieγ |. SeeSect. 3.1.2 for details

(ij ) ai = gi−22 edmi (e cm) Ref.

ee 0.0011596521859(38) de ≤ 1.6 × 10−27 PDG [170, 186]

μμ 11659208.0(5.4)(3.3)× 10−10 dμ ≤ 2.8 × 10−19 Muon g-2 Coll. [187, 188]

ττ −0.052< aτ < 0.013 (−2.2< dτ < 4.5)× 10−17 LEP2 [189], Belle [190]

(ij ) �iσμνeRjF

emμν expt. limit Ref.

eμ ≤3.4 × 10−11 ≤1.2 × 10−11 MEGA Coll. [180]

eτ ≤1.2 × 10−7 ≤1.1 × 10−7 BaBar [182]

μτ ≤8.4 × 10−8 ≤4.5 × 10−8 Belle, BaBar [181, 191]

�L2 = emliAμf j[iσμνqν

(AijLPL +AijRPR

)]fi + h.c.

(3.18)

where f is a Dirac (4-component) fermion. The radiativedecay fi → fj + γ proceeds at the rate Γ =m5

i e2/(16π)×

(|AijL |2 + |AijR |2) [178]. QED corrections to those decaysare unusually large and may reach as much as 15% [179].Bounds on the dimensionless coefficients Cijeγ and εijeγ can

be obtained by translating from AijL and AijR :

Cijeγ

m2NP

〈H 〉 = emi2AijR ,

Cji∗eγ

m2NP

〈H 〉 = emi2AijL . (3.19)

The experimental bounds on radiative lepton decays canbe used to set bounds on these off-diagonal Wilson coeffi-cients. The current experimental bounds are B(μ→ eγ ) <

1.2 × 10−11 [180], B(τ → μγ ) < 4.5 × 10−8 [181], andB(τ→ eγ ) < 1.1 × 10−7 [182].

For the off-shell photon, q2 �= 0, there exist additionalform factors,

�L = emliAμej[(gμν − qμqν

q2

)γν(BijL PL +BijR PR

)]ei

+ h.c., (3.20)

which induce contributions to the four-fermion operatorsto be discussed in the next subsections. These form fac-tors may be enhanced by a large factor compared to theon-shell photon form factors [184], ln(mNP/mli ), depend-ing on the nature of new physics. Therefore, those operatorsbecome relevant for constraining new physics in R-parity vi-olating SUSY [185] and in low-scale type-II see-saw mod-els [184].

3.1.2.2 Four-charged-lepton operators As before, to pres-ent the bounds on the dimensionless four-charged-fermion

coefficients in (3.14), we normalize them to the Fermi inter-actions:

Cijkl(n)��

m2NP

=−4GF√2ε

ijkl(n)��,

Cijklee

m2NP

=−4GF√2ε

ijklee ,

(3.21)Cilkj�e

m2NP

= 4GF√2ε

ijkl�e .

The current low energy constraints on the dimensionlessε’s are shown in Tables 4, 5, 6 and 7. The rows of the ta-bles are labeled by the flavor combination, and the columnby the Lorentz structure. The numbers given in this tablescorrespond to the best current experimental bound on thecoefficient of each operator, assuming it is the only non-zero coefficient present. The last column in the table liststhe experiment setting the bound. The compositeness searchlimits Λ@ LEP are at 95% C.L., the decay rate bounds at90% C.L.

Regarding the definition of the different coefficients wehave to make some comments. First, note the flavor indexpermutation between C�e and ε�e:

Cilkj

�e (�iel)(ek�j )=−1

2ε

ijkl�e

(�iγ

μ�j)(ekγμel). (3.22)

There are relations between the flavor indices of the differ-ent operators. For OLL = (eγ μPLe)(eγμPLe) and ORR =(eγ μPRe)(eγμPRe) we have

OijklPP = Oklij

PP , OijklPP = O∗jilk

PP , OijklPP = Oilkj

PP ,

(3.23)

by symmetry, Hermitian conjugation and Fierz rearrange-ment, respectively. Therefore, the constraints on eeμτ in thefirst two columns of Tables 4 to 7 apply to εeeμτ(n)xx , εμτee(n)xx ,

ε∗eeτμ(n)xx

, ε∗τμee(n)xx

, εeτμe(n)xx

, εμeeτ(n)xx

, ε∗τeeμ(n)xx

, and ε∗eμτe(n)xx

with (n)xx

Eur. Phys. J. C (2008) 57: 13–182 33

Table 4 Bounds on coefficients of flavor four-lepton operators, fromfour-charged-lepton processes. The number is the upper bound on thedimensionless operator coefficient εijkl (defined in (3.21)), arising fromthe measurement in the last column. The bound applies also to εklij .The second column is the bounds on εijkl

(3)��, and εijkl(1)�� [except in the

case of the bracketed limits, which are the upper bound on εijkl(1)�� and

2εijkl(1)��]. The third column is the bound on εijkl

(1)ee . The bounds in these

two columns apply also when the flavor indices are permuted to jilkand ilkj . The fourth column is the bound on εijkl

�e (which does not apply

to the flavor permutation ilkj , so this is listed with a line of its own).

The constraints in [brackets] apply to the two charged lepton–two

neutrino operator of the same flavor structure, and arise from lepton

universality in τ decays. See Sect. 3.1.2 for details

(ijkl) (eγ μPLe)(eγμPLe) (eγ μPRe)(eγμPRe) (eγμPLe)(eγμPRe) expt. limit Ref.

eeee (−1.8 −+2.8)× 10−3 (−1.8 −+2.8)× 10−3 (−2.4 −+4.9)× 10−3 Λ@LEP2 [194]

eeμμ (−7.2 −+5.2)× 10−3 (−7.8 −+5.8)× 10−3 (−9.0 −+9.6)× 10−3 Λ@LEP2 [193, 195]

eμμe (−7.2 −+5,2)× 10−3 (−7.8 −+5.8)× 10−3 1.3 × 10−2 Λ,RPV@LEP2 [193, 195]

eeττ (−7.3 −+13)× 10−3 (−8.0 −+15)× 10−3 (−1.2 −+1.8)× 10−2 Λ@LEP2 [193, 195]

τeeτ (−7.3 −+13)× 10−3 (−8.0 −+15)× 10−3 1.3 × 10−2 Λ,RPV@LEP2 [193, 195]

μμμμ ∼1 ∼1 ∼1 B(Z→ μμ)

μμττ ∼1 [0.0014] ∼1 ∼1 [0.01] B(Z→ μμ)

μττμ ∼1 [0.0014] ∼1 B(Z→ μμ)

ττττ ∼1 ∼1 ∼1 B(Z→ τ τ )

Table 5 Bounds on coefficients of four-lepton operators with �Lα =−�Lβ = 1. They apply also to flavor index permutations klij and ilkj ,except in the case of ττeμ, where the bound on τμeτ in the fourth col-

umn is from μ decay and is listed separately. See the caption of Table 4and Sect. 3.1.2 for further details

(ijkl) (eγ μPLe)(eγμPLe) (eγ μPRe)(eγμPRe) (eγμPLe)(eγμPRe) expt. limit

eeeμ 7.1 × 10−7 7.1 × 10−7 7.1 × 10−7 B(μ→ eee) < 10−12

eeeτ 7.8 × 10−4 7.8 × 10−4 7.8 × 10−4 B(τ→ eee) < 2 × 10−7

eeμτ 1.1 × 10−3 1.1 × 10−3 1.1 × 10−3 B(τ→ eeμ) < 1.9 × 10−7

μμeμ ∼1 ∼1 ∼1 B(Z→ eμ) < 1.7 × 10−6

μμeτ 1.1 × 10−3 1.1 × 10−3 1.1 × 10−3 B(τ→ μeμ) < 2.0 × 10−7

μμμτ 7.8 × 10−4 7.8 × 10−4 7.8 × 10−4 B(τ→ 3μ) < 1.9 × 10−7

ττeμ ∼1 [0.05] ∼1 ∼1 [0.05] B(Z→ eμ) < 1.7 × 10−6

τμeτ ∼1 [0.05] ∼1 [0.05] B(Z→ eμ) < 1.7 × 10−6

ττeτ ∼3 [0.05] ∼3 ∼3 [0.05] B(Z→ eτ ) < 9.8 × 10−6

τττμ ∼3 [0.05] ∼3 ∼3 [0.05] B(Z→ τ μ) < 1.2 × 10−5

Table 6 Bounds on coefficients of four-lepton operators with �Lα =�Lβ = 2. See the caption of Table 4 and Sect. 3.1.2 for details

(ijkl) (eγ μPLe)(eγμPLe) (eγ μPRe)(eγμPRe) (eγμPLe)(eγμPRe) expt. limit

eμeμ 3.0 × 10−3 3.0 × 10−3 2.0 × 10−3 (μe)↔ (eμ)

eτeτ [0.05] [0.05]μτμτ [0.05] [0.05]

Table 7 Bounds on coefficients of four-lepton operators with �Lα =�Lβ =− 12�Lρ . See the caption of Table 4 and Sect. 3.1.2 for details

(ijkl) (eγ μPLe)(eγμPLe) (eγ μPRe)(eγμPRe) (eγ μPLe)(eγμPRe) expt. limit

eμeτ 2.3 × 10−4 2.3 × 10−4 2.3 × 10−4 B(τ→ μee) < 1.1 × 10−7

μeμτ 2.6 × 10−4 2.6 × 10−4 2.6 × 10−4 B(τ→ eμμ) < 1.3 × 10−7

τeτμ [0.05] [0.05]

34 Eur. Phys. J. C (2008) 57: 13–182

equal to (3)��, (1)��, or (1)ee. Note, however that it is cal-culated assuming only one of these ε is non-zero. Similarly,the operator Oijkl

LR = (eiγμPLej )(ekγ μPRel), with coeffi-

cient εijkl�e , is related by Hermitian conjugation:

OijklLR = O∗jilk

LR , (3.24)

so again the bounds on εijkl�e apply to ε∗jilk�e . We can usually

apply also these bounds to εklij�e because the chirality of thefermion legs does not affect the matrix element squared, butεilkj

�e is bounded separately in the tables.The bounds from Z decays in Tables 4 and 5 are esti-

mated from the one loop penguin diagram obtained closingtwo of the legs of the four-fermion operator and couplingit with the Z [192]. These bounds would be more correctlyincluded by renormalization group mixing between the four-fermion operators and the Z–fermion–fermion operators dis-cussed in [169]. They are listed in the tables to indicate theexistence of a constraint. The bound can be applied to εiikl�e

and εijkk�e but it does not apply to εilki�e .Contact interaction bounds are usually quoted on the

scale Λ, where

εijklab

4GF√2

=± 1

1 + δ4π

Λ2, (3.25)

and δ = 1 for the operators OeeeeLL and Oeeee

RR of (3.14), 0 oth-erwise. Since our normalization does not have this factorof 2, we have a Feynman rule ε8GF/

√2 for these oper-

ators, and correspondingly stricter bounds on the ε’s. Thebounds are the same for εikki�e and εkiik�e . However, contactinteraction bounds are not quoted on operators of the form(eiγ

μPLej )(ej γμPRei), corresponding to εiijj�e . Such oper-ators are generated by sneutrino exchange in R-parity violat-ing SUSY, so we estimate the bound λ2/m2

ν< 4/(9 TeV2)

from the plotted constraints in [193], and impose 4|εijklab |GF/√

2< λ2/(2m2ν).

Many of the 4CL operators involving two τ ’s are poorlyconstrained. In some cases, see (3.11), (3.12), new physicsthat generates 4CL operators also induces (eiγ λP ej )×(νkγλLνl). The coefficients of operators of the form(μγ λP e)(νkγλLνl), (μγ λP τ)(νkγλLνl) or (eγ λP τ)×(νkγλLνl), are constrained from lepton universality mea-surements in μ and τ decays [196]. The decay rate τ →eiνkνl in the presence of the operators of (3.14), divided bythe SM prediction for τ→ eiντ νi , is(

1 − 2δkτ δil Re{εττ ii(1)�� + 2εττ ii(3)��

}+ 4mimτδkτ δil Re

{εττ ii�e

}

+ ∣∣εiτkl(1)��

∣∣2 + 4∣∣εiτkl(3)��

∣∣2 + ∣∣εiτkl�e

∣∣2). (3.26)

Within the experimental accuracy, the weak τ and μ de-cays verify lepton universality and agree with LEP precision

measurements of mW . Rough bounds on the ε’s can there-fore be obtained by requiring the new physics contributionto the decay rates to be less than the errors �B

B(τ→ eνν)=

0.05/17.84, �BB(τ → μνν) = 0.05/17.36. These are listed

in the tables in [brackets]. The bracketed limit in the sec-ond column applies to εijkl

(1)��; the bound on εijkl(3)�� is 1/2 the

quoted number. The limit on ετeτμ�e is from its contributionto μ→ eντ ντ .

Finally, we would like to remind the reader the variouscaveats to these four-fermion vertex bounds.

– The constraints are calculated “one operator at a time”.This is unrealistic; new physics is likely to induce manynon-renormalizable operators. In some cases, see (3.16),a symmetry in the new physics can cause cancellationssuch that it does not contribute to certain observables.

– The coefficients of the 4CL operators, and two ν–twocharged lepton (2ν2CL) operators may differ by a factorof few, because they are induced by the exchange of dif-ferent members of a multiplet, whose masses differ [197].

– The list of operators is incomplete. Perhaps some ofthe neglected operators give relevant constraints on newphysics. For instance, bounds from lepton universality onthe (H ∗�)γ μ∂μ(H�) operator [198] are relevant to extradimensional scenarios [199].

– Operators of dimension >6 are neglected. If the massscale of the new physics is ∼TeV, then higher dimensionoperators with Higgs VEVs [200] such as HHψψψψare not significantly suppressed.

3.1.2.3 Two lepton–two quark operators Once more, wenormalize the coefficients of the two lepton–two quark op-erators in (3.6) to the Fermi interactions:

Cijkl(n)�q

m2NP

=−4GF√2ε

ijkl(n)�q,

Cijkled

m2NP

=−4GF√2ε

ijkled ,

Cijkl�d

m2NP

= 4GF√2ε

ijkl�d ,

Cijkleu

m2NP

=−4GF√2ε

ijkleu ,

(3.27)

Cijkl�u

m2NP

=−4GF√2ε

ijkl�u ,

Cijkl�qS

m2NP

=−4GF√2ε

ijkl�qS,

Cijklqde

m2NP

=−4GF√2ε

ijklqde.

The main bounds on the dimensionless εs are given in Ta-bles 8 and 9. These numbers correspond to the best cur-rent experimental bound on the coefficient of each opera-tor, assuming it is the only non-zero coefficient present. Thebounds on ε�q in Table 8 apply both to ε(1)�q and ε(3)�q .These bounds have been obtained from the correspondingbounds on leptoquark couplings in Refs. [201, 202] that canbe checked for further details.

Eur. Phys. J. C (2008) 57: 13–182 35

Table 8 Bounds on coefficients of the left handed two quark–two lep-ton operators. Bound is the upper bound on the dimensionless operatorcoefficient εijkl (defined in (3.28)), arising from the experimental de-

termination of the observable in the next column. Bounds with a ∗ arealso valid under the exchange of the lepton indices

(eγ μPLe)(qγμPLq)

(ijkl) Bound on εijkl�q Observable (ijkl) Bound on εijkl

�q Observable

11 11 5.1 × 10−3 Rπ 22 11 5.1 × 10−3 Rπ

12 11 8.5 × 10−7 μ–e conversion on Ti 12 12∗ 2.9 × 10−7 B(K0L→ μe)

ij 12 4.5 × 10−6 B(K+→π+νν)B(K+→π0e+νe)

ij 22 1.0 Vcs

ij 13 3.6 × 10−3 Vub ij 23 4.2 × 10−2 Vcb

11 23 6.6 × 10−5 B(B+ → e+e−K+) 11 13 9.3 × 10−4 B(B+ → e+e−π+)22 23 5.4 × 10−5 B(B+ → μ+μ−K+) 22 13 1.4 × 10−3 B(B+ → μ+μ−π+)21 23∗ 4.5 × 10−3 B(B+ → e+μ−K+) 21 13∗ 3.9 × 10−5 B(B+ → e+μ−π+)12 23∗ 1.2 × 10−2 B(B0

s → μ+e−) 33 12 6.6 × 10−2 K–K

22 22 6.0 × 10−2 B(D+s →μ+νμ)

B(D+s →τ+ντ ) 33 22 6.0 × 10−2 B(D+

s →μ+νμ)B(D+

s →τ+ντ )32 23∗ 1.2 × 10−3 B(B+ → μ+τ−X+) 33 23 9.3 × 10−3 B(B+ → τ+τ−X+)

Table 9 Bounds on coefficients of the right handed vector and scalar 2quark-2 lepton operators. Bound is the upper bound on the dimension-less operator coefficient εijkl (defined in (3.28)), arising from the ex-

perimental determination of the observable in the next column. Boundswith a ∗ are also valid under the exchange of the lepton indices

(eγ μPRe)(qγμPRq)

(ijkl) Bound on εijkleu Observable (ijkl) Bound on εijkl

eu Observable

11 12 1.7 × 10−2 B(D+→π+e+e−)B(D0→π−e+νe)

21 12∗ 1.3 × 10−2 B(D+→π+μ−e+)B(D0→π−e+νe)

22 12 9.0 × 10−3 B(D+→π+μ+μ−)B(D0→π−e+νe)

33 12 0.19 B(D0 −D0)

(�PRe)(dPLq)

(ijkl) Bound on εijklqde Observable (ijkl) Bound on εijkl

qde Observable

11 11 1.5 × 10−7 Rπ 22 11 3.0 × 10−4 Rπ

12 11 5.1 × 10−3 B(π+ → μ+νe) 12 12∗ 2.1 × 10−8 B(K0L→ μ+e−)

11 12 2.7 × 10−8 B(K0L→ e+e−) 22 12 8.4 × 10−7 B(K0

L→ μ+μ−)

22 21 1.3 × 10−2 B(D+ → μ+νμ) 22 22 1.2 × 10−2 B(D+s →μ+νμ)

B(D+s →τ+ντ )

33 22 0.2 B(D+s →μ+νμ)

B(D+s →τ+ντ ) 33 13 2.5 × 10−3 B(B+ → τ+ντ )

11 13 9.0 × 10−5 B(B0 → e+e−) 12 13∗ 1.2 × 10−4 B(B0 → μ+e−)

13 13∗ 2.5 × 10−3 B(B0 → τ+e−) 23 13∗ 3.3 × 10−3 B(B0 → τ+μ−)

22 13 7.5 × 10−5 B(B0 → μ+μ−) 11 23 6.0 × 10−4 B(B0s → e+e−)

12 23∗ 2.1 × 10−4 B(B0s → μ+e−) 22 23 1.2 × 10−4 B(B0

s → μ+μ−)

3.2 Phenomenological parameterizations of quark andlepton Yukawa couplings

3.2.1 Quark sector

The quark Yukawa sector is described by the following La-grangian:

Lquark = ucRiY uijQjH + dcRiY dijQjH + h.c., (3.28)

where i, j = 1,2,3 are generation indices, Qi = (dLi, uLi)are the left handed quark doublets, ucR and dcR are the righthanded up and down quark singlets respectively, and H isthe Higgs field. On the other hand, Yu and Yd are complex3 × 3 matrices, which can be cast by means of a singularvalue decomposition as

Yu = V uRDuYV uL †,

(3.29)

Yd = V dRDdYV dL †.

36 Eur. Phys. J. C (2008) 57: 13–182

Here,DuY = diag(yu1 , yu2 , y

u3 ) is a diagonal matrix whose en-

tries can be chosen real and positive with yu1 < yu2 < y

u3 , and

similarly for DdY . V u,dR and V u,dL are 3 × 3 unitary matri-ces that depend on three real parameters and six phases. Theunitary matrices V u,dR can be absorbed in the definition ofthe right handed fields without any physical effect. In neu-tral currents the left rotations cancel out via the Glashow–Iliopoulos–Maiani (GIM) mechanism [203]. On the otherhand, the redefinition of the left handed fields produces fla-vor mixing in the charged currents. In the physical basiswhere both the up and down Yukawa couplings are simul-taneously diagonal, the charged current reads

Jμcc = ucLγ μ(1 − γ5)

2

(V uL

†V dL)dL. (3.30)

The matrix V uL†V dL can be generically written as V uL

†V dL =Φ1UCKMΦ2, whereΦ1,2 are diagonal unitary matrices (thus,containing only phases) that can be absorbed by appropri-ate redefinitions of the left handed fields. Finally, UCKM de-pends on three angles and one phase that cannot be removedby field redefinitions and accounts for the physical mixingbetween quark generations and the CP violation [204, 205].It is usually parameterized thus:

UCKM =⎛⎜⎝

c13c12 c13s12 s13e−iδ

−c23s12 − s23s13c12eiδ c23c12 − s23s13s12e

iδ s23c13

s23s12 − c23s13c12eiδ −s23c12 − c23s13s12e

iδ c23c13

⎞⎟⎠ ,

(3.31)

where sij = sin θij , cij = cos θij and δ is the CP-violatingphase. Experiments show a hierarchical structure in the off-diagonal entries of the CKM matrix: |Vub| � Vcb � Vus ,that can be well described by the following phenomeno-logical parameterization of the CKM matrix, proposed byWolfenstein [206]. It reads

UCKM =

⎛⎜⎜⎝

1 − λ2

2 λ Aλ3(ρ − iη)−λ 1 − λ2

2 Aλ2

Aλ3(1 − ρ − iη) −Aλ2 1

⎞⎟⎟⎠

+ O(λ4), (3.32)

where λ is determined with a very good precision in semi-leptonic K decays, giving λ 0.23, and A is measured insemileptonic B decays, giving A 0.82. The parameters ρand η are more poorly measured, although a rough estimateis ρ 0.1, η 0.3 [207].

3.2.2 Leptonic sector with Dirac neutrinos

A Dirac mass term for the neutrinos requires the existenceof three right handed neutrinos, which are singlets under thestandard model gauge group. In consequence, the leptonic

Lagrangian would contain in general a Majorana mass termfor the right handed neutrinos, which has to be forbidden byimposing exact lepton number conservation. Then the lep-tonic Lagrangian reads

Llep = ecRiY eijLjH + νcRiY νijLjH + h.c., (3.33)

where Li = (νLi, eLi) are the left handed lepton doubletsand ecR and νcR are respectively the right handed charged lep-ton and neutrino singlets. Analogously to the quark sector,the Yukawa couplings can be decomposed as

Y e = V eRDeYV eL†, (3.34)

Y ν = V νRDνYV νL†, (3.35)

where V e,νR do not have any physical effect, whereas theVe,νL have an effect in the charged current, that in the ba-

sis where the charged lepton and neutrino Yukawa couplingsare simultaneously diagonal reads

Jμcc = ecLγ μ(1 − γ5)

2

(V eL

†V νL)νL. (3.36)

As in the case of the quark sector, the matrix V eL†V νL de-

pends on three angles and six phases and can be expressedas V eL

†V νL = Φ1UPMNSΦ2. The matrices Φ1 and Φ2 canbe absorbed by appropriate redefinitions of the left handedfields, yielding a physical mixing matrix UPMNS [208, 209]that depends on three angles and one phase, and that canbe parameterized by the same structure as for the quarksector, (3.31). However, the values for the angles differsubstantially from the quark sector. The experimental val-ues that result from the global fit are sin2 θ12 = 0.26–0.36,sin2 θ23 = 0.38–0.63 and sin2 θ13 ≤ 0.025 at 2σ [210, 211].On the other hand, the CP-violating phase δ is completelyunconstrained by present experiments.

In the theory under discussion the total lepton numberL = Le + Lμ + Lτ is conserved, but the individual lep-ton flavors Ll , l = e,μ, τ , are not, and LFV processes likeμ− → e−γ decay are allowed. For the neutrino massesmνj ,j = 1,2,3, satisfying the existing upper limits obtained in3H β-decay experiments, mj < 2.3 eV, the μ− → e−γ de-cay branching ratio is given by [212]

B(μ→ eγ )= 3α

32π

∣∣∣∣∑j

UPMNSej UPMNS∗

μj

m2νj

M2W

∣∣∣∣2

, (3.37)

where MW is the W±-boson mass. Thus, the μ− → e−γdecay rate is suppressed by the factor (mj/MW)4 < 6.7 ×10−43, which renders it unobservable. The same conclusionis valid for all other LFV decays and reactions in the min-imal extension of the standard theory with light neutrinomasses we are considering. The only observable manifes-tation of the non-conservation of the lepton charges Ll inthis theory is the oscillations of neutrinos.

Eur. Phys. J. C (2008) 57: 13–182 37

3.2.3 Leptonic sector with Majorana neutrinos

Neutrino masses can also be accommodated in the standardmodel without extending the particle content, just by addinga dimension five operator to the leptonic Lagrangian [213]:

Llep = ecRiY eijLjH + 1

4κij (LiH)(LjH)+ h.c. (3.38)

with κ a 3 × 3 complex symmetric matrix that breaks ex-plicitly lepton number and that has dimensions of mass−1.Then, after the electroweak symmetry breaking, a Majoranamass term for neutrinos is generated:

mν = 1

2κ⟨H 0⟩2. (3.39)

This term can be diagonalized as mν = V νL∗DmνV νL†, so

that the charged current reads as in (3.36), with V eL†V νL =

Φ1UΦ2, where the matrix U has the form of the CKM ma-trix, (3.31). The matrix Φ1 containing three phases can beremoved by a redefinition of the left handed charged leptonfields. However, due to the Majorana nature of the neutri-nos, the matrixΦ2 cannot be removed and is physical, yield-ing a leptonic mixing matrix UPMNS = UΦ2 that is definedby three angles and three phases [214, 215], one associatedto U , the “Dirac phase”, and two associated to Φ2, the “Ma-jorana phases”.

In the leptonic Lagrangian given by (3.38) the origin ofthe dimension five operator remains open. In the rest of thissection, we shall review the heavy Majorana singlet (righthanded) neutrino mass mechanism (type I see-saw) [216–220] and the triplet Higgs mass mechanism (type II see-saw)[215, 221–224] as the possible origins of this effective op-erator. The third [225] tree level realization of the opera-tor (3.38) via triplet fermion (type III see-saw) [226] is dis-cussed in Sect. 4.1.

3.2.3.1 Type I see-saw In the presence of singlet righthanded neutrinos, the most general Lagrangian compatiblewith the standard model gauge symmetry reads

Llep = ecRiY eijLjH + νcRiY νijLjH − 1

2νcTRi Mij ν

cRj + h.c.,

(3.40)

where lepton number is explicitly broken by the Majoranamass term for the singlet right handed neutrinos.9 The see-saw mechanism is implemented when eig(M)� 〈H 0〉. Ifthis is the case, at low energies the right handed neutrinosare decoupled and the theory can be well described by the

9Here we explicitly assume three generations of singlet neutrinos. Forthe phenomenology of a large number of singlets as predicted by stringtheories, see [227, 228].

effective Lagrangian for Majorana neutrinos, (3.38), with[216–220]

κ = 2Y νTM−1Y ν. (3.41)

Working in the basis where the charged lepton Yukawa ma-trix and the right handed mass matrix are simultaneouslydiagonal, it can be checked that the complete Lagrangian,(3.40), contains fifteen independent real parameters and sixcomplex phases [229]. Of these, three correspond to thecharged lepton masses, three to the right handed masses,and the remaining nine real parameters and six phases, tothe neutrino Yukawa coupling. The independent parametersof the neutrino Yukawa coupling can be expressed in severalways. The most straightforward parameterization uses thesingular value decomposition of the neutrino Yukawa ma-trix:

Yν = V νRDνYV νL†, (3.42)

where DνY = diag(yν1 , yν2 , y

ν3 ), with yνi ≥ 0 and yν1 ≤ yν2 ≤

yν3 . On the other hand, V νL and V νR are 3 × 3 unitary matri-ces, that depend in general on three real parameters and sixphases. Both can be generically written asΦ1VΦ2, where Vhas the form of the CKM matrix and Φ1,2 are diagonal uni-tary matrices (thus, containing only phases). One can checkthat for V νR theΦ2 matrix can be absorbed into the definitionof V νL , so that

V νR =⎛⎜⎝eiα

R1

eiαR2

1

⎞⎟⎠

×⎛⎜⎜⎝

cR2 cR3 cR2 s

R3 sR2 e

−iδR

−cR1 sR3 − sR1 sR2 cR3 eiδR

cR1 cR3 − sR1 sR2 sR3 eiδ

RsR1 c

R2

sR1 sR3 − cR1 sR2 cR3 eiδ

R −sR1 cR3 − cR1 sR2 sR3 eiδR

cR1 cR2

⎞⎟⎟⎠.

(3.43)

Similarly, for VL the Φ1 matrix can be absorbed into thedefinition of L and eR , while keeping Ye diagonal and real.In consequence,

V νL =⎛⎜⎜⎝

cL2 cL3 cL2 s

L3 sL2 e

−iδL

−cL1 sL3 − sL1 sL2 cL3 eiδL

cL1 cL3 − sL1 sL2 sL3 eiδ

LsL1 c

L2

sL1 sL3 − cL1 sL2 cL3 eiδ

L −sL1 cL3 − cL1 sL2 sL3 eiδL

cL1 cL2

⎞⎟⎟⎠

×⎛⎜⎝eiα

L1

eiαL2

1

⎞⎟⎠ . (3.44)

Therefore, in this parameterization the independent para-meters in the Yukawa coupling can be identified with thethree Yukawa eigenvalues, yi , the three angles and threephases in VL, and the three angles and three phases in

38 Eur. Phys. J. C (2008) 57: 13–182

VR [229–231]. The requirement that the low energy phe-nomenology is successfully reproduced imposes constraintsamong these parameters. To be precise, the low energy lep-tonic Lagrangian depends just on the three charged leptonmasses and the six real parameters and three complex phasesof the effective neutrino mass matrix. In consequence, thereare still six real parameters and three complex phases thatare not determined by low energy neutrino data; this infor-mation about the high energy Lagrangian is “lost” in the de-coupling of the three right handed neutrinos and cannot berecovered just from neutrino experiments.

The ambiguity in the determination of the high energyparameters can be encoded in the three right handed neu-trino masses and an orthogonal complex matrix R definedas [232]

R =D−1√M

YνUPMNSD−1√m

⟨H 0⟩, (3.45)

so that the most general Yukawa coupling compatible withthe low energy data is given by:

Y ν =D√MRD√

mU†PMNS

⟨H 0⟩. (3.46)

It is straightforward to check that this equation indeed sat-isfies the seesaw formula, (3.41). In this expression, D√

m

andD√M

are diagonal matrices whose entries are the squareroots of the light neutrino and the right handed neutrinomasses, respectively, and UPMNS is the leptonic mixing ma-trix. It is customary to parameterize R in terms of three com-plex angles, θi :

R =

⎛⎜⎜⎝c2c3 −c1s3 − s1s2c3 s1s3 − c1s2c3

c2s3 c1c3 − s1s2s3 −s1c3 − c1s2s3

s2 s1c2 c1c2

⎞⎟⎟⎠ , (3.47)

up to reflections, where ci ≡ cos θi , si ≡ sin θi .Whereas the physical interpretation of the right handed

masses is very transparent, the meaning of R is more ob-scure. R can be interpreted as a dominance matrix in thesense that [233]

– R is an orthogonal transformation from the basis of theleft handed leptons mass eigenstates to the one of the righthanded neutrino mass eigenstates;

– if and only if an eigenvaluemi ofmν is dominated—in thesense already given before - by one right handed neutrinoeigenstate Nj , then |Rji | ≈ 1;

– if a light pseudo-Dirac pair is dominated by a heavypseudo-Dirac pair, then the corresponding 2 × 2 sectorin R is a boost.

An interesting limit of this dominance behavior is theseesaw model with two right handed neutrinos (2RHN)[234, 235]. In this limit, the parameterization (3.46) still

holds, with the substitutions D√M

= diag(M−11 ,M−1

2 ) and[236–239]

R =(

0 cos θ ξ sin θ0 − sin θ ξ cos θ

)(normal hierarchy), (3.48)

R =(

cos θ ξ sin θ 0− sin θ ξ cos θ 0

)(inverted hierarchy), (3.49)

with θ a complex parameter and ξ =±1 a discrete parame-ter that accounts for a discrete indeterminacy in R.

A third possible parameterization of the neutrino Yukawacoupling uses the Gram–Schmidt decomposition, in order tocast the Yukawa coupling as a product of a unitary matrixand a lower triangular matrix [240]:

Y ν =U�Y� =U�

⎛⎝y11 0 0y21 y22 0y31 y32 y33

⎞⎠ , (3.50)

where the diagonal elements of Y� are real. Three of thesix phases in U� can be absorbed into the definition of thecharged leptons. Therefore, the nine real parameters and thesix phases of the neutrino Yukawa coupling are identifiedwith the three angles and three phases in U� and the six realparameters and three phases in Y�.

In the SM extended with right handed neutrinos, thecharged lepton masses and the effective neutrino mass ma-trix are the only source of information about the leptonicsector. However, if supersymmetry is discovered, the struc-ture of the low energy slepton mass matrices would provideadditional information about the leptonic sector, providedthe mechanism of supersymmetry breaking is specified. As-suming that the slepton mass matrices are proportional to theidentity at the high energy scale, quantum effects induced bythe right handed neutrinos would yield at low energies a lefthanded slepton mass matrix with a complicated structure,whose measurement would provide additional informationabout the seesaw parameters [144, 145]. To be more spe-cific, in the minimal supersymmetric seesaw model the off-diagonal elements of the low energy left handed and righthanded slepton mass matrices and A-terms read, in the lead-ing log approximation [178]

(m2L

)ij− 1

8π2

(3m2

0 +A20

)Y νik

†Y νkj log

MX

Mk, (3.51)

(m2eR

)ij 0, (3.52)

(Ae)ij − 3

8π2A0YeY

νik

†Y νkj log

MX

Mk, (3.53)

where m0 and A0 are the universal soft supersymmetrybreaking parameters at high scale MX. Note that the diag-onal elements of those mass matrices include the tree levelsoft mass matrix, the radiative corrections from gauge and

Eur. Phys. J. C (2008) 57: 13–182 39

charged lepton Yukawa interactions, and the mass contribu-tions from F- and D-terms (which are different for chargedsleptons and sneutrinos). Therefore, the measurement at lowenergies of rare lepton decays, electric dipole moments andslepton mass splittings would provide information about thecombination

Cij ≡∑k

Y νik†Y νkj log

MX

Mk≡ (Y †

ν LYν)ij, (3.54)

where Lij = log MXMiδij .

Interestingly enough, C encodes precisely the additionalinformation needed to reconstruct the complete seesaw La-grangian from low energy observations [241, 242] (note inparticular that C is a Hermitian matrix that depends on sixreal parameters and three phases, which together with thenine real parameters and three phases of the neutrino massmatrix sum up to the independent fifteen real parameters andsix complex phases in Yν andM).

To determine Yν andM from the low energy observablesC and mν , it is convenient to define

Y ν = diag

(√logMX

M1,

√logMX

M2,

√logMX

M3

)Y ν,

(3.55)

Mk =Mk logMX

Mk,

so that the effective neutrino mass matrix and C now read

mν = Y νt diag(M−1

1 , M−12 , M−1

3

)Y ν⟨H 0u

⟩2,

(3.56)C = Y ν†Y ν,

where H 0u is the neutral component of the up-type Higgs

doublet. Using the singular value decomposition Y ν =V νRD

νY V

ν†L , one finds that V ν†

L and DνY could be straight-forwardly determined from C, since

C ≡ Y ν†Y ν = V †LD

2Y VL. (3.57)

On the other hand, frommν = Y νt D−1M Y

ν〈H 0u 〉2 and the sin-

gular value decomposition of Y ν ,

D−1Y V

∗LmνV

ν†L D

ν−1Y = V ν∗R D−1

M Vν†R , (3.58)

where the left hand side of this equation is known (mν is oneof our inputs, and V νL and DνY were obtained from (3.57)).

Therefore, V νR and DM can also be determined. This sim-ple procedure shows that starting from the low energy ob-servables mν and C it is possible to determine uniquely thematrices DM and Y ν = V νRDνY V ν†

L . Finally, inverting (3.56),the actual parameters of the Lagrangian Mk and Y ν can becomputed.

This procedure is particularly powerful in the case of thetwo right handed neutrino model, as the number of inde-pendent parameters involved (either at high energies or atlow energies) is drastically reduced. The matrix C definedin (3.54) depends in general on six moduli and three phases.However, since the Yukawa coupling depends in the 2RHNmodel on only three unknown moduli and one phase, so doesC, and consequently it is possible to obtain predictions onthe moduli of three C-matrix elements and the phases oftwo C-matrix elements. Namely, from (3.46) one obtains

U†CU =U†Y ν†Y νU =D√mR

†DMRD√m/⟨H 0u

⟩2, (3.59)

where we have written U ≡ UPMNS. Since m1 = 0 in the2RHN model,10 it follows that (U†CU)1i = 0, for i =1,2,3, leading to three relations among the elements in C.For instance, one could derive the diagonal elements in C interms of the off-diagonal elements:

C11 =−C∗12U

∗21 +C∗

13U∗31

U∗11

,

C22 =−C12U∗11 +C∗

23U∗31

U∗21

, (3.60)

C33 =−C13U∗11 +C23U

∗21

U∗31

.

The observation of these correlations would be non-trivialtests of the 2RHN model.

The relations for the phases arise from the hermiticityof C, since the diagonal elements in C have to be real. Tak-ing as the independent phase the argument of C12, one canderive from (3.60) the arguments of the remaining elements:

eiargC13 =[−i Im(C12U21U

∗11

)

±√|C13|2|U11|2|U31|2 −

[Im(C12U21U

∗11

)]2]

× [|C13|U31U∗11

]−1,

(3.61)

eiargC23 =[i Im

(C12U21U

∗11

)

±√|C23|2|U21|2|U31|2 −

[Im(C12U21U

∗11

)]2]

× [|C23|U31U∗21

]−1,

where the ± sign has to be chosen so that the eigenvaluesof C are positive. We conclude then that the C-matrix para-meters C12, |C13| and |C23| can be regarded as independentand can be used as an alternative parameterization of the

10Here we are assuming a neutrino spectrum with normal hierarchy. Inthe case with inverted hierarchy, the analysis is similar, using m3 = 0.

40 Eur. Phys. J. C (2008) 57: 13–182

2RHN model [243]. Together with the five moduli and thetwo phases of the neutrino mass matrix, we sum up to theeight moduli and the three phases necessary to reconstructthe high energy Lagrangian of the 2RHN model.

3.2.3.2 Type II seesaw The type II seesaw mechanism[215, 221–224] consists on adding to the SM particle con-tent a Higgs triplet

T =(

T 0 − 1√2T +

− 1√2T + −T ++

). (3.62)

Then, the leptonic potential compatible with the SM gaugesymmetry reads

Llep = ecRiY eijLjH + YTij LiT Lj + h.c. (3.63)

From this Lagrangian, it is apparent that the triplet T carrieslepton number −2. If the neutral component of the tripletacquires a VEV and breaks lepton number spontaneously ashappens in the Gelmini–Roncadelli model [224], the associ-ated massless majoron rules out the model. Therefore phe-nomenology suggests to break lepton number explicitly viathe triplet coupling to the SM Higgs boson [244]. The mostgeneral scalar potential involving one Higgs doublet and oneHiggs triplet reads

V = m2HH

†H + 1

2λ1(H †H

)2 +M2T T

†T + 1

2λ2(T †T

)2

+ λ3(H †H

)(T †T

)+μ′H †TH †, (3.64)

where the term proportional to μ′ breaks lepton numberexplicitly. The type II seesaw mechanism is implementedwhen MT � 〈H 0〉. Then the minimization of the scalar po-tential yields

⟨H 0⟩2 −m2

H

λ1 − 2μ2L//M

2T

,⟨T 0⟩ −μ′〈H 0〉2

M2T

, (3.65)

which produce Majorana masses for the neutrinos given by

mν = YT −μ′〈H 0〉2

M2T

. (3.66)

The Yukawa matrix YT has the same flavor structure as thenon-renormalizable operator κ defined in (3.38) for the ef-fective Lagrangian of Majorana neutrinos. Therefore, the pa-rameterization of the type II seesaw model is completelyidentical to that case.

Supersymmetric models with low scale triplet Higgseshave been extensively considered in studies of collider phe-nomenology [245–247]. The model [244] was first super-symmetrized in Ref. [248] as a possible scenario for lepto-genesis. The requirement of a holomorphic superpotential

implies introducing the triplets in a vector-like SU(2)W ×U(1)Y representation, as T ∼ (3,1) and T ∼ (3,−1). Therelevant superpotential terms are

1√2YijT LiT Lj +

1√2λ1H1TH1 + 1√

2λ2H2T H2

+MT T T +μH2H1, (3.67)

where Li are the SU(2)W lepton doublets and H1(H2) isthe Higgs doublet with hypercharge Y = −1/2(1/2). De-coupling the triplet at high scale at the electroweak scale theMajorana neutrino mass matrix is given by (v2 = 〈H2〉)

mijν = Y ijTv2

2λ2

MT. (3.68)

Note that in the supersymmetric case there is only one massparameter, MT , while the mass parameter μ′ of the non-supersymmetric version is absent.

The couplings YT also induce LFV in the slepton massmatrix m2

Lthrough renormalization group (RG) running

from MX to the decoupling scale MT [249]. In the leading-logarithm approximation those are given by (i �= j ):

(m2L

)ij≈ −1

8π2

(9m2

0 + 3A20

)(Y

†T YT

)ij

logMX

MT,

(m2eR

)ij≈ 0, (3.69)

(Ae)ij ≈ −9

16π2A0(YeY

†T YT

)ij

logMX

MT.

Phenomenological implications of those relations will bepresented in Sect. 5.

3.2.3.3 Renormalization of the neutrino mass matrix Tomake a connection between high scale parameters and lowscale observables one needs to consider renormalization ef-fects on neutrino masses and mixing. Below the scale wherethe dimension five operator is generated, the running of theneutrino mass matrix is governed by the renormalizationgroup (RG) equation of the coupling matrix κν , given by[250–253]

(4π)2d

d lnμκν = (4π)2Ag κν +Ce

((Y †e Ye)Tκν + κν Y †

e Ye),

(3.70)

where Ce = −3/2 for the SM and Ce = 1 for the MSSM.The first term does not affect the running of the neutrinomixing angles and CP violation phases; however, it affectsof course the running of the neutrino mass eigenvalues. The

Eur. Phys. J. C (2008) 57: 13–182 41

flavor universal factor Ag is given by

Ag =

⎧⎪⎨⎪⎩

−3α2(4π)+ λ+ 2 tr(3Y †u Yu + 3Y †

d Yd + Y †e Ye)

SM,

−2α1(4π)− 6α2(4π)+ tr(Y †u Yu) MSSM,

(3.71)

where λ denotes the Higgs self-coupling constant and αi =g2i /(4π), where g1 and g2 are the U(1)Y and SU(2) gauge

coupling constants, respectively.Due to the smallness of the tau–Yukawa coupling in the

SM, the mixing angles are not affected significantly by therenormalization group running below the generation scaleof the dimension five operator. However, if the neutrino

mass matrix mν = 〈φ〉22 κν is realized in the seesaw sce-

nario (type I), running effects above and between the see-saw scales can also lead to relevant running effects in theSM. Note that in the MSSM case the running of the mixingangles and CP violation phases can be large even below theseesaw scales due to the possible enhancement of the tau–Yukawa coupling by the factor (1 + tanβ2)1/2.

In order to understand generic properties of the RG evo-lution and to estimate the typical size of the RG effects, itis useful to consider RGEs for the leptonic mixing angles,CP phases and neutrino masses themselves, which can bederived from the RGE in (3.70). For example, below theseesaw scales, up to O(θ13) corrections, the evolution ofthe mixing angles in the MSSM is given by [254] (see also[255, 256])

dθ12

d lnμ= −y2

τ

32π2sin 2θ12s

223|m1e

iαM +m2|2�m2

21

, (3.72)

dθ13

d lnμ= y2

τ

32π2sin 2θ12 sin 2θ23

m3

�m231(1 + ζ )

× I (m1,m2, αM,βM, δ), (3.73)

dθ23

d lnμ= −y2

τ

32π2

sin 2θ23

�m231

[c2

12

∣∣m2eiβM +m3e

iαM∣∣2

+ s212|m1e

iβM+m3|21 + ζ

], (3.74)

where I (m1,m2, αM,βM, δ)≡m1 cos(βM − δ)− (1+ ζ )×m2 cos(αM − βM + δ) − ζm3 cos δ, sij = sin θij , cij =cos θij , and ζ = �m2

21/�m231. Here yτ denotes the tau–

Yukawa coupling, and one can safely neglect the contri-butions coming from the electron– and muon–Yukawa cou-plings. For the matrix P containing the Majorana phases, weuse the convention P = diag(1, eiαM/2, eiβM/2). In additionto the above formulae, formulae for the running of the CPphases have been derived [254]. For example, the running

of the Dirac CP-violating phase δ, observable in neutrinooscillation experiments, is given by

dδ

d lnμ= Cy2

τ

32π2

δ(−1)

θ13+ Cy

2τ

8π2δ(0) + O(θ13). (3.75)

The coefficients δ(−1) and δ(0) are omitted here and can befound in [254], where also formulae for the running of theMajorana CP phases and for the neutrino mass eigenvalues(mass squared differences) can be found. From (3.75), it canbe seen that the Dirac CP phase generically becomes moreunstable under RG corrections for smaller θ13.

In the seesaw scenario (type I), the SM or MSSM areextended by heavy right handed neutrinos and their super-partners, which are SM gauge singlets. Integrating them outbelow their mass scales MR yields the dimension five oper-ator for neutrino masses in the SM or MSSM. Above MR ,the neutrino Yukawa couplings are active, and the RGEs inthe MSSM above the scalesMR are

(4π)2dκν

d lnμ={−6

5α1(4π)− 6α2(4π)+ 2 tr

(Y †ν Yν)

+ 6 tr(Y †u Yu

)}κν +

(Y †e Ye)Tκν + κν

(Y †e Ye)

+ (Y †ν Yν)Tκν + κν

(Y †ν Yν), (3.76)

(4π)2dMR

d lnμ= 1

8π2

[(YνY

†ν

)MR +MR

(YνY

†ν

)T], (3.77)

(4π)2dYν

d lnμ= −Yν

[3

5α1(4π)+ 3α2(4π)

− tr(3Y †u Yu + Y †

ν Yν)

− 3Y †ν Yν − Y †

e Ye

]. (3.78)

For non-degenerate seesaw scales, a method for dealing withthe effective theories, where the heavy singlets are partlyintegrated out, can be found in [257]. Analytical formulaefor the running of the neutrino parameters above the seesawscales are derived in [258, 259]. The two loop beta functionscan be found in Ref. [260].

The running correction to the neutrino mass matrix andits effects on the related issue have been widely analyzed(see e.g. [250–279]). We shall summarize below some of thefeatures of RG running of the neutrino mixing parameters inthe MSSM (cf. (3.72)–(3.74)).

– The RG effects are enhanced for relatively large tanβ ,because the tau–Yukawa coupling becomes large.

– The mixing angles are comparatively stable with respectto the RG running in the case of normal hierarchical neu-trino mass spectrum, m1 � m2 � m3 even when tanβis large [261–267]. Nevertheless, the running effects can

42 Eur. Phys. J. C (2008) 57: 13–182

have important implications facing the high precision offuture neutrino oscillation experiments.

– For m1 � 0.05 eV and the case of tanβ � 10, the RGrunning effects can be rather large and the leptonic mixingangles can run significantly. Particularly, the RGE effectscan be very large for the solar neutrino mixing angle θ12

[261–267, 274, 275].– The solar neutrino mixing angle θ12 at MR depends

strongly on the Majorana phase αM [254, 267, 268, 275],which is the relative phase between m1 and m2, and playsvery important role in the predictions of the effectiveMajorana mass in (ββ)0ν -decay. The effect of RG run-ning for θ12 is smallest for the CP-conserving odd caseαM = ±π , while it is significant for the CP-conservingeven case αM = 0. For αM = 0 and tanβ ∼ 50, for in-stance, we have tan2 θ12(MR) � 0.5 × tan2 θ12(MZ) form1 � 0.02 eV.

– The RG running effect on θ12 due to the τ–Yukawa cou-pling always makes θ12(MZ) larger than θ12(MR) [267].This constrains the models which predict the value of so-lar neutrino mixing angle at MR , θ12(MR) > θ12(MZ).For example, the bi-maximal models are strongly re-stricted. However, the running effects due to the neutrinoYukawa couplings are free from this feature [257]. Thus,bi-maximal models can predict the correct value of neu-trino mixing angles with the neutrino Yukawa contribu-tions [269–272].

– The RG corrections to neutrino mixing angles dependstrongly on the deviation of the seesaw parameter matrixR (3.45) from identity [274]. For hierarchical light neu-trinos, m1 � 0.01 eV, tanβ � 30 and R non-trivial, thecorrection to θ23 and θ13 can be beyond their likely futureexperimental errors while θ12 is quite stable against theRG corrections [274].

– The correction to θ23 can be large when m1 and/or tanβare/is relatively large, e.g., (i) when m1 � 0.2 eV iftanβ � 10, and (ii) for anym1 and αM if tanβ � 40 [274,275].

– The RG corrections to sin θ13 can be relatively small, evenfor the large tanβ if m1 � 0.05 eV, and for any m1 �0.30 eV, if θ13(MZ)∼= 0 and αM ∼= 0 (with βM = δ = 0).For αM = π and tanβ ∼ 50 one can have sin θ13(MR)�0.10 for m1 � 0.08 eV even if sin θ13(MZ) = 0 [274,275].

– For tanβ � 30, the value of �m221(MR) depends strongly

on m1 in the interval m1 � 0.05 eV, and on αM , βM , δ,and s13 for m1 � 0.1 eV. The dependence of �m2

31(MR)

on m1 and the CP phases is rather weak, unless tanβ �40, m1 � 0.10 eV, and s13 � 0.05 [275].

– Some products of the neutrino mixing parameters, suchas s12c12c23(m1/m2 − eiαM ) are practically stable withrespect to RG running if one neglects the first and sec-ond generation charged lepton Yukawa couplings and s13

[268, 273, 275].

3.2.4 Quark–lepton complementarity

3.2.4.1 Golden complementarity Quark–lepton comple-mentarity [280–282] is based on the observation that θ12 +θC is numerically close to π/4. Here θ12 is the solar neutrinomixing angle and θC is the Cabibbo angle. For hierarchicallight neutrino masses this result is relatively stable againstthe renormalization effects [274]. To illustrate the idea wefirst review the model of exact golden complementarity.

Consider the following textures [283] for the light neu-trino Majorana mass matrix mν and for the charged leptonYukawa couplings Ye:

mν =⎛⎝

0 m 0m m 00 0 matm

⎞⎠ ,

(3.79)

Ye =⎛⎝λe 0 00 λμ/

√2 λτ /

√2

0 −λμ/√

2 λτ /√

2

⎞⎠ .

It just assumes some texture zeroes and some strict equal-ities among different entries. The mass eigenstates of theneutrino mass matrix are given by m1 =−m/ϕ, m2 =mϕ,m3 = matm, where ϕ = (1 + √

5)/2 = 1 + 1/ϕ ≈ 1.62 isknown as the golden ratio [284]. Thanks to its peculiar math-ematical properties this constant appears in various naturalphenomena, possibly including solar neutrinos. The threeneutrino mixing angles obtained from (3.79) are θatm = π/4,θ13 = 0 and, more importantly,

tan2 θ12 = 1/ϕ2 = 0.382, i.e. sin2 2θ12 = 4/5, (3.80)

in terms of the parameter sin2 2θ12 directly measured by vac-uum oscillation experiments, such as KamLAND. This pre-diction for θ12 is 1.4σ below the experimental best fit value.A positive measurement of θ13 might imply that the predic-tion for θ12 suffers an uncertainty up to θ13.

Those properties follow from the Z2 ⊗ Z′2 symmetry of

the neutrino mass matrix. Explicitly RmνRT =mν, where

R =⎛⎝−1/

√5 2/

√5 0

2/√

5 1/√

5 00 0 1

⎞⎠ , R′ =

⎛⎝

1 0 00 1 00 0 −1

⎞⎠ ,

(3.81)

and the rotations satisfy detR = −1, R · RT = 1 andR ·R = 1. The first Z2 is a reflection along the diagonalof the golden rectangle in the (1,2) plane, see Fig. 2. Thesecond Z′

2 is the L3 → −L3 symmetry. Those symmetriesallow contributions proportional to the identity matrix to beadded to mν. This property allows one to extend this typesymmetries to the quark sector.

Eur. Phys. J. C (2008) 57: 13–182 43

A seesaw model with singlet neutrinos satisfying theZ2 ⊗Z′

2 symmetry and giving rise to the mass matrix (3.79)is presented in [283].

Noticing that the golden prediction (3.80) satisfies withhigh accuracy the quark–lepton complementarity motivatesone to give a golden geometric explanation also to theCabibbo angle. SU(5) unification relates the down-quarkYukawa matrix Yd to Ye and suggests that the up-quarkYukawa matrix Yu is symmetric, like mν . One can there-fore assume that Yd is diagonal in the two first genera-tions and that Yu is invariant under a Z2 reflection describedby a matrix analogous to R in (3.81), but with the factors1 ↔ 2 exchanged. Figure 2 illustrates the geometrical mean-ing of two reflection axis (dashed lines): the up-quark re-flection is along the diagonal of the golden rectangle tiltedby π/4; note also the connection with the decomposition ofthe golden rectangle as an infinite sum of squares (‘goldenspiral’). Similarly to the neutrino case, this symmetry al-lows for two independent terms that can be tuned such thatmu�mc:

Yu = λ⎛⎝

1 0 00 1 00 0 1

⎞⎠+ λ√

5

⎛⎝−2 1 01 2 00 0 c

⎞⎠ . (3.82)

The second term fixes cot θC = ϕ3, as can be geometricallyseen from Fig. 2. We therefore have

sin2 2θC = 1/5 i.e. θ12 + θC = π/4 i.e.(3.83)

Vus = sin θC = (1 + ϕ6)−1/2 = 0.229.

This prediction is 1.9σ above the present best-fit value,sin θC = 0.2258 ± 0.0021. However, as the basic elements

Fig. 2 Geometrical illustration of the connection between the predic-tions for θ12 and θC and the golden rectangle. The two dashed linesare the reflection axis of the Z2 symmetry for the neutrino mass matrixand for the up quark mass matrix

of flavor presented here follow by construction from the2 × 2 submatrices, one naturally expects that the goldenprediction for Vus has an uncertainty at least comparableto |Vub| ∼ |Vtd | ∼ few × 10−3. Thus the numerical accu-racy is amazing. Should the 1.4σ discrepancy between thegolden prediction (3.80) and the experimental measurementhold after final SNO and KamLAND results, analogy withthe quark sector would allow one to predict the order of mag-nitude of neutrino mixing angle θ13.

Interestingly, similar predictions on the mixing anglesare obtained if some suitably chosen assumptions are madeon the properties of neutral currents of quarks and lep-tons [285].

3.2.4.2 Correlation matrix from S3 flavor symmetry in GUTOn more general phenomenological ground the quark–lepton complementarity [281, 282] can be described by thecorrelation matrix VM between the CKM and the PMNSmixing matrices,

VM =UCKMΩUPMNS, (3.84)

where Ω = diag(eiωi ) is a diagonal matrix. In the singletseesaw mechanism the correlation matrix VM diagonalizesthe symmetric matrix

C = mdiagD V νR

† 1

MV νR!m

diagD , (3.85)

where M is the heavy neutrino Majorana mass matrix andV νR diagonalizes the neutrino Dirac matrix mD from theright. In GUT models such as SO(10) or E6 we have in-triguing relations between the Yukawa coupling of the quarksector and the one of the lepton sector. For instance, in min-imal renormalizable SO(10) with Higgs in the 10, 126, and120, we have Ye ≈ YTd . In fact the flavor symmetry impliesthe structure of the Yukawa matrices: the equivalent entriesof Ye and Yd are usually of the same order of magnitude. Insuch a case one gets

UPMNS = (UCKM)†VM.

As a consequence, a S3 flavor permutation symmetry, softlybroken into S2, gives us the prediction of VM13 = 0 [286] andthe correlations between CP-violating phases and the mixingangle θ12 [287].

The six generators of the S3 flavor symmetry are the el-ements of the permutation group of three objects. The ac-tion of S3 on the fields is to permute the family label ofthe fields. In the following we shall introduce the S2 sym-metry with respect the second and third generations. TheS2 group is an Abelian one and swap the second fam-ily {μL, (νμ)L, sL, cL,μR, (νμ)R, sR, cR} with the third one{τL, (ντ )L, bL, tL, τR, (ντ )R, bR, tR}.

44 Eur. Phys. J. C (2008) 57: 13–182

Let us assume that there is an S3 flavor symmetry at highenergy, which is softly broken into S2 [84]. In this case, be-fore the S3 breaking all the Yukawa matrices have the fol-lowing structure:

Y =⎛⎝a b b

b a b

b b a

⎞⎠ , (3.86)

where a and b independent. The S3 symmetry implies that(1/

√3,1/

√3,1/

√3) is an eigenvector of our matrix in

(3.86). Moreover these kind of matrices have two equaleigenvalues. This gives us an undetermined mixing angle inthe diagonalizing mixing matrices.

When S3 is softly broken into S2, one gets

Y =⎛⎝a b b

b c d

b d c

⎞⎠ , (3.87)

with c ≈ a and d ≈ b. When S3 is broken the degener-acy is removed. In general the S2 symmetry implies that(0,1/

√2,−1/

√2) is an exact eigenvector of our matrix

(3.87). The fact that S3 is only softly broken into S2 al-lows us to say that (1/

√3,1/

√3,1/

√3) is still in a good

approximation an eigenvector of Y in (3.87). Then the mix-ing matrix that diagonalize from the right the Yukawa mix-ing matrix in (3.87) is given in good approximation by thetri-bi-maximal mixing matrix (2.11).

Let us now investigate the VM in this model. The massmatrix mD will have the general structure in (3.87). To bemore defined, let us assumed that there is an extra softlybroken Z2 symmetry under which the 1st and the 2nd fam-ilies are even, while the 3rd family is odd. This extra softlybroken Z2 symmetry gives us a hierarchy between the off-diagonal and the diagonal elements of mD , i.e. b, d � a, c.In fact if Z2 is exact both b and d are zero. For simplicity, weassume also a quasi-degenerate spectrum for the eigenvaluesof the Dirac neutrino matrix as in [288].

The right handed neutrino Majorana mass matrix is of theform

M =⎛⎝a b b′b c d

b′ d e

⎞⎠ . (3.88)

Because S3 is only softly broken into S2 we have that a ≈c ≈ e, and b ≈ b′ ≈ d . In this approximation the M matrixis diagonalized by a U of the form in (2.11). In this case wehave that mν is near to be S3 and S2 symmetric, then it isdiagonalized by a mixing matrix Uν near the tri-bi-maximalone given in (2.11). The C matrix is diagonalized by the mix-ing matrix VM = UνU. We obtain that VM is a rotation inthe (1,2) plane, i.e. it contains a zero in the (1,3) entry. Asshown in [288], it is possible to fit the CKM and the PMNSmixing matrix within this model.

3.3 Leptogenesis and cosmological observables

3.3.1 Basic concepts and results

CP violation in the leptonic sector can have profound cos-mological implications, playing a crucial role in the genera-tion, via leptogenesis, of the observed baryon number asym-metry of the universe [289]:

nB

nγ= (6.1+0.3

−0.2

)× 10−10. (3.89)

In the original framework a CP asymmetry is generatedthrough out-of-equilibriumL-violating decays of heavy Ma-jorana neutrinos [290] leading to a lepton asymmetry L �= 0.In the presence of sphaleron processes [291], which are(B+L)-violating and (B−L)-conserving, the lepton asym-metry is partially transformed to a baryon asymmetry.

The lepton number asymmetry resulting from the decayof heavy Majorana neutrinos, εNj , was computed by severalauthors [292–294]. The evaluation of εNj , involves the com-putation of the interference between the tree level diagramand one loop diagrams for the decay of the heavy MajorananeutrinoNj into charged leptons l±α (α = e,μ, τ ). Summingthe asymmetries εαNj over charged lepton flavor, one obtains

εNj =g2

MW2

∑α,k �=j

[Im((m

†D

)jα(mD)αk

(m

†DmD

)jk

)

× 1

16π

(I (xk)+

√xk

1 − xk)]

1

(m†DmD)jj

, (3.90)

where Mk denote the heavy neutrino masses, the variable

xk is defined as xk = Mk2

Mj2 and I (xk) = √

xk(1 + (1 +xk) log( xk1+xk )). From (3.90) it can be seen that, when onesums over all charged leptons, the lepton number asymme-try is only sensitive to the CP-violating phases appearing inm

†DmD in the basis where MR is diagonal. Note that this

combination is insensitive to rotations of the left-hand neu-trinos.

If the lepton flavors are distinguishable in the final state,it is the flavored asymmetries that are relevant [295–298].Below T ∼ 1012 GeV, the τ Yukawa interactions are fastcompared to the Hubble rate, so at least one flavor maybe distinguishable. The asymmetry in family α, generatedfrom the decay of the kth heavy Majorana neutrino dependson the combination [299] Im((m†

DmD)kk′(m∗D)αk(mD)αk′)

as well as on Im((m†DmD)k′k(m

∗D)αk(mD)αk′). Summing

over all leptonic flavors α the second term becomes real sothat its imaginary part vanishes and the first term gives riseto the combination Im((m†

DmD)jk(m†DmD)jk) that appears

in (3.90). Clearly, when one works with separate flavors thematrix UPMNS does not cancel out and one is lead to the

Eur. Phys. J. C (2008) 57: 13–182 45

interesting possibility of having viable leptogenesis even inthe case of R being a real matrix [300–303].

The simplest leptogenesis scenario corresponds to thecase of heavy hierarchical neutrinos where M1 is muchsmaller thanM2 andM3. In this limit, the asymmetries gen-erated byN2 andN3 are frequently ignored, because the pro-duction of N2 and N3 can be suppressed by kinematics (forinstance, they are not produced thermally, if the re-heat tem-perature after inflation is <M2,M3), and the asymmetriesfrom their decays are partially washed out [295, 304, 305].In this hierarchical limit, the εαN1

can be simplified into

εαN1− 3

16πv2

(Iα12M1

M2+ Iα13

M1

M3

), (3.91)

where

Iα1i ≡Im[(m†

D)1α(mD)αi(m†DmD)1i]

(m†DmD)11

. (3.92)

The flavor-summed CP asymmetry εN1 can be written interms of the parameterization equation (3.46) as

εN1 ≈− 3

8π

M1

v2

∑i m

2i Im(R2

1i )∑i mi |R1i |2 . (3.93)

In this case, obviously, leptogenesis demands non-zeroimaginary parts in the R matrix. It has an upper bound|εN1 |< εDIN1

where [306]

εDIN1= 3

8π

(m3 −m1)M1

v2, (3.94)

which is proportional toM1. So the requirement of generat-ing a sufficient baryon asymmetry gives a lower bound onM1 [306, 307]. Depending on the cosmological scenario,the range for minimal M1 varies from order 107 GeV to109 GeV [308, 309]. This bound does not move much withthe inclusion of flavor effects [296, 310, 311]. In a super-symmetric world there is an upper bound TRH < 108 GeVon the re-heating temperature of the universe from the pos-sible overproduction of gravitinos, the so called gravitinoproblem [312–315]. Together with the lower bound on M1

the gravitino problem puts severe constraints on supersym-metric thermal leptogenesis scenarios.

However, the upper bound (3.94) is based on the (nat-ural) assumption that higher order corrections suppressed byM1/M2, M1/M3 in (3.90) are negligible. This may not betrue as explicitly demonstrated in [316] in which neutrinomass model is presented realizing εN1 � εDIN1

. In such a caselow scale standard thermal leptogenesis consistent with thegravitino bound is possible also for hierarchical heavy neu-trinos.

Thermal leptogenesis is a rather involved thermodynami-cal non-equilibrium process and depends on additional para-meters and on the proper treatment of thermal effects [309].

In the simplest case, the Ni are hierarchical, and N1 decaysinto a combination of flavors which are indistinguishable.11

In this case, the baryon asymmetry only depends on four pa-rameters [306, 308, 318, 319]: the mass M1 of the lightestheavy neutrino, together with the corresponding CP asym-metry εN1 in its decay, as well as the rescaled N1 decay rate,or effective neutrino mass m1 defined as

m1 =∑α

(m

†D

)1α(mD)α1/M1, (3.95)

in the weak basis where MR is diagonal, real and positive.Finally, the baryon asymmetry depends also on the sum ofall light neutrino masses squared,m2 =m2

1+m22+m2

3, sinceit has been shown that this sum controls an important classof washout processes. If lepton flavors are distinguishable,the final baryon asymmetry depends on partial decay ratesmα1 and CP asymmetries εα1 .

The N1 decays in the early universe at temperaturesT ∼M1, producing asymmetries in the distinguishable finalstates. A particular asymmetry will survive once washoutby inverse decays go out of equilibrium. In the unfla-vored calculation (where lepton flavors are indistinguish-able), the fraction of the asymmetry that survives is of ordermin{1,H/Γ }, where the Hubble rate H and the N1 total de-cay rate Γ are evaluated at T =M1. This is usually writtenH/Γ =m∗/m1, where [320–322]

m∗ = 16π5/2

3√

5g

1/2∗v2

MPlanck 10−3 eV, (3.96)

andMPlanck is the Planck mass (MPlanck = 1.2× 1019 GeV),v = 〈φ0〉/√2 174 GeV is the weak scale and g∗ is theeffective number of relativistic degrees of freedom in theplasma and equals 106.75 in the SM case. In a flavored cal-culation, the fraction of a flavor asymmetry that survives canbe estimated in the same way, replacing Γ by the partial de-cay rate.

3.3.2 Implications of flavor effects

For a long time the flavor effects in thermal leptogene-sis were known [295] but their phenomenological implica-tions were considered only in specific neutrino flavor mod-els [235]. As discussed, in the single-flavor calculation, themost important parameters for thermal leptogenesis fromN1

decays are M1, m1, εN1 and the light neutrino mass scale.Including flavor effects gives this parameter space more di-mensions (M1, ε

α, mα1 ), but it can still be projected ontoM1,m space. For the readers convenience we summarize here

11This can occur above ∼1012 GeV, before the τ Yukawa interactionbecomes fast compared to the Hubble rate, or in the case where the N1decay rate is faster than the charged lepton Yukawa interactions [317].

46 Eur. Phys. J. C (2008) 57: 13–182

some general results on the implications of flavored lepto-genesis.

In the unflavored calculation, leptogenesis does not workfor degenerate light neutrinos with a mass scale above∼0.1 eV [323–326]. This bound does not survive in the fla-vored calculation, where models with a neutrino mass scaleup to the cosmological bound,

∑mν < 0.68 eV [327], can

be tuned to work [296, 317].Considering the scale of leptogenesis, flavored leptoge-

nesis works for M1 a factor of ∼3 smaller in the “inter-esting” region of m < matm. But the lower bound on M1,in the optimized m region, remains ∼109 GeV [310, 311].A smaller M1 could be possible for very degenerate lightneutrinos [296].

An important, but disappointing, observation in single-flavor leptogenesis was the lack of a model-independentconnection between CP violation for leptogenesis andPMNS phases. It was shown [328, 329] that thermal lep-togenesis can work with no CP violation in UPMNS, andconversely, that leptogenesis can fail in spite of phases inUPMNS. In the “flavoured” leptogenesis case, it is still truethat the baryon asymmetry is not sensitive to PMNS phases[330, 331] (leptogenesis can work for any value of thePMNS phases). However, interesting observations can bemade in classes of models [297, 300, 302, 331].

3.3.3 Other scenarios

We have presented a brief discussion of minimal thermalleptogenesis in the context of type I seesaw with hierarchicalheavy neutrinos. This scenario is the most popular one be-cause it is generic, supported by neutrino mass mechanismand, most importantly, it has predictions for the allowed see-saw parameter space, as described above. There are manyother scenarios in which leptogenesis may also be viable.

Resonant leptogenesis [293, 332] may occur when twoor more heavy neutrinos are nearly degenerate in mass andin this scenario the scale of the heavy neutrino masses canbe lowered whilst still being compatible with thermal lepto-genesis [332–335]. Heavy neutrinos of TeV scale or belowcould in principle be detected at large colliders [336]. In theseesaw context low scale heavy neutrinos may follow fromextra symmetry principles [334, 337–339]. Also, the SM ex-tensions with heavy neutrinos at TeV scale or below includeKaluza–Klein modes in models with extra dimensions or ex-tra matter content of little Higgs models.

Leptogenesis from the out-of-equilibrium decays of aHiggs triplet [244, 340, 341] is another viable scenario butrequires the presence of at least two triplets for non-zeroCP asymmetry. Despite the presence of gauge interactionsthe washout effects in this scenario are not drastically largerthan those in the singlet leptogenesis scenario [341]. Hybridleptogenesis from type I and type II seesaw can for instance

occur in SO(10) models [340, 342, 343]. In that case thereare twelve independent CP-violating phases.

“Soft leptogenesis” [344, 345] can work in a one genera-tional SUSY seesaw model because CP violation in this sce-nario comes from complex supersymmetry breaking terms.If the soft SUSY-breaking terms are of suitable size, there isenough CP violation in N–N∗ mixing to imply the observedasymmetry. Unlike non-supersymmetric triplet Higgs lepto-genesis, soft leptogenesis with a triplet scalar [341, 346] canalso work in the minimal supersymmetric model of type IIseesaw mechanism.

A very predictive supersymmetric leptogenesis scenariois obtained if the sneutrino is playing the role of inflaton[307, 347–350]. In this scenario the universe is dominatedby N . Relating N properties to neutrino masses via the see-saw mechanism implies a lower bound TRH > 106 GeV onthe re-heating temperature of the universe [349]. A connec-tion of this scenario with LFV is discussed in Sect. 5.2.

Dirac leptogenesis is another possibility considered in theliterature. In this case neutrinos are of Dirac type rather thanMajorana. In the original paper [351] two Higgs doubletswere required and their decays create the leptonic asym-metry. Recently some authors have studied the connectionbetween leptogenesis and low energy data with two Higgsdoublets [352].

Finally, let us mention that right handed neutrinos couldhave been produced non-thermally in the early universe, bydirect couplings to the inflation field. If this is the case,the constraints on neutrino parameters from leptogenesis de-pend on the details of the inflationary model [353–355].

For a recent overview of the present knowledge ofneutrino masses and mixing and what can be learnedabout physics beyond the standard model from the vari-ous proposed neutrino experiments, see [4] and referencestherein.

4 Organizing principles for flavor physics

4.1 Grand unified theories

Grand unification is an attempt to unify all known inter-actions but gravity in a single simple gauge group. It ismotivated in part by the arbitrariness of electromagneticcharge in the standard model. One has charge quantiza-tion in a purely non-Abelian theory, without an U(1) fac-tor, as in Schwinger’s original idea [356] of a SU(2) the-ory of electroweak interactions. The minimal gauge groupwhich unifies weak and strong interactions, SU(5) [357],automatically implies a quantized U(1) piece too. WhileDirac needed a monopole to achieve charge quantization[358], grand unification in turn predicts the existence ofmagnetic monopoles [359, 360]. Since it unifies quarks and

Eur. Phys. J. C (2008) 57: 13–182 47

leptons [361], it also predicts another remarkable phenom-enon: the decay of the proton. Here we are mostly interestedin GUT implications on the flavor structure of Yukawa ma-trices.

4.1.1 SU(5): the minimal theory

The 24 gauge bosons reduce to the 12 ones of the SM plusa SU(2) doublet, color triplet pair (Xμ,Yμ) (vector lepto-quarks), with Y = 5/6 (charges +4/3,+1/3) and their an-tiparticles. The 15 fermions of a single family in the SM fitin the 5F and 10F anomaly-free representations of SU(5),and the new super-weak interactions of leptoquarks withfermions are (α, β and γ are color indices):

L(X,Y ) = g5√2X(−4/3)αμ

× (eγ μdcα + dαγ μec − εαβγ ucβγ μuγ)

− g5√2Y (−1/3)αμ

× (νγ μdcα + uαγ μec + εαβγ ucβγ μdγ)

+ h.c., (4.1)

where all fermions above are explicitly left handed andψc ≡ CψT .

The exchange of the heavy gauge bosons leads to the ef-fective interactions suppressed by two powers of their massmX (mX mY due to SU(2)L symmetry), which preservesB − L, but breaks both B and L symmetries and leads to(d = 6) proton decay [213, 362]. From τP � 6 × 1033 yr[363], mX � 1015.5 GeV.

The Higgs sector consists of an adjoint 24H and a fun-damental 5H , the first breaks SU(5)→ SM, the latter com-pletes the symmetry breaking á la Weinberg–Salam. Now,5H = (T ,D), where T is a color triplet and D the usualHiggs SU(2)L doublet of the SM and so the Yukawa inter-actions in the matrix form

LY = 10Fyu10F5H + 5Fyd10F5∗H (4.2)

give the quark and lepton mass matrices

mu = yu〈D〉, md =mTe = yd〈D〉. (4.3)

Note the correlation between down quarks and chargedleptons [364], valid at the GUT scale, and impossible tobe true for all three generations. Actually, in the SM it iswrong for all of them. It can be corrected by an extra Higgs,45H [6], or higher dimensional non-renormalizable interac-tion [7].

From (4.2), one gets also the interactions of the triplet,which lead to proton decay and thus the triplet T must be

superheavy, mT � 1012 GeV. The enormous split betweenmT and mD mW can be achieved through the large scaleof the breaking of SU(5),

〈24H 〉 = vX diag(2,2,2,−3,−3), (4.4)

with m2X =m2

Y = 254 g

25v

2X . This fine-tuning is known as the

doublet–triplet problem. Whatever solution one may adopt,the huge hierarchy can be preserved in perturbation theoryonly by supersymmetry with low scale breaking of orderTeV.

The consistency of grand unification requires that thegauge couplings of the SM unify at a single scale, in atiny window 1015.5 �MGUT � 1018 GeV (lower limit fromproton decay, upper limit from perturbativity, i.e. to staybelow MPl). Here the minimal ordinary SU(5) theory de-scribed above fails badly, while the version with low en-ergy supersymmetry does great [365–368]. Actually, oneneeded a heavy top quark [368], with mt 200 GeV inorder for the theory to work. The same is needed in orderto achieve a radiative symmetry breaking of the SM gaugesymmetry, where only the Higgs doublet becomes tachy-onic [369, 370]. One can then define the minimal supersym-metric SU(5) GUT with the three families of fermions 10F

and 5F, and with 24H and 5H and 5H supermultiplets. Itpredicts md =mTe at MGUT, which works well for the thirdgeneration; the first two can be corrected by higher dimen-sional operators. Although this theory typically has a veryfast d = 5 [150, 371–374] proton decay [375], the higher di-mensional operators can easily make it in accord with exper-iments [376–378]. The main problem are massless neutri-nos, unless one breaks R-parity (whose approximate or ex-act conservation must be assumed in supersymmetric SU(5),contrary to some supersymmetric SO(10)). Other ways outinclude adding singlets, right handed neutrinos (type I see-saw [216–220]), or a 15H multiplet (type II see-saw [215,221–223]). In both cases their Yukawa are not connected tothe charged sector, so it is much more appealing to go toSO(10) theory, which unifies all fermions (of a single fam-ily) too, besides the interactions.

Before we move to SO(10), what about ordinary non-supersymmetric SU(5)? In order to have mν �= 0 and toachieve the unification of gauge couplings one can add ei-ther (a) 15H Higgs multiplet [379] or (b) 24F fermionic mul-tiplet [380]. The latter one is particularly interesting, since itleads to the mixing of the type I and type III see-saw [225,226], with the remarkable prediction of a light SU(2) fermi-onic triplet below TeV and MGUT ≤ 1016 GeV, which of-fers hope both for the observable see-saw at LHC and de-tectable proton decay in a future generation of experimentsnow planned [381].

These fermionic triplets TF would be produced in pairsthrough a Drell–Yan process. The production cross sectionfor the sum of all three possible final states, T +

F T−F , T +

F T0F

48 Eur. Phys. J. C (2008) 57: 13–182

and T −F T

0F , can be read from Fig. 42 of [382]: it is approxi-

mately 20 pb for 100 GeV triplet mass, and around 40 fb for500 GeV triplets. The triplets then decay into W or Z anda light lepton through the same Yukawa couplings that enterinto the seesaw.

The clearest signature would be the three charged leptondecay of the charged triplet, but it has only a 3% branchingratio. A more promising situation is the decay into two jetswith SM gauge boson invariant mass plus a charged lepton:this happens in approximately 23% of all decays. The signa-tures in this case is two same charge leptons plus two pairsof jets having theW or Z mass and peaks in the lepton-dijetmass. From the above estimates the cross section for suchevents is around 1 pb (2fb) for 100 (500) GeV triplet mass.Such signatures were suggested originally in L–R symmet-ric theories [383] but are quite generic of the seesaw mech-anism.

4.1.2 SO(10): the minimal theory of matter and gaugecoupling unification

There are a number of features that make SO(10) special:

– a family of fermions is unified in a 16 dimensional spino-rial representation; this in turn predicts the existence ofright handed neutrinos, making the implementation of thesee-saw mechanism almost automatic;

– L–R symmetry [361, 384–386] is a finite gauge transfor-mation in the form of charge conjugation. This is a conse-quence of both left handed fermions fL and its charged

conjugated counterparts (f c)L ≡ Cf TR residing in thesame representation 16F;

– in the supersymmetric version, the matter parity M =(−1)3(B−L), equivalent to the R-parity R =M(−1)2S , isa gauge transformation [387–389], a part of the centre Z4

of SO(10). In the renormalizable version of the theory itremains exact at all energies [390–392]. The lightest su-persymmetric partner (LSP) is then stable and is a naturalcandidate for the dark matter of the universe;

– its other maximal subgroup, besides SU(5) × U(1), isGPS = SU(2)L × SU(2)R × SU(4)C quark–lepton sym-metry of Pati and Salam, which plays an important role inrelating quark and lepton masses and mixings;

– the unification of gauge couplings can be achieved evenwithout supersymmetry (for a recent and complete workand references therein, see [393, 394]).

Fermions belong to the spinor representation 16F (foruseful reviews on spinors and SO(2N) group theory in gen-eral see [395–399]). From

16 × 16 = 10 + 120 + 126, (4.5)

the most general Yukawa sector in general contains 10H ,120H and 126H , respectively the fundamental vector repre-sentation, the three-index antisymmetric representation and

the five-index antisymmetric and anti-self-dual representa-tion. 126H is necessarily complex, supersymmetric or not;10H and 126H Yukawa matrices are symmetric in genera-tion space, while the 120H one is antisymmetric.

The decomposition of the relevant representations underGPS gives

16 = (2,1,4)+ (1,2, 4),10 = (2,2,1)+ (1,1,6),120 = (2,2,1)+ (3,1,6)+ (1,3,6)+ (2,2,15) (4.6)

+ (1,1,10)+ (1,1,10),

126 = (3,1,10)+ (1,3,10)+ (2,2,15)+ (1,1,6).The see-saw mechanism, whether type I or II, requires

126: it contains both (1,3,10) whose VEV gives a mass toνR (type I), and (3,1,10), which contains a color singlet,B − L = 2 field ΔL, that can give directly a small mass toνL (type II). In SU(5) language this is seen from the decom-position

126 = 1 + 5 + 15 + 45 + 50. (4.7)

The 1 of SU(5) belongs to the (1,3,10) of GPS and gives amass for νR , while 15 corresponds to the (3,1,10) and givesthe direct mass to νL.

126 can be a fundamental field, or a composite of two16H fields (for some realistic examples see for example[400–402]), or can even be induced as a two-loop effectiverepresentation built out of a 10H and two gauge 45 dimen-sional representations [403–405].

Normally the light Higgs is chosen to be the smallestone, 10H . Since 〈10H 〉 = 〈(2,2,1)〉 is a SU(4)C singlet,md = me follows immediately, independently of the num-ber of 10H . Thus we must add either 120H or 126H orboth in order to correct the bad mass relations. Both of thesefields contain (2,2,15), which VEV alone gives the relationme =−3mTd .

As 126H is needed anyway for the see-saw, it is naturalto take this first. The crucial point here is that in general(2,2,1) and (2,2,15) mix through 〈(1,3,10)〉 [222, 406]and thus the light Higgs is a mixture of the two. In otherwords, 〈(2,2,15)〉 in 126H is in general non-vanishing (insupersymmetry this is not automatic, but depends on theHiggs superfields needed to break SO(10) at MGUT or onthe presence of higher dimensional operators).

If one considers all the operators allowed by SO(10) forthe Yukawa couplings, there are too many model parame-ters, and so no prediction is really possible. One optionis to assume that the minimal number of parameters mustbe employed. It has been shown that 4 (3 of them non-renormalizable) operators are enough in models with 10 and45 Higgs representations only [8]. Although this is an impor-tant piece of information and it has been the starting point of

Eur. Phys. J. C (2008) 57: 13–182 49

a lot of model building, it is difficult to see a reason for someoperators (of different dimensions) to be present and othernot, without using some sort of flavor symmetry, so thesetype of models will not be considered in this subsection. Onthe other hand, a self consistent way of truncating the largenumber of SO(10) allowed operators without relying on ex-tra symmetries is to consider only the renormalizable ones.This is exactly what we shall assume.

In this case there are just two ways of giving mass to νR :by a nonzero VEV of the Higgs 126, or generate an effec-tive non-renormalizable operator radiatively [403]. We shallconsider in turn both of them.

4.1.2.1 Elementary 126H It is rather appealing that 10Hand 126H may be sufficient for all the fermion masses, withonly two sets of symmetric Yukawa coupling matrices. Themass matrices atMGUT are

md = vd10Y10 + vd126Y126, (4.8)

mu = vu10Y10 + vu126Y126, (4.9)

me = vd10Y10 − 3vd126Y126, (4.10)

mν =−mDM−1R mD +mνL, (4.11)

where

mD = vu10Y10 − 3vu126Y126, (4.12)

MR = vRY126, (4.13)

mνL = vLY126. (4.14)

These relations are valid at MGUT, so it is there that theirvalidity must be tested. The analysis done so far used theresults of renormalization group running fromMZ toMGUT

from [407, 408].The first attempts in fitting the mass matrices assumed the

domination of the type I seesaw. It was pioneered by treatingCP violation perturbatively in a non-supersymmetric frame-work [406], and later improved with a more detailed treat-ment of complex parameters and supersymmetric low en-ergy effective theory [409–411]. Nevertheless, these fits hadproblems to reproduce correctly the PMNS matrix parame-ters.

A new impetus to the whole program was given by theobservation that in case type II seesaw dominates (a wayto enforce it is to use a 54 dimensional Higgs representation[412]) the neutrino mass, an interesting relation in these typeof models between b–τ unification and large atmosphericmixing angle can be found [413–415]. The argument is verysimple and it can be traced to the relation [416]

mν ∝md −me, (4.15)

which follows directly from (4.8), (4.10) and (4.14), if onlythe second term (type II) in (4.11) is considered. Consid-ering only the heaviest two generations as an example andtaking the usually good approximation of small second gen-eration masses and small mixing angles, one finds all theelements of the right-hand side small except the 22 element,which is proportional to the difference of two big numbers,mb −mτ . Thus, a large neutrino atmospheric mixing angleis linked to the smallness of this 22 matrix element, and so tob–τ unification. Note that in these types of models b–τ uni-fication is no more automatic due to the presence of the 126,which breaks SU(4)C . It is, however, quite a good predictionof the RGE running in the case of low energy supersymme-try.

The numerical fitting was able to reproduce also a largesolar mixing angle both in case of type II [417, 418] ormixed seesaw [419], predicting also a quite large |Ue3| ≈0.16 mixing element, close to the experimental upper bound.The difficulty in fitting the CKM CP-violating phase in thefirst quadrant was overcome by new solutions found in [420,421], maintaining the prediction of large |Ue3| ≥ 0.1 matrixelement.

All these fittings were done assuming no constraints com-ing from the Higgs sector. Regarding it, it was found thatthe minimal supersymmetric model [422–424] has only 26model parameters [425], on top of the usual supersymmetrybreaking soft terms, as in the MSSM. When one considersthis minimal model, the VEVs in the mass formulae (4.8)–(4.14) are not completely arbitrary, but are connected by therestrictions of the Higgs sector. This has been first noticed in[426–428] showing a possible clash with the positive resultsof the unconstrained Yukawa sector studied in [420, 421].The issue has been pursued in [429], showing that in the re-gion of parameter space where the fermion mass fitting issuccessful, there are necessarily intermediate scale thresh-olds which spoil perturbativity of the RGE evolution of thegauge couplings.

To definitely settle the issue, two further checks shouldbe done. (a) The χ2 analysis used in the fitting procedureshould be implemented at MZ , not at MGUT. The point isin fact that while the errors at MZ are uncorrelated, theybecome strongly correlated after running to MGUT, due tothe large Yukawa coupling of top and possibly also of bot-tom, tau and neutrino. (b) Another issue is to consider alsothe effect of the possible increased gauge couplings on theYukawas. Only after these two checks will be done, this min-imal model could be ruled out.

A further important point is that in the case of VEVsconstrained by the Higgs sector one finds from the chargedfermion masses that the model predicts large tanβ 40,as confirmed by the last fits in [429]. In this regime theremay be sizable corrections to the “down” fermion mass ma-trices from the soft SUSY breaking parameters [430]; this

50 Eur. Phys. J. C (2008) 57: 13–182

brings into the game also the soft SUSY breaking sector,lowering somewhat the predictivity but relaxing the diffi-culty in fitting the experimental data. In this scenario pre-dictions on masses would become predictions on the softsector.

Some topics have to be still mentioned in connection withthe above: the important calculation of the mass spectrumand Clebsch–Gordan coefficients in SO(10) [399, 431–439],the doublet–triplet splitting problem [440, 441], the Higgsdoublet mass matrix [399, 433], the running of the gaugecouplings at two loops together with threshold corrections[434], and the study of proton decay [435, 442, 443].

What if this model turns out to be wrong? There are othermodels on the market. The easiest idea is to add a 120 di-mensional Higgs, that may also appear as a natural choice,being the last of the three allowed representations that cou-ple with fermions. There are three different ways of doingit considered in the literature: (a) take 120 as a small, non-leading, contribution, i.e. a perturbation to the previous for-mulae [444–446]; (b) consider 120 on an equal footing as10 and 126, but assume some extra discrete symmetry or realparameters in the superpotential, breaking CP spontaneously[447–450] (and suppressing in the first two references thedangerous d = 5 proton decay modes); (c) assume small 126contributions to the charged fermion masses [451–454].

Another limit is to forget the 10H altogether, as has beenproposed for non-supersymmetric theories [455]. The twogeneration study predicts a too small ratiomb/mτ ≈ 0.3, in-stead of the value 0.6 that one gets by straight running. Theidea is that this could get large corrections due to Dirac neu-trino Yukawas [456] and the effect of finite second genera-tion masses, as well as the inclusion of the first generationand CP-violating phases. This is worth pursuing for it pro-vides an alternative minimal version of SO(10), and after all,supersymmetry may not be there.

4.1.2.2 Radiative 126H The original idea [403] is thatthere is no 126H representation in the theory, but the sameoperator is generated by loop corrections. The representa-tion that breaks the rank of SO(10) is now 16H , which VEVwe callMΛ. Generically there is a contribution to the right-handed neutrino mass at two loops:

MR ≈(α

4π

)2 M2Λ

MGUT

MSUSY

MGUTY10, (4.16)

which is too small in low energy supersymmetry (low break-ing scale MSUSY) as well as non-supersymmetric theories(MSUSY =MGUT, but low intermediate scale MΛ requiredby gauge coupling unification). The only exception, pro-posed in [404], could be split supersymmetry [457, 458].

In the absence of 126H , the charged fermion masses mustbe given by only 10H and 120H [404], together with radia-tive corrections. The simplest analysis of the tree order two

generation case gives three interesting predictions-relations[405, 459]: (1) almost exact b–τ unification; (2) large at-mospheric mixing angle related to the small quark θbc mix-ing angle; (3) somewhat degenerate neutrinos. For a seriousnumerical analysis one needs to use the RGE for the caseof split supersymmetry, taking a very small tanβ < 1 to getan approximate b–τ unification [458]. One needs also somefine-tuning of the parameters to account for the small ra-tio MSUSY/MGUT ≤ 10−(3−4) required in realistic modelsto have gluinos decay fast enough [460].

4.2 Higher dimensional approaches

Recently, in the context of theories with extra spatial di-mensions, some new approaches toward the question ofSM fermion mass hierarchy and flavor structure have arisen[461–468]. For instance, the SM fermion mass spectrumcan be generated naturally by permitting the quark/leptonmasses to evolve with a power-law dependence on the massscale [465, 466]. The most studied and probably most at-tractive idea for generating a non-trivial flavor structure isthe displacement of various SM fermions along extra dimen-sion(s). This approach is totally different from the one dis-cussed in Sect. 2, as it is purely geometrical and thus doesnot rely on the existence of any novel symmetry in the shortdistance theory. The displacement idea applies to the scenar-ios with large flat [467] or small warped [468] extra dimen-sion(s), as we develop in the following subsections.

4.2.1 Large extra dimensions

In order to address the gauge hierarchy problem, a sce-nario with large flat extra dimensions has been proposed byArkani-Hamed, Dimopoulos and Dvali (ADD) [469–471],based on a reduction of the fundamental gravity scale downto the TeV scale. In this scenario, gravity propagates in thebulk whereas SM fields live on a 3-brane. One could as-sume that this 3-brane has a certain thickness L along anextra dimension (as for example in [472]). Then SM fieldswould feel an extra dimension of size L, exactly as in auniversal extra dimension (UED) model [473] (where SMfields propagate in the bulk) with one extra dimension ofsize L.12

In such a framework, the SM fermions can be localizedat different positions along this extra dimension L. Thenthe relative displacements of quark/lepton wave functionpeaks produce suppression factors in the effective four-dimensional Yukawa couplings. These suppression factors

12The constraint from electroweak precision measurements is R−1 �2–5 TeV, the one from direct search at LEP collider is L−1 � 5 TeVand the expected LHC sensitivity is about L−1 ∼ 10 TeV.

Eur. Phys. J. C (2008) 57: 13–182 51

being determined by the overlaps of fermion wave func-tions (getting smaller as the distance between wave func-tion peaks increases), they can vary with the fermion flavorsand thus induce a mass hierarchy. This mechanism was firstsuggested in [467] and its variations have been studied in[474–484].

Let us describe this mechanism more precisely. Thefermion localization can be achieved through either non-perturbative effects in string/M theory or field-theoreticalmethods. One field-theoretical possibility is to couple theSM fermion fields Ψi(xμ, x5) [i = 1, . . . ,3 being the fam-ily index and μ= 1, . . . ,4 the usual coordinate indexes] tofive dimensional scalar fields with VEV Φi(x5) dependingon the extra dimension (parameterized by x5).13 Indeed, chi-ral fermions are confined in solitonic backgrounds [485]. Ifthe scalar field profile behaves as a linear function of theform Φi(x5) = 2μ2x5 − mi around its zero-crossing pointx0i = mi/2μ2, the zero-mode of five dimensional fermion

acquires a Gaussian wave function of typical width μ−1

and centered at x0i along the x5 direction: Ψ (0)i (xμ, x5) =

Ae−μ2(x5−x0i )

2ψi(xμ), ψi(xμ) being the four-dimensional

fermion field and A = (2μ2/π)1/4 a normalization factor.Then the four-dimensional Yukawa couplings between thefive dimensional SM Higgs boson H and zero-mode fermi-ons, obtained by integration on x5 over the wall width L,14

SYukawa =∫d5x

√LκH(xμ, x5)Ψ

(0)i (xμ, x5)Ψ

(0)j (xμ, x5)

=∫d4x Yijh(xμ)ψi(xμ)ψj (xμ), (4.17)

are modulated by the following effective coupling constants,

Yij =∫dx5 κA

2e−μ2(x5−x0i )

2e−μ2(x5−x0

j )2

= κe−μ2

2 (x0i −x0

j )2. (4.18)

It can be considered as natural to have a five dimensionalYukawa coupling constant equal to

√Lκ , where the dimen-

sionless parameter κ is universal (in flavor and nature offermions) and of order unity, so that the flavor structureis mainly generated by the field localization effect throughthe exponential suppression factor in (4.18). The remarkablefeature is that, due to this exponential factor, large hierar-chies can be created among the physical fermion masses,even for all fundamental parameters mi of order of the sameenergy scale μ.

13Although we concentrate here on the case with only one extra dimen-sion, for simplicity, the mechanism can be directly extended to moreextra dimensions.14Here, the factor

√L compensates with the Higgs component

along x5, since the Higgs boson is not localized.

This mechanism can effectively accommodate all thedata on quark and charged lepton masses and mixings [486–488]. In case that right handed neutrinos are added to the SMso that neutrinos acquire Dirac masses (as those originatingfrom Yukawa couplings (4.17)), neutrino oscillation experi-ment results can also be reproduced [472]. The fine-tuning,arising there on relative x0

i parameters, turns out to be im-proved when neutrinos get Majorana masses instead [489](see also [235, 490]).

4.2.2 Small extra dimensions

Another type of higher-dimensional scenario solving thegauge hierarchy problem was suggested by Randall andSundrum (RS) [491, 492]. There, the unique extra dimen-sion is warped and has a size of order M−1

Pl (MPl being thereduced Planck mass:MPl = 2.44×1018 GeV) leading to aneffective gravity scale around the TeV. In the initial version,gravity propagates in the bulk and SM particles are all stuckon the TeV-brane. An extension of the original RS modelwas progressively proposed [493–497], motivated by its in-teresting features with respect to the gauge coupling unifi-cation [498–503] and dark matter problem [504, 505]. Thisnew set-up is characterized by the presence of SM fields,except the Higgs boson (to ensure that the gauge hierarchyproblem does not re-emerge), in the bulk.

In this RS scenario with bulk matter, a displacementof SM fermions along the extra dimension is also possi-ble [468]: the effect is that the effective four-dimensionalYukawa couplings are affected by exponential suppressionfactors, originating from the wave function overlaps be-tween bulk fermions and Higgs boson (confined on our TeV-brane). If the fermion localization depends on the flavor andnature of fermions, then the whole structure in flavor spacecan be generated by these wave function overlaps. In partic-ular, if the top quark is located closer to the TeV-brane thanthe up quark, then its overlap with the Higgs boson, and thusits mass after electroweak symmetry breaking, is larger rela-tively to the up quark (for identical five dimensional Yukawacoupling constants).

More precisely, the fermions can acquire different local-izations if each field Ψi(xμ, x5) is coupled to a distinct fivedimensional mass mi :

∫d4x

∫dx5

√G miΨiΨi , G being

the determinant of the RS metric. To modify the locationof fermions, the masses mi must have a non-trivial depen-dence on x5, like mi = sign(x5)cik, where ci are dimen-sionless parameters and 1/k is the curvature radius of anti-de Sitter space. Then the fields decompose as, Ψi(xμ, x5)=∑∞n=0ψ

(n)i (x

μ)f in(x5) [n labeling the tower of Kaluza–Klein (KK) excitations], admitting the following solutionfor the zero-mode wave function, f i0 (x5) = e(2−ci )k|x5|/Ni0,where Ni0 is a normalization factor.

52 Eur. Phys. J. C (2008) 57: 13–182

The Yukawa interactions with the Higgs boson H read

SYukawa =∫d5x

√G(Y(5)ij HΨ+iΨ−j + h.c.

)

=∫d4xMij ψ

(0)Li ψ

(0)Rj + h.c.+ · · · . (4.19)

The Y (5)ij are the five dimensional Yukawa coupling con-stants and the dots stand for KK mass terms. The fermionmass matrix is obtained after integrating:

Mij =∫dx5

√GY

(5)ij Hf

i0 (x5)f

j

0 (x5). (4.20)

The Y (5)ij can be chosen almost universal so that the quark/lepton mass hierarchies are mainly governed by the over-lap mechanism. Large fermion mass hierarchies can be pro-duced for fundamental mass parameters mi all of order ofthe unique scale of the theory k ∼MPl.

With this mechanism, the quark masses and CKM mix-ing angles can be effectively accommodated [506–508], aswell as the lepton masses and PMNS mixing angles in bothcases where neutrinos acquire Majorana masses (via eitherdimension five operators [509] or the see-saw mechanism[510]) and Dirac masses (see [511], and [512, 513] for orderunity Yukawa couplings leading to mass hierarchies essen-tially generated by the geometrical mechanism).

4.2.3 Sources of FCNC in extra dimension scenarios

GIM-violating FCNC effects in extra dimension scenariosmay appear both from tree level and from loop effects.

At tree level FCNC processes can be induced by ex-changes of KK excitations of neutral gauge bosons. Theneutral current action of the effective four-dimensional cou-pling, between SM fermions ψ(0)i (x

μ) and KK excitations

of any neutral gauge boson A(n)μ (xμ), reads in the interac-tion basis

SNC = gSML

∫d4x

∞∑n=1

ψ(0)Li γ

μC(n)Lijψ(0)Lj A

(n)μ + {L↔R}.

(4.21)

Therefore, FCNC interactions can be induced by the non-universality of the effective coupling constants gSM

L/R×Ci(n)0between KK modes of the gauge fields and the three SMfermion families (which have different locations along x5).

At the loop level, KK fermion excitations may invali-date the GIM cancellation, as discussed e.g. in [511, 514]for �±α → �±β γ . Indeed, these excitations have KK masseswhich are not negligible (and thus not quasi-degenerate infamily space) compared to mW± . The GIM mechanism isalso invalidated by the loop contributions of the KK W±(n)

modes which couple (KK level by level), e.g. to leptons inthe four-dimensional theory, via an effective mixing matrixof type V eff

MNS = Ul†L C(n)L UνL being non-unitary due to thenon-universality of

C(n)L ≡ diag(C1(n)m ,C2(n)

m ,C3(n)m

). (4.22)

In this diagonal matrix, Ci (n)m quantifies the wave func-tion overlap along the extra dimension between the W±(n)[n ≥ 1] and exchanged (mth level KK) fermion f im(x5)

[i = {1,2,3} being the generation index] (see below formore details).

The GIM mechanism for leptons can be clearly restoredif the three coefficientsCi(n)m as well as the three KK fermionmasses mi(m)KK are equal to each other, i.e. are universal withrespect to i = {1,2,3} (KK level by level) [515]. Within thequark sector, on the other hand, the top quark mass cannotbe totally neglected relatively to the KK up-type quark ex-citation scales, leading to a mass shift of the KK top quarkmode from the rest of the KK up-type quark modes and re-moving the degeneracy among three family masses of the upquark excitations at fixed KK level (with regard to mW±(n) ).Moreover, this means that the Yukawa interaction with theHiggs boson induces a substantial mixing of the top quarkKK tower members among themselves [481, 516].

For example, the data on b→ sγ (receiving a contribu-tion from the exchange of aW±(n) [n= 0,1, . . . ] gauge fieldand an up quark, or its KK excitations, at one loop-level) canbe accommodated in the RS model withm(W±(1)) 1 TeV,as shown in [515] using numerical methods for the diagonal-ization of a large dimensional mass matrix and taking intoaccount the top quark mass effects described previously.

4.2.4 Mass bounds on Kaluza–Klein excitations

In this subsection we develop constraints on the KK gaugeboson masses derived from the tree level FCNC effect de-scribed above. Our purpose is to determine whether theseconstraints still allow the KK gauge bosons to be sufficientlylight to imply potentially visible signatures at LHC.

4.2.4.1 Large extra dimensions Let us consider the gener-ic framework of a flat extra dimension, with a large size L,along which gravity as well as gauge bosons propagate. TheSM fermions are located at different points of the fifth di-mension, so that their mass hierarchy can be interpreted interm of the geometrical mechanism described in details inSect. 4.2.1. In such a framework the exchange of the KKexcitations of the gluon can bring important contributionsto the K0–K0 mixing (�F = 2) at tree level. Indeed, theKK gluon can couple the d quark with the s quark, if theselight down-quarks are displaced along the extra dimension.The obtained KK contribution to the mass splitting �mK in

Eur. Phys. J. C (2008) 57: 13–182 53

the kaon system depends on the KK gluon coupling betweenthe s and d quarks (which is fixed by quark locations) andmainly on the mass of the first KK gluon M(1)KK. Assumingthat the s, d quark locations are such that the ms , md massvalues are reproduced, the obtained �mK and also |εK | aresmaller than the associated experimental values for, respec-tively,

M(1)KK � 25 TeV, and M

(1)KK � 300 TeV, (4.23)

as found by the authors of [517]. The same bound comingfrom the D0 meson system is weaker.

In the lepton sector the experimental upper limit on thebranching ratio B(μ → eee) imposes typically the con-straint [517]

M(1)KK � 30 TeV, (4.24)

since the exchange of the KK excitations of the electroweakneutral gauge bosons contributes to the decay μ→ eee.

To conclude, we stress that if the extra dimensions treatfamilies in a non-universal way (which could explain thefermion mass hierarchy), the indirect bounds from FCNCphysics like the ones in (4.23)–(4.24) force the mass of theKK gauge bosons to be far from the collider reach. As amatter of fact, the LHC will be able to probe the KK excita-tions of gauge bosons only up to 6–7 TeV [518–521] in thepresent context.

4.2.4.2 Small extra dimensions In the context of the RSmodel with SM fields in the bulk, described in Sect. 4.2.2,the exchange of KK excitations of neutral gauge bosons (likee.g. the first Z0 excitation: Z(1)) also contributes to FCNCprocesses at tree level [468, 507, 522–526] since these KKstates possess FC couplings if the different families of fermi-ons are displaced along the warped extra dimension. Thereexist some configurations of fermion locations, pointed outin [513], which simultaneously reproduce all quark/leptonmasses and mixing angles via the wave function effectsand lead to amplitudes of FCNC reactions [lα → lβ lγ lγ ,Z0 → lαlβ , P 0−P 0 mixing of a generic meson P ,μ–e con-version, K0 → lαlβ and K+ → π+νν] compatible with thecorresponding experimental constraints even for light neu-tral KK gauge bosons:

M(1)KK � 1 TeV. (4.25)

The explanation of this result is the following. If the SMfermions with different locations are localized typicallyclose to the Planck-brane, they have quasi-universal cou-plings Ci(n)0 [cf. (4.21)] with the KK gauge bosons whichhave a wave function almost constant along the fifth dimen-sion near the Planck-brane. Therefore, small FC couplingsare generated in the physical basis for these fermions lead-ing to the weak bound (4.25). The fermions from the third

family, associated to heavy flavors, cannot be localized ex-tremely close to the Planck-brane since their wave functionoverlap with the Higgs boson [confined on the TeV-brane]must be large in order to generate high effective Yukawacouplings. Nevertheless, this is compensated by the fact thatphenomenological FCNC constraints are usually less severein the third generation sector.

As a result, the order of lower limits on M(1)KK com-ing from the considerations on both fermion mass data andFCNC processes can be as low as TeV. From the purely the-oretical point of view, the favored order of magnitude forM(1)KK is O(1) TeV which corresponds to a satisfactory solu-

tion for the gauge hierarchy problem. From the model build-ing point of view one has to rely on an appropriate extensionof the RS model insuring that, for light KK masses, the devi-ations of the electroweak precision observables do not con-flict with the experimental results. The existing RS exten-sions, like the scenarios with brane-localized kinetic termsfor fermions [527] and gauge bosons [528] (see [529, 530]for the localized gauge boson kinetic terms and [531] for thefermion ones), or the scenarios with an extended gauge sym-metry (see [532–534] for different fermion charges underthis broken symmetry), allowM(1)KK to be as low as ∼3 TeV.In such a case, one can expect a direct detection of the KKexcited gauge bosons at LHC.

4.3 Minimal flavor violation in the lepton sector

4.3.1 Motivations and basic idea

Within the SM the dynamics of flavor-changing transitionsis controlled by the structure of fermion mass matrices. Inthe quark sector, up and down quarks have mass eigen-values which are up to 105 times smaller than the elec-troweak scale, and mass matrices which are approximatelyaligned. This results in the effective CKM and GIM suppres-sions of charged and neutral flavor violating interactions, re-spectively. Forcing this connection between the low energyfermion mass matrices and the flavor-changing couplings tobe valid also beyond the SM, leads to new-physics scenarioswith a high level of predictivity (in the flavor sector) and anatural suppression of flavor-changing transitions. The latterachievement is a key ingredient to maintain a good agree-ment with experiments in models where flavored degrees offreedom are expected around the TeV scale.

This is precisely the idea behind the minimal flavor vi-olation principle [535–537]. It is a fairly general hypothe-sis that can be implemented in strongly-interacting theories[535], low energy supersymmetry [536, 537], multi-Higgs[537, 538] and GUT [539] models. In a model indepen-dent formulation, the MFV construction consists in iden-tifying the flavor symmetry and symmetry breaking struc-ture of the SM and enforce it in a more general effec-tive theory (written in terms of SM fields and valid above

54 Eur. Phys. J. C (2008) 57: 13–182

the electroweak scale). In the quark sector this procedureis unambiguous: the largest group of flavor changing fieldtransformations commuting with the gauge group is Gq =SU(3)QL × SU(3)uR × SU(3)dR , and this group is brokenonly by the Yukawa couplings. The invariance of the SM La-grangian under Gq can be formally recovered elevating theYukawa matrices to spurion fields with appropriate transfor-mation properties under Gq . The hypothesis of MFV statesthat these are the only spurions breaking Gq also beyondthe SM. Within the effective theory formulation, this im-plies that all the higher dimensional operators constructedfrom SM and Yukawa fields must be (formally) invariantunder Gq . The consequences of this hypothesis in the quarksector have been extensively analyzed in the literature (seee.g. Refs. [540, 541]). Without entering into the details, wecan state that the MFV hypothesis provides a plausible ex-planation of why no new-physics effects have been observedso far in the quark sector.

Apart from arguments based on the analogy with quarks,and despite the scarce experimental information, the defini-tion of a minimal lepton flavor violation (MLFV) principle[542] is demanded by a severe fine-tuning problem in LFVdecays of charged leptons. Within a generic effective theoryapproach, the radiative decays li → lj γ proceed through thefollowing gauge-invariant operator

δRLij

Λ2LFV

H †eiRσσρL

jLFσρ, (4.26)

where δRLij are the generic flavor-changing couplings andΛLFV denotes the cut-off of the effective theory. In the ab-sence of a specific flavor structure, it is natural to expectδRLij = O(1). In this case the experimental limit for μ→ eγ

implies ΛLFV > 105 TeV, in clear tension with the expec-tation of new degrees of freedom close to the TeV scale inorder to stabilize the Higgs sector of the SM.

The implementation of a MFV principle in the leptonsector is not as simple as in the quark sector. The problemis that the neutrino mass matrix itself cannot be accommo-dated within the renormalizable part of the SM Lagrangian.The most natural way to describe neutrino masses, explain-ing their strong suppression, is to assume they are Majoranamass terms suppressed by the heavy scale of lepton num-ber violation (LNV). In other words, neutrino masses aredescribed by a non-renormalizable interaction of the typeequation (3.4) suppressed by the scale ΛLNV � v = |〈H 〉|.This implies that we have to face a two scale problem (pre-sumably with the hierarchy ΛLNV � ΛLFV) and that weneed some additional hypothesis to identify the irreducibleflavor-symmetry breaking structures. As we shall illustratein the following, we can choose whether to extend or notthe field content of the SM. The construction of the effec-tive theory based on one of these realizations of the MLFV

hypothesis can be viewed as a general tool to exploit the ob-servable consequences of a specific (minimalistic) hypothe-sis about the irreducible sources of lepton-flavor symmetrybreaking.

4.3.2 MLFV with minimal field content

The lepton field content is the SM one: three left handeddoublets LiL and three right handed charged lepton singletseiR . The flavor symmetry group is Gl = SU(3)LL × SU(3)eRand we assume the following flavor symmetry breaking La-grangian

LSym.Br. = −Y ije eiR(H †L

jL

)

− 1

2ΛLNVκijν(LciL τ2H

)(HT τ2L

jL

)+ h.c.

→ − vY ije eiRejL −v2

2ΛLNVκijν ν

ciL ν

jL + h.c. (4.27)

Here the two irreducible sources of LFV are the coefficientof dimension five LNV operator (κijν ) and the charged leptonYukawa coupling (Ye), transforming respectively as (6,1)and (3,3) under Gl . An explicit realization of this scenario isprovided by the so-called triplet see-saw mechanism (or see-saw of type II). This approach has the advantage of beinghighly predictive, but it differs in an essential way from theMFV hypothesis in the quark sector since one of the basicspurion originates from a non-renormalizable coupling.

Having identified the irreducible sources of flavor sym-metry breaking and their transformation properties, we canclassify the non-renormalizable operators suppressed by in-verse powers of ΛLFV which contribute to flavor violatingprocesses. These operators must be invariant combinationsof SM fields and the spurions Ye and κν . The complete list ofthe leading operators contributing to LFV decays of chargedleptons is given in Refs. [542, 543]. The case of the radia-tive decays li → lj γ is particularly simple since there areonly two dimension six operators (operators with a structureas in (3.4), with Fσρ replaced by the stress tensors of theU(1)Y and SU(2)L gauge groups, respectively). The MLFVhypothesis forces the flavor-changing couplings of these op-erators to be a spurion combination transforming as (3,3)under Gl :(δRLmin

)ij∝ (Yeκ†

ν κν)ij+ · · · (4.28)

where the dots denote terms with higher powers of Ye or κν .Up to the overall normalization, this combination can becompletely determined in terms of the neutrino mass eigen-values and the PMNS matrix. In the basis where Ye is diag-onal we can write,

Eur. Phys. J. C (2008) 57: 13–182 55

(Yeκ

†ν κν)i �=j = mli

v

(Λ2

LNV

v4UPMNSm

2νU

†PMNS

)

i �=j

→ mli

v

Λ2LNV

v4

[(UPMNS)i2(UPMNS)

∗j2�m

2sol

± (UPMNS)i3(UPMNS)∗j3�m

2atm

], (4.29)

where �m2atm and �m2

sol denote the squared mass differ-ences deduced from atmospheric- and solar-neutrino data,and +/− correspond to normal/inverted hierarchy, respec-tively. The overall factor Λ2

LNV/v2 implies that the absolute

normalization of LFV rates suffers of a large uncertainty.Nonetheless, a few interesting conclusions can still be drawn[542].

– The LFV decay rates are proportional toΛ4LNV/Λ

4LFV and

could be detected only in presence of a large hierarchybetween these two scales. In particular, B(μ→ eγ ) >

10−13 only if ΛLNV > 109ΛLFV.– Ratios of similar LFV decay rates, such as B(μ →eγ )/B(τ → μγ ), are free from the normalization am-biguity and can be predicted in terms of neutrino massesand PMNS angles: violations of these predictions wouldunambiguously signal the presence of additional sourcesof lepton-flavor symmetry breaking. One of these predic-tion is the 10−2–10−3 enhancement of B(τ → μγ ) ver-sus B(μ→ eγ ) shown in Fig. 3. Given the present andnear-future experimental prospects on these modes, thismodest enhancement implies that the μ→ eγ search ismuch more promising within this framework.

– Ratios of LFV transitions among the same two fami-lies (such as μ→ eγ versus μ→ 3e or τ → μγ vsτ → 3μ and τ → μee) are determined by known phasespace factors and ratios of various Wilson coefficients.As data will become available on different lepton flavorviolating processes, if the flavor patter is consistent withthe MLFV hypothesis, from these ratios it will be pos-

Fig. 3 Bli→lj γ ≡ Γ (li → lj γ )/Γ (li → lj νi νj ) for μ → eγ andτ→ μγ as a function of sin θ13 in the MLFV framework with minimalfield content [542]. The normalization of the vertical axis correspondsto ΛLNV/ΛLFV = 1010. The shading is due to different values of thephase δ and the normal/inverted spectrum

sible to disentangle the contributions of different opera-tors.

– A definite prediction of the MLFV hypothesis is thatthe rates for decays involving light hadrons (π0 → μe,KL→ μe, τ→ μπ0, . . .) are exceedingly small.

4.3.3 MLFV with extended field content

In this scenario we assume three heavy right handed Ma-jorana neutrinos in addition to the SM fields. As a conse-quence, the maximal flavor group becomes Gl × SU(3)νR .In order to minimize the number of free parameters (or tomaximize the predictivity of the model), we assume that theMajorana mass term for the right handed neutrinos is pro-portional to the identity matrix in flavor space: (MR)ij =MR × δij . This mass term breaks SU(3)νR to O(3)νR and isassumed to be the only source of LNV (MR ↔ΛLNV).

Once the field content of model is extended, there arein principle many alternative options to define the irre-ducible sources of lepton flavor symmetry breaking (see e.g.Ref. [544] for an extensive discussion). However, this spe-cific choice has two important advantages: it is predictiveand closely resemble the MFV hypothesis in the quark sec-tor. The νR are the counterpart of right handed up quarksand, similarly to the quark sector, the symmetry breakingsources are two Yukawa couplings of (3.40). An explicitexample of MLFV with extended field content is the min-imal supersymmetric standard model with degenerate righthanded neutrinos.

The classification of the higher dimensional operators inthe effective theory proceeds as in the minimal field con-tent case. The only difference is that the basic spurions arenow Yν and Ye , transforming as (3,1,3) and (3,3,1) underGl ×O(3)νR , respectively. The determination of the spurionstructures in terms of observable quantities is more involvedthan in the minimal field content case. In general, invert-ing the see-saw relation allows us to express Yν in terms ofneutrino masses, PMNS angles and an arbitrary complex-orthogonal matrix R of (3.45) [232]. Exploiting the O(3)νRsymmetry of the MLFV Lagrangian, the real orthogonal partof R can be rotated away. We are then left with a Hermitian-orthogonal matrix H [545] which can be parameterized interms of three real parameters (φi ) which control the amountof CP violation in the right handed sector:

Yν = M1/2R

vH(φi)m

1/2diagU

†PMNS. (4.30)

With this parameterization for Yν the flavor changing cou-pling relevant to li → lj γ decays reads

δRLext ∝ Ye(Y †ν Yν)

→ me

v

(MR

v2UPMNSm

1/2diagH

2m1/2diagU

†PMNS

). (4.31)

56 Eur. Phys. J. C (2008) 57: 13–182

In the CP-conserving limit H → I and the phenomenolog-ical predictions turns out to be quite similar to the minimalfield content scenario [542]. In particular, all the general ob-servations listed in the previous section remain valid. In thegeneral case, i.e. for H �= I , the predictivity of the modelis substantially weakened. However, in principle some in-formation about the matrix H can be extracted by study-ing baryogenesis through leptogenesis in the MLFV frame-work [546].

4.3.4 Leptogenesis

On general grounds, we expect that the tree-level degener-acy of heavy neutrinos is lifted by radiative corrections. Thisallows the generation of a lepton asymmetry in the interfer-ence between tree-level and one loop decays of right handedneutrinos. Following the standard leptogenesis scenario, weassume that this lepton asymmetry is later communicatedto the baryon sector through sphaleron effects and that sat-urates the observed value of the baryon asymmetry of theuniverse.

The most general form of the νR mass splittings allowedwithin the MLFV framework has the following form:

�MR

MR= cν

[YνY

†ν +

(YνY

†ν

)T ]

+ c(1)νν[YνY

†ν YνY

†ν +

(YνY

†ν YνY

†ν

)T ]

+ c(2)νν[YνY

†ν

(YνY

†ν

)T ]+ c(3)νν[(YνY

†ν

)TYνY

†ν

]

+ cνl[YνY

†e YeY

†ν +

(YνY

†e YeY

†ν

)T ]+ · · · .Even without specifying the value of the ci , this form allowsus to derive a few general conclusions [546].

– The term proportional to cν does not generate a CPVasymmetry, but sets the scale for the mass splittings: theseare of the order of magnitude of the decay widths, realiz-ing in a natural way the condition of resonant leptogene-sis.

– The right amount of leptogenesis can be generated evenwith Ye = 0, if all the φi are non-vanishing. However,since Yν ∼√

MR , for low values ofMR (� 1012 GeV) theasymmetry generated by the cνl term dominates. In thiscase ηB is typically too small to match the observed valueand has a flat dependence on MR . At MR � 1012 GeVthe quadratic terms c(i)νν dominate, determining an approx-imate linear growth of ηB with MR . These two regimesare illustrated in Fig. 4.

As demonstrated in Ref. [546], baryogenesis through lep-togenesis is viable in MLFV models. In particular, assum-ing a loop hierarchy between the ci (as expected in a per-turbative scenario) and neglecting flavor-dependent effectsin the Boltzmann equations (one-flavor approximation ofRef. [547]), the right size of ηB is naturally reached forMR � 1012 GeV. As discussed in Ref. [301] (see also [303]),this lower bound can be weakened by the inclusion of flavor-dependent effects in the Boltzmann equations and/or by thetanβ-enhancement of Ye occurring in two-Higgs doubletmodels.

From the phenomenological point of view, an importantdifference with respect to the CP-conserving case is the factthat non-vanishing φi change the predictions of the LFV de-cays, typically producing an enhancement of the B(μ→eγ )/B(τ → μγ ) ratio or the both decays separately [545].For MR � 1012 GeV their effect is moderate and the CP-conserving predictions are recovered. The other importantinformation following from the leptogenesis analysis is the

Fig. 4 Baryon asymmetry (ηB )as a function of the right handedneutrino mass scale (MR) forcνl = 0 (dots) and cνl �= 0(crosses) in the MLFVframework with extended fieldcontent [546]

Eur. Phys. J. C (2008) 57: 13–182 57

fact that the largeMR regime is favored. Assuming ΛLFV tobe close to the TeV scale, the MR regime favored by lepto-genesis favors a μ→ eγ rate within the reach of the MEGexperiment [548].

4.3.5 GUT implementation

Once we accept the idea that flavor dynamics obeys a MFVprinciple, both in the quark and in the lepton sector, it is in-teresting to ask if and how this is compatible with a grandunified theory (GUT), where quarks and leptons sit in thesame representations of a unified gauge group. This ques-tion has recently been addressed in [539], considering theexemplifying case of SU(5)gauge.

Within SU(5)gauge, the down-type singlet quarks (dciR)and the lepton doublets (LiL) belong to the 5 representa-tion; the quark doublet (QiL), the up-type (uciR) and leptonsinglets (eciR) belong to the 10 representation, and finally theright handed neutrinos (νiR) are singlet. In this frameworkthe largest group of flavor transformation commuting withthe gauge group is GGUT = SU(3)5 × SU(3)10 × SU(3)1,which is smaller than the direct product of the quark andlepton groups discussed before (Gq × Gl). We should there-fore expect some violations of the MFV+MLFV predictionseither in the quark or in the lepton sector or in both.

A phenomenologically acceptable description of the lowenergy fermion mass matrices requires the introduction ofat least four irreducible sources of GGUT breaking. Fromthis point of view the situation is apparently similar to thenon-unified case: the four GGUT spurions can be put in one-to-one correspondence with the low energy spurions Yu, Yd ,Ye, and Yν . However, the smaller flavor group does not al-low the diagonalization of Yd and Ye (which transform in thesame way under GGUT) in the same basis. As a result, twoadditional mixing matrices can appear in the expressions forflavor changing rates: C = V TeRVdL and G = V TeLVdR . Thehierarchical texture of the new mixing matrices is knownsince they reduce to the identity matrix in the limit YTe = Yd .Taking into account this fact, and analyzing the structureof the allowed higher-dimensional operators, a number ofreasonably firm phenomenological consequences can be de-duced [539]:

– There is a well defined limit in which the standard MFVscenario for the quark sector is fully recovered: MR �1012 GeV and small tanβ (in a two-Higgs doublet case).For MR ∼ 1012 GeV and small tanβ , deviations fromthe standard MFV pattern can be expected in rare K de-cays but not in B physics. Ignoring fine-tuned scenarios,MR � 1012 GeV is excluded by the present constraintson quark FCNC transitions. Independently from the valueofMR , deviations from the standard MFV pattern can ap-pear both in K and in B physics for tanβ �mt/mb .

– Contrary to the non-GUT MFV framework, the rate forμ→ eγ (and other LFV decays) cannot be arbitrarilysuppressed by lowering the average mass MR of theheavy νR . This fact can easily be understood by lookingat the flavor structure of the relevant effective couplings,which now assume the following form:

δRLGUT = c1YeY

†ν Yν + c2YuY

†u Ye + c3YuY

†u Y

Td + · · · .

(4.32)

In addition to the terms involving Yν ∼ √MR already

present in the non-unified case, the GUT group al-lows also MR-independent terms involving the quarkYukawa couplings. The latter become competitive forMR � 1012 GeV and their contribution is such that forΛLFV � 10 TeV the μ→ eγ rate is above 10−13 (i.e.within the reach of MEG [548]).

– Improved experimental information on τ→ μγ and τ→eγ would be a powerful tool in discriminating the rel-ative size of the standard MFV contributions versus thecharacteristic GUT-MFV contributions due to the differ-ent hierarchy pattern among τ → μ, τ → e, and μ→ e

transitions.

5 Phenomenology of theories beyond the standardmodel

5.1 Flavor violation in non-SUSY models directly testableat LHC

5.1.1 Multi-Higgs doublet models

The arbitrariness of quark masses, mixing and CP violationin the standard model stems from the fact that gauge invari-ance does not constrain the flavor structure of Yukawa inter-actions. In the SM neutrinos are strictly massless. No neu-trino Dirac mass term can be introduced, due to the absenceof right handed neutrinos and no Majorana mass terms canbe generated, due to exact B − L conservation. Since neu-trinos are massless, there is no leptonic mixing in the SM,which in turn leads to separate lepton flavor conservation.Therefore, the recent observation of neutrino oscillations isevidence for physics beyond the SM. Fermion masses, mix-ing and CP violation are closely related to each other andalso to the Higgs sector of the theory.

It has been shown that gauge theories with fermions,but without scalar fields, do not break CP symmetry [549].A scalar (Higgs) doublet is used in the SM to break boththe gauge symmetry and generate gauge boson masses aswell as fermion masses through Yukawa interactions. Thisis known as the Higgs mechanism, which was proposed byseveral authors [550–553]. It predicts the existence of one

58 Eur. Phys. J. C (2008) 57: 13–182

neutral scalar Higgs particle—the Higgs boson. In the SMwhere a single Higgs doublet is introduced, it is not possi-ble to have spontaneous CP violation since any phase in thevacuum expectation value can be eliminated by rephasingthe Higgs field. Furthermore, in the SM it is also not possi-ble to violate CP explicitly in the Higgs sector since gaugeinvariance together with renormalizability restrict the Higgspotential to have only quadratic and quartic terms and her-miticity constrains both of these to be real. Thus, CP viola-tion in the SM requires the introduction of complex Yukawacouplings.

The scenario of spontaneous CP and T violation has thenice feature of putting the breakdown of discrete symmetrieson the same footing as the breaking of the gauge symmetry,which is also spontaneous in order to preserve renormaliz-ability. A simple extension of the Higgs sector that may giverise to spontaneous CP violation requires the presence of atleast two Higgs doublets, and was introduced by Lee [554].

If one introduces two Higgs doublets, it is possible tohave either explicit or spontaneous CP breaking. Explicit CPviolation in the Higgs sector arises due to the fact that in thiscase there are gauge invariant terms in the Lagrangian whichcan have complex coefficients. Note however that the pres-ence of complex coefficients does not always lead to explicitCP breaking.

Extensions of the SM with extra Higgs doublets are verynatural since they keep the ρ parameter at tree level equal toone [555]. In multi-Higgs systems there are in general, addi-tional sources of CP violation in the Higgs sector [556]. Themost general renormalizable polynomial consistent with theSU(2)×U(1)× SU(3)c model with nd Higgs doublets, φi ,may be written as

Lφ = Yabφ†aφb +Zabcd

(φ†aφb)(φ†c φd), (5.1)

where repeated indices are summed. Hermiticity of Lφ im-plies:

Y ∗ab = Yba; Z∗

abcd = Zbadc. (5.2)

Furthermore, by construction it is obvious that:

Zabcd = Zcdab. (5.3)

In models with more than one Higgs doublet, one has thefreedom to make Higgs-basis transformations (HBT) that donot change the physical content of the model, but do changeboth the quadratic and the quartic coefficients. Coefficientsthat are complex in one Higgs basis may become real in an-other basis. Furthermore, a given model may have complexquartic coefficients in one Higgs basis, while they may allbecome real in another basis, with only the quadratic coef-ficients now complex, thus indicating that in that particular

model CP is only softly broken. Such Higgs-basis transfor-mations leave the Higgs kinetic energy term invariant andare of the form:

φaHBT−→ φ′a = Vaiφi, φ†

a

HBT−→ (φ′)†a = V ∗ai(φ

′)†i , (5.4)

where V is an nd × nd unitary matrix acting in the space ofHiggs doublets. In [557] conditions for a given Higgs po-tential to violate CP at the Lagrangian level, expressed interms of CP-odd Higgs-basis invariants, were derived. Theseconditions are expressed in terms of couplings of the un-broken Lagrangian, therefore they are relevant even at highenergies, where the SU(2) × U(1) symmetry is restored.This feature renders them potentially useful for the study ofbaryogenesis. The derivation of these conditions follows thegeneral method proposed in [558] and already mentioned inprevious sections. The method consists of imposing invari-ance of the Lagrangian under the most general CP transfor-mation of the Higgs doublets, which is a combination of asimple CP transformation for each Higgs field with a Higgs-basis transformation:

φaCP−→Waiφ

∗i ; φ†

a

CP−→W ∗aiφ

Ti . (5.5)

Here W is an nd × nd unitary matrix operating in Higgsdoublets space.

A set of necessary and sufficient conditions for CP in-variance in the case of two Higgs doublets have been de-rived [557]:

I1 ≡ Tr[Y ZY Z − Z ZY Y ] = 0,

I2 ≡ Tr[Y Z2 Z − Z Z2 Y ] = 0,(5.6)

where all matrices inside the parenthesis are 2 × 2 matri-ces. In the general case these are nd × nd matrices, and aredefined by:

(ZY )ij ≡ ZijmnYmn; Zij ≡ Zijmm;(Z2)ij ≡ ZipnmZmnpj ; Zij ≡ Zimmj

(5.7)

CP-odd HBT invariants are also useful [557] to find outwhether, in a given model, there is hard or soft CP breaking.One may also construct CP-odd weak basis invariants, in-volving vi ≡ 〈0|φ0

i |0〉, i.e., after spontaneous gauge symme-try breaking has occurred [559, 560]. Further discussions onHiggs-basis independent methods for the two-Higgs-doubletmodel can be found in [561–564].

So far, we have considered CP violation at the La-grangian level in models with multi-Higgs doublets, i.e., ex-plicit CP violation. It is also possible to derive criteria [565]to verify whether CP and T in a given model are sponta-neously broken. Under T the Higgs fields φj transform as

T φjT−1 =Ujkφk, (5.8)

Eur. Phys. J. C (2008) 57: 13–182 59

where U is a unitary matrix which may mix the scalar dou-blets. If no extra symmetries beyond SU(2) × U(1) arepresent in the Lagrangian, U reduces to a diagonal ma-trix possibly with phases. Invariance of the vacuum underT leads to the following condition:

〈0|φ0j |0〉 =U∗

jk〈0|φ0k |0〉∗. (5.9)

Therefore, a set of vacua lead to spontaneous T, CP viola-tion if there is no unitary matrix U satisfying (5.8) and (5.9)simultaneously.

Most of the previous discussion dealt with the generalcase of n-Higgs doublets. We analyze now the case of twoHiggs doublets, where the most general gauge invariantHiggs potential can be explicitly written as

VH2 = m1φ†1φ1 + peiϕφ†

1φ2 + pe−iϕφ†2φ1 +m2φ

†2φ2

+ a1(φ

†1φ1)2 + a2

(φ

†2φ2)2 + b(φ†

1φ1)(φ

†2φ2)

+ b′(φ†1φ2)(φ

†2φ1)+ c1e

iθ1(φ

†1φ1)(φ

†2φ1)

+ c1e−iθ1(φ†

1φ1)(φ

†1φ2)+ c2e

iθ2(φ

†2φ2)(φ

†2φ1)

+ c2 e−iθ2(φ†

2φ2)(φ

†1φ2)+ deiδ(φ†

1φ2)2

+ de−iδ(φ†2φ1)2, (5.10)

where mi , p, ai , b, b′, ci , and d are real and all phases areexplicitly displayed. It is clear that this potential contains anexcess of parameters. With the appropriate choice of Higgsbasis some of these may be eliminated, without loss of gen-erality, leaving eleven independent parameters [569–571].The Higgs sector contains five spinless particles: three neu-tral and a pair of charged ones, usually denoted by h,H (CPeven), A (CP odd) (or if CP is violated h1,2,3) and H±.

In general, models with two Higgs doublets have treelevel Higgs-mediated flavor changing neutral currents(FCNC). This is a problem in view of the present strin-gent experimental limits on FCNC. In order to solve thisproblem the concept of natural flavor conservation (NFC)was introduced by imposing extra symmetries on the La-grangian. These symmetries constrain the Yukawa couplingsof the neutral scalars in such a way that the resulting neu-tral currents are diagonal. Glashow and Weinberg [566] andPaschos [567] have shown that the only way to achieve NFCis to ensure that only one Higgs doublet gives mass to quarksof a given charge.

In the case of two Higgs doublets the simplest solution toavoid FCNC is to require invariance of the Lagrangian underthe following transformation of the Z2 type:

φ1 −→ φ1, φ2 −→−φ2,(5.11)

dR −→ dR, uR −→−uR,

where dR (uR) denote the right handed down (up) quarks;all other fields remain unchanged.

It is clear from (5.10) that this symmetry eliminates ex-plicit CP violation in the Higgs sector, since the only term ofthe Higgs potential with a phase that survives is the one withcoefficient d , moreover a HBT of the form φ1 −→ eiδ/2φ1,φ2 −→ φ2, eliminates the phase from the Higgs potential.Furthermore, it can be shown that this symmetry also elimi-nates the possibility of having spontaneous CP violation.

In conclusion, models with two Higgs doublets and ex-act NFC cannot give rise to spontaneous CP violation. Ex-plicit CP violation in this case requires complex Yukawacouplings leading to the Kobayashi–Maskawa mechanismwith no additional source of CP violation through neutralscalar Higgs boson exchange. An interesting alternative sce-nario in the case of two Higgs doublets was considered in[568] with no NFC. Here CP violating Higgs FCNC are nat-urally suppressed through a permutation symmetry which issoftly broken, still allowing for spontaneous CP violation.

Three Higgs doublet models have been considered in anattempt to introduce CP violation in an extension of the SMwith NFC [566] in the Higgs sector. It was shown that in-deed, in such models it is possible to violate CP in the Higgssector either at the Lagrangian level [572] or spontaneously[573–575].

It is also possible to generate spontaneous CP violationwith only one additional Higgs singlet [576], but in this caseat least one isosinglet vectorial quark is required in order togenerate a non-trivial phase at low energies in the Cabibbo–Kobayashi–Maskawa matrix. Such models may provide asolution to the strong CP problem of the type proposed byNelson [577, 578] and Barr [579] as well as a common ori-gin to all CP violations [580, 581] including the generationof the observed baryon asymmetry of the Universe. The factthat the SM cannot provide the observed baryon asymmetry[582–587], provides yet another reason to study an enlargedHiggs sector.

A lot of work has been done by many authors on possibleextensions of the Higgs sector and their implications bothfor the hadronic and the leptonic sectors at the existing andfuture colliders, see e.g. [588]. Among the simplest multi-Higgs models are the two Higgs Doublet Models (2HDM)which have been analyzed in detail in many different real-izations. The need to avoid potentially dangerous tree levelHiggs FCNC has led to the consideration of different vari-ants of this model with a certain discrete Z2 symmetry im-posed.

In the Type-I 2HDM the Z2 discrete symmetry imposedon the Lagrangian is such that only one of the Higgs dou-blets couples to quarks and leptons. A very well knownfermiophobic Higgs boson may arise in such model [589–591]. Another example is the Inert Doublet Model, with anunbroken discrete Z2 symmetry which forbids one Higgs

60 Eur. Phys. J. C (2008) 57: 13–182

doublet to couple to fermions and to get a non-zero VEV[592, 593]. Physical particles related to such doublets arecalled “inert” particles, the lightest is stable and contributesto the Dark Matter density. In [594], the naturalness problemhas been addressed in the framework of an Inert DoubletModel with a heavy (SM-like) Higgs boson. In this con-text Dark Matter may be composed of neutral inert Higgsbosons. Predictions are given for multilepton events withmissing transverse energy at the LHC, and for the direct de-tection of dark matter.

The Type-II 2HDM allows one of the Higgs doublet tocouple only to the right-handed up quarks while the otherHiggs doublet can only couple to right handed down-typequarks and charged leptons. This is achieved by the intro-duction of an appropriate Z2 symmetry, analogous to theone in (5.12). The Higgs sector of the MSSM model can beviewed as a particular realization of Type-II models but withadditional constraints required by supersymmetry. Variousscenarios are possible for these models—with and withoutdecoupling of heavy Higgs particles [570, 571, 595].

Type-III 2HDM are models where, unlike in models ofType-I and II, NFC is not imposed on the Yukawa interac-tions. This class of models has in general scalar mediatedFCNC at tree level. Various schemes have been proposedto suppress these currents, including the ad-hoc assumptionthat FCNC couplings are approximately given by the geo-metric mean of the Yukawa couplings of the two generations[596]. A very interesting alternative [597] is to have an exactsymmetry of the Lagrangian which constrains FCNC cou-plings to be related in an exact way to the elements of theCKM matrix in such a way that FCNC are non-vanishingbut naturally suppressed by the smallness of CKM mixing.Another example of Type III 2HDM is the Top Two HiggsDoublet Model which was first proposed in [598], and re-cently analyzed in detail in [599]. In this framework a dis-crete symmetry is imposed allowing only the top quark tohave Yukawa couplings to one of the doublets while all otherquarks and leptons have Yukawa couplings to the other dou-blet.

Lepton flavor violation is a feature common to manypossible extensions of the SM. It can occur both throughcharged and neutral currents. The possibility of having lep-ton flavor violation in extensions of the SM, has been con-sidered long before the discovery of neutrino masses [600,601]. For example, in the case of multi-Higgs doublet mod-els, it has been pointed out that even for massless neutrinoslepton flavor can be violated [602, 603]. In the context ofthe minimal extension of the SM, necessary to accommo-date neutrino masses, where only right handed neutrinos areincluded LFV effects are extremely small. It is well knownthat the effects of LFV can be large in supersymmetry.

CLEO submitted recently a paper [604] where the ratio ofthe tauonic and muonic branching fractions is examined for

the three Υ (1S,2S,3S) states. Agreement with expectationsfrom lepton universality is found. The conclusion is that lep-ton universality is respected within the current experimentalaccuracy which is roughly 10%. However there is tendencyfor the tauonic branching fraction to turn out systematicallylarger than the muonic at a few per cent level.

5.1.2 Low scale singlet neutrino scenarios

In the pre-LHC era neutrino oscillations have provided someof the most robust evidence for physics beyond the SM.There are many open questions in this field; why is the ab-solute mass scale for the neutrinos so small with respect tothe other SM particles? what is this mass scale? why is thepattern of mixing so different from the quark sector? If na-ture has chosen the singlet seesaw scenario [216–220] as ananswer to those questions we face the prospect of never be-ing able to produce the heavy neutrinos at a collider. Never-theless, several extensions of this minimal see-saw scenariocontain heavy neutrinos at or around the TeV scale, theseinclude models based around the group E6 [605, 606] andalso in SO(10) models [403].

Furthermore, even within the usual see-saw scenario, theobserved nearly maximal mixing pattern of the light neutri-nos requires further explanation. Flavor symmetries are of-ten invoked as possible reasons for the almost tri-bi-maximalstructure of the PMNS mixing matrix [607]. It is also possi-ble that the small magnitude of the light neutrino masses isdue to an approximate symmetry, allowing the right handedneutrinos to be as light as O(200 GeV) [337].

TeV scale right handed neutrinos can also arise in radia-tive mechanisms of neutrino mass generation. Generically,in these models a tree-level neutrino mass is forbidden orsuppressed by a symmetry but small neutrino masses mayarise through loops sensitive to symmetry breaking effects[225, 608]. Indeed, several supersymmetric realizations ofradiative mechanisms contain TeV scale right handed neu-trinos linked to the scale of supersymmetry breaking [609,610].

5.1.2.1 Heavy neutrinos accessible to the LHC A low,electroweak-scale mass is not sufficient to imply that heavyneutrinos could be produced and detected at the LHC. Theymust have a large enough coupling (mixing) with other SMfields so that experiments will be able to distinguish theirproduction and decay from SM background processes. Inthis review we concentrate on the case where heavy neu-trino production and decay occurs through mixing with SMfields only. Quantitatively, we can consider a generalizationof the Langacker–London parameters, Ωll′ , defined as

Ωll′ = δll′ −3∑i=1

BliB∗l′i =

(3+nR)∑i=4

BliB∗l′i , (5.12)

Eur. Phys. J. C (2008) 57: 13–182 61

where l, l′ = e,μ, τ and Bli is the full 3 × (3 + nR) neu-trino mixing matrix taking into account all (3 light and nRheavy) neutrinos. The 3×3 matrix Bli where i = 1, . . . ,3 isa good approximation to the usual PMNS matrix andΩll′ es-sentially measures the deviation from unitarity of the PMNSmatrix.

The Ωll′ are constrained by precision electroweak data[611] and the following upper limits have been set at 90%C.L.

Ωee ≤ 0.012, Ωμμ ≤ 0.0096, Ωττ ≤ 0.016.

(5.13)

In addition, the off-diagonal elements of Ωll′ are con-strained by limits on lepton flavor violating processes suchas τ,μ→ eγ and τ,μ→ eee and μ→ e conversion innuclei [514, 612]. These limits are rather model dependentbut for MR �MW and mD �MW (where mD is the Diraccomponent of the neutrino mass matrix), the present upperbounds are [182]

|Ωeμ| ≤ 0.0001, |Ωeτ | ≤ 0.02, |Ωμτ | ≤ 0.02.

(5.14)

It has been pointed out that a heavy Majorana neutrino(N ) may be produced via a DY type of mechanism at hadroncolliders [608, 613–617], pp→W+∗ → �+N , where N→�+W−, leading to lepton number violation by 2. Most of theprevious studies were concentrated on the ee mode, whichwould result in a too week signal to be appreciable due to therecent very stringent bound |VeN |2/mN < 5×10−8 GeV−1,from the absence of the neutrinoless double beta decay. Ithas been recently proposed to search for the unique andclean signal, μ±μ±+2 jets at the LHC [617]. It was con-cluded that a search at the LHC with an integrated luminos-ity of 100 fb−1 can be sensitive to a mass range ofmN ∼ 10–400 GeV at a 2σ level, and up to 250 GeV at a 5σ level. Ifthis type of signal could be established, it would be evenfeasible to consider the search for CP violation in the heavyMajorana sector [618].

A recent analysis [619] studied more background proc-esses including some fast detector simulations. In particu-lar, the authors claimed a large background due to the fakedleptons bb→ μ+μ+. The search sensitivity is thus reducedto 175 GeV at a 5σ level. However, the background esti-mate for processes such as bb+n-jet has large uncertaintiesdue to QCD perturbative calculations and kinematical ac-ceptance. More studies remain to be done for a definitiveconclusion.

5.1.2.2 Low scale model with successful baryogenesis Asa more detailed example satisfying the constraints of (5.14)we consider a model potentially accessible to colliders,

where MR 250 GeV which has been shown to success-fully explain the baryon asymmetry of the Universe [337].

Leptogenesis has been discussed in Sect. 3.3.1. Low scaleleptogenesis scenario would be possible with nearly degen-erate heavy neutrinos, where self-energy effects on the lep-tonic asymmetries become relevant [293, 294]. In this casethe CP asymmetry in the heavy neutrino decays can beresonantly enhanced [332], to the extent that the observedbaryon asymmetry can be explained with heavy neutrinos aslight as the electroweak scale [335, 337].

We shall consider a model with right handed neutrinoswhich transform under an SO(3) flavor symmetry. Ignoringeffects from the neutrino Yukawa couplings this symmetryis assumed to be exact at some high scale, e.g. the GUTscale MGUT. This restricts the form of the heavy Majorananeutrino mass matrix atMGUT

MR = 1mN + δMS, (5.15)

where δMS = 0 at MGUT. This form has also been consid-ered in a class of “minimal flavor violating” models of thelepton sector [542] and naturally provides nearly degenerateheavy neutrinos compatible with resonant leptogenesis.

All other fields are singlets under this SO(3) flavor sym-metry and so the neutrino Yukawa couplings will breakSO(3) explicitly. We can still choose heavy neutrino Yukawacouplings Y ν so that a subgroup of the SO(3)× U(1)Le ×U(1)Lμ ×U(1)Lτ flavor symmetry present without the neu-trino Yukawa couplings remains unbroken. In this casea particular flavor direction can be singled out leavingSO(2)U(1) unbroken. This residual U(1) symmetry actsto prevent the light Majorana neutrinos from acquiring amass. The form of the neutrino Yukawa couplings can bewritten

Y νT =⎛⎝

0 ae−iπ/4 aeiπ/4

0 be−iπ/4 beiπ/4

0 ce−iπ/4 ceiπ/4

⎞⎠+ δY ν. (5.16)

The residual U(1) symmetry is broken both by small SO(3)breaking effects in the heavy Majorana mass matrix, δMS ,and by small effects parameterized by δY ν in the Yukawacouplings. Although we shall not consider the specific originof these effects, δMS could arise through renormalizationgroup running for example.

In [337], a specific model was considered where mN =250 GeV and which successfully explained the baryonasymmetry of the Universe. One of either a, b or c was con-strained to be small to allow a single lepton flavor asymme-try (and subsequently a baryon asymmetry) to be generatedat T ∼ 250 GeV. The other two parameters could be as largeas O(10−2). This scenario has the features necessary for amodel to be visible at the LHC; heavy neutrinos with masses

62 Eur. Phys. J. C (2008) 57: 13–182

around O(1 TeV) and sufficient mixing between these neu-trinos and the light neutrinos to allow them to be producedfrom a vector boson. Specifically

Ωee = |a|2v2

m2N

, Ωμμ = |b|2v2

m2N

, Ωττ = |c|2v2

m2N

,

(5.17)

where v = 246 GeV is the vacuum expectation value of theHiggs field.

It should be noted that in this model the heavy neutrinosproduced at the LHC would be linked indirectly with themechanism providing light neutrinos with small masses. Thelight neutrinos acquire masses directly through the mecha-nism responsible for breaking the flavor symmetries. How-ever, studying the properties of the heavy neutrinos acces-sible to the LHC would allow us to better understand theunderlying symmetry protecting light neutrinos from largemasses and may give us insight into the observed pattern oflarge mixing. In addition, further knowledge of heavy neu-trinos seen at the LHC, for example small couplings withone or more lepton flavors or large, resonantly enhancedCP violation, would provide us with further information onpossible explanations for the baryon asymmetry of the Uni-verse.

5.1.3 Lepton flavor violation from the mirror leptons inlittle Higgs models

Little Higgs models [620–624] offer an alternative route tothe solution of the little hierarchy problem. One of the mostattractive models of this class is the littlest Higgs model[625] with T-parity (LHT) [626–628], where the discretesymmetry forbids tree-level corrections to electroweak ob-servables, thus weakening the electroweak precision con-straints [629]. Under this new symmetry the particles havedistinct transformation properties, that is, they are eitherT-even or T-odd. The model is based on a two-stage sponta-neous symmetry breaking occurring at the scale f and theelectroweak scale v. Here the scale f is taken to be largerthan about 500 GeV, which allows to expand expressions inthe small parameter v/f . The additionally introduced gaugebosons, fermions and scalars are sufficiently light to be dis-covered at LHC and there is a dark matter candidate [630].Moreover, the flavor structure of the LHT model is richerthan the one of the SM, mainly due to the presence of threedoublets of mirror quarks and three doublets of mirror lep-tons and their weak interactions with the ordinary quarksand leptons, as discussed in [631–633].

Now, it is well known that in the SM the FCNC processesin the lepton sector, like �i → �jγ and μ→ eee, are verystrongly suppressed due to tiny neutrino masses. In particu-lar, the branching ratio for μ→ eγ in the SM amounts to at

most 10−54, to be compared with the present experimentalupper bound, 1.2 × 10−11 [180], and with the one that willbe available within the next two years, ∼10−13 [634, 635].Results close to the SM predictions are expected within theLH model without T-parity, where the lepton sector is iden-tical to the one of the SM and the additional O(v2/f 2) cor-rections have only minor impact on this result. Similarly thenew effects on (g − 2)μ turn out to be small [636, 637].

A very different situation is to be expected in the LHTmodel, where the presence of new flavor violating interac-tions and of mirror leptons with masses of order 1 TeV canchange the SM expectations by up to 45 orders of magni-tude, bringing the relevant branching ratios for lepton fla-vor violating (LFV) processes close to the bounds availablepresently or in the near future.

5.1.3.1 The model A detailed description of the LHTmodel can be found in [638], where also a complete setof Feynman rules has been derived. Here we just want tostate briefly the ingredients needed for the analysis of LFVdecays.

The T-odd gauge boson sector consists of three heavy“partners” of the SM gauge bosons

W±H , ZH , AH , (5.18)

with masses given to lowest order in v/f by

MWH = gf, MZH = gf, MAH = g′f√5. (5.19)

The T-even fermion sector contains, in addition to theSM fermions, the heavy top partner T+. On the other hand,the T-odd fermion sector [631] consists of three generationsof mirror quarks and leptons with vectorial couplings underSU(2)L ×U(1)Y , that are denoted by(uiH

diH

),

(νiH

�iH

)(i = 1,2,3). (5.20)

To first order in v/f the masses of up- and down-type mirrorfermions are equal. Naturally, their masses are of order f .In the analysis of LFV decays, except for KL,S → μe,KL,S → π0μe, Bd,s → �i�j and τ→ �π, �η, �η′, only mir-ror leptons are relevant.

As discussed in detail in [632], one of the importantingredients of the mirror sector is the existence of fourCKM-like unitary mixing matrices, two for mirror quarks(VHu,VHd) and two for mirror leptons (VHν,VH�), that arerelated via

V†HuVHd = VCKM, V

†HνVH� = V †

PMNS. (5.21)

An explicit parameterization of VHd and VH� in termsof three mixing angles and three complex (non-Majorana)phases can be found in [633].

Eur. Phys. J. C (2008) 57: 13–182 63

The mirror mixing matrices parameterize flavor violatinginteractions between SM fermions and mirror fermions thatare mediated by the heavy gauge bosons W±

H , ZH and AH .The matrix notation indicates which of the light fermions ofa given electric charge participates in the interaction.

In the course of the analysis of charged LFV decays itis useful to introduce the following quantities (i = 1,2,3)[639]:

χ(μe)i = V ∗ie

H� ViμH�, χ

(τe)i = V ∗ie

H� ViτH�,

χ(τμ)i = V ∗iμ

H� ViτH�,

(5.22)

that govern μ → e, τ → e and τ → μ transitions, re-spectively. Analogous quantities in the mirror quark sector(i = 1,2,3) [638, 641],

ξ(K)i = V ∗is

Hd VidHd, ξ

(d)i = V ∗ib

Hd VidHd,

ξ(s)i = V ∗ib

Hd VisHd,

(5.23)

are needed for the analysis of the decays KL,S → μe,KL,S → π0μe and Bd,s → �i�j .

As an example, the branching ratio for the μ→ eγ decaycontains the χ(μe)i factors introduced in (5.22) via the shortdistance function [639]

D′μeodd = 1

4

v2

f 2

∑i

(χ(μe)i

(D′

0(yi)−7

6E′

0(yi)

− 1

10E′

0(y′i )

)), (5.24)

where yi = (m�Hi/MWH )2, y′i = ayi with a = 5/ tan2 θW ,and explicit expressions for the functions D′

0,E′0 can be

found in [642].The new parameters of the LHT model, relevant for the

study of LFV decays, are

f, m�H1, m�H2, m�H3, θ�12, θ�13, θ�23,

δ�12, δ�13, δ�23

(5.25)

and the ones in the mirror quark sector that can be probed byFCNC processes inK and B meson systems, as discussed indetail in [638, 641]. Once the new heavy gauge bosons andmirror fermions will be discovered and their masses mea-sured at the LHC, the only free parameters of the LHT modelwill be the mixing angles θ�ij and the complex phases δ�ij ofthe matrix VH�, that can be determined with the help of LFVprocesses. Analogous comments apply to the determinationof VHd parameters in the quark sector (see [638, 641] fordetails on K and B physics in the LHT model).

5.1.3.2 Results LFV processes in the LHT model have forthe first time been discussed in [643], where the decays

�i → �jγ have been considered. Further, the new contri-butions to (g − 2)μ in the LHT model have been calcu-lated by these authors. In [639, 640] the analysis of LFVin the LHT model has been considerably extended, and in-cludes the decays �i → �jγ , μ→ eee, the six three bodyleptonic decays τ− → �−i �

+j �

−k , the semileptonic decays

τ→ �π, �η, �η′ and the decays KL,S → μe, KL,S → π0μe

and Bd,s → �i�j that are flavor violating both in the quarkand lepton sector. Moreover, μ–e conversion in nuclei andthe flavor conserving (g − 2)μ have been studied. Further-more, a detailed phenomenological analysis has been per-formed in that paper, paying particular attention to variousratios of LFV branching ratios that will be useful for a cleardistinction of the LHT model from the MSSM.

In contrast to K and B physics in the LHT model, wherethe SM contributions constitute a sizable and often the dom-inant part, the T-even contributions to LFV observablesare completely negligible due to the smallness of neutrinomasses and the LFV decays considered are entirely gov-erned by mirror fermion contributions.

In order to see how large these contributions can pos-sibly be, it is useful to consider first those decays forwhich the strongest constraints exist. Therefore Fig. 5 showsB(μ→ eee) as a function of B(μ→ eγ ), obtained from ageneral scan over the mirror lepton parameter space, withf = 1 TeV. It is found that in order to fulfill the presentbounds, either the mirror lepton spectrum has to be quasi-degenerate or the VH� matrix must be very hierarchical.Moreover, as shown in Fig. 6, even after imposing the con-straints on μ→ eγ and μ→ eee, the μ–e conversion ratein Ti is very likely to be found close to its current bound,and for some regions of the mirror lepton parameter spaceeven violates this bound.

The existing constraints on LFV τ decays are still rela-tively weak, so that they presently do not provide a usefulconstraint on the LHT parameter space. However, as seenin Table 10, most branching ratios in the LHT model can

Fig. 5 Correlation between B(μ→ eγ ) and B(μ→ eee) in the LHTmodel (upper dots) [639]. The lower dots represent the dipole contri-bution to μ→ eee separately, which, unlike in the LHT model, is thedominant contribution in the MSSM. The grey region is allowed by thepresent experimental bounds

64 Eur. Phys. J. C (2008) 57: 13–182

Fig. 6 R(μTi → eTi) as afunction of B(μ→ eγ ), afterimposing the existingconstraints on μ→ eγ andμ→ eee [639]. The grey regionis allowed by the presentexperimental bounds

Table 10 Upper bounds on LFV τ decay branching ratios in the LHTmodel, for two different values of the scale f , after imposing the con-straints on μ→ eγ and μ→ eee [639]. For f = 500 GeV, also the

bounds on τ → μπ,eπ have been included. The current experimentalupper bounds are also given. The bounds in [183] have been obtainedby combining Belle [646, 647] and BaBar [182, 648] results

Decay f = 1000 GeV f = 500 GeV exp. upper bound

τ→ eγ 8 × 10−10 1 × 10−8 9.4 × 10−8 [183]

τ→ μγ 8 × 10−10 2 × 10−8 1.6 × 10−8 [183]

τ− → e−e+e− 7 × 10−10 2 × 10−8 2.0 × 10−7 [644]

τ− → μ−μ+μ− 7 × 10−10 3 × 10−8 1.9 × 10−7 [644]

τ− → e−μ+μ− 5 × 10−10 2 × 10−8 2.0 × 10−7 [645]

τ− → μ−e+e− 5 × 10−10 2 × 10−8 1.9 × 10−7 [645]

τ− → μ−e+μ− 5 × 10−14 2 × 10−14 1.3 × 10−7 [644]

τ− → e−μ+e− 5 × 10−14 2 × 10−14 1.1 × 10−7 [644]

τ→ μπ 2 × 10−9 5.8 × 10−8 5.8 × 10−8 [183]

τ→ eπ 2 × 10−9 4.4 × 10−8 4.4 × 10−8 [183]

τ→ μη 6 × 10−10 2 × 10−8 5.1 × 10−8 [183]

τ→ eη 6 × 10−10 2 × 10−8 4.5 × 10−8 [183]

τ→ μη′ 7 × 10−10 3 × 10−8 5.3 × 10−8 [183]

τ→ eη′ 7 × 10−10 3 × 10−8 9.0 × 10−8 [183]

reach the present experimental upper bounds, in particularfor low values of f , and are very interesting in view of newexperiments taking place in this and the coming decade.

The situation is different in the case of KL→ μe, KL→π0μe and Bd,s → �i�k , due to the double GIM suppressionin the quark and lepton sectors. E.g. B(KL→ μe) can reachvalues of at most 3 × 10−13 which is still one order of mag-nitude below the current bound, and KL→ π0μe is even bytwo orders of magnitude smaller. Still, measuring the ratesfor KL→ μe and KL→ π0μe would be desirable, as, dueto their sensitivity to Re(ξ (K)i ) and Im(ξ (K)i ) respectively,these decays can shed light on the complex phases presentin the mirror quark sector.

While the possible huge enhancements of LFV branch-ing ratios in the LHT model are clearly interesting, such ef-fects are common to many other NP models, such as theMSSM, and therefore cannot be used to distinguish thesemodels. However, correlations between various branching

ratios should allow a clear distinction of the LHT modelfrom the MSSM. While in the MSSM [169, 175, 242, 649,650] the dominant role in decays with three leptons in the fi-nal state and in μ–e conversion in nuclei is typically playedby the dipole operator, in [639] it is found that this operatoris basically irrelevant in the LHT model, where Z0-penguinand box diagram contributions are much more important. Ascan be seen in Table 11 and also in Fig. 5 this implies a strik-ing difference between various ratios of branching ratios inthe MSSM and in the LHT model and should be very usefulin distinguishing these two models. Even if for some decaysthis distinction is less clear when significant Higgs contri-butions are present [169, 175, 650], it should be easier thanthrough high energy processes at LHC.

Another possibility to distinguish different NP modelsthrough LFV processes is given by the measurement ofμ→ eγ with polarized muons. Measuring the angular dis-tribution of the outgoing electrons, one can determine the

Eur. Phys. J. C (2008) 57: 13–182 65

Table 11 Comparison of various ratios of branching ratios in the LHT model and in the MSSM without and with significant Higgs contributions[639]

Ratio LHT MSSM (dipole) MSSM (Higgs)

B(μ− → e−e+e−)/B(μ→ eγ ) 0.4–2.5 ∼6 × 10−3 ∼6 × 10−3

B(τ− → e−e+e−)/B(τ → eγ ) 0.4–2.3 ∼1 × 10−2 ∼1 × 10−2

B(τ− → μ−μ+μ−)/B(τ → μγ ) 0.4–2.3 ∼2 × 10−3 0.06–0.1

B(τ− → e−μ+μ−)/B(τ → eγ ) 0.3–1.6 ∼2 × 10−3 0.02–0.04

B(τ− → μ−e+e−)/B(τ → μγ ) 0.3–1.6 ∼1 × 10−2 ∼1 × 10−2

B(τ− → e−e+e−)/B(τ− → e−μ+μ−) 1.3–1.7 ∼5 0.3–0.5

B(τ− → μ−μ+μ−)/B(τ− → μ−e+e−) 1.2–1.6 ∼0.2 5–10

R(μTi → eTi)/B(μ→ eγ ) 0.01–100 ∼5 × 10−3 0.08–0.15

size of left and right handed contributions separately [651].In addition, detecting also the electron spin would yield in-formation on the relative phase between these two contribu-tions [652]. We recall that the LHT model is peculiar in thisrespect as it does not involve any right handed contribution.

On the other hand, the contribution of mirror leptons to(g − 2)μ, being a flavor conserving observable, is negli-gible [639, 643], so that the possible discrepancy betweenSM prediction and experimental data [653] cannot be cured.This should also be contrasted with the MSSM with largetanβ and not too heavy scalars, where those correctionscould be significant, thus allowing to solve the possible dis-crepancy between SM prediction and experimental data.

5.1.3.3 Conclusions We have seen that LFV decays openup an exciting playground for testing the LHT model. In-deed, they could offer a very clear distinction between thismodel and supersymmetry. Of particular interest are the ra-tios B(�i → eee)/B(�i → eγ ) that are O(1) in the LHTmodel but strongly suppressed in supersymmetric modelseven in the presence of significant Higgs contributions. Sim-ilarly, finding the μ–e conversion rate in nuclei at the samelevel as B(μ→ eγ ) would point into the direction of LHTphysics rather than supersymmetry.

5.1.4 Low scale triplet Higgs neutrino mass scenarios inlittle Higgs models

An important open issue to address in the context of lit-tle Higgs models is the origin of non-zero neutrino masses[654–658]. The neutrino mass mechanism which naturallyoccurs in these models is the triplet Higgs mechanism [244]which employs a scalar with the SU(2)L × U(1)Y quan-tum numbers T ∼ (3,2). The existence of such a multi-plet in some versions of the little Higgs models is a directconsequence of global symmetry breaking which makes theSM Higgs light. For example, in the minimal littlest Higgsmodel [625], the triplet Higgs with non-zero hyperchargeoccurs from the breaking of global SU(5) down to SO(5)

symmetry as one of the Goldstone bosons. Its mass MT ∼gsf, where gs < 4π is a model dependent coupling constantin the weak coupling regime [659], is therefore predicted tobe below the cut-off scale Λ, and could be within the massreach of LHC. The present lower bound for the invariantmass of T is set by Tevatron toMT ≥ 136 GeV [660, 661].

Although the triplet mass scale is of order O(1) TeV, theobserved neutrino masses can be obtained naturally. Due tothe specific quantum numbers the triplet Higgs boson cou-ples only to the left-chiral lepton doublets Li ∼ (2,−1),i = e,μ, τ, via the Yukawa interactions of (3.63) and to theSM Higgs bosons via (3.64). Those interactions induce lep-ton flavor violating decays of charged leptons which havenot been observed. The most stringent constraint on theYukawa couplings comes from the upper limit on the tree-level decay μ→ eee and is15 Y eeT Y

eμT < 3×10−5(M/TeV)2

[662, 663]. Experimental bounds on the tau Yukawa cou-plings are much less stringent. The hierarchical light neu-trino masses imply Y eeT , Y

eμT � Y ττT consistently with the

direct experimental bounds.Non-zero neutrino masses and mixing is presently the

only experimentally verified signal of new physics beyondthe SM. In the triplet neutrino mass mechanism [244] pre-sented in Sect. 3.2.3.2 the neutrino masses are given by

(mν)ij = Y ijT vT , (5.26)

where vT is the induced triplet VEV of (3.65). It is nat-ural that the smallness of neutrino masses is explained bythe smallness of vT . In the little Higgs models this can beachieved by requiring the Higgs mixing parameter μ�MT ,which can be explained, for example, via shining of explicitlepton number violation from extra dimensions as shown inRefs. [664, 665], or if the triplet is related to the Dark Energyof the Universe [666, 667]. Models with additional (approx-imate) T-parity [626] make the smallness of vT technically

15In little Higgs models with T-parity there exist additional sources offlavor violation from the mirror fermion sector [639, 643] discussed inthe previous subsection.

66 Eur. Phys. J. C (2008) 57: 13–182

natural (if the T-parity is exact, vT must vanish). In that caseYT vT ∼ O(0.1) eV while the Yukawa couplings Y can beof order charged lepton Yukawa couplings of the SM. As aresult, the branching ratio of the decay T →WW is negli-gible. We also remind that vT contributes to the SM obliquecorrections, and the precision data fit T < 2 × 10−4 [668]sets an upper bound vT ≤ 1.2 GeV on that parameter.

Notice the particularly simple connection between theflavor structure of light neutrinos and the Yukawa couplingsof the triplet via (5.26). Therefore, independently of theoverall size of the Yukawa couplings, one can predict theleptonic branching ratios of the triplet from neutrino oscil-lations. For the normally hierarchical light neutrino massesneutrino data implies negligible T branching fractions toelectrons and B(T ++ → μ+μ+) ≈ B(T ++ → τ+τ+) ≈B(T ++ → μ+τ+) ≈ 1/3. Those are the final state signa-tures predicted by the triplet neutrino mass mechanism forcollider experiments.

At LHC T ++ can be produced singly and in pairs. Thecross section of the single T ++ production via the WW fu-sion process [662] qq → q ′q ′T ++ scales as ∼ v2

T . In thecontext of the littlest Higgs model this process, followedby the decays T ++ →W+W+, was studied in Refs. [669–671]. The detailed ATLAS simulation of this channel shows[671] that in order to observe an 1 TeV T ++, one musthave vT > 29 GeV. This is in conflict with the precisionphysics bound vT ≤ 1.2 GeV as well as with the neutrinodata. Therefore the WW fusion channel is not experimen-tally promising for the discovery of doubly charged Higgs.

On the other hand, the Drell–Yan pair production process[662, 672–678]

pp→ T ++T −−

is not suppressed by any small coupling and its cross sec-tion is known up to next to leading order [674] (possibleadditional contributions from new physics such as ZH arestrongly suppressed and we neglect those effects here). Fol-lowed by the lepton number violating decays T ±± → �±�±,this process allows to reconstruct T ±± invariant mass fromthe same charged leptons rendering the SM backgroundto be very small in the signal region. If one also assumesthat neutrino masses come from the triplet Higgs interac-tions, one fixes the T ±± leptonic branching ratios. This al-lows to test the triplet neutrino mass model at LHC. The pureMonte Carlo study of this scenario shows [677] that T ++up to the mass 300 GeV is reachable in the first year of LHC

(�

†L �

†R

)( m2L(1 + δLL) (A∗ −μ tanβ)m� +mLmRδLR

(A−μ∗ tanβ)m� +mLmRδLR† m2R(1 + δRR)

)(�L

�R

)

(L = 1 fb−1) and T ++ up to the mass 800 GeV is reach-able for the luminosity L= 30 fb−1. Including the Gaussianmeasurement errors to the Monte Carlo the correspondingmass reaches become [677] 250 GeV and 700 GeV, respec-tively. The errors of those estimates of the required luminos-ity for discovery depend strongly on the size of statisticalMonte Carlo sample of the background processes.

5.2 Flavor and CP violation in SUSY extensions of the SM

Supersymmetric models provide the richest spectrum of lep-ton flavor and CP-violating observables among all mod-els. They are also among the best studied scenarios of newphysics beyond the standard model. In this Section we re-view phenomenologically most interesting aspects of someof the supersymmetric scenarios.

5.2.1 Mass insertion approximation and phenomenology

In the low energy supersymmetric extensions of the SMthe flavor and CP-violating interactions would originatefrom the misalignment between fermion and sfermion masseigenstates. Understanding why all these processes arestrongly suppressed is one of the major problems of lowenergy supersymmetry, the supersymmetric flavor and CPproblem. The absence of deviations from the SM predic-tions in LFV and CPV (and other flavor changing processesin the quark sector) experiments suggests the presence ofa quite small amount of fermion-sfermion misalignment.From the phenomenological point of view those effectsare most easily described by the mass insertion approxima-tion.

The relevant one loop amplitudes can be exactly writ-ten in terms of the general mass matrix of charginos andneutralinos, resulting in quite involved expressions. To ob-tain simple approximate expressions, it is convenient to usethe so-called mass insertion method [145, 679]. This is aparticularly convenient method since, in a model indepen-dent way, the tolerated deviation from alignment is quan-tified by the upper limits on the mass insertion δ’s, de-fined as the small off-diagonal elements in terms of whichsfermion propagators are expanded, normalized with an av-erage sfermion mass, δij =Δij /m2

f. They are of four types:

δLL, δRR , δRL and δLR , according to the chiralities of thecorresponding partner fermions. We shall adopt here theusual convention for the slepton mass matrix in the basiswhere the lepton mass matrix m� is diagonal:

Eur. Phys. J. C (2008) 57: 13–182 67

where mL, mR, are respectively the average real masses ofthe left handed and right handed sleptons and A containsonly the diagonal entries the trilinear matrices at the elec-troweak scale. Notice that these flavor diagonal left–rightmixing are always present in any MSSM and play a veryimportant role in LFV processes. In this way, our δLR con-tain only the off-diagonal elements of the trilinear matrices.This definition is then slightly different from the original de-finition in Refs. [145, 680]. The deviations from universalityare then all gathered in the different δ matrices.

Each element in these δ matrices can be tested by ex-periment. Searches for the decay �i → �jγ provide boundson the absolute values of the off-diagonal (flavor violat-ing) |δLLij |, |δRRij |, |δLRij | and |δRLij |, while measurements ofthe lepton EDM (MDM), parameters and their CP-violatingphases, also provide limits on the imaginary (real) partof combinations of flavor violating δ’s, δLLij δ

LRji , δLRij δ

RRji ,

δLLij δRRji and δLRij δ

LRji . Many authors have addressed the is-

sue of the bounds on these misalignment parameters and

phases in the sleptonic sector [680]. Following [163] wepresent the current limits on μ→ eγ and we analyze the im-pact of the planned experimental improvements on τ→ μγ .In the basis where Y� is diagonal, and in the mass insertionapproximation, the branching ratio of the process reads

B(�i → �jγ ) = 10−5 ×B(�i → �j νj νi)M4W

m4L

× tan2 β∣∣δLLij

∣∣2FSUSY, (5.27)

where FSUSY = O(1) is a function of supersymmetricmasses including both chargino and neutralino exchange(see e.g., [163], and references therein). We focus for def-initeness on the mSUGRA scenario, also assuming gaug-ino and scalar universality at the gauge coupling unificationscale and fixing μ as required by the radiative electroweaksymmetry breaking.

As for LFV, Figs. 7 and 8 display the upper bounds on the|δ|’s in the (M1,mR) plane, where M1 and mR are the bino

Fig. 7 Upper limits on δ12’s in mSUGRA. HereM1 and mR are the bino and right-slepton masses, respectively

Fig. 8 Upper limits on δ23’s in mSUGRA. HereM1 and mR are the bino and right-slepton masses, respectively

68 Eur. Phys. J. C (2008) 57: 13–182

and right-slepton masses, respectively. Deviations from themSUGRA assumptions can be estimated by means of rela-tively simple analytical expressions. In Figs. 7 and 8 we cansee that the bounds on δRRji depend strongly and are prac-tically absent for some values of M1 and mR . This fact isdue to a destructive interference between the bino and bino–higgsino amplitudes [163]. On the contrary, the limits onδLLji are robust because of a constructive interference be-tween the chargino and bino amplitudes. A weaker boundon δRR12 on the cancellation regions can be obtained combin-ing the experimental information from the decays μ→ eγ ,μ→ eee and μ–e conversion in nuclei [165, 681]. Thepresent limits on μ→ eγ provide interesting constraints onthe related δ’s. As will be discussed in the following, thepresent sensitivity already allow to test these δ’s at the levelof the radiative effects. Such a sensitivity could hopefully bereached also in future experiments on τ→ μγ .

Another issue is the origin of the CP-violating phasesin the leptonic EDMs. Unless the sparticle masses are in-creased above several TeVs, the phases in the flavor diag-onal elements of the slepton left–right mass matrices (inthe lepton flavor basis), in the parameters μ and Ai ofsupersymmetric models, have to be quite small, and thisconstitutes the so-called supersymmetric CP problem. Forthe bounds on the sources of CPV also associated to FV,like e.g. Im(δLLij δ

RRji )ee and so on, we refer to the plots in

Ref. [163].

5.2.2 Lepton flavor violation from RGE effects in SUSYseesaw model

5.2.2.1 Predictions from flavor models Consider first thepossibility that flavor and CP are exact symmetries of thesoft supersymmetry breaking sector defined at the appropri-ate cutoff scale Λ (to be identified with the Planck scale forsupergravity, the messenger mass for gauge mediation, etc).If below this scale there are flavor and CP-violating Yukawainteractions, it is well-known that in the running down tomSUSY they will induce a small amount of flavor and CPviolation in sparticle masses.

The Yukawa interactions associated to the fermion mass-es and mixing of the SM clearly violate any flavor and CPsymmetries. However, with the exception of the third gener-ation Yukawa couplings, all the entries in the Yukawa ma-trices are very small and the radiatively induced misalign-ment in the sfermion mass matrices turns out to be negligi-ble. The Yukawa interactions of heavy states beyond the SMcoupling to the SM fermions induce misalignments propor-tional to a proper combination of their Yukawa couplingstimes lnmF/Λ, where mF represents the heavy state massscale. This is the case for the seesaw interactions of the righthanded neutrinos [144, 145] and/or the GUT interactions ofthe heavy colored triplets [682, 683] (those eventually ex-changed in diagrams inducing proton decay). Notice that

the observation of large mixing in light neutrino masses,may suggest the possibility that also the seesaw interactionscould significantly violate flavor- and potentially also CP,in particular in view of the mechanism of leptogenesis. Re-markably, for sparticle masses not exceeding the TeV, theseesaw and colored-triplet induced radiative contributions tothe LFV decays and lepton EDM might be close to or evenexceed the present or planned experimental limits. Clearly,these processes constitute an important constraint on seesawand/or GUT models.

For instance, in a type I seesaw model in the low energybasis where charged leptons are diagonal, the ij element ofthe left handed slepton mass matrix provides the dominantcontribution in the decay �i → �jγ . Assuming, for the sakeof simplicity, an mSUGRA spectrum at Λ =MPl, one ob-tains at the leading log [178]:

δLLij = (m2ij )LL

m2L

=− 1

8π2

3m20 +A2

0

m2L

Cij ,

Cij ≡∑k

Yν∗kiYνkj ln

MPl

Mk,

(5.28)

where m0 and A0 are respectively the universal scalarmasses and trilinear couplings at MPl, m2

L is an averageleft handed slepton mass and Mk the mass of the righthanded neutrino with k = 1,2,3. An experimental limiton B(�i → �jγ ) corresponds to an upper bound on |Cij |[38, 233]. For μ→ eγ and τ → μγ this bound is shown inFig. 9 as a function of the right handed selectron mass.

The seesaw model dependence resides in Cij . Notice thatin the fundamental theory at high energy, the size of Cij isdetermined both by the Yukawa eigenvalues and the large-ness of the mixing angles of VR,VL, the unitary matriceswhich diagonalize Yν (in the basis whereMR and Ye are di-agonal): VRYνVL = Y (diag)

ν . The left handed misalignmentbetween neutrino and charged lepton Yukawa’s is given byVL and, due to the mild effect of the logarithm in Cij , infirst approximation VL itself diagonalizes Cij . If we con-sider hierarchical Yν eigenvalues, Y3 > Y2 > Y1, the contri-butions from k = 1,2 in (5.28) can in first approximation beneglected with respect to the contribution from the heaviesteigenvalue (k = 3):

|Cij | ≈ |VLi3VLj3|Y 23 log(MPl/M3). (5.29)

Taking supersymmetric particle masses around the TeVscale, it has been shown that many seesaw models predict|Cμe| and/or |Cτμ| close to the experimentally accessiblerange. Let us consider the predictions for the seesaw-RGEinduced contribution to τ → μγ and μ→ eγ in the flavormodels discussed previously.

Eur. Phys. J. C (2008) 57: 13–182 69

Fig. 9 Upper limit on C32 andC21 for the experimentalsensitivities displayed [38]

The present experimental bound on τ → μγ is notvery strong but nevertheless promising. In models with“lopsided” Yν , one has VL32VL32 ≈ 1/2, hence |Cτμ| =O(4 × Y 2

3 ), for M3 4 × 1015 GeV. This is precisely thecase for the U(1) flavor model discussed in Sect. 2.2, whereY3 ≈ εnc3 with ε ≈ 0.22 (the Cabibbo angle). For this model,planned τ → μγ searches could thus be successful if theheaviest right handed neutrino has null charge, nc3 = 0. Onthe contrary, in models with small VL23 mixing, like in thenon-Abelian models discussed previously, the seesaw-RGEinduced effect is below the experimental sensitivity.

The present experimental bound on μ→ eγ is alreadyvery severe in constraining |Cμe|. For instance, if VL ≈VCKM, one obtains Cμe = O(10−3 × Y 2

3 ). As can be seenfrom Fig. 9, VL could in future be tested at a CKM-level ifY3 =O(1) [164]. The predictions for μ→ eγ are howeververy model dependent. For the simple U(1) flavor modelof Sect. 2.2, the mixings of VL are of the same order ofmagnitude as those of UPMNS and one expects |Cμe| =O(8 × ε2nc3+1): if nc3 = 0 the prediction exceeds the experi-mental limit, which is respected only with nc3 ≥ 1 [684]. Onthe contrary, the non-Abelian models discussed previouslyhave Y3 ∼ 1, but the VL-mixings are sufficiently small tosuppress the seesaw-RGE induced effect below the presentexperimental level [164].

5.2.2.2 Parameter dependence for degenerate heavy neutri-nos Equation (5.28) indicates that LFV in the minimal su-persymmetric seesaw model depends on soft supersymmetrybreaking masses as well as on the seesaw parameters. Thelatter can be parameterized via the heavy and light neutrinomasses, the light neutrino mixing matrix and the orthogonalmatrix R of (3.45). The three complex mixing angles para-meterizing R can be written as θj = xj + iyj , j = 1,2,3.For the following numerical examples we use the mSUGRApoint SPS1a [685] for SUSY breaking masses.

In the case of degenerate heavy neutrino masses, Mi =MR (i = 1,2,3), and real R, the R dependence in (5.28)and hence also in B(li → lj γ ) drops out. However, if R iscomplex, the LFV observables have more freedom since thedependence on yi can be as significant as the MR depen-dence, as Fig. 10 shows. For small |yi |, the change in Y †

ν Yν

is approximately

ΔR(Y †ν Yν) ≈ UPMNS diag(

√mi )

(R†R − 1

)

× diag(√mi )U

†PMNS, (5.30)

while the renormalization effects on the soft supersymmetrybreaking masses can be estimated via [273]

m8L 0.5m2

0M21/2

(m2

0 + 0.6M21/2

)2, (5.31)

where M1/2 is the universal gaugino mass at high scale. Incertain cases, the leading logarithmic approximation fails, aspointed out in [238, 273, 275, 686].

Equation (5.30) implies three features seen in Fig. 10:

(i) Compared to the case of degenerate light neutrinomasses, the y dependence in the hierarchical case isweaker.

(ii) Observables like (5.27) are larger in the case of com-plex R than in the case of real R. For a givenMR , evensmall values of y can enhance a process by orders ofmagnitude.

(iii) In contrast to the real R case, where B(li → lj γ ) fordegenerate light neutrinos is always larger than for hi-erarchical light neutrinos, the relative magnitude can bereversed for complex R.

To examine the parameter dependence of rare decays atlarge |yi | > 0.1, we extend the above analysis to the casewhere the yi are independent of one another. For randomvalues of all parameters in their full ranges, the typical be-

70 Eur. Phys. J. C (2008) 57: 13–182

Fig. 10 (Color online) Degenerate heavy neutrinos: LFV branching ratio versus |yi | = y for fixed Mi =MR = 1012 GeV in mSUGRA scenarioSPS1a for hierarchical (dark red) and degenerate (light green) light neutrino masses. The xi are scattered over 0< xi < 2π

Fig. 11 Degenerate heavy neutrinos: LFV branching ratios versus

MR

√y2

1 + y22 + y2

3 , for light neutrinos. The yi are scattered logarith-

mically in the range 10−5 < |yi | < 1 (independently of one another)

andMR is scattered logarithmically in the range 1010 <MR <MGUT.

The xi are scattered over 0< xi < 2π

havior

∣∣(Y †ν LYν

)jk

∣∣2 ∝{M2R(C1y

21 +C2y

22 +C3y

23) deg. νL

M2R hier. νL

(j �= k), (5.32)

is found, with Ci = O(1), slightly dependent on j, k. Thisbehavior can be seen in Fig. 11 for degenerate light neutri-nos. Thus for large |yi | all rare decays may be of a similarorder of magnitude. For hierarchical light neutrinos, a simi-lar behavior is observed, but versusM2

R only.

5.2.2.3 Parameter dependence for hierarchical heavy neu-trinos Hierarchical spectrum of heavy Majorana neutrinos,M1 � M2 � M3, is well motivated by the arguments oflight neutrino mass and mixing generation and leptogen-esis. Requiring successful thermal leptogenesis puts addi-tional constraints on the seesaw parameters and constrainsthe LFV observables [329]. This is the approach we takein this subsection. In particular, the relation (3.93) impliesa lower bound on M1 [306], e.g., if ε1 > 10−6, then M1 >

5 × 109 GeV. Furthermore, to allow for thermal productionof right handed neutrinos after inflation, one has to exclude

M1 > 1011 GeV, at least in simple scenarios. Otherwise a toohigh re-heating temperature would lead to an overabundanceof gravitinos, whose decays into energetic photons can spoilbig bang nucleosynthesis. Details of leptogenesis have beendescribed in Sect. 3.3.1.

Assuming hierarchical light neutrinos with �m2sol <

m23 < �m

2atm, the condition to reproduce the experimental

baryon asymmetry, ηB = (6.3 ± 0.3) × 10−10, puts con-straints on M1 and the R matrix [687]. This is illustratedin Fig. 12 in the M1–x2 plane. For M1 < 1011 GeV, x2 hasto approach the values 0,π,2π . A similar behavior is ob-served in theM1–x3 plane.

Taking M1 = 1010 GeV and x2 ≈ x3 ≈ n · π , experi-mental bounds on B(μ→ eγ ) can be used to constrainthe heavy neutrino scale, here represented by the heaviestright handed neutrino mass M3, as shown in the right plotof Fig. 12. Quantitatively, the present bound on B(μ→eγ ) already constrains M3 to be smaller than ≈1013 GeV,while the MEG experiment at PSI is sensitive to M3 ≤O(1012) GeV. If no signal is observed it will be difficultto test the type I seesaw model considered here at future col-liders.

Eur. Phys. J. C (2008) 57: 13–182 71

Fig. 12 Hierarchical heavy neutrinos: Region in the plane (x2,M1)

consistent with the generation of the baryon asymmetry ηB = (6.3 ±0.3) × 10−10 via leptogenesis (left). [Right] B(μ → eγ ) versusM3| cos2 θ2| in mSUGRA scenario SPS1a, for M1 = 1010 GeV and

x2 ≈ x3 ≈ n · π . All other seesaw parameters are scattered in theirallowed ranges for hierarchical light and heavy neutrinos. The solid(dashed) line indicates the present (expected future) experimental sen-sitivity

Fig. 13 The branching ratios of the LFV decays μ→ e+ γ and τ →μ+ γ versus m1 in the cases of complex and real matrix R with α =0;π/2;π . The three parameters describing the matrix R [275, 545]are generated randomly. The SUSY parameters are tanβ = 10, m0 =100 GeV, m1/2 = 250 GeV, A0 =−100 GeV, and the neutrino mixing

parameters are �m2� = 8.0 × 10−5 eV2, �m2atm = 2.2 × 10−3 eV2,

tan2 θ� = 0.4, tan2 θatm = 1, and sin θ13 = 0.0. The neutrino massspectrum at MZ is assumed to be with normal hierarchy, m1(MZ) <

m2(MZ) <m3(MZ). The right handed neutrino mass spectrum is takento be degenerate asM1 =M2 =M3 = 2 × 1013 GeV [275]

5.2.2.4 Effects of renormalization of light neutrino masseson LFV The RG running of the neutrino parameterscan have an important impact on lepton flavor violatingprocesses in MSSM extended by right handed neutrinos.In this example we assume universal soft SUSY break-ing terms at GUT scale and degenerate heavy neutrinoswith mass MR. The running effects below MR are rel-atively small when tanβ is smaller than 10 and/or m1

is much smaller than 0.05 eV. Because the combinations12c12c23(m1 − m2e

iαM ), where we use the notation ofSect. 3.2.3.3, is practically stable against the RG running,and this combination is the dominant term of (Y †

ν Yν)21 whenαM = 0, θ13 = 0 and R∗ = R are satisfied, the running ef-fect on LFV can be neglected in this case [275]. In gen-eral, (Y †

ν Yν)21 and B(μ→ e + γ ) can depend strongly on

θ13 and RG running has to be taken into account [686,688]. Note that due to RG running, the value of θ13 atMR differs from 0, even if θ13 = 0 is assumed at low en-ergy [275].

In many cases, the running of the neutrino parameterscan significantly affect the prediction of the LFV branch-ing ratios. In particular, for 0.05 � m1 � 0.30 eV, 30 �tanβ � 50, the predicted μ→ e + γ and τ → e + γ de-cay branching ratios, B(μ→ e + γ ) and B(τ → e + γ ),can be enhanced by the effects of the RG running of θij andmj by 1 to 3 orders of magnitude if π/4 � αM � π , whileB(τ→ μ+γ ) can be enhanced by up to a factor of 10 [275].The effects of the running of the neutrino mixing parame-ters of B(μ→ e + γ ) and B(τ → e + γ ) are illustrated inFig. 13.

72 Eur. Phys. J. C (2008) 57: 13–182

5.2.3 Correlations between LFV observables and colliderphysics

5.2.3.1 Correlations of LFV rare decays Equations (5.27)and (5.28) imply correlations between different LFV ob-servables. In addition to the correlations between differentclasses of LFV observables in the same flavor mixing chan-nels, the assumed LFV mechanism induces also correlationsamong the |(mL)2ij |2 and hence among observables of dif-ferent flavor mixing channels. In this framework, the ratiosof the branching ratios are approximately independent ofSUSY parameters:

B(τ → μγ )

B(μ→ eγ )∝ |(Y †

ν LYν)23|2|(Y †ν LYν)12|2

. (5.33)

Thus the measurement of the ratio between the decay ratesof the different LFV channels can provide unique informa-tion on the flavor structure of the lepton sector. The ratiosof interest, such as (5.33), can exhibit, for instance, strongdependence on CP-violating parameters in neutrino Yukawacouplings [691] especially in the case of quasi-degenerateheavy RH neutrinos. As a consequence such correlationshave been widely studied (see, e.g., [38, 178, 238, 242, 249,329, 334, 649, 691, 693] and the references quoted therein).

Consequently, bounds on one LFV decay channel (process)will limit the parameter space of the LFV mechanism andthus lead to bounds on the other LFV decay channels(processes). In Fig. 14, the correlation induced by the type Iseesaw mechanism between B(μ→ eγ ) and B(τ → μγ )

is shown, and the bounds induced by the former on thelatter can be easily read off. Interestingly, these boundsdo not depend on whether hierarchical or quasi-degenerateheavy and light neutrinos are assumed. The present and fu-ture prospective bounds are summarized in Table 12. Notethat the present upper bound on B(μ → eγ ) implies astronger constraint on B(τ → μγ ) than its expected futurebound.

The above results were derived in the simplifying caseof a real R matrix. For complex R with |yi | < 1 there isno significant change with respect to the results in Table 12in the case of hierarchical heavy and hierarchical light neu-trinos due to the weak R dependence of B(μ→ eγ ) andB(τ→ μγ ). However, for quasi-degenerate light neutrinos,B(τ → μγ ) is lowered by roughly one order of magnitude,somewhat spoiling the overlap of all scenarios observed inFig. 14.

In Fig. 15, we display the correlation between B(μ→eγ ) and B(τ → μγ ) for complex R and some fixed valuesof MR in the case of quasi-degenerate RH neutrino masses

Fig. 14 B(τ→ μγ ) versus B(μ→ eγ ), in mSUGRA scenario SPS1awith neutrino parameters scattered within their experimentally allowedranges [689]. For quasi-degenerate heavy neutrino masses, both hi-erarchical (triangles) and quasi-degenerate (diamonds) light neutrinomasses are considered with real R and 1011 < MR < 1014.5 GeV.

In the case of hierarchical heavy and light neutrino masses (stars),the xi are scattered over their full ranges 0 < xi < 2π and the yiand Mi are scattered within the bounds demanded by leptogenesisand perturbativity. Also indicated are the present experimental boundsB(μ→ eγ ) < 1.2 × 10−11 and B(τ→ μγ ) < 6.8 × 10−8 [191, 690]

Table 12 Present and expected future bounds on B(μ→ eγ ) from experiment, and bounds on B(τ → μγ ) from (i) experiment (ii) the bound onB(μ→ eγ ) together with correlations from the SUSY type I seesaw mechanism

B(μ→ eγ ) (exp.) B(τ→ μγ ) (exp.) B(τ→ μγ )a

Present 1.2 × 10−11 6.8 × 10−8 10−9

Future 10−14 10−9 10−12

afrom B(μ→ eγ ) (exp.) and SUSY seesaw

Eur. Phys. J. C (2008) 57: 13–182 73

Fig. 15 The correlation between B(μ→ eγ ) and B(τ → μγ ) for quasi-degenerate heavy neutrinos and light neutrino mass spectrum of normalhierarchical (left panel) and inverted hierarchical (right panel) type

Fig. 16 Correlation of LFV LC processes and rare decays in the eμ-channel (left) and the μτ -channel (right). The seesaw parameters are scatteredas in Fig. 14. The mSUGRA scenarios used are (from left to right): SPS1a, G′ (eμ) and C′, B′, SPS1a, I′ (μτ )

and a normal and inverted hierarchical light neutrino massspectrum. We note that, as Fig. 15 suggests, B(τ → μγ )

is almost independent of the CP violating parameters andphases respectively in R and U , while the dependence ofB(μ→ eγ ) on the CP-violating quantities is much stronger.This is reflected, in particular, in the fact that for a fixedMR ,B(τ → μγ ) is practically constant while B(μ→ μγ ) canchange by 2–3 orders of magnitude.

If the μ→ eγ and τ→ μγ decays will be observed, theratio of interest can give unique information on the origin ofthe lepton flavor violation.

5.2.3.2 LFV rare decays and linear collider processes Inhigh energy e+e− colliders [692], feasible tests of LFV areprovided by the processes e+e− → l−a l+b → l−i l

+j + 2χ0

1 .Analogously to (5.27), one can derive the approximate ex-pression [695]

σ(e+e− → l−i l

+j + 2χ0

1

)

≈ |(δmL)2ij |2m2lΓ 2l

σ(e+e− → l−i l

+i + 2χ0

1

), (5.34)

for the production cross section in the limit of small slep-ton mass corrections. By comparing (5.27) with (5.34), it isimmediately apparent that the linear collider (LC) processesare flavor-correlated with the rare decays considered previ-ously. These correlations are shown in Fig. 16 for the twomost important channels.

This observation implies that once the SUSY parametersare known, a measurement of, e.g., B(μ→ eγ ) will lead toa prediction for σ(e+e− → μe+2χ0

1 ). Quite obviously, thisprediction will be independent of the specific LFV mecha-nism (seesaw or other). Figure 16 also demonstrates that theuncertainties in the neutrino parameters nicely drop out ex-cept at large cross sections and branching ratios.

74 Eur. Phys. J. C (2008) 57: 13–182

In the previous results we have assumed a specific choiceof the as yet unknown mSUGRA parameters. The results ofa more systematic study of the model dependence are visual-ized in Fig. 17 by contour plots for σ(e+e− → μ+e−+2χ0

1 )

and B(μ→ eγ ) in the m0–m1/2 plane with the remainingmSUGRA parameters fixed.

Fig. 17 Contours of the polarized cross section σ(e+e− →μ+e− + 2χ0

1 ) (solid) and B(μ→ eγ ) (dashed) in the m0–m1/2 plane.The remaining mSUGRA parameters are A0 = 0 GeV, tanβ = 5,sign(μ)=+. The energy and beam polarizations are

√see = 1.5 TeV,

Pe− = +0.9, Pe+ = +0.7. The neutrino oscillation parameters arefixed at their central values as given in [689], the lightest neutrino massm1 = 0 and all complex phases are set to zero, and the degenerate righthanded neutrino mass scale is MR = 1014 GeV. The shaded (red) ar-eas are already excluded by mass bounds from various experimentalsparticle searches

5.2.3.3 LFV rare decays and LHC processes At the LHC,a feasible test of LFV is provided by squark and gluino pro-duction, followed by cascade decays of squarks and gluinosvia neutralinos and sleptons [696, 697]:

pp→ qaqb, gqa, gg,

qa(g)→ χ02 qa(g),

χ02 → lαlβ,

lα → χ01 lβ,

(5.35)

where a, b run over all squark mass eigenstates, includ-ing antiparticles, and α,β are slepton (lepton) mass (flavor)eigenstates, including antiparticles. LFV can occur in thedecay of the second lightest neutralino and/or the slepton,resulting in different lepton flavors, α �= β . The total crosssection for the signature l+α l−β +X can then be written as

σ(pp→ l+α l−β +X)

=[∑a,b

σ (pp→ qaqb)×B(qa → χ0

2 qa)

+∑a

σ (pp→ qag)×(B(qa → χ0

2 qa)

+B(g→ χ02g))+ σ(pp→ gg)×B(g→ χ0

2g)]

×B(χ02 → l+α l−β χ

01

), (5.36)

where X can involve jets, leptons and LSPs produced bylepton flavor conserving decays of squarks and gluinos, aswell as low energy proton remnants. The LFV branching ra-tio B(χ0

2 → l+α l−β χ01 ) is for example calculated in [698] in

the framework of model-independent MSSM slepton mix-ing. In general, it involves a coherent summation over allintermediate slepton states.

Just as for the linear collider discussed in the previoussection, we can correlate the expected LFV event rates atthe LHC with LFV rare decays. This is shown in Fig. 18 for

Fig. 18 (Color online) Correlation of the number of χ02 → μ+e−χ0

1events per year at the LHC and B(μ→ eγ ) in mSUGRA scenarioC′ (m0 = 85 GeV, m1/2 = 400 GeV, A0 = 0 GeV, tanβ = 10 GeV,signμ = +) for the case of hier. νR/L (blue stars), deg. νR /hier. νL

(red boxes) and deg. νR/L (green triangles). The respective neutrinoparameter scattering ranges are as in Fig. 14. An integrated LHC lu-minosity of 100 fb−1 is assumed. The current limit on B(μ→ eγ ) isdisplayed by the vertical line

Eur. Phys. J. C (2008) 57: 13–182 75

the event rates N(χ02 → μ+e−χ0

1 ) and N(χ02 → τ+μ−χ0

1 ),respectively, originating from the cascade reactions (5.35).Both are correlated with B(μ→ eγ ), yielding maximumrates of around 102–3 per year for an integrated luminosityof (100 fb−1) in the mSUGRA scenario C′, consistent withthe current limit on B(μ→ eγ ).

As in the linear collider case, the correlation is approxi-mately independent of the neutrino parameters, but highlydependent on the mSUGRA parameters. This is contem-plated further in Fig. 19, comparing the sensitivity of the sig-nature N(χ0

2 → μ+e−χ01 ) at the LHC with B(μ→ eγ ) in

the m0–m1/2 plane. As for the linear collider, LHC searchescan be competitive with the rare decay experiments forsmall m0 ≈ 200 GeV. Tests in the large-m0 region are againseverely limited by collider kinematics.

Up to now we have considered LFV in the class of type ISUSY seesaw model described in Sect. 3.2.3.1, which isrepresentative of models of flavor mixing in the left handedslepton sector only. However, it is instructive to analyze gen-eral mixing in the left and right handed slepton sector, inde-pendent of any underlying model for slepton flavor viola-tion. The easiest way to achieve this is by assuming mixingbetween two flavors only, which can be parameterized by amixing angle θL/R and a mass difference (�m)L/R betweenthe sleptons, in the case of left/right handed slepton mixing,respectively.16 In particular, the left/right handed selectron

Fig. 19 Contours of the number of χ02 → μ+e−χ0

1 events at the LHCwith an integrated luminosity of 100 fb−1 (solid) and of B(μ→ eγ )

in the m0–m1/2 plane. The remaining mSUGRA and neutrino oscilla-tion parameters are as in Fig. 17. The shaded (red) areas are alreadyexcluded by mass bounds from various experimental sparticle searches

16Note that this is different to the approach in [698], where the sleptonmass matrix elements are scattered randomly.

and smuon sector is then diagonalized by

(l1

l2

)=U ·

(eL/R

μL/R

), with

U =(

cos θL/R sin θL/R− sin θL/R cos θL/R

), (5.37)

and a mass difference ml2− m

l1= (�m)L/R between

the slepton mass eigenvalues.17 The LFV branching ratioB(χ0

2 → μ+e−χ01 ) can then be written in terms of the mix-

ing parameters and the flavor conserving branching ratio

B(χ02 → e+e−χ0

1 ) as

B(χ0

2 → μ+e−χ01

)

= 2 sin2 θL/R cos2 θL/R(�m)2L/R

(�m)2L/R + Γ 2l

×B(χ02 → e+e−χ0

1

), (5.38)

where Γl

is the average width of the two sleptons involved.Maximal LFV is thus achieved by choosing θL/R = π/4and (�m)L/R � Γ

l. For definiteness, we use (�m)L/R =

0.5 GeV. The results of this calculation can be seen inFig. 20, which shows contour plots of N(χ0

2 → μ+e−χ01 )

in the m0–m1/2 plane for maximal left and right handedslepton mixing, respectively. Also displayed are the corre-sponding contours of B(μ→ eγ ). We see that the presentbound B(μ→ eγ ) = 10−11 still permits sizable LFV sig-nal rates at the LHC. However, B(μ → eγ ) < 10−14

would exclude the observation of such an LFV signal atthe LHC.

5.2.4 Impact of θ13 on LFV in SUSY seesaw

In this subsection we present the results of the LFV tauand muon decays within the SUSY singlet-seesaw con-text. Specifically, we consider the constrained minimal su-persymmetric standard model (CMSSM) extended by threeright handed neutrinos, νRi and their corresponding SUSYpartners, νRi (i = 1,2,3), and use the seesaw mechanismfor the neutrino mass generation. We include the predic-tions for the branching ratios (BRs) of two types of LFVchannels, lj → liγ and lj → 3li , and compare them withthe present bounds and future experimental sensitivities. Wefirst analyze the dependence of the BRs with the most rel-evant SUSY-seesaw parameters, and we then focus on theparticular sensitivity to θ13, which we find specially inter-esting on the light of its potential future measurement. We

17In case of left handed mixing, the mixing angle θL and the massdifference (�m)L are also used to describe the sneutrino sector.

76 Eur. Phys. J. C (2008) 57: 13–182

Fig. 20 Contours of the events per year N(χ02 → μ+e−χ0

1 ) at theLHC with an integrated luminosity of 100 fb−1 in the m0–m1/2plane (solid lines). The remaining mSUGRA parameters are: A0 =−100 GeV, tanβ = 10, sign(μ) = +. The left and right panels are

for maximal eLμL and eRμR mixing (θ = π/4, �m = 1 GeV), re-spectively. For comparison, B(μ→ eγ ) is shown by dashed lines. Theshaded (red) areas are forbidden by mass bounds from various experi-mental sparticle searches

further study the constraints from the requirement of suc-cessfully producing the Baryon Asymmetry of the Universevia thermal leptogenesis, which is another appealing featureof the SUSY-seesaw scenario. We conclude with the impactthat a potential measurement of the leptonic mixing angleθ13 can have on LFV physics.

Regarding the technical aspects of the computation of thebranching ratios, the most relevant points are (for details, see[649, 686]:

– It is a full one loop computation of BRs, i.e., we includeall contributing one loop diagrams with the SUSY par-ticles flowing in the loops. For the case of lj → liγ theanalytical formulas can be found in [178, 649]. For thecase lj → 3li the complete set of diagrams (includingphoton-penguin, Z-penguin, Higgs-penguin and box di-agrams) and formulae are given in [649].

– The computation is performed in the physical basis for allSUSY particles entering in the loops. In other words, wedo not use the mass insertion approximation (MIA).

– The running of the CMSSM-seesaw parameters from theuniversal scale MX down to the electroweak scale is per-formed by numerically solving the full one loop renor-malization group equations (RGEs) (including the ex-tended neutrino sector) and by means of the public For-tran Code SPheno2.2.2. [699]. More concretely, we do notuse the Leading Log Approximation (LLog).

– The light neutrino sector parameters that are used in

mD =√m

diagN R

√m

diagν U

†MNS are those evaluated at the

seesaw scale mR . That is, we start with their low energy

values (taken from data) and then apply the RGEs to runthem up to mR .

– We have added to the SPheno code extra subroutines thatcompute the LFV rates for all the lj → liγ and lj → 3lichannels. We have also included additional subroutinesto: Implement the requirement of successful baryogene-sis (which we define as having nB/nγ ∈ [10−10,10−9])via thermal leptogenesis in the presence of upper boundson the reheat temperature; Implement the requirement ofcompatibility with present bounds on lepton electric di-pole moments: EDMeμτ � (6.9 × 10−28, 3.7 × 10−19,4.5 × 10−17) e cm.

In what follows we present the main results for the caseof hierarchical heavy neutrinos. We also include a compari-son with present bounds on LFV rates [180, 182, 191, 644,700] and their future sensitivities [694, 701–706]. For hi-erarchical heavy neutrinos, the BRs are mostly sensitive tothe heaviest mass mN3 , tanβ , θ1 and θ2 (using the R para-meterization of [232]). The other input seesaw parametersmN1 , mN2 and θ3 play a secondary role since the BRs do notstrongly depend on them. The dependence on mN1 and θ3appears only indirectly, once the requirement of a success-ful generation of baryon asymmetry of the universe (BAU)is imposed. We shall comment more on this later.

We display in Fig. 21 the predictions for B(μ→ eγ ) andB(τ → μγ ) as a function of mN3 , for a specific choice ofthe other input parameters. This figure clearly shows thestrong sensitivity of the BRs to mN3 . In fact, the BRs varyby as much as six orders of magnitude in the explored rangeof 5 × 1011 ≤ mN3 ≤ 5 × 1014 GeV. Notice also that for

Eur. Phys. J. C (2008) 57: 13–182 77

Fig. 21 On the left, B(μ→ eγ ) as a function of mN3 for SPS 1a,with mν1 = 10−5 eV and mν1 = 10−3 eV (times, dots, respectively),and θ13 = 0◦, 5◦ (blue/darker, green/lighter lines). Baryogenesis is en-abled by the choice θ2 = 0.05e0.2i (θ1 = θ3 = 0). On the upper horizon-tal axis we display the associated value of (Yν)33. A dashed (dotted)horizontal line denotes the present experimental bound (future sensi-

tivity). On the right, B(τ → μγ ) as a function of mN3 for SPS5, withmν1 = 10−3 eV and θ2 = 0.05e0.2i (θ1 = θ3 = 0◦). The predictions forθ13 = 0◦,5◦ are superimposed one on the top of the other. The uppercurve is obtained using the LLog approximation and the lower one isthe full RGE prediction. The dashed (dotted) horizontal line denotesthe present experimental bound (future sensitivity)

the largest values of mN3 considered, the predicted ratesfor μ→ eγ enter into the present experimental reach andonly into the future experimental sensitivity for τ → μγ .It is also worth mentioning that by comparing our full re-sults with the LLog predictions, we find that the LLog ap-proximation dramatically fails in some cases. In particular,for the SPS5 point, the LLog predictions overestimate theBRs by about four orders of magnitude. For the other pointsSPS4, SPS1a,b and SPS2 the LLog estimate is very sim-ilar to the full result, whereas for SPS3 it underestimatesthe full computation by a factor of three. In general, the di-vergence of the LLog and the full computation occurs forlow M0 and large M1/2 [238, 273] and/or large A0 values[686]. The failure of the LLog is more dramatic for SUSYscenarios with large A0. Figure 21 also shows that whilein some cases (as for instance SPS1a) the behavior of theBR withmN3 does follow the expected LLog approximation(BR ∼ (mN3 logmN3)

2), there are other scenarios where thisis not the case. A good example is SPS5. It is also worthcommenting on the deep minima of B(μ→ eγ ) appearingin Fig. 21 for the lines associated with θ13 = 0◦. These min-ima are induced by the effect of the running of θ13, shiftingit from zero to a negative value (or equivalently θ13 > 0 andδ = π ). In the LLog approximation, they can be understoodas a cancellation occurring in the relevant quantity Y †

ν LYν ,with Lij = log(MX/mNi )δij . Most explicitly, the cancella-tion occurs between the terms proportional to mN3L33 andmN2L22 in the limit θ13(mR)→ 0− (with θ1 = θ3 = 0). Thedepth of these minima is larger for smaller mν1 , as is visiblein Fig. 21.

Regarding the tanβ dependence of the BRs we obtainthat, similar to what was found for the degenerate case, the

BR grow as tan2 β . The hierarchy of the BR predictionsfor the several SPS points is dictated by the correspondingtanβ value, with a secondary role being played by the givenSUSY spectra. We find again the following generic hierar-chy: BSPS4 >BSPS1b � BSPS1a >BSPS3 � BSPS2 >BSPS5.

In what concerns to the θi dependence of the BRs, wehave found that they are mostly sensitive to θ1 and θ2. TheBRs are nearly constant with θ3. As has been shown in [649],the predictions for B(μ→ eγ ), B(μ→ 3e), B(τ → μγ )

and B(τ → eγ ) are above their corresponding experimen-tal bound for specific values of θ1. Particularly, the LFVmuon decay rates are well above their present experimen-tal bounds for most of the θ1 explored values. Notice alsofor SPS4 that the predicted B(τ → μγ ) rates are very closeto the present experimental reach even at θ1 = 0 (that is,R = 1). We have also explored the dependence with θ2 andfound similar results (not shown here), with the appearanceof pronounced dips at particular real values of θ2 with theB(μ→ eγ ), B(μ→ 3e) and B(τ → μγ ) predictions beingabove the experimental bounds for some θ2 values.

We next address the sensitivity of the LFV BRs to θ13. Wefirst present the results for the simplest R = 1 case and thendiscuss how this sensitivity changes when moving from thiscase towards the more general case of complex R, takinginto account additional constraints from the requirement ofa successful BAU.

For R = 1, the predictions of the BRs as functions of θ13

in the experimentally allowed range of θ13, 0◦ ≤ θ13 ≤ 10◦are illustrated in Fig. 22. In this figure we also include thepresent and future experimental sensitivities for all channels.We clearly see that the BRs of μ→ eγ , μ→ 3e, τ → eγ

78 Eur. Phys. J. C (2008) 57: 13–182

Fig. 22 B(μ→ eγ ) and B(μ→ 3e) as a function of θ13 (in degrees), for SPS 1a (dots), 1b (crosses), 2 (asterisks), 3 (triangles), 4 (circles) and5 (times). A dashed (dotted) horizontal line denotes the present experimental bound (future sensitivity)

and τ → 3e are extremely sensitive to θ13, with their pre-dicted rates varying many orders of magnitude along the ex-plored θ13 interval. In the case of μ→ eγ this strong sen-sitivity was previously pointed out in Ref. [707]. The otherLFV channels, τ → μγ and τ → 3μ (not displayed here),are nearly insensitive to this parameter. The most importantconclusion from Fig. 22 is that, for this choice of parameters,the predicted BRs for both muon decay channels, μ→ eγ

and μ→ 3e, are clearly within the present experimentalreach for several of the studied SPS points. The most strin-gent channel is manifestly μ→ eγ where the predicted BRsfor all the SPS points are clearly above the present experi-mental bound for θ13 � 5◦. With the expected improvementin the experimental sensitivity to this channel, this wouldhappen for θ13 � 1◦.

In addition to the small neutrino mass generation, the see-saw mechanism offers the interesting possibility of baryo-genesis via leptogenesis [290]. Thermal leptogenesis is anattractive and minimal mechanism to produce a successfulBAU with rates which are compatible with present data,nB/nγ ≈ (6.10 ± 0.21) × 10−10 [327]. In the supersym-metric version of the seesaw mechanism, it can be success-

fully implemented if provided that the following conditionscan be satisfied. Firstly, Big Bang Nucleosynthesis grav-itino problems have to be avoided, which is possible, for in-stance, for sufficiently heavy gravitinos. Since we considerthe gravitino mass as a free parameter, this condition canbe easily achieved. In any case, further bounds on the re-heat temperature TRH still arise from decays of gravitinosinto lightest supersymmetric particles (LSPs). In the case ofheavy gravitinos and neutralino LSPs masses into the range100–150 GeV (which is the case of the present work), oneobtains TRH � 2 × 1010 GeV. In the presence of these con-straints on TRH, the favored region by thermal leptogenesiscorresponds to small (but non-vanishing) complex R-matrixangles θi . For vanishing UMNS CP phases the constraintson R are basically |θ2|, |θ3| � 1 rad(mod π ). Thermal lep-togenesis also constrains mN1 to be roughly in the range[109 GeV,10 × TRH] (see also [309, 311]). In the presentwork we have explicitly calculated the produced BAU in thepresence of upper bounds on the reheat temperature TRH.We have furthermore set as “favored BAU values” thosethat are within the interval [10−10,10−9], which containsthe WMAP value, and choose the value of mN1 = 1010 GeV

Eur. Phys. J. C (2008) 57: 13–182 79

in some of our plots. Similar studies of the constraints fromleptogenesis on LFV rates have been done in [239].

Concerning the EDMs, which are clearly non-vanishingin the presence of complex θi , we have checked that all thepredicted values for the electron, muon and tau EDMs arewell below the experimental bounds. In the following wetherefore focus on complex but small θ2 values, leading tofavorable BAU, and study its effects on the sensitivity to θ13.Similar results are obtained for θ3, but for shortness are notshown here.

Figure 23 shows the dependence of the most sensitive BRto θ13, B(μ→ eγ ), on |θ2|. We consider two particular val-ues of θ13, θ13 = 0◦, 5◦ and choose SPS 1a. Motivated fromthe thermal leptogenesis favored θ2-regions [686], we take0 � |θ2| � π/4, with arg θ2 = {π/8,π/4,3π/8}. We dis-play the numerical results, considering mν1 = 10−5 eV andmν1 = 10−3 eV, while for the heavy neutrino masses we takemN = (1010,1011,1014) GeV. There are several importantconclusions to be drawn from Fig. 23. Let us first discuss thecase mν1 = 10−5 eV. We note that one can obtain a baryonasymmetry in the range 10−10 to 10−9 for a considerableregion of the analyzed |θ2| range. Notice also that there isa clear separation between the predictions of θ13 = 0◦ andθ13 = 5◦, with the latter well above the present experimen-tal bound. This would imply an experimental impact of θ13,in the sense that the BR predictions become potentially de-tectable for this non-vanishing θ13 value. With the plannedMEG sensitivity [701], both cases would be within exper-imental reach. However, this statement is strongly depen-dent on the assumed parameters, in particular mν1 . For in-stance, a larger value of mν1 = 10−3 eV, illustrated on theright panel of Fig. 23, leads to a very distinct situation re-garding the sensitivity to θ13. While for smaller values of|θ2| the branching ratio displays a clear sensitivity to hav-ing θ13 equal or different from zero (a separation larger than

two orders of magnitude for |θ2| � 0.05), the effect of θ13

is diluted for increasing values of |θ2|. For |θ2| � 0.3 theB(μ→ eγ ) associated with θ13 = 5◦ can be even smallerthan for θ13 = 0◦. This implies that in this case, a poten-tial measurement of B(μ→ eγ ) would not be sensitiveto θ13. Whether or not a SPS 1a scenario would be disfa-vored by current experimental data on B(μ→ eγ ) requiresa careful weighting of several aspects. Even though Fig. 23suggests that for this particular choice of parameters onlyvery small values of θ2 and θ13 would be in agreement withcurrent experimental data, a distinct choice of mN3 (e.g.mN3 = 1013 GeV) would lead to a rescaling of the estimatedBRs by a factor of approximately 10−2. Although we do notdisplay the associated plots here, in the latter case nearly theentire |θ2| range would be in agreement with experimentaldata (in fact the points which are below the present MEGAbound on Fig. 23 would then lie below the projected MEGsensitivity). Regarding the other SPS points, which are notshown here, we find BRs for SPS 1b comparable to those ofSPS 1a. Smaller ratios are associated with SPS 2, 3 and 5,while larger (more than one order of magnitude) BRs occurfor SPS 4.

Let us now address the question of whether a joint mea-surement of the BRs and θ13 can shed some light on exper-imentally unreachable parameters, like mN3 . The expectedimprovement in the experimental sensitivity to the LFV ra-tios supports the possibility that a BR could be measuredin the future, thus providing the first experimental evidencefor new physics, even before its discovery at the LHC. Theprospects are especially encouraging regarding μ→ eγ ,where the experimental sensitivity will improve by at leasttwo orders of magnitude. Moreover, and given the impres-sive effort on experimental neutrino physics, a measurementof θ13 will likely also occur in the future [708–717]. Given

Fig. 23 B(μ → eγ ) as a function of |θ2|, for arg θ2 ={π/8,π/4,3π/8} (dots, times, diamonds, respectively) and θ13 = 0◦,5◦ (blue/darker, green/lighter lines). We take mν1 = 10−5 (10−3) eV,

on the left (right) panel. In all cases black dots represent points associ-ated with a disfavored BAU scenario and a dashed (dotted) horizontalline denotes the present experimental bound (future sensitivity)

80 Eur. Phys. J. C (2008) 57: 13–182

that, as previously emphasized, μ→ eγ is very sensitiveto θ13, whereas this is not the case for B(τ → μγ ), andthat both BRs display the same approximate behavior withmN3 and tanβ , we now propose to study the correlationbetween these two observables. This optimizes the impactof a θ13 measurement, since it allows us to minimize theuncertainty introduced from not knowing tanβ and mN3 ,and at the same time offers a better illustration of the un-certainty associated with the R-matrix angles. In this case,the correlation of the BRs with respect to mN3 means that,for a fixed set of parameters, varying mN3 implies that thepredicted point (B(τ → μγ ), B(μ→ eγ )) moves along aline with approximately constant slope in the B(τ → μγ )–B(μ→ eγ ) plane. On the other hand, varying θ13 leads to adisplacement of the point along the vertical axis.

In Fig. 24, we illustrate this correlation for SPS 1a,choosing distinct values of the heaviest neutrino mass, andwe scan over the BAU-enabling R-matrix angles (setting θ3to zero) as

0 � |θ1|� π/4, −π/4 � arg θ1 � π/4,

0 � |θ2|� π/4, 0 � arg θ2 � π/4,

mN3 = 1012,1013,1014 GeV.

(5.39)

We consider the following values, θ13 = 1◦, 3◦, 5◦ and10◦, and only include in the plot the BR predictions whichallow for a favorable BAU. Other SPS points have also beenconsidered but they are not shown here for brevity (see[686]). We clearly observe in Fig. 24 that for a fixed valueof mN3 , and for a given value of θ13, the dispersion aris-ing from a θ1 and θ2 variation produces a small area ratherthan a point in the B(τ → μγ –B(μ→ eγ ) plane. The dis-persion along the B(τ → μγ ) axis is of approximately oneorder of magnitude for all θ13. In contrast, the dispersionalong the B(μ→ eγ ) axis increases with decreasing θ13,

ranging from an order of magnitude for θ13 = 10◦, to overthree orders of magnitude for the case of small θ13 (1◦).From Fig. 24 we can also infer that other choices ofmN3 (forθ13 ∈ [1◦,10◦]) would lead to BR predictions which wouldroughly lie within the diagonal lines depicted in the plot.Comparing these predictions for the shaded areas along theexpected diagonal “corridor”, with the allowed experimentalregion, allows us to conclude about the impact of a θ13 mea-surement on the allowed/excludedmN3 values. The most im-portant conclusion from Fig. 24 is that for SPS 1a, and forthe parameter space defined in (5.39), an hypothetical θ13

measurement larger than 1◦, together with the present ex-perimental bound on the B(μ→ eγ ), will have the impactof excluding values of mN3 � 1014 GeV. Moreover, with theplanned MEG sensitivity, the same θ13 measurement canfurther constrain mN3 � 3 × 1012 GeV. The impact of anyother θ13 measurement can be analogously extracted fromFig. 24.

As a final comment let us add that, remarkably, within aparticular SUSY scenario and scanning over specific θ1 andθ2 BAU-enabling ranges for various values of θ13, the com-parison of the theoretical predictions for B(μ→ eγ ) andB(τ→ μγ ) with the present experimental bounds allows usto set θ13-dependent upper bounds on mN3 . Together withthe indirect lower bound arising from leptogenesis consider-ations, this clearly provides interesting hints on the value ofthe seesaw parametermN3 . With the planned future sensitiv-ities, these bounds would further improve by approximatelyone order of magnitude. Ultimately, a joint measurement ofthe LFV branching ratios, θ13 and the sparticle spectrumwould be a powerful tool for shedding some light on other-wise unreachable SUSY seesaw parameters. It is clear fromall this study that the interplay between LFV processes andfuture improvement in neutrino data is challenging for thesearches of new physics.

Fig. 24 (Color online)Correlation betweenB(μ→ eγ ) and B(τ → μγ ) asa function of mN3 , for SPS 1a.The areas displayed representthe scan over θi as given in(5.39). From bottom to top, thecolored regions correspond toθ13 = 1◦, 3◦, 5◦ and 10◦ (red,green, blue and pink,respectively). Horizontal andvertical dashed (dotted) linesdenote the experimental bounds(future sensitivities)

Eur. Phys. J. C (2008) 57: 13–182 81

5.2.5 LFV in the CMSSM with constrained sequentialdominance

Sequential Dominance (SD) [140, 141, 143] representsclasses of neutrino models where large lepton mixing an-gles and small hierarchical neutrino masses can be readilyexplained within the seesaw mechanism. To understand howSequential Dominance works, we begin by writing the righthanded neutrino Majorana mass matrix MRR in a diagonalbasis as MRR = diag(MA,MB,MC). We furthermore writethe neutrino (Dirac) Yukawa matrix λν in terms of (1,3) col-umn vectors Ai, Bi, Ci as Yν = (A,B,C) using left–rightconvention. The term for the light neutrino masses in the ef-fective Lagrangian (after electroweak symmetry breaking),resulting from integrating out the massive right handed neu-trinos, is

Lνeff =(νTi Ai)(A

Tj νj )

MA+ (ν

Ti Bi)(B

Tj νj )

MB

+ (νTi Ci)(C

Tj νj )

MC, (5.40)

where νi (i = 1,2,3) are the left handed neutrino fields. Se-quential dominance then corresponds to the third term beingnegligible, the second term subdominant and the first termdominant:

AiAj

MA� BiBj

MB� CiCj

MC. (5.41)

In addition, we shall shortly see that small θ13 and almostmaximal θ23 require that

|A1| � |A2| ≈ |A2|. (5.42)

Without loss of generality, then, we shall label the dom-inant right handed neutrino and Yukawa couplings as A,the subdominant ones as B , and the almost decoupled (sub-subdominant) ones asC. Note that the mass ordering of righthanded neutrinos is not yet specified. Again without loss ofgenerality we shall order the right handed neutrino massesasM1 <M2 <M3, and subsequently identifyMA,MB,MCwithM1,M2,M3 in all possible ways. LFV in some of theseclasses of SD models has been analyzed in [718]. Tri-bi-maximal neutrino mixing corresponds to the choice for ex-ample [719], sometimes referred to as constrained sequen-tial dominance (CSD):

Yν =

⎛⎜⎜⎝

0 beiβ2 c1

−aeiβ3 beiβ2 c2

aeiβ3 beiβ2 c3

⎞⎟⎟⎠ . (5.43)

When dealing with LFV it is convenient to work in the ba-sis where the charged lepton mass matrix is diagonal. Letus now discuss the consequences of charged lepton correc-tions with a CKM-like structure, for the neutrino Yukawamatrix with CSD. By CKM-like structure we mean thatVeL is dominated by a 1–2 mixing θ , i.e. that its elements(VeL)13, (VeL)23, (VeL)31 and (VeL)32 are very small com-pared to (VeL)ij (i, j = 1,2). After re-diagonalizing thecharged lepton mass matrix, Yν in (5.43) becomes trans-formed as Yν → VeLYν . In the diagonal charged lepton massbasis the neutrino Yukawa matrix therefore becomes

Yν =⎛⎝asθ e

−iλeiβ3 b(cθ − sθ e−iλ)eiβ2 (c1cθ − c2sθ e−iλ)

−acθ eiβ3 b(cθ + sθ eiλ)eiβ2 (c1sθ eiλ + c2cθ )

aeiβ3 beiβ2 c3

⎞⎠ .

(5.44)

After ordering MA,MB,MC according to their size,there are six possible forms of Yν obtained from permut-ing the columns, with the convention always being that thedominant one is labeled by A, and so on. In particular thethird column of the neutrino Yukawa matrix could be A, Bor C depending on which of MA, MB or MC is the heavi-est. If the heaviest right handed neutrino mass is MA thenthe third column of the neutrino Yukawa matrix will con-sist of the (re-ordered) first column of (5.44) and assumingY ν33 ∼ 1 we conclude that all LFV processes will be deter-mined approximately by the first column of (5.44). Similarlyif the heaviest right handed neutrino mass is MB then weconclude that all LFV processes will be determined approx-imately by the second column of (5.44). Note that in bothcases the ratios of branching ratios are independent of theunknown Yukawa couplings which cancel, and only dependon the charged lepton angle θ , which in the case of tri-bi-maximal neutrino mixing is related to the physical reactorangle by θ13 = θ/

√2 [719, 720]. Also note that λ = δ − π

where δ is the standard PDG CP-violating oscillation phase.The results for these two cases are shown in Fig. 25 [721].The third caseM3 =MC is less predictive.

5.2.6 Decoupling of one heavy neutrino and cosmologicalimplications

The supersymmetric seesaw model involves many free para-meters. In order to correlate the model predictions for LFVprocesses one has to resort to some supplementary hypothe-ses. Here we discuss the consequences of the assumptionthat one of the heavy singlet neutrinos (not necessarily theheaviest one) decouples from the see-saw mechanism [238].

If the light neutrino masses are hierarchical, in whichcase the effects of the renormalization group (RG) running[722] of κ are negligible, at least 3 arguments support thisassumption. The first one is the naturalness of the see-saw

82 Eur. Phys. J. C (2008) 57: 13–182

Fig. 25 Ratios of branching ratios of LFV processes �i → �j γ inCSD for M3 = MA (left panel) and M3 = MB (right panel) withright handed neutrino masses M1 = 108 GeV, M2 = 5 × 108 GeV andM3 = 1014 GeV. The solid lines show the (naive) prediction, fromthe MI and LLog approximation and with RG running effects ne-glected, while the dots show the explicit numerical computation (usingSPheno2.2.2. [699] extended by software packages for LFV BRs and

neutrino mass matrix running [649, 686]) with universal CMSSM pa-rameters chosen as m0 = 750 GeV, m1/2 = 750 GeV, A0 = 0 GeV,tanβ = 10 and sign(μ) = +1. While the ratios do not significantlydepend on the choice of the SUSY model, since the model-dependencehas canceled out, they show a pronounced dependence on θ13 (and δ)in the case ofM3 =MA (andM3 =MB )

mechanism. Large mixing angles are not generic for hierar-chical light neutrino masses (for a review, see [723]). Theyare natural only for special patterns of the matrix κ . One is alarge hierarchy between one and the remaining two terms inthe sum in (3.41) [38, 140, 233, 724]. This is what we calldecoupling (one term hierarchically smaller) or dominance(hierarchically larger). Seesaw with only two heavy singletneutrinos [234, 235] is the limiting case of the decouplingof N3 with M3 →∞ and Y 3A

ν → 0. The immediate conse-quence of decoupling ismν1 �mν2 (κ has rank 2 if there areonly 2 terms in the sum in (3.41)). Similarly, for dominanceone has mν2 �mν3 .

Secondly, decoupling of the lightest singlet neutrino al-leviates the gravitino problem of leptogenesis which inthe see-saw models of neutrino masses appears to be themost natural mechanism for producing the observed baryonasymmetry of the Universe18 (BAU). As the Universe coolsdown leptonic asymmetries (subsequently converted intobaryon asymmetry through sphaleron transitions) Yα ≡(nα − nα)/s �= 0 (where nα and nα are the flavor α lep-ton and antilepton number densities, respectively and s isthe entropy density) are produced in the decays of N1. Thefinal magnitudes of Yα are proportional to the decay asym-metries ε1α , (which in turn are proportional to the heavyneutrino masses) and crucially depend on the processeswhich wash out the asymmetries generated by the N1 de-cays. The efficiency of these processes depends on the pa-

18See [290]; for a review of leptogenesis, see [324]; for a discussion offlavor effects in leptogenesis see, e.g. [296]; for recent analyses of thegravitino problem, see, e.g., [725].

rameters m1α = ∑A |R1AU

∗αA|2mνA , where U ≡ UPMNS

and it is the smallest (i.e. leptogenesis is most efficient)for m1α in the meV range (assuming vanishing density ofN1 after re-heating and strongly hierarchical spectrum ofMA). If it is N1 which is decoupled, there are essentiallyno lower bounds on m1α and M1, hence also the re-heatingtemperature TRH, already of order 109 GeV are sufficient[311, 726] (see, however, e.g., [316]) to reproduce the ob-served BAU.19

Finally, one heavy singlet neutrino NA must be decou-pled if its superpartner, NA, plays the role of the inflaton field[347]. In such a scenario the (s)neutrino mass MA must be[349] 2 × 1013 GeV and the re-heating temperature follow-ing inflaton decay is given by TRH ∼ √

mAMPl(MA/〈H 〉).Requiring TRH < 106 GeV (the gravitino problem) thenimplies mνA ≤∑α mAα < 10−17 eV. In this scenario, theleptonic asymmetries must be produced non-thermally inthe inflaton decay. Decoupling of N1 is favored because ifit is N2 or N3 which is the inflaton the produced asym-metry may be subsequently washed out during the decaysof N1.

The assumption that NA effectively decouples from theseesaw mechanism or thatNA effectively dominates the see-saw mechanism translates into one of the following forms

19In contrast, for N2 or N3 decoupled, the washout is much strongerand M1 has to be >∼ 1010 GeV. This requires TRH leading to a muchlarger dangerous gravitino production [727]. Lower TRH is in this casepossible only if N1 and N2 are sufficiently degenerate [728].

Eur. Phys. J. C (2008) 57: 13–182 83

of R:

Rdec Π(A)⎛⎝

1 0 00 z p

0 ∓p ±z

⎞⎠ or

Rdom Π(A)⎛⎝∓p ±z 0z p 00 0 1

⎞⎠ ,

(5.45)

where z,p are complex numbers satisfying z2 +p2 = 1 andΠ(A) denotes permutation of the rows of R. Both conditionscan be simultaneously satisfied for R =Π(A) · 1, known assequential dominance (for a review, see, e.g. [729]).

In the framework considered, violation of the leptonic fla-vor is transmitted from the neutrino Yukawa couplings Yνto the slepton mass matrices through the RG corrections.Branching ratios of LFV decays are well described by asingle mass-insertion approximation via (5.27) and (5.28).Since decoupling of N1 is best motivated we discuss the re-sults for LFV only in this case.20

The matrixR has then the first of the patterns displayed in(5.45) withΠ(1) = 1. The discussion simplifies if a technicalassumption that mν3M2 <mν2M3 is made. (m2

L)32 relevantfor τ→ μγ then reads

(m2L

)32 ≈ κmν3M3U33U

∗23

〈H 〉2

[(|z|2 + S|p|2)+ ρU∗22

U∗23x

+ ρU32

U33x∗ + ρ2U32U

∗22

U33U∗23

(S|z|2 + |p|2)

], (5.46)

where ρ =√mν2/mν3 ∼ 0.4, S =M2(1 +�l2/�t)/M3 ∼M2/M3 and x = Sp∗z−z∗p. For (m2

L)A1 relevant for �A→eγ we get:

(m2L

)A1 ≈ κmν3M3UA3U

∗12

〈H 〉2

[U∗

13

U∗12

(|z|2 + S|p|2)+ ρx

+ ρ2UA2

UA3

(S|z|2 + |p|2)

]. (5.47)

Analysis of the expressions (5.46) and (5.47) leads to anumber of conclusions [238]. Firstly, the branching ratios ofthe LFV decays depend (apart from the scales of soft su-persymmetry breaking and the value of tanβ) mostly on the

20Results for N2 decoupled are the same as for decoupled N1 (includ-ing sub-leading effects if M1 takes the numerical value of M2). Thesame is true also for N3 decoupled (including the case with only 2heavy singlet neutrinos) if M2 is numerically the same as M3 for de-coupled N1. However, if N3 is the inflaton the LFV decays have therates too low to be observed. In addition, if N3 decouples due to itsvery large mass its large Yukawa can, for mν1/mν3 > M2/M3, stilldominate the LFV effects which are then practically unconstrained bythe oscillation data; some constraints can then be obtained from thelimits on the electron EDM [730].

mass of the heaviest of the two un-decoupled singlet neu-trinos (in this case N3). Secondly, for fixed M3, they de-pend strongly on the magnitude and phase of R32, mildlyon the undetermined element U13 of the light neutrino mix-ing matrix and, in addition, on the Majorana phases of Uwhich cannot be measured in oscillation experiments [545].The latter dependence is mild for B(τ → μγ ) but can leadto strong destructive interference either in B(μ→ eγ ) orin B(τ → eγ ) decreasing them by several orders of mag-nitude. The interference effects are seen in Fig. 26(a), (b)where the predicted ranges (resulting from varying the un-known Majorana phases) of B(μ→ eγ ) are shown as afunction of |R32| for M1 appropriate for the sneutrino in-flation scenario, three different values of arg(R32) and form0 = 100 GeV, M1/2 = 500 GeV and tanβ = 10, consis-tent with the dark matter abundance [731]. Results for othervalues of these parameters can be obtained by appropriaterescalings using (5.27). For comparison, for selected val-ues of R23, we also indicate the ranges of B(μ → eγ )

resulting from generic form of the matrix R (constrainedonly by the conditions 0 < R12,R13 < 1.5 and Re(YABν ),

Im(YABν ) < 10).The bulk of the predicted values of B(μ→ eγ ) shown

in Fig. 26(a) and (b) exceed the current experimental limit.Since M1/2 = 500 GeV leads to masses of the third gen-eration squarks above 1 TeV, suppressing B(μ→ eγ ) byincreasing the SUSY breaking scale conflicts with the sta-bility of the electroweak scale. Moreover, as discussed in[238] in the scenario considered here generically B(τ →μγ )/B(μ→ eγ )∼ 0.1. Thus the observation τ→ μγ withB >∼ 10−9, accessible to future experiments would excludethis scenario.

For completeness, in Fig. 26(c) we also show pre-dictions for B(μ → eγ ) in the case of N3 dominance.(m2L)21 is in this case controlled mainly by |U13|. More-

over, B(τ → μγ )/B(μ→ eγ )∼ max(|U13|2, ρ4S2), whileB(τ → eγ )/B(μ→ eγ )∼ 1 allowing for experimental testof this scenario (cf. [732]). The limits R32 → 0 in panels aand b and or R21 → 0 in panel c correspond to pure sequen-tial dominance.

In conclusion, the well motivated assumption about thedecoupling/dominance of one heavy singlet neutrino sig-nificantly constrains the predictions for the LFV processesin supersymmetric model. The forthcoming experimentsshould be able to verify this assumption and, in conse-quence, to test an interesting class of neutrino mass models.

5.2.7 Triplet seesaw mechanism and lepton flavor violation

In this subsection we intend to discuss the aspect of lowscale LFV in rare decays arising in the context of the tripletseesaw mechanism of Sect. 3.2.3.2. We consider both non-SUSY and SUSY versions of it. The flavor structure of the

84 Eur. Phys. J. C (2008) 57: 13–182

Fig. 26 (Color online) Predicted ranges of B(μ → eγ ) for(M1,M2,M3) = (2,3,50) × 1013 GeV, m0 = 100 GeV, M1/2 =500 GeV and tanβ = 10, for the decoupling of N1 and U13 = 0(panel a) or U13 = 0.1i (panel b). Yellow ranges show the possible

variation for arbitrary form of R with arg(R32) = 0. Lower (upper)pairs of lines in the panel c show similar ranges for N3 dominance forU13 = 0 (0.1i). The current experimental bound of 1.2 × 10−11 [180]is also shown

Fig. 27 Diagrams thatcontribute to the decay�j → �iγ through the exchangeof the triplet scalars

(high energy) Yukawa matrix YT of (3.63) is the same asthat of the (low energy) neutrino mass matrix mν . There-fore, in the triplet seesaw scenario the neutrino mass matrix(containing nine real parameters), which can be tested in thelow energy experiments, is directly linked to the symmet-ric matrix YT (containing also nine real parameters), mod-ulo the ratioM2

T /μ′, see (3.66). This feature has interesting

implications for LFV [249]. Collider phenomenology of thelow scale triplet was discussed in Sect. 5.1.4 The triplet La-grangian also induces LFV decays of the charged leptonsthrough the one loop exchange of the triplet states.

The diagrams relevant for the LFV radiative decays �j →�iγ (see e.g., [733, 734]) are depicted in Fig. 27. DenotingUPMNS = V · diag(1, eiφ1, eiφ2), where, φ1,2 are the Majo-rana phases, those imply the following flavor structure:

(Y

†T YT

)ij=(M2T

μ′v2

)2(m†νmν

)ij

=(M2T

μ′v2

)2[V(mDν)2V †]

ij, (5.48)

where i, j = e,μ, τ are family indices. Therefore, theamount of LFV is directly and univocally expressed in termsof the low energy neutrino parameters. In particular, LFVdecays depend only on 7 independent neutrino parameters(there is no dependence on the Majorana phases φi ). No-tice that this simple flavor structure is peculiar of the triplet

seesaw case, which represents a concrete and explicit real-ization of the ‘minimal flavor violation’ hypothesis [536] inthe lepton sector [542]. Indeed, according to the latter, thelow energy SM Yukawa couplings are the only source ofLFV. This is not generically the case for the seesaw mech-anism realized through the exchange of the so-called ‘righthanded’ neutrinos, where the number of independent para-meters of the high energy flavor structures is twice more thatof the mass matrix mν .

Finally, the parametric dependence of the dipole ampli-tude in Fig. 27 is

Dij ≈ (Y†T YT )ij

16π2M2T

= (m†νmν)ij

16π2v4

(MT

μ′

)2

. (5.49)

From the present experimental bound on B(μ→ eγ ) <

1.2 × 10−11 [180], one infers the bound μ′ > 10−10MT

[comparable limit is obtained from B(τ → μγ )]. We canpush further our discussion considering the relative size ofLFV in different family sectors:

(Y†T YT )τμ

(Y†T YT )μe

≈ [V (mDν )2V †]τμ[V (mDν )2V †]μe ,

(5.50)(Y

†T YT )τe

(Y†T YT )μe

≈ [V (mDν )2V †]τe[V (mDν )2V †]μe .

Eur. Phys. J. C (2008) 57: 13–182 85

These ratios depend only on the neutrino parameters, whiledo not depend on details of the model, such as the massscales MT , or μ′. By taking the present best fit values ofthe neutrino masses and mixing angles [211, 735] providedby the analysis of the experimental data, those ratios can beexplicitly expressed as

(Y†T YT )τμ

(Y†T YT )μe

≈(�m2

A

�m2S

)sin 2θ23

sin 2θ12 cos θ23∼ 40,

(5.51)(Y

†T YT )τe

(Y†T YT )μe

≈− tan θ23 ∼−1,

where �m2A(�m

2S) is the squared-mass difference relevant

for the atmospheric (solar) neutrino oscillations. These re-sults hold for θ13 = 0 and for either hierarchical, quasi-degenerate or inverted hierarchical neutrino spectrum (formore details see [249, 736]). It is immediate to translate theabove relations into model-independent predictions for ra-tios of LFV processes:

B(τ → μγ )

B(μ→ eγ )≈((Y

†T YT )τμ

(Y†T YT )μe

)2B(τ→ μντ νμ)

B(μ→ eνμνe)∼ 300,

(5.52)B(τ→ eγ )

B(μ→ eγ )≈((Y

†T YT )τe

(Y†T YT )μe

)2B(τ→ eντ νe)

B(μ→ eνμνe)∼ 0.2.

Now we focus upon the supersymmetric version of thetriplet seesaw mechanism. (Just recall just that in the su-persymmetric case there is only one mass parameter, MT ,while the mass parameter μ′ of the non-supersymmetric ver-sion is absent from the superpotential and its role is taken byλ2MT .) Regarding the aspect of LFV, in this case we haveto consider besides the diagrams of Fig. 27 also the relatedones with each particle in the loop replaced by its superpart-ner (�k → �k, T → T ). Such additional contributions wouldcancel those in Fig. 27 in the limit of exact supersymme-try. In the presence of soft supersymmetry breaking (SSB)the cancellation is only partial and the overall result for thecoefficient of the dipole amplitude behaves like

Dij ≈ (Y†T YT )ij

16π2

m2

M4T

∼ (m†νmν)ij

16π2(λ2v22)

2

m2

M2T

, (5.53)

which is suppressed with respect to the non-supersymmetricresult (5.49) for MT > m∼ O(102 GeV) (m denotes an av-erage soft breaking mass parameter). In the supersymmet-ric version of the triplet seesaw mechanism flavor violationcan also be induced by renormalization effects via (3.69)(the complete set of RGEs of the MSSM with the tripletstates have been computed in [249]). Thus in SUSY modelthe LFV processes can occur also in the case of very heavytriplet. In that case the relevant flavor structure responsible

for LFV is again Y †T YT for which we have already noticed

its unambiguous dependence on the neutrino parameters in(5.48). Clearly, we find that analogous ratios as in (5.50)hold also for the LFV entries of the soft breaking parame-ters, e.g.,

(m2L)τμ

(m2L)μe

≈ [V (mDν )2V †]τμ[V (mDν )2V †]μe ,

(5.54)(m2L)τe

(m2L)μe

≈ [V (mDν )2V †]τe[V (mDν )2V †]μe .

Such SSB flavor violating mass parameters induce extracontributions to the LFV processes. For example, the radia-tive decays �j → �iγ receive also one loop contributionswith the exchange of the charged-sleptons/neutralinos andsneutrinos/charginos, where the slepton masses (m2

L)ij are

the source of LFV. The relevant dipole terms have a para-metric dependence of the form

Dij ≈ g2

16π2

(m2L)ij

m4tanβ

≈ g2

16π2

(m†νmν)ij

(λ2v22)

2

M2T

m2log

(MG

MT

)tanβ. (5.55)

Notice the inverted dependence on the ratio m/MT with re-spect to the triplet exchange contribution. Due to this fea-ture, the MSSM sparticle induced contributions (5.55) tendsto dominate over the one induced by the triplet exchange. Inthis case, analogous ratios as in (5.52) can be derived, i.e.,

B(τ → μγ )

B(μ→ eγ )≈((m2L)τμ

(m2L)μe

)2B(τ→ μντ νμ)

B(μ→ eνμνe)∼ 300,

(5.56)B(τ → eγ )

B(μ→ eγ )≈((m2L)τe

(m2L)μe

)2B(τ → eντ νe)

B(μ→ eνμνe)∼ 0.2.

(For more details see [249].)The presence of extra SU(2)W triplet states at intermedi-

ate energy spoils the successful gauge coupling unificationof the MSSM. A simple way to recover gauge coupling uni-fication is to introduce more states X, to complete a certainrepresentation R—such that R = T +X—of some unifyinggauge group G, G⊃ SU(3)× SU(2)W ×U(1)Y . In generalthe Yukawa couplings of the states X are related to those ofthe triplet partners T . Indeed, this is generally the case inminimal GUT models. In this case RG effects generates notonly lepton-flavor violation but also closely correlated fla-vor violation in the quark sector (due to the X-couplings).An explicit scenario with G= SU(5) where both lepton andquark flavor violation arise from RG effects was discussedin Ref. [249]. In Sect. 5.3.2 we review a supersymmetricSU(5) model for the triplet seesaw scenario.

86 Eur. Phys. J. C (2008) 57: 13–182

5.3 SUSY GUTs

5.3.1 Flavor violation in the minimal supersymmetricSU(5) seesaw model

In this section we review flavor- and/or CP-violating phe-nomena in the minimal SUSY SU(5) GUT, in which theright handed neutrinos are introduced to generate neutrinomasses by the type-I seesaw mechanism [216–220]. Here, itis assumed that the Higgs doublets in this MSSM are em-bedded in 5- and 5-dimensional SU(5) multiplets. Rich fla-vor structure is induced even in those minimal particle con-tents. The flavor violating SUSY breaking terms for the righthanded squarks and sleptons are generated by the GUT in-teraction, while those are suppressed in the MSSM (+νR)under the universal scalar mass hypothesis for the SUSYbreaking terms.

The Yukawa interactions for quarks and leptons and theMajorana mass terms for the right handed neutrinos in thismodel are given by the following superpotential,

W = 1

4YuijΨiΨjH +√

2YdijΨiΦjH

+ Y νijΦiNjH + 1

2MNijNiNj , (5.57)

where Ψ and Φ are for 10- and 5-dimensional multiplets,respectively, and N is for the right handed neutrinos. H (H )is 5- (5-) dimensional Higgs multiplets. After removing theunphysical degrees of freedom, the Yukawa coupling con-stants in (5.57) are given as follows,

Yuij = VkiYukeiϕuk Vkj ,Y dij = Ydi δij , (5.58)

Y νij = eiϕdi U!ij Yνj .

Here, Yu, Yd, Yν denote diagonal Yukawa couplings, ϕuiand ϕdi (i = 1–3) are CP-violating phases inherent in theSUSY SU(5) GUT (

∑i ϕui =

∑i ϕdi = 0). The unitary ma-

trix V is the CKM matrix in the extension of the SM tothe SUSY SU(5) GUT, and each unitary matrices U andV have only a phase. When the Majorana mass matrix forthe right handed neutrinos is real and diagonal in the basisof (5.59), U is the PMNS matrix measured in the neutrinooscillation experiments and the light neutrino mass eigen-values are given as mνi = Y 2

νi〈H2〉2/MNi , in whichMNi are

the diagonal components.The colored Higgs multiplets Hc and Hc are introduced

inH andH as SU(5) partners of the Higgs doubletsHf andHf in the MSSM, respectively, and they have new flavorviolating interactions. Equation (5.57) is represented by thefields in the MSSM as follows,

W =WMSSM+N

+ 1

2VkiYuke

iϕuk VkjQiQjHc + YuiVij eiϕdj U iEjHc+ Ydi e−iϕdi QiLiHc + e−iϕui V !ij Ydj UiDjHc

+ eiϕdi U!ij YνjDiNjHc. (5.59)

Here, the superpotential in the MSSM with the right handedneutrinos is

WMSSM+N = VjiYujQiUjHf +YdiQiDiHf +YdiLiEiHf+U!ijYνj LiNjHf +MijNiNj . (5.60)

The flavor violating interactions, which are absent in theMSSM, emerge in the SUSY SU(5) GUT due to existenceof the colored Higgs multiplets. The colored Higgs interac-tions are also baryon-number violating [150, 371], and thenproton decay induced by the colored Higgs exchange is a se-rious problem, especially in the minimal SUSY SU(5) GUT[374]. However, the constraint from the proton decay de-pends on the detailed structure in the Higgs sector, and itis also loosened by global symmetries, such as the Peccei–Quinn symmetry and the U(1)R symmetry. Thus, we mayignore the constraint from the proton decay while we adoptthe minimal Yukawa structure in (5.57).

The sfermion mass terms get sizable corrections by thecolored Higgs interactions, when the SUSY-breaking termsin the MSSM are generated by dynamics above the col-ored Higgs masses. In the minimal supergravity scenario theSUSY breaking terms are supposed to be given at the re-duced Planck mass scale (MG). In this case, the flavor vio-lating SUSY breaking mass terms at low energy are inducedby the radiative correction, and they are qualitatively givenin a flavor basis as

(m2uL

)ij −Vi3V !j3

Y 2b

(4π)2(3m2

0 +A20

)

×(

2 logM2G

M2Hc

+ logM2Hc

MSUSY2

),

(m2uR

)ij −e−iϕuij V !i3Vj3

2Y 2b

(4π)2(3m2

0 +A20

)log

M2G

M2Hc

,

(m2dL

)ij −V !3iV3j

Y 2t

(4π)2(3m2

0 +A20

)

×(

3 logM2G

M2Hc

+ logM2Hc

M2SUSY

), (5.61)

(m2dR

)ij −eiϕdij U!ikUjk

Y 2νk

(4π)2(3m2

0 +A20

)log

M2G

M2Hc

,

(m2lL

)ij −UikU!jk

f 2νk

(4π)2(3m2

0 +A20

)log

M2G

M2Nk

,

Eur. Phys. J. C (2008) 57: 13–182 87

(m2eR

)ij −eiϕdij V3iV

!3j

3Y 2t

(4π)2(3m2

0 +A20

)log

M2G

M2Hc

,

with i �= j , where ϕuij ≡ ϕui −ϕuj and ϕdij ≡ ϕdi −ϕdi andMHc is the colored Higgs mass. Here, MSUSY is the SUSY-breaking scale in the MSSM, m0 and A0 are the universalscalar mass and trilinear coupling, respectively, in the mini-mal supergravity scenario. Yt is the top quark Yukawa cou-pling constant while Yb is for the bottom quark. The off-diagonal components in the right handed squarks and slep-ton mass matrices are induced by the colored Higgs inter-actions, and they depend on the CP-violating phases in theSUSY SU(5) GUT with the right handed neutrinos [737].

One of the important features of the SUSY GUTs is thecorrelation between the leptonic and hadronic flavor viola-tions [738, 739]. From (5.62), we get a relation

(m2dR

)23 eiϕd23

(m2lL

)!23 ×

(log

M2G

M2Hc

/log

M2G

M2N3

). (5.62)

The right handed bottom-strange squark mixing may betested in the B factory experiments since it affects Bs–Bsmixing, CP asymmetries in the b–s penguin processes suchas Bd → φKs , and the mixing induced CP asymmetry inBd →Msγ . (See Chap. 2.) The relation in (5.62) impliesthat the deviations from the standard model predictions inthe b–s transition processes are correlated with B(τ → μγ )

in the SUSY SU(5) GUT. We may test the model in the Bfactories.

In Fig. 28 we show the CP asymmetry in Bd → φKs(SφKS ) and B(τ → μγ ) as an example of the correlation.Here, we assume the minimal supergravity hypothesis forthe SUSY breaking terms. See the caption and Ref. [738]for the input parameters and the details of the figure. It isfound that SφKS and B(τ → μγ ) are correlated and a largedeviation from the standard model prediction for SφKS isnot possible due to the current bound on B(τ → μγ ) in theSUSY SU(5) GUT.

In (5.62), we take the SU(5)-symmetric Yukawa interac-tions given in (5.57), while they fails to explain the fermionmass relations in the first and second generations. We haveto extend the minimal model by introducing non-trivialHiggs or matter contents or the higher-dimensional opera-tors including SU(5)-breaking Higgs field. These extensions

may affect the prediction for the sfermion mass matrices.However, the relation in (5.62) is rather robust when the neu-trino Yukawa coupling constant of the third generation is aslarge as those for the top and bottom quarks and the largemixing in the atmospheric neutrino oscillation comes fromthe lopside structure of the neutrino Yukawa coupling.

Another important feature of the SUSY GUTs is that boththe left and right handed squarks and sleptons have flavormixing terms. In this case, the hadronic and leptonic electricdipole moments (EDMs) are generated due to the flavor vi-olation, and they may be large enough to be observed in thefuture EDM measurements [740]. A diagram in Fig. 29(a)generates the electron EDM even at one loop level, whenthe relative phase between the left and right handed slep-ton mixing terms is non-vanishing. While this contributionis suppressed by the flavor violation, it is compensated bya heavier fermion mass, that is, mτ . Similar diagrams inFig. 29(b) contribute to quark EDMs and chromo-electric di-pole moments (CEDM), which induce the hadronic EDMs.

Fig. 28 B(τ→ μγ ) as a function of the CP asymmetry in Bd → φKs(SφKS ) for fixed gluino masses mg = 400, 600, 800, and 1000 GeV.Here, tanβ = 10, 200 GeV<m0 <1 TeV, A0 = 0, mντ = 5× 10−2 eV,MN3 = 5× 1014 GeV, and U32 = 1/

√2. ϕd23 is taken for the deviation

of SφKS from the SM prediction to be maximum. The current experi-mental bound on B(τ → μγ ) [647] is also shown in the figure

Fig. 29 (a) Diagrams thatgenerate electron EDM and(b) those that generate EDMsand CEDMs of the ith quarkdue to flavor violation insfermion mass terms

88 Eur. Phys. J. C (2008) 57: 13–182

Fig. 30 CEDMs for the strange (dcs ) and down quarks (dcd ) as func-tions of the right handed tau neutrino mass, MN3 . Here, MHc = 2 ×1016 GeV, mντ = 0.05 eV, U23 = 1/

√2, and U13 = 0.2, 0.02, and

0.002. For the MSSM parameters, we take m0 = 500 GeV, A0 = 0,

mg = 500 GeV and tanβ = 10. The CP phases ϕdi are taken for theCEDMs to be maximum. The upper bounds on the strange and downquark CEDMs from the mercury atom and neutron EDMs are shownin the figures

The EDM measurements are important to probe the interac-tion of the SUSY SU(5) GUT.

In Fig. 30 the CEDMs for strange (dcs ) and down quarks(dcd ) are shown as functions of the right handed tau neutrinomass in the SUSY SU(5) GUT with the right handed neu-trinos. See the caption and Ref. [741] for the input parame-ters. The mercury atom EDM, which is a diamagnetic atom,is sensitive to quark CEDMs via the nuclear force, whilethe neutron EDM depends on them in addition to the quarkEDMs. (The evaluation of the hadronic EDMs from the ef-fective operators at the parton level is reviewed in Sect. 9.1and also Ref. [742].) The strange quark contribution to themercury atom EDM is suppressed by the strange quarkmass. On the other hand, it is argued in Refs. [743, 744] thatthe strange quark component in nucleon is not negligible andthe strange quark CEDM may give a sizable contribution tothe neutron EDM. It implies that we may probe the differ-ent flavor mixings by measurements of the various hadronicEDMs, though the evaluation of the hadronic EDMs still haslarge uncertainties.

It is argued that the future measurements of neutron anddeuteron EDMs may reach to levels of ∼10−28 e cm and∼10−29 e cm, respectively. When the sensitivity of deuteronEDM is established, we may probe the new physics to thelevel of dcs ∼ 10−28 e cm and dcd ∼ dcu ∼ 10−30 e cm [849].The future measurements for the EDMs will give great im-pacts on the SUSY SU(5) GUT with the right handed neu-trinos.

5.3.2 LFV in the minimal SU(5) GUT with triplet seesaw

In this section we discuss phenomenology of the minimalSU(5) GUT which incorporates the triplet seesaw mecha-nism, previously presented in Sects. 3.2.3.2 and 5.2.7. Re-view of more general class of GUT models also including

triplet Higgs has been given in Sect. 4.1. In GUTs based onSU(5) there is no natural place for incorporating singlet neu-trinos. From this point of view SU(5) presents some advan-tage for implementing the triplet seesaw mechanism. In par-ticular, a very predictive scenarios can be obtained in the su-persymmetric case [249, 736, 745]. The triplet states T (T )fit into the 15 (15) representation, 15 = S + T + Z with S,T and Z transforming as S ∼ (6,1,− 2

3 ), T ∼ (1,3,1),Z ∼(3,2, 1

6 ) under SU(3)× SU(2)L×U(1)Y (the 15 decompo-sition is obvious). We shall briefly show that it is also pos-sible to relate not only neutrino mass parameters and LFV(as shown in Sect. 5.2.7) but also sparticle and Higgs spectraand electroweak symmetry breakdown [736, 745]. For thispurpose, consider that the SU(5)model conserves B−L, sothat the relevant superpotential reads

WSU(5) = 1√2(Y155 15 5 + λ5H 15 5H )+ Y5 5 5H 10

+ Y1010 10 5H +M55H 5H + ξX 15 15, (5.63)

where the multiplets are understood as 5 = (dc,L), 10 =(uc, ec,Q) and the Higgs doublets fit with their coloredpartners t and t , like 5H = (t,H2), 5H = (t ,H1) and X isa singlet superfield. The B − L quantum numbers are thecombination Q + 4

5Y where Y are the hypercharges andQ10 = 1

5 ,Q5 =− 35 ,Q5H (5H )

=− 25 (

25 ),Q15 = 6

5 ,Q15 = 45 ,

QX =−2. Both the scalar SX and auxiliary FX componentsof the superfield X are assumed to acquire a VEV throughsome unspecified dynamics in the hidden sector. Namely,while 〈SX〉 only breaks B −L, 〈FX〉 breaks both SUSY andB−L. These effects are parameterized by the superpotentialmass term M1515 15, where M15 = ξ 〈SX〉, and the bilinearSSB term −B15M1515 15, with B15M15 =−ξ 〈FX〉. The 15and 15 states act, therefore, as messengers of both B − Land SUSY breaking to the MSSM observable sector. Once

Eur. Phys. J. C (2008) 57: 13–182 89

SU(5) is broken to the SM group we find, below the GUTscaleMG,

W =W0 +WT +WS,Z,

W0 = YeecH1L+ YddcH1Q+ YuucQH2 +μH2H1,

WT = 1√2(YT LT L+ λH2T H2)+MT T T ,

WS,Z = 1√2YSd

cSdc + YZdcZL+MZZZ +MSSS.

(5.64)

Here, W0 denotes the MSSM superpotential,21 the termWT is responsible for neutrino mass generation [cf. (3.67)],while the couplings and masses of the colored fragmentsS and Z are included in WS,Z . It is also understood thatMT = MS = MZ ≡ M15. At the decoupling of the heavystates S,T ,Z we obtain at tree-level the neutrino masses,given by (3.68) and at the quantum level all SSB mass para-meters of the MSSM via gauge and Yukawa interactions. Atone loop level, only the trilinear scalar couplings, the gaug-ino masses and the Higgs bilinear mass term BH are gener-ated:

Ae = 3BT16π2

Ye(Y

†T YT + Y †

ZYZ), Au = 3BT

16π2|λ|2Yu,

Ad = 2BT16π2

(YZY

†Z + 2YSY

†S

)Yd, (5.65)

Ma = 7BT16π2

g2a, BH = 3BT

16π2|λ|2.

The scalar mass matrices instead are generated at the two-loop level and receive both gauge-mediated contributionsproportional to Cfa g4

a (Cfa is the quadratic Casimir of thef -particle) and Yukawa-mediated ones of the form Y

†pYp

(p = S,T ,Z). The former piece is the flavor blind contribu-tion, which is proper of the gauge-mediated scenarios [151,152, 746–751], while the latter ones constitute the flavor vi-olating contributions transmitted to the SSB terms by theYukawa’s YS,T ,Z . These contributions are mostly relevantfor the mass matrices m2

Land m2

dc. For example,

m2L=(BT

16π2

)2[21

10g4

1 +21

2g4

2 −(

27

5g2

1 + 21g22

)Y

†T YT

−(

21

15g2

1 + 9g22 + 16g2

3

)Y

†ZYZ

+ 18(Y

†T YT

)2 + 15(Y

†ZYZ

)2 + 3Tr(Y

†T YT

)Y

†T YT

+ 12Y †ZYSY

†S YZ + 3Tr

(Y

†ZYZ

)Y

†ZYZ

21This should be regarded as an effective approach where the Yukawamatrices Yd,Ye,Yu include SU(5)-breaking effects needed to repro-duce a realistic fermion spectrum.

+ 9Y †T Y

TZ Y

∗ZYT + 9

(Y

†T YT Y

†ZYZ + h.c.

)

+ 3Y †T Y

Te Y

∗e YT + 6Y †

ZYdY†d YZ

]. (5.66)

Since the flavor structure of m2L

is proportional to YT (andto YZ which is SU(5)-related to YT ), it can be expressed interms of the neutrino parameters [cf. (3.68)] and so the rel-ative size of LFV in different leptonic families is predictedaccording to the results of (5.54).

All the soft masses have the same scaling behavior m∼BT /(16π2) which demands B > 10 TeV to fulfill the natu-ralness principle. This scenario appears very predictive sinceit contains only three free parameters: the triplet mass MT ,the effective SUSY breaking scale BT and the coupling con-stant λ. The parameter space is then constrained by the ex-perimental bounds on the Higgs boson mass, the B(μ→eγ ), the sfermion masses, and the requirement of radia-tive electroweak symmetry breaking. The phenomenologi-cal predictions more important and relevant for LHC, theB-factories [694] the incoming MEG experiment [548], theSuper Flavor factory [752] or the PRISM/PRIME exper-iment at J-PARC [753], concern the sparticle and Higgsboson spectra and the LFV decays. Regarding the spec-trum, the gluino is the heaviest sparticle while, in mostof the parameter space, �1 is the lightest. In the exam-ple shown in Fig. 31 the squark and slepton masses lie inthe ranges 700–950 GeV and 100–300 GeV, respectively.The gluino mass is about 1.3 TeV. The chargino massesare mχ±1

∼ 320 GeV and mχ±2∼ 450–550 GeV. Moreover,

mχ01∼ 190 GeV, mχ0

2≈mχ±1 and mχ0

3,4≈mχ±2 . These mass

ranges are within the discovery reach of the LHC.The Higgs sector is characterized by a decoupling regime

with a light SM-like Higgs boson (h) with mass in the range110–120 GeV which is testable in the near future at LHC(mainly through the decay into 2 photons). The remainingthree heavy states (H,A and H±) have mass mH,A,H± ≈450–550 GeV (again, for BT = 20 TeV). All the spectra in-crease almost linearly with BT .

Figure 32(b) shows instead several LFV processes: μ→eX,μ→ e conversion in nuclei, τ→ eY and τ→ μY (X =γ, ee, Y = γ, ee,μμ). One observe that e.g., the behav-ior of the radiative-decay branching ratios is in agreementwith the estimates given in (5.56) for θ13 = 0. For θ13 = 0.2one obtains instead that B(τ → μγ )/B(μ→ eγ ) ∼ 2 andB(τ → eγ )/B(μ→ eγ ) ∼ 0.1 (the full analysis can befound in Ref. [736]). The other LFV processes shown arealso correlated to the radiative ones in a model-independentway [736, 745]. The analysis shows that the future experi-mental sensitivity will allow to measure at most B(μ→ γ ),B(μ→ 3e), B(τ → μγ ) and CR(μ→ e Ti) for tiny θ13. Inparticular, being B(τ → μγ )/B(μ→ eγ ) ∼ 300, B(τ →μγ ) is expected not to exceed 3 × 10−9, irrespective ofthe type of neutrino spectrum. Therefore τ → μγ falls into

90 Eur. Phys. J. C (2008) 57: 13–182

Fig. 31 (Color online) Sparticle and Higgs spectrum for BT =20 TeV. Left panel: squark masses, mu (black solid line), m

d(red

dashed) and the gluino mass (blue dash-dotted). In the inner plot tanβand μ are shown as obtained by the electroweak symmetry breaking

conditions. Right panel: the masses of the charged sleptons, the sneu-trinos, the charginos, the neutralinos and the Higgs bosons as the labelsindicate

the LHC capability. All the decays τ → �i�k�k would haveB < O(10−11). The predictions for the LFV branching ra-tios in the present scenario are summarized in Table 13.

Finally, such supersymmetric SU(5) framework with a15, 15 pair may be realized in contexts based on string in-spired constructions [754, 755].

5.3.3 LFV from a generic SO(10) framework

The spinorial representation of the SO(10), given by a 16-dimensional spinor, can accommodate all the SM model par-ticles as well as the right handed neutrino. As discussedin Sect. 4.1, the product of two 16 matter representationscan only couple to 10, 120 or 126 representations, whichcan be formed by either a single Higgs field or a non-renormalizable product of representations of several Higgsfields. In either case, the Yukawa matrices resulting from thecouplings to 10 and 126 are complex symmetric, whereasthey are antisymmetric when the couplings are to the 120.Thus, the most general SO(10) superpotential relevant tofermion masses can be written as

WSO(10) = Y 10ij 16i 16j 10 + Y 126

ij 16i 16j 126

+ Y 120ij 16i 16j 120, (5.67)

where i, j refer to the generation indices. In terms of the SMfields, the Yukawa couplings relevant for fermion masses are

given by [756, 757]:

16 16 10 ⊃ 5(uuc + ννc)+ 5

(ddc + eec),

16 16 126 ⊃ 1νc νc + 15νν + 5(uuc − 3ννc

)

+ 45(ddc − 3eec

),

16 16 120 ⊃ 5ννc + 45uuc + 5(ddc + eec)

+ 45(ddc − 3eec

),

(5.68)

where we have specified the corresponding SU(5) Higgsrepresentations for each of the couplings and all the fermi-ons are left handed fields. The resulting up-type quarks andneutrinos’ Dirac mass matrices can be written as

mu =M510 +M5

126 +M45120, (5.69)

mνD =M510 − 3M5

126 +M5120. (5.70)

A simple analysis of the fermion mass matrices in theSO(10)model, as detailed in (5.70) leads us to the followingresult: At least one of the Yukawa couplings in Y ν = v−1

u mνD

has to be as large as the top Yukawa coupling [164]. Thisresult holds true in general, independently of the choice ofthe Higgses responsible for the masses in (5.69), (5.70), pro-vided that no accidental fine-tuned cancellations of the dif-ferent contributions in (5.70) are present. If contributionsfrom the 10’s solely dominate, Y ν and Yu would be equal.If this occurs for the 126’s, then Y ν =−3Yu [395]. In caseboth of them have dominant entries, barring a rather pre-cisely fine-tuned cancellation between M5

10 and M5126 in

(5.70), we expect at least one large entry to be present in Y ν .A dominant antisymmetric contribution to top quark mass

Eur. Phys. J. C (2008) 57: 13–182 91

Fig. 32 Branching ratios of several LFV processes as a function of λ.The left (right) vertical line indicates the lower bound on λ imposed byrequiring perturbativity of the Yukawa couplings YT,S,Z when m1 = 0

(0.3) eV [normal-hierarchical (quasi-degenerate) neutrino mass spec-trum]. The regions in green (grey) are excluded by the m

�1> 100 GeV

constraint (perturbativity requirement when m1 = 0)

Table 13 Expectations for thevarious LFV processesassumingB(μ→ eγ )= 1.2 × 10−11. Theresults in parenthesis apply tothe case of theinverted-hierarchical neutrinospectrum, whenever these aredifferent from those obtained forthe normal-hierarchical andquasi-degenerate ones

Decay mode Prediction for branching ratio

s13 = 0 s13 = 0.2

τ− → μ−γ 3 × 10−9 2(3)× 10−11

τ− → e−γ 2 × 10−12 1(3)× 10−12

μ− → e−e+e− 6 × 10−14 6 × 10−14

τ− → μ−μ+μ− 7 × 10−12 4(6)× 10−14

τ− → μ−e+e− 3 × 10−11 2(3)× 10−13

τ− → e−e+e− 2 × 10−14 1(3)× 10−14

τ− → e−μ+μ− 3 × 10−15 2(4)× 10−15

μ→ e; Ti 6 × 10−14 6 × 10−14

92 Eur. Phys. J. C (2008) 57: 13–182

due to the 120 Higgs is phenomenologically excluded, sinceit would lead to at least a pair of heavy degenerate up quarks.Apart from sharing the property that at least one eigenvalueof bothmu andmνD has to be large, for the rest it is clear from(5.69) and (5.70) that these two matrices are not alignedin general, and hence we may expect different mixing an-gles appearing from their diagonalization. This freedom isremoved if one sticks to particularly simple choices of theHiggses responsible for up quark and neutrino masses.

Therefore, we see that the SO(10) model with only twoten-plets would inevitably lead to small mixing in Y ν . Infact, with two Higgs fields in symmetric representations,giving masses to the up sector and the down sector sep-arately, it would be difficult to avoid the small CKM-likemixing in Y ν . We shall call this case the CKM case. Fromhere, the following mass relations hold between the quarkand leptonic mass matrices at the GUT scale:22

Yu = Y ν; Yd = Y e. (5.71)

In the basis where charged lepton masses are diagonal, wehave

Y ν = V TCKMYuDiagVCKM. (5.72)

The large couplings in Y ν ∼ O(Yt ) induce significant off-diagonal entries in m2

Lthrough the RG evolution between

MGUT and the scale of the right handed Majorana neutrinos,MRi . The induced off-diagonal entries relevant to lj → li , γ

are of the order of:

(m2L

)i �=j ≈−3m2

0 +A20

8π2Y 2t VtiVtj ln

MGUT

MR3

+ O(Y 2c

),

(5.73)

where Vij are elements of VCKM, and i, j flavor indices. Inthis expression, the CKM angles are small but one wouldexpect the presence of the large top Yukawa coupling tocompensate such a suppression. The required right handedneutrino Majorana mass matrix, consistent with both the ob-served low energy neutrino masses and mixing as well aswith CKM-like mixing in Y ν is easily determined from theseesaw formula defined at the scale of right handed neutri-nos.

The B(li → lj γ ) are now predictable in this case. Con-sidering mSUGRA boundary conditions and taking tanβ =40, we obtain that reaching a sensitivity of O(10−13–10−14), as planned by the MEG experiment at PSI, forB(μ→ eγ ) would allow us to probe the SUSY spectrum

22Clearly this relation cannot hold for the first two generations of downquarks and charged leptons. One expects, small corrections due to non-renormalizable operators or suppressed renormalizable operators [6] tobe invoked.

Fig. 33 Contour plot of B(μ→ e, γ ) in the plane of the GUT-scaleuniversal scalar and gaugino masses, (m0,M1/2), at A0 = 0 in theCKM high tanβ case. Note that while the plane is presently uncon-strained, the planned MEG experiment sensitivity of O(10−13–10−14)

will be able to probe it in the (m0,mg)� 1 TeV region

completely up to m0 = 1200 GeV,M1/2 = 400 GeV (noticethat this corresponds to gluino and squark masses of order1 TeV). This clearly appears from Fig. 33, which shows theB(μ→ eγ ) contour plot in the (m0,M1/2) plane. Thus, insummary, though the present limits on B(μ→ e, γ ) wouldnot induce any significant constraints on the supersymmetrybreaking parameter space, an improvement in the limit to∼O(10−13–10−14), as foreseen, would start imposing non-trivial constraints especially for the large tanβ region.

To obtain mixing angles larger than CKM angles, asym-metric mass matrices have to be considered. In general, it issufficient to introduce asymmetric textures either in the upsector or in the down sector. In the present case, we assumethat the down sector couples to a combination of Higgs rep-resentations (symmetric and antisymmetric)23 Φ , leading toan asymmetric mass matrix in the basis where the up sectoris diagonal. As we shall see below, this would also requirethat the right handed Majorana mass matrix be diagonal inthis basis. We have:

WSO(10) = 1

2Yu,νii 16i 16i 10u + 1

2Yd,eij 16i 16jΦ

+ 1

2YRii 16i 16i 126,

where the 126, as before, generates only the right handedneutrino mass matrix. To study the consequences of theseassumptions, we see that at the level of SU(5), we have

23The couplings of the Higgs fields in the superpotential can beeither renormalizable or non-renormalizable. See [758] for a non-renormalizable example.

Eur. Phys. J. C (2008) 57: 13–182 93

WSU(5) = 1

2Yuii10i 10i 5u + Y νii 5i 1i 5u

+ Ydij10i 5j 5d + 1

2MRii 1i 1i,

where we have decomposed the 16 into 10 + 5 + 1 and 5u

and 5d are components of 10u and Φ respectively. To havelarge mixing ∼UPMNS in Y ν we see that the asymmetric ma-trix Yd should now give rise to both the CKM mixing as wellas PMNS mixing. This is possible if

V TCKMYdUTPMNS = YdDiag. (5.74)

Therefore the 10 that contains the left handed down-quarks would be rotated by the CKM matrix whereas the5 that contains the left handed charged leptons would be ro-tated by the UPMNS matrix to go into their respective massbases [737, 758–760]. Thus we have the following relationsin the basis where charged leptons and down quarks are di-agonal:

Yu = VCKMYuDiagV

TCKM, (5.75)

Y ν = UPMNSYuDiag. (5.76)

Using the seesaw formula of (3.41) and (5.76), we have

MR = Diag

{m2u

mν1

,m2c

mν2

,m2t

mν3

}. (5.77)

We now turn our attention to lepton flavor violation in thiscase. The branching ratio, B(μ→ e, γ ) would now dependon

[Y νY νT

]21 = Y 2

t Uμ3Ue3 + Y 2c Uμ2Ue2 + O

(h2u

). (5.78)

It is clear from the above that in contrast to the CKM case,the dominant contribution to the off-diagonal entries de-pends on the unknown magnitude of the element Ue3 [761].If Ue3 is close to its present limit ∼0.14 [762] (or at leastlarger than (Y 2

c /Y2t )Ue2 ∼ 4 × 10−5), the first term on the

RHS of the (5.78) would dominate. Moreover, this wouldlead to large contributions to the off-diagonal entries in theslepton masses with Uμ3 of O(1). Thus, we have

(m2L

)21 ≈−3m2

0 +A20

8π2Y 2t Ue3Uμ3 ln

MGUT

MR3

+ O(Y 2c

).

(5.79)

This contribution is larger than the CKM case by a factorof (Uμ3Ue3)/(VtdVts)∼ O(102). This would mean about afactor 104 times larger than the CKM case in B(μ→ e, γ ).Such enhancement with respect to the CKM case is clearlyshown in the scatter plots of Fig. 34, where the CKM case iscompared with the PMNS case with Ue3 = 0.07. The aimof the figure is to show the capability of MEG to probethe region of mSUGRA parameter space accessible to theLHC. In fact, the plots show the value of B(μ→ e, γ )

obtained by scanning the parameter space in the large re-gion (0<m0 < 5 TeV, 0<M1/2 < 1.5 TeV, −3m0 <A0 <

+3m0, sign(μ)), and then keeping the points which give atleast one squark lighter than 2.5 TeV (so roughly accessibleto the LHC). We see that in this “LHC accessible” regionthe maximal case (with Ue3 = 0.07) is already excluded bythe MEGA limit (B(μ→ e, γ ) < 1.2 × 10−11), and there-fore MEG will constrain the parameter space far beyond theLHC sensitivity for this case. If Ue3 is very small, i.e. eitherzero or �(Y 2

c /Y2t )Ue2 ∼ 4× 10−5, the second term ∝ Y 2

c in(5.78) would dominate, thus giving a strong suppression tothe branching ratio. This could be not true, once RG effects

Fig. 34 Scatter plots of B(μ→ e, γ ) (left) and B(τ → μγ ) (right)versus M1/2 for tanβ = 40, both for the (maximal) PMNS case with|Ue3| = 0.07 and the (minimal) CKM case. The plots were obtained by

scanning the SUSY parameter space in the LHC accessible region (seethe text for details)

94 Eur. Phys. J. C (2008) 57: 13–182

on Ue3 itself [258, 722] are taken into account. The pointis that the PMNS boundary condition (5.76) is valid at highscale. Thus, it is necessary to evolve the neutrino masses andmixing from the low energy scale, where measurements areperformed, up to high energy. Such effect turns out to be notnegligible in case of low energy Ue3 � 10−3, giving a highenergy constant enhancement of order O(10−3) [688]. Theconsequence is that the term in (5.78) ∝ Y 2

t always dom-inates, giving a contribution to the branching ratio largerthan the CKM case (which turns out to be really a “mini-mal” case) and bringing the most of the parameter space inthe realm of MEG even for very small low energy values ofUe3 [688, 763].

The τ → μ transitions are instead Ue3-independentprobes of SUSY, whose importance was first pointed outin Ref. [732]. The off-diagonal entry in this case is givenby:

(m2L

)32 ≈−3m2

0 +A20

8π2Y 2t Uμ3Uτ3 ln

MGUT

MR3

+ O(Y 2c

).

(5.80)

In the τ → μγ decay the situation is at the moment simi-larly constrained with respect to μ→ eγ , if Ue3 happens tobe very small. The main difference is that B(τ → μγ ) doesnot depend on the value of Ue3, so that τ → μγ will be apromising complementary channel with respect to μ→ eγ .As far as Beauty factories [191, 644, 764] are concerned,we see from Fig. 34, that even with the present bound itis possible to rule out part of LHC accessible region in thePMNS high tanβ regime; the planned accuracy of the Su-perKEKB [694] machine ∼O(10−8) will allow to test muchof high tanβ region and will start probing the low tanβPMNS case, with a sensitivity to soft masses as high as(m0,mg)� 900 GeV. The situation changes dramatically ifone takes into account the possibility of a Super Flavor fac-tory: taking the sensitivity of the most promising τ → μγ

process to ∼O(10−9), the PMNS case will be nearly ruledout in the high tanβ regime and severely constrained in thelow tanβ one; as for the CKM case we would enter the re-gion of interest.

Let’s finish with some remarks. Suppose that the LHCdoes find signals of low energy supersymmetry, then grandunification becomes a very appealing scenario, because ofthe successful unification of gauge couplings driven by theSUSY partners. Among SUSY-GUT models, an SO(10)framework is much favored as it is the ‘minimal’ GUT tohost all the fermions in a single representation and it ac-counts for the smallness of the observed neutrino masses bynaturally including the see-saw mechanism. In the above wehave addressed the issue by a generic benchmark analysis,within the ansatz that there is no fine-tuning in the neutrinoYukawa sector. We can state that LFV experiments should

be able to tell us much about the structure of such a SUSY-GUT scenario. If they detect LFV processes, by their rateand exploiting the interplay between different experiments,we would be able to get hints of the structure of the un-known neutrinos’ Yukawa’s. On the contrary, in the case thatboth MEG and a future Super Flavor factory happen not tosee any LFV process, only two possibilities should be left:(i) the minimal mixing, low tanβ scenario; (ii) mSUGRASO(10) see-saw without fine-tuned Yν couplings is not a vi-able framework of physics beyond the standard model.

Actually one should remark that LFV experiments willbe able to falsify some of above scenarios even in regionsof the mSUGRA parameter space that are beyond the reachof LHC experiments. In this sense, the power of LFV ex-periments of testing/discriminating among different SUSY-GUTs models results very interesting and highly comple-mentary to the direct searches at the LHC.

5.3.4 LFV, QFV and CPV observables in GUTs and theircorrelations

In a SUSY grand unified theory (GUT), quarks and lep-tons sit in same multiplets and are transformed ones intothe others through GUT symmetry transformations. If theenergy scale where the SUSY breaking terms are transmit-ted to the visible sector is larger then the GUT scale, as inthe case of gravity mediation, such breaking terms, and inparticular the sfermion mass matrices, will have to respectthe underlying GUT symmetry. Hence, as already discussedin Sect. 5.3.1, the quark–lepton unification seeps also intothe SUSY breaking soft sector. If the soft SUSY breakingterms respect boundary conditions which are subject to theGUT symmetry to start with, we generally expect the pres-ence of relations among the (bilinear and trilinear) scalarterms in the hadronic and leptonic sectors [165, 739]. Suchrelations hold true at the (superlarge) energy scale where thecorrect symmetry of the theory is the GUT symmetry. Afterits breaking, the mentioned relations will undergo correc-tions which are computable through the appropriate RGE’swhich are related to the specific structure of the theory be-tween the GUT and the electroweak scale (for instance, newYukawa couplings due to the presence of RH neutrinos act-ing down to the RH neutrino mass scale, presence of a sym-metry breaking chain with the appearance of new symme-tries at intermediate scales, etc.). As a result of such a com-putable running, we can infer the correlations between thesoftly SUSY breaking hadronic and leptonic MIs at the lowscale where FCNC tests are performed. Moreover, given thata common SUSY soft breaking scalar term of Lsoft at scalesclose to MPlanck can give rise to RG-induced δq ’s and δl’sat the weak scale, one may envisage the possibility to makeuse of the FCNC constraints on such low energy δ’s to inferbounds on the soft breaking parameters of the original su-pergravity Lagrangian (Lsugra). Indeed, for each scalar soft

Eur. Phys. J. C (2008) 57: 13–182 95

Table 14 Links betweenvarious transitions betweenup-type, down-type quarks andcharged leptons for SU(5). m2

f

refers to the average mass forthe sfermion f ,

m2Qavg

=√m2Qm2dc

and

m2Lavg

=√m2Lm2ec

Relations at weak-scale Boundary conditions atMGUT

(δuij )RR ≈ (m2ec/m2

uc)(δlij )RR m2

uc(0)=m2

ec(0)

(δqij )LL ≈ (m2

ec/m2

Q)(δlij )RR m2

Q(0)=m2

ec(0)

(δdij )RR ≈ (m2L/m2

dc)(δlij )LL m2

dc(0)=m2

L(0)

(δdij )LR ≈ (m2Lavg/m2

Qavg)(mb/mτ )(δ

lij )!LR Aeij =Adji

parameter of Lsugra one can ascertain whether the hadronicor the leptonic corresponding bound at the weak scale yieldsthe strongest constraint at the large scale [165].

Let us consider the scalar soft breaking sector of theMSSM:

−Lsoft = m2QiiQ

†i Qi +m2

uciiuc!

i uci +m2

eciiec!

i eci

+m2dciidc!

i dci +m2

LiiL

†i Li +m2

H1H

†1H1

+m2H2H

†2H2 +Auij Qi ucjH2 +Adij Qi dcjH1

+Aeij Li ecjH1 +(Δlij)LLL

†i Lj +

(Δeij)RRec!

i ecj

+ (Δqij)LLQ

†i Qj +

(Δuij)RRuc!

i ucj

+ (Δdij)RRdc!

i dcj +

(Δeij)LReL!i ecj

+ (Δuij)LRuL!i ucj +

(Δdij)LRdL!

i dcj + · · · ,

(5.81)

where we have explicitly written down the various off-diagonal entries of the soft SUSY breaking matrices. Con-sider now that SU(5) is the relevant symmetry at the scalewhere the above soft terms firstly show up. Then, taking intoaccount that matter is organized into the SU(5) representa-tions 10 = (q,uc, ec) and 5 = (l, dc), one obtains the fol-lowing relations

m2Q =m2

ec=m2

uc=m2

10,

m2dc

=m2L =m2

5, (5.82)

Aeij =Adji .These equations for matrices in flavor space lead to relationsbetween the slepton and squark flavor violating off-diagonalentries Δij . These are:

(Δuij)LL

= (Δuij)RR

= (Δdij)LL

= (Δlij)RR, (5.83)

(Δdij)RR

= (Δlij)LL, (5.84)

(Δdij)LR

= (Δlji)LR

= (Δlij)!RL. (5.85)

These GUT correlations among hadronic and leptonic scalarsoft terms are summarized in the second column of Table 14.

Table 15 Links between various transitions between up-type, down-type quarks and charged leptons for PS/SO(10) type models

Relations at weak-scale Boundary conditions atMGUT

(δuij )RR ≈ (m2ec/m2

uc)(δlij )RR m2

uc(0)=m2

ec(0)

(δqij )LL ≈ (m2

L/m2

Q)(δlij )LL m2

Q(0)=m2

L(0)

Assuming that no new sources of flavor structure are presentfrom the SU(5) scale down to the electroweak scale, apartfrom the usual SM CKM one, one infers the relations in thefirst column of Table 14 at low scale. Here we have takeninto account that due to their different gauge couplings “av-erage” (diagonal) squark and slepton masses acquire differ-ent values at the electroweak scale.

Two comments are in order when looking at Table 14.First, the boundary conditions on the sfermion masses at theGUT scale (last column in Table 14) imply that the squarkmasses are always going to be larger at the weak scale com-pared to the slepton masses due to the participation of theQCD coupling in the RGEs. As a second remark, noticethat the relations between hadronic and leptonic δ MI in Ta-ble 14 always exhibit opposite “chiralities”, i.e. LL inser-tions are related to RR ones and vice-versa. This stems fromthe arrangement of the different fermion chiralities in SU(5)five- and ten-plets (as it clearly appears from the final col-umn in Table 14). This restriction can easily be overcome ifwe move from SU(5) to left-right symmetric unified mod-els like SO(10) or the Pati–Salam (PS) case (we exhibit thecorresponding GUT boundary conditions and δ MI at theelectroweak scale in Table 15).

So far we have confined the discussion within the sim-ple SU(5) model, without the presence of any extra parti-cles like right handed (RH) neutrinos. In the presence of RHneutrinos, one can envisage of two scenarios [164]: (a) witheither very small neutrino Dirac Yukawa couplings and/orvery small mixing present in the neutrino Dirac Yukawa ma-trix, (b) Large Yukawa and large mixing in the neutrino sec-tor. In the latter case, (5.83)–(5.85) are not valid at all scalesin general, as large RGE effects can significantly modify thesleptonic flavor structure while keeping the squark sectoressentially unmodified; thus essentially breaking the GUTsymmetric relations. In the former case where the neutrinoDirac Yukawa couplings are tiny and do not significantly

96 Eur. Phys. J. C (2008) 57: 13–182

Fig. 35 Left four panels: allowed region for (δd23)RR using constraints as indicated. Right four panels: the same for (δd12)RR . For the parameterspace considered, please see the text

modify the sleptonic flavor structure, the GUT symmetricrelations are expected to be valid at the weak scale. How-ever, in both cases it is possible to say that there exists abound on the hadronic δ parameters of the form [739]:

∣∣(δdij)RR

∣∣≥m2L

m2dc

∣∣(δlij)LL

∣∣. (5.86)

The situation is different if we try to translate the bound fromquark to lepton MIs. An hadronic MI bound at low energyleads, after RGE evolution, to a bound on the correspond-ing grand unified MI atMGUT, applying both to slepton andsquark mass matrices. However, if the neutrino Yukawa cou-plings have sizable off-diagonal entries, the RGE runningfrom MGUT to MW could still generate a new contributionto the slepton MI that exceeds this GUT bound. Thereforehadronic bounds cannot be translated to leptons unless wemake some additional assumptions on the neutrino Yukawamatrices. On general grounds, given that SM contributionsin the lepton sector are absent and that the branching ratiosof leptonic processes constrain only the modulus of the MIs,it turns out that all the MI bounds arising from the leptonsector are circles in the Re(δdij )AB–Im(δdij )AB plane and arecentered at the origin.

In the following the effect of leptonic bounds on thequark mass insertions are reviewed, following the resultspresented in [165], where constraints on δs were stud-ied scanning the mSUGRA parameter space in the ranges:M1/2 ≤ 160 GeV, m0 ≤ 380 GeV, |A0| ≤ 3m0 and 5 <tanβ < 15. For instance, in presence of a (Δd23)LR at the

GUT scale, this would have effects both in the τ → μγ

and b→ sγ decays. Using (δd23)LR � (mb/mτ )(m2l/m2

q)×

(δl23)RL, a bound on (δl23)RL from the τ→ μγ decay trans-lates into a bound on (Δd23)LR (neglecting the effects ofneutrino Yukawa’s the inequality transforms into equality).Thus, leptonic processes set a bound on the SUSY contribu-tions to B(B→Xsγ ). However, it turns out that the presentleptonic bounds have no effect on the (δd23)LR couplings.This is due both to the existence of strong hadronic boundsfrom b→ sγ and CP asymmetries and to the relatively weakleptonic bounds here.

Similarly, in presence of a (Δd23)RR at the GUT scale,the corresponding MIs at the electroweak scale are (δd23)RRand (δl23)LL that contribute to �MBs and τ → μγ respec-tively (the impact of (Δd23)RR on b→ sγ and b→ s�+�− isnot relevant because of the absence of interference betweenSUSY and SM contributions). In Fig. 35 the allowed valuesof Re(δd23)RR and Im(δd23)RR with the different constraintsare shown. The leptonic constraints are quite effective asthe bound on the B(τ → μγ ) from B-factories is alreadyvery stringent, while the recent measurement of �MBs isless constraining. The plots correspond to 5 < tanβ < 15,thus, the absolute bound on (δl23)LL is set by tanβ = 5 andit scales with tanβ as (δl23)LL ∼ (5/ tanβ).24

24Sizable SUSY contributions to �MBs are still possible from theHiggs sector in the large tanβ regime both within [765–767] and alsobeyond [768] the Minimal Flavor Violating framework. However, forthe considered parameter space, the above effects are completely neg-ligible.

Eur. Phys. J. C (2008) 57: 13–182 97

As in the LR sector, in the LL one, there is no apprecia-ble improvement from the inclusion of leptonic constraints.In fact, τ→ μγ is not effective to constrain (δl23)RR , i.e. theleptonic MI related to (δd23)LL in this SUSY-GUTs scheme,in large portions of the parameter space because of strongcancellations among amplitudes. The analysis of the con-straints on the different (δd13) MIs gives similar results tothat of the (δd23) MIs. In this case, the hadronic constraintscome mainly from �MBd and the different CP asymmetriesmeasured at B-factories, while the leptonic bounds are dueto the decay τ→ eγ .

Coming to the 1–2 sector, let’s see, as an example, theallowed values of Re(δd12)RR and Im(δd12)RR . In this case, asit appears from Fig. 35, leptonic constraints, already usingthe present limit on B(μ→ eγ ), are competitive and con-strain the direction in which the constraint from εK is noteffective (upper left plot). Similarly in the LR sector, evenif the hadronic bounds coming from ε′/ε are quite stringent,the bounds from μ→ eγ are even more effective, while theLL sector results less constrained by leptonic processes, asan effect of the cancellations that μ→ eγ decay suffers inthe RR leptonic sector.

5.4 R-parity violation

5.4.1 Introduction

In supersymmetric extensions of the standard model (SM),baryon and lepton numbers are no longer automatically pro-tected. This is the main reason for introducing R-parity.R-parity is associated with a Z2 subgroup of the group ofcontinuous U(1) transformations acting on the gauge su-perfields and the two chiral doublet Higgs superfields Hdand Hu, with their definition extended to quark and lep-ton superfields so that quarks and leptons carry R = 0and squarks and sleptons R = ±1. One can express R-parity in terms of spin S, baryon B and lepton L num-ber [769]:

R-parity = (−1)2S (−1)3(B−L). (5.87)

Taking into account the important phenomenological dif-ferences between models with and without R-parity, it isworth studying if and how R-parity can be broken. Oneof the main reasons to introduce R-parity is avoiding pro-ton decay. However there are in principle other discrete orcontinuous symmetries that can protect proton decay whileallowing for some R-parity violating couplings. In the ab-sence of R-parity, R-parity odd terms allowed by renormal-izability and gauge invariance [150] must be included inthe superpotential of the minimal supersymmetric standardmodel,

WRp = μiHuLi + 1

2λijkLiLjE

ck + λ′ijkLiQjDck

+ 1

2λ′′ijkUci D

cjD

ck, (5.88)

where there is summation over the generation indicesi, j, k = 1,2,3, and summation over gauge indices is un-derstood. One has for example LiLjEck ≡ (εabLai Lbj )Eck =(NiEj − EiNj )E

ck and Uci D

cjD

ck ≡ εαβγ U

αci D

βcj D

γc

k ,where a, b = 1,2 are SU(2)L indices, α,β, γ = 1,2,3 areSU(3)C indices, and εab and εαβγ are totally antisymmetrictensors (with ε12 = ε123 = +1). Gauge invariance enforcesantisymmetry of the λijk couplings in their first two indicesand antisymmetry of the λ′′ijk couplings in their last two in-dices,

λijk =−λjik, (5.89)

λ′′ijk =−λ′′ikj . (5.90)

The bilinear terms μiHuLi in (5.88) can be rotated awayfrom the superpotential upon suitably redefining the leptonand Higgs superfields. However, in the presence of genericsoft supersymmetry breaking terms of dimension two, bilin-ear R-parity violation will reappear. The fact that one canmake μi = 0 in (5.88) does not mean that the Higgs–leptonmixing associated with bilinear R-parity breaking is unphys-ical, but rather that there is not a unique way of parameteriz-ing it. If R-parity is violated in the leptonic sector, no quan-tum numbers differentiate between lepton and Higgs super-fields, and they consequently mix with each other [770].The R-parity violation in the baryonic sector does not implylepton flavor violation, and we do not consider such optionhere.

A general consequence of R-parity violation is that unlessthe relevant couplings are negligibly small, the supersym-metric model does not have a dark matter candidate. Thusexperimental studies on dark matter will also shed light onR-parity violation.

5.4.2 Limits on couplings

Limits on R-parity violating couplings can be obtained bydirect searches at colliders or requiring that the R-parity vi-olating contribution to a given observable does not exceedthe limit imposed by the precision of the experimental mea-surement.

On the collider side R-parity violation implies the pos-sibility of the creation, decay or exchange of single sparti-cles, thus allowing new decay channels. For example, evenfor relatively small R-parity violating interactions, the decayof the lightest supersymmetric particle will lead to colliderevents departing considerably from the characteristic miss-ing momentum signal of R-parity conserving theories. In ab-sence of definite theoretical predictions for the values of the

98 Eur. Phys. J. C (2008) 57: 13–182

45 independent trilinear Yukawa couplingsΛ (λijk , λ′ijk andλ′′ijk), it is necessary in practice to assume a strong hierar-chy among the couplings. A simplifying assumption widelyused for the search at colliders is to postulate the existenceof a single dominant R-parity violating coupling. When dis-cussing specific bounds, it is necessary to choose a definitebasis for quark and lepton superfields. Often it is understoodthat the single coupling dominance hypothesis applies in themass eigenstate basis. It can be more natural to apply thishypothesis in the weak eigenstate basis when dealing withmodels in which the hierarchy among couplings originatesfrom a flavor theory. In this case, a single process allowsone to constrain several couplings, provided one has someknowledge of the rotations linking the weak eigenstate andmass eigenstate bases. Indirect bounds from loop processestypically lead to bounds on the products of two most impor-tant R-parity violating couplings, or on the sum of productsof two couplings. The limits on single dominant couplings,and on products of couplings, as well as a more completelist of references, are collected in [771].

5.4.3 Spontaneous R-parity breaking

The spontaneous breaking of R-parity is characterized byan R-parity invariant Lagrangian leading to non-vanishingVEVs for some R-parity odd scalar field, which in turn gen-erates R-parity violating terms. Such a spontaneous break-down of R-parity generally also entails the breaking ofthe global U(1) lepton number symmetry L which im-plies the existence of a massless Nambu–Goldstone realpseudoscalar boson J , the majoron. Another light scalarparticle, denoted ρ, generally accompanies the majoronin the supersymmetric models. If the U(1) symmetry isalso explicitly broken by interaction terms in the La-grangian, both of these particles acquire finite masses. Themost severe constraints on the models with a spontaneousR-parity breaking, arise in the cases where the majoron car-ries electroweak gauge charges and hence is coupled to theZ bosons and to quarks and leptons. The non-singlet compo-nents contribute to theZ boson invisible width by an amountof one-half that a single light neutrino, δΓ Zinv/6 83 MeV.To suppress the non-singlet components one must allow ei-ther for sufficiently small sneutrino VEVs, vL/MZ � 1, orfor some large hierarchy of scales between vL and the VEVparameters associated with additional electroweak singletscalar fields [772].

However, it is not necessary that models with sponta-neous R-parity violation have a majoron. Models withouta majoron include a class of models with triplet Higgses,where B − L is a gauge symmetry, which is necessarilyspontaneously broken unless effects of non-renormalizableterms or some additional new fields are included [773]. Aninteresting experimental signal in these models may be a

relatively light doubly charged scalar, which decays domi-nantly to same charge leptons (not necessarily of the samegeneration) [774]. Another possibility for a model with-out a majoron is a model where the lepton number is bro-ken by two units explicitly, in which case the spontaneousbreaking by one unit (which breaks the R-parity) does notlead to a majoron [775]. The interactions in spontaneouslyR-parity breaking models through the lepton number vio-lation closely resemble explicitly R-parity breaking modelswith only bilinear R-parity violation. In the case of sponta-neous breaking, the parameters which are free in the modelwith only bilinear couplings are related to each other viathe sneutrino vacuum expectation value (VEV). Thus a con-straint from one process affects availability of the otherprocesses. Example bounds for such a model can be foundin [776].

It is worth emphasizing that choosing single couplingdominance in the case of spontaneous breaking is not possi-ble and in this sense, the models with spontaneous breakingare more predictive than those without.

5.4.4 Neutrino sector

The presence of non-zero couplings λijk , λ′ijk or bilinearR-parity violating parameters implies the generation of neu-trino masses and mixing [777]. This is an interesting featureof R-parity violating models, but it can also be a problem,since the contribution of R-parity violating couplings mayexceed by orders of magnitude the experimental bounds.Two types of contributions can be distinguished: tree-levelor loop contributions.

The tree-level contributions are due to bilinear R-parityviolation terms which induce a mixing between neutrinosand neutralinos [778]. This gives a massive neutrino state attree level. When quantum corrections are included, all threeneutrinos acquire a mass. The tree-level contribution arisingfrom the neutrino–neutralino mixing can be understood, inthe limit of small neutrino–neutralino mixing, as a sort ofseesaw mechanism, in which the neutral gauginos and hig-gsinos play the role of the right handed neutrinos.

The loop contributions are induced by the trilinearR-parity violating couplings λijk and λ′ijk and by bilinearR-parity violating parameters [779]. If bilinear R-parity vio-lation is strongly suppressed one can concentrate on the dia-grams involving trilinear R-parity violating couplings only.The trilinear couplings λijk and λ′ijk contribute to each en-try of the neutrino mass matrix through the lepton–sleptonand quark–squark loops. The neutrino mass matrix dependstherefore on a large number of trilinear R-parity violatingcouplings. In order to obtain a predictive model, one hasto make assumptions on the structure of the trilinear cou-plings. In general, however, the bilinear R-parity violationcontribution cannot be neglected. The presence of bilinear

Eur. Phys. J. C (2008) 57: 13–182 99

terms drastically modifies the calculation of one loop neu-trino masses. The neutrino mass matrix receives contribu-tions already at tree level, as discussed above, and moreoverin addition to the lepton–slepton and quark–squark loops,one loop diagrams involving insertions of bilinear R-parityviolating masses or slepton VEVs must be considered. Oneshould note that the bilinear R-parity violating terms, if notsuppressed, give too large loop contributions to neutrinomasses.

The scenario known as bilinear R-parity violation(BRpV) corresponds to the explicit introduction of the threemass parameters μi in the first term in (5.88), without re-ferring to their origin, and assuming that all the trilinear pa-rameters are zero. The μi terms introduce tree-level mixingbetween the Higgs and lepton superfields. Therefore, theyviolate R-parity and lepton number, and contribute to thebreaking of the SU(2) symmetry by the induction of sneu-trino vacuum expectation values vi . As it was mentionedbefore, the mixing between neutralinos and neutrinos leadsto an effective tree-level neutrino mass matrix of the form,

m0ijν = M1g

2 +M2g′2

4 detMχ0ΛiΛj , (5.91)

where the parameters Λi = μvi + μivd are proportional tothe sneutrino vacuum expectation values in the basis wherethe μi terms are removed from the superpotential. Due to thesymmetry of this mass matrix, only one neutrino acquires amass. Once quantum corrections are included, this symme-try is broken, and the effective neutrino mass matrix takesthe form [780],

mijν =AΛiΛj +B(Λiεj +Λjεi)+Cεiεj . (5.92)

If the tree-level contribution dominates, as for example inSUGRA models with low values of tanβ , the atmosphericmass scale is given at tree level, and the solar mass scaleis generated at one loop, explaining the hierarchy betweenthem. Most of the time, the dominant loop in SUGRA isthe one formed with bottom quarks and squarks, followedin importance by loops with charginos and neutralinos. Inthe tree-level dominance case the atmospheric mixing angleis well approximated by tan2 θatm =Λ2

2/Λ23, and the reactor

angle by tan2 θ13 =Λ21/(Λ

22 +Λ2

3). In this case, the small-ness of the reactor angle is achieved with a small value ofΛ1, and the maximal mixing in the atmospheric sector witha similar value forΛ2 andΛ3. Supergravity scenarios wheretree-level contribution does not dominate can also be found[781], in which case the previous approximations for the an-gles are not valid.

5.4.5 Lepton flavor violating processes at low energies

Many processes, which are either rare or forbidden in theR-parity conserving model, become possible when interac-tions following from the superpotential WRp in (5.88) are

available. These interactions include tree-level couplings be-tween different lepton or quark generations, as well as tree-level couplings between leptons and quarks, or leptons andHiggses.

In addition to the trilinear couplings λ and λ′, bilinearcouplings or spontaneous R-parity breaking contribute to thelepton flavor violating processes mentioned below throughmixing.

For references about this section, see Ref. [771].

– li → lj γ , li → lj lklm, and μ–e-conversion, and semi-leptonic decays of τ -leptons. The rare decays of leptonsto lighter leptons are excellent probes of new physics,because they do not involve any hadronic uncertain-ties. Both the lepton flavor violating trilinear λ- and λ′-type couplings give rise to LFV decays li → lj γ (looplevel process with ν–l, ν–l, or q–q ′ in the loop), li →lj lklm (tree-level process via ν or l), as well as for μ–e-conversion. In these processes, two non-vanishingΛ cou-plings are needed and usual approach is to assume a dom-inant product of two couplings, when determining boundson couplings. In the μ–e-conversion, certain pairs of cou-plings can be probed only in the loop-level process, me-diated by virtual γ or Z, which are logarithmically en-hanced compared to μ→ eγ [185]. The hadronic contri-butions to the μ–e-conversion in nuclei make the theoret-ical error larger than in the decays without hadrons. Therelatively large mass of τ allows for new semileptonicdecay modes for τ . The bounds from these processesvary between Λ ∼ O(10−4–10−1) for 100 GeV fermionmasses, and they scale as mass2.

The experimental accuracies of the processes men-tioned above are expected to increase considerably in thecoming years.

– Leptonic and semileptonic decays of hadrons and topquarks. R-parity violating couplings λ′ijk allow for lep-ton flavor violating decays of hadrons, e.g. KL→ e±μ∓,Bd → μ+τ−, K+ → π+νiνj [782], as well as semilep-tonic LFV top decays, e.g. t→ τ+b, if kinematically al-lowed. The sensitivity on the couplings is restricted bythe theoretical uncertainties in hadronic contributions. For100 GeV sfermions, the bounds are Λ∼ O(10−4–10−1).

5.4.6 Anomalous muon magnetic moment aμ and electronelectric dipole moment

Λ couplings affect leptons also through contributions to di-pole moments. The experimental measurement of aμ is quiteprecise. The theoretical calculation of the standard modelcontribution to aμ contains still uncertainty, which preventsexact comparison with measurement. The contribution ofR-parity violation on aμ is small, and constrained by tinyneutrino masses.

100 Eur. Phys. J. C (2008) 57: 13–182

Contribution from complex Λ to electron EDM could belarge for large phases. The one loop contribution involvingboth bilinear and trilinear couplings is sizable for electronEDM, while one loop terms with only trilinear terms aresuppressed by neutrino masses.

5.4.7 Collider signatures

The main advantage of collider studies compared to the lowenergy probes is that the particles can be directly produced,and thus their masses and couplings can be experimentallymeasured.

A major difference between R-parity conserving andbreaking models from the detection point of view is theamount of missing energy. If R-parity is violated, the super-symmetric particles decay to the SM particles leaving lit-tle or no missing energy. Decays of sparticles through λ-and λ′-type couplings lead to multi-lepton final states, andλ′ and λ′′ to multi-jet final states. Sparticles can decay firstvia the R-parity conserving couplings to the lightest super-symmetric particle (LSP), which then decays via R-parityviolating couplings. If e.g. a neutralino is the LSP, it may bea cascade decay product of a sfermion, chargino, or a heav-ier neutralino. Thus typically one gets a larger number ofjets or leptons in the final state in R-parity violating than inthe R-parity conserving decay. The sparticles can also de-cay directly to the standard model fermions via λ, λ′, or λ′′couplings. Assuming all the supersymmetric particles decayinside the detector, a consequence of the decay of the LSP isthat the amount of missing energy when R-parity is violatedis considerably lower than in the R-parity conserving case,and only neutrinos carry the missing energy. When R-parityis violated, the LSP is not stable and need not be neutral.If then the coupling through which the LSP decays is sup-pressed, a long lived possibly charged particle appears, leav-ing a heavily ionizing, easily detectable charged track in thedetector.

A simplifying assumption for the search strategy at col-liders is to postulate the existence of a single dominantR-parity violating coupling. In case a non-vanishing cou-pling does exist with a magnitude leading to distinct phe-nomenology at colliders, a direct sensitivity to a long-livedLSP might be provided by the observation of displacedvertices in an intermediate range of coupling values up toO(10−5–10−4). For largerΛ values the presence of R-parityviolating supersymmetry will become manifest through thedecay of short-lived sparticles produced by pair via gaugecouplings. A possible search strategy in such cases consistsof neglecting R-parity violating contributions at productionin non-resonant processes. This is valid provided that theinteraction strength remains sufficiently small compared toelectromagnetic or weak interaction strengths, for Λ valuestypically below O(10−2–10−1). In a similar or larger range

of couplings values, R-parity violation could show up at col-liders via single resonant or non-resonant production of su-persymmetric particles.

For bilinear or spontaneous breaking, the lightest super-symmetric particle decays through mixing with the corre-sponding Rp = +1 particle. If the LSP is neutralino orchargino, it decays through mixing with neutrino or chargedlepton, and if the LSP is a slepton it decays through mix-ing with the Higgs bosons, e.g. stau mixes with chargedHiggs. Assuming that neutralino is the LSP, the dominantdecay mode of stau is to tau and neutralino. Through mix-ing the charged Higgs has then a branching ratio to tau andneutralino. Thus the detection of R-parity violation includesprecise measurement of the branching ratios of particles.

The main signature of BRpV is the decay of the neu-tralino, which decays 100% of the time into R-parityand lepton number violating modes. If squarks and slep-tons are heavy and the neutralino is heavier than thegauge bosons, the neutralino decays into on-shell gaugebosons and leptons: χ0

1 →W∓�±i ,Zνi . If the gauge bosonsare produced off-shell, then the decay modes are χ0

1 →qq ′�±i , �

∓j νj �

±i , qqνi, �

±j �

∓j νi, νj νj νi . When sfermions

cannot be neglected, the decay channels are the same, butsquarks and sleptons contribute as intermediate particles[783]. In this model, very useful quantities are formed withratios of branching ratios, since they can be directly linkedto R-parity violating parameters and neutrino observables.We have for example,

B(χ01 → qq ′μ)

B(χ01 → qq ′τ)

≈ Λ22

Λ23

≈ tan2 θatm, (5.93)

where the last approximation is valid in the tree-level dom-inance scenario. In this way, collider and neutrino measure-ments, coming from very different experiments, can be con-trasted.

Detection possibilities and extraction of limits depend alot on the specific model and on the collider type and energy.On general grounds a lepton–hadron collider provides bothleptonic and baryonic quantum numbers in the initial stateand is therefore suited for searches involving λ′. In e+p col-lisions, the production of ujL squarks of the j th generationvia λ′1j1 is especially interesting as it involves a valence dquark of the incident proton. In contrast, for e−p collisionswhere charge conjugate processes are accessible, the λ′11kcouplings become of special interest as they allow for theproduction, involving a valence u quark, of dkR squarks ofthe kth generation.

The excluded regions of the parameter space for R-parityviolating scenarios have been worked out from the data atLEP, HERA and Tevatron, see e.g. [784–789]. In the fol-lowing we shall concentrate on the search possibilities at theLHC.

Eur. Phys. J. C (2008) 57: 13–182 101

5.4.8 Hadron colliders

In hadron colliders the λ or λ′ couplings can provide a viablesignal. In many SUSY scenarios neutralinos and charginosare among the lightest supersymmetric particles. Their pairproduction or associated production of χ±

1 χ0 via gauge cou-

plings and decay via λ or λ′ couplings may lead to a tri-lepton signal from each particle, providing a clean signature.One should notice that if the couplings are small, the vertexmay be displaced which makes the analysis more compli-cated. With small enough couplings the lightest neutralino,if LSP, decays outside the detector.

If kinematically possible, gluinos and squarks are copi-ously produced at hadron colliders. The NLO cross sectionhas been calculated in [790]. For mq > mg > mcL , the pro-duction with decay via λ′121 �= 0 was studied at CDF. Alsocoupling λ′13k from t pair production at CDF and λ′ cou-plings from χ0

1 decay at D0 have been investigated.When R-parity is violated, the supersymmetric particles

can be produced singly, and thus they can be producedas resonances through R-parity violating interactions. In ahadron–hadron collider this allows one to probe for res-onances in a wide mass range because of the continuousenergy distribution of the colliding partons. This produc-tion mode requires non-negligible R-parity violating cou-pling. If a single R-parity violating coupling is dominant,the exchanged supersymmetric particle may decay throughthe same coupling involved in its production, giving a twofermion final state. It is also possible that the decay of theresonant SUSY particle goes through gauge interactions,giving rise to a cascade decay.

The resonant production of sneutrinos and charged slep-tons (via λ′ couplings) has been investigated at hadron col-liders [791–795]. The production of a charged lepton withneutralino leads to a like-sign dilepton signature via λ′ cou-plings. The production of a charged lepton with a charginoin the resonant sneutrino case decay leads to a tri-lepton finalstate via λ′ couplings. The χ0

1 , χ±1 , ν masses can be recon-

structed using the tri-lepton signal.Single production is possible also in two-body processes

without resonance [796]. Sfermion production with a gaugeboson has been studied in either via λ or λ′-coupling. (Theprocess qiqj → W−νk or qiqj → W+ lkL can get contri-bution also from resonant production, but e.g. in SUGRAml−mν = cos 2βm2

W and resonance production is not kine-matically viable.) Similarly via λ′ or λ′′ gluino can be pro-duced with a lepton or quark, respectively. Sneutrino pro-duction with two associated jets may also provide a de-tectable signal [797].

Resonant production of squarks can occur via λ′′-typecouplings, leading eventually to jets in the final states. Al-though the cross sections can be considerable for theseprocesses, the backgrounds in hadronic colliders are large,

and the processes seem difficult to study [798]. In specialcircumstances the backgrounds can be small, e.g. for stopproduction in di dj → t1 → bχ+

1 , with χ+i → liνi χ

0i (here

it is assumed mt1 > mχ+1> mχ0

1, mtop > mχ0

1). Then for

λ′′3jk , mχ01

is stable [799, 800]. Also single gluino produc-

tion, didj → gt via resonant stop production has a goodsignal to background ratio for λ′′3jk = O(0.1) [801].

With the t t production cross section of the order of800 pb, the LHC can be considered a top quark factory, with∼108 top quarks being produced per year, assuming an in-tegrated luminosity of 100 fb−1. This statistics allows forprecise studies of top quark physics, in particular, for mea-surements of rare RpV decays. A simulation of the signaland background using ATLFAST [802], to take into accountthe experimental conditions prevailing at the ATLAS detec-tor [803], was made for a top quark decaying through a λ′�31coupling to t→ χ0�d , assuming only one slepton gives theleading contribution as an intermediate state [804].

The importance of treating the top quark production anddecay simultaneously gg → t χ0�d , rather than Γ (gg →t t)B(t → χ0�d), was shown. The latest approach can un-derestimate the cross section by a factor of a few units, de-pending on the slepton mass. The reason is that the sleptonforces the top quark to be off-shell, becoming the resonanceitself, as can be appreciated from χ0� mass invariant distri-butions.

Two scenarios were chosen for the neutralino decay,χ0 → bdνe and χ0 → cde, the last one assuming a largestop–scharm mixing. The sensitivity of the LHC is presentedas the significance S/

√B as a function of λ′131, for slep-

ton masses 150 and 200 GeV. The channel t → χ0ed →cdeed is more promising with exclusion limits at 2σ c.l.for λ′ > 0.03 and observation at 5σ c.l. for λ′ > 0.05,with these values slightly increasing for heavier sleptons.The t → χ0ed → bdνeed channel is observable only fora 150 GeV slepton mass. The significance is reduced toλ′ > 0.08 at 2σ and λ′ > 0.15 at 5σ level.

Since a λ′�33 ∼ hbε�/μ trilinear term is generated inBRpV when the ε� term is removed from the superpoten-tial, we can see that the above exclusion limits for λ′ are notsignificant in BRpV, probing only values of ε� parametersmuch larger than what is needed for neutrino oscillations.

5.5 Higgs-mediated lepton flavor violation insupersymmetry

If neutrinos are massive, one would expect LFV transitionsin the Higgs sector through the decay modes H 0 → li lj me-diated at one loop level by the exchange of the W bosonsand neutrinos. However, as for the μ→ eγ and the τ→ μγ

case, also the H 0 → li lj rates are GIM suppressed. In asupersymmetric (SUSY) framework the situation is com-pletely different. Besides the previous contributions, super-symmetry provides new direct sources of flavor violation,

102 Eur. Phys. J. C (2008) 57: 13–182

namely the possible presence of off-diagonal soft terms inthe slepton mass matrices and in the trilinear couplings[144]. In practice, flavor violation would originate from anymisalignment between fermion and sfermion mass eigen-states. LFV processes arise at one loop level through theexchange of neutralinos (charginos) and charged sleptons(sneutrinos). The amount of the LFV is regulated by a Super-GIM mechanism that can be much less severe than in thenon-supersymmetric case [144]. Another potential source ofLFV in models such as the minimal supersymmetric stan-dard model (MSSM) could be the Higgs sector, in fact, ex-tensions of the SM containing more than one Higgs dou-blet generally allow flavor violating couplings of the neutralHiggs bosons. Such couplings, if unsuppressed, will leadto large flavor-changing neutral currents in direct opposi-tion to experiments. The MSSM avoid these dangerous cou-plings at the tree level segregating the quark and Higgs fieldsso that one Higgs (Hu) can couple only to up-type quarkswhile the other (Hd) couples only to d-type. Within unbro-ken supersymmetry this division is completely natural, infact, it is required by the holomorphy of the superpotential.However, after supersymmetry is broken, couplings of theform QUcHd and QDcHu are generated at one loop [430].In particular, the presence of a non-zero μ term, coupledwith SUSY breaking, is enough to induce non-holomorphicYukawa interactions for quarks and leptons. For large tanβvalues the contributions to d-quark masses coming fromnon-holomorphic operator QDcHu can be equal in size tothose coming from the usual holomorphic operatorQDcHddespite the loop suppression suffered by the former. This isbecause the operator itself gets an additional enhancementof tanβ .

As shown in Ref. [805] the presence of these loop in-duced non-holomorphic couplings also leads to the ap-pearance of flavor-changing couplings of the neutral Higgsbosons. These new couplings generate a variety of flavor-changing processes such as B0 → μ+μ−, B0–B0 etc. [537].Higgs-mediated FCNC can have sizable effects also in thelepton sector [806, 807]: given a source of non-holomorphiccouplings, and LFV among the sleptons, Higgs-mediatedLFV is unavoidable. These effects have been widely dis-cussed in the recent literature both in a generic 2HDM[808, 809] and in supersymmetry [807, 810] frameworks.Through the study of many LFV processes as �i → �j �k�k[806, 807], τ → �jη [169, 810], �i → �jγ [175, 650],μN→ eN [811], Φ0 → �j �k [174] (with �i = τ,μ, �j,k =μ,e, Φ = h0,H 0,A0) or the cross section of the μN→ τX

reaction [812].

5.5.1 LFV in the Higgs sector

SM extensions containing more than one Higgs doublet gen-erally allow flavor violating couplings of the neutral Higgs

bosons with fermions. Such couplings, if unsuppressed, willlead to large flavor-changing neutral currents in direct oppo-sition to experiments. The possible solution to this probleminvolves an assumption about the Yukawa structure of themodel. A discrete symmetry can be invoked to allow a givenfermion type to couple to a single Higgs doublet, and in suchcase FCNC’s are absent at tree level. In particular, when asingle Higgs field gives masses to both types of fermions theresulting model is referred as 2HDM-I. On the other hand,when each type of fermion couples to a different Higgs dou-blet the model is said 2HDM-II.

In the following, we shall assume a scenario where thetype-II 2HDM structure is not protected by any symmetryand is broken by loop effects (this occurs, for instance, inthe MSSM).

Let us consider the Yukawa interactions for charged lep-tons, including the radiatively induced LFV terms [806]:

−L lRiYliH1Li + lRi(YliΔ

ijL + YljΔijR

)H2Lj + h.c.,

(5.94)

where H1 and H2 are the scalar doublets, lRi are lepton sin-glet for right handed fermions,Lk denote the lepton doubletsand Ylk are the Yukawa couplings.

In the mass eigenstate basis for both leptons and Higgsbosons, the effective flavor violating interactions are de-scribed by the four dimension operators [806]:

−L (2G2F

) 14mli

c2β

(ΔijL li

RljL +ΔijR l

i

LljR

)

× (cβ−αh0 − sβ−αH 0 − iA0)

+ (8G2F

) 14mli

c2β

(ΔijL li

RνjL +ΔijRνiLl

j

R

)H± + h.c.,

(5.95)

where α is the mixing angle between the CP-even Higgsbosons h0 and H0, A0 is the physical CP-odd boson, H±are the physical charged Higgs-bosons and tβ is the ratioof the vacuum expectation value for the two Higgs (wherewe adopt the notation, cx, sx = cosx, sinx and tx = tanx).Irrespective to the mechanism of the high energy theoriesgenerating the LFV, we treat the ΔijL,R terms in a model in-

dependent way. In order to constrain the ΔijL,R parameters,we impose that their contributions to LFV processes do notexceed the experimental bounds [175, 650].

On the other hand, there are several models with a spe-cific ansatz about the flavor-changing couplings. For in-stance, the famous multi-Higgs-doublet models proposed byCheng and Sher [596] predict that the LFV couplings of allthe neutral Higgs bosons with the fermions have the formHfifj ∼√

mimj .In supersymmetry, the Δij terms are induced at one loop

level by the exchange of gauginos and sleptons, provided

Eur. Phys. J. C (2008) 57: 13–182 103

a source of slepton mixing. In the so mass insertion (MI)approximation, the expressions of ΔijL,R are given by

ΔijL = − α1

4πμM1δ

ijLLm

2L

×[I′(M2

1 ,m2R,m

2L

)+ 1

2I′(M2

1 ,μ2,m2

L

)]

+ 3

2

α2

4πμM2δ

ijLLm

2LI

′(M2

2 ,μ2,m2

L

), (5.96)

ΔijR = α1

4πμM1m

2RδijRR

[I′(M2

1 ,μ2,m2

R

)− (μ↔mL)],

(5.97)

respectively, where μ is the Higgs mixing parameter, M1,2

are the gaugino masses and m2L(R)

stands for the left–left(right–right) slepton mass matrix entry. The LFV mass in-sertions (MIs), i.e. δ3�

XX = (m2�)

3�XX/m

2X (X = L,R), are the

off-diagonal flavor changing entries of the slepton mass ma-trix. The loop function I

′(x, y, z) is such that I

′(x, y, z) =

dI (x, y, z)/dz, where I (x, y, z) refers to the standard threepoint one loop integral which has mass dimension-2

I3(x, y, z)= xy log(x/y)+ yz log(y/z)+ zx log(z/x)

(x − y)(z− y)(z− x) .

(5.98)

The above expressions, i.e. (5.96), (5.97), depend only onthe ratio of the SUSY mass scales and they do not decou-ple for large mSUSY. As first shown in Ref. [174], both ΔijRand ΔijL couplings suffer from strong cancellations in cer-tain regions of the parameter space due to destructive in-terferences among various contributions. For instance, from(5.97) it is clear that, in theΔijR case, such cancellations hap-pen if μ=mL.

In the SUSY see-saw model, in the mass insertion ap-proximation, one obtains specific values for δijLL dependingon the assumptions on the flavor mixing in Yν [164, 707].If the latter is of CKM size, δ21(31)

LL 3 × 10−5 and δ32LL

10−2, while in the case of the observed neutrino mixing, tak-ing Ue3 = 0.07 at about half of the current CHOOZ bound,we get δ21(31)

LL 10−2 and δ32LL 10−1.

5.5.2 Phenomenology

In order to constrain the ΔijL,R parameters, we impose thattheir contributions to LFV processes as li → lj lklk and li →lj γ do not exceed the experimental bounds. At tree level,Higgs exchange contribute only to �i → �j �k�k , τ → �jη

and μN → eN . On the other hand, a one loop Higgs ex-change leads to the LFV radiative decays �i → �jγ . In thefollowing, we report the expression for the branching ratiosof the above processes.

5.5.2.1 �i → �jγ The �i → �jγ process can be gener-ated by the one loop exchange of Higgs and leptons. How-

ever, the dipole transition implies three chirality flips: two inthe Yukawa vertices and one in the lepton propagator. Thisstrong suppression can be overcome at higher order level.Going to two loop level, one has to pay the typical price ofg2/16π2 but one can replace the light fermion masses fromYukawa vertices with the heavy fermion (boson) masses cir-culating in the second loop. In this case, the virtual Higgsboson couples only once to the lepton line, inducing theneeded chirality flip. As a result, the two loop amplitude canprovide the major effects. Naively, the ratio between the twoloop fermionic amplitude and the one loop amplitude is

A(2-loop)fli→lj γA

1-loopli→lj γ

∼ αem

4π

m2f

m2li

log

(m2f

m2H

),

where mf =mb,mτ is the mass of the heavy fermion circu-lating in the loop. We remind that in a Model II 2HDM (asSUSY) the Yukawa couplings between neutral Higgs bosonsand quarks are Htt ∼mt/ tanβ and Hbb∼mb tanβ . Sincethe Higgs mediated LFV is relevant only at large tanβ ≥ 30,it is clear that the main contributions arise from the τ and bfermions and not from the top quark. So, in this framework,τ → lj γ does not receive sizable two loop effects by heavyfermionic loops, contrary to the μ→ eγ case.

However, the situation can drastically change when a Wboson circulates in the two loop Barr–Zee diagrams. Bear-ing in mind that HW+W− ∼ mW and that pseudoscalarbosons do not couple to aW pair, it turns out thatA(2-loop)W

li→lj γ /

A(2-loop)fli→lj γ ∼ m2

W/(m2f tanβ); thus, two loop W effects

are expected to dominate, as it is confirmed numerically[650, 808].

As final result, the following approximate expressionholds [175, 650]:

B(�i → �jγ )

B(�i → �j νj ντ )

3

2

αel

π

(m2�i

m2A

)2

t6βΔ2ij

{δm

mAlogm2�i

m2A

+ 1

6

+ αelπ

[m2W

m2�i

F (aW )

tβ−∑f=b,τ

Nf q2f

m2f

m2�i

(logm2f

m2�i

+ 2

)

− Nc4

(q2t

mtμ

tβm2�i

s2θt h(xtH )

− q2b

mbAb

m2�i

s2θbh(x

bH)

)]}2

3

2

α3el

π3Δ2

21t4β

(m4W

M4H

)(F(aW )

)2, (5.99)

where δm = (mH − mA) ∼ O(m2Z/mA0). The terms of

the first row of (5.99) refer to one loop effects and their

104 Eur. Phys. J. C (2008) 57: 13–182

role is non-negligible only in τ decays. It turns out thatpseudoscalar and scalar one loop amplitudes have oppositesigns, so, as we have mA mH , they cancel each other to avery large extent. Since these cancellations occur, two loopeffects can become important or even dominant. The twoterms of the second row of (5.99) refer to two loop Barr–Zee effects induced byW and fermionic loops, respectively,while the last row of (5.99) is relative two loop Barr–Zee ef-fects with a squark loop in the second loop. As regards thesquark loop effects, it is very easy to realize that they arenegligible compared to W effects. In fact, it is well knownthat Higgs mediated LFV can play a relevant or even a dom-inant role compared to gaugino mediated LFV provided thatslepton masses are not below the TeV scale while maintain-ing the Higgs masses at the electroweak scale (and assuminglarge tβ values). In this context, it is natural to assume squarkmasses at least of the same order as the slepton masses (atthe TeV scale). So, in the limit where x

fH=m2

f/m2

H � 1,

the loop function h(xfH) is such that (logx

fH+5/3)/6x

fH

thus, even for maximum squark mixing angles θt,b

, namelyfor s2θ

t,b= sin 2θ

t,b 1, and large Ab and μ terms, two loop

squark effects remain much below the W effects, as it isstraightforward to check by (5.99).

As a final result the main two loop effects are providedby the exchange of a W boson, with the loop functionF(aW ) ∼ 35

16 (logaW )2 for aW = m2W/m

2H � 1. It is note-

worthy that one and two loop amplitudes have the samesigns. In addition, two loops effects dominate in large por-tions of the parameter space, specially for large mH values,where the mass splitting δm=mH −mA decreases to zero.

5.5.2.2 �i → �j �k�k The li → lj lklk process can be me-diated by a tree level Higgs exchange [806, 807]. However,up to one loop level, li → lj lklk gets additional contributionsinduced by li → lj γ

∗ amplitudes [175, 650]. It is worth not-ing that the Higgs mediated monopole (chirality conserving)and dipole (chirality violating) amplitudes have the sametan3 β dependence. This has to be contrasted to the non-Higgs contributions. For instance, within SUSY, the gauginomediated dipole amplitude is proportional to tanβ while themonopole amplitude is tanβ independent. The expressionfor the Higgs mediated li → lj lklk can be approximated inthe following way [175, 650]:

B(τ → lj lklk)

B(τ → lj νj ντ )

m2τm

2lk

32m4A

Δ2τj tan6 β[3 + 5δjk]

+ αel3π

(logm2τ

m2lk

− 3

)B(τ → lj γ )

B(τ→ lj νj ντ ), (5.100)

where we have disregarded subleading monopole effects.

5.5.2.3 μN → eN The μ → e conversion in Nucleiprocess can be generated by a scalar operator through thetree level Higgs exchange [811]. Moreover, at one looplevel, additional contributions induced by li → lj γ

∗ ampli-tudes arise [175]; however they are subleading [175]. Fi-nally, the following expression for B(μAl → eAl) is de-rived [811]:

B(μAl→ eAl) 1.8 × 10−4 m7μm

2p

v4m4hωAlcaptΔ2

21t6β, (5.101)

where ωAlcapt 0.7054 × 106 s−1. We observe that B(μ→3e) is completely dominated by the photonic μ→ eγ ∗ di-pole amplitude so that B(μ→ eee) αemB(μ→ eγ ). Onthe other hand, tree level Higgs mediated contributions arenegligible because suppressed by the electron mass throughthe H(A)ee ∼ me coupling. On the contrary, μN → eN isnot suppressed by the light constituent quark mu and mdbut only by the nucleon masses, because the Higgs-bosoncoupling to the nucleon is shown to be characterized by thenucleon mass using the conformal anomaly relation [811].In particular, the most important contribution turns out tocome from the exchange of the scalar Higgs boson H whichcouples to the strange quark [811].

In fact, the coherent μ–e conversion process, wherethe initial and final nuclei are in the ground state, is ex-pected to be enhanced by a factor of O(Z) (where Zis the atomic number) compared to incoherent transitionprocesses. Since the initial and final states are the same,the elements 〈N |pp|N〉 and 〈N |nn|N〉 are nothing but theproton and the neutron densities in a nucleus in the non-relativistic limit of nucleons. In this limit, the other matrixelements 〈N |pγ5p|N〉 and 〈N |nγ5n|N〉 vanish. Therefore,in the coherent μ–e conversion process, the dominant con-tributions come from the exchange of H , not A [811].

Moreover, we know that μ→ eγ ∗ (chirality conserving)monopole amplitudes are generally subdominant comparedto (chirality flipping) dipole effects [175]. Note also that,the enhancement mechanism induced by Barr–Zee type di-agrams is effective only for chirality flipping operators so,in the following, we shall disregard chirality conserving oneloop effects.

5.5.2.4 τ→ μP (P = π,η,η′) Now we consider the im-plications of virtual Higgs exchange for the decays τ →μP , where P is a neutral pseudoscalar meson (P = π,η,η′)[169, 810]. Since we assume CP conservation in the Higgssector, only the exchange of the A Higgs boson is relevant.Moreover, in the large tanβ limit, only the A couplings todown-type quarks are important. These can be written as

−i(√2GF)1/2 tanβA(ξdmddRdL + ξsms sRsL

+ ξbmbbRbL)+ h.c. (5.102)

Eur. Phys. J. C (2008) 57: 13–182 105

The parameters ξd, ξs, ξb are equal to one at tree level, butthey can significantly deviate from this value because ofhigher order corrections proportional to tanβ [537, 805],generated by integrating out superpartners. In the limit ofquark flavor conservation, each ξq (q = d, s, b) has theform ξq = (1 +Δq tanβ)−1, where Δq appears in the loop-generated term −hqΔqH 0∗

2 qcq + h.c. [537, 805]. At ener-

gies below the bottom mass, the b-quark can be integratedout so the bilinear −imbbcb+h.c. is effectively replaced by

the gluon operator Ω = g2s

64π2 εμνρσGaμνG

aρσ , where gs and

Gaμν are the SU(3)C coupling constant and field strength, re-spectively [169]. In the limit in which the processes τ→ 3μand τ → μη are both dominated by Higgs exchange, thesedecays are related as [169]:

B(τ→ lj η)

B(τ → lj νj ντ ) 9π2

(f 8η m

2η

m2Amτ

)2(1 − m

2η

m2τ

)2

×[ξs + ξb

3

(1 +√

2f 0η

f 8η

)]2

Δ23j tan6 β,

where m2η/m

2τ 9.5 × 10−2 and the relevant decay con-

stants are f 0η ∼ 0.2fπ , f 8

η ∼ 1.2fπ and fπ ∼ 92 MeV. In theabove expression, both the contribution of the (bottom-loopinduced) gluon operatorΩ and the factors ξq were included.

For ξs ∼ ξb ∼ 1, it turns out thatB(τ− → μ−η)/B(τ− →μ−μ+μ−) 5, but it could also be a few times largeror smaller than that, depending on the actual values ofξs, ξb . Finally, let us compare τ → μη′ and τ → μπ withτ → μη in the limit of Higgs exchange domination. Bothratios are suppressed, although for different reasons. Theratio B(τ → μπ)/B(τ → μη) is small because it is para-metrically suppressed by m4

π/m4η ∼ 10−2. The ratio B(τ →

μη′)/B(τ → μη), which seems to be O(1), is much smallerbecause the singlet and octet contributions to τ → μη′ tendto cancel against each other [169].

These results, combined with the present bound on τ →μη, imply that the Higgs mediated contribution to B(τ →μη′) and B(τ→ μπ) can reach O(10−9) [169].

5.5.2.5 Higgs → μτ The LFV Higgs → μτ decays andthe related phenomenology have been extensively investi-gated in [174]. Concerning the Higgs boson decays, wehave [174]

B(A→ μ+τ−

)= tan2 β(|ΔL|2 + |ΔR|2

)B(A→ τ+τ−

),

(5.103)

where we have approximated 1/c2β tan2 β since non-

negligible effects can only arise in the large tanβ limit. IfA is replaced with H [or h] in (5.103), the r.h.s. should alsobe multiplied by a factor (cβ−α/sα)2 [or (sβ−α/cα)2]. Werecall that B(A→ μτ) can reach values of order 10−4. The

same holds for the ‘non-standard’ CP-even Higgs boson (ei-ther H or h, depending on mA).

We now make contact with the physical observable, i.e.the B(Φ0 → μ+τ−), and discuss the phenomenological im-plications. We outline some general features of B(Φ0 →μ+τ−) at large tanβ and the prospects for these decay chan-nels at the Large Hadron Collider (LHC) and other collid-ers. Let us discuss the different Higgs bosons, as reported in[174], assuming for definiteness tanβ ∼ 50, |50Δ|2 ∼ 10−3

(Δ = ΔL or ΔR) and an integrated luminosity of 100 fb−1

at LHC.If Φ0 denotes one of the ‘non-standard’ Higgs bosons,

we have CΦ 1 and B(Φ0 → τ+τ−)∼ 10−1, so B(Φ0 →μ+τ−) ∼ 10−4. The main production mechanisms at LHCare bottom-loop mediated gluon fusion and associated pro-duction with bb, which yield cross sections σ ∼ (103,102,

20) pb for mA ∼ (100,200,300) GeV, respectively. Thecorresponding numbers of Φ0 → μ+τ− events are about(104,103,2 × 102). These estimates do not change muchif the bottom Yukawa coupling Yb is enhanced (suppressed)by radiative corrections, since in this case the enhancement(suppression) of σ would be roughly compensated by thesuppression (enhancement) of B(Φ0 → μ+τ−).

If Φ0 denotes the other (more ‘standard model-like’)Higgs boson, the factor CΦ · B(Φ0 → τ+τ−) strongly de-pends on mA, while the production cross section at LHC,which is dominated by top-loop mediated gluon fusion, isσ ∼ 30 pb. For mA ∼ 100 GeV we may have CΦ ·B(Φ0 →τ+τ−)∼ 10−1 and B(Φ0 → μ+τ−)∼ 10−4, which wouldimply ∼300 μ+τ− events. The number of events is generi-cally smaller for large mA since CΦ scales as 1/m4

A, consis-tently with the expected decoupling of LFV effects for sucha Higgs boson.

The above discussion suggests that LHC may offer goodchances to detect the decays Φ0 → μτ , especially in thecase of non-standard Higgs bosons. This indication shouldbe supported by a detailed study of the background. At Teva-tron the sensitivity is lower than at LHC because both theexpected luminosity and the Higgs production cross sec-tions are smaller. The number of events would be smallerby a factor 102–103. A few events may be expected also ate+e− or μ+μ− future colliders, assuming integrated lumi-nosities of 500 and 1 fb−1, respectively. At a μ+μ− col-lider an enhancement may occur for the non-standard Higgsbosons if radiative corrections strongly suppress Yb , sincein this case both the resonant production cross section [σ ∼(4π/m2

A)B(Φ0 → μ+μ−)] and the LFV branching ratios

B(Φ0 → μ+τ−) would be enhanced. As a result, for lightmA, hundreds of μ+τ− events could occur.

5.5.2.6 μN→ τX Higgs mediated LFV effects can havealso relevant impact on the cross section of the μN → τX

106 Eur. Phys. J. C (2008) 57: 13–182

reaction [812]. The contribution of the Higgs boson media-tion to the differential cross section μ−N → τ−X is givenby [812]

d2σ

dx dy=∑q

xfq(x)

{|CL|2q

(1 − Pμ

2

)+ |CR|2q

(1 + Pμ

2

)}

× s

8πy2, (5.104)

where the function fq(x) is the PDF for q-quarks, Pμ isthe incident muon polarization such that Pμ = +1 and −1correspond to the right and left handed polarization, respec-tively, and s is the centre-of-mass (CM) energy. The para-meters x and y are defined as x ≡Q2/2P · q , y ≡ 2P · q/s,in the limit of massless tau leptons, where P is the four mo-mentum of the target, q is the momentum transfer, and Qis defined asQ2 ≡−q2. As seen in (5.104), experimentally,the form factors of ChHL and CAL (ChHR and CAR ) can be se-lectively studied by using purely left handed (right handed)incident muons. In SUSY models such as the MSSM withheavy right handed neutrinos, LFV is radiatively induceddue to the left handed slepton mixing, which only affectsChHL and CAL . Therefore, in the following, we focus only onthose ChHL and CAL couplings.

The magnitudes of the effective couplings are con-strained by the current experimental results of searches forLFV processes of tau decays. Therefore, both couplings aredetermined by the one that is more constrained, namely thepseudo-scalar coupling. It is constrained by the τ → μη

decay (B(τ → μη) < 3.4 × 10−7). Then the constraint isgiven on the s-associated scalar and pseudo-scalar couplingsby

(∣∣CAL∣∣2)s≤ 10−9 [GeV−4]×B(τ→ μη). (5.105)

The largest values of ChHL and CAL can be realized withmSUSY ∼ O(1) TeV and the higgsino mass μ∼ O(10) TeV[169, 810].

The cross sections of the μN→ τX reaction in the DISregion is evaluated for the maximally allowed values of theeffective couplings as a reference. They are plotted in Fig. 36for different quark contributions as a function of the muonbeam energy in the laboratory frame. For the PDF, CTEQ6Lhas been used. The target N is assumed to be a proton. Fora nucleus target, the cross section would be higher, approx-imately by the number of nucleons in the target. The crosssection sharply increases above Eμ ∼ 50 GeV in Fig. 36.This enhancement comes from the b-quark contribution inaddition to the d- and s-quark contributions which is en-hanced by a factor of mb/ms over the s-quark contribution.The cross section is enhanced by one order of magnitudewhen the muon energy changes from 50 to 100 GeV. Typi-cally, for Eμ = 100 GeV and Eμ = 300 GeV, the cross sec-tion is 10−4 and 10−3 fb, respectively. With an intensity of

Fig. 36 Cross section of the μ−N→ τ−X DIS process as a functionof the muon energy for the Higgs mediated interaction [812]. It is as-sumed that the initial muons are purely left handed. CTEQ6L is usedfor the PDF

1020 muons per year and a target mass of 100 g/cm2, about104 (102) events could be expected for σ(μN → τX) =10−3 (10−5) fb, which corresponds to Eμ = 300 (50) GeVfrom Fig. 36. This would provide good potential to improvethe sensitivity by four (two) orders of magnitude from thepresent limit from τ→ μη decay, respectively. Such a muonintensity could be available at a future muon collider and aneutrino factory.

5.5.3 Correlations

The numerical results shown in Figs. 37 and 38 allow us todraw several observations [175, 650].

– τ → lj γ has the largest branching ratios except for a re-gion around mH ∼ 700 GeV where strong cancellationsamong two loop effects reduce their size.25 The follow-ing approximate relations are found:

B(τ→ lj γ )

B(τ→ lj η)

(δm

mAlogm2τ

m2A

+ 1

6+ αelπ

(m2W

m2τ

)F(aW )

tanβ

)2

≥ 1,

where the last relation is easily obtained by using theapproximation for F(z). If two loop effects were dis-regarded, then we would obtain B(τ → lj γ )/B(τ →lj η) ∈ (1/36,1) for δm/mA ∈ (0,10%). Two loop con-

25For a detailed discussion about the origin of these cancellations andtheir connection with non-decoupling properties of two loopW ampli-tude, see Ref. [808].

Eur. Phys. J. C (2008) 57: 13–182 107

Fig. 37 Branching ratios of various τ → μ and τ → e LFV processes versus the Higgs boson mass mH in the decoupling limit as reported in[650]. X = γ,μμ, ee, η

Fig. 38 Left: branching ratios of μ→ eγ , μ→ eee and μAl→ eAl

in the Higgs mediated LFV case versus the Higgs boson mass mh[175]. Right: branching ratios of μ→ eγ , μ→ eee and μAl→ eAl

in the gaugino mediated LFV case versus a common SUSY massmSUSY [175]. In the figure we set tβ = 50 and δ21

LL = 10−2

tributions significantly enhance B(τ→ lj γ ) specially forδm/mA→ 0.

– In Fig. 37 non-negligible mass splitting δm/mA effectscan be visible at low mH regime through the bands ofthe τ → lj γ and τ → lj ee processes. These effects tendto vanish with increasing mH as is correctly reproducedin Fig. 37. τ → ljμμ does not receive visible effects byδm/mA terms being dominated by the tree level Higgsexchange.

– As is shown in Fig. 37 B(τ → lj γ ) is generally largerthan B(τ → ljμμ); their ratio is regulated by the follow-

ing approximate relation:

B(τ → lj γ )

B(τ→ ljμμ) 36

3 + 5δjμ

B(τ → lj γ )

B(τ→ lj η)≥ 36

3 + 5δjμ,

where the last relation is valid only out of the cancellationregion. Moreover, from the above relation it turns out that

B(τ→ lj η)

B(τ→ ljμμ) 36

3 + 5δjμ.

108 Eur. Phys. J. C (2008) 57: 13–182

If we relax the condition ξs,b = 1, B(τ → lj η) can getvalues few times smaller or bigger than those in Fig. 37.

– It is noteworthy that a tree level Higgs exchange predictsthat B(τ → lj ee)/B(τ → ljμμ)∼m2

e/m2μ while, at two

loop level, we obtain (out of the cancellation region):

B(τ→ lj ee)

B(τ→ ljμμ) 0.4

3 + 5δjμ

B(τ→ lj γ )

B(τ→ lj η)≥ 0.4

3 + 5δjμ.

Let us underline that, in the cancellation region, the lowerbound of B(τ → lj ee) is given by the monopole contri-butions. So, in this region, B(τ→ lj ee) is much less sup-pressed than B(τ → lj γ ).

– The approximate relations among μAl→ eAl, μ→ eγ

and μ→ eee branching ratios are

B(μ→ eγ )

B(μAl→ eAl) 102

(F(aW )

tanβ

)2

,

(5.106)B(μ→ eee)

B(μ→ eγ ) αel.

In the above equations we retained only dominant twoloop effects arising fromW exchange. The exact behaviorfor the examined processes is reported in Fig. 38 wherewe can see that μ→ eγ gets the largest branching ratioexcept for a region around mH ∼ 700 GeV where strongcancellations among two loop effects sink its size.

The correlations among the rates of the above processesare an important signature of the Higgs-mediated LFV andallow us to discriminate between different SUSY scenarios.In fact, it is well known that, in a supersymmetric frame-work, besides the Higgs mediated LFV transitions, we havealso LFV effects mediated by the gauginos through loops ofneutralinos (charginos)–charged sleptons (sneutrinos). Onthe other hand, the above contributions have different decou-pling properties regulated by the mass of the heaviest scalarmass (mH ) or by the heaviest mass in the slepton gauginoloops (mSUSY). In principle, themSUSY andmH masses maybe unrelated, so we can always proceed by considering onlythe Higgs mediated effects (assuming a relatively light mHand an heavy mSUSY) or only the gaugino mediated contri-butions (if mH is heavy). In the following, we are interestedto make a comparison between Higgs and gaugino mediatedLFV effects. In order to make the comparison as simple aspossible, let us consider the simple case where all the SUSYparticles are degenerate. In this case, it turns out that

Δ21L ∼ α2

24πδ21LL,

B(�i → �jγ )

B(�i → �j νj νi)

∣∣∣∣Gauge

= 2αel75π

(1 + 5

4tan2 θW

)2( m4W

m4SUSY

)∣∣δijLL∣∣2t2β,

B(�i → �jγ )

B(�i → �j νj νi)

∣∣∣∣Higgs

10α3el

π3

(α2

24π

)2(m4W

M4H

)(logm2W

M2H

)4∣∣δijLL∣∣2t4β. (5.107)

In Fig. 38 we report the branching ratios of the examinedprocesses as a function of the heaviest Higgs boson massmH (in the Higgs LFV mediated case) or of the commonSUSY mass mSUSY (in the gaugino LFV mediated case).We set tβ = 50 and we consider the PMNS scenario asdiscussed above so that (δ21

LL)PMNS 10−2. Sub-leadingcontributions proportional to (δ23

LL(RR)δ31RR(LL))PMNS were

neglected since, in the PMNS scenario, it turns out that(δ23LL(RR)δ

31RR(LL))PMNS/(δ

21LL)PMNS 10−3 [707]. As we

can see from Fig. 38, Higgs mediated effects start beingcompetitive with the gaugino mediated ones when mSUSY isroughly one order of magnitude larger than the Higgs massmH . Moreover, we stress that, both in the gaugino and in theHiggs mediated cases, μ→ eγ gets the largest effects. Inparticular, within the PMNS scenario, it turns out that Higgsmediated B(μ→ eγ ) ∼ 10−11 when mH ∼ 200 GeV andtβ = 50, that is just closed to the present experimental reso-lution.

The correlations among different processes predicted inthe gaugino mediated case are different from those predictedin the Higgs mediated case. For instance, in the gaugino me-diated scenario, B(τ → lj lklk) gets the largest contributionsby the dipole amplitudes that are tanβ enhanced with re-spect to all other amplitudes resulting in a precise ratio withB(τ→ lj γ ), namely

B(�i → �j �k�k)

B(�i → �jγ )

∣∣∣∣Gauge

αel3π

(logm2τ

m2lk

− 3

) αel, (5.108)

B(τ → �j ee)

B(τ → �jμμ)

∣∣∣∣Gauge

log m

2τ

m2e− 3

log m2τ

m2μ− 3

5. (5.109)

Moreover, in the large tanβ regime, one can find the simpletheoretical relations

B(μ− e in Ti)

B(μ→ eγ )

∣∣∣∣Gauge

αel. (5.110)

If some ratios different from the above were discovered, thenthis would be clear evidence that some new process is gen-erating the �i → lj transition, with Higgs mediation being aleading candidate.

5.5.4 Conclusions

We have reviewed the allowed rates for Higgs-mediatedLFV decays in a SUSY framework. In particular, we have

Eur. Phys. J. C (2008) 57: 13–182 109

analyzed the decay modes of the τ,μ lepton, namely �i →�j �k�k , �i → �jγ , τ → lj η and μN → eN . We have alsodiscussed the LFV decay modes of the Higgs bosons Φ→�i�j (Φ = h0,H 0,A0) so as the impact of Higgs mediatedLFV effects on the cross section of the μN→ τX reaction.Analytical relations and correlations among the rates of theabove processes have been established at the two loop levelin the Higgs boson exchange. The correlations among theprocesses are a precise signature of the theory. In this respectexperimental improvements in all the decay channels of theτ lepton would be very welcome. In conclusion, the Higgs-mediated contributions to LFV processes can be within thepresent or upcoming attained experimental resolutions andprovide an important opportunity to detect new physics be-yond the standard model.

5.6 Tests of unitarity and universality in the lepton sector

5.6.1 Deviations from unitarity in the leptonic mixingmatrix

The presence of physics beyond the SM in the leptonic sec-tor can generate deviations from unitarity in the mixing ma-trix. This is analogous to what happens in the quark sector,where the search for deviations from unitarity of the CKMmatrix is considered a sensitive way to look for new physics.

In the leptonic sector a clear example of non-unitarityis given by the see-saw mechanism [216–220]. To generatenaturally small neutrino masses, new heavy particles—righthanded neutrinos—are added, singlet under the SM gaugegroup. Thus a Yukawa coupling for neutrinos can be writ-ten, as well as Majorana masses for the new heavy fields.The mass matrix of the complete theory is now an enlargedmass matrix (5 × 5 at least), whose diagonalization leads tosmall Majorana neutrino masses. The non-unitarity of the3× 3 leptonic mixing matrix can now be understood simplyby observing that it is a sub-matrix of a bigger one which isunitary, since the complete theory must conserve probabili-ties.

Another way to see this is looking at the effective the-ory we obtain once the heavy fields are integrated out. Theunique dimension five operator is the well-known Weinbergoperator [213] which generates neutrino masses when theelectroweak symmetry is broken. Masses are naturally smallsince they are suppressed by the mass M of the heavy par-ticles which have been integrated out: mν ∼ v2/M , where vis the Higgs VEV. If we go on in the expansion in effectiveoperators, we obtain only one dimension six operator whichrenormalizes the kinetic energy of neutrinos. Once we per-form a field redefinition to go into a mass basis with canon-ical kinetic terms, a non-unitary mixing matrix is obtained[813]. In minimal models deviations from unitarity gener-

ated in this way are very suppressed, since the dimensionsix operator is proportional to v2/M2. However, in more so-phisticated versions of this mechanism like double (or in-verse) see-saw [814] the suppression can be reduced with-out affecting the smallness of neutrinos masses and avoid-ing any fine-tuning of Yukawa couplings. In terms of effec-tive operators, this means that it is possible to “decouple”the dimension five operator from the dimension six, permit-ting small neutrino masses and not so small unitarity devia-tions.

Usually the elements of the leptonic mixing matrix aremeasured using neutrino oscillation experiments assumingunitarity. No information can be extracted from electroweakdecays on the individual matrix elements, due to the im-possibility of detecting neutrino mass eigenstates. This isquite different from the way of measuring the CKM ma-trix elements. Here oscillations are important too, but sincequark mass eigenstates can be tagged, direct measurementsof the matrix elements can be made using electroweak de-cays.

The situation changes if we relax the hypothesis of uni-tarity of the leptonic mixing matrix. Electroweak decays ac-quire now an important meaning, since they can be used toconstrain deviations from unitarity. Consider as an exam-ple the decay W → lνl . The decay rate is modified as fol-lows: Γ = ΓSM(NN

†)ll , where N is the non-unitary lep-tonic mixing matrix and ΓSM is the SM decay rate. This,and other electroweak processes, can therefore be used toobtain information on (NN†)ll . Moreover, lepton flavor vi-olating processes like μ→ 3e or μ–e conversion in nucleican occur, while rare lepton decays like li → lj γ can be en-hanced, permitting to constrain the off-diagonal elements of(NN†). Finally, universality violation effects are produced,even if the couplings are universal: for example the branch-ing ratio of π decay (see Sect. 6) is now proportional to(NN†)ee/(NN

†)μμ.In Ref. [815] all these processes have been considered,

a global fit has been performed and the matrix |(NN†)| hasbeen determined (90% C.L.):

∣∣NN†∣∣≈⎛⎝

0.994 ± 0.005 <7.0 × 10−5 <1.6 × 10−2

<7.0 × 10−5 0.995 ± 0.005 <1.0 × 10−2

<1.6 × 10−2 <1.0 × 10−2 0.995 ± 0.005

⎞⎠ .

(5.111)

Similar bounds can be inferred for |N†N |, leading to theconclusion that deviations from unitarity in the leptonic mix-ing matrix are experimentally constrained to be smaller thanfew percent. Notice however that these bounds apply to a3× 3 mixing matrix, i.e. they constrain deviations from uni-

110 Eur. Phys. J. C (2008) 57: 13–182

tarity induced by higher energy physics which has been in-tegrated out.26

However, since on the contrary the quark sector decayscan only constrain the elements of |(NN†)|, to determinethe individual elements of the leptonic mixing matrix, oscil-lation experiments are needed. In Ref. [815] neutrino oscil-lation physics is reconsidered in the case in which the mix-ing matrix is not unitary. The main consequence of this isthat the flavor basis is no longer orthogonal, which givesrise to two physical effects:

– “zero distance” effect, i.e. flavor conversion in neutrinooscillations at L= 0: Pνανβ (E,L= 0)A· ∝ |(NN†)βα|2;

– non-diagonal matter effects.

With the resulting formulas for neutrino oscillations, a fit topresent oscillation experiments is performed, in order to de-termine the individual matrix elements. As in the standardcase, no information at all is available on phases (four orsix, depending on the nature—Dirac or Majorana—of theneutrinos), since appearance experiments would be needed.However the moduli of matrix elements can be determined,but now they are all independent, so that the free parame-ters are nine instead of three. The elements of the e-row canbe constrained using the data from CHOOZ [816], Kam-LAND [817] and SNO [818], together with the informa-tion on �m2

23 resulting from an analysis of K2K [819]. Incontrast, less data are available for the μ-row: only thosecoming from K2K and Super-Kamiokande [820] on at-mospheric neutrinos, and only |Nμ3| and the combination|Nμ1|2 + |Nμ2|2 can be determined. No information at all isavailable on the τ -row. The final result is the following (3σranges):

|N | =⎛⎝

0.75–0.89 0.45–0.66 <0.34[(|Nμ1|2 + |Nμ2|2)1/2 = 0.57–0.86

]0.57–0.86

? ? ?

⎞⎠ .

(5.112)

Notice that, without assuming unitarity, only half of theelements can be determined from oscillation experimentsalone. Adding the information from near detectors at NO-MAD [821], KARMEN [822], BUGEY [823] and MINOS[824], which put bounds on |(NN†)αβ |2 by measuring the“zero distance” effect, the degeneracy in the μ-row can besolved, but the τ -row is still unknown.

In order to determine/constrain all the elements of theleptonic mixing matrix without assuming unitarity, data on

26They do not apply for instance to the case of light sterile neutrinos,where the low energy mixing matrix is larger. Indeed in this case theywould be included in the sum over all light mass eigenstates containedinside (NN†)ll and unitarity would be restored.

oscillations must be combined with data from decays. Thefinal result is

|N | =⎛⎝

0.75–0.89 0.45–0.65 <0.200.19–0.55 0.42–0.74 0.57–0.820.13–0.56 0.36–0.75 0.54–0.82

⎞⎠ , (5.113)

which can be compared to the one obtained with standardanalysis [825] where similar bounds are found.

It would be good to be able to determine the elements ofthe mixing matrix with oscillation experiments alone, per-mitting thus a “direct” test of unitarity. This would be forinstance a way to detect light sterile neutrinos [826]. Thiscould be possible exploring the appearance channels for in-stance at future facilities under discussion, such as Super-Beams [713, 827–829], β-Beams [830] and Neutrino Facto-ries [831, 832], where the τ -row and phases could be mea-sured. Moreover, near detectors at neutrino factories couldalso improve the bounds on (NN†)eτ and (NN†)μτ byabout one order of magnitude. All this information, com-ing from both decays and oscillation experiments, will beimportant not only to detect new physics, but even to dis-criminate among different scenarios.

5.6.2 Lepton universality

High precision electroweak tests (HPET) represent a power-ful tool to probe the SM and, hence, to constrain or obtainindirect hints of new physics beyond it. A typical and rel-evant example of HPET is represented by the Lepton Uni-versality (LU) breaking. Kaon and pion physics are obviousgrounds where to perform such tests, for instance in the wellstudied π�2 (π → �ν�) and K�2 (K → �ν�) decays, wherel = e or μ.

Unfortunately, the relevance of these single decay chan-nels in probing the SM is severely hindered by our theoret-ical uncertainties, which still remain at the percent level (inparticular due to the uncertainties on non-perturbative quan-tities like fπ and fK ). This is what prevents us from fullyexploiting such decay modes in constraining new physics, inspite of the fact that it is possible to obtain non-SM contri-butions which exceed the high experimental precision whichhas been achieved on those modes.

On the other hand, in the ratios Rπ and RK of the elec-tronic and muonic decay modes Rπ = Γ (π→ eν)/Γ (π→μν) and RK = Γ (K→ eν)/Γ (K→ μν), the hadronic un-certainties cancel to a very large extent. As a result, the SMpredictions of Rπ and RK are known with excellent accu-racy [833] and this makes it possible to fully exploit the greatexperimental resolutions on Rπ [834] and RK [834, 835] toconstrain new physics effects. Given our limited predictivepower on fπ and fK , deviations from the μ–e universalityrepresent the best hope we have at the moment to detect newphysics effects in π�2 and K�2.

Eur. Phys. J. C (2008) 57: 13–182 111

The most recent NA48/2 result on RK :

Rexp.K = (2.416 ± 0.043stat. ± 0.024syst.)× 10−5 NA48/2,

which will further improve with current analysis, signifi-cantly improves on the previous PDG value:

Rexp.K = (2.44 ± 0.11)× 10−5.

This is to be compared with the SM prediction, which reads

RSMK = (2.472 ± 0.001)× 10−5.

The details of the experimental measurement of RK are pre-sented in Sect. 6.2 of this report. Denoting by �re−μNP thedeviation from μ–e universality in RK due to new physics,i.e.,

RK =RSMK

(1 +�re−μNP

), (5.114)

the NA48/2 result requires (at the 2σ level):

−0.063 ≤�re−μNP ≤ 0.017 NA48/2.

In the following, we consider low energy minimal SUSYextensions of the SM (MSSM) with R parity as the sourceof new physics to be tested by RK [836]. The question weintend to address is whether SUSY can cause deviationsfrom μ–e universality in Kl2 at a level which can be probedwith the present attained experimental sensitivity, namely atthe percent level. We shall show that (i) it is indeed possi-ble for regions of the MSSM to obtain �re−μNP of O(10−2)

and (ii) such large contributions to K�2 do not arise fromSUSY lepton flavor conserving (LFC) effects, but, rather,from LFV ones.

At first sight, this latter statement may seem rather puz-zling. The K→ eνe and K→ μνμ decays are LFC and onecould expect that it is through LFC SUSY contributions af-fecting differently the two decays that one obtains the domi-nant source of lepton flavor non-universality in SUSY. Onthe other hand, one can easily guess that, whenever newphysics intervenes in K → eνe and K → μνμ to create adeparture from strict SM μ–e universality, these new contri-butions will be proportional to the lepton masses; hence, itmay happen (and, indeed, this is what occurs in the SUSYcase) that LFC contributions are suppressed with respectto the LFV ones by higher powers of the first two genera-tions lepton masses (it turns out that the first contributionsto �re−μNP from LFC terms arise at the cubic order in m�,with � = e,μ). A second, important reason for such resultis that among the LFV contributions to RK one can selectthose which involve flavor changes from the first two leptongenerations to the third one with the possibility of pickingup terms proportional to the tau–Yukawa coupling whichcan be large in the large tanβ regime (the parameter tanβ

denotes the ratio of Higgs vacuum expectation values re-sponsible for the up- and down-quark masses, respectively).Moreover, the relevant one loop induced LFV Yukawa inter-actions are known [806] to acquire an additional tanβ factorwith respect to the tree level LFC Yukawa terms. Thus, theloop suppression factor can be (partially) compensated inthe large tanβ regime.

Finally, given the NA48/2 RK central value below theSM prediction, one may wonder whether SUSY contribu-tions could have the correct sign to account for such an ef-fect. Although the above mentioned LFV terms can only addpositive contributions to RK (since their amplitudes cannotinterfere with the SM one), it turns out that there exist LFCcontributions arising from double LFV mass insertions (MI)in the scalar lepton propagators which can destructively in-terfere with the SM contribution. We shall show that thereexist regions of the SUSY parameter space where the to-tal RK arising from all such SM and SUSY terms is indeedlower than RSM

K .Finally, we also discuss the potentiality of τ–μ(e) uni-

versality breaking in τ decays to probe new physics ef-fects.

5.6.2.1 μ–e universality in π → �ν and K → �ν decaysDue to the V –A structure of the weak interactions, theSM contributions to π�2 and K�2 are helicity suppressed;hence, these processes are very sensitive to non-SM effects(such as multi-Higgs effects) which might induce an effec-tive pseudoscalar hadronic weak current.

In particular, charged Higgs bosons (H±) appearingin any model with two Higgs doublets (including theSUSY case) can contribute at tree level to the aboveprocesses. The relevant four-Fermi interaction for the de-cay of charged mesons induced by W± and H± has thefollowing form:

4GF√2Vud

[(uγμPLd)

(lγ μPLν

)

− tan2 β

(mdml

m2H±

)(uPRd)(lPLν)

], (5.115)

where PR,L = (1 ± γ5)/2. Here we keep only the tanβ en-hanced part of the H±ud coupling, namely the md tanβterm. The decays M → lν (being M the generic meson)proceed via the axial-vector part of the W± coupling andvia the pseudoscalar part of the H± coupling. Then, oncewe implement the PCACs

〈0|uγμγ5d∣∣M−⟩= ifMpμM,

〈0|uγ5d∣∣M−⟩=−ifM m2

M

md +mu ,(5.116)

112 Eur. Phys. J. C (2008) 57: 13–182

we easily arrive at the amplitude

MM→lν = GF√2Vu(d,s)fM

[ml −ml tan2 β

×(

md

md +mu)m2M

m2H±

]l(1 − γ5)ν. (5.117)

We observe that the SM term is proportional to ml becauseof the helicity suppression while the charged Higgs term isproportional toml because of the Yukawa coupling. The treelevel partial width is given by [806]

Γ(M− → l−ν

)= G2F

8π|Vu(d,s)|2f 2

MmMm2l

(1− m2

l

m2M

)×rM,

(5.118)

where

rM =[

1 − tan2 β

(md,s

mu +md,s)m2M

m2H±

]2

, (5.119)

and where mu is the mass of the up quark while ms,d standsfor the down-type quark mass of theM meson (M =K,π ).From (5.119) it is evident that such tree level contribu-tions do not introduce any lepton flavor dependent correc-tion. The first SUSY contributions violating the μ–e uni-versality in π → �ν and K → �ν decays arise at the oneloop level with various diagrams involving exchanges of(charged and neutral) Higgs scalars, charginos, neutrali-nos and sleptons. For our purpose, it is relevant to divideall such contributions into two classes: (i) LFC contribu-tions where the charged meson M decays without FCNCin the leptonic sector, i.e. M→ �ν�; (ii) LFV contributionsM → �iνk , with i and k referring to different generations(in particular, the interesting case will be for i = e,μ, andk = τ ).

5.6.2.2 The lepton flavor conserving case One-loop cor-rections to Rπ and RK include box, wave function renor-malization and vertex contributions from SUSY particle ex-change. The complete calculation of the μ decay in theMSSM [837] can be easily applied to meson decays.

The dominant diagrams containing one loop correctionsto the lWνl vertex have the following suppression factors(compared to the tree level graph):

–g2

216π2

m2l

m2W

tanβm2W

m2h

for loops with hW±l exchange (with

h=H 0, h0),

–g2

216π2

m2l

m2W

tan2 βm2W

m2h

for loops with hH±l exchange (with

h=H 0, h0 and A0),

–g2

216π2

m2W

M2SUSY

for loops generated by charginos/neutralinos

and sleptons.

For dominant box contributions we have the following esti-mates:

–g2

216π2

mdmlM2 tan2 β for boxes with hW±l or Z0H±l ex-

change (where M is the heavier mass circulating in theloop),

–g2

216π2 (

mdmlmWMH± )

2 tan4 β for boxes with hH±l ,

–g2

216π2

m2W

M2SUSY

for loops generated by charginos/neutralinos

and sleptons (where MSUSY is the heavier mass circulat-ing in the loop).

To get a feeling of the order of magnitude of the above con-tributions let us show the explicit expression of the dominantHiggs contributions to the lWνl vertex [837]:

�re−μSUSY = α2

32π

m2μ

M2W

tan2 β(−2 + I(A0,H±)

+ c2αI(H 0,H±)+ s2

αI(h0,H±)),

where

I (1,2)= 1

2

m21 +m2

2

m21 −m2

2

logm2

1

m22

,

and α is the mixing angle in the CP-even Higgs sector. Evenif we assume tanβ = 50 and arbitrary relations among theHiggs boson masses we get a value for �re−μSUSY ≤ 10−6

much below the actual experimental resolution. In addition,in the large tanβ limit, α→ 0 and mA0 ∼mH 0 ∼mH± and�r

e−μSUSY tends to vanish. The charginos/neutralinos sleptons

(le,μ) contributions to �re−μSUSY are of the form

�re−μSUSY ∼ α2

4π

(m2μ − m2

e

m2μ + m2

e

)m2W

M2SUSY

.

The degeneracy of slepton masses (in particular those ofthe first two generations) severely suppresses these contribu-tions. Even if we assume a quite large mass splitting amongslepton masses (at the 10% level for instance) we end upwith �re−μSUSY ≤ 10−4. For the box-type non-universal con-tributions, we find similar or even more suppressed effectscompared to those we have studied. So, finally, it turns outthat all these LFC contributions yield values of �re−μK SUSYwhich are much smaller than the percent level required bythe achieved experimental sensitivity.

On the other hand, one could wonder whether the quan-tity �re−μSUSY can be constrained by the pion physics. In prin-ciple, the sensitivity could be even higher: from

Rexp.π = (1.230 ± 0.004)× 10−4 PDG,

and by making a comparison with the SM prediction

RSMπ = (1.2354 ± 0.0002)× 10−4,

Eur. Phys. J. C (2008) 57: 13–182 113

one obtains (at the 2σ level)

−0.0107 ≤�re−μNP ≤ 0.0022.

Unfortunately, even in the most favorable cases, �re−μSUSY re-mains much below its actual experimental upper bound.

In conclusion, SUSY effects with flavor conservation inthe leptonic sector can differently contribute to theK→ eνeandK→ μνμ decays, hence inducing μ–e non-universalityin RK , however such effects are still orders of magnitudebelow the level of the present experimental sensitivity onRK . The same conclusions hold for Rπ .

5.6.2.3 The lepton flavor violating case It is well knownthat models containing at least two Higgs doublets gener-ally allow for flavor violating couplings of the Higgs bosonswith the fermions [838]. In the MSSM such LFV couplingsare absent at tree level. However, once non-holomorphicterms are generated by loop effects (so called HRS correc-tions [430]) and given a source of LFV among the slep-tons, Higgs-mediated (radiatively induced)H�i�j LFV cou-plings are unavoidable [806, 807]. These effects have beenwidely discussed through the study of several processes,namely τ → �j �k�k [806, 807], τ → μη [810], μ–e con-version in nuclei [811], B → �j τ [807], H → �j �k [174]and �i → �jγ [650].

Moreover, it has been shown [839] that Higgs-mediatedLFV couplings generate a breaking of μ–e universality inpurely leptonic π± and K± decays.

One could naively think that SUSY effects in the LFVchannels M → �iνk are further suppressed with respect tothe LFC ones. On the contrary, charged Higgs mediatedSUSY LFV contributions, in particular in the kaon decaysinto an electron or a muon and a tau neutrino, can be stronglyenhanced. The quantity which now accounts for the devia-tion from the μ–e universality reads

RLFVπ,K =

∑i Γ (π(K)→ eνi)∑i Γ (π(K)→ μνi)

, i = e,μ, τ,

with the sum extended over all (anti)neutrino flavors (exper-imentally one determines only the charged lepton flavor inthe decay products).

The dominant SUSY contributions to RLFVπ,K arise from

the charged Higgs exchange. The effective LFV Yukawacouplings we consider are (see Fig. 39)

�H±ντ → g2√2

mτ

MWΔ3lR tan2 β, �= e,μ. (5.120)

A crucial ingredient for the effects we are going to discussis the quadratic dependence on tanβ in the above coupling:one power of tanβ comes from the trilinear scalar couplingin Fig. 39, while the second one is a specific feature of theabove HRS mechanism.

Fig. 39 Contribution to the effective ντ �RH+ coupling

The Δ3�R terms are induced at one loop level by the ex-

change of bino (see Fig. 39) or bino–higgsino and sleptons.Since the Yukawa operator is of dimension four, the quan-tities Δ3�

R depend only on ratios of SUSY masses, henceavoiding SUSY decoupling. In the so called MI approxima-tion the expression of Δ3�

R is given by:

Δ3�R α1

4πμM1m

2Rδ

3�RR

[I ′(M2

1 ,μ2,m2

R

)− (μ↔mL)],

(5.121)

where μ is the Higgs mixing parameter, M1 is the bino (B)mass and m2

L(R) stands for the left–left (right–right) slepton

mass matrix entry. The LFV MIs, i.e. δ3�XX = (m2

�)3�XX/m

2X

(X = L,R), are the off-diagonal flavor changing entriesof the slepton mass matrix. The loop function I

′(x, y, z)

is such that I′(x, y, z) = dI (x, y, z)/dz, where I (x, y, z)

refers to the standard three point one loop integral whichhas mass dimension-2.

Making use of the LFV Yukawa coupling in (5.120), itturns out that the dominant contribution to �re−μNP reads[839]

RLFVK RSM

K

[1 +

(m4K

M4H

)(m2τ

m2e

)∣∣Δ31R

∣∣2 tan6 β

]. (5.122)

In (5.122) terms proportional toΔ32R are neglected given that

they are suppressed by a factor m2e/m

2μ with respect to the

term proportional to Δ31R .

Taking Δ31R 5 × 10−4 (by means of a numerical analy-

sis, it turns out that Δ3�R ≤ 10−3 [174]), tanβ = 40 and

MH = 500 GeV we end up with RLFVK RSM

K (1 + 0.013).We see that in the large (but not extreme) tanβ regime andwith a relatively heavy H±, it is possible to reach contribu-tions to �re−μK SUSY at the percent level thanks to the possibleLFV enhancements arising in SUSY models.

Turning to pion physics, one could wonder whether theanalogous quantity �re−μπ SUSY is able to constrain SUSY

LFV. However, the correlation between �re−μπ SUSY and

�re−μK SUSY:

�re−μπ SUSY

(md

mu +md)2(

m4π

m4k

)�r

e−μK SUSY, (5.123)

114 Eur. Phys. J. C (2008) 57: 13–182

clearly shows that the constraints on �re−μK SUSY force

�re−μπ SUSY to be much below its actual experimental upper

bound.

5.6.2.4 On the sign of �re−μSUSY The above SUSY domi-

nant contribution to �re−μNP arises from LFV channels in theK → eν mode, hence without any interference effect withthe SM contribution. Thus, it can only increase the valueof RK with respect to the SM expectation. On the otherhand, the recent NA48/2 result exhibits a central value lowerthan RSM

K (and, indeed, also lower than the previous PDGcentral value). One may wonder whether SUSY could ac-count for such a lower RK . Obviously, the only way it canis through terms which, contributing to the LFC K → lνlchannels, can interfere (destructively) with the SM contribu-tion. We already commented that SUSY LFC contributionsare subdominant. However, one can envisage the possibilityof making use of the large LFV contributions to give riseto LFC ones through double LFV MI that, as a final effect,preserves the flavor.

To see this point explicitly, we report the corrections tothe LFC H±�ν� vertices induced by LFV effects

�H±ν�→ g2√2

m�

MWtanβ

(1 + mτ

m�Δ��RL tanβ

), (5.124)

where Δ��RL is generated by the same diagram as in Fig. 39but with an additional δ3�

LL MI in the sneutrino propagator.In the MI approximation, ��RL is given by

Δ��RL − α1

4πμM1m

2Lm

2Rδ�3RRδ

3�LLI

′′(M21 ,m

2L,m

2R

),

(5.125)

where I ′′(x, y, z)= d2I (x, y, z)/dy dz. In the large sleptonmixing case, Δ��RL terms are of the same order of Δ3�

R .27

These new effects modify the previous RLFVK expression in

the following way [839]:

RLFVK RSM

K

[∣∣∣∣1 − m2K

M2H

mτ

meΔ11RLt

3β

∣∣∣∣2

+(m4K

M4H

)(m2τ

m2e

)∣∣Δ31R

∣∣2 tan6 β

]. (5.126)

In the above expression, besides the contributions reportedin (5.122), we also included the interference between SMand SUSY LFC terms (arising from a double LFV source).Setting the parameters as in the example of the above sectionand if Δ11

RL = 10−4 we get RLFVK RSM

K (1 − 0.032), that is

27Im(δ13RRδ

31LL) is strongly constrained by the electron electric dipole

moment [163]. However, sizable contributions to RLFVK can still be in-

duced by Re(δ13RRδ

31LL).

just within the expected experimental resolution reachableby NA48/2 once all the available data will be analyzed. Fi-nally, we remark that the above effects do not spoil the pionphysics constraints.

The extension of the above results to B → �ν [766] isobtained with the replacement mK → mB , while for theD → �ν case m2

K → (ms/mc)m2D . In the most favorable

scenarios, taking into account the constraints from LFV τdecays [650], spectacular order-of-magnitude enhancementsfor Re/τB and O(50%) deviations from the SM in Rμ/τB areallowed [766]. There exists a stringent correlation betweenRe/τB and Re/μK so that

Re/τB

[rH + m

4B

m4K

�re−μK SUSY

]≤ 2 × 102. (5.127)

In particular, it turns out that �re−μK SUSY is much more ef-

fective to constrain Re/τB Γ (B→ eντ ) than LFV tau decayprocesses.

5.6.2.5 Lepton universality in M → �ν versus LFV τ de-cays Obviously, a legitimate worry when witnessing sucha huge SUSY contribution through LFV terms is whetherthe bounds on LFV tau decays, like τ → eX (with X =γ,η,μμ), are respected [650]. Higgs mediated B(τ →�jX) and �re−μK SUSY have exactly the same SUSY depen-dence; hence, we can compute the upper bounds of the rel-evant LFV tau decays which are obtained for those valuesof the SUSY parameters yielding �re−μK SUSY at the percentlevel.

The most sensitive processes to Higgs mediated LFV inthe τ lepton decay channels are τ → μ(e)η, τ → μ(e)μμ

and τ→ μ(e)γ . The related branching ratios are [650]

B(τ→ lj η)

B(τ → lj νj ντ ) 18π2

(f 8η m

2η

mτ

)2(1 − m

2η

m2τ

)2

×( |Δ3j |2 tan6 β

m4A

), (5.128)

wherem2η/m

2τ 9.5×10−2 and the relevant decay constant

is f 8η ∼ 110 MeV,

B(τ→ lj γ )

B(τ → lj νj ντ ) 10

(αel

π

)3

tan4 β|Δτj |2

×[mW

mAlog

(m2W

m2A

)]4

, (5.129)

B(τ → ljμμ)

B(τ → lj νj ντ ) m

2τm

2μ

32

( |Δ3j |2 tan6 β

m4A

)

× (3 + 5δjμ), (5.130)

Eur. Phys. J. C (2008) 57: 13–182 115

where |Δ3j |2 = |Δ3jL |2 + |Δ3j

R |2. It is straightforward tocheck that, in the large tanβ regime, B(τ → lj η) andB(τ → lj γ ) are of the same order of magnitude [650] andthey are dominant compared to B(τ→ ljμμ).28

Given that �re−μK SUSY and B(τ → ljX) have the same

SUSY dependence, once we saturate the�re−μK SUSY value (atthe % level), the upper bounds on B(τ → ljX) (allowed by|Δ31R |2) are automatically predicted. We find that

B(τ→ lj γ ) ∼ B(τ→ lj η) 10−2( |Δ3j |2 tan6 β

m4A

)

10−8�re−μK SUSY. (5.131)

So, employing the constraints for �re−μK SUSY at the %level, we obtain the desired upper bounds: B(τ → eη),

B(τ → eγ )≤ 10−10. Given the experimental upper boundson the LFV τ lepton decays [764], we conclude that it ispossible to saturate the upper bound on �re−μK SUSY while re-maining much below the present and expected future upperbounds on such LFV decays. There exist other SUSY con-tributions to LFV τ decays, like the one loop neutralino-charged slepton exchanges, for instance, where there is a di-rect dependence on the quantities δ3j

RR . Given that the exist-ing bounds on the leptonic δRR involving transitions to thethird generation are rather loose [681], it turns out that alsothese contributions fail to reach the level of experimentalsensitivity for LFV τ decays.

5.6.2.6 e–μ universality in τ decays Studying the τ–μ–euniversality in the leptonic τ decays is an interesting labora-tory for search for physics beyond the SM. In the SM the τdecay partial width for the leptonic modes is

Γ(τ→ lνlντ (γ )

)

= G2Fm

5τ

192π3f(m2l /m

2τ

)

×[

1 + 3

5

m2τ

M2W

][1 + α(mτ )

π

(25

4− π2

)], (5.132)

where f (x)= 1 − 8x + 8x3 − x4 − 12x2 logx is the leptonmass correction and the last two factors are corrections fromthe nonlocal structure of the intermediate W± boson propa-gator and QED radiative corrections respectively. The Fermiconstant GF is determined by the muon life-time

GF ≡Gμ = (1.16637 ± 0.00002)× 10−5 GeV−2, (5.133)

28It is remarkable that �re−μK SUSY ∼ |Δ31R |2 while B(τ → eX) ∼

|Δ31L |2 + |Δ31

R |2 (with X = η,γ or μμ). In practice, �re−μK SUSY is sen-sitive only to RR-type LFV terms in the slepton mass matrix whileB(τ→ eX) does not distinguish between left and right sectors.

and absorbs all the remaining electroweak radiative (loop)corrections.

The main source of non-universal contributions would bethe tree level contribution from the charged Higgs boson(mass dependent couplings) and different slepton masses ofthe μ, τ and e sleptons exchanged in the one loop induced�–W–ν� vertex. On the other hand, as discussed in previoussections, the last contribution can provide a correction thatcan be at most as large as 10−4 (in the limiting case of verylight sleptons and gauginos ∼MW ), very far for the actualand forthcoming experimental resolutions. However, differ-ently from theM→ �ν case, a tree level charged Higgs ex-change breaks the lepton universality and it provides a con-tribution that we are going to discuss.

The deviations from τ − μ − e universality can beconveniently discussed by studying the ratios Gτ,e/Gμ,e ,Gτ,μ/Gμ,e and Gτ,μ/Gτ,e , given by the ratios of the cor-responding branching fractions. With the highly accurateexperimental result for the Gμ,e , the first two ratios are es-sentially a direct measure of non-universality in the corre-sponding tau decays. When the statistical error of futureexperiments will become negligible, the main problem forachieving maximum precision will be to reduce the system-atic errors. One may expect that certain systematic errorswill be canceled in the ratio Gτ,μ/Gτ,e .

The 2004 world averaged data for the leptonic τ decaymodes and τ life-time are [834, 840]

Be|exp = (17.84 ± 0.06)%,

Bμ|exp = (17.37 ± 0.06)%, (5.134)

ττ = (290.6 ± 1.1)× 10−15 s.

Note that the relative errors of the above measured quantitiesare of the 0.34–0.38%, the biggest being for the life-time.One can parameterize a possible beyond the SM contribu-tion by a quantity Δl (l = e,μ), defined as

Bl = Bl |SM(1 +Δl). (5.135)

Including the W -propagator effect and QED radiative cor-rections, the following results for the branching ratios in theSM are obtained [840]:

Be|SM = (17.80 ± 0.07)%,

Bμ|SM = (17.32 ± 0.07)%.(5.136)

Together with the experimental data this leads to the fol-lowing 95% C.L. bounds on Δl , for the electron and muondecay mode, respectively [840]:

(−0.80 ≤Δe ≤ 1.21)%,

(−0.76 ≤Δμ ≤ 1.27)%.(5.137)

116 Eur. Phys. J. C (2008) 57: 13–182

One can see that the negative contributions are constrainedmore strongly that the positive ones. A tree level chargedHiggs exchange leads to the following contribution [841]:

Γ W±+H±

= Γ W±[

1 − 2mlmτ tan2 β

M2H±

ml

mτκ

(m2l

m2τ

)+ m

2τm

2l tan4 β

4M4H±

]

Γ W±[

1 − 1.15 × 10−3(

200 GeV

MH±

)2( tanβ

50

)2],

(5.138)

where κ(x)= g(x)f (x)

0.94 with g(x)= 1+9x−9x2 −x3 +6x(1 + x) ln(x). In the above expression, the second termcomes from the interference with the SM amplitude and it ismuch more important than the last one, which is suppressedby a factor m2

τ tan2 β/8M2H± .

For the future precision of Gτ,μ and Gτ,e measurementsof order 0.1% (Gμ,e is known with 0.002% precision) theonly effect that eventually can be observed is the slightlysmaller value of Gτ,μ as compared to Gτ,e and Gμ,e . Ifmeasured, such effect would mean a rather precise informa-tion about MSSM: large tanβ ≥ 40 and smallMH± ∼ 200–300 GeV.

5.6.2.7 Conclusions High precision electroweak tests,such as deviations from the SM expectations of the lep-ton universality breaking, represent a powerful tool to probethe SM and, hence, to constrain or obtain indirect hints ofnew physics beyond it. Kaon and pion physics are obviousgrounds where to perform such tests, for instance in the wellstudied π → �ν� and K → �ν� decays, where l = e or μ.In particular, a precise measurement of the flavor conserv-ing K→ �ν� decays may shed light on the size of LFV innew physics. μ–e non-universality in K�2 is quite effectivein constraining relevant regions of SUSY models with LFV.A comparison with analogous bounds coming from τ LFVdecays shows the relevance of the measurement of RK toprobe LFV in SUSY. Moreover, τ–μ–e universality in theleptonic τ decays is an additional interesting laboratory forsearching for physics beyond the SM.

5.7 EDMs from RGE effects in theories with low energysupersymmetry

EDMs probe new physics in general and in particular lowenergy supersymmetry. For definiteness and for simplic-ity, we focus here on lepton EDMs, as they are free fromthe theoretical uncertainties associated to the calculation ofhadronic matrix elements. After a brief review of the con-straints on slepton masses we discuss here a specific kindof sources of CPV, those induced radiatively by the Yukawainteractions of the heavy particles present in see-saw and/or

grand unified models. It has been emphasized that these in-teractions could lead to LFV decays, in particular μ→ eγ ,at an observable rate; it is then natural to wonder whetherthis is also the case for EDMs.

As shown in Sect. 3, LFV decays, EDMs and additionalcontribution to MDMs all have a common origin, the di-mension five dipole operator possibly induced by some newphysics beyond the SM:

Ld=5 = 1

2ψRiAijψLjσ

μνFμν + h.c., (5.139)

B(�i → �jγ )∝ |Aij |2, δa�i =2m�ie

ReAii,

(5.140)d�i = ImAii.

If induced at one loop, this amplitude displays a quadraticsuppression with respect to the new physics mass scale,MNP, and a linear dependence on the non-dimensional cou-pling Γ NP encoding the pattern of F and CP violations (inthe basis where the charged lepton mass matrix is diagonal):

Aij ≈ em�i

(4π)2Γ NPij

M2NP

. (5.141)

For low energy supersymmetry, the loops involve ex-change of gauginos and sleptons, so that Γ NP is proportionalto the misalignment between leptons and sleptons, conve-niently described by the flavor violating (FV) δ’s of the massinsertion approximation. It is well known that the flavor con-serving (FC) μ and a terms are potentially a very importantsource of CPV. In the expansion in powers of the FV δ’s,they indeed contribute to d�i at zero order:

Im(Aii) = fμm�i Arg(μ)+ fam�i Im(ai)

+ fLLRR Im(δLLm�δ

RR)ii+ · · · , (5.142)

where the various f represent supersymmetric loop func-tions and can be found for instance in [163]. Notice that thecontribution arising at second order in the FV δ’s could beeven more important than the FC one, as happens for in-stance if CPV is always associated to FV.

Assuming no cancellations between the amplitudes, wefirst review briefly some limits considering for definite-ness the mSUGRA scenario with tanβ = 10 and sleptonmasses in the range suggested by gμ. The strong impactof μ→ eγ on δLL has been emphasized previously, whereit was stressed that the impact of de is also remarkablein constraining the FC sources of CPV: argμ ≤ 2 × 10−3,Imae/mR ≤ 0.2. As for the other FV source in (5.142), oneobtains Im(δLLm�δRR)ee/mτ ≤ 10−5. The planned sensi-tivity dμ ≤ 10−23 e cm would also give interesting bounds:argμ ≤ 10−1, Im(δLLm�δRR)μμ/mτ ≤ 10−1. Notice that,

Eur. Phys. J. C (2008) 57: 13–182 117

due to the lepton mass scaling law of the μ-term contribu-tion, the present bound on argμ from de implies that theμ-term contribution to dμ cannot exceed 2 × 10−25 e cm,below the planned projects. A positive measure of dμ wouldthus signal a different source of CPV, i.e. the aμ-term or theFV contribution. In the following we take real μ.

The ai -terms and the FV δ’s at low energy can be thoughtas the sum of two contributions. The first is already presentat the Planck scale where soft masses are defined; we as-sume that this contribution is absent because of some inhi-bition mechanism, as could happen in supergravity. The sec-ond contribution is induced radiatively running from high tolow energies by the Yukawa couplings of heavy particles29

that potentially violate F and CP. Since LFV experimentsare testing this radiatively induced misalignment, in the fol-lowing we shall consider what happens for EDMs, begin-ning with the pure see-saw model and then adding a grandunification scenario, where heavy colored Higgs triplets arepresent to complete the Higgs doublets representations (inSU(5) for instance they complete the 5 and 5). Notice thatthese triplets are important as in supersymmetric theoriesproton decays mainly through their exchange.

Consider first the case of degenerate right handed neutri-nos with massM . One can solve approximately the RGE byexpanding in powers of ln(Λ/M)/(4π)2, i.e. the log of theratio of the two scales between which the neutrino Yukawacouplings Yν are present times the corresponding loop fac-tor suppression. For LFV decays δLLij is induced at 1st order

and is proportional to the combination (Y †ν Yν)ij . In partic-

ular, μ→ eγ constrains (Y †ν Yν)21 to be small and this has

a strong impact on see-saw models. To obtain an imaginarypart for EDMs, one needs a non-hermitian combination ofYukawa couplings, which can be found only at 4th order:Im(Y †

ν Yν[Y †ν Yν,Y

†� Y�]Y †

ν Yν)ii . Such a contribution is negli-gibly small with respect to the present and planned experi-mental sensitivities.

Allowing for a non-degenerate spectrum of right handedneutrinos, EDMs get strongly enhanced while LFV decaysnot. The latter are simply modified by taking into accountthe different mass thresholds:

δLLij ∝∑k

Ck, Ck ≡ Y †ν ik ln

Λ

MkYνkj . (5.143)

On the contrary for EDMs the see-saw induced FC andFV contributions—coming respectively from Imai [229,242, 842, 843] and Im(δRRm�δLL)ii [843, 844]—arise at2nd and 3rd order and are proportional to the combinations[684]:

Im(ai)∝∑k>k′

ln(Mk/Mk′)

ln(Λ/Mk′)Im(CkCk

′)ii,

29The SM fermion Yukawa couplings induce negligible effects.

Im(δRRm�δ

LL)ii∝∑k>k′

lnk

k′ Im(Ckm2

�Ck′)ii,

where lnk

k′ is a logarithmic function. The FV contributiongenerically dominates for tanβ � 10. Without going in thedetails of this formulae, we just display some representa-tive upper estimates for the see-saw induced EDMs, con-sidering for definiteness the gμ region with tanβ = 20.The see-saw induced dμ is below the planned sensitivity,dSSμ � 10−25 e cm. On the contrary for de it could be at hand,dSSe � 0.5×10−27 e cm; the expectation is however stronglymodel dependent and usually see-saw models that satisfy thebound from μ→ eγ predict a much smaller value [684].The possibility of large de and its correlation with leptogen-esis is discussed in [329].

Perspectives are much more interesting if there is alsoa stage of grand unification. In minimal supersymmetricSU(5), the Higgs triplet Yukawa couplings contribute tothe RGE-running for energies larger than their mass scaleMT ∼MGUT. For LFV, δRR is generated at 1st order and isproportional to a combination of the up quark Yukawa cou-plings [683]:

δRRij ∝ (YTu Y ∗u

)ij

lnΛ

MT. (5.144)

Due to the weaker experimental bounds on δRR , this contri-bution is not very significant. On the other hand δLL is notchanged, as also happens to the FC contribution to EDMs[843]. The FV contribution to EDMs is on the contrarystrongly enhanced: it arises at 2nd order (also for degenerateright handed neutrinos) and is proportional to:

Im(δRRm�δ

LL)ii∝ Im

(Cm�Y

Tu Y

∗u

)ii

lnΛ

MT. (5.145)

As a result, considering for definiteness the gμ region withtanβ = 20 and the representative values for triplet and righthanded neutrino masses MT = 2 × 1016 GeV and M3 =1015 GeV, the induced dμ is still below planned, dSS5

μ �5 × 10−25 e cm, but the induced de could exceed by muchthe present limit: dSS5

e � 10−25 e cm. In turn this means thatIm(e−iβC13) � 0.1 (β being the angle of the unitarity tri-angle), which has of course an impact on see-saw models.Further details can be found in [843, 845]. Notice howeverthat, in addition to the problems with light fermion masses,minimal supersymmetric SU(5) is generically considered tobe ruled out by proton decay induced by Higgs triplet ex-change.

More realistic GUTs like SO(10) succeed in suppress-ing the proton decay rate by introducing more Higgs tripletsand enforcing a peculiar structure for their mass matrix.What are the expectation for de in this case? Consider whathappens in a semirealistic SO(10) model [845], where in

118 Eur. Phys. J. C (2008) 57: 13–182

Fig. 40 (Color online) The predictions for τp→Kν and de are shownas a function of r for the degenerate (flat blue) and close to pseudoDirac (red) cases by taking mT = 1017 GeV, Λ = 2 × 1018 GeV and

maximal CPV phase for de . The supersymmetric parameters tanβ = 3,M1 = 200 GeV, mR = 400 GeV, have been selected. The shaded (grey)regions are excluded experimentally. See [845] for more details

addition to the three 16 fermion representations we intro-duce a couple of 10H ’s containing the Higgs doublets andtriplets, 10uH = (HuD,HuT )+ (H uD, H uT ), 10dH = (HdD,HdT )+(H dD, H

dT ). Up and down quark fermion masses arise when

the doublets HuD and H dD acquire a VEV; in particular Yν =Yu, Y� = Yd , and also the triplet Yukawa couplings are fullydetermined in terms of Yu and Yd . As for the mass matricesof the Higgses, the doublets are diagonal in this basis, whilethe triplets are a priori undetermined:

(H dD H uD

)(e.w. 00 MGUT

)(HuD

HdD

),

(H dT H uT

)MT

(HuT

HdT

).

(5.146)

Let consider two limiting cases for the pattern of the tripletmass matrix, diagonal degenerate and close to pseudo-Dirac:

MdegT =

(1 00 1

)mT , M

cpDT =

(0 11 r

)mT , (5.147)

where r is a small real parameter, r < 1, and the exactpseudo-Dirac form corresponds to the limit r → 0. Noticethat the close to pseudo-Dirac form is naturally obtained inthe Dimopoulos–Wilczek mechanism to solve the doublet–triplet splitting problem. The prediction for proton life-timedisplays a strong dependence on the structure of MT , andonly the pseudo-Dirac form is allowed, as can be seen inFig. 40 (there is an intrinsic ambiguity due to GUT phases,so that the prediction is in between the dotted and solidcurves). For EDMs on the contrary the Higgs triplets con-tribution to RGE is cumulative and, due to the log, mildlysensitive to the triplet mass matrix structure. In the case ofO(1) CPV phase (a small phase would be unnatural in thiscontext), de would exceed the present bound for the valuesof supersymmetric parameters selected in Fig. 40. Plannedsearches will be a fortiori more constraining. The impact ofthese results go beyond the essential model described above.Indeed, the week points of the model, like the fermion mass

spectra, could be addressed without changing by much theexpectations for de. It is remarkable that EDMs turns outto be complementary to proton decay in constraining super-symmetric GUTs.

In the above model one obtains the relation dμ/de ∼|Vts/Vtd |2 ≈ 25, so that the prediction for dμ is below theplanned sensitivity. However, there are GUT models wherethis is not the case. For instance a significant dμ is obtainedin L-R symmetric guts [846].

5.7.1 Electron–neutron EDM correlations in SUSY

One of the questions we would like to address is whethernon-zero EDM signals can constitute indirect evidence forsupersymmetry. Supersymmetric models contain additionalsources of CP violation compared to the SM, which induceconsiderable and usually too large EDMs (Fig. 41). In typi-cal (but not all) SUSY models, the same CP-violating sourceinduces both hadronic and leptonic EDMs such that theseare correlated. The most important source is usually the CP-phase of the μ-term and, in certain non-universal scenarios,the gaugino phases. The CP-phases of theA-terms generallylead to smaller contributions.

Typical SUSY models lead to |dn|/|de| ∼ O(10)–O(100). Thus, if both the neutron and the electron EDMsare observed, and this relation is found, it can be viewed asa clue pointing towards supersymmetry.

Since generic SUSY models suffer from the “SUSY CPproblem”, EDMs should be analyzed in classes of modelswhich allow for their suppression. These include modelswith either small CP phases or heavy spectra. dn–de EDMcorrelations have been analyzed in mSUGRA, the decou-pling scenario with 2 heavy sfermion generations, and split

Fig. 41 One loop EDMcontributions

Eur. Phys. J. C (2008) 57: 13–182 119

Fig. 42 EDM correlation in mSUGRA

SUSY [847]. Assuming that the SUSY CP phases are all ofthe same order of magnitude at the GUT scale, one finds

mSUGRA: de ∼ 10−1dn,

split SUSY: de ∼ 10−1dn,

decoupling: de ∼(10−1–10−2

)dn.

These results are insensitive to tanβ and order one varia-tions in the mass parameters. The de/dn ratio is dominatedby the factor me/mq ∼ 10−1, although different diagramscontribute to de and dn.

An example of the dn–de correlation in mSUGRA is pre-sented in Fig. 42. There m0,m1/2, |A| are varied randomlyin the range [200 GeV, 1 TeV], tanβ = 5 and the phase ofthe μ-term φμ is taken to be in the range [−π/500,π/500].The effect of the phase of the A-terms, φA, is negligible aslong is it is of the same order of magnitude as φμ at the GUTscale. Clearly, the relation dn/de ∼ 10 holds for essentiallyall parameter values.

As the next step, we would like to see how stable thesecorrelations are. One might expect that breaking universal-ity at the GUT scale would completely invalidate the aboveresults. To answer this question, we study a non-universalMSSM parameterized by

msquark, mslepton, M3, M1 =M2, |A|,φμ, φA, φM3

(5.148)

at the GUT scale. The mass parameters are varied randomlyin the range [200 GeV, 2 TeV] and the phases in the range[−π/300,π/300], tanβ = 5. We find that although the cor-relation is not as precise as in the mSUGRA case, about 90%of the points satisfy the relation dn/de ∼ 10–100 (Fig. 43).In most of the remaining 10%, 104 > dn/de > 102, whicharise when the gluino phase dominates. The reason for thecorrelation is that in most cases φμ is significant and inducesboth dn and de . Apart from the factor mq/me , the SUSYEDM diagrams are comparable as long as there are no largemass hierarchies in the SUSY spectrum. This means that theEDM correlation survives to a large extent, although it ispossible to violate it in certain cases.

Fig. 43 EDM correlation in non-universal SUSY models

It is instructive to compare the SUSY EDM ‘prediction’to those of other models. Start with the standard model. TheSM background due to the CKM phase is very small, prob-ably beyond the experimental reach. The neutron EDM canalso be induced by the QCD θ -term,

dn ∼ 3 × 10−16θ e cm, (5.149)

which does not affect the electron EDM. Thus, one hasdn� de.

In extra dimensional models, usually there are no extrasources of CP violation and the EDM predictions are similarto the SM values. Two Higgs doublet models have additionalsources of CP violation, however, the leading EDM contri-butions appear at 2 or 3 loops such that the typical EDMvalues are significantly smaller than those in SUSY models.

To conclude, we find that typical SUSY models predict|dn|/|de| ∼ O(10)–O(100). Thus, if

de > dn (5.150)

or

de� dn (5.151)

is found, common SUSY scenarios would be disfavored,although such relations could still be obtained in baroqueSUSY models.

It is interesting to consider SUSY GUT model, whereCP phases in the neutrino Yukawa couplings contributes tohadronic EDMs. For instance, in SU(5) SUSY GUT withright handed neutrinos, not only large mixing but also CP-violating phases in neutrino sector give significant contribu-tion to the mixing and CP phases in the right handed scalardown sector. Though 1–2 mixing in the neutrino Yukawacoupling is strongly restricted by the B(μ→ eγ ), 2–3 mix-ing in the neutrino Yukawa couplings can be significantlylarge and this case is interesting in B physics. Large 2–3

120 Eur. Phys. J. C (2008) 57: 13–182

mixing with CP violation in neutrino sector may give a sig-nificant contribution not only to the B(τ → μγ ) but also tocolor EDM of s quark which may affect [848, 849] neutronand Hg EDM.

5.7.2 EDMs in split supersymmetry

Supersymmetry breaking terms involve many new sourcesof CP violation. Particularly worrisome are the phases asso-ciated with the invariants arg(A∗Mg) and arg(A∗B), whereA and B represent the usual trilinear and bilinear soft termsandMg the gaugino masses. Such phases survive in the uni-versal limit in which all the flavor structure originates fromthe SM Yukawa’s. If these phases are of order one, the elec-tron and neutron EDMs induced at one loop by gaugino-sfermion exchange are typically (barring accidental cancel-lations [848, 850, 851]) a couple of orders of magnitudeabove the limits [852–855], a difficulty which is known asthe supersymmetric CP problem.

Different remedies are available to this problem mak-ing the one loop sfermion contribution to the EDMs smallenough, each with its pros and cons. One remedy is to haveheavy enough sfermions (say heavier than 50–100 TeV to beon the safe side). Gauginos and higgsinos are not requiredto be heavy, and can be closer to the electroweak scale, thuspreserving the supersymmetric solution to the dark matterproblem and gauge coupling unification. This is the “Split”limit of the MSSM [457, 458, 856]. In this limit, the heavysfermions suppress the dangerous one loop contributions toa negligible level. Nevertheless, some phases survive belowthe sfermion mass scale and, if they do not vanish for an ac-cidental or a symmetry reason, they give rise to EDMs thatare safely below the experimental limits, but sizable enoughto be well within the sensitivity of the next generation ofexperiments [847, 856–858]. Such contributions only ariseat the two-loop level, since the new phases appear in thegaugino–higgsino sector, which is not directly coupled tothe SM fermions.

Besides the large EDMs, a number of additional unsat-isfactory issues, all related to the presence of TeV scalars,plague the MSSM. The number of parameters exceeds 100;flavor changing neutral current processes are also one ortwo orders of magnitudes above the experimental limitsin most of the wide parameter space; in the context of agrand unified theory, the proton decay rate associated tosfermion-mediated dimension five operators is ruled out bythe Super-Kamiokande limit, at least in the minimal ver-sion of the supersymmetric SU(5) model; in the supergrav-ity context, another potential problem comes from the grav-itino decay, whose rate is slow enough to interfere with pri-mordial nucleosynthesis. While none of those issues is ofcourse deadly—remedies are well known for each of them—it should be noted that the split solution of the supersymmet-ric CP problem also solves all of those issues at once. At the

same time, it gives rise to a predictive framework, character-ized by a rich, new phenomenology, mostly determined interms of only 4 relevant parameters. Of course the price tobe paid to make the sfermions heavy is the large fine-tuning(FT) necessary to reproduce the Higgs mass, which exac-erbates the FT problem already present in the MSSM. Thiscould be hard to accept, or not, depending on the interpre-tation of the FT problem, the two extreme attitudes being(i) ignoring the problem, as long as the tuning is not muchworse than permille and (ii) accepting a tuning in the Higgsmass as we accept the tuning of the cosmological constant,as in split supersymmetry. The second possibility can in turnbe considered as a manifestation of an anthropic selectionprinciple [859–863].

Before moving the quantitative discussion of the effect,let us note that the pure gaugino–higgsino contribution tothe EDMs, dominant in split supersymmetry and possiblynear the experimental limit, might also be important in thenon-split case, depending on the mechanism invoked to pushthe one loop sfermion contribution below the experimentallimit.

5.7.2.1 Sources of CP violation in the split limit Belowthe heavy sfermion mass scale, denoted generically by m,the MSSM gauginos and higgsinos, together with the SMfields constitute the field content of the model. The only in-teractions of gauginos and higgsinos besides the gauge onesare

−L =√2(guH

†W aTaHu + g′uYHuH †BHu

+ gdH †c W

aTaHd + g′dYHdH †c BHd

)+ h.c., (5.152)

where the Higgs–higgsino–gaugino couplings gu, gd , g′u,g′d can be expressed in terms of the gauge couplings andtanβ through the matching with the supersymmetric gaugeinteractions at the scale m, Hc = iσ2H

∗, Ta are the SU(2)generators, and YHu = −YHd = 1/2. CP violating phasescan enter the effective Lagrangian below the sfermion massscale m through the μ-parameter, the gaugino masses Mi ,i = 1,2,3, or the couplings gu, gd , g′u, g′d (besides of coursethe Yukawa couplings, not relevant here). Only three combi-nations of the phases of the above parameters are physical, ina basis in which the Higgs VEV is in its usual form, 〈H 〉 =(0, v)T , with v positive. The three combinations are φ1 =arg(g′∗u g′∗d M1μ), φ2 = arg(g∗ug∗dM2μ), ξ = arg(gug∗d g′d g′∗u ).Actually, the parameters above are not independent them-selves. The tree-level matching with the full theory abovem gives in fact arg(gu) = arg(g′u), arg(gd) = arg(g′d). As aconsequence, the phase ξ vanishes, thus leaving only two in-dependent phases. Moreover, if the phases ofM1 andM2 areequal, as in most models of supersymmetry breaking, thereis actually only one CP invariant: φ2 = arg(g∗ug∗dM2μ).

Eur. Phys. J. C (2008) 57: 13–182 121

Fig. 44 Two loop contributionsto the light SM fermion EDMs.The third diagram is for adown-type fermion f

In terms of mass eigenstates, the relevant interactions are

−L = g

cWχ+i γ

μ(GRijPR +GLijPL

)χ+j Zμ

+[gχ+i γ

μ(CRijPR +CLijPL

)χ0j W

+μ

+ g√2χ+i

(DRijPR +DLijPL

)χ+j h+ h.c.

], (5.153)

where

GLij = ViW+cW+V †W+j + Vih+u ch+u V

†h+u j,

(5.154)−GRij ∗ =UiW−cW−U†

W−j +Uih−d ch−d U†h−d j,

CLij =−ViW+N∗

jW3+ 1√

2Vih+uN∗jh0u,

(5.155)

CRij =−U∗iW−NjW3

− 1√2U∗ih−dNjh0d

,

gDRij = g∗uVih+u UjW− + g∗dViW+Ujh−d,

(5.156)DL = (DR)†.

In (5.154), cf = T3f − s2WQf (s2

W ≡ sin2 θW ) is the neutralcurrent coupling coefficient of the fermion f and, accord-ingly, cW± = ± cos2 θW , ch+u ,h−d

= ±(1/2 − s2W). The ma-

trices U , V , N diagonalize the complex chargino and neu-tralino mass matrices, M+ = UTMD+V , M0 = NTND0 N ,where MD+ = Diag(M+

1 ,M+2 ) ≥ 0, MD0 = Diag(M0

1 , . . . ,

M04 )≥ 0 and

M+ =(M2 guv

gdv μ

),

M0 =

⎛⎜⎜⎝

M1 0 −g′dv/

√2 g′uv/

√2

0 M2 gdv/√

2 −guv/√

2−g′dv/

√2 gdv/

√2 0 −μ

g′uv/√

2 −guv/√

2 −μ 0

⎞⎟⎟⎠ .

(5.157)

5.7.2.2 Two loop contributions to EDMs Fermion EDMsare generated only at two loops, since charginos and neu-tralinos, which carry the information on CP violation, are

only coupled to gauge and Higgs bosons. Three diagramscontribute to the EDM of the light SM fermion f at the two-loop level. They are induced by the effective γ γ h, γZh,and γWW effective couplings and are shown in Fig. 44. TheEDM df of the fermion f is then given by [857], where

df = dγHf + dZHf + dWWf , (5.158)

dγH

f = eQf α2

4√

2π2s2W

Im(DRii) mfM+

i

MWm2H

fγH(r+iH)

(5.159)

dZHf = e(T3fL − 2s2WQf )α

2

16√

2π2c2Ws

4W

Im(DRijG

Rji −DLijGLji

),

× mfM+i

MWm2H

fZH(rZH , r

+iH , r

+jH

), (5.160)

dWWf = eT3fLα2

8π2s4W

Im(CLijC

R∗ij

)mfM+i M

0j

M4W

× fWW(r+iW , r

0jW

). (5.161)

In (5.159) a sum over indices i, j is understood, Qf is thecharge of the fermion f , T3fL is the third component of theweak isospin of the fermion’s left handed component. Also,rZH = (MZ/mH )2, r+iH = (M+

i /mH )2, r+iW = (M+

i /MW)2,

r0iW = (M0

i /MW)2, where mH is the Higgs mass, and the

loop functions are given by

fγH (r)=∫ 1

0

dx

1 − x j(

0,r

x(1 − x)), (5.162)

fZH (r, r1, r2)= 1

2

∫ 1

0

dx

x(1 − x)j(r,xr1 + (1 − x)r2x(1 − x)

),

(5.163)

fWW(r1, r2)=∫ 1

0

dx

1 − x j(

0,xr1 + (1 − x)r2x(1 − x)

). (5.164)

Their analytic expressions can be found in Ref. [857]. Thesymmetric loop function j (r, s) is defined recursively by

j (r)= r log r

r − 1, j (r, s)= j (r)− j (s)

r − s . (5.165)

Equation (5.159) hold at the chargino mass scale M+.The neutron EDM is determined as a function of the down

122 Eur. Phys. J. C (2008) 57: 13–182

and up quark dipoles at a much lower scale μ, at which

dq(μ)= ηQCDdq(M+), ηQCD =

[αs(M

+)αs(μ)

]γ /2b,

(5.166)

where the β-function coefficient is b = 11 − 2nq/3 andnq is the number of effective light quarks. The anomalous-dimension coefficient is γ = 8/3. For αs(MZ) = 0.118 ±0.004 and μ = 1 GeV (the scale of the neutron mass),the value of ηQCD is 0.75 for M+ = 1 TeV and 0.77 forM+ = 200 GeV. We expect an uncertainty of about 5% fromnext-to-leading order effects. This result [857] gives a QCDrenormalization coefficient about a factor of 2 smaller thanusually considered [864], and it agrees with the recent find-ings of Ref. [865].

The neutron EDM can be expressed in terms of the quarkEDMs using QCD sum rules [866, 867]:

dn = (1 ± 0.5)f 2πm

2π

(mu +md)(225 MeV)3

(4

3dd − 1

3du

),

(5.167)

where fπ ≈ 92 MeV and we have neglected the contri-bution of the quark chromo-electric dipoles, which doesnot arise at the two-loop level in the heavy-squark masslimit. Since dd and du are proportional to the correspond-ing quark masses, dn depends on the light quark massesonly through the ratio mu/md , for which we take the valuemu/md = 0.553 ± 0.043.

It is instructive to consider the limit Mi,μ�MZ,mH

which simplifies the EDM dependence on the CP-violatinginvariants |gugd/M2μ| sinφ2 and |g′ug′d/M1μ| sinφ1. Theterms depending on the second invariant are actually sup-pressed, so that both the electron and neutron EDM aremostly characterized by a single invariant even in the casein which the phases of M1 and M2 are different. The rela-tive importance of the three contributions to df in (5.158)can be estimated to leading order in log(M2μ/m

2H ) from

dZHf

dγH

f

≈ (T3fL − 2s2WQf )(3 − 4s2

W)

8c2WQf

;(5.168)

dWWf

dγH

f

≈− T3fL

8s2WQf

(M2 = μ).

Numerically, (5.168) gives dZHe ≈ 0.05dγHe , dWWe ≈−0.3dγHe and dZHn ≈ dγHn , dWWn ≈ −0.7dγHn . These sim-ple estimates show the importance of the ZH contributionto the neutron EDM.

5.7.2.3 Numerical results Let us consider a standard uni-fied framework for the gaugino masses at the GUT scale.Using the RGEs given in Refs. [458, 868], the parame-ters in (5.159) can be expressed in terms of the (single)phase φ ≡ φ2 and four positive parameters M2, μ (evalu-ated at the low energy scale), tanβ , and the sfermion massscale m. In first approximation, the dipoles depend on βand φ through an overall factor sin 2β sinφ. The overallsfermion scale m enters only logarithmically through theRGE equations for gu,d , g′u,d . The numerical results for theelectron and neutrino EDMs can then conveniently be pre-sented in the M2–μ plane by setting sin 2β sinφ = 1 (it isthen sufficient to multiply the results by sin 2β sinφ) and,for example, m = 109 GeV. Figure 45 shows the predic-tion for the electron EDM, the neutron EDM, and their ra-tio dn/de . The red thick line corresponds to the present ex-perimental limits de < 1.6 × 10−27 e cm [186], while thelimit dn < 3×10−26 e cm [869] does not impose a constrainton the parameters shown in Fig. 45.

An interesting test of split supersymmetry can be pro-vided by a measurement of both the electron and the neu-tron EDMs. Indeed, in the ratio dn/de the dependence onsinφ, tanβ and m approximately cancels out. Nevertheless,because of the different loop functions associated with thedifferent contributions, the ratio dn/de varies by O(100%)when M2 and μ are varied in the range spanned in the fig-ures. Still, the variation of dn/de is comparable to the the-oretical uncertainty in (5.167), and is significantly smallerthan the variation in the ordinary MSSM prediction, evenin the case of universal phases [847].30 On the other hand,the usual tight correlation between the electron and muonEDMs, dμ/de =mμ/me persists.

6 Experimental tests of charged lepton universality

Lepton universality postulates that lepton interactions donot depend explicitly on lepton family number other thanthrough their different masses and mixings. Whereas there islittle doubt about the universality of electric charge there arescenarios outside the standard model in which lepton univer-sality is violated in the interactions with W and Z bosons.Violations may also have their origin in non-SM contribu-tions to the transition amplitudes. Such apparent violationsof lepton universality can be expected in various particle de-cays:

– in W , Z and π decay resulting from R-parity violatingextensions to the MSSM [870, 871],

30Note that the ZH contribution is missing in the analysis of the splitsupersymmetry case in Ref. [847], which leads to a somewhat strongercorrelation between de and dn.

Eur. Phys. J. C (2008) 57: 13–182 123

Fig. 45 Prediction for dn, de ,and their ratio dn/de . In thecontour plots we have chosentanβ = 1, sinφ = 1, andm= 109 GeV. The results fordn and de scale approximatelylinearly with sin 2β sinφ, whilethe ratio is fairly independent oftanβ , sinφ and m. The red thickline corresponds to the presentexperimental limitde < 1.6 × 10−27 e cm [186].Note that the uncertainty in dn isa factor of a few. The scatterplot shows dn values whenM1,3and μ are varied in the range[200 GeV,1 TeV], mh in[100 GeV,300 GeV] and the CPphase in the range [−π,π]

– in W decay resulting from charged Higgs bosons [872,873],

– in π decay resulting from box diagrams involving non-degenerate sleptons [874],

– in K decay resulting from LFV contributions in SUSY[839] (see Sect. 5.6),

– in Υ decay resulting from a light Higgs boson [875],– in π and K decay from scalar interactions [876], en-

hanced by the strong chiral suppression of the SM ampli-tude for decays into eνe. Since these contributions resultin interference terms with the SM amplitude the devia-tions scale with the mass M of the exchange particle like1/M2 rather than 1/M4 as may be expected naively.

We assume the V –A Lorentz structure of the chargedweak current, and parameterize universality violations byallowing for different strengths of the couplings of the in-dividual lepton flavors: 31

L =∑l=e,μ,τ

gl√2Wμνlγ

μ

(1 − γ5

2

)l + h.c. (6.1)

Experimental limits have recently been compiled byLoinaz et al. [877]. Results are shown in Table 16.

Following the notation of Ref. [877] one may parame-terize the violations by gl ≡ g(1 − εl/2). After introducing

31Still more general violations lead to deviations from the 1−γ5 struc-ture of the weak interaction.

Table 16 Limits on lepton universality from various processes. Oneshould keep in mind that violations may affect the various tests differ-ently so which constraint is best depends on the mechanism. Hypothet-ical non-V –A contributions, for example, would lead to larger effectsin decay modes with stronger helicity suppression such as π→ eν andK→ eν. Adapted from Ref. [877]. The ratios estimated from tau de-cays are re-calculated using PDG averages, as described in the text

Decay mode Constraint

W → e νe (gμ/ge)W = 0.999 ± 0.011

W → μ νμ (gτ /ge)W = 1.029 ± 0.014

W → τ ντ

μ→ e νe νμ (gμ/ge)τ = 1.0002 ± 0.0020

τ→ e νe ντ (gτ /ge)τμ = 1.0012 ± 0.0023

τ→ μ νμ ντ

π→ e νe (gμ/ge)π = 1.0021 ± 0.0016

π→ μ νμ (gτ /ge)τπ = 1.0030 ± 0.0034

τ→ π ντ

K→ e νe (gμ/ge)K = 1.024 ± 0.020

K→ μ νμ (gτ /gμ)Kτ = 0.979 ± 0.017

τ→K ντ

Δll′ ≡ εl − εl′ the various experimental limits on deviationsfrom lepton universality can be compared (see Fig. 46).

It is very fortunate that for most decay modes new dedi-cated experiments are being prepared. In the following sub-

124 Eur. Phys. J. C (2008) 57: 13–182

Fig. 46 Experimental constraints on violations of lepton universal-ity from (a) W decay, (b) τ decay, (c) π and K decay and (d) thecombination of (a)–(c). Parameters are defined in the text. The ±1σ

bands are indicated. The shaded areas correspond to 68% and 90%confidence levels. Results from the analysis in Ref. [877]

sections the status and prospects of these experimental testsof lepton universality are presented.

6.1 π decay

In lowest order the decay width of π → lνl (l = e,μ) isgiven by:

Γ treeπ→lνl =

g2l g

2udV

2ud

256π

f 2π

M4W

m2l mπ

(1 − m2

l

m2π

)2

. (6.2)

By taking the branching ratio the factors affected by hadron-ic uncertainties cancel:

Rtreee/μ ≡

Γ treeπ→eνΓ treeπ→μν

=(ge

gμ× me

mμ× 1 −m2

e/m2π

1 −m2μ/m

2π

)2

.

Radiative corrections lower this result by 3.74(3)% [878]when assuming that final states with additional photons are

included. Within the SM (i.e. ge = gμ) this leads to:

RSMe/μ = 1.2354(2)× 10−4. (6.3)

Two experiments [879, 880] contribute to the present worldaverage for the measured value:

Rexpe/μ = 1.231(4)× 10−4. (6.4)

As a result μe universality has been tested at the level:(gμ/ge)π = 1.0021(16).

Measurements of Re/μ are based on the analysis of e+energy and time delay with respect to the stopping π+.The decay π → eν is characterized by Ee+ = 0.5mπc2 =69.3 MeV and an exponential time distribution following thepion life-time τπ = 26 ns. In the case of the π→ μν decaythe 4 MeV muons, which have a range of about 1.4 mm inplastic scintillator, can be kept inside the target and are mon-itored by the observation of the subsequent decay μ→ eνν,which is characterized by Ee+ < 0.5mμc2 = 52.3 MeV, and

Eur. Phys. J. C (2008) 57: 13–182 125

a time distribution which first grows according to the pionlife-time and then falls with the muon life-time. A majorsystematic error is introduced by uncertainties in the lowenergy tail of the π → eν(γ ) energy spectrum in the re-gion below 0.5mμc2. This tail fraction typically amounts to≈1%. The low energy tail can be studied by suppressingthe π → μ→ e chain by the selection of early decays andby vetoing events in which the muon is observed in the tar-get signal. Suppression factors of typically 10−5 have beenobtained. A study of this region is also interesting, since itmight reveal the signal from a heavy sterile neutrino [881].

Although the two experiments contributing to the presentworld average of Re/μ reached very similar statistical andsystematic errors there were some significant differences.The TRIUMF experiment [880] made use of a single largeNaI(Tl) crystal as main positron detector, with an energy res-olution of 5% (fwhm) and a solid angle acceptance of 2.9%of 4π sr. The PSI experiment [879] used a setup of 132 iden-tical BGO crystals with 99.8% of 4π sr acceptance and anenergy resolution of 4.4% (fwhm). A large solid angle re-duces the low energy tail of π→ eν(γ ) events but may alsointroduce a high energy tail for μ→ eννγ .

Two new experiments have been approved recently aim-ing at a reduction of the experimental uncertainty by an or-der of magnitude. First results may be expected in the year2009.

– At PSI [882] the 3π sr CsI calorimeter built for a determi-nation of the π+ → π0e+ν branching ratio will be used.Large samples of π→ eν decays have been recorded par-asitically in the past which were used as normalization forπ+ → π0e+ν with an accuracy of<0.3%, i.e. the level ofthe present experimental uncertainty of Re/μ. The setupwas also used for the most complete studies of the radia-tive decays π→ eνγ [883] and μ→ eννγ [884] done sofar. Based on this experience an improvement in precisionfor Re/μ by almost an order of magnitude is expected.

– At TRIUMF [885] a single large NaI(Tl) detector will beused again. The detector is similar in size to the one usedin the previous experiment but has significantly better en-ergy resolution. The crystal will be surrounded by CsI de-tectors to reduce the low energy tail of the π → eν re-sponse function. By reducing the distance between targetand positron detector the geometric acceptance will be in-creased by an order of magnitude.

6.2 K decay

Despite the poor theoretical control over the meson decayconstants, ratios of leptonic decay widths of pseudoscalarmesons such as RK ≡ Γ (K → eν)/Γ (K → μν) can bepredicted with high accuracy, and have been traditionally

considered as tests of the V –A structure of weak interac-tions through their helicity suppression and of μ–e univer-sality. The standard model predicts [878]:

RK(SM)= (2.472 ± 0.001)× 10−5 (6.5)

to be compared with the world average [170] of publishedRK measurements:

RK(exp)= (2.44 ± 0.11)× 10−5. (6.6)

As mentioned above the strong helicity suppression ofΓ (K→ eν) makes RK sensitive to physics beyond the SM.As discussed in detail in Sect. 5.6.2.3 lepton flavor violat-ing contributions predicted in SUSY models may lead to adeviation of RK from the SM value in the percent range.Such contributions, arising mainly from charged Higgs ex-change with large lepton flavor violating Yukawa couplings,do not decouple if SUSY masses are large and exhibit astrong dependence on tanβ . For large (but not extreme) val-ues of this parameter, not excluded by other measurements,the interference between the SM amplitude and a doublelepton-flavor violating contribution could produce a −3%effect. Other experimental constraints such as those fromRπ or lepton flavor violating τ decays were shown in [839]not to be competitive with those from RK in this scenario.

6.2.1 Preliminary NA48 results for RK

In the original NA48/2 proposal [886] the measurement ofK leptonic decays was not considered interesting enough tobe mentioned. Nevertheless, triggers for such decays wereimplemented during the 2003 run. Since these were not veryselective they had to be highly down-scaled. The data stillcontain about 4000 Ke2 decays which is more than fourtimes the previous world sample. In the analysis of thesedata [887] ∼15% background due to misidentified Kμ2 de-cays was observed (see below). The preliminary result waspresented at the HEP2005 Europhysics conference in Lis-bon [888]:

RK(exp)= (2.416 ± 0.043stat ± 0.024syst)× 10−5, (6.7)

marginally consistent with the SM value. While the uncer-tainty in this result is dominated by the statistical error, theunoptimized Ke2 trigger and the lack of a sufficiently largecontrol sample resulted in a ±0.8% uncertainty.

During 2004 a 56 hours special run with simplified trig-ger logic at ∼1/4 nominal beam intensity was performed,dedicated to the collection of semileptonic K± decays fora measurement of |Vus |. About 4000 Ke2 decays were ex-tracted from these data. The preliminary result for RK isconsistent with the 2003 value with similar uncertainty al-though the trigger efficiencies were better known.

The NA48 apparatus includes the following subsystemsrelevant for the RK measurement

126 Eur. Phys. J. C (2008) 57: 13–182

– a magnetic spectrometer, composed of four drift cham-bers and a dipole magnet (MNP33)

– a scintillator hodoscope consisting of two planes seg-mented into vertical and horizontal strips, providing a fastlevel-1 (L1) trigger for charged particles

– a liquid krypton electromagnetic calorimeter (LKr) withan L1 trigger system.

In the analysis of the 2003-04 data Ke2 decays were se-lected using two main criteria:

– 0.95< E/pc < 1.05 where E is the energy deposited inLKr and p is the momentum measured with the magneticspectrometer.

– the missing mass MX must be zero within errors, as ex-pected for a neutrino.

The main background resulted from misidentified Kμ2 de-cays. The E/pc distribution of muons has a tail which ex-tends to E/pc∼ 1 and the observed fraction of muons with0.95 < E/pc < 1.05 is ∼5 × 10−6. Kμ2 background waspresent for p > 25 GeV/c where theMX resolution providedby the magnetic spectrometer was insufficient to separateKe2 from Kμ2 decays.

6.2.2 A new measurement of Γ (K → eν)/Γ (K → μν) atthe SPS

During the Summer of 2007 NA62, the evolution of theNA48 experiment, has accumulated more than 100K Ke2decays. For this run the spectrometer momentum resolutionwas improved by increasing the MNP33 momentum kickfrom 120 to 263 MeV/c.Ke2 decays are selected by requiring signals from the two

hodoscope planes (denoted byQ1) and an energy depositionof at least 10 GeV in the LKr calorimeter. This trigger hasan efficiency >0.99 for electron momenta p > 15 GeV/c.The same down-scaled Q1 trigger was used to collect Kμ2

decays. The beam intensity was adjusted to obtain a totaltrigger rate of 104 Hz, which saturates the data acquisitionsystem.

Figure 47 shows the M2X versus momentum distribution

for Ke2 and Kμ2 decays for the 2004 data, together withthe predicted distributions for the 2004 run and for the 2007run, as obtained from a Monte Carlo simulation (for Kμ2

decays the electron mass is assigned to the muon). In the2007 run, for electron momenta up to 35 GeV/c the Kμ2

contamination to the Ke2 signal is reduced to a negligiblelevel thanks to the improved spectrometer momentum reso-lution (see Fig. 48(a)). Using a lower limit of 15 GeV/c forthe electron momentum, and taking into account the detec-tor acceptance, this means that ∼43% of the Ke2 events willbe kinematically background free (see Fig. 48(b)).

The fraction of Kμ2 faking Ke2 decays was measured atall momenta in parallel with data taking. For this purpose

Fig. 47 Distributions of M2X versus p for Ke2 and Kμ2 decays. In

the MX calculation the electron mass is assumed for both processes:(a) measured data from the 2004 run, (b) Monte Carlo predictions for2004 conditions, (c) Monte Carlo predictions for the conditions ex-pected in 2007

Fig. 48 (a) Kμ2 contamination in the Ke2 sample. (b) Simulated mo-mentum distributions of genuine electrons from Ke2 decay (full his-togram), and of fake electrons from Kμ2 decays (dashed histogram)

a ∼5 cm thick lead plate was inserted between the two ho-doscope planes covering six 6.5 cm wide vertical hodoscopecounters. The requirement that charged particles traverse thelead without interacting helps to select a pure sample ofKμ2

decay for which the muon E/pc distribution can be directlymeasured for the evaluation of theKμ2 contamination to theKe2 signal. Table 17 lists the relevant parameters describingthe running conditions both for the 2004 and 2007 runs.

Eur. Phys. J. C (2008) 57: 13–182 127

Table 17 Comparison of the 2004 and 2007 running conditions

2004 2007 2004 2007

Acceptance (mr2) 0.36 × 0.36 0.18 × 0.18 SPS duty cycle (s/s) 4.8 / 16.8 4.8 / 16.8

�Ω (sr) 4 × 10−7 1 × 10−7 live time (days) 2.1 100

�p/p effective (%) ±3 ±2.5 nr. of pulses 1.08 × 104 3 × 105

RMS (%) ∼3.0 ∼1.8 Protons per pulse 2.5 × 1011 1.5 × 1012

TRIM3 x′ (mr) 0 ±0.3 beam momentum (GeV/c) ≈60 ≈75

pT (MeV/c) 0 ±22.5 Triggers/pulse 45,000 48,000

MNP33 x′ (mr) ±2.0 ±3.5 Good Ke2/pulse ∼0.37 ∼0.5

pT (MeV/c) ±120 ±263 Good Ke2 (total) 4000 >100,000

The overall statistical error, which includes the statisticaluncertainty on the background measurement, is expected tobe 0.3%. The uncertainty in the trigger efficiency will be re-duced to less than ±0.2%. The data collected in 2007 willprovide a measurement of RK with a total uncertainty (sta-tistical and systematic errors combined in quadrature) of lessthan ±0.5%.

6.3 τ decay

There are two ways to test lepton universality in chargedweak interactions using τ decays:

– the universality of all three couplings can be tested bycomparing the rates of the decays τ → μνν, τ → eνν

and μ→ eνν, and– gτ /gμ can be extracted by comparing τ → πν and π→μν.

When comparing the experimental constraints one shouldkeep in mind the complementarity of these two tests.Whereas the purely leptonic decay modes are mediated bya transversely polarized W , the semileptonic modes involvelongitudinal polarization.

6.3.1 Leptonic τ decays

The decay width of �i → �f νν including radiative correc-tions is given by [889]:

Γ (�i → �f νν)=g2�ig2�f

32m2W

m5�i

192π3(1 +C�i�f ), (6.8)

where (1+C�i�f )= f (x)(1+ 35

m2�i

M2W

)(1+ α(m�i )

2π ( 254 −π2))

combines weak and radiative corrections and f (x) = 1 −8x + 8x3 − x4 − 12x2 lnx with x ≡m2

�f/m2

�i.

Electron–muon universality could thus be tested at the0.2% level using:

gμ

ge=√B(τ → μνν)

B(τ → eνν)

(1 +Cτe)(1 +Cτμ) = 1.0002 ± 0.0020, (6.9)

where Cτe = −0.004 and Cτμ = −0.0313 are the correc-tions from (6.8). The values of the branching ratios of lep-tonic τ decays are taken from [170] and are based mostly onmeasurements from LEP experiments. e–τ universality hasbeen verified with similar precision:

gτ

ge=√(1 +Cμe)(1 +Cτμ)

τμ

ττ

(mμ

mτ

)5

B(τ → μνν)

= 1.0012 ± 0.0023, (6.10)

where Γ (�i → �f νν) = B(�i → �f νν)/τ�i has been usedand Cμe =−0.0044. The measurement of μ–τ universalitycan then be derived from gτ /ge and gμ/ge , giving gτ /gμ =1.0010 ± 0.0023.

The measurements used in above formulas are relativelyold [170], and no input from BaBar or Belle is used. Themeasurements of leptonic branching fractions were done bythe LEP experiments in the course of the runs at or nearthe Z0 resonance [170]. The τ+τ− events were selectedvia their topology, and the τ decay products were requiredto pass particle identification, using information from thecalorimetry, tracking devices, time projection chambers andmuon systems. The largest uncertainty on the measurementof tau branching ratios was statistical, with systematics lim-itations arising from the simulation and from particle identi-fication.

The measurement of the τ life-time [170] comes fromLEP experiments as well. Due to the large

√s, each τ in

the event has a large boost and travels 90 µm in average.However, as there is nothing but τ ’s produced in each event,their production vertex is unknown and has to be estimatedaveraging over other events or by minimizing the sum ofimpact parameters of both τ ’s decay products.

The most accurate published measurement of the τ mass[890] was done by the BES experiment, through an energyscan of the τ+τ− production cross section in e+e− colli-sions around the threshold region. The collision energy scalewas calibrated with J/ψ and ψ(2S) resonances, with a pre-cision of 0.25 MeV.

128 Eur. Phys. J. C (2008) 57: 13–182

Therefore the major contributions to the uncertainties onthe ratios gτ /ge and gμ/ge are:

– the τ leptonic branching fractions (0.3%), and– the τ life-time (0.34%).

In the calculation above, the measurements of leptonic τdecays are taken as independent. However, there are com-mon sources of systematic uncertainties such as uncertain-ties on track reconstruction, number of τ decays registeredby an experiment and so on. If one measured the branch-ing ratio B(τ → eνν)/B(τ → μνν) directly in one exper-iment, as was done by ARGUS and CLEO and as is donefor pion decays as well, most uncertainties would cancel.Taking the PDG average on the branching ratio [170] oneobtains gμ/ge = 1.0028 ± 0.0055.

The following improvements can be expected in the fu-ture. The KEDR experiment is working, like BES, at theτ -pair production threshold. They plan to measure mτ witha 0.15 MeV accuracy. A preliminary result, with accuracycomparable to BES’s measurement, is available [891]. BothBaBar and Belle have accumulated large statistics of τ+τ−events and should be able to perform measurements of lep-tonic and semileptonic τ decays, as well as to improve themeasurement of the τ life-time. While the collected τ sam-ple is much larger than at LEP, there are still significantuncertainties remaining on luminosity, tracking and parti-cle identification. If the ratio of decay fractions is measured,then only the particle identification uncertainties will re-main. Currently the electron and muon identification uncer-tainties for both BaBar and Belle are around 1–2%. At theB-factories the τ boost in the c.m frame is much smallerthan at LEP, and in addition the energies of the e+ and e−beams are not the same. This leads to significant differencesin the technique of the life-time measurement. In particularthe 3-dimensional reconstruction of the trajectories of thedecay products is poor and only the impact parameter in theplane transverse to the beams, multiplied by the polar angleof the total momentum vector of 3-prong τ decay products,can be used [892]. While the statistics allows for a very ac-curate measurement, the work focuses on understanding thealignment of the vertex detector and the systematics in thereconstruction of the impact parameter. The measurement ofthe τ mass can also be done at the B-factories. Belle has pre-sented a mass measurement analyzing the kinematic limit ofthe invariant mass of 3-prong τ decays [893]. This measure-ment is however less precise than those of BES or KEDR.

If one takes into account recent preliminary measure-ments of the τ mass from the KEDR experiment [891] andof the life-time from BaBar [892], the determination of τ–euniversality changes slightly to gτ /ge = 1.0021 ± 0.0020.

6.3.2 Hadronic τ decays

Another way to test τ–μ universality is to compare the de-cay rates for τ→ πν and π→ μν:

g2τ

g2μ

= B(τ → πν)

B(π→ μν)

τπ

ττ

2mτm2μ

m3π

(m2π −m2

μ

m2τ −m2

π

)2

(1 +Cτπ),

(6.11)

where Cτπ =−(1.6+0.9−1.4)10−3 [833, 894].

Taking measurements from Ref. [170] one obtainsgτ /gμ = 0.9996 ± 0.037. Here the main uncertainties comefrom

– τ → πν decay (1%), where the dominant contribution isdue to τ→ ππ0ν contamination and π0 reconstruction,

– the τ life-time (0.34%), and– the hadronic correction (0.1%).

Again, no results from the B factories are available yet,and one should expect that the large τ samples collected byBaBar and Belle will allow a significant improvement, incase the understanding of particle identification will be im-proved.

7 CP violation with charged leptons

There are two powerful motivations for probing CP symme-try in lepton decays:

– The discovery of CP asymmetries in B decays that areclose to 100% in a sense ‘de-mystifies’ CP violation. Forit established that complex CP phases are not intrinsi-cally small and can be close to 90 degrees even. This de-mystification would be completed, if CP violation werefound in the decays of leptons as well.

– We know that CKM dynamics, which is so successfulin describing quark flavor transitions, is not relevant tobaryogenesis. There are actually intriguing arguments forbaryogenesis being merely a secondary effect driven byprimary leptogenesis [895]. To make the latter less spec-ulative, one has to find CP violation in dynamics of theleptonic sector.

The strength of these motivations has been well recognizedin the community, as can be seen from the planned experi-ments to measure CP violation in neutrino oscillations andthe ongoing heroic efforts to find an electron EDM. Yetthere are other avenues to this goal as well that certainlyare at least as difficult, namely to probe CP symmetry inmuon and τ decays. Those two topics are addressed belowin Sects. 7.1 and 7.2. There are also less orthodox probes,namely attempts (i) to extract an EDM for τ leptons from

Eur. Phys. J. C (2008) 57: 13–182 129

e+e− → τ+τ−, (ii) to search for a T-odd correlation in po-larized ortho-positronium decays and (iii) to measure themuon transverse polarization in K+ → μ+νπ0 decays. Itis understood that the standard model does not produce anobservable effect in any of these three cases or the other oneslisted above (except for τ± → νKSπ

±, as described below).Concerning topic (i), one has to understand that one is

searching for a CP-odd effect in an electromagnetic pro-duction process unlike in τ decays, which are controlled byweak forces.

In [e+e−]OP → 3γ , topic (ii), one can construct variousT-odd correlations or integrated moments between the spinvector �SOP of polarized ortho-positronium and the momenta�ki of two of the photons that define the decay plane:

AT odd =⟨�SOP · (�k1 × �k2)

⟩,

ACP = ⟨(�SOP · �k1)(�SOP · (�k1 × �k2)

)⟩.

(7.1)

– The moment AT odd is P and CP even, yet T odd. Ratherthan by CP or T violation in the underlying dynamics itis generated by higher order QED processes. It has beenconjectured [896] that the leading effect is formally oforder α relative to the decay width due to the exchangeof a photon between the two initial lepton lines. From itone has to remove the numerically leading contribution,which has to be absorbed into the bound state wave func-tion. The remaining contribution is presumably at the sub-permille level. Alternatively AT odd can be generated atorder α2—or at roughly the 10−5 level—through the in-terference of the lowest order decay amplitude with onewhere a fermion loop connects two of the photon lines.

– On the other hand the moment ACP is odd under T aswell as under P and in particular CP. Final state interac-tions cannot generate a CP-odd moment with CP invari-ant dynamics. Observing ACP �= 0 thus unambiguouslyestablishes CP violation. The present experimental up-per bound is around few percent; it seems feasible, seeSect. 7.4, to improve the sensitivity by more than threeorders of magnitude, i.e. down to the 10−5 level! Thecaveat arises at the theoretical level: with the ‘natural’scale for weak interference effects in positronium givenbyGF m

2e ∼ 10−11, one needs a dramatic enhancement to

obtain an observable effect.

Discussing topic (iii)—the muon transverse polarizationin Kμ3 decays—under the heading of CP violation in theleptonic sector will seem surprising at first. Yet a general,though hand waving argument, suggests that the highly sup-pressed direct CP violation in nonleptonic �S = 1—as ex-pressed through ε′—rules against an observable signal evenin the presence of new physics—unless the latter has a spe-cial affinity for leptons. The present status of the data andfuture plans are discussed in Sect. 7.3.

7.1 μ decays

The muon decay μ− → e−νeνμ and its ‘inverse’ νμe− →μ−νe are successfully described by the ‘V –A’ interaction,which is a particular case of the local, derivative-free, lep-ton number conserving, four-fermion interaction [897]. The‘V –A’ form and the nature of the neutrinos (νe and νe) havebeen determined by experiment [898–900].

The observables—energy spectra, polarizations and an-gular distributions—may be parameterized in terms of thedimensionless coupling constants gγεμ and the Fermi cou-pling constant GF. The matrix element is

M = 4GF√2

∑γ=S,V,Tε,μ=R,L

gγεμ〈eε|Γ γ∣∣(νe)n

⟩⟨(νμ)m

∣∣Γγ |μμ〉. (7.2)

We use here the notation of Fetscher et al. [898, 901] whoin turn use the sign conventions and definitions of Scheck[902]. Here γ = S,V,T indicate a (Lorentz) scalar, vec-tor, or tensor interaction, and the chirality of the electronor muon (right or left handed) is labeled by ε,μ= R,L. Thechiralities n and m of the νe and the νμ are determined bygiven values of γ , ε and μ. The 10 complex amplitudes gγεμand GF constitute 19 independent parameters to be deter-mined by experiment. The ‘V –A’ interaction correspondsto gV

LL = 1, with all other amplitudes being 0.Experiments show the interaction to be predominantly

of the vector type and left handed [gVLL.0.96(90% C.L.)]

with no evidence for other couplings. The measurement ofthe muon life-time yields the most precise determination ofthe Fermi coupling constant GF, which is presently knownwith a relative precision of 8 × 10−6 [903, 904]. Continuedimprovement of this measurement is certainly an importantgoal [905], since GF is one of the fundamental parametersof the standard model.

7.1.1 T invariance in μ decays

PT2 —the component of the decay positron polarizationwhich is transverse to the positron momentum and the muonpolarization—is T odd and due to the practical absence of astrong or electromagnetic final state interaction it probes Tinvariance. A second-generation experiment has been per-formed at PSI by the ETH Zürich–Cracow–PSI Collabora-tion [906]. They obtained, for the energy averaged trans-verse polarization component:

〈PT2〉 = (−3.7 ± 7.7stat. ± 3.4syst.)× 10−3. (7.3)

7.1.2 Future prospects

The precision on the muon life-time can presumably be in-creased over the ongoing measurements by one order of

130 Eur. Phys. J. C (2008) 57: 13–182

magnitude [903]. Improvement in measurements of the de-cay parameters seems more difficult. The limits there arenot given by the muon rates which usually are high enoughalready (≈3 × 108 s−1 at the µE1 beam at PSI, for exam-ple), but rather by effects like positron depolarization inmatter or by the small available polarization (<7%) of theelectron targets used as analysers. The measurement of thetransverse positron polarization might be improved with asmaller phase space (lateral beam dimension of a few mil-limetres or better). This experiment needs a pulsed beamwith high polarization.

7.2 CP violation in τ decays

The betting line is that τ decays—next to the electron EDMand ν oscillations—provide the best stage to search for man-ifestations of CP breaking in the leptonic sector. There ex-ists a considerable literature on the subject started by dis-cussions on a tau-charm factory more than a decade ago[907–910] and attracting renewed interest recently [911–914] stressing the following points:

– There are many more channels than in muon decays mak-ing the constraints imposed by CPT symmetry much lessrestrictive.

– The τ lepton has sizable rates into multi-body finalstates. Due to their non-trivial kinematics asymmetriescan emerge also in the final state distributions, wherethey are likely to be significantly larger than in the in-tegrated widths. The channel KL → π+π−e+e− can il-lustrate this point. It commands only the tiny branchingratio of 3× 10−7. The forward-backward asymmetry 〈A〉in the angle between the π+π− and e+e− planes consti-tutes a CP odd observable. It has been measured by KTeVand NA48 to be truly large, namely about 13%, althoughit is driven by the small value of |εK | ∼ 0.002. I.e., onecan trade branching ratio for the size an CP asymmetry.

– New physics in the form of multi-Higgs models can con-tribute on the tree-level like the SM W exchange.

– Some of the channels should exhibit enhanced sensitivityto new physics.

– Having polarized τ leptons provides a powerful handle onCP asymmetries and control over systematics.

These features will be explained in more detail below. Itseems clear that such measurements can be performed onlyin e+e− annihilation, i.e. at the B factories running nowor better still at a Super-Flavor factory, as discussed in theWorking Group 2 report. There one has the added advantagethat one can realistically obtain highly polarized τ leptons:This can be achieved directly by having the electron beamlongitudinally polarized or more indirectly even with unpo-larized beams by using the spin alignment of the producedτ pair to ‘tag’ the spin of the τ under study by the decay ofthe other τ like τ→ νρ.

7.2.1 τ→ νKπ

The most promising channels for exhibiting CP asymmetriesare τ− → νKSπ

−, νK−π0 [910]:

– Due to the heaviness of the lepton and quark flavors theyare most sensitive to non-minimal Higgs dynamics whilebeing Cabibbo suppressed in the SM.

– They can show asymmetries in the final state distribu-tions.

The SM does generate a CP asymmetry in τ decays thatshould be observable. Based on known physics one can reli-ably predict a CP asymmetry [911]:

Γ (τ+ →KSπ+ν)− Γ (τ− →KSπ

−ν)Γ (τ+ →KSπ+ν)+ Γ (τ− →KSπ−ν)

= (3.27 ± 0.12)× 10−3 (7.4)

due to KS ’s preference for antimatter over matter. Strictlyspeaking, this prediction is more general than the SM: nomatter what produces the CP impurity in the KS wave func-tion, the effect underlying (7.4) has to be present, while ofcourse not affecting τ∓ → νK∓π0.

To generate a CP asymmetry, one needs two different am-plitudes contribute coherently. This requirement is satisfied,since the Kπ system can be produced from the (QCD) vac-uum in a vector and scalar configuration with form factorsFV and FS , respectively. Both are present in the data, withthe vector component (mainly in the form of the K∗) domi-nant as expected [915]. Within the SM, there does not arise aweak phase between them on an observable level, yet it canreadily be provided by a charged Higgs exchange in non-minimal Higgs models, which contributes to FS .

A few general remarks on the phenomenology might behelpful to set the stage. For a CP violation in the underlyingweak dynamics to generate an observable asymmetry in par-tial widths or energy distributions one needs also a relativestrong phase between the two amplitudes:

Γ(τ− → νK−π0)− Γ (τ+ → νK+π0)

∝ Im(FHF

∗V

)ImgHg

∗W, (7.5)

d

dEKΓ(τ− → νK−π0)− d

dEKΓ(τ+ → νK+π0)

∝ Im(FHF

∗V

)ImgHg

∗W, (7.6)

where FH denotes the Higgs contribution to FS and gH itsweak coupling. This should not represent a serious restric-tion, since the Kπ system is produced in a mass range withseveral resonances. If on the other hand one is searching fora T-odd correlation like

OT ≡ ⟨�στ · ( �pK × �pπ)⟩, (7.7)

Eur. Phys. J. C (2008) 57: 13–182 131

then CP violation can surface even without a relative strongphase

OT ∝ Re(FHF

∗V

)ImgHg

∗W . (7.8)

Yet there is a caveat: final state interactions can generateT-odd moments even from T invariant dynamics, when onehas

OT ∝ Im(FHF

∗V

)RegHg

∗W . (7.9)

Fortunately one can differentiate between the two scenar-ios of (7.8), (7.9) at a B or a Super-Flavor factory, whereone can compare directly the T-odd moments for the CP-conjugate pair τ+ and τ−:

OT(τ+) �=OT

(τ−) =⇒ CP violation! (7.10)

A few numerical scenarios might illuminate the situa-tion: a Higgs amplitude 1% or 0.1% the strength of theSM W -exchange amplitude—the former [latter] contribut-ing [mainly] to FS [FV ]—is safely in the ‘noise’ of presentmeasurements of partial widths; yet it could conceivablycreate a CP asymmetry as large 1% or 0.1%, respectively.More generally a CP-odd observable in a SM allowedprocess is merely linear in a new physics amplitude, sincethe SM provides the other amplitude. On the other handSM forbidden transitions—say lepton flavor violation as inτ → μγ—have to be quadratic in the new physics ampli-tude.

CP odd ∝ ∣∣T ∗SMTNP

∣∣ vs. LFV ∝ |TNP|2. (7.11)

Probing CP symmetry at the 0.1% level in τ → νKπ thushas roughly the same sensitivity for a new physics amplitudeas searching for B(τ→ μγ ) at the 10−8 level.

CLEO has undertaken a pioneering search for a CP asym-metry in the angular distribution of τ → νKSπ placing anupper bound of a few percent [916].

7.2.2 Other τ decay modes

It appears unlikely that analogous asymmetries could be ob-served in the Cabibbo allowed channel τ → νππ , yet de-tailed studies of τν3π/4π look promising, also because themore complex final state allows us to form T-odd correla-tions with unpolarized τ leptons; yet the decays of polarizedτ might exhibit much larger CP asymmetries [912].

Particular attention should be paid to τ → νK2π , whichhas potentially very significant additional advantages:

– One can interfere vector with axial vector K2π configu-rations.

– The larger number of kinematical variables and of specificchannels should allow more internal cross checks of sys-tematic uncertainties like detection efficiencies for posi-tive vs. negative particles.

7.3 Search for T violation in K+ → π0μ+ν decay

The transverse muon polarization in K+ → π0μ+ν decay,PT , is an excellent probe of T violation, and thus of physicsbeyond the standard model. Most recently the E246 experi-ment at the KEK proton synchrotron has set an upper boundof |PT | ≤ 0.0050 (90% C.L.). A next generation experi-ment is now being planned for the high intensity acceleratorJ-PARC which is aiming at more than one order of magni-tude improvement in the sensitivity with σ(PT )∼ 10−4.

7.3.1 Transverse muon polarization

A non-zero value for the transverse muon polarization (PT )in the three body decay K → πμν (Kμ3) violates T con-servation with its T-odd correlation [917]. Over the lastthree decades dedicated experiments have been carried outin search for a non-zero PT . Unlike other T-odd chan-nels in e.g. nuclear beta decays, PT in Kμ3 has the ad-vantage that final state interactions (FSI), which may in-duce a spurious T-odd effect, are very small. With only onecharged particle in the final state the FSI contribution orig-inates only in higher loop effects and has been shown to besmall. The single photon exchange contribution from two-loop diagrams was estimated more than twenty years agoas P FSI

T ≤ 10−6 [918]. Quite recently two-photon exchangecontributions have been studied [919]. The average value ofP FSIT over the Dalitz plot was calculated to be less than 10−5.

An important feature of a PT study is the fact that thecontribution from the standard model (SM) is practicallyzero. Since only a single element of the CKM matrix Vus isinvolved for the semileptonic Kμ3 decay in the SM, no CPviolation appears in first order. The lowest order contributioncomes from radiative corrections to the uγμ(1 − γ5)sW

μ

vertex, and this was estimated to be less than 10−7 [920].Therefore, non-zero PT in the range of 10−3–10−4 wouldunambiguously imply the existence of a new physics contri-bution [920].

Sizable PT can be accommodated in multi-Higgs doubletmodels through CP violation in the Higgs sector [921–928].PT can be induced due to interference between chargedHiggs exchange (FS , FP ) and W exchange (FV , FA) asshown in Fig. 49. It is conceivable that the coupling ofcharged Higgs fields to leptons is strongly enhanced rela-tive to the coupling to the up-type quarks [929] which wouldlead to an experimentally detectable PT of O(10−3). Thus,PT could reveal a source of CP violation that escapes detec-tion in K→ 2π,3π [920].

A number of other models also allow PT at an observ-able level without conflicting with other experimental con-straints, and experimental limits on PT could thus constrainthose models. Among them SUSY models with R-parity

132 Eur. Phys. J. C (2008) 57: 13–182

Fig. 49 Two interferingdiagrams inducing PT in themulti-Higgs model (fromRef. [920])

Fig. 50 E246 setup using the superconducting toroidal spectrometer. The elaborate detector system [933] consists of an active target (to monitorstopping K+), a large-acceptance CsI(Tl) barrel (to detect π0), tracking chambers (to track μ+), and muon polarimeters (to measure PT )

breaking [930] and a SUSY model with squark family mix-ing [931] should be mentioned. A recent paper [932] dis-cusses a generic effective operator leading to a PT expres-sion in terms of a cut-off scaleΛ and the Wilson coefficientsCS and CT .

7.3.2 KEK E246 experiment

The most recent and highest precision PT experiment wasperformed at the KEK proton synchrotron. The experi-ment used a stopped K+ beam with an intensity of ∼105/sand a setup with a superconducting toroidal spectrometer(Fig. 50). Data were taken between 1996 and 2000 for a to-tal of 5200 hours of beam time. The determination of themuon polarization was based on a measurement of the de-cay positron azimuthal asymmetry in a longitudinal mag-netic field using “passive polarimeters”. Thanks to (i) thestopped beam method which enabled total coverage of thedecay phase space and hence a forward/backward symmet-ric measurement with respect to the π0 direction and (ii) the

rotational-symmetric structure of the toroidal system, sys-tematic errors could be substantially suppressed.

The T-odd asymmetry was deduced using a double ratioscheme as

AT = (Afwd −Abwd)/2, (7.12)

where the fwd(bwd) asymmetry was calculated using the“clockwise” and “counter-clockwise” positron emissionrates Ncw and Nccw as

Afwd(bwd) =Ncw

fwd(bwd) −Nccwfwd(bwd)

Ncwfwd(bwd) +Nccw

fwd(bwd). (7.13)

PT was then deduced using

PT =AT /{α〈cos θT 〉

}′ (7.14)

with α the analyzing power and 〈cos θT 〉 the average kine-matic attenuation factor. The final result was [934]

PT =−0.0017 ± 0.0023(stat)± 0.0011(syst), (7.15)

Eur. Phys. J. C (2008) 57: 13–182 133

Table 18 Goal of the J-PARCTREK experiment comparedwith the E246 result

E246 @ KEK-PS TREK @ J-PARC

Detector SC toroidal spectrometer E246-upgraded

Proton beam energy 12 GeV 30 GeV

Proton intensity 1.0 × 1012/s 6 × 1013/s

K+ intensity 1.0 × 105/s 3 × 106/s

Run time ∼2.0 × 107 s 1.0 × 107 s

σ(PT )stat 2.3 × 10−3 ∼1.0 × 10−4

σ(PT )syst 1.1 × 10−3 <1.0 × 10−4

Im ξ =−0.0053 ± 0.0071(stat)± 0.0036(syst), (7.16)

corresponding to the upper limits of |PT | < 0.0050 (90%C.L.) and | Im ξ | < 0.016 (90% C.L.), respectively. HereIm ξ is the physics parameter proportional to PT afterremoval of the kinematic factor. This result constrainedthe three-Higgs doublet model parameter in the way of| Im(α1γ

∗1 )|< 544(MH1/ GeV)2, as the most stringent con-

straint on this parameter. Systematic errors were investi-gated thoroughly, although the total size was smaller thanhalf of the statistical error. There were two items that couldnot be canceled out by any of the two cancellation mech-anisms of the 12-fold azimuthal rotation and π0-fwd/bwd:the effect from the decay plane rotation, θz and the misalign-ment of the muon magnetic field, δz, which should both beeliminated in the next generation J-PARC experiment.

7.3.3 The proposed J-PARC E06 (TREK) experiment

A new possible PT experiment, E06 (TREK), at J-PARCis aiming at a sensitivity of σ(PT ) ∼ 10−4. J-PARC is ahigh intensity proton accelerator research complex now un-der construction in Japan with the first beam expected in2008. In the initial phase of the machine, the main syn-chrotron will deliver a 9 µA proton beam at 30 GeV. A lowmomentum beam of 3×106 K+ per second will be availablefor stopped K+; this is about 30 times the beam intensityused for E246. Essentially the same detector concept willbe adopted; namely the combination of a stopped K+ beamand the toroidal spectrometer, because this system has theadvantage of suppressing systematic errors by means of thedouble ratio measurement scheme. However, the E246 setupwill be upgraded significantly. The E246 detector will beupgraded in several parts so as to accommodate the highercounting rate and to better control the systematics. The ma-jor planned upgrades are the following:

– The muon polarimeter will become an active polarime-ter, providing the muon-decay vertex and the positrontrack, leading to an essentially background-free muon de-cay measurement, with an increased positron acceptanceand analyzing power.

– New dipole magnets will be added, improving the fielduniformity and the alignment accuracy.

– The electronics and readout of the CsI(Tl) E246 calorime-ter will be replaced to maximize the counting rate, fullyexploiting the intrinsic crystal speed.

– The tracking system and the active target will be im-proved for higher resolution and higher decay-in-flightbackground rejection.

As a result, 20 times higher sensitivity to PT will be ob-tained after a one year run. The systematic errors will becontrolled with sufficient accuracy and a final experimen-tal error of ∼10−4 will be attained (see Table 18). A fulldescription of the experiment can be found in the proposal[935].

It is now proposed to run for net 107s corresponding toroughly one year of J-PARC beam-time under the abovementioned beam condition. This would yield 2.4×109 goodK+μ3 events in the π0-fwd/bwd regions, providing an esti-

mate of σ(PT )stat = 1.35 × 10−4. The inclusion of otherπ0 regions, enabled by the adoption of the active polarime-ter, would bring the statistical sensitivity further down tothe 10−4 level. The dominant systematic errors is expectedto arise from the misalignment of the polarimeter and themuon magnetic field; this will be determined from data, andMonte Carlo studies indicate a residual systematics at the10−4 level.

It is proposed to run TREK in the early stage of J-PARCoperation. The experimental group has already started rel-evant R&D for the upgrades after obtaining scientific ap-proval, and the exact schedule will be determined after fund-ing is granted.

7.4 Measurement of CP violation in ortho-positroniumdecay

CP violation in the o-Ps decay can be detected by an ac-curate measurement of the angular correlation between theo-Ps spin �SOP and the momenta of the photons from the o-Psdecay [936], as shown in (7.1). It is useful then to write themeasurable quantity:

N(cos θ)=N0(1 +CCP cos θ), (7.17)

134 Eur. Phys. J. C (2008) 57: 13–182

with the CP violation amplitude parameter, CCP, differentfrom zero, if CP violating interactions take part in the o-Psdecay. In this equation, N(cos θ) is the number of eventswith a measured value cos θ ± |�(cos(θ))| (hereafter, forthe sake of simplicity, it will be referred to as the cos θ value,intending that this is measured with an uncertainty, depend-ing on the spatial resolution of the detector). Here cos θ isdefined as the product of cos θ1, the cosine of the angle be-tween the �SOP and the unit vector in the direction of highestenergy photon k1, and cos θn, the cosine of the angle be-tween the �SOP and the unit vector in the direction perpen-dicular to the o-Ps decay plane, n [937].

The measured distribution N(cos θ) should show anasymmetry given by N(cos θ+) − N(cos θ−) = 2N0CCP ×cos θ , for cos θ+ = − cos θ− = cos θ . The quantity CCP canbe determined by measuring the rate of events N+ for agiven cos θ+ = cos θ and N− for cos θ− = − cos θ . In prac-tice, N+ is the number of events in which k2 forms an anglewith k1 smaller than π , and the o-Ps spin forms an angle θnsmaller than π/2 with the perpendicular to the o-Ps decayplane. In the N− events, k2 forms an angle 2π − θ12 with k1

and the �SOP forms an angle π − θn with the normal to theo-Ps decay plane. In other terms, in the N− events the per-pendicular to the o-Ps decay plane is reversed with respectto the N+ events, by flipping the direction of k2 specularlywith respect to k1. Then the measurement of the asymmetry

A= (N+ −N−)(N+ +N−)

= CCP cos θ (7.18)

allows us to derive the experimental value of CCP.The measurement of the asymmetry A implies that cos θ

in (7.18) is a well defined quantity in the experiment. Inturn, this implies that the o-Ps spin direction is defined. Thisdirection can be selected using an external magnetic field�B , which aligns the o-Ps spin parallel (m = 1), perpendic-ular (m= 0) or antiparallel (m=−1) to the field direction.The magnetic field, in addition, perturbs and mixes the twom= 0 states (one for the para-Ps and the other for the o-Ps).Thus, two new states are possible for the Ps system: theperturbed singlet and the perturbed triplet states, both withm= 0. Their life-times depend on the �B field intensity. Theperturbed singlet state has a life-time shorter than 1 ns (as forthe unperturbed singlet state of the para-Ps), which is not rel-evant in the measurement described here, because too shortcompared to the typical detector time resolution of 1 ns. Forvalues of | �B| of few kGauss, the perturbed o-Ps life-timecan be substantially reduced [938] with respect to the unper-turbed value of about 142 ns [939]. Thanks to this effect, itis possible to separate the m = 0 from the m = ±1 states,by measuring the o-Ps decay time. This is the time betweenthe positron emission (by e.g., a 22Na positron source) andthe detection of the o-Ps decay photons. The Ps is formed in

a target region, where SiO2 powder is used as target mater-ial. The value of the | �B| field that maximizes the decay timeseparation between m = 0 and m = ±1 states is found tobe B = 4 kGauss, corresponding to a m= 0 perturbed o-Pslife-time of 30 ns.

The measurement of the asymmetry A is performed inthe following way. The direction and intensity of the �B fieldare fixed. The k1 and k2 detectors are also fixed. In thisway cos θ has a well defined value. For each event, the Psdecay time and the energies of the three photons from theo-Ps decay are measured. The off-line analysis requires thehighest energy photon in the k1 detector to be within an en-ergy range �E1 = Emax

1 − Emin1 . The second highest en-

ergy photon must be recorded in the k2 detector within anenergy range �E2 = Emax

2 − Emin2 . Then the N+ and N−

events are counted to determine the asymmetry in (7.18).The measurement of the asymmetry A in both the perturbedstates (selected imposing short decay time, e.g., between 10and 60 ns) and unperturbed states (selected imposing longdecay time, between 60 and 170 ns) allows one to elimi-nate the time-independent systematics [937]. Other system-atics, which are time-dependent, do not cancel out with thismethod and determine the final uncertainty on the CCP mea-surement.

An improved detector with superior spatial and energyresolution, as compared to [937], is sketched in Fig. 51. Itconsists of a barrel of BGO crystals with the o-Ps form-ing region at its centre. The crystal signals are read out byavalanche photodiodes (APD), as the detector must work inthe magnetic field. Improved spatial and angular resolutionis obtained thanks to the smaller size of the crystal face ex-posed to the photons, 3×3 cm2, and the larger barrel radius,42 cm. Note that such a detector could also be used effi-ciently for PET scanning, combined with NMR diagnostic.

Fig. 51 Schematic view of the BGO crystal barrel calorimeter used todetect the photons from the o-Ps decay: left, detector front view anddefinition of the k1 and k2 vectors; right, detector side view, showingalso the direction of the magnetic field �B

Eur. Phys. J. C (2008) 57: 13–182 135

This possibility makes this detector a valuable investmentalso for applications in nuclear medical imaging.

With this detector configuration and a simulation of thedetector response, the precision to be reached in the mea-surement of the CCP parameter has been evaluated. A sim-ilar analysis was used for the event selection as describedin [938], except that no veto is needed in the present config-uration, thanks to the good spatial resolution of the proposedcrystals. Various uncertainties affect the CCP measurement.The time-dependent uncertainties on the asymmetry A areinduced mainly by the two-photon background, which af-fect more strongly the events with shorter decay time, aswell as by the inhomogeneity of the o-Ps formation re-gion, which affect the measurement of the o-Ps decay time.For high event statistics (at least 1012 selected three photonevents) the following contributions to the asymmetry mea-surement were found: �Astat ∼ 10−6, �Asyst(2γ bkgd) ∼10−6, �Asyst(o-Ps formation) ∼ 2 × 10−6 resulting into atotal uncertainty: �Astat+syst ∼ 2.5 × 10−6. Being �CCP

related to the asymmetry total uncertainty by the relation�CCP =�Astat+syst/Q [937] with Q, the analyzing power,evaluated to be ∼0.5 for this detector configuration, the totaluncertainty on the CCP parameter is �CCP ∼ 5 × 10−6.

Although this precision is not sufficient to measure theexpected standard model CCP value of order of 10−9, itis suitable to discover CP-violating terms in the orderof 10−5, which if detected would be signal of unexpectednew physics beyond the standard model.

8 LFV experiments

Mixing of leptonic states with different family number asobserved in neutrino oscillations does not necessarily im-ply measurable branching ratios for LFV processes involv-ing the charged leptons. In the standard model the rates ofLFV decays are suppressed relative to the dominant family-number conserving modes by a factor (δmν/mW)4 whichresults in branching ratios which are out of reach experimen-tally. Note that a similar family changing quark decay suchas b→ sγ does obtain a very significant branching ratio ofO(10−4) due to the large top mass.

As has been discussed in great detail in this report, inalmost any further extension to the standard model such assupersymmetry, grand unification or extra dimensions ad-ditional sources of LFV appear. For each scenario a largenumber of model calculations can be found in the literatureand have been reviewed in previous sections, with predic-tions that may well be accessible experimentally. Improvedsearches for charged LFV thus may either reveal physics be-yond the SM or at least lead to a significant reduction inparameter space allowed for such exotic contributions.

Charged LFV processes, i.e. transitions between e, μ,and τ , might be found in the decay of almost any weakly de-caying particle. Although theoretical predictions generallydepend on numerous unknown parameters these uncertain-ties tend to cancel in the relative strengths of these modes.Once LFV in the charged lepton sector were found, the com-bined information from many different experiments wouldallow us to discriminate between the various interpretations.Searches have been performed in μ, τ , π , K , B , D, W andZ decay. Whereas highest experimental sensitivities werereached in dedicated μ and K experiments, τ decay startsto become competitive as well.

8.1 Rare μ decays

LFV muon decays include the purely leptonic modes μ+ →e+γ and μ+ → e+e+e−, as well as the semileptonic μ–econversion in muonic atoms and the muonium–antimuoniumoscillation. The present experimental limits are listed in Ta-ble 19.

Whereas most theoretical models favor μ+ → e+γ , thismode has a disadvantage from an experimental point of viewsince the sensitivity is limited by accidental e+γ coinci-dences and muon beam intensities have to be reduced nowalready. Searches for μ–e conversion, on the other hand, arelimited by the available beam intensities and large improve-ments in sensitivity may still be achieved.

All recent results for μ+ decays were obtained with “sur-face” muon beams containing muons originating in the de-cay of π+’s that stopped very close to the surface of the pionproduction target, or “subsurface” beams from pion decays

Table 19 Present limits on rareμ decays Mode Upper limit (90% C.L.) Year Exp./Lab. Ref.

μ+ → e+γ 1.2 × 10−11 2002 MEGA / LAMPF [180, 940]

μ+ → e+e+e− 1.0 × 10−12 1988 SINDRUM I / PSI [700]

μ+e− ↔ μ−e+ 8.3 × 10−11 1999 PSI [941]

μ−Ti → e−Ti 6.1 × 10−13 1998 SINDRUM II / PSI [942]

μ−Ti → e+Ca∗ 3.6 × 10−11 1998 SINDRUM II / PSI [943]

μ−Pb → e−Pb 4.6 × 10−11 1996 SINDRUM II / PSI [944]

μ−Au → e−Au 7 × 10−13 2006 SINDRUM II / PSI [945]

136 Eur. Phys. J. C (2008) 57: 13–182

just below that region. Such beams are superior to conven-tional pion decay channels in terms of muon stop densityand permit the use of relatively thin (typically 10 mg/cm2)foils to stop the beam. Such low-mass stopping targets arerequired for the ultimate resolution in positron momentumand emission angle, minimal photon yield, or the efficientproduction of muonium in vacuum.

8.1.1 μ→ eγ

Neglecting the positron mass the 2-body decay μ+ → e+γof muons at rest is characterized by

Eγ =Ee =mμc2/2 = 52.8 MeV,

Θeγ = 180◦,

tγ = te,where t is the time of emission from the target, and Θ theopening angle between positron and photon. All μ→ eγ

searches performed during the past three decades were lim-ited by accidental coincidences between a positron from nor-mal muon decay,μ→ eνν, and a photon produced in the de-cay of another muon, either by bremsstrahlung or by e+e−annihilation in flight. This background dominates by far theintrinsic background from radiative muon decay μ→ eννγ .Accidental eγ coincidences can be suppressed by testing thethree conditions listed above. The vertex constraint resultingfrom the ability to trace back positrons and photons to anextended stopping target can further reduce background. At-tempts have been made to suppress accidental coincidencesby observing the low energy positron associated with the

photon, but with minimal success. High muon polarization(Pμ) could help if one would limit the solid angle to ac-cept only positrons and photons (anti)parallel to the muonspin since their rate is suppressed by the factor 1 − Pμ forantiparallel emission at E = mμc2/2 but the reduced solidangle would have to be compensated by increased beam in-tensity which would raise the background again.

The most sensitive search to date was performed by theMEGA Collaboration at the Los Alamos Meson Physics Fa-cility (LAMPF) which established an upper limit (90% C.L.)on B(μ→ eγ ) of 1.2× 10−11 [180, 940]. The MEG exper-iment [946, 947] at PSI, aims at a single-event sensitivityof ∼10−13–10−14, and began commissioning in early 2007.A straightforward improvement factor of more than an or-der of magnitude in suppression of accidental backgroundresults from the DC muon beam at PSI, as opposed to thepulsed LAMPF beam which had a macro duty cycle of 7.7%.Another order of magnitude improvement is achieved by su-perb time resolution (≈0.15 ns FWHM on tγ − te).

The MEG setup is shown in Fig. 52. The spectrometermagnet makes use of a novel “COBRA” (COnstant Bend-ing RAdius) design which results in a graded magnetic fieldvarying from 1.27 T at the centre to 0.49 T at both ends.This field distribution not only results in a constant projectedbending radius for the 52.8 MeV positron, for polar emissionangles θ with | cos θ |< 0.35, but also sweeps away positronswith low longitudinal momenta much faster than a constantfield as used by MEGA. This design significantly reducesthe instantaneous rates in the drift chambers.

The drift chambers are made of 12.5 µm thin foils sup-ported by C-shaped carbon fibre frames which are out of the

Fig. 52 Side and end views of the MEG setup. The magnetic field isshaped such that positrons are quickly swept out of the tracking re-gion thus minimizing the load on the detectors. The cylindrical 0.8 m3

single-cell LXe detector is viewed from all sides by 846 PMTs im-

mersed in the LXe allowing the reconstruction of photon energy, time,conversion point and direction and the efficient rejection of pile-up sig-nals

Eur. Phys. J. C (2008) 57: 13–182 137

Fig. 53 Installing one of thetiming counters into theCOBRA magnet during the pilotrun with the positronspectrometer at the end of 2006.The large ring is one of twoHelmholtz coils used tocompensate the COBRA strayfield at the locations of thephotomultipliers of the LXedetector

way of the positrons. The foils have “vernier” cathode padswhich permit the measurement of the trajectory coordinatealong the anode wires with an accuracy of about 500 µm.

There are two timing counters at both ends of the mag-net (see Fig. 53), each of which consists of a layer of plasticscintillator fibers and 15 plastic scintillator bars of dimen-sions 4× 4× 90 cm3. The fibers give hit positions along thebeam axis and the bars measure positron timings with a pre-cision of σ = 40 ps. The counters are placed at large radii soonly high energy positrons reach them, giving a total rate ofa few 104/s for each bar.

High-strength Al-stabilized conductor for the magnet coilmakes the magnet as thin as 0.20 X0 radially, so that 85%of 52.8 MeV gamma rays traverse the magnet without in-teraction before entering the gamma detector placed out-side the magnet. Whereas MEGA used rather inefficient pairspectrometers to detect the photon, MEG developed a novelliquid Xe scintillation detector, shown in Fig. 52. By view-ing the scintillation light from all sides the electromagneticshower induced by the photon can be reconstructed whichallows for a precise measurement of the photon conversionpoint [948]. Special PMTs that work at LXe temperature(−110◦C), persist under high pressures and are sensitive tothe VUV scintillation light of LXe (λ≈ 178 nm) have beendeveloped in collaboration with Hamamatsu Photonics. Toidentify and separate pile-up efficiently, fast waveform digi-tizing is used for all the PMT outputs.

The performance of the detector was measured with aprototype. The results are shown in Table 20. First data tak-ing with the complete setup took place during the secondhalf of 2007. A sensitivity of O(10−13) for the 90% C.L. up-per limit in case no candidates are found should be reachedafter two years.

8.1.1.1 Beyond MEG Ten times larger surface muon ratesthan used by MEG can be achieved at PSI today already butthe background suppression would have to be improved by

Table 20 Performance of a prototype of the MEG LXe detector atEγ = 53 MeV

Observable Resolution (σ )

energy 1.2%

time 65 ps

conversion point ≈4 mm

two orders of magnitude. Accidental backgroundNacc scaleswith the detector resolutions as

Nacc ∝�Ee ·�t · (�Eγ ·�Θeγ ·�xγ )2 ·A−1T ,

with xγ the coordinate of the photon trajectory at the targetand AT the target area. Here it is assumed that the photoncan be traced back to the target with an uncertainty which issmall compared to AT . Since the angular resolution is dic-tated by the positron multiple scattering in the target this canbe written:

Nacc ∝�Ee ·�t · (�Eγ ·�xγ )2 · dTAT,

with dT the target thickness. When using a series of n targetfoils each of them could have a thickness of dT /n and thebeam would still be stopped. Since the area would increaselike n · AT the background could be reduced in proportionwith 1/n2:

Nacc ∝�Ee ·�t · (�Eγ ·�xγ )2 · dT /nn ·AT ,

so a geometry with ten targets, 1 mg/cm2 each, would leadto the required background suppression.

8.1.2 μ→ 3e

As has been discussed above the sensitivity of μ→ eγ

searches is limited by background from accidental coinci-

138 Eur. Phys. J. C (2008) 57: 13–182

dences between a positron and a photon originating in the in-dependent decays of two muons. Similarly, searches for thedecay μ→ 3e suffer from accidental coincidences betweenpositrons from normal muon decay and e+e− pairs origi-nating from photon conversions or scattering of positronsoff atomic electrons (Bhabha scattering). For this reason themuon beam should be continuous on the time scale of themuon life-time and longer. In addition to the obvious con-straints on relative timing and total energy and momentum,which can be applied in μ→ eγ searches as well, there arepowerful constraints on vertex quality and location to sup-press the accidental background. Since the final state con-tains only charged particles the setup may consist of a mag-netic spectrometer without the need for an electromagneticcalorimeter with its limited performance in terms of energyand directional resolution, rate capability, and event defini-tion in general. On the other hand, of major concern are thehigh rates in the tracking system of a μ→ 3e setup whichhas to stand the load of the full muon decay spectrum.

8.1.2.1 The SINDRUM I experiment The present experi-mental limit, B(μ→ 3e) < 1 × 10−12 [700], was publishedway back in 1988. Since no new proposals exist for this de-cay mode we shall analyse the prospects of an improvedexperiment with this SINDRUM experiment as a point ofreference. A detailed description of the experiment may befound in Ref. [949].

Data were taken during six months using a 25 MeV/csubsurface beam. The beam was brought to rest with a rateof 6×106 μ+/s in a hollow double-cone foam target (length220 mm, diameter 58 mm, total mass 2.4 g). SINDRUMI is a solenoidal spectrometer with a relatively low mag-netic field of 0.33 T corresponding to a transverse momen-tum threshold around 18 MeV/c for particles crossing thetracking system. This system consisted of five cylindricalMWPCs concentric with the beam axis. Three-dimensionalspace points were found by measuring the charges inducedon cathode strips oriented ±45◦ relative to the sense wires.

Gating times were typically 50 ns. The spectrometer ac-ceptance for μ→ 3e was 24% of 4π sr (for a constanttransition-matrix element) so the only place for a significantimprovement in sensitivity would be the beam intensity.

Figure 54 shows the time distribution of the recordede+e+e− triples. Apart from a prompt contribution of cor-related triples one notices a dominant contribution fromaccidental coincidences involving low-invariant-mass e+e−pairs. Most of these are explained by Bhabha scattering ofpositrons from normal muon decay μ→ eνν. The acciden-tal background thus scales with the target mass, but it is notobvious how to reduce this mass significantly below the 11mg/cm2 achieved in this search.

Figure 55 shows the vertex distribution of prompt events.One should keep in mind that most of the uncorrelated

Fig. 54 Relative timing of e+e+e− events. The two positrons arelabeled low and high according to the invariant mass when com-bined with the electron. One notices a contribution of correlatedtriples in the centre of the distribution. These events are mainlyμ→ 3eνν decays. The concentration of events along the diagonal isdue to low-invariant-mass e+e− pairs in accidental coincidence witha positron originating in the decay of a second muon. The e+e− pairsare predominantly due to Bhabha scattering in the target

Fig. 55 Spatial distribution ofthe vertex fitted to prompte+e+e− triples. One clearlynotices the double-cone target

Eur. Phys. J. C (2008) 57: 13–182 139

Fig. 56 Total momentumversus total energy for threeevent classes discussed in thetext. The line shows thekinematic limit (withinresolution) defined byΣ | �pc| + |Σ �pc| ≤mμc2 for anymuon decay. The enhancementin the distribution of correlatedtriples below this limit is due tothe decay μ→ 3eνν

triples contain e+e− pairs coming from the target and theirvertex distribution will thus follow the target contour aswell. This 1-fold accidental background is suppressed by theratio of the vertex resolution (couple of mm2) and the targetarea. There is no reason, other than the cost of the detectionsystem, not to choose a much larger target. Such an increasemight also help to reduce the load on the tracking detectors.Better vertex resolution would help as well. At these low en-ergies tracking errors are dominated by multiple scatteringin the first detector layer but it should be possible to gain bybringing it closer to the target.

Finally, Fig. 56 shows the distribution of total momen-tum versus total energy for three classes of events, (i) uncor-related e+e+e− triples, (ii) correlated e+e+e− triples, and(iii) simulated μ→ 3e decays. The distinction between un-correlated and correlated triples has been made on the basisof relative timing and vertex as discussed above.

8.1.2.2 How to improve? What would a μ → 3e set-up look like that would aim at a single-event sensitiv-ity around 10−16, i.e., would make use of a beam ratearound 1010 μ+/s? The SINDRUM I measurement wasbackground-free at the level of 10−12 with a beam of 0.6 ×107 μ+/s. Taking into account that background would haveset in at 10−13, the increased stop rate would raise the back-ground level to ≈10−10, so six orders of magnitude in back-ground reduction would have to be achieved. Increasingthe target size and improving the tracking resolution shouldbring two orders of magnitude from the vertex requirementalone. Since the dominant sources of background are acci-dental coincidences between two decay positrons (one ofwhich undergoes Bhabha scattering) the background ratescales with the momentum resolution squared. Assumingan improvement by one order of magnitude, i.e., from the≈10% FWHM obtained by SINDRUM I to ≈1% for a newsearch, one would gain two orders of magnitude from theconstraint on total energy alone. The remaining factor 100would result from the test on the collinearity of the e+ andthe e+e− pair.

As mentioned in Ref. [949] a dramatic suppression ofbackground could be achieved by requiring a minimal open-

ing angle (typically 30◦) for both e+e− combinations. De-pending on the mechanism for μ→ 3e, such a cut might,however, lead to a strong loss in μ→ 3e sensitivity as well.

Whereas background levels may be under control, thequestion remains whether detector concepts can be devel-oped that work at the high beam rates proposed. A largemodularity will be required to solve problems of patternrecognition.

8.1.3 μ–e conversion

When negatively charged muons stop in matter they quicklyform muonic atoms which reach their ground states in atime much shorter than the life-time of the atom. Muonicatoms decay mostly through muon decay in orbit (MIO)μ−(A,Z) → e−νμνe(A,Z) and nuclear muon capture(MC) μ−(A,Z)→ νμ(A,Z − 1)∗ which in lowest ordermay be interpreted as the incoherent sum of elementaryμ−p→ nνμ captures. The MIO rate decreases slightly forincreasing values of Z (down to 85% of the free muon ratein the case of muonic gold) due to the increasing muon bind-ing energy. The MC rate at the other hand increases roughlyproportional toZ4. The two processes have about equal ratesaround Z = 12.

When the hypotheticalμ–e conversion leaves the nucleusin its ground state the nucleons act coherently, boosting theprocess relative to the incoherent processes with exited finalstates. The resulting Z dependence has been studied by sev-eral authors [950–953]. For Z � 40 all calculations predict aconversion probability relative to the MC rate which followsthe linear rise with Z expected naively. The predictions may,however, deviate by factors 2–3 at higher Z values.

As a result of the two-body final state the electrons pro-duced in μ–e conversion are mono-energetic and their en-ergy is given by:

Eμe =mμc2 −Bμ(Z)−R(A), (8.1)

where Bμ(Z) is the atomic binding energy of the muonand R is the atomic recoil energy for a muonic atom withatomic number Z and mass number A. In first approxima-tion Bμ(Z)∝Z2 and R(A)∝A−1.

140 Eur. Phys. J. C (2008) 57: 13–182

8.1.3.1 Background Muon decay in orbit (MIO) consti-tutes an intrinsic background source which can only besuppressed with sufficient electron energy resolution. Theprocess predominantly results in electrons with energyEMIO

below mμc2/2, the kinematic endpoint in free muon de-

cay, with a steeply falling high energy component reachingup to Eμe. In the endpoint region the MIO rate varies as(Eμe − EMIO)

5 and a resolution of 1–2 MeV (FWHM) issufficient to keep MIO background under control. Since theMIO endpoint rises at lower Z great care has to be taken toavoid low-Z contaminations in and around the target.

Another background source is due to radiative muon cap-ture (RMC) μ−(A,Z)→ γ (A,Z − 1)∗νμ after which thephoton creates an e+e− pair either internally (Dalitz pair) orthrough γ → e+e− pair production in the target. The RMCendpoint can be kept below Eμe for selected isotopes.

Most low energy muon beams have large pion contam-inations. Pions may produce background when stopping inthe target through radiative pion capture (RPC) which takesplace with a probability of O(10−2). Most RPC photonshave energies above Eμe. As in the case of RMC these pho-tons may produce background through γ → e+e− pair pro-duction. There are various strategies to cope with RPC back-ground:

– One option is to keep the total number of π− stoppingin the target during the live time of the experiment be-low 104−5. This can be achieved with the help of a mod-erator in the beam exploiting the range difference betweenpions and muons of given momentum or with a muon stor-age ring exploiting the difference in life-time.

– Another option is to exploit the fact that pion capturetakes place at a time scale far below a nanosecond. Thebackground can thus be suppressed with a beam counter

in front of the experimental target or by using a pulsedbeam selecting only delayed events.

Cosmic rays (electrons, muons, photons) are a copioussource of electrons with energies around ≈100 MeV. Withthe exception of γ → e+e− pair production in the targetthese events can be recognized by an incoming particle. Inaddition, passive shielding and veto counters above the de-tection system help to suppress this background.

8.1.3.2 SINDRUM II The present best limits (see Ta-ble 19) have been measured with the SINDRUM II spec-trometer at PSI. Most recently a search was performed on agold target [945]. In this experiment (see Fig. 57) the pionsuppression is based on the factor of two shorter range ofpions as compared to muons at the selected momentum of52 MeV/c. A simulation using the measured range distri-bution shows that about one in 106 pions cross an 8 mmthick CH2 moderator. Since these pions are relatively slow99.9% of them decay before reaching the gold target whichis situated some 10 m further downstream. As a result pionstops in the target have been reduced to a negligible level.What remains are radiative pion capture in the degrader andπ− → e−νe decay in flight shortly before entering the de-grader. The resulting electrons may reach the target wherethey can scatter into the solid angle acceptance of the spec-trometer. O(10) events are expected with a flat energy dis-tribution between 80 and 100 MeV. These events are peakedin forward direction and show a time correlation with thecyclotron rf signal. To cope with this background two eventclasses have been introduced based on the values of polarangle and rf phase. Fig. 58 shows the corresponding mo-mentum distributions.

Fig. 57 Plan view of the SINDRUM II experiment. The 1 MW590 MeV proton beam hits the 40 mm carbon production target (topleft of the figure). The πE5 beam line transports secondary particles(π,μ, e) emitted in the backward direction to a degrader situated at the

entrance of a solenoid connected axially to the SINDRUM II spectrom-eter. Inset a shows the momentum dispersion at the position of the firstslit system. Inset b shows a cross section of the beam at the position ofthe beam focus

Eur. Phys. J. C (2008) 57: 13–182 141

Fig. 58 Momentum distributions of electrons and positrons for two event classes described in the text. Measured distributions are compared withthe results of simulations of muon decay in orbit and μ–e conversion

The spectra show no indication for μ–e conversion. Thecorresponding upper limit on

Bμe ≡ Γ(μ−Au → e−Aug.s.

)/Γcapture

(μ−Au

)

< 7 × 10−13 90% C.L. (8.2)

has been obtained with the help of a likelihood analysis ofthe momentum distributions shown in Fig. 58 taking intoaccount muon decay in orbit, μ–e conversion, a contribu-tion taken from the observed positron distribution describingprocesses with intermediate photons such as radiative muoncapture and a flat component from pion decay in flight orcosmic ray background.

8.1.3.3 New initiatives Based on a scheme originally de-veloped during the eighties for the Moscow Meson Factory[954] μe-conversion experiments are being considered bothin the USA and in Japan. The key elements are:

– A pulsed proton beam allows one to remove pion back-ground by selecting events in a delayed time window. Pro-ton extinction factors below 10−9 are needed.

– A large acceptance capture solenoid surrounding the pionproduction target leads to a major increase in muon flux.

– A bent solenoid transporting the muons to the experi-mental target results in a significant increase in momen-tum transmission compared to a conventional quadrupolechannel. A bent solenoid not only removes neutral parti-cles and photons but also separates electric charges.

Unfortunately, the MECO proposal at BNL [955] de-signed along these lines was stopped because of the highcosts. Presently the possibilities are studied to perform aMECO-type of experiment at Fermilab (mu2e). There isgood hope that a proton beam with the required characteris-tics can be produced with minor modifications to the exist-ing accelerator complex which will become available after

the Tevatron stops operation in 2009. A letter of intent is inpreparation.

Further improvements are being considered for an ex-periment at J-PARC. To fully exploit the life-time differ-ence to suppress pion induced background the separationhas to occur in the beam line rather than after the muon hasstopped since the life-time of the muonic atom may be sig-nificantly shorter than the 2.2 µs of the free muon. For thispurpose a muon storage ring PRISM (Phase Rotated IntenseSlow Muon source, see Fig. 59) is being considered [956]which makes use of large-acceptance fixed-field alternating-gradient (FFAG) magnets. A portion of the PRISM-FFAGring is presently under construction as r&d project. As thename suggests the ring is also used to reduce the momentumspread of the beam (from ≈30% to ≈3%) which is achievedby accelerating late muons and decelerating early muons inRF electric fields. The scheme requires the construction ofa pulsed proton beam [957] a decision about which has notbeen made yet. The low momentum spread of the muonsallows for the use of a relatively thin target which is an es-sential ingredient for high resolution in the momentum mea-surement with the PRIME detector [704, 705].

Table 21 lists the μ− stop rates and single-event sensitiv-ities for the various projects discussed above.

8.2 Searches for lepton flavor violation in τ decays

Highest sensitivities to date are achieved at the B-factoriesand further improvements are to be expected. At the LHCthe modes with three charged leptons in the final state suchas τ → 3μ could be sufficiently clean to reach even highersensitivity. Studies have been performed for LHCb [159]and CMS (see below).

8.2.1 B-factories

Present generation B-factories operating around the Υ (4S)resonance also serve as τ -factories, because the production

142 Eur. Phys. J. C (2008) 57: 13–182

Fig. 59 Layout ofPRISM/PRIME. Theexperimental target is situated atthe entrance of the 180◦ bentsolenoid that transports decayelectrons to the detectionsystem. See text for furtherexplanations

Table 21 μ–e conversion searches

Project Lab Status Ep [GeV] pμ [MeV/c] μ− stops [s−1] Sa

SINDRUM II PSI finished 0.6 52 ± 1 107 2 × 10−13

MECO BNL canceled 8 45 ± 25 1011 2 × 10−17

mu2e FNAL under study 8 45 ± 25 0.6 × 1010 4 × 10−17

PRISM/PRIME J-PARC LOI 40 68 ± 3 1012 5 × 10−19

aSingle-event sensitivity: value of Bμe corresponding to an expectation of one observed event

cross sections σbb = 1.1 nb and στ+τ− = 0.9 nb are quitesimilar at centre-of-mass energy near 10.58 GeV. BaBar andBelle have thus been able to reach the highest sensitivity tolepton flavor violating tau decays.

Many theories beyond the standard model allow forτ± → �±γ and τ± → �±�∓�± decays, where �− = e−,μ−,at the level of ∼O(10−10–10−7). Examples are

– SM with additional heavy right handed Majorana neutri-nos or with left handed and right handed neutral isosin-glets [958];

– mSUGRA models with right handed neutrinos introducedvia the see-saw mechanism [242, 959];

– supersymmetric models with Higgs exchange [174, 807]or SO(10) symmetry [164, 960];

– technicolour models with non-universal Z′ exchange[961].

Large neutrino mixing could induce large mixing be-tween the supersymmetric partners of the leptons. Whilesome scenarios predict higher rates for τ± → μ±γ de-

cays, others, for example with inverted mass hierarchy forthe sleptons [242], predict higher rates for τ± → e±γ de-cays.

Semi-leptonic neutrino-less decays involving pseudo-scalar mesons like τ± → �±P 0, where P 0 = π0, η, η′ maybe enhanced over τ± → �±�∓�± decays in supersymmet-ric models, for example, arising out of exchange of CP-oddpseudo-scalar neutral Higgs boson, which are further en-hanced by color factors associated with these decays. Thelarge coupling of Higgs at the ss vertex enhances final statescontaining the η meson, giving a prediction of B(τ± →μ±η) : B(τ± → μ±μ∓μ±) : B(τ± → μ±γ )= 8.4 : 1 : 1.5[810]. Some models with heavy Dirac neutrinos [198, 962],two Higgs doublet models, R-parity violating supersym-metric models, and flavor changing Z′ models with non-universal couplings [963] allow for observable parameterspace of new physics [964], while respecting the existingexperimental bounds at the level of ∼O(10−7).

Eur. Phys. J. C (2008) 57: 13–182 143

Fig. 60 Transverse and longitudinal views of a simulated τ→ μγ event in the BaBar detector. The second tau decays to eνν

Fig. 61 mEC vs. �E for simulated τ → μγ events as reconstructedin the BaBar detector. The tails of distributions are due to initial stateradiation and photon energy reconstruction effects. Latter causes alsothe shift in 〈�E〉

8.2.1.1 Search strategy In the clean e+e− annihilation en-vironment, the decay products of two taus produced are wellseparated in space as illustrated in Fig. 60.

As shown in Fig. 61 neutrino-less τ -decays have twocharacteristic features:

– the measured energy of τ daughters is close to half thecentre-of-mass energy,

– the total invariant mass of the daughters is centeredaround the mass of the τ lepton.

While for ��� modes the achieved mass resolution is excel-lent, the resolution (σ ) of the �γ final state improves from∼20 to 9 MeV by assigning the point of closest approach ofthe muon trajectory to the e+e− collision axis as the decayvertex and by using a kinematic fit with the μγ CM energyconstrained to

√s/2 [191]. The energy resolution is typi-

cally 45 MeV with a long tail due to radiation.The principal sources of background are radiative QED

(di-muon or Bhabha) and continuum (qq) events as wellas τ+τ− events with a mis-identified standard model de-cay mode. There is also some irreducible contribution fromτ+τ− events with hard initial state radiation in which one of

the τ ’s decays into a mode with the same charged particle asthe signal. For example, τ→ μνν decays accompanied by ahard γ is an irreducible background in the τ→ μγ search.

The general strategy to search for the neutrino-less de-cays is to define a signal region, typically of size ∼2σ , inthe energy–mass plane of the τ daughters and to reduce thebackground expectation from well-known CM processes in-side the signal region by optimizing a set of selection crite-ria:

– the missing momentum is consistent with the zero-masshypothesis

– the missing momentum points inside the acceptance ofthe detector

– the second tau is found with the correct invariant mass– minimal opening angle between two tau decay products– minimal value for the highest momentum of any recon-

structed track– particle identification

The analyses are performed in a blind fashion by excludingevents in the region of the signal box until all optimizationsand systematic studies of the selection criteria have beencompleted. The cut values are optimized using control sam-ples, data sidebands and Monte Carlo extrapolation to thesignal region to yield the lowest expected upper limit underthe no-signal hypothesis. The measured mEC vs. �E distri-bution for the τ → μγ search after applying the constraintslisted above is shown in Fig. 62.

For the τ± → �±P 0 searches, the pseudo-scalar mesons(P 0) are reconstructed in the following decay modes: π0 →γ γ for τ± → �±π0, η→ γ γ and η→ π+π−π0 (π0 →γ γ ) for τ± → �±η, and η′ → π+π−η(η→ γ γ ) and η′ →ρ0γ for τ± → �±η′.

144 Eur. Phys. J. C (2008) 57: 13–182

8.2.1.2 Experimental results from BaBar and Belle By thebeginning of 2007 BaBar and Belle had recorded integratedluminosities of L ∼ 400 and 700 fb−1, respectively, whichcorresponds to a total of ∼109 τ -decays. Analysis of thesedata samples is still ongoing and published results includeonly part of the data analysed. No signal has yet been ob-served in any of the probed channels and some limits andthe corresponding integrated luminosities are summarizedin Table 22. Frequentest upper limits have been calculatedfor the combination of the two experiments [183] using thetechnique of Cousins and Highland [965] following the im-plementation of Barlow [966].

8.2.1.3 Projection of limits to higher luminosities B(τ± →μ±γ )and B(τ± → μ±μ∓μ±) have been lowered by fiveorders of magnitude over the past twenty-five years. Fur-ther significant improvements in sensitivity are expectedduring the next five years. Depending upon the nature ofbackgrounds contributing to a given search, two extremescenarios can be envisioned in extrapolating to higher lumi-nosities:

Fig. 62 Measured distribution of mEC vs. �E for τ → μγ recon-structed by BaBar [191]. The shaded region taken from Fig. 61 con-tains 68% of the hypothetical signal events

– If the expected background is kept below O(1) events,while maintaining the same efficiency B90

UL ∝ 1/L if nosignal events would be observed. In τ± → μ±μ∓μ±searches, for example, the backgrounds are still quite lowand the irreducible backgrounds are negligible even forprojected Super B-factories.

– If there is background now already and no reduction couldbe achieved in the future measurements B90

UL ∝ 1/√

L.

The√

L scaling is, however, unduly pessimistic since theanalyses improve steadily. Better understanding of the na-ture of the backgrounds will lead to a more effective separa-tion of signal and background.

The τ± → μ±γ searches suffer from significant back-ground from both μ+μ− and τ+τ− events and to a lesserextend from qq production. While one can expect to re-duce these backgrounds with continued optimization withmore luminosity at the present day B-factories, much ofthe background from τ+τ− events is irreducible comingfrom τ → μνν decays accompanied by initial state radia-tion. This source represents about 20% of the total back-ground in the searches performed by the BaBar experiment[191] and it is conceivable that an analysis can be developedthat reduces all but this background with minimal impact onthe efficiency. One could also include new selection criteriasuch as a cut on the polar angle of the photon which couldreduce the radiative “irreducible” background by 85% witha 40% loss of signal efficiency. Table 23 summarizes the fu-ture sensitivities for various LFV decay modes.

In order to further reduce the impact of irreducible back-grounds at a future Super B-factory experiment, one canconsider what is necessary to improve the mass resolutionof the, e.g., μ–γ system. Currently, this resolution is lim-ited by the γ angular resolution. Therefore improvementsmight be expected if the granularity of the electromagneticcalorimeter is increased.

Table 22 Integrated luminosities and observed upper limits on the branching fractions at 90% C.L. for selected LFV tau decays by BaBar andBelle

Channel BaBar Belle

L (fb−1) BUL (10−8) Ref. L (fb−1) BUL (10−8) Ref.

τ± → e±γ 232 11 [182] 535 12 [181]

τ± → μ±γ 232 6.8 [191] 535 4.5 [181]

τ± → �±�∓�± 92 11–33 [644] 535 2–4 [967]

τ± → e±π0 339 13 [648] 401 8.0 [968]

τ± → μ±π0 339 11 [648] 401 12 [968]

τ± → e±η 339 16 [648] 401 9.2 [968]

τ± → μ±η 339 15 [648] 401 6.5 [968]

τ± → e±η′ 339 24 [648] 401 16 [968]

τ± → μ±η′ 339 14 [648] 401 13 [968]

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8.2.2 CMS

So far, only τ → μ transitions have been studied sincemuons are more easily identified and the CMS triggerthresholds for muons are generally lower than for electrons.The τ → μγ channel was studied in the past [969] both forCMS and for ATLAS but found not to be competitive withthe prospects at the B-factories. The τ → 3μ channel looksmore promising and will be discussed below.

8.2.2.1 τ production at the LHC It is planned to operatethe LHC in three different phases. After a commissioningphase the LHC will be ramped up to an initial luminosityof L = 1032 cm−2 s−1 followed by a low luminosity phase(L = 2 × 1033 cm−2 s−1). A high luminosity phase withL = 1034 cm−2 s−1 will start in 2010 and last for a periodof several years. The integrated luminosity per year will be10–30 and 100–300 fb−1 for the low and high luminosityphases, respectively [970].

The rate of τ leptons produced was estimated with thehelp of PYTHIA 6.227 using the parton distribution func-tion CTEQ5L. The results are shown in Table 24. Duringthe low luminosity phase assuming an integrated luminosityof only 10 fb−1 per year about 1012 τ leptons will be pro-duced within the CMS detector. The dominant productionsources of τ leptons at the LHC are the Ds and various Bmesons. The W and the Z production sources will provideconsiderably less τ leptons per year, but at higher energieswhich is an advantage for the efficient detection of their de-cay products (see below).

8.2.2.2 τ → 3μ detection A key feature of CMS is a 4 Tmagnetic field, which ensures the measurement of charged-particle momenta with a resolution of σpT/pT = 1.5% for

Table 23 Expected 90% C.L. upper limits on LFV τ decays with75 ab−1 assuming no signal is found and reducible backgrounds aresmall (∼O(1) events) and the irreducible backgrounds scale as 1/L

Decay mode Sensitivity

τ→ μγ 2 × 10−9

τ→ eγ 2 × 10−9

τ→ μμμ 2 × 10−10

τ→ eee 2 × 10−10

τ→ μη 4 × 10−10

τ→ eη 7 × 10−10

10 GeV muons [970] using a four-station muon system.A silicon pixel detector and tracker allow to reconstructsecondary vertices with a resolution of about 50 µm [971]and help to improve the muon reconstruction. Furthermore,CMS has an electromagnetic calorimeter (ECAL) composedof PbWO4 and a copper scintillator hadronic calorime-ter (HCAL). As a result of the high magnetic field andthe amount of material that has to be crossed only muonswith pT > 3 GeV/c are accepted. The reconstruction effi-ciency varies between ≈70% at 5 GeV [972] and ≈98% at100 GeV/c [970].

The two levels of the CMS trigger system are called“level 1” (L1) and “high level” (HLT). The triggers relevantfor this analysis are the dedicated single and di-muon trig-gers. For the low luminosity phase it is planned to use as pT

thresholds for single muons 14 GeV/c at L1 and 19 GeV/cfor the HLT. The thresholds for the di-muon trigger will be3 GeV/c at L1 and 7 GeV/c for the HLT.

Most τ → 3μ events produced via W and Z decays willbe accepted by the present triggers. Unfortunately, the lowpT of the muons from the decays of τ ’s originating in Dsor B decay result in a very low trigger efficiency (Fig. 63).Dedicated trigger algorithms with improved efficiency arepresently being studied.

To improve the identification of low pT muons a newmethod is currently under development combining the en-ergy deposit in the ECAL, HCAL and the number of recon-structed muon track segments in the muon systems. The in-

Fig. 63 pT distributions of the leading and next-to-leading muon fromthe decay τ→ 3μ at CMS. The indicated trigger thresholds for the lowluminosity phase are clearly too high for the efficient detection of theseevents

Table 24 Number of τ leptons per year produced during the low-luminosity phase of the LHC

Production channel W → τντ γ /Z→ ττ B0 → τX B± → τX Bs → τX Ds → τX

Nτ /10 fb−1 1.7 × 108 3.2 × 107 4.0 × 1011 3.8 × 1011 7.9 × 1010 1.5 × 1012

146 Eur. Phys. J. C (2008) 57: 13–182

Fig. 64 Invariant massdistribution from the simulationof τ → 3μ events

variant mass distribution of reconstructed τ → 3μ events isshown in Fig. 64. The resolution is about 24 MeV/c2, whichensures a good capability to reduce background events.

8.2.2.3 Background and expected sensitivity The mainsources of muons are decays of D and B mesons whichare copiously produced at LHC energies. A previous study[973] suggested that these background events can be sup-pressed by appropriate selection criteria. The probability tomisidentify an event from pile-up is small and cosmic rayscan be rejected by timing. Due to the high momentum of themuons from direct W and Z decays, the contribution to thebackground is negligible [974].

One rare decay that can mimic the signal is Ds → φμνμ

followed by a decay φ→ μμ. This background can be re-duced by an invariant mass cut around the φ mass. Radia-tive φ decay φ→ μμγ survives this cut since the photonusually remains undetected. These radiative decays and anyother heavy meson decays may be suppressed using sec-ondary vertex properties and isolation criteria and by explor-ing the three-muon angular distributions. These studies arein progress.

Predictions of the achievable sensitivity are available inan older CMS Note [973]. In case no signal is observed theexpected upper limit on the τ → 3μ branching ratio at 95%C.L. for theW source is 7.0×10−8 (3.8×10−8) for 10 fb−1

(30 fb−1) of collected data. For the Z source a limit of 3.4×10−7 and for the B meson source a limit of 2.1 × 10−7 wasderived assuming an integrated luminosity of 30 fb−1. TheDs source was not studied in this early paper.

Potentially including the muons fromD and B meson de-cays may lead to significant improvements of the sensitivity.Further studies are necessary to make firm predictions.

8.3 B0d,s → e±μ∓

The present limits B(B0d → eμ) < 1.7 × 10−7 [975] deter-

mined by Belle and B(B0s → eμ) < 6.1 × 10−6 [976] from

CDF are of interest since they place bounds on the massesof two Pati–Salam leptoquarks [361] (see below). Both mea-surements are almost background free so significant im-provements should be expected from these experiments.These decay modes have similarities with the K0

L→ μe de-cay for which an upper limit of 4.7 × 10−12 exists [977].

The prospects of a more sensitive search have been stud-ied for the LHCb experiment [978]. Although backgroundlevels are higher, this is more than compensated by the im-proved single-event sensitivity. The event selection closelyfollows that of the B0

s → μ+μ− decay. The dominant back-grounds come from (i) events in which two b hadrons de-cay into leptons combining to a fake vertex and (ii) fromtwo-body charmless hadronic decays when the two hadronsare misidentified as leptons. Signal and background areseparated on the basis of particle identification, invariantmass (σ(mB) = 50 MeV/c2), transverse momenta, properdistance and the isolation of the B0 candidate from theother decay products. See Ref. [978] for details. Simula-tion shows that for an integrated luminosity of 2 fb−1 thetotal background can be reduced to ≈80 events with a se-lection efficiency of 1.4%. Assuming no signal would befound the 90% C.L. upper limits would be 1.6 × 10−8 and6.5 × 10−8 for B(B0

d → e±μ∓) and B(B0s → e±μ∓), re-

spectively. These values correspond to 90% C.L. lower lim-its on the leptoquark mass and mixings of 90×Fdmix TeV and

65 × F smix TeV, where Fd,smix are factors taking into accountgeneration mixing within the model. Present limits are 50and 21 TeV, respectively (see Fig. 65).

Eur. Phys. J. C (2008) 57: 13–182 147

Fig. 65 90% C.L. limits onB(B0

d → eμ) (left panel) andB(B0

s → eμ) (right panel) andthe corresponding lower limitson the products of Pati–Salamleptoquark mass and mixing.Present results are comparedwith results projected for LHCbfor an integrated luminosity of2 fb−1 in case no signal wouldbe observed. Dashed regionsindicate the theoreticaluncertainties in the relationbetween the variables

8.4 In flight conversions

Lepton Flavor Violation could manifest itself in the conver-sion of high energy muons into tau leptons when scatter-ing on nucleons in a fixed target configuration [979]. Muonscan be produced much more copiously than tau leptons soμ→ τ conversions could be more sensitive than neutrino-less τ→ μX decays. When considering the effective leptonflavor violating four-fermion couplings, tau decays mainlyinvolve light quarks, so heavy quark couplings are onlyloosely constrained [980]. In SUSY models, muon to tauconversion could be greatly enhanced by Higgs mediationat energies where heavy quarks contribute [812].

Within the context of this workshop the experimental fea-sibility of such experiment has been investigated. The crosssection for mu to tau lepton conversion on target has beenestimated to be at most 550 ab [980] for 50 GeV muons, us-ing an effective model independent interpretation of the taudecay LFV constraints [964] based on the 2000 data [981].By rescaling the upper limit on B(τ→ μπ+π−) to the cur-rent value [982, 983], one obtains an upper limit at 90% C.L.on the cross section of 4.7 ab. This value scales roughly lin-early with the muon energy. In the context of the MSSM,the experimental data available in 2004 constrained the crosssection in the range from 0.1 ab to 1 ab for muon energiesfrom 100 to 300 GeV [812].

The following assumptions were made to assess the ex-perimental feasibility:

– the goal is a sensitivity to the cross section correspondingto 1/10 of the present limits from tau decay, collecting atleast thousand events per year;

– the active target consists of 330 planes of 300 µm thicksilicon, with either strip or pixel readout;

– the target has transverse dimensions corresponding to anarea of 1 m2 and the beam is distributed homogeneouslyover the target.

As a consequence, 3.75 × 1019 muons/yr are neededwhich, assuming a 10% duty cycle and an effective data tak-ing year of 107 s, corresponds to 3.75×1013 muons/s (peak)and 3.75 × 1012 muons/s (average).

Using the LEPTO 6.5.1 generator [984] deep inelasticmuon scattering off nucleons was studied. The amount ofpower dissipated in the target is sustainable, and the interac-tion rate is 0.6 interactions per 25 ns, which is comparableto LHC experiments. Radiation levels and occupancy in thesilicon active target appear to be tractable, provided pixelreadout is used.

When requiring momentum transfer above 2 GeV and in-variant mass of the hadronic final state above 3 GeV an ef-fective interaction cross section of 47 nb was found. Thisvalue reduces to 15 nb when applying the level 0 triggerrequirement of at least 60 GeV of hadronic energy which re-sults in a rate of 7.7 MHz. The amount of data that needs tobe extracted from the tracker for further event selection canprobably be handled at such rate.

Unfortunately it appears that the required muon flux isincompatible with the operation of calorimeters as trigger-ing and detecting devices. Assuming an LHCb-like electro-magnetic calorimeter with a 2.6 cm thick lead absorber andan integration time of 25 ns, and assuming that electronsfrom muon decay travel unscreened for 4 m before encoun-tering the electro-magnetic calorimeter, three high energyelectrons per 25 ns integration time reach the calorime-ter, preventing any effective way of triggering on electrons.Assuming an LHCb-like hadronic calorimeter structured intowers consisting of 75 layers including 13 × 13 cm2 scin-tillating pads and 16 mm of iron each, each tower will de-tect 25 TeV of equivalent hadronic energy for each 25 ns ofintegration time just because of the muon flux energy loss.The Poisson fluctuation of the number of muons will inducea fluctuation in the detected hadronic energy per tower ofabout 200 GeV, preventing the use of the hadronic calorime-ter as a trigger for μN→ τX.

In conclusion, the idea of using an intense but trans-versely spread muon flux to produce and detect LFV muonconversions to tau leptons does not appear feasible in thispreliminary study, mainly because it does not appear possi-ble to operate calorimeters at these rates.

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9 Experimental studies of electric and magnetic dipolemoments

9.1 Electric dipole moments

We review here the current status and prospects of thesearches for fundamental EDMs, a flavor diagonal signal ofCP violation. At the non-relativistic level, the EDM d de-termines a coupling of the spin to an external electric field,H ∼ d �E · �S. Searches for intrinsic EDMs have a long his-tory, stretching back to the prescient work of Purcell andRamsey who used the neutron EDM as a test of parityin nuclear physics. At the present time, there are two pri-mary motivations for anticipating a nonzero EDM at or nearcurrent sensitivity levels. Firstly, a viable mechanism forbaryogenesis requires a new CP-odd source, which if tiedto the electroweak scale necessarily has important implica-tions for EDMs. The second is that CP-odd phases appearquite generically in models of new physics introduced forother reasons, e.g. in supersymmetric models. Indeed, it isonly the limited field content of the SM which limits the ap-pearance of CP-violation to the CKM phase and θQCD. Thelack of any observation of a nonzero EDM has, on the flip-side, provided an impressive source of constraints on newphysics, and there is now a lengthy body of literature on theconstraints imposed, for example, on the soft breaking sectorof the MSSM. Generically, the EDMs ensure that new CP-odd phases in this sector are at most of O(10−1–10−2), atuning that appears rather unwarranted given the O(1) valueof the CKM phase.

The strongest current EDM constraints are shown forthree characteristic classes of observables in Table 25, andwill be discussed in detail in the following.

We summarize first the details of the EDM constraints,and the induced bounds on a generic class of CP-odd opera-tors normalized at 1 GeV, commenting on how the next gen-eration of experiments will impact significantly on the levelof sensitivity in all sectors. We then turn to a brief discus-sion of some of the constraints on new physics that ensuefrom these bounds. More detailed discussions of phenom-enology of EDMs is given in the first half of this report (seee.g. Sect. 5.7).

9.1.1 CP-odd operators and electric dipole moments

We shall briefly review the relevant formulae for the ob-servable EDMs in terms of CP-odd operators normalized at

1 GeV. Including the most significant flavor diagonal CP-odd operators (see e.g. [742]) up to dimension six, the cor-responding effective Lagrangian takes the form,

L1 GeVeff = g2

s

32π2θGaμνG

μν,a − i

2

∑i=e,u,d,s

diψi(Fσ)γ5ψi

− i

2

∑i=u,d,s

di ψigs(Gσ)γ5ψi

+ 1

3wf abcGaμνG

νβ,bGμ,cβ

+∑i,j=e,q

Cij (ψiψi)(ψj iγ5ψj )+ · · · . (9.1)

The θ -term, as it has a dimensionless coefficient, is partic-ularly dangerous leading to the strong CP problem and inwhat follows we shall invoke the axion mechanism [986] toremove this term.

The physical observables can be conveniently separatedinto three main categories, depending on the physical mech-anisms via which an EDM can be generated: EDMs ofparamagnetic atoms and molecules; EDMs of diamagneticatoms; and the neutron EDM. The inheritance pattern forthese three classes is represented schematically in Fig. 66and, while the experimental constraints on the three classesof EDMs differ by several orders of magnitude, it is impor-tant that the actual sensitivity to the operators in (9.1) turnsout to be quite comparable in all cases. This is due to vari-ous enhancements or suppression factors which are relevantin each case, primarily associated with various violationsof “Schiff shielding”—the non-relativistic statement that anelectric field applied to a neutral atom must necessarily bescreened and thus remove any sensitivity to the EDM.

9.1.2 EDMs of paramagnetic atoms

For paramagnetic atoms, Schiff shielding is violated by rel-ativistic effects which can in fact be very large. One hasroughly [987],

dpara(de)∼ 10α2Z3de, (9.2)

which for large atoms such as Thallium amounts to a hugeenhancement of the field seen by the electron EDM (see e.g.

Table 25 Current constraintswithin three representativeclasses of EDMs

Class EDM Current bound Ref.

Paramagnetic 205Tl |dTl|< 9 × 10−25 e cm [186]

Diamagnetic 199Hg |dHg|< 2 × 10−28 e cm [985]

Nucleon n |dn|< 3 × 10−26 e cm [869]

Eur. Phys. J. C (2008) 57: 13–182 149

Fig. 66 A schematic plot of the hierarchy of scales between the lep-tonic and hadronic CP-odd sources and three generic classes of ob-servable EDMs. The dashed lines indicate generically weaker depen-

dencies in SUSY models. The current situation is given on the left,while on the right we show the dependencies of several classes of next-generation experiments

[987, 988]), which counteracts the apparently lower sensi-tivity of the Tl EDM bound,

dTl =−585de − 43 GeV × eCsingletS . (9.3)

We have also included here the most relevant CP-oddelectron–nucleon interaction, namely CSeiγ5eNN , whichin turn is related to the semileptonic four-fermion operatorsin (9.1).

9.1.3 EDMs of diamagnetic atoms

For diamagnetic atoms, Schiff shielding is instead violatedby the finite size of the nucleus and differences in the distri-bution of the charge and the EDM. However, this is a rathersubtle effect,

ddia ∼ 10Z2(RN/RA)2dq , (9.4)

and the suppression by the ratio of nuclear to atomic radii,RN/RA, generally leads to a suppression of the sensitivityto the nuclear EDM, parameterized to leading order by theSchiff moment S, by a factor of 103 (see e.g. [987, 988]).Thus, although the apparent sensitivity to the Hg EDM isorders of magnitude stronger than for the Tl EDM, bothexperiments currently have comparable sensitivity to vari-ous CP-odd operators and thus play a very complementaryrole. Combining the atomic dHg(S), nuclear S(gπNN), and

QCD g(1)πNN(dq), components of the calculation [742, 988],we have

dHg = 7 × 10−3e(du − dd )+ 10−2de + O(CS,Cqq), (9.5)

where the overall uncertainty is rather large, a factor of 2–3,due to significant cancellations between various contribu-

tions. A valuable feature of dHg is its sensitivity to the tripletcombination of color EDM operators dq .

9.1.4 Neutron EDM

The neutron EDM measurement is of course not sensitiveto the above atomic enhancement/suppression factors and,using the results obtained using QCD sum rule techniques[742] (see also [849, 989, 990] for alternative chiral ap-proaches), wherein under Peccei–Quinn relaxation of theaxion the contribution of sea-quarks is also suppressed atleading order:

dn = (1.4 ± 0.6)(dd − 0.25du)

+ (1.1 ± 0.5)e(dd + 0.5du)

+ 20 MeV × ew+ O(Cqq). (9.6)

Note that the proportionality to dq〈qq〉 ∼mq〈qq〉 ∼ f 2πm

2π

removes any sensitivity to the poorly known absolute valueof the light quark masses.

9.1.5 Future developments

The experimental situation is currently very active, and anumber of new EDM experiments, as detailed in this re-port, promise to improve the level of sensitivity in all threeclasses by one-to-two orders of magnitude in the comingyears. These include: new searches for EDMs of polarizableparamagnetic molecules, which aim to exploit additional po-larization effects enhancing the effective field seen by theunpaired electron by a remarkable factor of up to 105, andare therefore primarily sensitive to the electron EDM; newsearches for the EDM of the neutron in cryogenic systems;

150 Eur. Phys. J. C (2008) 57: 13–182

and also proposed searches for EDMs of charged nuclei andions using storage rings. This latter technique clearly aims toavoid the effect of Schiff shielding and enhance sensitivity tothe nuclear EDM and its hadronic constituents. A schematicsummary of how a number of these new experiments willbe sensitive to the set of CP-odd operators is exhibited inFig. 66.

9.1.6 Constraints on new physics

Taking the existing bounds, and the formulae above, we ob-tain the following set of constraints on the CP-odd sourcesat 1 GeV (assuming an axion removes the dependence on θ ),∣∣∣∣de + e(26 MeV)2

(3Ced

md+ 11

Ces

ms+ 5Ceb

mb

)∣∣∣∣< 1.6 × 10−27e cm from dTl, (9.7)

∣∣(dd − du)+ O(ds, de,Cqq,Cqe)∣∣< 2 × 10−26 e cm

from dHg, (9.8)∣∣e(dd + 0.56du)+ 1.3(dd − 0.25du)+ O(ds ,w,Cqq)

∣∣< 2 × 10−26 e cm from dn, (9.9)

where the additional O(· · · ) dependencies are known lessprecisely, but may not always be sub-leading in particularmodels. The precision of these results varies from 10–15%for the Tl bound, to around 50% for the neutron bound, andto a factor of a few for Hg. It is remarkable to note that, ac-counting for the naive mass-dependence df ∝mf , all theseconstraints are of essentially the same order of magnitudeand thus highly complementary. Constraints obtained in thehadronic sector using other calculational techniques differsomewhat but generally give results consistent with thesewithin the quoted precision.

The application of these constraints to models of newphysics has many facets and is discussed in several specificcases elsewhere in this report. We shall limit our attentionhere to just a few simple examples relevant to the motiva-tions noted above.

9.1.7 The SUSY CP-problem

It is now rather well-known that a generic spectrum ofsoft SUSY-breaking parameters in the MSSM will generateEDMs via one loop diagrams [852] that violate the exist-ing bounds by one-to-two orders of magnitude leading tothe SUSY CP problem. The situation is summarized ratherschematically in Fig. 67.

In many respects the situation is better described by theamount of fine tuning of the MSSM spectrum that is requiredto avoid these leading order contributions, and by how muchthe ability to avoid the EDM constraints is limited by sec-ondary constraints from numerous, and more robust, two

Fig. 67 Constraints on the CMSSM phases θA and θμ from a com-bination of the three most sensitive EDM constraints, dn, dTl and dHg,forMSUSY = 500 GeV, and tanβ = 3 (from [742]). The region allowedby EDM constraints is at the intersection of all three bands aroundθA = θμ = 0

loop contributions [991] and four-fermion sources [992].Indeed, if we consider two extreme cases: (i) the 2HDM,where all SUSY fermions and sfermions are very heavy; and(ii) split SUSY, where all SUSY scalars are very heavy, onefinds that while one loop EDMs are suppressed, two loopcontributions are already very close to the current bounds[857, 858, 992]. This bodes well for the ability of next-generation experiments to provide a comprehensive test oflarge SUSY phases at the electroweak scale, regardless ofthe detailed form of the SUSY spectrum.

9.1.8 Constraints on new SUSY thresholds

If SUSY is indeed discovered at the LHC, but with no sign ofphases in the soft sector, one may instead consider the abil-ity of EDMs to detect new supersymmetric CP-odd thresh-olds. At dimension five there are several R-parity conserv-ing operators, besides those well-known examples associ-ated with neutrino masses and baryon and lepton numberviolation [150, 371]. Writing the relevant dimension five su-perpotential as [993, 994]

�W = yh

ΛhHdHuHdHu +

Yqeijkl

Λqe(UiQj )EkLl

+ Yqqijkl

Λqq(UiQj )(DkQl)

+ Yqqijkl

Λqq

(Uit

AQj)(Dkt

AQl), (9.10)

one finds that order-one CP-odd coefficients with a genericflavor structure, particularly for the semileptonic operators,are probed by the sensitivity of dTl and dHg at the remarkablelevel of Λ ∼ 108 GeV [993, 994]. This is comparable to,or better than, the corresponding sensitivity of lepton-flavorviolating observables.

Eur. Phys. J. C (2008) 57: 13–182 151

9.1.9 Constraints on minimal electroweak baryogenesis

As noted above, one of the primary motivations for antici-pating nonzero EDMs at or near the current level of sensitiv-ity is through the need for a viable mechanism of baryogene-sis. This is clear in essentially all baryogenesis mechanismsthat are tied to the electroweak scale. As a simple illustra-tion, one can consider a minimal extension of the SM Higgssector [995–997],

Ldim 6 = 1

Λ2

(H †H

)3 + Zuij

Λ2CP

(H †H

)Uci HQj

+ Zdij

Λ2CP

(H †H

)Dci H

†Qj

+ Zeij

Λ2CP

(H †H

)Eci H

†Lj . (9.11)

The first term is required to induce a sufficiently strong firstorder electroweak phase transition, while the remaining op-erators provide the additional source (or sources) of CP-violation, where we have assumed consistency with the prin-ciple of minimal flavor violation. Modified Higgs couplingsof this kind, including CP-violating effects, are currently thesubject of significant research within collider physics, rele-vant to the LHC in particular [588], making EDM probes ofmodels of this kind quite complementary.

As discussed in [997], such a scenario can reproducethe required baryon-to-entropy ratio, ηb = 8.9 × 10−11,while remaining consistent with the EDM bounds, pro-vided the thresholds and the Higgs mass are quite low, e.g.400 GeV<Λ, ΛCP < 800 GeV. The EDMs in this case aregenerated at the two loop level, and it is clear that an im-provement in EDM sensitivity by an order of magnitudewould provide a conclusive test of minimal mechanisms ofthis form.

9.2 Neutron EDM

The neutron electric dipole moment is sensitive to manysources of CP violation. Most famously, it constrains QCD(the “strong CP problem”), but it also puts tight con-straints on supersymmetry and other physics models be-yond the standard model. The standard model predictionof ∼10−32 e cm is a factor of 106 below existing limits, soany convincing signal within current or foreseen sensitivityranges will be a clear indication of physics beyond the SM.

All current nEDM experiments use NMR techniques tosearch for electric-field induced changes in the Larmor pre-cession frequency of bottled ultracold neutrons. Recent re-sults from a room-temperature apparatus at ILL yielded anew limit of |dn|< 2.9×10−26 e cm (90% C.L.) which rules

out many “natural” varieties of SUSY. Several new experi-ments hope to improve on this limit: two of these involvenew cryogenic techniques that promise an eventual increasein sensitivity by two orders of magnitude. First results, at thelevel of ∼10−27 e cm, are to be expected within about fouryears.

9.2.1 ILL

A measurement of the neutron EDM was carried out at theILL between 1996 and 2002, by a collaboration from theUniversity of Sussex, the Rutherford Appleton Laboratory,and the ILL itself. The final published result provided a limitof |dn| < 2.9 × 10−26 e cm (90% C.L.) [869]. This repre-sents a factor of two improvement beyond the intermediateresult [998] and almost a factor of four beyond the resultsexisting prior to this experiment [999, 1000]. The collabo-ration, which has now expanded to include Oxford Univer-sity and the University of Kure, has designed and developed“CryoEDM”, a cryogenic version of the experiment that isexpected to achieve two orders of magnitude improvementin sensitivity. Construction and initial testing are underwayat the time of writing.

Experimental technique The room-temperature measure-ment was carried out using stored ultracold neutrons (i.e.having energies �200 neV) from the ILL reactor. The Ram-sey technique of separated oscillatory fields was used to de-termine the Larmor precession frequency of the neutronswithin �B and �E fields. The signature of an EDM is a fre-quency shift proportional to any change in the applied elec-tric field.

The innovative feature of this experiment was the use ofa cohabiting atomic-mercury magnetometer [1001]. Spin-polarized Hg atoms shared the same volume as the neutrons,and the measurement of their precession frequency provideda continuous high-resolution monitoring of the magneticfield drift: prior to this, such drift entirely dominated the tiny�E-field induced frequency changes that were sought.

Systematics Analysis of the data revealed a new sourceof systematic error, which, as the problem of B-field drifthad been virtually eliminated, became potentially the dom-inant error. Its origins lay in a geometric-phase (GP) effect[1002]—an unfortunate collusion between any small appliedaxial �B-field gradient and the component of �B induced inthe particle’s rest frame by the Lorentz transformation of theelectric field. This GP effect induced a frequency shift pro-portional to �E, and hence a false EDM signal. In fact, the Hgmagnetometer itself was some 50 times more susceptible tothis effect than were the neutrons, so the introduction of themagnetometer brought the GP systematic with it.

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This effect was overcome by careful measurement of theneutron-to-Hg frequency ratios for both polarities of mag-netic field, in order to determine the point nominally cor-responding to zero applied axial B-field gradient, as wellas by a series of auxiliary measurements to pin down smallcorrections due to local dipole [1003] and quadrupole fields(as well as the Earth’s rotation). The final result thereforeremained statistically limited.

The experiment is now complete and, as will be discussedbelow, the equipment will be used for further studies by an-other collaboration based largely at the PSI.

Still another collaboration, led by the PNPI in Russia,is developing a new room-temperature nEDM apparatus,which they plan to run at ILL. It is also intended to reacha sensitivity of ∼10−27 e cm, to be achieved in part by theuse of multiple back-to-back measurement chambers withopposing electric fields to cancel some systematic errors.

Cryogenic experiments overview It has been known forseveral decades [1004] that 8.9 Å neutrons incident on su-perfluid 4He at 0.5 K will down-scatter, transferring their en-ergy and momentum to the helium and becoming ultracoldneutrons (UCN) in the process. This so-called super-thermalUCN source provides a much higher flux than is availablesimply from the low energy tail of the Maxwell distribution.In addition, the immersion of the apparatus in a bath of liq-uid helium should allow for the provision of stronger elec-tric fields than could be sustained in vacuo. The other twovariables that contribute to the figure of merit for this experi-ment, namely the polarization and the NMR coherence time,should also be improved: the incident cold neutron beam canbe very highly polarized, and the polarization remains intactduring the down-scattering process; and the improved uni-formity of magnetic field attainable with superconductingshields and coil will reduce depolarization during storage,while losses from up-scattering will be much reduced due tothe cryogenic temperatures of the walls of the neutron stor-age vessels.

ILL CryoEDM experiment status The majority of the ap-paratus for the cryoEDM experiment has been installed atILL, and testing is underway. UCN production via this su-perthermal mechanism has been demonstrated [1005], andthe solid-state UCN detectors developed by the collabora-tion have also been shown to work well [1006]. At the timeof writing, there are still some hardware problems to beresolved, in particular with components in and around theRamsey measurement chamber. A high precision scan of themagnetic field was carried out in 2007, and measurementswere made of the neutron polarization. An initial HV sys-tem will be installed in spring 2008. By the end of 2008,the system is expected to have a statistical sensitivity of∼10−27 e cm.

Future plans In order to achieve optimum sensitivity, anumber of improvements will need to be made:

– The superconducting magnetic shielding requires addi-tional protective “end caps” to shield fully the ends of thesuperconducting solenoid.

– The current measurement chamber only has two cells: onewith HV applied, and one at ground as a control. It isplanned to upgrade to a four-cell chamber, with the HVapplied to the central electrode, in order to be able to carryout simultaneous measurements with electric fields in op-posite directions. As well as canceling several potentialsystematic errors, this will reduce the statistical uncer-tainty by doubling the number of neutrons counted.

– The ILL is preparing a new beam line with six times thecurrently available intensity of 8.9 Å neutrons, and wishesto transfer the experiment to that beam line in 2009. Fund-ing for these improvements is expected to be contingenton successful running of the existing apparatus.

A sensitivity of ∼2 × 10−28 e cm should be achievablewithin two to three years of running at the new beam line.

9.2.2 PSI

The present best limit for the neutron electric dipole mo-ment (EDM), |dn| < 2.9 × 10−26 e cm [869], was obtainedby the Sussex/Rutherford/ILL Collaboration from measure-ments at the ILL source for ultracold neutrons [1007], asdiscussed in the previous section. The experiment is at thispoint statistically limited and also facing systematic chal-lenges not far away [869, 1002, 1003]. In order to make fur-ther progress, both, statistical sensitivity and control of sys-tematics, have to be improved. Gaining in statistics requiresnew sources for ultracold neutrons (UCN). These can be in-tegrated into the experiment as for the new cryogenic EDMsearches, delivering UCN in superfluid helium, or a multi-purpose UCN source, delivering UCN in vacuum. This highintensity UCN source is presently under construction at thePaul Scherrer Institut in Villigen, Switzerland [1008]. It isexpected to become operational towards the end of 2008 andto deliver UCN densities of more than 1000 cm−3 to typicalexperiments, i.e. almost two orders of magnitude more thanpresently available.

The in-vacuum technique will be pushed to its limits, de-livering first results in about 4 years. The following steps areplanned by a sizable international collaboration [1009]:

– While the new UCN source is under construction the col-laboration operates and improves the apparatus of the for-mer Sussex/RAL/ILL Collaboration at ILL Grenoble. Inorder to better control the systematic issues, the magneticfield and its gradients will be monitored and stabilized us-ing an array of laser optically pumped Cs-magnetometers

Eur. Phys. J. C (2008) 57: 13–182 153

[1010, 1011]. An order of magnitude improvement com-pared to todays field fluctuations over the typical mea-surement times of 100–1000 s is certainly feasible. It isalso necessary to improve the sensitivity of the Hg co-magnetometer [1012]. Other improvements of the systemare with regard to UCN polarization and detection as wellas upgrading the data acquisition system. The hardwareefforts are accompanied by a full simulation of the sys-tem.

– It is planned to move the apparatus from ILL to PSI to-wards the end of 2008 in order to be ready for data takingfor about two years, 2009 and 2010. In addition to theimprovements of phase I, an external magnetic field sta-bilization system and a temperature stabilization are en-visaged. Furthermore, work on developing a second co-magnetometer using a hyper-polarized noble gas speciesis ongoing and might further improve the systematics con-trol. In case of a successful development, also the re-placement of the Hg system together with an increase ofthe electric field strength may become possible. In anycase, a factor of 5 gain in sensitivity is expected from thehigher UCN intensity, corresponding to a limit of about5–6× 10−27 e cm in case the EDM is not found. In paral-lel to the described activities, the design of a new experi-mental apparatus will start in 2007. After a major designeffort in 2008, set-up of the new apparatus will start in2009.

– The new experiment will be an optimized version of theroom-temperature in-vacuum approach. Another order ofmagnitude gain in sensitivity will be obtained by a consid-erable increase of the statistics due to a larger experimen-tal volume (×√

5), a better adaption to the UCN source(×√

2), longer running time (×√3) and by an improve-

ment of the electric field strength (×2). Completion ofthe new experimental apparatus is anticipated for end of2010, and data taking planned for 2011–2014.

The features of the experiment include

– continued use of the successful Ramsey-technique withUCN in vacuum and the apparatus at room-temperature,

– increased sensitivity due to much larger UCN statisticsat the new PSI source, larger experimental volume, bet-ter polarization product and possibly larger electric fieldstrength,

– application of a double neutron chamber system,– improved magnetic field control and stabilization with

multiple laser optically pumped Cs-magnetometers, and– an improved co-magnetometry system.

As another very strong source for UCN is currently un-der construction at the FRMII in Munich, in the long run andfor the optimum conditions for the experiment, the collabo-ration will have the opportunity to choose between PSI andFRMII.

9.2.3 SNS

A sizable US Collaboration [1013] is planning to develop acryogenic experiment, following an early concept by Goluband Lamoreaux [1014]. It will be based at the SNS 1.4 MWspallation source at Oak Ridge. A fundamental neutronphysics beam line is under construction, which will includea double monochromator to select 8.9 Å neutrons for UCNproduction in liquid helium.

In this experiment, spin-polarized 3He will be used bothas a magnetometer and as a neutron detector. The precessionof the 3He can, in principle, be detected with SQUID mag-netometers. Meanwhile, the cross section for the absorptionreaction n + 3He → p + 3H + 764 keV is negligible for atotal spin J = 1, but very large (∼5 Mb) for J = 0. In con-sequence, a scintillation signal from this reaction will be de-tected with a beat frequency corresponding to the differencebetween the Larmor precession frequencies of the neutronsand the 3He.

An application for funding to construct this experimentis currently under review. Extensive tests are underway tostudy, for example, the electric fields attainable in liquid he-lium, the 3He spin relaxation time and the diffusion of 3Hein 4He. If construction goes according to plan, commission-ing will be in approximately 2013, with results followingprobably four or five years later. The ultimate sensitivity willbe below 10−28 e cm.

9.3 Deuteron EDM

A new concept of investigating the EDM of bare nuclei inmagnetic storage rings has been developed by the storagering EDM Collaboration (SREC) over the past several years.The latest version of the methods analyzed turns out to bevery sensitive for light (bare) nuclei and promises the bestEDM experiment for θQCD, quark and quark-color EDMs.

The search for hadronic EDMs has been dominated bythe search for a neutron EDM and nuclear Schiff moments inheavy diamagnetic atoms, such as 199Hg. The latter dependon nuclear theory to relate the measured Schiff moment tothe underlying CP violating interaction.

The sensitive ‘traditional’ EDM experiments are, so far,all performed on electrically neutral systems, such as theneutron, atoms, or molecules. A strong electric field is im-posed, together with a weak magnetic field, and using NMRtechniques, a change of the Larmor precession frequency islooked for. The application of strong electric fields precludesa straightforward use of this technique on charged particles.These particles would accelerate out of the setup, leavinglittle time to make an accurate measurement.

Attempt to search for an EDM on simple nuclear sys-tems, such as the proton or deuteron, when part of an atom,are severely hindered by shielding. This so-called Schiff-screening precludes an external electric field to penetrate

154 Eur. Phys. J. C (2008) 57: 13–182

to the nucleus. Due to rearrangement of the atomic elec-trons, the net effect of the electric field on the nucleus is es-sentially zero. Known loop-holes include relativistic effects,non-electric components in the binding of the electrons, andan extended size of the nucleus. None of these loopholesare sufficiently strong to allow a sensitive measurement ona light atom. For hydrogen atoms, the atomic EDM result-ing from a nuclear EDM is down by some seven orders ofmagnitude.

Nevertheless, light nuclei, and the deuteron in particular,are attractive to search for hadronic EDMs because of theirrelatively simple structure. Moreover, a novel experimentaltechnique, using the motional electric field experienced bya relativistic particle when traversing a magnetic field, makeit possible to directly search for EDM on charged systems,such as the (bare) deuteron.

9.3.1 Theoretical considerations

The deuteron is the simplest nucleus. It consists of a weaklybound proton and neutron in a predominantly 3S1 state, witha small admixture of the D-state. From a theoretical pointof view, the deuteron is especially attractive, because it isthe simplest system in which the P-odd, T-odd nucleon–nucleon (NN) interaction contributes to an EDM. Moreover,the deuteron properties are well understood, so reliable andprecise calculations are possible.

In [1015], a framework is presented that could serve asa starting point for the microscopic calculation of complexsystems. The most general form of the interaction, basedonly on symmetry considerations, contains ten P- and T-oddmeson-nucleon coupling constants for the lightest pseudo-scalar and vector mesons (π , ρ, η and ω).

This P-odd, T-odd interaction induces a P -wave admix-ture to the deuteron wave function. It is this admixture thatleads to an EDM. Since the proton and neutron that make upthe deuteron may also have an EDM, a disentanglement ofone- and two-body contributions,

dD d(1)D + d(2)D (9.12)

to the EDM is necessary to uncover the underlying structureof the P-odd T-odd physics.

The two-body component is predominantly due to the po-larization effect, and shows little model dependence for allleading high-quality potentials. Additional contributions ar-rive from meson exchange.

The one body contribution is simply the sum of the pro-ton and neutron EDMs. The nucleon EDM has a wide vari-ety of sources, as already discussed for the neutron. Thereexists no good model to describe the non-perturbative dy-namics of bound quarks. A commonly used method is toevaluate hadronic loop diagrams, containing mesonic and

baryonic degrees of freedom. Within the framework pre-sented in [1015], the EDMs for the proton, neutron anddeuteron are found (reproducing only the pion dependence),

dp =−0.05g(0)π + 0.03g(1)π + 0.14g(2)π + · · · ,dn =+0.14g(0)π − 0.14g(2)π + · · · ,dD =+0.09g(0)π + 0.23g(1)π + · · · .

(9.13)

These dependences clearly show the complementarity ofthese three particles.

The magnitudes of the coupling constants can be calcu-lated for several viable sources of CP violation. In the stan-dard model, there is room for CP violation via the so-calledθ parameter. In the case of the nucleons, one has the relation

dn −dp 3 × 10−16θ e cm, (9.14)

which yields the severe constraint θ < 1 × 10−10. For thedeuteron, one finds

dD −10−16θ e cm. (9.15)

At the level of dD 10−29 e cm, one probes θ at the levelof 10−13. Since θ contributes differently to the neutron andthe deuteron, it is clear that both experiments are comple-mentary. Indeed. the prediction

dD/dn =−1/3 (9.16)

provides a beautiful check as to whether θ is the sourceof the observed EDMs, should both be measured. In fact,measurement of the EDMs of the proton, deuteron and 3Hewould allow to verify if they satisfy the relation

dD : dp : d3He 1 : 3 : −3. (9.17)

Here, it was assumed that 3He has properties very similar tothe neutron, which provides most of the spin.

Generic supersymmetric models contain a plethora ofnew particles, which may be discovered at LHC, and newCP-violating phases. Following the work by Lebedev etal. [1016] and the review by Pospelov and Ritz [742], wefind that SUSY loops give rise to ordinary quark EDMs,dq , as well as quark-color EDMs, dq . For the neutron anddeuteron one finds (with the color EDM part divided inisoscalar and isovector parts)

dn 1.4(dd − 0.25du)+ 0.83e(dd + du)+ 0.27e(dd − du),

dD (dd + du)− 0.2e(dd + du)+ 6e(dd − du),(9.18)

and similar relations for e.g. the mercury EDM. The isovec-tor part is limited to |ec(dd − du)|< 2 × 10−26 e cm by thepresent limit on the 199Hg atom. The experimental bound

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on the neutron suggests that |e(dd + du)|< 4× 10−26 e cm,assuming the isoscalar contribution to be dominant. Also inthis case, the deuteron and neutron show complementarity.This is in particular in their sensitivity to the isovector con-tribution, which is 20 times larger for the deuteron.

The large sensitivity to new physics (see e.g.[1016]) andthe relative simplicity of calculating the nuclear wave func-tion, make it clear that small nuclei hold great discovery po-tential and should therefore be vigorously pursued.

9.3.2 Experimental approach

All sensitive EDM searches are performed on neutral sys-tems, which are (essentially) at rest. The only exceptionis the proposed use of molecular ions (HfF+ and ThF+)[1017], but also for this experiment, the motion of the mole-cules is not crucial.

In the recent past, several novel techniques have beenproposed to use the motional electric field sensed by a parti-cle moving through a magnetic field at relativistic velocities.The evolution of the spin orientation for a spin-1/2 particlein an electromagnetic field ( �E, �B) is described by the so-called Thomas or BMT equation [1018]. The spin preces-sion vector �Ω , relative to the momentum of the particle, isgiven by [1019]

�Ω = e

m

[a �B +

(a − 1

γ 2 − 1

)�β × �E

+ η2

(�E + �β × �B − γ

γ + 1�β( �β · �E)

)](9.19)

with �μ = 2(1 + a)(e/m)�S and �d = η/2(e/m)�S. It was as-sumed that �β · �B = 0. The first two terms between bracketswill be referred to as ωa , whereas the last one will be re-ferred to as ωη.

For fast particles, the electric field in the rest frame of theparticle is dominated by �β × �B . For commonplace storagerings, this field can exceed the size of a static electric fieldmade in the laboratory by more than an order of magnitude,thus giving the storage ring method a distinct advantage.

In a homogeneous magnetic field, �ωa ∝ �B and �ωη ∝�β × �B are orthogonal, leading to a small tilt in the preces-sion plane and an second order increase in the precessionfrequency. Although this was used to set a limit on the muonEDM [188, 1020], it does not allow for a sensitive search.

The application of a radially oriented electric field Er toslow down ωa and thus to increase the tilt, was proposed in[1021]. For a field strength

Er = aβ

1 − (1 + a)β2Bz (9.20)

the spin of an originally longitudinally polarized beam re-mains aligned with the momentum at all times. In this case

�β · �E = 0, and thus

�Ω = e

m

η

2( �E + �β × �B). (9.21)

The EDM thus manifests itself as a precession of the spinaround the motional electric field �E∗ = γ [ �E + �β × �B], i.e.as a growing vertical polarization component parallel to �B .This approach can be used for all particles with a small mag-netic anomaly, so that the necessary electric field strengthremains feasible. Concept experiments, employing this tech-nique, have been worked out for the muon [1022–1024] andthe deuteron [1025]. Other candidate particles have beenidentified as well (see e.g. [1026]).

A third, most sensitive approach is reminiscent of themagnetic resonance technique introduced by Rabi [1027].The spin is allowed to precess under the influence of a di-pole magnetic field. In the original application, an oscillat-ing magnetic field oriented perpendicular to the driving fieldis applied. By scanning the oscillation frequency, a reso-nance will be observed when the frequency of the oscillatingfield matches the spin precession frequency.

In this application, the oscillating magnetic field are re-placed by an oscillating electric field [1028]. When at reso-nance, the electric field coherently interacts with the electricdipole moment. As a consequence, the polarization compo-nent along the magnetic field oscillates in the case of a siz-able EDM. In practice, only the onset of the first oscillationcycle will be visible in the form of a slow growth of the ver-tical polarization, proportional to the EDM.

The oscillating electric field is obtained by modulatingthe velocity of the deuterons circulating in a magnetic field,setting up a so-called synchrotron oscillation. For a time de-pendent velocity β(t) = β0 + δβ(t) generated by an oscil-lating longitudinal electric field ERF(t) and a constant mag-netic field B , the spin evolution follows from

�Ω = e

m

[{aB + η

2�β0 × �B

}

+ η2

{δ �β(t)× �B − β2γ

γ + 1�ERF(t)

}]

≡ �Ω0 + �δΩ(t). (9.22)

The first term yields spin precession about �Ω0, with-out affecting the polarization parallel to it. For δβ(t) =δβ cos(ωt + ψ), and Bδβ � β2γ /(γ + 1)ERF, the paral-lel polarization component is given by

dP‖dt

e

mP◦ηδβB cos(�ωt +�φ), (9.23)

with �ω≡Ω0 −ω and �φ ≡ φ−ψ . The beam is assumedto have a longitudinal polarization P0 at injection time. For�ω = 0 the vertical polarization will grow continuously at

156 Eur. Phys. J. C (2008) 57: 13–182

a rate proportional to the EDM. Maximum sensitivity is ob-tained for �φ = 0 or π , whereas for �φ = π/2 or 3π/2there is no sensitivity to the EDM. The latter will prove use-ful in controlling systematic errors. At the same time, theradial polarization component is given by

P⊥ P0 sin(Ω0t + φ). (9.24)

This polarization component can be incorporated in a feed-back cycle, to phase-lock the velocity modulation to the spinprecession, i.e. to guarantee�ω= 0 and�φ constant. In ad-dition, observation of Ω0 allows to measure or stabilize themagnetic field.

From (9.23) and (9.24), the main design criteria are eas-ily derived, several of which are common to all other EDMexperiments. They include

– high initial polarization P0;– large field strength Eeff ∝ (δβB);– close control over the resonance conditions �ω and

phase �φ;– long spin coherence time P◦(t);– long synchrotron coherence time δβ(t);– sensitive method for independent observation of P‖

and P⊥.

The parameters of the current concept deuteron EDMring are presented in Table 26. Coherent synchrotron os-cillation can be obtained by a set of two RF cavities, oneoperating at a harmonic of the revolution frequency to bringthe beam close to the spin-synchrotron resonance, and a sec-ond operating at the resonance frequency to create a forcedoscillation.

The statistical reach of the experiment is determined bythe number of particles used to determine the polarization,

Table 26 Parameters of the concept deuteron EDM storage ring.

Parameter Symbol Design value

Deuteron momentum pD 1500 MeV/c

Magnetic field strength B 2 T

Bending radius ρ 2.5 m

Length of each straight section l 5 m

Orbit length L 26 m

Momentum compaction αp 1

Cyclotron period tc 137 ns

Spin precession period ts 660 ns

Spin coherence time τs 1000 s

Motional electric field E∗/γ 375 MV/m

Synchrotron amplitude δβ/β 1%

Synchrotron harmonic h 40

Particles per fill N 1012

Initial polarization P◦ 0.9

EDM precession rate @ d = 10−26 e cm ωη 1 µrad/s

as well as the analyzing power of the polarimeter. The mostefficient way to probe the deuteron polarization at the en-ergy considered is by nuclear scattering. To obtain high effi-ciency, conventional techniques, in which a target is insertedinto the beam are unsuitable. Instead, slow extraction of thebeam onto a thick analyzer target is necessary. Slow extrac-tion could be realized by exciting a weak beam resonance,or alternatively, by Coulomb scattering off a thin gas jet. Thethickness of the analyzer target is optimized to yield maxi-mum efficiency, which may reach the percent level.

The EDM will create a left-right asymmetry in the scat-tered particle rate, whose initial rate of growth is propor-tional to the EDM. False signals from, e.g., oscillating radialmagnetic fields in the ring will be mitigated by varying thelattice parameters. This will change the systematic error am-plitude, while leaving the EDM signal unchanged. Variousfeatures of the ring design and bunches with opposite EDMsignals will be used to reduce the impact of other systematiceffects.

The expected very high observability of most of the fieldimperfections in the experiment comes from the combina-tion of gross amplification of the original perturbations inthe control bunches, and observation and correction of theamplified parasitic growth of the vertical polarization com-ponent. This growth is many orders of magnitude more sen-sitive to ring imperfections than any other beam parameter.Preliminary studies shows no unmanageable sources of sys-tematic errors at the level of the expected statistical uncer-tainty of 10−29 e cm.

There is currently great interest in EDM experiments be-cause of their potential to find new physics complementaryto and even reaching beyond that which can be found at fu-ture accelerators (LHC and beyond). The new approach de-scribed here would be the most sensitive experiment for themeasurement of several possible sources of EDMs in nu-cleons and nuclei for the foreseeable future, if systematicuncertainties can be controlled.

9.4 EDM of deformed nuclei: 225Ra

In the nuclear sector, the strongest EDM limits have been setby cell measurements which restrict the EDM of 199Hg to<2.1×10−28 e cm. A promising avenue for extending thesesearches is to take advantage of the large enhancements inthe atomic EDM predicted for octupole deformed nuclei.One such case is 225Ra, which is predicted to be two to threeorders of magnitude more sensitive to T-violating interac-tions than 199Hg. The next generation EDM search aroundlaser-cooled and trapped 225Ra is being developed by theArgon group. They have demonstrated transverse cooling,Zeeman slowing, and capturing of 225Ra and 226Ra atoms ina magneto-optical trap (MOT). They have measured manyof the transition frequencies, life-times, hyperfine splittings

Eur. Phys. J. C (2008) 57: 13–182 157

and isotope shifts of the critical transitions. This new devel-opment should enable them to launch a new generation ofnuclear EDM searches. The combination of optical trappingand the use of octupole deformed nuclei should extend thereach of a new EDM search by two orders of magnitude.A non-zero EDM in diamagnetic atoms is expected to bemost sensitive to a chromo-electric induced EDM effect.

Radium-225 is an especially good case for the search ofthe EDM because it has a relatively long life-time (t1/2 =14.9 d), has spin 1/2 which eliminates systematic effectsdue to electric quadrupole coupling, is available in rela-tively large quantities from the decay of the long-lived 229Th(t1/2 = 7300 yr), and has a well-established octupole nature.The octupole deformation enhances parity doubling of theenergy levels. For example, the sensitivity to T-odd, P-oddeffects in 225Ra is expected to be a factor of approximately400 larger than in 199Hg, which has been used by previoussearches to set the lowest limit (<2 × 10−28 e cm) so far onthe atomic EDM. The 14.9-day half-life for 225Ra is suffi-ciently long that measurements can be performed and sys-

tematics can be checked without resorting to an accelerator-based experiment. Nevertheless, if a 225Ra beam facilitywere available for this experiment, approximately a hundredtimes more atoms could be produced which could have theimpact of improving the sensitivity by yet another order ofmagnitude.

Laser cooling and trapping of 225Ra atoms was devel-oped in preparation of an EDM search. The laser trap allowsone to collect and store the radioactive 225Ra atoms that areotherwise too rare to be used for the search with conven-tional atomic-beam or vapor-cell type methods. Moreover,an EDM measurement on atoms in a laser trap would bene-fit from the advantages of high electric field, long coherencetime, and a negligible so-called “v×E” systematic effect.

The Argon group has demonstrated a magneto opticaltrap (MOT) of Ra atoms by using the 7s2 1S0 → 7s7p3P1 transition as the primary trapping transition, and 7s6d3D1 → 7s7p 1P1 as the re-pump transition (see Fig. 68).They used a Ti:Sapphire ring laser system to generate the714 nm light to excite the 7s2 1S0 → 7s7p 3P1 transition.

Fig. 68 Atomic level structureof radium-225 indicating thecycling transition at 714 nm andthe re-pump transition at1428 nm. The values in boxesindicate the relative transitionprobabilities

158 Eur. Phys. J. C (2008) 57: 13–182

The primary leak channel from this two-level quasi-cyclingsystem is the decay from 7s7p 3P1 to 7s6d 3D1, from whichthe atoms were pumped back to the ground-level via the7s6d 3D1 → 7s7p 1P1 transition followed by a spontaneousdecay from 7s7p 1P1 back to the ground-level. The re-pumpwas induced by laser light at 1428.6 nm generated by a diodelaser. This re-pump transition can be excited for an averageof 1400 times before the atom leaks to other metastable lev-els. Therefore, with the re-pump in place, an atom can cyclefor an average of 3.5 × 107 times and stay in the MOT forat least 30 s before it leaks to dark levels. Here the MOTis used only to capture the atoms; the trapped atoms wouldthen be transferred to an optical dipole trap for storage andmeasurement. They plan to achieve a life-time of 300 s inthe dipole trap.

The ultimate goal of the present series of measurementsis to provide a measurement that is comparable in sensitivityto the atomic EDM experiment for 199Hg. Because of theenhancement from the octupole deformation of 225Ra, themeasurement would then be more than two orders of mag-nitude more sensitive to T-violating effects in the nucleusthan that of the 199Hg experiment. The immediate goal overthe next two years is to provide an initial atomic EDM limitof ∼1 × 10−26 e cm. Thereafter, the plan is to improve theexperiment until the ultimate goal is achieved.

9.5 Electrons bound in atoms and molecules

9.5.1 Theoretical aspects

We discuss here permanent EDMs of diatomic molecules in-duced by the EDM of the electron and by P- and T-odd e–Nneutral currents. In heavy molecules the effective electricfield Eeff on unpaired electron(s) is many orders of mag-nitude higher than the external laboratory field required topolarize the molecule. As a result, the EDM of such mole-cules is strongly enhanced. The exact value of the enhance-ment factor is very sensitive to relativistic effects and to elec-tronic correlations. In recent years several methods to calcu-late Eeff were suggested and reliable results were obtainedfor a number of molecules.

The study of a non-relativistic electron in a stationarystate immediately leads to the zero energy shift δε for anatom in the external field E0 induced by the electron EDMde = deσ . Indeed, the average acceleration 〈a〉 = 0, so theaverage force −e〈E〉 = 0. Therefore, δε = −de · 〈E〉 = 0.This statement is known as Schiff theorem. In the relativis-tic case, the position-dependence of the Lorentz contractionof the electron EDM leads instead to a net overall atomicEDM [1029]. Even though 〈E〉 = 0, it still can be (and in-deed is) the case that 〈de · E〉 �= 0, if de is not spatially uni-form. Taking account of the fact that the length-contractedvalue of de is NOT spatially uniform for an electron inside

the Coulomb field of an atom exactly reproduces the formof the enhancement factor.

Reliable calculations of atomic energy shifts are easierwith the relativistic EDM Hamiltonian for the Dirac elec-tron, which automatically turns to zero in the non-relativisticapproximation [1030]:

Hd = 2de

(0 00 σ

)·E ∼= 2de

(0 00 σ

)·Eint. (9.25)

This Hamiltonian is singular at the origin and we ne-glected the external field E0. Using (9.25) it is straightfor-ward to show that the induced EDM of the heavy atom dat isof the order of 10α2Z3de, where Z is the number of protonsin the nucleus. If Z ∼ 102 the atomic enhancement factorkat ≡ dat/de ∼ 103. This estimate holds for atoms with anunpaired electron with j = 1

2 . For higher angular momen-tum j the centrifugal barrier strongly suppresses dat.

Atomic EDM can be also induced by a scalar P, T-odde–N neutral current [1030]:

HS = i Gα21/2

ZkSγ0γ5n(r), (9.26)

where G is Fermi constant, γi are Dirac matrices, n(r) isthe nuclear density normalized to unity, and ZkS = ZkS,p+NkS,n is the dimensionless coupling constant for a nucleuswith Z protons and N neutrons. Atomic EDMs inducedby the interactions (9.25), (9.26) are obviously sensitiveto relativistic corrections to the wave function. Numericalcalculations also show their sensitivity to correlation ef-fects. For example, the Dirac–Fock calculation for Tl givesdTl =−1910de while the final answer within all order many-body perturbation theory is dTl = −585de (see Ref. [1030]for details). Note that the present limit on the electron EDMfollows from the experiment with Tl [186].

The internal electric field in a polar molecule, Emol ∼e

R2o∼ 109 V/cm, is 4–5 orders of magnitude larger than the

typical laboratory field in an atomic EDM experiment. Thisfield is directed along the molecular axis and is averaged tozero by the rotation of the molecule. The molecular axis canbe polarized in the direction of the external electric field E0.One usually needs the field E0 ∼ 104V/cm to fully polarizethe heavy diatomic molecule. The corresponding molecularenhancement factor is kmol ∼ kat × Emol

E0∼ 104kat.

For closed-shell molecules all electrons are coupled andthe net EDM is zero. Therefore one needs a molecule withat least one unpaired electron. Such molecules have nonzeroprojection Ω of electronic angular momentum on the mole-cular axis. Again, as in the case of atoms, for the moleculeswith one unpaired electron the largest enhancement corre-sponds to Ω = 1

2 . The centrifugal barrier leads to strongsuppression of the factor kmol for higher values of Ω . On

Eur. Phys. J. C (2008) 57: 13–182 159

Table 27 Calculated values of parameters Eeff and WS from (9.27) for diatomic molecules. The question marks reflect the uncertainty in theknowledge of the ground state

Molecule State Ω Eeff (109 V/cm) WS (kHz) Ref.

BaF ground 1/2 −7.5 ± 0.8 −12 ± 1 [1031, 1032]

YbF ground 1/2 −25 ± 3 −44 ± 5 [1032, 1033]

HgF ground 1/2 −100 ± 15 −190 ± 30 [1034]

HgH ground 1/2 −79 −144 [1034]

PbF ground 1/2 +29 +55 [1034]

PbO metastable 1 −26 [1035]

HI+ ground 3/2 −4 [1036]

PtH+ ground (?) 3 20 [1037]

HfF+ metastable (?) 1 24 [1038]

the other hand, such molecules can be polarized in a muchweaker external field.

For strong external fieldE0 the factor kmol depends onE0

and it is more practical to define an effective electric field onthe electron Eeff so, that the P, T-odd energy shift for a fullypolarized molecule is equal to:

δεP,T =Eeffde + 12WSkS, (9.27)

where two terms correspond to interactions (9.25) and(9.26). Calculated values of Eeff and WS for a number ofmolecules are listed in Table 27.

An EDM experiment is currently going on with YbFmolecules. This molecule has a ground state with Ω = 1

2 .The P, T-odd parameters (9.27) were calculated with differ-ent methods by several groups, and estimates of the system-atic uncertainty are available. Several other molecules andmolecular ions have been suggested for the search for elec-tron EDM including PbO, PbF, HgH, and PtH+. PbF andHgH have Ω = 1

2 and calculations are similar to the YbFcase. The ground state of PbO has closed shells and the ex-periment is done on the metastable state with two unpairedelectrons and Ω = 1. Here electronic correlations are muchstronger and calculations are more difficult.

Finally, molecular ions like PtH+ are less studied andeven their ground states are not known exactly. It is antic-ipated that such ions can be trapped and a long coherencetime for the EDM experiment can be achieved. Recentlythe first estimates of the effective field for PtH+ and severalother molecular ions were reported [1037]. These estimatesare based on non-relativistic molecular calculations. Properrelativistic molecular calculations for these ions may be ex-tremely challenging.

9.5.2 Experimental aspects

Over a dozen different experiments searching for the elec-tron electric dipole moment that are under way or planned

will be reviewed here. At present the experimental upperlimit on de is [186]: |de| ≤ 1.6 × 10−27 e cm, where e isthe unit of electronic charge.

Most of this work is being done in small groups on uni-versity campuses. These experiments employ a wide rangeof technologies and conceptual approaches. Many of the lat-est generation of experiments promise two or more orders ofmagnitude improvement in statistical sensitivity, and mosthave means to suppress systematic errors well beyond thoseobtained in the previous generation.

To detect de, most experiments rely on the energy shift�E = −de · �E upon application of �E to an electron. Un-til recently, most experimental searches for de used gas-phase paramagnetic atoms or molecules and employed thestandard methods of atomic, molecular, and optical physics(laser and rf spectroscopy, optical pumping, atomic andmolecular beams or vapor cells, etc.) in order to directlymeasure the energy shift �E. Recently, another class of ex-periments has been actively pursued, in which paramagneticatoms bound in a solid are studied. Here the principles arerather different than for the gas-phase experiments, and tech-niques are more similar to those used in condensed matterphysics (magnetization and electric polarization of macro-scopic samples, cryogenic methods, etc.). We discuss thesetwo classes of experiments separately.

9.5.2.1 A simple model experiment using gas-phase atomsor molecules Experimental searches for de using gas-phase atoms or molecules share many broad features. Eachconsists of a state selector, where the initial spin state ofthe system is prepared; an interaction interval in which thesystem evolves for a time τ in an electric field �E (and of-ten a magnetic field B ‖ �E as well); and a detector to de-termine the final state of the spin. To understand the es-sential features, we consider a simple model that is read-ily adapted to describe most realistic experimental condi-tions. In this model, an “atom” of spin 1/2 with enhance-ment factor R, containing an unpaired electron with spin

160 Eur. Phys. J. C (2008) 57: 13–182

magnetic moment μ and EDM de . The spin is initially pre-pared to lie along x, i.e., is in the eigenstate |χx+〉 of spinalong x: |ψ0〉 = |χx+〉 ≡ 1√

2( 1

1). During the interaction in-

terval the spin precesses about �E = Ez and B = Bz, in thexy plane, by angle 2φ = −(deRE + μB)τ/�. At time τthe quantum state has then evolved to |ψ〉 = 1√

2( e

−iφeiφ). Fi-

nally, the detector measures the probability that the result-ing spin state lies along y. This is determined by the over-lap of the wave function |ψ〉 with |χy+〉 ≡ 1√

2( 1

1). Hence

the signal S from N detected atoms observed in time τ isS =N |〈χy+|ψ〉|2 =N cos2 φ.

The angle φ is the sum of a large term φ1 =−μBτ/(2�)

and an extremely small term φ2 = −deREτ/(2�). To iso-late φ2 one observes S for �E and B both parallel and an-tiparallel. Reversing �E · B changes the relative sign of φ1

and φ2 and thus changes S; the largest change in S occursby choosing B such that φ1 = ±π/4. With this choice, wehave S± ≡ S(�E ·B><0)= N

2 (1± 2φ2). The minimum uncer-tainty in determination of the phase φ2 in time τ , due to shotnoise, is δφ2 = √

1/N . If the experiment is repeated T/τtimes for a total time of observation T, the statistical uncer-tainty in de is δde =√

1/N0√

1/T τ |�/Eeff|, where we usedRE = Eeff. In practice, other “technical” noise sources cansignificantly increase this uncertainty, particularly fluctua-tions in the magnetic field. Hence, careful magnetic shield-ing is required in all EDM experiments.

9.5.2.2 Systematic errors The EDM is revealed by a termin the signal proportional to a P, T-odd pseudoscalar suchas �E · B . False terms of the same apparent form can ap-pear even without P, T violation through a variety of exper-imental imperfections. The most dangerous effects appearwhen B depends on the sign of �E, which can occur in sev-eral ways. For example, leakage currents flowing throughinsulators separating the electric field electrodes can gener-ate an undesired magnetic field BL. Also, if the atoms ormolecules have a non-zero velocity v, a motional magneticfield Bmot = 1

c�E × v exists in addition to the applied mag-

netic field B; along with various other imperfections in thesystem, this effect can lead to systematic errors. A relatedsystematic effect involves geometric phases, which appearif the direction of the quantization axis (often determined byB total = B +Bmot) varies between the state selector and theanalyzer [1030].

A variety of approaches are employed to deal with theseand other systematics. Aside from leakage currents, mostsystematics depend on a combination of two or more imper-fections in the experiment (i.e. misaligned or stray fields);these can be isolated by deliberately enhancing one im-perfection and looking for a change in the EDM signal.Some experiments utilize, in addition to the atoms of in-terest, additional species as so-called “co-magnetometers”.These co-magnetometer species (e.g., paramagnetic atoms

with low R) are chosen to have negligible or small enhance-ment factors, but retain sensitivity to magnetic systematicssuch as those mentioned above.

In paramagnetic molecule experiments, issues with sys-tematic effects are somewhat different. Here the ratioEeff/Eext is enhanced, and relative sensitivity to magneticsystematics is correspondingly reduced. The �E × v effect iseffectively eliminated by the large tensor Stark effect [1039]typically found in molecular states. The saturation of themolecular polarization |P | (and hence Eeff) leads to a well-understood non-linear dependence of the EDM signal onEext that can discriminate against certain systematics. Con-versely, the extreme electric polarizability leads to a varietyof new effects, such as a dependence of the magnetic mo-ment μ on Eext, and geometric phase induced systematicerrors related to variations in the direction of �Eext.

9.5.2.3 Experiments with gas-phase atoms and molecules• The Berkeley thallium atomic beam experiment

This experiment gives the best current limit on de. In itsfinal version [186], two pairs of vertical counter-propagatingatomic beams, each consisting of Tl (Z = 81,RTl = −585[1040]) and Na (Z = 11,RNa = 0.32), were employed (seeFig. 69). Spin alignment and rotation of the 62P1/2(F = 1)state of Tl and the 32S1/2(F = 2,F = 1) states of Na wereaccomplished, respectively, by laser optical pumping and byatomic beam magnetic resonance with separated oscillatingrf fields of the Ramsey type. Detection was achieved viaalignment-sensitive laser induced fluorescence. In the inter-action region, with length ≈1 meter, the side-by-side atomicbeams were exposed to nominally identical B fields, butopposite �E fields of ≈120 kV/cm. This provided common-mode rejection of magnetic noise and control of some sys-tematic effects. Average thermal velocities corresponded toan interaction time τ ≈ 2.3 ms (1 ms) for Tl (Na) atoms. Useof counter-propagating atomic beams served to cancel all buta very small remnant of the �E × v effect. Various auxiliarymeasurements, including use of Na as a co-magnetometer,further reduced this remnant and isolated the geometricphase effect. E and leakage currents were measured usingauxiliary measurements based on the observable quadraticStark effect in Tl. About 5.2×1013 photo-electrons of signalper up/down beam pair were collected by the fluorescencedetectors. The final result is de = (6.9 ± 7.4)× 10−28 e cm,which yields the limit |de| ≤ 1.6 × 10−27 e cm (90% conf.).• Cesium vapor cell experiments

An experiment to search for de in a vapor cell of Cs(Z = 55; RCs = 115 [1041]) was reported by L. Hunterand co-workers [1042] at Amherst in 1989. The method isbeing revisited in a present-day search by led by M. Ro-malis at Princeton [1043]. The Amherst experiment wascarried out with two glass cells, one stacked on the otherin the z direction. Nominally equal and opposite �E fields

Eur. Phys. J. C (2008) 57: 13–182 161

Fig. 69 Schematic diagram of the Berkeley thallium experi-ment [186], not to scale. Laser beams for state selection and analysisat 590 nm (for Na) and 378 nm (for Tl) are perpendicular to the page,with indicated linear polarizations. The diagram shows the up-goingatomic beams active

were applied in the two cells. The cells were filled withCs, as well as N2 buffer gas to minimize Cs spin relax-ation. Circularly polarized laser beams, directed along x,were used for spin polarization via optical pumping. Mag-netic field components in all three directions were reducedto less than 10−7 G. Thus precession of the atomic polar-ization in the xy plane was nominally due to �E alone. Thefinal spin orientation was monitored by a probe laser beamdirected along y. The effective interaction time was the spinrelaxation time τ ≈ 15 ms. The signals were the intensitiesof the probe beams transmitted through each cell. A non-zero EDM would have been indicated by a dependence ofthese signals on the rotational invariant J · (σ × �E)τ , whereσ ,J were the pump and probe circular polarizations, re-spectively. The most important sources of possible system-atic error were leakage currents and imperfect reversal of �E.The result was de = (−1.5 ± 5.5 ± 1.5)× 10−26 e cm.

In the new experiment at Princeton, each cell also con-tains 129Xe at high pressure. Cs polarization is transferredto the 129Xe nuclei by spin exchange collisions. Under cer-

tain conditions this coupling can also give rise to a self-compensation mechanism, where slow changes in compo-nents of magnetic field transverse to the initial polarizationaxis are nearly canceled by interaction between the alkalielectron spin and the noble gas nuclear spin. This leaves onlya signal proportional to an anomalous interaction that doesnot scale with the magnetic moments—for example, interac-tion of de with Eeff. This mechanism (which is understoodin some detail [1044, 1045]) has the potential to reduce boththe effect of magnetic noise, and some systematic errors.• Experiments with laser-cooled atoms

Laser-cooled atoms offer significant advantages for elec-tron EDM searches. The low velocities of cold atoms yieldlong interaction times, and also suppress �E×v effects. How-ever, these techniques typically yield small numbers of de-tectable atoms, and magnetic noise must be controlled atunprecedented levels. New systematics due to, e.g., electricforces on atoms and/or perturbations due to trapping fields(see e.g. [1046]) can appear.

Experiments based on atoms trapped in an optical latticehave been proposed by a number of investigators [1047–1049]. Two such experiments, similar in their design, arecurrently being developed: one led by D.S. Weiss at Penn-sylvania State University and another led by D. Heinzen atthe University of Texas. Both plan to use Cs atoms to de-tect de, along with Rb atoms (Z = 37, RRb = 25) as a co-magnetometer. The Texas apparatus consists of two side-by-side far-off-resonance optical dipole traps, each in a vertical1-D lattice configuration. These traps are placed in nomi-nally equal and opposite �E fields and a common B field ofseveral mG parallel to �E. To load the atoms into the opticallattice, cold atomic beams from 2D magneto-optical trapsexterior to the shields will be captured with optical molassesbetween the �E-field plates. The electric field plates will beconstructed from glass coated with a transparent, conductiveindium tin oxide layer.

We are aware of two other EDM experiments based onlaser-cooled atoms. One employing a slow “fountain”, inwhich Cs atoms are launched upwards and then fall backdown due to gravity, has been proposed and developed byH. Gould and co-workers at the Lawrence Berkeley Na-tional Laboratory [1050]. Another, using 210Fr (τ = 3.2min;Z = 87,RFr = 1150), has been proposed and is being devel-oped by a group at the Research Center of Nuclear Physics(RCNP), Osaka University, Japan [1051].• The YbF experiment

E.A. Hinds and co-workers [1052–1054] at Imperial Col-lege, London have developed a molecular beam experi-ment for investigation of de using YbF. Figure 70 showsthe relevant energy level structure of the X2Σ+

1/2(v = 0,

N = 0) J = 1/2 ground state of a 174YbF molecule.174Yb has nuclear spin IYb = 0, while IF = 1/2; hence theJ = 1/2 state has two hyperfine components, F = 1 and

162 Eur. Phys. J. C (2008) 57: 13–182

Fig. 70 Schematic diagram, not to scale, of the hyperfine structureof the X2Σ electronic state of 174YbF in the lowest vibrational androtational level. Δ is the tensor Stark shift. δ is the shift caused by thecombination of the Zeeman effect and the effect of de in �Eeff

F = 0, separated by 170 MHz. An external electric field�Eext along z with magnitude Eext = 8.3 kV/cm correspondsto Eeff ≈ 13 GV/cm [1033, 1052–1054], which splits theF = 1,mF = ±1 levels by 2deEeff. In this external field,the level F = 1,mF = 0 is shifted downward relative tomF =±1 by an amountΔ= 6.7 MHz due to the large tensorStark shift associated with the molecular electric dipole.

In the experiment, a cold beam of YbF molecules is gen-erated by chemical reactions within a supersonic expan-sion of Ar or Xe carrier gas. Laser optical pumping re-moves all F = 1 state molecules, leaving only F = 0 re-maining in the beam. Next, a 170 MHz rf magnetic fieldalong x excites molecules from F = 0 to the coherent su-perposition |ψ〉 = 1√

2|F = 1,mF = 1〉 + 1√

2|1,−1〉. While

flying through the central interaction region of length 65 cm,the beam is exposed to parallel electric and magnetic fields(±E,±B)z (B ∼ 0.1 mGauss). Next, an rf field drives eachF = 1 molecule back to F = 0. Because of the phase shift2φ developed in the central region, the final population ofF = 0 molecules is proportional to cos2 φ. These F = 0molecules are detected by laser induced fluorescence in theprobe region.

The most significant systematic errors in this experimentare expected to arise from variation in the direction andmagnitude of �E along the beam axis. If the direction of �Echanges in an absolute sense, a geometric phase could begenerated, and if �E changes relative to B , the magnetic pre-cession phase φ1, proportional to �Eext · B/|�Eext|, could beaffected. A preliminary result of the YbF experiment [1052–1054], published in 2002, is de = (−0.2±3.2)×10−26 e cm.Many significant improvements have been made since 2002,and it is likely that this experiment will yield a much moreprecise result in the near future.• The PbO experiment

A search for de using the metastable a(1)3Σ1 state ofPbO is being carried out at Yale [1055–1057]. The a(1) statehas a relatively long natural life-time: τ [a(1)] = 82(2) µs,

and can be populated in large numbers using laser excitationin a vapor cell. In this state, the level of total (rotational +electronic) angular momentum J = 1 contains two closely-spaced “Ω doublet” states of opposite parity, denoted as e−and f+. An external electric field �Eext = Eextz mixes e−and f+ states with the same value of M , yielding molecu-lar states with equal but opposite electrical polarization P .The degree of polarization |P | ≈ 1 for Eext � 10 V/cm.When |P | = 1 the effective molecular field is calculated tobe Eeff ∼= 26 GV/cm [1035]. The opposite molecular polar-ization in the twoΩ-doublet levels leads to a sign differencein the EDM-induced energy shift between these two levels.This difference provides an excellent opportunity for effec-tive control of systematic errors, since comparison of the en-ergy shifts in the upper and lower states acts as an “internalco-magnetometer” requiring only minor changes in experi-mental parameters to monitor.

The Yale experiment is carried out in a cell contain-ing PbO vapor, consisting of an alumina body supportingtop and bottom gold foil electrodes, and flat sapphire win-dows on all 4 sides. The electric field �Eext = Eextz is quiteuniform over a large cylindrical volume (diameter 5 cm,height 4 cm), and is chosen in the range 30–90 V/cm. Themagnetic field Bz is chosen in the range 50–200 mG. Thecell is enclosed in an oven mounted in a vacuum cham-ber. At the operating temperature 700 C, the PbO densityis nPbO ≈ 4 × 1013 cm−3.

A state with simultaneously well-defined spin and elec-trical polarization is populated as follows. A pulsed laserbeam with z linear polarization excites the transition X[J =0+] → a(1)[J = 1−,M = 0]. (X is the electronic groundstate of PbO.) Following the laser pulse a Raman transi-tion is driven by two microwave beams. The first, with xlinear polarization, excites the upward 28.2 GHz transitiona(1)[J = 1−,M = 0] → a(1)[J = 2+,M = ±1]. The sec-ond, with z linear polarization and detuned to the red or bluewith respect to the first by 20–60 MHz, drives the downwardtransition a(1)[J = 2+,M =±1]→ a(1)[J = 1,M =±1].The net result is that about 50% of the J = 1−,M = 0 mole-cules are transferred to a coherent superposition ofM =±1levels in a single desired Ω-doublet component. The subse-quent spin precession (due to E and B) is detected by ob-serving the frequency of quantum beats in the fluorescencethat accompanies spontaneous decay to the X state. The sig-nature of a non-zero EDM is a term in the quantum beatfrequency that is proportional to �Eext · B and that changessign when one switches from one Ω-doublet component tothe other.

The present experimental configuration is sufficient toyield statistical uncertainty comparable to the present limiton de in a reasonable integration time of a few weeks. How-ever, large improvements can be made in a next generationof the experiment. In the new scheme, detection will be ac-complished via absorption of a resonant microwave probe

Eur. Phys. J. C (2008) 57: 13–182 163

beam tuned to the 28.2 GHz transition described above.With this method, the signal-to-noise ratio is linearly pro-portional to the path length of the probe beam in the PbO va-por. In a second generation experiment the cell can be made∼10 times longer than it is now, and the probe beam canpass through the cell multiple times by using suitable mir-rors. Improvements in sensitivity of up to a factor of 3000over the current generation are envisioned.• Other molecule experiments

E. Cornell and co-workers at the Joint Institute for Labo-ratory Astrophysics (Boulder, Colorado) have proposed anexperiment [1058] to search for de in the 3Δ1 electronicstate of the molecular ion HfF+. The premise is to take ad-vantage of the long spin coherence times typical for trappedion experiments with atoms, along with the large effectiveelectric field acting on de in a molecule. Preliminary cal-culations [1036] suggest that the 3Δ1 state is a low-lyingmetastable state with very small Ω-doublet splittings; as inPbO, this state could thus be polarized by small externalelectric fields (�10 V/cm) to yield Eeff ≈ 18 GV/cm [1037].To search for de, electron-spin-resonance spectroscopy, us-ing the Ramsey method, is to be performed in the presenceof rotating electric and magnetic fields. The electric field po-larizes the ions and its rotation prevents them from being ac-celerated out of the trap. As in PbO, use of both upper andlower Ω-doublet components will yield opposite signs ofthe EDM signal, but nearly identical signals due to system-atic effects. However, this experiment has the unique disad-vantage that it is impossible to reverse the electric field: inthe laboratory frame it must always point inward toward thetrap center.

N. Shafer-Ray and co-workers at Oklahoma Universityhave proposed an experiment to search for de in the ground2Π1/2 electronic state of PbF [1059, 1060]. The proposedscheme is similar to the YbF experiment, and the value ofEeff is also approximately the same as for YbF. The primaryadvantage of PbF is that its electric field-dependent mag-netic moment should vanish when a suitable, large externalelectric field E0 ≈ 67 kV/cm is applied [1059, 1060]. Thiscould dramatically reduce magnetic field-related systematicerrors.

9.5.2.4 Experiments with solid-state samples Recently,S. Lamoreaux [1061] revived an old idea of F. Shapiro[1062] to search for de by applying an electric field �Eext

to a solid sample with unpaired electron spins. If de �= 0,at sufficiently low temperature the sample can acquire sig-nificant spin-polarization and thus a detectable magnetiza-tion along the axis of �Eext. Lamoreaux pointed out that useof modern magnetometric techniques and materials (suchas Gd3Ga5O12: gadolinum gallium garnet, or GGG) couldyield impressive sensitivity to de. GGG has a number ofattractive properties. Its resistivity is so high (>1016 Ohm-cm for T < 77 K) that it can support large applied electric

fields (�Eext ≈ 10 kV/cm) with very small leakage currents.Moreover, the ion of interest in GGG, Gd3+ (Z = 64) hasa non-negligible enhancement factor [1063]. A complemen-tary experiment is being done by L. Hunter and co-workers[1064] at Amherst College. Here, a strong external magneticfield is applied to the ferrimagnetic solid Gd3−xYxFe5O12

(gadolinium yttrium iron garnet, or GdYIG), thus caus-ing substantial polarization of the Gd3+ electron spins. Ifde �= 0, this results in electric charge polarization of thesample, and thus a voltage developed across the sample thatreverses with applied magnetic field.

The basic theoretical considerations that must be takeninto account to estimate the expected signals [1065] in thesesolid-state experiments include the same types of calcula-tions needed for free atoms. In addition, however, it is neces-sary to construct models for the modification of atomic elec-tron orbitals in the solid material, as well as the response ofthe material to the EDM-induced perturbation of the heavyparamagnetic atom. The results of the calculations are as fol-lows. When all Gd spins are polarized in the GdIG sample,the resulting macroscopic electric field across the sample isE = 0.7 × 10−10(de/10−27 e cm) V/cm. A similar calcula-tion can be used to determine the degree of spin polarizationof GGG upon application of an external electric field [1061].An externally applied electric field of 10 kV/cm yields aneffective electric field E∗ = −�E/de = 3.6 × 105 V/cmacting on the EDM ([1065]; see also [1066]). The result-ing magnetization M of the sample is simply related toits magnetic susceptibility χ : M = χdE∗/μa , where μa isthe magnetic moment of a Gd3+ ion. Using the standardexpression for χ(T ) in a paramagnetic sample, one findsM ≈ 8nGd(deE

∗)/(kBT ). Here kB is the Boltzmann con-stant and T is the sample temperature. This yields a mag-netic fluxΦ = 4πMS over an area S of an infinite flat sheet.In a recent development [1066], Lamoreaux has pointed outthat this type of electrically induced spin polarization can beamplified in a system that is super-paramagnetic, so that itsmagnetic susceptibility χ is extremely large. It appears thatGdIG (GdYIG with x = 0) has this property at sufficientlylow magnetic field. If so, the sensitivity of a magnetizationmeasurement in GdIG at T = 4 K could be similar to that ofGGG at much lower temperatures, greatly simplifying therequired experimental techniques.• The Indiana GGG experiment

C.Y. Liu of Indiana University has devised a prototypeexperiment [1067, 1068] in which two GGG disks, 4 cmin diameter and of thickness ≈1 cm, are sandwiched be-tween three planar electrodes. High voltages are applied sothat the electric fields in the top and bottom samples are inthe same direction. If de �= 0, a magnetic field similar to adipole field should be generated, and this is to be detectedby a flux pickup coil located in the central ground plane.The latter is designed as a planar gradiometer with 3 con-centric loops, arranged to sum up the returning flux and to

164 Eur. Phys. J. C (2008) 57: 13–182

reject common-mode magnetic fluctuations. As the electricfield polarization is modulated, the gradiometer detects thechanging flux and feeds it to a SQUID sensor. The entireassembly is immersed in a liquid helium bath.

The EDM sensitivity of the prototype experiment is esti-mated to be δde ≈ 4 × 10−26 e cm. Although this falls shortof the ultimate desired sensitivity of 10−30 e cm, the proto-type experiment is useful as a learning tool for solving somebasic technical problems. At Indiana, a second-generationexperiment is also being planned, which will operate atmuch lower temperatures (≈10–15 mK), and will employlower-noise SQUID magnetometers. However, questions re-main as to the nature of the magnetic susceptibility χ ofGGG at such low temperatures.

Some thought has gone into possible systematic effectsin this system. Although crystals with inversion symmetrysuch as GGG and GdIG should not exhibit a linear mag-netoelectric effect [1069], crystal defects and substitutionalimpurities can spoil this ideal. Furthermore a quadratic mag-netoelectric effect does exist, and to avoid systematic errorsarising from it, good control of electric field reversal is re-quired.• The Amherst GdYIG experiment

GdYIG is ferrimagnetic, and both Gd3+ ions and Fe lat-tices contribute to its magnetization M . Their contributionsare generally of opposite sign, but at moderately low temper-atures T the Fe component is roughly constant while the Gdcomponent changes rapidly with T . There exists a “compen-sation” temperature TC where the Gd and Fe magnetizationscancel each other, and the net magnetization M vanishes.For T > TC(< TC), M is dominated by Fe (Gd). The Gdcontribution to M can be reduced by replacing some Gd3+ions with non-magnetic Y3+. With x the average number ofY ions per unit cell, (so that 3-x is the average number of Gdions per unit cell), the compensation temperature becomesTC = [290 − 115(3 − x)] K. This dependence of TC on xis exploited in the Amherst GdYIG experiment. A toroidalsample is employed, consisting of two half-toroids, each inthe shape of the letter C. One “C” has x = 1.35 with a cor-responding TC = 103 K. The other “C” has x = 1.8 with acorresponding TC = 154 K. These are joined together withcopper foil electrodes at the interface. At T = 127 K, themagnetizations of the 2 “C’s” are identical, but their Gdmagnetizations are nominally opposite. When a magneticfield H is applied to the sample with a toroidal current coil,all Gd spins are nominally oriented toward the same copperelectrode. Thus EDM signals from C1 and C2 add construc-tively. However below 103 K (above 154 K) the Gd mag-netization is parallel (antiparallel) to M in both C’s, whichresults in cancellation of one EDM signal by the other. Dataare acquired by observing the voltage difference A (B) be-tween the two foil electrodes for positive (negative) polar-ity of the applied magnetic field H . An EDM should be re-

vealed by the appearance of an asymmetry d = A− B thathas a specific temperature dependence, as described above.

A large spurious effect has been seen that mimics anEDM signal when T < 180 K, but which deviates grosslyfrom expectations for T > 180 K. This effect, which is as-sociated with a component of magnetization that does notreverse with H , has so far frustrated efforts to realize thefull potential of the GdYIG experiment. The best limit thathas been achieved so far is [1064]: de < 5 × 10−24 e cm.

9.6 Muon EDM

The best direct upper limits for an electric dipole moment(EDM) of the muon come from the experiments measur-ing the muon anomalous magnetic moment (g − 2). TheCERN experiment obtained 1.1 × 10−18 e cm (95% C.L.)[1020] and the preliminary limit from Brookhaven is 2.8 ×10−19 e cm [188]. Assuming lepton universality, the electronEDM limit of de < 2.2× 10−27 e cm [186] can be scaled bythe electron to muon mass ratio, in order to obtain an indi-rect limit of dμ < 5 × 10−25 e cm. However, viable modelsexist in which the simple linear mass scaling does not ap-ply and the value for the muon EDM could be pushed upto values in the 10−22 e cm region (see, e.g., [846, 1070–1072]). In order for experimental searches to become suffi-ciently sensitive, dedicated efforts are needed. Several yearsago, a letter of intent for a dedicated experiment at JPARC[1073] was presented, proposing a new sensitive “frozenspin” method [1021, 1022]: The anomalous magnetic mo-ment precession of the muon spin in a storage ring can becompensated by the application of a radial electric field, thusfreezing the spin; a potential electric dipole moment wouldlead to a rotation of the spin out of the orbital plane and thusan observable up-down asymmetry which increases withtime. The projected sensitivity of the proposed experiment(0.5 GeV/c muon momentum, 7 m ring radius) is 10−24–10−25 e cm. Recently it has been pointed out that there isno immediate advantage from working at high muon mo-menta and a sensitive approach with a very compact setup(125 MeV/c muon momentum, 0.42 m ring radius) was out-lined [1024]. Already at an existing beam line, such as theµE1 beam at PSI, a measurement with a sensitivity of bet-ter than dμ ∼ 5× 10−23 e cm within one year of data takingappears feasible. The estimates for the sensitivity assume anoperation in a “one-muon-per-time” mode and the experi-ment would appear to be statistically limited. With an im-proved muon accumulation and injection scheme, the sensi-tivity could be further increased [1074]. Thus the compactstorage ring approach at an existing facility could bring theproof of principle for the frozen spin technique and coverthe next 3–4 orders of magnitude in experimental sensitivityto a possible muon EDM.

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9.7 Muon g− 2

In his famous 1928 paper [1075–1077] Dirac pointed outthat the interaction of an electron with external electric andmagnetic fields may have two extra terms where “the twoextra terms

eh

c(σ,H)+ i eh

cρ1(σ,E), (9.28)

. . . when divided by the factor 2m can be regarded as theadditional potential energy of the electron due to its new de-gree of freedom.” These terms represent the magnetic di-pole (Dirac) moment and electric dipole moment interac-tions with the external fields.

In modern notation, for the magnetic dipole moment ofthe muon we have:

uμ

[eF1(q2)γβ + ie

2mμF2(q2)σβδqδ

]uμ, (9.29)

where F1(0)= 1, and F2(0)= aμ.The magnetic dipole moment of a charged lepton can dif-

fer from its Dirac value (g = 2) for several reasons. Recallthat the proton’s g-value is 5.6 (ap = 1.79), a manifestationof its quark-gluon internal structure. On the other hand, theleptons appear to have no internal structure, and the mag-netic dipole moments are thought to deviate from 2 throughradiative corrections, i.e. resulting from virtual particlesthat couple to the lepton. We should emphasize that theseradiative corrections need not be limited to the standardmodel particles. While the current experimental uncertaintyof ±0.5 ppm on the muon anomaly is 770 times larger thanthat on the electron anomaly [1078], the former is far moresensitive to the effects of high mass scales. In the lowest or-der diagram where mass effects appear, the contribution ofheavy virtual particles with massM scales as (mlepton/M)

2,giving the muon a factor of (mμ/me)2 43000 increase insensitivity over the electron.

9.7.1 The standard model value of the anomalous magneticmoment

The standard model value of a lepton’s anomalous magneticmoment (the anomaly)

a� ≡ (gs − 2)

2

has contributions from three different sets of radiativeprocesses: quantum electrodynamics (QED)—with loopscontaining leptons (e,μ, τ ) and photons; hadronic—withhadrons in vacuum polarization loops; and weak—withloops involving the bosons W,Z, and Higgs:

aSM� = aQED

� + ahadronic� + aweak

� . (9.30)

The QED contribution has been calculated up to the lead-ing five-loop corrections [1079]. The dominant “Schwingerterm” [1080, 1081] a(2) = α/2π , is shown diagrammaticallyin Fig. 71(a). Examples of the hadronic and weak contribu-tions are given in Fig. 71(b)–(d).

The hadronic contribution cannot be calculated directlyfrom QCD, since the energy scale (mμc2) is very low, al-though Blum has performed a proof of principle calculationon the lattice [1082–1084]. Fortunately, dispersion theorygives a relationship between the vacuum polarization loopand the cross section for e+e− → hadrons,

aμ(Had;1)=(αmμ

3π

)2 ∫ ∞

4m2π

ds

s2K(s)R(s), (9.31)

where

R ≡ σtot(e+e− → hadrons

)/σtot

(e+e− → μ+μ−) (9.32)

and experimental data are used as input [1085, 1086].The standard model value of the muon anomaly has re-

cently been reviewed [1085], and the latest values of thecontributions are given in Table 28. The sum of these con-tributions, adding experimental and theoretical errors inquadrature, gives

aSM(06)μ = 11 659 1785 (61)× 10−11, (9.33)

Fig. 71 The Feynman graphs for: (a) lowest order QED (Schwinger)term; (b) lowest order hadronic correction; (c) and (d) lowest orderelectroweak terms. The * emphasizes that in the loop the muon isoff-shell. With the known limits on the Higgs mass, the contributionfrom the single Higgs loop is negligible

Table 28 Standard modelcontributions to the muonanomalous magnetic dipolemoment, aμ. All values aretaken from Ref. [1085]

QED 116 584 718.09 ± 0.145loops ± 0.08α ± 0.04masses ×10−11

Hadronic (lowest order) aμ[HVP(06)] = 6901 ± 42exp ± 19rad ± 7QCD ×10−11

Hadronic (higher order) aμ[HVPh.o.] = −97.9 ± 0.9exp ± 0.3rad ×10−11

Hadronic (light-by-light) aμ[HLLS] = 110 ± 40 ×10−11

Electroweak aμ[EW ] = 154 ± 2 ± 1 ×10−11

166 Eur. Phys. J. C (2008) 57: 13–182

which should be compared with the experimental world av-erage [187]

aexpμ = 11 659 2080 (63)× 10−11. (9.34)

One finds �aμ = 295(88)× 10−11, a 3.4σ difference. It isclear that both the theoretical and the experimental uncer-tainty should be reduced to clarify whether there is a truediscrepancy or a statistical fluctuation. We shall discuss po-tential improvements to the experiment below.

9.7.2 Measurement of the magnetic dipole moment

The measured value of the muon anomaly has a 0.46 ppmstatistical uncertainty and a 0.28 ppm systematic uncer-tainty, which are combined in quadrature to obtain the to-tal error of 0.54 ppm. To significantly improve the measuredvalue, both errors must be reduced. We first discuss the ex-perimental technique, and then the systematic errors.

In all but the first experiments by Garwin et al. [1087]the measurement of the magnetic anomaly made use of thespin rotation in a magnetic field relative to the momentumrotation:

�ωS = −qg �B2m

− q �Bγm(1 − γ ),

�ωC = − q �Bmγ, (9.35)

�ωa ≡ �ωS − �ωC =−(g − 2

2

)q �Bm

=−aμ q�Bm.

A series of three beautiful experiments at CERN culminatedin a 7.3 ppm measure of aμ[1088]. In the third CERN exper-iment, a new technique was developed based on the observa-tion that electrostatic quadrupoles could be used for vertical

focusing. With the velocity transverse to the magnetic field( �β · �B = 0), the spin precession formula becomes

�ωa =− qm

[aμ �B −

(aμ − 1

γ 2 − 1

) �β × �Ec

]. (9.36)

For γmagic = 29.3 (p = 3.09 GeV/c), the second term van-ishes so ωa becomes independent of the electric field andthe precise knowledge of the muon momentum. Also knowl-edge of the muon trajectories to determine the average mag-netic field becomes less critical, which reduces the uncer-tainty in B .

This technique was used also in experiment E821 at theBrookhaven National Laboratory Alternating Gradient Syn-chrotron (AGS) [187, 1085]. The AGS proton beam is usedto produce a beam of pions that decay to muons in an 80 mpion decay channel. Muons with pmagic are brought into thestorage ring and stored using a fast muon kicker. Calorime-ters, placed on the inner radius of the storage ring measureboth the energy and arrival time of the decay electrons. Sincethe highest energy electrons are emitted antiparallel to themuon spin the rate of high energy electrons is modulated bythe spin precession frequency:

N(t,Eth)=N0(Eth)e−t/γ τ [1+A(Eth) cos

(ωat+φ(Eth)

)].

(9.37)

The time spectrum for electrons with E > Eth = 1.8 GeVis shown in [187] Fig. 72. The value of ωa is obtained fromthese data using the five parameter function (9.37) as a start-ing point, but many additional small effects must be takeninto account [187, 1085].

The magnetic field is measured with nuclear magneticresonance (NMR) probes, and tied through calibration to the

Fig. 72 The time spectrum of109 positrons with energygreater than 1.8 GeV from theY2000 run. The endpoint energyis 3.1 GeV. The time interval foreach of the diagonal “wiggles”is given on the right

Eur. Phys. J. C (2008) 57: 13–182 167

Larmor frequency of the free proton [187]. The anomaly isdetermined from

aμ = ωa/ωp

λ− ωa/ωp = Rλ− R , (9.38)

where the tilde on ωa indicates that the measured muonprecession frequency has been adjusted for any necessary(small) corrections, such as the pitch and radial electric fieldcorrections [1085], and λ= μμ/μp is the ratio of the muonto proton magnetic moments.

9.7.3 An improved g− 2 experiment

One of the major features of an upgraded experiment wouldbe a substantially increased flux of muons into the storagering. The BNL beam [187] took forward muons from piondecays, and selected muons 1.7% below the pion momen-tum. With this scheme, approximately half of the injectedbeam consisted of pions. An upgraded experiment wouldneed to quadruple the quadrupoles in the pion decay chan-nel, to increase the beam line acceptance. To decrease thehadron flash at injection one would need to go further awayfrom the pion momentum. Alternatively one could increasethe pion momentum to 5.32 GeV/c so that backward de-cays would produce muons at the magic momentum. Thenthe pion flash would be completely eliminated, which wouldsignificantly reduce the systematic error from gain instabili-ties.

The inflector magnet that permits the beam to enter thestorage ring undeflected would need to be replaced, since thepresent model loses half of the beam through multiple scat-tering in material across the beam channel. The fast muonkicker would also need to be improved. With the significantincrease in beam, the detectors would have to be segmented,new readout electronics would be needed, and a better mea-sure of lost muons would also be needed.

To reduce the magnetic field systematic errors, significanteffort will be needed to improve on the tracking of the fieldwith time, and the calibration procedure used to tie the NMRfrequency in the probes to the free proton Larmor frequency[187].

While there are technical issues to be resolved, thepresent technique—magic γ , electrostatic focusing, uniformmagnetic field—could be pushed to below 0.1 ppm. To gofurther would probably require a new technique. One pos-sibility discussed by Farley [1089] would be to use muonsat much higher energy, say 15 GeV, which would increasethe number of precessions that can be observed. The storagering would consist of a small number of discrete magnetswith uniform field and edge focusing and the field averagedover the orbit would be independent of orbit radius (parti-cle momentum). The averaged field could be calibrated byinjecting polarized protons and observing the proton g − 2precession.

Acknowledgements The authors of this report are grateful to allthe additional workshop participants who contributed with their pre-sentations during the Working Group meetings: A. Baldini, W. Bertl,G. Colangelo, F. Farley, L. Fiorini, G. Gabrielse, J.R. Guest, A. Ho-ecker, J. Hosek, P. Iaydijev, Y. Kuno, S. Lavignac, D. Leone, A. Luc-cio, J. Miller, W. Morse, H. Nishiguchi, Y. Orlov, E. Paoloni, A. Pilaft-sis, W. Porod, N. Ramsey, S. Redin, W. Rodejohann, M.-A. Sanchis-Lozano, N. Shafer-Ray, A. Soni, A. Strumia, G. Venanzoni, T. Ya-mashita, Z. Was, H. Wilschut.

This work was supported in part by the Marie Curie research train-ing network “HEPTOOLS” (MRTN-CT-2006-035505).

References

1. T. Akesson et al., Eur. Phys. J. C 51, 421 (2007).arXiv:hep-ph/0609216

2. A. Blondel et al. (eds.), ECFA/CERN studies of a Euro-pean neutrino factory complex. CERN-2004-002, ECFA-04-230, April 2004

3. V. Barger et al., arXiv:0705.4396 [hep-ph]4. R.N. Mohapatra et al., Rep. Prog. Phys. 70, 1757 (2007).

arXiv:hep-ph/05102135. C.D. Froggatt, H.B. Nielsen, Nucl. Phys. B 147, 277 (1979)6. H. Georgi, C. Jarlskog, Phys. Lett. B 86, 297 (1979)7. J.R. Ellis, M.K. Gaillard, Phys. Lett. B 88, 315 (1979)8. G. Anderson, S. Raby, S. Dimopoulos, L.J. Hall, G.D. Stark-

man, Phys. Rev. D 49, 3660 (1994). arXiv:hep-ph/93083339. N. Irges, S. Lavignac, P. Ramond, Phys. Rev. D 58, 035003

(1998). arXiv:hep-ph/980233410. J.K. Elwood, N. Irges, P. Ramond, Phys. Rev. Lett. 81, 5064

(1998). arXiv:hep-ph/980722811. M. Leurer, Y. Nir, N. Seiberg, Nucl. Phys. B 398, 319 (1993).

arXiv:hep-ph/921227812. M. Dine, R.G. Leigh, A. Kagan, Phys. Rev. D 48, 4269 (1993).

arXiv:hep-ph/930429913. Y. Nir, N. Seiberg, Phys. Lett. B 309, 337 (1993).

arXiv:hep-ph/930430714. M. Leurer, Y. Nir, N. Seiberg, Nucl. Phys. B 420, 468 (1994).

arXiv:hep-ph/931032015. D.B. Kaplan, M. Schmaltz, Phys. Rev. D 49, 3741 (1994).

arXiv:hep-ph/931128116. A. Pomarol, D. Tommasini, Nucl. Phys. B 466, 3 (1996)

arXiv:hep-ph/950746217. C.D. Carone, L.J. Hall, H. Murayama, Phys. Rev. D 53, 6282

(1996). arXiv:hep-ph/951239918. R. Barbieri, G.R. Dvali, L.J. Hall, Phys. Lett. B 377, 76 (1996).

arXiv:hep-ph/951238819. E. Dudas, C. Grojean, S. Pokorski, C.A. Savoy, Nucl. Phys. B

481, 85 (1996). arXiv:hep-ph/960638320. R. Barbieri, L.J. Hall, S. Raby, A. Romanino, Nucl. Phys. B

493, 3 (1997). arXiv:hep-ph/961044921. P. Binetruy, S. Lavignac, P. Ramond, Nucl. Phys. B 477, 353

(1996). arXiv:hep-ph/960124322. P. Binetruy, S. Lavignac, S.T. Petcov, P. Ramond, Nucl. Phys. B

496, 3 (1997). arXiv:hep-ph/961048123. Y. Nir, R. Rattazzi, Phys. Lett. B 382, 363 (1996).

arXiv:hep-ph/960323324. R. Barbieri, L.J. Hall, A. Romanino, Nucl. Phys. Proc. Suppl. A

52, 141 (1997)25. C.D. Carone, L.J. Hall, Phys. Rev. D 56, 4198 (1997).

arXiv:hep-ph/970243026. R. Barbieri, L.J. Hall, A. Romanino, Phys. Lett. B 401, 47

(1997). arXiv:hep-ph/970231527. R. Barbieri, L. Giusti, L.J. Hall, A. Romanino, Nucl. Phys. B

550, 32 (1999). arXiv:hep-ph/9812239

168 Eur. Phys. J. C (2008) 57: 13–182

28. L.J. Hall, N. Weiner, Phys. Rev. D 60, 033005 (1999).arXiv:hep-ph/9811299

29. R. Barbieri, L.J. Hall, A. Romanino, Nucl. Phys. B 551, 93(1999). arXiv:hep-ph/9812384

30. A. Aranda, C.D. Carone, R.F. Lebed, Phys. Lett. B 474, 170(2000). arXiv:hep-ph/9910392

31. R. Barbieri, P. Creminelli, A. Romanino, Nucl. Phys. B 559, 17(1999). arXiv:hep-ph/9903460

32. Z. Berezhiani, A. Rossi, Nucl. Phys. B 594, 113 (2001).arXiv:hep-ph/0003084

33. M.C. Chen, K.T. Mahanthappa, Phys. Rev. D 62, 113007(2000). arXiv:hep-ph/0005292

34. A. Aranda, C.D. Carone, R.F. Lebed, Phys. Rev. D 62, 016009(2000). arXiv:hep-ph/0002044

35. A. Aranda, C.D. Carone, P. Meade, Phys. Rev. D 65, 013011(2002). arXiv:hep-ph/0109120

36. E. Ma, G. Rajasekaran, Phys. Rev. D 64, 113012 (2001).arXiv:hep-ph/0106291

37. S.F. King, G.G. Ross, Phys. Lett. B 520, 243 (2001).arXiv:hep-ph/0108112

38. S. Lavignac, I. Masina, C.A. Savoy, Phys. Lett. B 520, 269(2001). arXiv:hep-ph/0106245

39. R.G. Roberts, A. Romanino, G.G. Ross, L. Velasco-Sevilla,Nucl. Phys. B 615, 358 (2001). arXiv:hep-ph/0104088

40. J.L. Chkareuli, C.D. Froggatt, H.B. Nielsen, Nucl. Phys. B 626,307 (2002). arXiv:hep-ph/0109156

41. K.S. Babu, E. Ma, J.W.F. Valle, Phys. Lett. B 552, 207 (2003).arXiv:hep-ph/0206292

42. G.G. Ross, L. Velasco-Sevilla, Nucl. Phys. B 653, 3 (2003).arXiv:hep-ph/0208218

43. Y. Nir, G. Raz, Phys. Rev. D 66, 035007 (2002). arXiv:hep-ph/0206064

44. H.K. Dreiner, H. Murayama, M. Thormeier, Nucl. Phys. B 729,278 (2005). arXiv:hep-ph/0312012

45. S.F. King, G.G. Ross, Phys. Lett. B 574, 239 (2003).arXiv:hep-ph/0307190

46. G.G. Ross, L. Velasco-Sevilla, O. Vives, Nucl. Phys. B 692, 50(2004). arXiv:hep-ph/0401064

47. W. Grimus, A.S. Joshipura, S. Kaneko, L. Lavoura,M. Tanimoto, J. High Energy Phys. 0407, 078 (2004).arXiv:hep-ph/0407112

48. G.L. Kane, S.F. King, I.N.R. Peddie, L. Velasco-Sevilla, J. HighEnergy Phys. 0508, 083 (2005). arXiv:hep-ph/0504038

49. Z. Berezhiani, F. Nesti, J. High Energy Phys. 0603, 041 (2006).arXiv:hep-ph/0510011

50. G. Altarelli, F. Feruglio, Nucl. Phys. B 741, 215 (2006).arXiv:hep-ph/0512103

51. O. Vives, arXiv:hep-ph/050407952. S. Abel, D. Bailin, S. Khalil, O. Lebedev, Phys. Lett. B 504, 241

(2001). arXiv:hep-ph/001214553. J.L. Diaz-Cruz, J. Ferrandis, Phys. Rev. D 72, 035003 (2005).

arXiv:hep-ph/050409454. P.H. Chankowski, K. Kowalska, S. Lavignac, S. Pokorski, Phys.

Rev. D 71, 055004 (2005). arXiv:hep-ph/050107155. W. Grimus, L. Lavoura, J. High Energy Phys. 0508, 013 (2005).

arXiv:hep-ph/050415356. I. de Medeiros Varzielas, G.G. Ross, Nucl. Phys. B 733, 31

(2006). arXiv:hep-ph/050717657. I. de Medeiros Varzielas, S.F. King, G.G. Ross, arXiv:hep-ph/

060704558. I. Masina, C.A. Savoy, Nucl. Phys. B 755, 1 (2006). arXiv:

hep-ph/060310159. I. Masina, C.A. Savoy, Phys. Lett. B 642, 472 (2006).

arXiv:hep-ph/060609760. S.F. King, M. Malinsky, Phys. Lett. B 645, 351 (2007).

arXiv:hep-ph/0610250

61. C. Hagedorn, M. Lindner, R.N. Mohapatra, J. High EnergyPhys. 0606, 042 (2006). arXiv:hep-ph/0602244

62. T. Appelquist, Y. Bai, M. Piai, Phys. Lett. B 637, 245 (2006).arXiv:hep-ph/0603104

63. L. Ferretti, S.F. King, A. Romanino, J. High Energy Phys. 0611,078 (2006). arXiv:hep-ph/0609047

64. F. Feruglio, C. Hagedorn, Y. Lin, L. Merlo, arXiv:hep-ph/0702194

65. E. Ma, arXiv:hep-ph/040907566. E. Ma, Phys. Rev. D 43, 2761 (1991)67. N.G. Deshpande, M. Gupta, P.B. Pal, Phys. Rev. D 45, 953

(1992)68. M. Frigerio, S. Kaneko, E. Ma, M. Tanimoto, Phys. Rev. D 71,

011901 (2005). arXiv:hep-ph/040918769. E. Ma, Mod. Phys. Lett. A 17, 2361 (2002). arXiv:hep-ph/

021139370. C. Luhn, S. Nasri, P. Ramond, arXiv:hep-th/070118871. E. Ma, Mod. Phys. Lett. A 21, 1917 (2006). arXiv:hep-ph/

060705672. W. Grimus, L. Lavoura, Phys. Lett. B 572, 189 (2003). arXiv:

hep-ph/030504673. E. Ma, arXiv:hep-ph/061202274. Y. Koide, Lett. Nuovo Cimento 34, 201 (1982)75. E. Ma, arXiv:hep-ph/070101676. Y. Yamaguchi, Phys. Lett. 9, 281 (1964)77. J. Kubo, A. Mondragon, M. Mondragon, E. Rodriguez-

Jauregui, Prog. Theor. Phys. 109, 795 (2003) [Erratum: Prog.Theor. Phys. 114, 287 (2005)]. arXiv:hep-ph/0302196

78. J. Kubo, H. Okada, F. Sakamaki, Phys. Rev. D 70, 036007(2004). arXiv:hep-ph/0402089

79. S.L. Chen, M. Frigerio, E. Ma, Phys. Rev. D 70, 073008(2004) [Erratum: Phys. Rev. D 70, 079905 (2004)].arXiv:hep-ph/0404084

80. L. Lavoura, E. Ma, Mod. Phys. Lett. A 20, 1217 (2005).arXiv:hep-ph/0502181

81. T. Teshima, Phys. Rev. D 73, 045019 (2006). arXiv:hep-ph/0509094

82. Y. Koide, Phys. Rev. D 73, 057901 (2006). arXiv:hep-ph/0509214

83. R.N. Mohapatra, S. Nasri, H.B. Yu, Phys. Lett. B 639, 318(2006). arXiv:hep-ph/0605020

84. S. Morisi, M. Picariello, Int. J. Theor. Phys. 45, 1267 (2006).arXiv:hep-ph/0505113

85. S. Kaneko, H. Sawanaka, T. Shingai, M. Tanimoto, K.Yoshioka, Prog. Theor. Phys. 117, 161 (2007). arXiv:hep-ph/0609220

86. F. Feruglio, Y. Lin, arXiv:0712.1528 [hep-ph]87. E. Ma, arXiv:hep-ph/061201388. O. Felix, A. Mondragon, M. Mondragon, E. Peinado, arXiv:

hep-ph/061006189. P.F. Harrison, D.H. Perkins, W.G. Scott, Phys. Lett. B 530, 167

(2002). arXiv:hep-ph/020207490. X.G. He, A. Zee, Phys. Lett. B 560, 87 (2003). arXiv:hep-ph/

030109291. N. Cabibbo, Phys. Lett. B 72, 333 (1978)92. L. Wolfenstein, Phys. Rev. D 18, 958 (1978)93. E. Ma, Phys. Rev. D 70, 031901 (2004). arXiv:hep-ph/040419994. G. Altarelli, F. Feruglio, Nucl. Phys. B 720, 64 (2005).

arXiv:hep-ph/050416595. K.S. Babu, X.G. He, arXiv:hep-ph/050721796. E. Ma, Phys. Rev. D 72, 037301 (2005). arXiv:hep-ph/050520997. A. Zee, Phys. Lett. B 630, 58 (2005). arXiv:hep-ph/050827898. E. Ma, Phys. Rev. D 73, 057304 (2006). arXiv:hep-ph/051113399. X.G. He, Y.Y. Keum, R.R. Volkas, J. High Energy Phys. 0604,

039 (2006). arXiv:hep-ph/0601001100. B. Adhikary, B. Brahmachari, A. Ghosal, E. Ma, M.K. Parida,

Phys. Lett. B 638, 345 (2006). arXiv:hep-ph/0603059

Eur. Phys. J. C (2008) 57: 13–182 169

101. B. Adhikary, A. Ghosal, Phys. Rev. D 75, 073020 (2007).arXiv:hep-ph/0609193

102. F. Yin, Phys. Rev. D 75, 073010 (2007). arXiv:0704.3827 [hep-ph]

103. G. Altarelli, F. Feruglio, Y. Lin, Nucl. Phys. B 775, 31 (2007).arXiv:hep-ph/0610165

104. X.G. He, Nucl. Phys. Proc. Suppl. 168, 350 (2007). arXiv:hep-ph/0612080

105. E. Ma, Mod. Phys. Lett. A 17, 289 (2002). arXiv:hep-ph/0201225

106. E. Ma, Mod. Phys. Lett. A 17, 627 (2002). arXiv:hep-ph/0203238

107. K.S. Babu, T. Kobayashi, J. Kubo, Phys. Rev. D 67, 075018(2003). arXiv:hep-ph/0212350

108. S.L. Chen, M. Frigerio, E. Ma, Nucl. Phys. B 724, 423 (2005).arXiv:hep-ph/0504181

109. M. Hirsch, A. Villanova del Moral, J.W.F. Valle, E. Ma, Phys.Rev. D 72, 091301 (2005) [Erratum: Phys. Rev. D 72, 119904(2005)]. arXiv:hep-ph/0507148

110. E. Ma, Mod. Phys. Lett. A 20, 1953 (2005). arXiv:hep-ph/0502024

111. E. Ma, Mod. Phys. Lett. A 20, 2601 (2005). arXiv:hep-ph/0508099

112. E. Ma, Mod. Phys. Lett. A 20, 2767 (2005). arXiv:hep-ph/0506036

113. E. Ma, H. Sawanaka, M. Tanimoto, Phys. Lett. B 641, 301(2006). arXiv:hep-ph/0606103

114. E. Ma, Mod. Phys. Lett. A 21, 2931 (2006). arXiv:hep-ph/0607190

115. E. Ma, Mod. Phys. Lett. A 22, 101 (2007). arXiv:hep-ph/0610342

116. L. Lavoura, H. Kuhbock, Mod. Phys. Lett. A 22, 181 (2007).arXiv:hep-ph/0610050

117. I. de Medeiros Varzielas, S.F. King, G.G. Ross, Phys. Lett. B644, 153 (2007). arXiv:hep-ph/0512313

118. S. Morisi, M. Picariello, E. Torrente-Lujan, Phys. Rev. D 75,075015 (2007). arXiv:hep-ph/0702034

119. Y. Koide, arXiv:hep-ph/0701018120. M. Hirsch, A.S. Joshipura, S. Kaneko, J.W.F. Valle,

arXiv:hep-ph/0703046121. P.D. Carr, P.H. Frampton, arXiv:hep-ph/0701034122. M.C. Chen, K.T. Mahanthappa, arXiv:0705.0714 [hep-ph]123. P.H. Frampton, T.W. Kephart, arXiv:0706.1186 [hep-ph]124. T. Kobayashi, H.P. Nilles, F. Ploger, S. Raby, M. Ratz, Nucl.

Phys. B 768, 135 (2007). arXiv:hep-ph/0611020125. E. Ma, Fizika B 14, 35 (2005). arXiv:hep-ph/0409288126. C. Hagedorn, M. Lindner, F. Plentinger, Phys. Rev. D 74,

025007 (2006). arXiv:hep-ph/0604265127. Y. Kajiyama, J. Kubo, H. Okada, Phys. Rev. D 75, 033001

(2007). arXiv:hep-ph/0610072128. P.H. Frampton, T.W. Kephart, Phys. Rev. D 51, 1 (1995).

arXiv:hep-ph/9409324129. K.S. Babu, J. Kubo, Phys. Rev. D 71, 056006 (2005).

arXiv:hep-ph/0411226130. J. Kubo, Phys. Lett. B 622, 303 (2005). arXiv:hep-ph/0506043131. S.L. Chen, E. Ma, Phys. Lett. B 620, 151 (2005).

arXiv:hep-ph/0505064132. E. Ma, Phys. Lett. B 632, 352 (2006). arXiv:hep-ph/0508231133. Y. Cai, H.B. Yu, Phys. Rev. D 74, 115005 (2006). arXiv:hep-ph/

0608022134. H. Zhang, arXiv:hep-ph/0612214135. Y. Koide, arXiv:0705.2275 [hep-ph]136. M. Schmaltz, Phys. Rev. D 52, 1643 (1995). arXiv:hep-ph/

9411383137. W. Grimus, L. Lavoura, J. High Energy Phys. 0601, 018 (2006).

arXiv:hep-ph/0509239

138. W. Grimus, L. Lavoura, arXiv:hep-ph/0611149139. S.M. Barr, Phys. Rev. D 65, 096012 (2002). arXiv:hep-ph/

0106241140. S.F. King, Phys. Lett. B 439, 350 (1998). arXiv:hep-ph/

9806440141. S.F. King, Nucl. Phys. B 562, 57 (1999). arXiv:hep-ph/

9904210142. S.F. King, Nucl. Phys. B 576, 85 (2000). arXiv:hep-ph/

9912492143. S.F. King, J. High Energy Phys. 0209, 011 (2002).

arXiv:hep-ph/0204360144. F. Borzumati, A. Masiero, Phys. Rev. Lett. 57, 961 (1986)145. L.J. Hall, V.A. Kostelecky, S. Raby, Nucl. Phys. B 267, 415

(1986)146. S. Ferrara, L. Girardello, F. Palumbo, Phys. Rev. D 20, 403

(1979)147. P. Fayet, Phys. Lett. B 69, 489 (1977)148. E. Witten, Nucl. Phys. B 188, 513 (1981)149. S. Dimopoulos, H. Georgi, Nucl. Phys. B 193, 150 (1981)150. S. Weinberg, Phys. Rev. D 26, 287 (1982)151. L. Alvarez-Gaume, M. Claudson, M.B. Wise, Nucl. Phys. B

207, 96 (1982)152. G.F. Giudice, R. Rattazzi, Phys. Rep. 322, 419 (1999).

arXiv:hep-ph/9801271153. E. Cremmer, S. Ferrara, L. Girardello, A. Van Proeyen, Nucl.

Phys. B 212, 413 (1983)154. S.K. Soni, H.A. Weldon, Phys. Lett. B 126, 215 (1983)155. V.S. Kaplunovsky, J. Louis, Phys. Lett. B 306, 269 (1993).

arXiv:hep-th/9303040156. A. Brignole, L.E. Ibanez, C. Munoz, Nucl. Phys. B 422,

125 (1994) [Erratum: Nucl. Phys. B 436, 747 (1995)].arXiv:hep-ph/9308271

157. P.H. Chankowski, O. Lebedev, S. Pokorski, Nucl. Phys. B 717,190 (2005). arXiv:hep-ph/0502076

158. S.P. Martin, arXiv:hep-ph/9709356159. M. Shapkin, Nucl. Phys. Proc. Suppl. 169, 363 (2007)160. G.G. Ross, O. Vives, Phys. Rev. D 67, 095013 (2003).

arXiv:hep-ph/0211279161. S.F. King, I.N.R. Peddie, G.G. Ross, L. Velasco-Sevilla,

O. Vives, J. High Energy Phys. 0507, 049 (2005).arXiv:hep-ph/0407012

162. S. Antusch, S.F. King, M. Malinsky, arXiv:0708.1282 [hep-ph]163. I. Masina, C.A. Savoy, Nucl. Phys. B 661, 365 (2003).

arXiv:hep-ph/0211283164. A. Masiero, S.K. Vempati, O. Vives, Nucl. Phys. B 649, 189

(2003). arXiv:hep-ph/0209303165. M. Ciuchini, A. Masiero, P. Paradisi, L. Silvestrini, S.K. Vem-

pati, O. Vives, arXiv:hep-ph/0702144166. K. Hagiwara, S. Matsumoto, D. Haidt, C.S. Kim, Z. Phys.

C 64, 559 (1994) [Erratum: Z. Phys. C 68, 352 (1995)].arXiv:hep-ph/9409380

167. W. Buchmuller, D. Wyler, Nucl. Phys. B 268, 621 (1986)168. Z. Berezhiani, A. Rossi, Phys. Lett. B 535, 207 (2002).

arXiv:hep-ph/0111137169. A. Brignole, A. Rossi, Nucl. Phys. B 701, 3 (2004). arXiv:

hep-ph/0404211170. W.M. Yao et al. (Particle Data Group), J. Phys. G 33, 1 (2006)171. M.S. Bilenky, A. Santamaria, Nucl. Phys. B 420, 47 (1994).

arXiv:hep-ph/9310302172. L. Lavoura, Eur. Phys. J. C 29, 191 (2003). arXiv:hep-ph/

0302221173. T. Inami, C.S. Lim, Prog. Theor. Phys. 65, 297 (1981) [Erratum:

Prog. Theor. Phys. 65, 1772 (1981)]174. A. Brignole, A. Rossi, Phys. Lett. B 566, 217 (2003).

arXiv:hep-ph/0304081175. P. Paradisi, J. High Energy Phys. 0608, 047 (2006).

arXiv:hep-ph/0601100

170 Eur. Phys. J. C (2008) 57: 13–182

176. F. Jegerlehner, arXiv:hep-ph/0703125177. G.A. Gonzalez-Sprinberg, A. Santamaria, J. Vidal, Nucl. Phys.

B 582, 3 (2000). arXiv:hep-ph/0002203178. J. Hisano, T. Moroi, K. Tobe, M. Yamaguchi, Phys. Rev. D 53,

2442 (1996). arXiv:hep-ph/9510309179. A. Czarnecki, E. Jankowski, Phys. Rev. D 65, 113004 (2002).

arXiv:hep-ph/0106237180. M.L. Brooks et al. (MEGA Collaboration), Phys. Rev. Lett. 83,

1521 (1999). arXiv:hep-ex/9905013181. K. Hayasaka et al. (Belle Collaboration), arXiv:0705.0650

[hep-ex]182. B. Aubert et al. (BaBar Collaboration), Phys. Rev. Lett. 96,

041801 (2006). arXiv:hep-ex/0508012183. S. Banerjee, arXiv:hep-ex/0702017184. M. Raidal, A. Santamaria, Phys. Lett. B 421, 250 (1998).

arXiv:hep-ph/9710389185. K. Huitu, J. Maalampi, M. Raidal, A. Santamaria, Phys. Lett. B

430, 355 (1998). arXiv:hep-ph/9712249186. B.C. Regan, E.D. Commins, C.J. Schmidt, D. DeMille, Phys.

Rev. Lett. 88, 071805 (2002)187. G.W. Bennett et al. (Muon g-2 Collaboration), Phys. Rev. D 73,

072003 (2006). arXiv:hep-ex/0602035188. R. McNabb (Muon g-2 Collaboration), arXiv:hep-ex/

0407008189. J. Abdallah et al. (DELPHI Collaboration), Eur. Phys. J. C 35,

159 (2004). arXiv:hep-ex/0406010190. K. Inami et al.(Belle Collaboration), Phys. Lett. B 551, 16

(2003). arXiv:hep-ex/0210066191. B. Aubert et al. (BaBar Collaboration), Phys. Rev. Lett. 95,

041802 (2005). arXiv:hep-ex/0502032192. S. Davidson, C. Pena-Garay, N. Rius, A. Santamaria, J. High

Energy Phys. 0303, 011 (2003). arXiv:hep-ph/0302093193. M. Acciarri et al. (L3 Collaboration), Phys. Lett. B 489, 81

(2000). arXiv:hep-ex/0005028194. D. Bourilkov, Phys. Rev. D 64, 071701 (2001). arXiv:hep-ph/

0104165195. G. Abbiendi et al. (OPAL Collaboration), Eur. Phys. J. C 33,

173 (2004). arXiv:hep-ex/0309053196. A. Pich, J.P. Silva, Phys. Rev. D 52, 4006 (1995). arXiv:

hep-ph/9505327197. S. Bergmann, Y. Grossman, D.M. Pierce, Phys. Rev. D 61,

053005 (2000). arXiv:hep-ph/9909390198. M.C. Gonzalez-Garcia, J.W.F. Valle, Mod. Phys. Lett. A 7, 477

(1992)199. A. De Gouvea, G.F. Giudice, A. Strumia, K. Tobe, Nucl. Phys.

B 623, 395 (2002). arXiv:hep-ph/0107156200. Z. Berezhiani, R.S. Raghavan, A. Rossi, Nucl. Phys. B 638, 62

(2002). arXiv:hep-ph/0111138201. S. Davidson, D.C. Bailey, B.A. Campbell, Z. Phys. C 61, 613

(1994). arXiv:hep-ph/9309310202. M. Herz, arXiv:hep-ph/0301079203. S.L. Glashow, J. Iliopoulos, L. Maiani, Phys. Rev. D 2, 1285

(1970)204. N. Cabibbo, Phys. Rev. Lett. 10, 531 (1963)205. M. Kobayashi, T. Maskawa, Prog. Theor. Phys. 49, 652 (1973)206. L. Wolfenstein, Phys. Rev. Lett. 51, 1945 (1983)207. M. Battaglia et al., hep-ph/0304132208. B. Pontecorvo, Sov. Phys. JETP 7, 172 (1958) [Zh. Eksp. Teor.

Fiz. 34, 247 (1957)]209. Z. Maki, M. Nakagawa, S. Sakata, Prog. Theor. Phys. 28, 870

(1962)210. M. Maltoni, T. Schwetz, M.A. Tortola, J.W.F. Valle, New J.

Phys. 6, 122 (2004). arXiv:hep-ph/0405172211. G.L. Fogli, E. Lisi, A. Marrone, A. Palazzo, Prog. Part. Nucl.

Phys. 57, 742 (2006). arXiv:hep-ph/0506083

212. S.T. Petcov, Sov. J. Nucl. Phys. 25, 340 (1977) [Yad. Fiz. 25,641 (1977); Errata: Yad. Fiz. 25, 698 (1977), Yad. Fiz. 25, 1336(1977)]

213. S. Weinberg, Phys. Rev. Lett. 43, 1566 (1979)214. S.M. Bilenky, J. Hosek, S.T. Petcov, Phys. Lett. B 94, 495

(1980)215. J. Schechter, J.W.F. Valle, Phys. Rev. D 22, 2227 (1980)216. P. Minkowski, Phys. Lett. B 67, 421 (1977)217. T. Yanagida, in Proceedings of the Workshop on the Baryon

Number of the Universe and Unified Theories, Tsukuba, Japan,13–14 February 1979

218. M. Gell-Mann, P. Ramond, R. Slansky, in Supergravity, ed. byP. van Nieuwenhuizen, D.Z. Freedman (North Holland, Ams-terdam, 1979)

219. S.L. Glashow, NATO Adv. Study Inst. Ser. B Phys. 59, 687(1979)

220. R.N. Mohapatra, G. Senjanovic, Phys. Rev. Lett. 44, 912 (1980)221. M. Magg, C. Wetterich, Phys. Lett. B 94, 61 (1980)222. G. Lazarides, Q. Shafi, C. Wetterich, Nucl. Phys. B 181, 287

(1981)223. R.N. Mohapatra, G. Senjanovic, Phys. Rev. D 23, 165 (1981)224. G.B. Gelmini, M. Roncadelli, Phys. Lett. B 99, 411 (1981)225. E. Ma, Phys. Rev. Lett. 81, 1171 (1998). arXiv:hep-ph/9805219226. R. Foot, H. Lew, X.G. He, G.C. Joshi, Z. Phys. C 44, 441 (1989)227. W. Buchmuller, K. Hamaguchi, O. Lebedev, S. Ramos-

Sanchez, M. Ratz, Phys. Rev. Lett. 99, 021601 (2007).arXiv:hep-ph/0703078

228. J.R. Ellis, O. Lebedev, arXiv:0707.3419 [hep-ph]229. J.R. Ellis, J. Hisano, S. Lola, M. Raidal, Nucl. Phys. B 621, 208

(2002). arXiv:hep-ph/0109125230. S. Pascoli, S.T. Petcov, W. Rodejohann, Phys. Rev. D 68,

093007 (2003). arXiv:hep-ph/0302054231. J.A. Casas, A. Ibarra, F. Jimenez-Alburquerque, arXiv:hep-ph/

0612289232. J.A. Casas, A. Ibarra, Nucl. Phys. B 618, 171 (2001). arXiv:

hep-ph/0103065233. S. Lavignac, I. Masina, C.A. Savoy, Nucl. Phys. B 633, 139

(2002). arXiv:hep-ph/0202086234. P.H. Frampton, S.L. Glashow, T. Yanagida, Phys. Lett. B 548,

119 (2002). arXiv:hep-ph/0208157235. M. Raidal, A. Strumia, Phys. Lett. B 553, 72 (2003).

arXiv:hep-ph/0210021236. A. Ibarra, G.G. Ross, Phys. Lett. B 575, 279 (2003).

arXiv:hep-ph/0307051237. A. Ibarra, G.G. Ross, Phys. Lett. B 591, 285 (2004).

arXiv:hep-ph/0312138238. P.H. Chankowski, J.R. Ellis, S. Pokorski, M. Raidal, K. Turzyn-

ski, Nucl. Phys. B 690, 279 (2004)239. S.T. Petcov, W. Rodejohann, T. Shindou, Y. Takanishi, Nucl.

Phys. B 739, 208 (2006). arXiv:hep-ph/0510404240. T. Endoh, T. Morozumi, T. Onogi, A. Purwanto, Phys. Rev. D

64, 013006 (2001) [Erratum: Phys. Rev. D 64, 059904 (2001)].arXiv:hep-ph/0012345

241. S. Davidson, A. Ibarra, J. High Energy Phys. 0109, 013 (2001).arXiv:hep-ph/0104076

242. J.R. Ellis, J. Hisano, M. Raidal, Y. Shimizu, Phys. Rev. D 66,115013 (2002). arXiv:hep-ph/0206110

243. A. Ibarra, J. High Energy Phys. 0601, 064 (2006).arXiv:hep-ph/0511136

244. E. Ma, U. Sarkar, Phys. Rev. Lett. 80, 5716 (1998).arXiv:hep-ph/9802445

245. K. Huitu, J. Maalampi, M. Raidal, Nucl. Phys. B 420, 449(1994). arXiv:hep-ph/9312235

246. K. Huitu, J. Maalampi, M. Raidal, Phys. Lett. B 328, 60 (1994).arXiv:hep-ph/9402219

247. M. Raidal, P.M. Zerwas, Eur. Phys. J. C 8, 479 (1999).arXiv:hep-ph/9811443

Eur. Phys. J. C (2008) 57: 13–182 171

248. T. Hambye, E. Ma, U. Sarkar, Nucl. Phys. B 602, 23 (2001).arXiv:hep-ph/0011192

249. A. Rossi, Phys. Rev. D 66, 075003 (2002). arXiv:hep-ph/0207006

250. K.S. Babu, C.N. Leung, J.T. Pantaleone, Phys. Lett. B 319, 191(1993)

251. P.H. Chankowski, Z. Płuciennik, Phys. Lett. B 316, 312 (1993)252. S. Antusch, M. Drees, J. Kersten, M. Lindner, M. Ratz, Phys.

Lett. B 519, 238 (2001)253. S. Antusch, M. Drees, J. Kersten, M. Lindner, M. Ratz, Phys.

Lett. B 525, 130 (2002)254. S. Antusch, J. Kersten, M. Lindner, M. Ratz, Nucl. Phys. B 674,

401 (2003)255. P.H. Chankowski, W. Krolikowski, S. Pokorski, Phys. Lett. B

473, 109 (2000)256. J.A. Casas, J.R. Espinosa, A. Ibarra, I. Navarro, Nucl. Phys. B

573, 652 (2000)257. S. Antusch, J. Kersten, M. Lindner, M. Ratz, Phys. Lett. B 538,

87 (2002)258. S. Antusch, J. Kersten, M. Lindner, M. Ratz, M.A. Schmidt, J.

High Energy Phys. 0503, 024 (2005)259. J.W. Mei, Phys. Rev. D 71, 073012 (2005)260. S. Antusch, M. Ratz, J. High Energy Phys. 0207, 059 (2002)261. N. Haba, N. Okamura, M. Sugiura, Prog. Theor. Phys. 103, 367

(2000)262. N. Haba, Y. Matsui, N. Okamura, M. Sugiura, Eur. Phys. J. C

10, 677 (1999)263. N. Haba, N. Okamura, Eur. Phys. J. C 14, 347 (2000)264. J.R. Ellis, S. Lola, Phys. Lett. B 458, 310 (1999)265. J.A. Casas, J.R. Espinosa, A. Ibarra, I. Navarro, Nucl. Phys. B

556, 3 (1999)266. J.A. Casas, J.R. Espinosa, A. Ibarra, I. Navarro, Nucl. Phys. B

569, 82 (2000)267. T. Miura, T. Shindou, E. Takasugi, Phys. Rev. D 66, 093002

(2002)268. N. Haba, Y. Matsui, N. Okamura, Eur. Phys. J. C 17, 513 (2000)269. S. Antusch, J. Kersten, M. Lindner, M. Ratz, Phys. Lett. B 544,

1 (2002)270. T. Miura, T. Shindou, E. Takasugi, Phys. Rev. D 68, 093009

(2003)271. S. Kanemura, K. Matsuda, T. Ota, T. Shindou, E. Takasugi,

K. Tsumura, Phys. Rev. D 72, 093004 (2005)272. T. Shindou, E. Takasugi, Phys. Rev. D 70, 013005 (2004)273. S.T. Petcov, S. Profumo, Y. Takanishi, C.E. Yaguna, Nucl. Phys.

B 676, 453 (2004)274. J.R. Ellis, A. Hektor, M. Kadastik, K. Kannike, M. Raidal, Phys.

Lett. B 631, 32 (2005). arXiv:hep-ph/0506122275. S.T. Petcov, T. Shindou, Y. Takanishi, Nucl. Phys. B 738, 219

(2006)276. H.B. Nielsen, Y. Takanishi, Nucl. Phys. B 636, 305 (2002)277. R. Gonzalez Felipe, F.R. Joaquim, B.M. Nobre, Phys. Rev. D

70, 085009 (2004)278. G.C. Branco, R. Gonzalez Felipe, F.R. Joaquim, B.M. Nobre,

Phys. Lett. B 633, 336 (2006)279. J.W. Mei, Z.Z. Xing, Phys. Rev. D 70, 053002 (2004)280. S.T. Petcov, A.Y. Smirnov, Phys. Lett. B 322, 109 (1994).

arXiv:hep-ph/9311204281. M. Raidal, Phys. Rev. Lett. 93, 161801 (2004). arXiv:hep-ph/

0404046282. H. Minakata, A.Y. Smirnov, Phys. Rev. D 70, 073009 (2004).

arXiv:hep-ph/0405088283. Y. Kajiyama, M. Raidal, A. Strumia, arXiv:0705.4559 [hep-ph]284. Euclid, Elements, Book 6, Definition 3. (A straight line is said

to have been cut in extreme and mean ratio when, as the wholeline is to the greater segment, so is the greater to the less)

285. Q. Duret, B. Machet, arXiv:0705.1237 [hep-ph]

286. B.C. Chauhan, M. Picariello, J. Pulido, E. Torrente-Lujan,arXiv:hep-ph/0605032

287. M. Picariello, arXiv:hep-ph/0611189288. F. Caravaglios, S. Morisi, arXiv:hep-ph/0611078289. C.L. Bennett et al. (WMAP Collaboration), Astrophys. J. Suppl.

148, 1 (2003). arXiv:astro-ph/0302207290. M. Fukugita, T. Yanagida, Phys. Lett. B 174, 45 (1986)291. V.A. Kuzmin, V.A. Rubakov, M.E. Shaposhnikov, Phys. Lett. B

155, 36 (1985)292. J. Liu, G. Segre, Phys. Rev. D 48, 4609 (1993). arXiv:

hep-ph/9304241293. M. Flanz, E.A. Paschos, U. Sarkar, Phys. Lett. B 345, 248

(1995) [Erratum: Phys. Lett. B 382, 447 (1996)]. arXiv:hep-ph/9411366

294. L. Covi, E. Roulet, F. Vissani, Phys. Lett. B 384, 169 (1996).arXiv:hep-ph/9605319

295. R. Barbieri, P. Creminelli, A. Strumia, N. Tetradis, Nucl. Phys.B 575, 61 (2000). arXiv:hep-ph/9911315

296. A. Abada, S. Davidson, F.X. Josse-Michaux, M. Losada,A. Riotto, J. Cosmol. Astropart. Phys. 0604, 004 (2006).arXiv:hep-ph/0601083

297. E. Nardi, Y. Nir, E. Roulet, J. Racker, J. High Energy Phys.0601, 164 (2006). arXiv:hep-ph/0601084

298. A. Abada, S. Davidson, A. Ibarra, F.X. Josse-Michaux, M.Losada, A. Riotto, J. High Energy Phys. 0609, 010 (2006).arXiv:hep-ph/0605281

299. T. Fujihara, S. Kaneko, S. Kang, D. Kimura, T. Morozumi,M. Tanimoto, Phys. Rev. D 72, 016006 (2005). arXiv:hep-ph/0505076

300. S. Pascoli, S.T. Petcov, A. Riotto, arXiv:hep-ph/0609125301. G.C. Branco, A.J. Buras, S. Jager, S. Uhlig, A. Weiler,

arXiv:hep-ph/0609067302. G.C. Branco, R. Gonzalez Felipe, F.R. Joaquim, arXiv:hep-ph/

0609297303. S. Uhlig, arXiv:hep-ph/0612262304. M. Plumacher, Z. Phys. C 74, 549 (1997). arXiv:hep-ph/

9604229305. G. Engelhard, Y. Grossman, E. Nardi, Y. Nir, arXiv:hep-ph/

0612187306. S. Davidson, A. Ibarra, Phys. Lett. B 535, 25 (2002). arXiv:

hep-ph/0202239307. K. Hamaguchi, H. Murayama, T. Yanagida, Phys. Rev. D 65,

043512 (2002). arXiv:hep-ph/0109030308. W. Buchmuller, P. Di Bari, M. Plumacher, Nucl. Phys. B 643,

367 (2002). arXiv:hep-ph/0205349309. G.F. Giudice, A. Notari, M. Raidal, A. Riotto, A. Strumia, Nucl.

Phys. B 685, 89 (2004). arXiv:hep-ph/0310123310. S. Blanchet, P. Di Bari, arXiv:hep-ph/0607330311. S. Antusch, A.M. Teixeira, arXiv:hep-ph/0611232312. J.R. Ellis, J.E. Kim, D.V. Nanopoulos, Phys. Lett. B 145, 181

(1984)313. J.R. Ellis, D.V. Nanopoulos, S. Sarkar, Nucl. Phys. B 259, 175

(1985)314. J.R. Ellis, D.V. Nanopoulos, K.A. Olive, S.J. Rey, Astropart.

Phys. 4, 371 (1996). arXiv:hep-ph/9505438315. T. Moroi, arXiv:hep-ph/9503210316. M. Raidal, A. Strumia, K. Turzynski, Phys. Lett. B 609, 351

(2005) [Erratum: Phys. Lett. B 632, 752 (2006)]. arXiv:hep-ph/0408015

317. S. Blanchet, P. Di Bari, G.G. Raffelt, arXiv:hep-ph/0611337318. W. Buchmuller, P. Di Bari, M. Plumacher, Phys. Lett. B 547,

128 (2002). arXiv:hep-ph/0209301319. S. Davidson, arXiv:hep-ph/0304120320. E.W. Kolb, M.S. Turner, Front. Phys. 69, 1 (1990)321. W. Fischler, G.F. Giudice, R.G. Leigh, S. Paban, Phys. Lett. B

258, 45 (1991)

172 Eur. Phys. J. C (2008) 57: 13–182

322. W. Buchmuller, T. Yanagida, Phys. Lett. B 302, 240 (1993)323. W. Buchmuller, P. Di Bari, M. Plumacher, Nucl. Phys. B 665,

445 (2003). arXiv:hep-ph/0302092324. W. Buchmuller, P. Di Bari, M. Plumacher, Ann. Phys. 315, 305

(2005). arXiv:hep-ph/0401240325. W. Buchmuller, P. Di Bari, M. Plumacher, New J. Phys. 6, 105

(2004). arXiv:hep-ph/0406014326. T. Hambye, Y. Lin, A. Notari, M. Papucci, A. Strumia, Nucl.

Phys. B 695, 169 (2004). arXiv:hep-ph/0312203327. WMAP Collaboration, D.N. Spergel et al., arXiv:astro-ph/

0603449328. G.C. Branco, T. Morozumi, B.M. Nobre, M.N. Rebelo, Nucl.

Phys. B 617, 475 (2001). arXiv:hep-ph/0107164329. J.R. Ellis, M. Raidal, Nucl. Phys. B 643, 229 (2002).

arXiv:hep-ph/0206174330. S. Davidson, J. Garayoa, F. Palorini, N. Rius, arXiv:0705.1503

[hep-ph]331. S. Antusch, S.F. King, A. Riotto, J. Cosmol. Astropart. Phys.

0611, 011 (2006). arXiv:hep-ph/0609038332. A. Pilaftsis, Phys. Rev. D 56, 5431 (1997). arXiv:hep-ph/

9707235333. A. Pilaftsis, Nucl. Phys. B 504, 61 (1997). arXiv:hep-ph/

9702393334. J.R. Ellis, M. Raidal, T. Yanagida, Phys. Lett. B 546, 228

(2002). arXiv:hep-ph/0206300335. A. Pilaftsis, T.E.J. Underwood, Nucl. Phys. B 692, 303 (2004).

arXiv:hep-ph/0309342336. F. del Aguila, J.A. Aguilar-Saavedra, R. Pittau, J. Phys. Conf.

Ser. 53, 506 (2006). arXiv:hep-ph/0606198337. A. Pilaftsis, T.E.J. Underwood, Phys. Rev. D 72, 113001 (2005).

arXiv:hep-ph/0506107338. W. Buchmuller, C. Greub, Nucl. Phys. B 381, 109 (1992)339. J. Gluza, Phys. Rev. D 66, 010001 (2002). arXiv:hep-ph/

0201002340. T. Hambye, G. Senjanovic, Phys. Lett. B 582, 73 (2004).

arXiv:hep-ph/0307237341. T. Hambye, M. Raidal, A. Strumia, Phys. Lett. B 632, 667

(2006). arXiv:hep-ph/0510008342. S. Antusch, S.F. King, Phys. Lett. B 597, 199 (2004).

arXiv:hep-ph/0405093343. S. Antusch, Phys. Rev. D 76, 023512 (2007). arXiv:0704.1591

[hep-ph]344. Y. Grossman, T. Kashti, Y. Nir, E. Roulet, Phys. Rev. Lett. 91,

251801 (2003). arXiv:hep-ph/0307081345. G. D’Ambrosio, G.F. Giudice, M. Raidal, Phys. Lett. B 575, 75

(2003). arXiv:hep-ph/0308031346. G. D’Ambrosio, T. Hambye, A. Hektor, M. Raidal, A. Rossi,

Phys. Lett. B 604, 199 (2004). arXiv:hep-ph/0407312347. H. Murayama, H. Suzuki, T. Yanagida, J. Yokoyama, Phys. Rev.

Lett. 70, 1912 (1993)348. H. Murayama, T. Yanagida, Phys. Lett. B 322, 349 (1994).

arXiv:hep-ph/9310297349. J.R. Ellis, M. Raidal, T. Yanagida, Phys. Lett. B 581, 9 (2004).

arXiv:hep-ph/0303242350. S. Antusch, M. Bastero-Gil, S.F. King, Q. Shafi, Phys. Rev. D

71, 083519 (2005). arXiv:hep-ph/0411298351. K. Dick, M. Lindner, M. Ratz, D. Wright, Phys. Rev. Lett. 84,

4039 (2000). arXiv:hep-ph/9907562352. D. Atwood, S. Bar-Shalom, A. Soni, Phys. Lett. B 635, 112

(2006). arXiv:hep-ph/0502234353. T. Asaka, K. Hamaguchi, M. Kawasaki, T. Yanagida, Phys. Lett.

B 464, 12 (1999). arXiv:hep-ph/9906366354. G.F. Giudice, M. Peloso, A. Riotto, I. Tkachev, J. High Energy

Phys. 9908, 014 (1999). arXiv:hep-ph/9905242355. M. Endo, F. Takahashi, T.T. Yanagida, Phys. Rev. D 74, 123523

(2006). arXiv:hep-ph/0611055

356. J.S. Schwinger, Ann. Phys. 2, 407 (1957)357. H. Georgi, S.L. Glashow, Phys. Rev. Lett. 32, 438 (1974)358. P.A.M. Dirac, Phys. Rev. 74, 817 (1948)359. G. ’t Hooft, Nucl. Phys. B 79, 276 (1974)360. A.M. Polyakov, JETP Lett. 20, 194 (1974) [Pisma Zh. Eksp.

Teor. Fiz. 20, 430 (1974)]361. J.C. Pati, A. Salam, Phys. Rev. D 10, 275 (1974)362. F. Wilczek, A. Zee, Phys. Rev. Lett. 43, 1571 (1979)363. K. Kobayashi et al. (Super-Kamiokande Collaboration), Phys.

Rev. D 72, 052007 (2005). arXiv:hep-ex/0502026364. A.J. Buras, J.R. Ellis, M.K. Gaillard, D.V. Nanopoulos, Nucl.

Phys. B 135, 66 (1978)365. S. Dimopoulos, S. Raby, F. Wilczek, Phys. Rev. D 24, 1681

(1981)366. L.E. Ibanez, G.G. Ross, Phys. Lett. B 105, 439 (1981)367. M.B. Einhorn, D.R.T. Jones, Nucl. Phys. B 196, 475 (1982)368. W.J. Marciano, G. Senjanovic, Phys. Rev. D 25, 3092 (1982)369. K. Inoue, A. Kakuto, H. Komatsu, S. Takeshita, Prog. Theor.

Phys. 68, 927 (1982) [Erratum: Prog. Theor. Phys. 70, 330(1983)]

370. L. Alvarez-Gaume, J. Polchinski, M.B. Wise, Nucl. Phys. B221, 495 (1983)

371. N. Sakai, T. Yanagida, Nucl. Phys. B 197, 533 (1982)372. J. Hisano, H. Murayama, T. Yanagida, Nucl. Phys. B 402, 46

(1993). arXiv:hep-ph/9207279373. V. Lucas, S. Raby, Phys. Rev. D 55, 6986 (1997). arXiv:

hep-ph/9610293374. T. Goto, T. Nihei, Phys. Rev. D 59, 115009 (1999). arXiv:

hep-ph/9808255375. H. Murayama, A. Pierce, Phys. Rev. D 65, 055009 (2002).

arXiv:hep-ph/0108104376. B. Bajc, P. Fileviez Perez, G. Senjanovic, Phys. Rev. D 66,

075005 (2002). arXiv:hep-ph/0204311377. B. Bajc, P. Fileviez Perez, G. Senjanovic, arXiv:hep-ph/

0210374378. D. Emmanuel-Costa, S. Wiesenfeldt, Nucl. Phys. B 661, 62

(2003). arXiv:hep-ph/0302272379. I. Dorsner, P.F. Perez, Nucl. Phys. B 723, 53 (2005).

arXiv:hep-ph/0504276380. B. Bajc, G. Senjanovic, arXiv:hep-ph/0612029381. R.J. Wilkes, arXiv:hep-ex/0507097382. K. Cheung, C.W. Chiang, Phys. Rev. D 71, 095003 (2005).

arXiv:hep-ph/0501265383. W.Y. Keung, G. Senjanovic, Phys. Rev. Lett. 50, 1427 (1983)384. R.N. Mohapatra, J.C. Pati, Phys. Rev. D 11, 2558 (1975)385. G. Senjanovic, R.N. Mohapatra, Phys. Rev. D 12, 1502 (1975)386. G. Senjanovic, Nucl. Phys. B 153, 334 (1979)387. R.N. Mohapatra, Phys. Rev. D 34, 3457 (1986)388. A. Font, L.E. Ibanez, F. Quevedo, Phys. Lett. B 228, 79 (1989)389. S.P. Martin, Phys. Rev. D 46, 2769 (1992). arXiv:hep-ph/

9207218390. C.S. Aulakh, K. Benakli, G. Senjanovic, Phys. Rev. Lett. 79,

2188 (1997). arXiv:hep-ph/9703434391. C.S. Aulakh, A. Melfo, G. Senjanovic, Phys. Rev. D 57, 4174

(1998). arXiv:hep-ph/9707256392. C.S. Aulakh, A. Melfo, A. Rasin, G. Senjanovic, Phys. Lett. B

459, 557 (1999). arXiv:hep-ph/9902409393. N.G. Deshpande, E. Keith, P.B. Pal, Phys. Rev. D 46, 2261

(1993)394. N.G. Deshpande, E. Keith, P.B. Pal, Phys. Rev. D 47, 2892

(1993). arXiv:hep-ph/9211232395. R.N. Mohapatra, B. Sakita, Phys. Rev. D 21, 1062 (1980)396. F. Wilczek, A. Zee, Phys. Rev. D 25, 553 (1982)397. R. Slansky, Phys. Rep. 79, 1 (1981)398. P. Nath, R.M. Syed, Nucl. Phys. B 618, 138 (2001).

arXiv:hep-th/0109116

Eur. Phys. J. C (2008) 57: 13–182 173

399. C.S. Aulakh, A. Girdhar, Int. J. Mod. Phys. A 20, 865 (2005).arXiv:hep-ph/0204097

400. C.H. Albright, S.M. Barr, Phys. Rev. Lett. 85, 244 (2000).arXiv:hep-ph/0002155

401. K.S. Babu, J.C. Pati, F. Wilczek, Nucl. Phys. B 566, 33 (2000).arXiv:hep-ph/9812538

402. T. Blazek, S. Raby, K. Tobe, Phys. Rev. D 60, 113001 (1999).arXiv:hep-ph/9903340

403. E. Witten, Phys. Lett. B 91, 81 (1980)404. B. Bajc, G. Senjanovic, Phys. Lett. B 610, 80 (2005).

arXiv:hep-ph/0411193405. B. Bajc, G. Senjanovic, Phys. Rev. Lett. 95, 261804 (2005).

arXiv:hep-ph/0507169406. K.S. Babu, R.N. Mohapatra, Phys. Rev. Lett. 70, 2845 (1993).

arXiv:hep-ph/9209215407. H. Fusaoka, Y. Koide, Phys. Rev. D 57, 3986 (1998).

arXiv:hep-ph/9712201408. C.R. Das, M.K. Parida, Eur. Phys. J. C 20, 121 (2001).

arXiv:hep-ph/0010004409. K. Matsuda, Y. Koide, T. Fukuyama, Phys. Rev. D 64, 053015

(2001). arXiv:hep-ph/0010026410. K. Matsuda, Y. Koide, T. Fukuyama, H. Nishiura, Phys. Rev. D

65, 033008 (2002) [Erratum: Phys. Rev. D 65, 079904 (2002)].arXiv:hep-ph/0108202

411. T. Fukuyama, N. Okada, J. High Energy Phys. 0211, 011(2002). arXiv:hep-ph/0205066

412. H.S. Goh, R.N. Mohapatra, S. Nasri, Phys. Rev. D 70, 075022(2004). arXiv:hep-ph/0408139

413. B. Bajc, G. Senjanovic, F. Vissani, arXiv:hep-ph/0110310414. B. Bajc, G. Senjanovic, F. Vissani, Phys. Rev. Lett. 90, 051802

(2003). arXiv:hep-ph/0210207415. B. Bajc, G. Senjanovic, F. Vissani, Phys. Rev. D 70, 093002

(2004). arXiv:hep-ph/0402140416. B. Brahmachari, R.N. Mohapatra, Phys. Rev. D 58, 015001

(1998). arXiv:hep-ph/9710371417. H.S. Goh, R.N. Mohapatra, S.P. Ng, Phys. Lett. B 570, 215

(2003). arXiv:hep-ph/0303055418. H.S. Goh, R.N. Mohapatra, S.P. Ng, Phys. Rev. D 68, 115008

(2003). arXiv:hep-ph/0308197419. B. Dutta, Y. Mimura, R.N. Mohapatra, Phys. Rev. D 69, 115014

(2004). arXiv:hep-ph/0402113420. S. Bertolini, M. Malinsky, Phys. Rev. D 72, 055021 (2005).

arXiv:hep-ph/0504241421. K.S. Babu, C. Macesanu, Phys. Rev. D 72, 115003 (2005).

arXiv:hep-ph/0505200422. T.E. Clark, T.K. Kuo, N. Nakagawa, Phys. Lett. B 115, 26

(1982)423. C.S. Aulakh, R.N. Mohapatra, Phys. Rev. D 28, 217 (1983)424. D.G. Lee, R.N. Mohapatra, Phys. Rev. D 51, 1353 (1995).

arXiv:hep-ph/9406328425. C.S. Aulakh, B. Bajc, A. Melfo, G. Senjanovic, F. Vissani, Phys.

Lett. B 588, 196 (2004). arXiv:hep-ph/0306242426. C.S. Aulakh, arXiv:hep-ph/0506291427. B. Bajc, A. Melfo, G. Senjanovic, F. Vissani, Phys. Lett. B 634,

272 (2006). arXiv:hep-ph/0511352428. C.S. Aulakh, S.K. Garg, arXiv:hep-ph/0512224429. S. Bertolini, T. Schwetz, M. Malinsky, Phys. Rev. D 73, 115012

(2006). arXiv:hep-ph/0605006430. L.J. Hall, R. Rattazzi, U. Sarid, Phys. Rev. D 50, 7048 (1994).

arXiv:hep-ph/9306309431. X.G. He, S. Meljanac, Phys. Rev. D 41, 1620 (1990)432. D.G. Lee, Phys. Rev. D 49, 1417 (1994)433. B. Bajc, A. Melfo, G. Senjanovic, F. Vissani, Phys. Rev. D 70,

035007 (2004). arXiv:hep-ph/0402122434. C.S. Aulakh, A. Girdhar, Nucl. Phys. B 711, 275 (2005).

arXiv:hep-ph/0405074

435. T. Fukuyama, A. Ilakovac, T. Kikuchi, S. Meljanac, N. Okada,Eur. Phys. J. C 42, 191 (2005). arXiv:hep-ph/0401213

436. T. Fukuyama, A. Ilakovac, T. Kikuchi, S. Meljanac, N. Okada,J. Math. Phys. 46, 033505 (2005). arXiv:hep-ph/0405300

437. T. Fukuyama, A. Ilakovac, T. Kikuchi, S. Meljanac, N. Okada,Phys. Rev. D 72, 051701 (2005). arXiv:hep-ph/0412348

438. C.S. Aulakh, arXiv:hep-ph/0501025439. C.S. Aulakh, Phys. Rev. D 72, 051702 (2005)440. D.G. Lee, R.N. Mohapatra, Phys. Lett. B 324, 376 (1994).

arXiv:hep-ph/9310371441. K.S. Babu, I. Gogoladze, Z. Tavartkiladze, arXiv:hep-ph/

0612315442. H.S. Goh, R.N. Mohapatra, S. Nasri, S.P. Ng, Phys. Lett. B 587,

105 (2004). arXiv:hep-ph/0311330443. T. Fukuyama, A. Ilakovac, T. Kikuchi, S. Meljanac, N. Okada,

J. High Energy Phys. 0409, 052 (2004). arXiv:hep-ph/0406068444. S. Bertolini, M. Frigerio, M. Malinsky, Phys. Rev. D 70, 095002

(2004). arXiv:hep-ph/0406117445. W.M. Yang, Z.G. Wang, Nucl. Phys. B 707, 87 (2005).

arXiv:hep-ph/0406221446. B. Dutta, Y. Mimura, R.N. Mohapatra, Phys. Lett. B 603, 35

(2004). arXiv:hep-ph/0406262447. B. Dutta, Y. Mimura, R.N. Mohapatra, Phys. Rev. Lett. 94,

091804 (2005). arXiv:hep-ph/0412105448. B. Dutta, Y. Mimura, R.N. Mohapatra, Phys. Rev. D 72, 075009

(2005). arXiv:hep-ph/0507319449. W. Grimus, H. Kuhbock, arXiv:hep-ph/0607197450. W. Grimus, H. Kuhbock, arXiv:hep-ph/0612132451. C.S. Aulakh, arXiv:hep-ph/0602132452. L. Lavoura, H. Kuhbock, W. Grimus, arXiv:hep-ph/0603259453. C.S. Aulakh, arXiv:hep-ph/0607252454. C.S. Aulakh, S.K. Garg, arXiv:hep-ph/0612021455. B. Bajc, A. Melfo, G. Senjanovic, F. Vissani, Phys. Rev. D 73,

055001 (2006). arXiv:hep-ph/0510139456. F. Vissani, A.Y. Smirnov, Phys. Lett. B 341, 173 (1994).

arXiv:hep-ph/9405399457. N. Arkani-Hamed, S. Dimopoulos, J. High Energy Phys. 0506,

073 (2005). arXiv:hep-th/0405159458. G.F. Giudice, A. Romanino, Nucl. Phys. B 699, 65 (2004) [Er-

ratum: Nucl. Phys. B 706, 65 (2005)]. arXiv:hep-ph/0406088459. B. Bajc, AIP Conf. Proc. 805, 326 (2006). arXiv:hep-ph/

0602166460. A. Arvanitaki, C. Davis, P.W. Graham, A. Pierce, J.G. Wacker,

Phys. Rev. D 72, 075011 (2005). arXiv:hep-ph/0504210461. K. Yoshioka, Mod. Phys. Lett. A 15, 29 (2000). arXiv:

hep-ph/9904433462. M. Bando, T. Kobayashi, T. Noguchi, K. Yoshioka, Phys. Rev.

D 63, 113017 (2001). arXiv:hep-ph/0008120463. A. Neronov, Phys. Rev. D 65, 044004 (2002). arXiv:gr-qc/

0106092464. N. Arkani-Hamed, L.J. Hall, D.R. Smith, N. Weiner, Phys. Rev.

D 61, 116003 (2000). arXiv:hep-ph/9909326465. K.R. Dienes, E. Dudas, T. Gherghetta, Phys. Lett. B 436, 55

(1998). arXiv:hep-ph/9803466466. K.R. Dienes, E. Dudas, T. Gherghetta, Nucl. Phys. B 537, 47

(1999). arXiv:hep-ph/9806292467. N. Arkani-Hamed, M. Schmaltz, Phys. Rev. D 61, 033005

(2000). arXiv:hep-ph/9903417468. T. Gherghetta, A. Pomarol, Nucl. Phys. B 586, 141 (2000).

arXiv:hep-ph/0003129469. N. Arkani-Hamed, S. Dimopoulos, G.R. Dvali, Phys. Lett. B

429, 263 (1998). arXiv:hep-ph/9803315470. I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos, G.R. Dvali,

Phys. Lett. B 436, 257 (1998). arXiv:hep-ph/9804398471. N. Arkani-Hamed, S. Dimopoulos, G.R. Dvali, Phys. Rev. D 59,

086004 (1999). arXiv:hep-ph/9807344

174 Eur. Phys. J. C (2008) 57: 13–182

472. G. Barenboim, G.C. Branco, A. de Gouvea, M.N. Rebelo, Phys.Rev. D 64, 073005 (2001). arXiv:hep-ph/0104312

473. T. Appelquist, H.C. Cheng, B.A. Dobrescu, Phys. Rev. D 64,035002 (2001). arXiv:hep-ph/0012100

474. M.V. Libanov, S.V. Troitsky, Nucl. Phys. B 599, 319 (2001).arXiv:hep-ph/0011095

475. J.M. Frere, M.V. Libanov, S.V. Troitsky, Phys. Lett. B 512, 169(2001). arXiv:hep-ph/0012306

476. J.M. Frere, M.V. Libanov, S.V. Troitsky, J. High Energy Phys.0111, 025 (2001). arXiv:hep-ph/0110045

477. M.V. Libanov, E.Y. Nougaev, J. High Energy Phys. 0204, 055(2002). arXiv:hep-ph/0201162

478. G.R. Dvali, M.A. Shifman, Phys. Lett. B 475, 295 (2000).arXiv:hep-ph/0001072

479. P.Q. Hung, Phys. Rev. D 67, 095011 (2003). arXiv:hep-ph/0210131

480. D.E. Kaplan, T.M.P. Tait, J. High Energy Phys. 0006, 020(2000). arXiv:hep-ph/0004200

481. D.E. Kaplan, T.M.P. Tait, J. High Energy Phys. 0111, 051(2001). arXiv:hep-ph/0110126

482. M. Kakizaki, M. Yamaguchi, Prog. Theor. Phys. 107, 433(2002). arXiv:hep-ph/0104103

483. C.H. Chang, W.F. Chang, J.N. Ng, Phys. Lett. B 558, 92 (2003).arXiv:hep-ph/0301271

484. S. Nussinov, R. Shrock, Phys. Lett. B 526, 137 (2002).arXiv:hep-ph/0101340

485. R. Jackiw, C. Rebbi, Phys. Rev. D 13, 3398 (1976)486. E.A. Mirabelli, M. Schmaltz, Phys. Rev. D 61, 113011 (2000).

arXiv:hep-ph/9912265487. G.C. Branco, A. de Gouvea, M.N. Rebelo, Phys. Lett. B 506,

115 (2001). arXiv:hep-ph/0012289488. P.Q. Hung, M. Seco, Nucl. Phys. B 653, 123 (2003).

arXiv:hep-ph/0111013489. J.M. Frere, G. Moreau, E. Nezri, Phys. Rev. D 69, 033003

(2004). arXiv:hep-ph/0309218490. H.V. Klapdor-Kleingrothaus, U. Sarkar, Phys. Lett. B 541, 332

(2002). arXiv:hep-ph/0201226491. M. Gogberashvili, Int. J. Mod. Phys. D 11, 1635 (2002).

arXiv:hep-ph/9812296492. L. Randall, R. Sundrum, Phys. Rev. Lett. 83, 3370 (1999).

arXiv:hep-ph/9905221493. Y. Grossman, M. Neubert, Phys. Lett. B 474, 361 (2000).

arXiv:hep-ph/9912408494. H. Davoudiasl, J.L. Hewett, T.G. Rizzo, Phys. Lett. B 473, 43

(2000). arXiv:hep-ph/9911262495. A. Pomarol, Phys. Lett. B 486, 153 (2000). arXiv:hep-ph/

9911294496. S. Chang, J. Hisano, H. Nakano, N. Okada, M. Yamaguchi,

Phys. Rev. D 62, 084025 (2000). arXiv:hep-ph/9912498497. B. Bajc, G. Gabadadze, Phys. Lett. B 474, 282 (2000).

arXiv:hep-th/9912232498. A. Pomarol, Phys. Rev. Lett. 85, 4004 (2000). arXiv:hep-ph/

0005293499. L. Randall, M.D. Schwartz, J. High Energy Phys. 0111, 003

(2001). arXiv:hep-th/0108114500. L. Randall, M.D. Schwartz, Phys. Rev. Lett. 88, 081801 (2002).

arXiv:hep-th/0108115501. W.D. Goldberger, I.Z. Rothstein, Phys. Rev. D 68, 125011

(2003). arXiv:hep-th/0208060502. K.W. Choi, I.W. Kim, Phys. Rev. D 67, 045005 (2003).

arXiv:hep-th/0208071503. K. Agashe, A. Delgado, R. Sundrum, Ann. Phys. 304, 145

(2003). arXiv:hep-ph/0212028504. K. Agashe, G. Servant, Phys. Rev. Lett. 93, 231805 (2004).

arXiv:hep-ph/0403143505. K. Agashe, G. Servant, J. Cosmol. Astropart. Phys. 0502, 002

(2005). arXiv:hep-ph/0411254

506. S.J. Huber, Q. Shafi, Phys. Lett. B 498, 256 (2001). arXiv:hep-ph/0010195

507. S.J. Huber, Nucl. Phys. B 666, 269 (2003). arXiv:hep-ph/0303183

508. S. Chang, C.S. Kim, M. Yamaguchi, Phys. Rev. D 73, 033002(2006). arXiv:hep-ph/0511099

509. S.J. Huber, Q. Shafi, Phys. Lett. B 544, 295 (2002). arXiv:hep-ph/0205327

510. S.J. Huber, Q. Shafi, Phys. Lett. B 583, 293 (2004). arXiv:hep-ph/0309252

511. S.J. Huber, Q. Shafi, Phys. Lett. B 512, 365 (2001).arXiv:hep-ph/0104293

512. G. Moreau, J.I. Silva-Marcos, J. High Energy Phys. 0601, 048(2006). arXiv:hep-ph/0507145

513. G. Moreau, J.I. Silva-Marcos, J. High Energy Phys. 0603, 090(2006). arXiv:hep-ph/0602155

514. A. Ilakovac, A. Pilaftsis, Nucl. Phys. B 437, 491 (1995). arXiv:hep-ph/9403398

515. C.S. Kim, J.D. Kim, J.h. Song, Phys. Rev. D 67, 015001 (2003).arXiv:hep-ph/0204002

516. F. del Aguila, M. Perez-Victoria, J. Santiago, Phys. Lett. B 492,98 (2000). arXiv:hep-ph/0007160

517. A. Delgado, A. Pomarol, M. Quiros, J. High Energy Phys. 0001,030 (2000). arXiv:hep-ph/9911252

518. T.G. Rizzo, J.D. Wells, Phys. Rev. D 61, 016007 (2000).arXiv:hep-ph/9906234

519. I. Antoniadis, K. Benakli, M. Quiros, Phys. Lett. B 460, 176(1999). arXiv:hep-ph/9905311

520. P. Nath, Y. Yamada, M. Yamaguchi, Phys. Lett. B 466, 100(1999). arXiv:hep-ph/9905415

521. T.G. Rizzo, Phys. Rev. D 61, 055005 (2000). arXiv:hep-ph/9909232

522. G. Burdman, Phys. Lett. B 590, 86 (2004). arXiv:hep-ph/0310144

523. K. Agashe, G. Perez, A. Soni, Phys. Rev. Lett. 93, 201804(2004). arXiv:hep-ph/0406101

524. K. Agashe, G. Perez, A. Soni, Phys. Rev. D 71, 016002 (2005).arXiv:hep-ph/0408134

525. K. Agashe, A.E. Blechman, F. Petriello, Phys. Rev. D 74,053011 (2006). arXiv:hep-ph/0606021

526. K. Agashe, G. Perez, A. Soni, arXiv:hep-ph/0606293527. F. del Aguila, M. Perez-Victoria, J. Santiago, J. High Energy

Phys. 0302, 051 (2003). arXiv:hep-th/0302023528. M. Carena, T.M.P. Tait, C.E.M. Wagner, Acta Phys. Pol. B 33,

2355 (2002). arXiv:hep-ph/0207056529. M. Carena, E. Ponton, T.M.P. Tait, C.E.M. Wagner, Phys. Rev.

D 67, 096006 (2003). arXiv:hep-ph/0212307530. M. Carena, A. Delgado, E. Ponton, T.M.P. Tait, C.E.M. Wagner,

Phys. Rev. D 68, 035010 (2003). arXiv:hep-ph/0305188531. M. Carena, A. Delgado, E. Ponton, T.M.P. Tait, C.E.M. Wagner,

Phys. Rev. D 71, 015010 (2005). arXiv:hep-ph/0410344532. K. Agashe, A. Delgado, M.J. May, R. Sundrum, J. High Energy

Phys. 0308, 050 (2003). arXiv:hep-ph/0308036533. K. Agashe, R. Contino, L. Da Rold, A. Pomarol, Phys. Lett. B

641, 62 (2006). arXiv:hep-ph/0605341534. A. Djouadi, G. Moreau, F. Richard, arXiv:hep-ph/0610173535. R.S. Chivukula, H. Georgi, Phys. Lett. B 188, 99 (1987)536. L.J. Hall, L. Randall, Phys. Rev. Lett. 65, 2939 (1990)537. G. D’Ambrosio, G.F. Giudice, G. Isidori, A. Strumia, Nucl.

Phys. B 645, 155 (2002). hep-ph/0207036538. A.V. Manohar, M.B. Wise, Phys. Rev. D 74, 035009 (2006).

hep-ph/0606172539. B. Grinstein, V. Cirigliano, G. Isidori, M.B. Wise, Nucl. Phys.

B 763, 35 (2007). hep-ph/0608123540. A.J. Buras, P. Gambino, M. Gorbahn, S. Jager, L. Silvestrini,

Phys. Lett. B 500, 161 (2001). hep-ph/0007085

Eur. Phys. J. C (2008) 57: 13–182 175

541. M. Bona et al. (UTfit Collaboration), J. High Energy Phys.0603, 080 (2006). hep-ph/0509219

542. V. Cirigliano, B. Grinstein, G. Isidori, M.B. Wise, Nucl. Phys.B 728, 121 (2005). hep-ph/0507001

543. V. Cirigliano, B. Grinstein, Nucl. Phys. B 752, 18 (2006).hep-ph/0601111

544. S. Davidson, F. Palorini, Phys. Lett. B 642, 72 (2006).hep-ph/0607329

545. S. Pascoli, S.T. Petcov, C.E. Yaguna, Phys. Lett. B 564, 241(2003). hep-ph/0301095

546. V. Cirigliano, G. Isidori, V. Porretti, Nucl. Phys. B 763, 228(2007). hep-ph/0608123

547. S. Blanchet, P. Di Bari, J. Cosmol. Astropart. Phys. 0606, 023(2006). hep-ph/0603107

548. M. Grassi (MEG Collaboration), Nucl. Phys. Proc. Suppl. 149,369 (2005)

549. W. Grimus, M.N. Rebelo, Phys. Rep. 281, 239 (1997).arXiv:hep-ph/9506272

550. F. Englert, R. Brout, Phys. Rev. Lett. 13, 321 (1964)551. G.S. Guralnik, C.R. Hagen, T.W.B. Kibble, Phys. Rev. Lett. 13,

585 (1964)552. P.W. Higgs, Phys. Lett. 12, 132 (1964)553. P.W. Higgs, Phys. Rev. 145, 1156 (1966)554. T.D. Lee, Phys. Rev. D 8, 1226 (1973)555. W. Bernreuther, O. Nachtmann, Eur. Phys. J. C 9, 319 (1999).

arXiv:hep-ph/9812259556. G.C. Branco, L. Lavoura, J.P. Silva, CP Violation (Clarendon,

Oxford, 1999)557. G.C. Branco, M.N. Rebelo, J.I. Silva-Marcos, Phys. Lett. B 614,

187 (2005). arXiv:hep-ph/0502118558. J. Bernabeu, G.C. Branco, M. Gronau, Phys. Lett. B 169, 243

(1986)559. L. Lavoura, J.P. Silva, Phys. Rev. D 50, 4619 (1994).

arXiv:hep-ph/9404276560. F.J. Botella, J.P. Silva, Phys. Rev. D 51, 3870 (1995).

arXiv:hep-ph/9411288561. S. Davidson, H.E. Haber, Phys. Rev. D 72, 035004 (2005) [Erra-

tum: Phys. Rev. D 72, 099902 (2005)]. arXiv:hep-ph/0504050562. J.F. Gunion, H.E. Haber, Phys. Rev. D 72, 095002 (2005).

arXiv:hep-ph/0506227563. I.P. Ivanov, Phys. Lett. B 632, 360 (2006). arXiv:

hep-ph/0507132564. H.E. Haber, D. O’Neil, Phys. Rev. D 74, 015018 (2006).

arXiv:hep-ph/0602242565. G.C. Branco, J.M. Gerard, W. Grimus, Phys. Lett. B 136, 383

(1984)566. S.L. Glashow, S. Weinberg, Phys. Rev. D 15, 1958 (1977)567. E.A. Paschos, Phys. Rev. D 15, 1966 (1977)568. G.C. Branco, M.N. Rebelo, Phys. Lett. B 160, 117 (1985)569. L. Lavoura, Phys. Rev. D 50, 7089 (1994). arXiv:hep-ph/

9405307570. I.F. Ginzburg, M. Krawczyk, Phys. Rev. D 72, 115013 (2005).

arXiv:hep-ph/0408011571. J.F. Gunion, H.E. Haber, Phys. Rev. D 67, 075019 (2003).

arXiv:hep-ph/0207010572. S. Weinberg, Phys. Rev. Lett. 37, 657 (1976)573. G.C. Branco, Phys. Rev. Lett. 44, 504 (1980)574. G.C. Branco, Phys. Rev. D 22, 2901 (1980)575. G.C. Branco, A.J. Buras, J.M. Gerard, Nucl. Phys. B 259, 306

(1985)576. L. Bento, G.C. Branco, P.A. Parada, Phys. Lett. B 267, 95

(1991)577. A.E. Nelson, Phys. Lett. B 136, 387 (1984)578. A.E. Nelson, Phys. Lett. B 143, 165 (1984)579. S.M. Barr, Phys. Rev. Lett. 53, 329 (1984)580. G.C. Branco, P.A. Parada, M.N. Rebelo, arXiv:hep-ph/0307119

581. Y. Achiman, Phys. Lett. B 599, 75 (2004). arXiv:hep-ph/0403309

582. M.B. Gavela, P. Hernandez, J. Orloff, O. Pene, C. Quimbay,Nucl. Phys. B 430, 382 (1994). arXiv:hep-ph/9406289

583. P. Huet, E. Sather, Phys. Rev. D 51, 379 (1995). arXiv:hep-ph/9404302

584. G.W. Anderson, L.J. Hall, Phys. Rev. D 45, 2685 (1992)585. W. Buchmuller, Z. Fodor, T. Helbig, D. Walliser, Ann. Phys.

234, 260 (1994). arXiv:hep-ph/9303251586. K. Kajantie, M. Laine, K. Rummukainen, M.E. Shaposhnikov,

Nucl. Phys. B 466, 189 (1996). arXiv:hep-lat/9510020587. L. Fromme, S.J. Huber, M. Seniuch, J. High Energy Phys. 0611,

038 (2006). arXiv:hep-ph/0605242588. E. Accomando et al., arXiv:hep-ph/0608079589. L. Brucher, R. Santos, Eur. Phys. J. C 12, 87 (2000).

arXiv:hep-ph/9907434590. C. Delaere (LEP Collaboration), PoS HEP2005, 331 (2006)591. A.G. Akeroyd, M.A. Diaz, F.J. Pacheco, Phys. Rev. D 70,

075002 (2004). arXiv:hep-ph/0312231592. N.G. Deshpande, E. Ma, Phys. Rev. D 18, 2574 (1978)593. Q.H. Cao, E. Ma, G. Rajasekaran, arXiv:0708.2939 [hep-ph]594. R. Barbieri, L.J. Hall, V.S. Rychkov, Phys. Rev. D 74, 015007

(2006). arXiv:hep-ph/0603188595. I.F. Ginzburg, M. Krawczyk, P. Osland, Nucl. Instrum. Methods

A 472, 149 (2001). arXiv:hep-ph/0101229596. T.P. Cheng, M. Sher, Phys. Rev. D 35, 3484 (1987)597. G.C. Branco, W. Grimus, L. Lavoura, Phys. Lett. B 380, 119

(1996). arXiv:hep-ph/9601383598. A.K. Das, C. Kao, Phys. Lett. B 372, 106 (1996). arXiv:

hep-ph/9511329599. E. Lunghi, A. Soni, arXiv:0707.0212 [hep-ph]600. T.P. Cheng, L.F. Li, Phys. Rev. Lett. 38, 381 (1977)601. S.M. Bilenky, S.T. Petcov, B. Pontecorvo, Phys. Lett. B 67, 309

(1977)602. J.D. Bjorken, S. Weinberg, Phys. Rev. Lett. 38, 622 (1977)603. G.C. Branco, Phys. Lett. B 68, 455 (1977)604. D. Besson et al. (CLEO Collaboration), Phys. Rev. Lett. 98,

052002 (2007). arXiv:hep-ex/0607019605. R.N. Mohapatra, J.W.F. Valle, Phys. Rev. D 34, 1642 (1986)606. S. Nandi, U. Sarkar, Phys. Rev. Lett. 56, 564 (1986)607. P.F. Harrison, D.H. Perkins, W.G. Scott, Phys. Lett. B 458, 79

(1999). arXiv:hep-ph/9904297608. A. Pilaftsis, Z. Phys. C 55, 275 (1992). arXiv:hep-ph/9901206609. N. Arkani-Hamed, L.J. Hall, H. Murayama, D.R. Smith,

N. Weiner, Phys. Rev. D 64, 115011 (2001). arXiv:hep-ph/0006312

610. F. Borzumati, Phys. Rev. D 64, 053005 (2001). arXiv:hep-ph/0007018

611. S. Bergmann, A. Kagan, Nucl. Phys. B 538, 368 (1999).arXiv:hep-ph/9803305

612. A. Ioannisian, A. Pilaftsis, Phys. Rev. D 62, 066001 (2000).arXiv:hep-ph/9907522

613. D.A. Dicus, D.D. Karatas, P. Roy, Phys. Rev. D 44, 2033 (1991)614. A. Datta, M. Guchait, A. Pilaftsis, Phys. Rev. D 50, 3195

(1994). arXiv:hep-ph/9311257615. F.M.L. Almeida, Y.D.A. Coutinho, J.A. Martins Simoes,

M.A.B. do Vale, Phys. Rev. D 62, 075004 (2000).arXiv:hep-ph/0002024

616. O. Panella, M. Cannoni, C. Carimalo, Y.N. Srivastava, Phys.Rev. D 65, 035005 (2002). arXiv:hep-ph/0107308

617. T. Han, B. Zhang, Phys. Rev. Lett. 97, 171804 (2006).arXiv:hep-ph/0604064

618. S. Bray, J.S. Lee, A. Pilaftsis, arXiv:hep-ph/0702294619. F. del Aguila, J.A. Aguilar-Saavedra, R. Pittau, arXiv:hep-ph/

0703261620. N. Arkani-Hamed, A.G. Cohen, H. Georgi, Phys. Rev. Lett. 86,

4757 (2001). arXiv:hep-th/0104005

176 Eur. Phys. J. C (2008) 57: 13–182

621. H.C. Cheng, C.T. Hill, S. Pokorski, J. Wang, Phys. Rev. D 64,065007 (2001). arXiv:hep-th/0104179

622. N. Arkani-Hamed, A.G. Cohen, H. Georgi, Phys. Lett. B 513,232 (2001). arXiv:hep-ph/0105239

623. M. Schmaltz, D. Tucker-Smith, Annu. Rev. Nucl. Part. Sci. 55,229 (2005). arXiv:hep-ph/0502182

624. M. Perelstein, Prog. Part. Nucl. Phys. 58, 247 (2007).arXiv:hep-ph/0512128

625. N. Arkani-Hamed, A.G. Cohen, E. Katz, A.E. Nelson, J. HighEnergy Phys. 0207, 034 (2002). arXiv:hep-ph/0206021

626. S. Chang, J. High Energy Phys. 0312, 057 (2003).arXiv:hep-ph/0306034

627. H.C. Cheng, I. Low, J. High Energy Phys. 0309, 051 (2003).arXiv:hep-ph/0308199

628. H.C. Cheng, I. Low, J. High Energy Phys. 0408, 061 (2004).arXiv:hep-ph/0405243

629. J. Hubisz, P. Meade, A. Noble, M. Perelstein, J. High EnergyPhys. 0601, 135 (2006). arXiv:hep-ph/0506042

630. J. Hubisz, P. Meade, Phys. Rev. D 71, 035016 (2005).arXiv:hep-ph/0411264

631. I. Low, J. High Energy Phys. 0410, 067 (2004). arXiv:hep-ph/0409025

632. J. Hubisz, S.J. Lee, G. Paz, J. High Energy Phys. 0606, 041(2006). arXiv:hep-ph/0512169

633. M. Blanke, A.J. Buras, A. Poschenrieder, S. Recksiegel,C. Tarantino, S. Uhlig, A. Weiler, Phys. Lett. B 646, 253 (2007).arXiv:hep-ph/0609284

634. S. Yamada, Nucl. Phys. Proc. Suppl. 144, 185 (2005)635. http://meg.web.psi.ch/636. S.C. Park, J.h. Song, arXiv:hep-ph/0306112637. R. Casalbuoni, A. Deandrea, M. Oertel, J. High Energy Phys.

0402, 032 (2004). arXiv:hep-ph/0311038638. M. Blanke, A.J. Buras, A. Poschenrieder, S. Recksiegel,

C. Tarantino, S. Uhlig, A. Weiler, J. High Energy Phys. 0701,066 (2007). arXiv:hep-ph/0610298

639. M. Blanke, A.J. Buras, B. Duling, A. Poschenrieder,C. Tarantino, arXiv:hep-ph/0702136

640. P.Q. Hung, Phys. Lett. B 659, 585 (2008). arXiv:0711.0733[hep-ph]

641. M. Blanke, A.J. Buras, A. Poschenrieder, C. Tarantino, S. Uh-lig, A. Weiler, J. High Energy Phys. 0612, 003 (2006).arXiv:hep-ph/0605214

642. A.J. Buras, arXiv:hep-ph/9806471643. S.R. Choudhury, A.S. Cornell, A. Deandrea, N. Gaur, A. Goyal,

Phys. Rev. D 75, 055011 (2007). arXiv:hep-ph/0612327644. B. Aubert et al. (BaBar Collaboration), Phys. Rev. Lett. 92,

121801 (2004). arXiv:hep-ex/0312027645. Y. Yusa et al. (Belle Collaboration), Phys. Lett. B 589, 103

(2004). arXiv:hep-ex/0403039646. Y. Enari et al.(Belle Collaboration), Phys. Lett. B 622, 218

(2005). arXiv:hep-ex/0503041647. K. Abe et al. (Belle Collaboration), arXiv:hep-ex/0609049648. B. Aubert et al.(BaBar Collaboration), Phys. Rev. Lett. 98,

061803 (2007). arXiv:hep-ex/0610067649. E. Arganda, M.J. Herrero, Phys. Rev. D 73, 055003 (2006).

arXiv:hep-ph/0510405650. P. Paradisi, J. High Energy Phys. 0602, 050 (2006).

arXiv:hep-ph/0508054651. Y. Kuno, Y. Okada, Phys. Rev. Lett. 77, 434 (1996).

arXiv:hep-ph/9604296652. Y. Farzan, arXiv:hep-ph/0701106653. A. Czarnecki, W.J. Marciano, A. Vainshtein, Phys. Rev. D 67,

073006 (2003) [Erratum: Phys. Rev. D 73, 119901 (2006)].arXiv:hep-ph/0212229

654. J. Lee, arXiv:hep-ph/0504136655. T. Han, H.E. Logan, B. Mukhopadhyaya, R. Srikanth, Phys.

Rev. D 72, 053007 (2005). arXiv:hep-ph/0505260

656. F. del Aguila, M. Masip, J.L. Padilla, Phys. Lett. B 627, 131(2005). arXiv:hep-ph/0506063

657. S.R. Choudhury, N. Gaur, A. Goyal, Phys. Rev. D 72, 097702(2005). arXiv:hep-ph/0508146

658. A. Abada, G. Bhattacharyya, M. Losada, Phys. Rev. D 73,033006 (2006). arXiv:hep-ph/0511275

659. G.F. Giudice, C. Grojean, A. Pomarol, R. Rattazzi, arXiv:hep-ph/0703164

660. D. Acosta et al. (CDF Collaboration), Phys. Rev. Lett. 93,221802 (2004). arXiv:hep-ex/0406073

661. V.M. Abazov et al. (D0 Collaboration), Phys. Rev. Lett. 93,141801 (2004). arXiv:hep-ex/0404015

662. K. Huitu, J. Maalampi, A. Pietila, M. Raidal, Nucl. Phys. B 487,27 (1997). arXiv:hep-ph/9606311

663. C.X. Yue, S. Zhao, arXiv:hep-ph/0701017664. E. Ma, M. Raidal, U. Sarkar, Phys. Rev. Lett. 85, 3769 (2000).

arXiv:hep-ph/0006046665. E. Ma, M. Raidal, U. Sarkar, Nucl. Phys. B 615, 313 (2001).

arXiv:hep-ph/0012101666. E. Ma, U. Sarkar, Phys. Lett. B 638, 356 (2006). arXiv:

hep-ph/0602116667. N. Sahu, U. Sarkar, arXiv:hep-ph/0701062668. G. Marandella, C. Schappacher, A. Strumia, Phys. Rev. D 72,

035014 (2005). arXiv:hep-ph/0502096669. T. Han, H.E. Logan, B. McElrath, L.T. Wang, Phys. Rev. D 67,

095004 (2003). arXiv:hep-ph/0301040670. T. Han, H.E. Logan, L.T. Wang, J. High Energy Phys. 0601, 099

(2006). arXiv:hep-ph/0506313671. G. Azuelos et al., Eur. Phys. J. C 39S2, 13 (2005). arXiv:

hep-ph/0402037672. J.F. Gunion, C. Loomis, K.T. Pitts, in Proceedings of 1996

DPF/DPB Summer Study on New Directions for High-EnergyPhysics (Snowmass 96), Snowmass, CO, 25 June–12 July 1996,p. LTH096. arXiv:hep-ph/9610237

673. J.F. Gunion, J. Grifols, A. Mendez, B. Kayser, F.I. Olness, Phys.Rev. D 40, 1546 (1989)

674. M. Muhlleitner, M. Spira, Phys. Rev. D 68, 117701 (2003).arXiv:hep-ph/0305288

675. A.G. Akeroyd, M. Aoki, Phys. Rev. D 72, 035011 (2005).arXiv:hep-ph/0506176

676. T. Rommerskirchen, T. Hebbeker, J. Phys. G 33, N47 (2007)677. A. Hektor, M. Kadastik, M. Muntel, M. Raidal, L. Rebane,

arXiv:0705.1495 [hep-ph]678. T. Han, B. Mukhopadhyaya, Z. Si, K. Wang, arXiv:0706.0441

[hep-ph]679. F. Gabbiani, A. Masiero, Nucl. Phys. B 322, 235 (1989)680. F. Gabbiani, E. Gabrielli, A. Masiero, L. Silvestrini, Nucl. Phys.

B 477, 321 (1996). arXiv:hep-ph/9604387681. P. Paradisi, J. High Energy Phys. 0510, 006 (2005).

arXiv:hep-ph/0505046682. R. Barbieri, L.J. Hall, Phys. Lett. B 338, 212 (1994).

arXiv:hep-ph/9408406683. R. Barbieri, L.J. Hall, A. Strumia, Nucl. Phys. B 445, 219

(1995). arXiv:hep-ph/9501334684. I. Masina, C.A. Savoy, Phys. Rev. D 71, 093003 (2005).

arXiv:hep-ph/0501166685. B.C. Allanach et al., in Proceedings of the APS/DPF/DPB Sum-

mer Study on the Future of Particle Physics (Snowmass 2001),ed. by N. Graf, Snowmass, CO, 30 June–21 July 2001, p. P125.arXiv:hep-ph/0202233

686. S. Antusch, E. Arganda, M.J. Herrero, A.M. Teixeira, J. HighEnergy Phys. 0611, 090 (2006). arXiv:hep-ph/0607263

687. F. Deppisch, H. Päs, A. Redelbach, R. Rückl, Phys. Rev. D 73,033004 (2006). arXiv:hep-ph/0511062

688. L. Calibbi, A. Faccia, A. Masiero, S.K. Vempati, arXiv:hep-ph/0610241

Eur. Phys. J. C (2008) 57: 13–182 177

689. M. Maltoni, T. Schwetz, M.A. Tortola, J.W.F. Valle, Phys. Rev.D 68, 113010 (2003). arXiv:hep-ph/0309130

690. S. Eidelman et al. (PDG Collaboration), Phys. Lett. B 592, 1(2004). arXiv:hep-ph/0310053

691. S.T. Petcov, T. Shindou, Phys. Rev. D 74, 073006 (2006).arXiv:hep-ph/0605151

692. G. Weiglein et al. (LHC/LC Study Group), Phys. Rep. 426, 47(2006). arXiv:hep-ph/0410364

693. F. Deppisch, H. Päs, A. Redelbach, R. Rückl, Y. Shimizu,arXiv:hep-ph/0206122

694. A.G. Akeroyd et al. (SuperKEKB Physics Working Group),arXiv:hep-ex/0406071

695. F. Deppisch, H. Päs, A. Redelbach, R. Rückl, Y. Shimizu, Phys.Rev. D 69, 054014 (2004). arXiv:hep-ph/0310053

696. K. Agashe, M. Graesser, Phys. Rev. D 61, 075008 (2000).arXiv:hep-ph/9904422

697. Yu.M. Andreev, S.I. Bityukov, N.V. Krasnikov, A.N. Toropin,arXiv:hep-ph/0608176

698. A. Bartl, K. Hidaka, K. Hohenwarter-Sodek, T. Kernreiter,W. Majerotto, W. Porod, Eur. Phys. J. C 46, 783 (2006).arXiv:hep-ph/0510074

699. W. Porod, Comput. Phys. Commun. 153, 275 (2003).arXiv:hep-ph/0301101

700. U. Bellgardt et al. (SINDRUM Collaboration), Nucl. Phys. B299, 1 (1988)

701. S. Ritt (MEGA Collaboration), on the web page http://meg.web.psi.ch/docs/talks/s_ritt/mar06_novosibirsk/ritt.ppt

702. T. Iijima, talk given at the 6th Workshop on a Higher Luminos-ity B Factory, KEK, Tsukuba, Japan, November 2004

703. J. Aysto et al., arXiv:hep-ph/0109217704. The PRIME Working Group, Search for the μ–e Conversion

Process at an Ultimate sensitivity of the order of 1018 withPRISM, unpublished

705. LOI to J-PARC 50-GeV PS, LOI-25, http://psux1.kek.jp/jhf-np/LOIlist/LOIlist.html

706. Y. Kuno, Nucl. Phys. Proc. Suppl. 149, 376 (2005)707. A. Masiero, S.K. Vempati, O. Vives, New J. Phys. 6, 202 (2004).

arXiv:hep-ph/0407325708. E. Ables et al. (MINOS Collaboration), Fermilab-proposal-

0875709. G.S. Tzanakos (MINOS Collaboration), AIP Conf. Proc. 721,

179 (2004)710. M. Komatsu, P. Migliozzi, F. Terranova, J. Phys. G 29, 443

(2003). arXiv:hep-ph/0210043711. P. Migliozzi, F. Terranova, Phys. Lett. B 563, 73 (2003).

arXiv:hep-ph/0302274712. P. Huber, J. Kopp, M. Lindner, M. Rolinec, W. Winter, J. High

Energy Phys. 0605, 072 (2006). arXiv:hep-ph/0601266713. Y. Itow et al., arXiv:hep-ex/0106019714. A. Blondel, A. Cervera-Villanueva, A. Donini, P. Huber,

M. Mezzetto, P. Strolin, arXiv:hep-ph/0606111715. P. Huber, M. Lindner, M. Rolinec, W. Winter, arXiv:hep-ph/

0606119716. J. Burguet-Castell, D. Casper, E. Couce, J.J. Gomez-

Cadenas, P. Hernandez, Nucl. Phys. B 725, 306 (2005).arXiv:hep-ph/0503021

717. J.E. Campagne, M. Maltoni, M. Mezzetto, T. Schwetz, arXiv:hep-ph/0603172

718. T. Blazek, S.F. King, Nucl. Phys. B 662, 359 (2003). arXiv:hep-ph/0211368

719. S.F. King, J. High Energy Phys. 0508, 105 (2005)arXiv:hep-ph/0506297

720. S. Antusch, S.F. King, Phys. Lett. B 631, 42 (2005).arXiv:hep-ph/0508044

721. S. Antusch, S.F. King, arXiv:0709.0666 [hep-ph]722. P.H. Chankowski, S. Pokorski, Int. J. Mod. Phys. A 17, 575

(2002). arXiv:hep-ph/0110249

723. G. Altarelli, F. Feruglio, Phys. Rep. 320, 295 (1999)724. A.Y. Smirnov, Phys. Rev. D 48, 3264 (1993). arXiv:hep-ph/

9304205725. J. Pradler, F.D. Steffen, Phys. Rev. D 75, 023509 (2007).

arXiv:hep-ph/0608344726. S. Davidson, J. High Energy Phys. 0303, 037 (2003).

arXiv:hep-ph/0302075727. P.H. Chankowski, K. Turzynski, Phys. Lett. B 570, 198 (2003).

arXiv:hep-ph/0306059728. A. Pilaftsis, Int. J. Mod. Phys. A 14, 1811 (1999). arXiv:

hep-ph/9812256729. S.F. King, Rep. Prog. Phys. 67, 107 (2004). arXiv:hep-ph/

0310204730. F.R. Joaquim, I. Masina, A. Riotto, arXiv:hep-ph/0701270731. J.R. Ellis, K.A. Olive, Y. Santoso, V.C. Spanos, Phys. Lett. B

565, 176 (2003). arXiv:hep-ph/0303043732. T. Blazek, S.F. King, Phys. Lett. B 518, 109 (2001).

arXiv:hep-ph/0105005733. F. Cuypers, S. Davidson, Eur. Phys. J. C 2, 503 (1998).

arXiv:hep-ph/9609487734. E.J. Chun, K.Y. Lee, S.C. Park, Phys. Lett. B 566, 142 (2003).

arXiv:hep-ph/0304069735. A. Strumia, F. Vissani, arXiv:hep-ph/0606054736. F.R. Joaquim, A. Rossi, arXiv:hep-ph/0607298737. T. Moroi, Phys. Lett. B 493, 366 (2000)738. J. Hisano, Y. Shimizu, Phys. Lett. B 565, 183 (2003)739. M. Ciuchini, A. Masiero, L. Silvestrini, S.K. Vempati, O. Vives,

Phys. Rev. Lett. 92, 071801 (2004)740. S. Dimopoulos, L.J. Hall, Phys. Lett. B 344, 185 (1995)741. J. Hisano, M. Kakizaki, M. Nagai, Y. Shimizu, Phys. Lett. B

604, 216 (2004)742. M. Pospelov, A. Ritz, Ann. Phys. 318, 119 (2005)743. V.M. Khatsimovsky, I.B. Khriplovich, A.R. Zhitnitsky, Z. Phys.

C 36, 455 (1987)744. A.R. Zhitnitsky, Phys. Rev. D 55, 3006 (1997)745. F.R. Joaquim, A. Rossi, arXiv:hep-ph/0604083746. M. Dine, W. Fischler, M. Srednicki, Nucl. Phys. B 189, 575

(1981)747. S. Dimopoulos, S. Raby, Nucl. Phys. B 192, 353 (1981)748. C.R. Nappi, B.A. Ovrut, Phys. Lett. B 113, 175 (1982)749. S. Dimopoulos, S. Raby, Nucl. Phys. B 219, 479 (1983)750. M. Dine, A.E. Nelson, Phys. Rev. D 48, 1277 (1993).

arXiv:hep-ph/9303230751. M. Dine, A.E. Nelson, Y. Shirman, Phys. Rev. D 51, 1362

(1995). arXiv:hep-ph/9408384752. M.A. Giorgi et al. (SuperB Group,), INFN Roadmap Report,

March 2006753. Y. Mori et al., (PRISM/PRIME Working Group), LOI

at J-PARC 50-GeV PS, LOI-25. http://psux1.kek.jp/~jhf-np/LOIlist/LOIlist.html

754. M. Cvetic, I. Papadimitriou, G. Shiu, Nucl. Phys. B 659, 193(2003) [Erratum:Nucl. Phys. B 696, 298 (2004)]

755. M. Cvetic, P. Langacker, arXiv:hep-th/0607238756. R. Barbieri, D.V. Nanopoulos, G. Morchio, F. Strocchi, Phys.

Lett. B 90, 91 (1980)757. P. Langacker, Phys. Rep. 72, 185 (1981)758. D. Chang, A. Masiero, H. Murayama, Phys. Rev. D 67, 075013

(2003). arXiv:hep-ph/0205111759. T. Moroi, J. High Energy Phys. 0003, 019 (2000).

arXiv:hep-ph/0002208760. N. Akama, Y. Kiyo, S. Komine, T. Moroi, Phys. Rev. D 64,

095012 (2001). arXiv:hep-ph/0104263761. J. Sato, K. Tobe, T. Yanagida, Phys. Lett. B 498, 189 (2001).

arXiv:hep-ph/0010348762. M. Apollonio et al.(CHOOZ Collaboration), Phys. Lett. B 466,

415 (1999). arXiv:hep-ex/9907037

178 Eur. Phys. J. C (2008) 57: 13–182

763. L. Calibbi, A. Faccia, A. Masiero, S.K. Vempati, Phys. Rev. D74, 116002 (2006). arXiv:hep-ph/0605139

764. K. Abe et al. (Belle Collaboration), Phys. Rev. Lett. 92, 171802(2004). arXiv:hep-ex/0310029

765. A.J. Buras, P.H. Chankowski, J. Rosiek, L. Slawianowska,Nucl. Phys. B 659, 3 (2003). arXiv:hep-ph/0210145

766. G. Isidori, P. Paradisi, Phys. Lett. B 639, 499 (2006).arXiv:hep-ph/0605012

767. E. Lunghi, W. Porod, O. Vives, Phys. Rev. D 74, 075003 (2006).arXiv:hep-ph/0605177

768. J. Foster, K.i. Okumura, L. Roszkowski, Phys. Lett. B 641, 452(2006). arXiv:hep-ph/0604121

769. G.R. Farrar, P. Fayet, Phys. Lett. B 76, 575 (1978)770. L.J. Hall, M. Suzuki, Nucl. Phys. B 231, 419 (1984)771. R. Barbier et al., Phys. Rep. 420, 1 (2005). arXiv:hep-ph/

0406039772. D. Comelli, A. Masiero, M. Pietroni, A. Riotto, Phys. Lett. B

324, 397 (1994). arXiv:hep-ph/9310374773. R. Kuchimanchi, R.N. Mohapatra, Phys. Rev. D 48, 4352

(1993). arXiv:hep-ph/9306290774. K. Huitu, J. Maalampi, Phys. Lett. B 344, 217 (1995).

arXiv:hep-ph/9410342775. R. Kitano, K.y. Oda, Phys. Rev. D 61, 113001 (2000).

arXiv:hep-ph/9911327776. M. Frank, K. Huitu, Phys. Rev. D 64, 095015 (2001).

arXiv:hep-ph/0106004777. J.R. Ellis, G. Gelmini, C. Jarlskog, G.G. Ross, J.W.F. Valle,

Phys. Lett. B 150, 142 (1985)778. S. Davidson, M. Losada, Phys. Rev. D 65, 075025 (2002).

arXiv:hep-ph/0010325779. K.S. Babu, R.N. Mohapatra, Phys. Rev. Lett. 64, 1705 (1990)780. M. Hirsch, M.A. Diaz, W. Porod, J.C. Romao, J.W.F. Valle,

Phys. Rev. D 62, 113008 (2000) [Erratum: Phys. Rev. D 65,119901 (2002)]. arXiv:hep-ph/0004115

781. M.A. Diaz, C. Mora, A.R. Zerwekh, Eur. Phys. J. C 44, 277(2005). arXiv:hep-ph/0410285

782. A. Deandrea, J. Welzel, M. Oertel, J. High Energy Phys. 0410,038 (2004). arXiv:hep-ph/0407216

783. W. Porod, M. Hirsch, J. Romao, J.W.F. Valle, Phys. Rev. D 63,115004 (2001). arXiv:hep-ph/0011248

784. A. Heister et al.(ALEPH Collaboration), Eur. Phys. J. C 31, 1(2003). arXiv:hep-ex/0210014

785. P. Abreu et al.(DELPHI Collaboration), Phys. Lett. B 500, 22(2001). arXiv:hep-ex/0103015

786. P. Achard et al. (L3 Collaboration), Phys. Lett. B 524, 65(2002). arXiv:hep-ex/0110057

787. B. Abbott et al. (D0 Collaboration), Phys. Rev. D 62, 071701(2000). arXiv:hep-ex/0005034

788. S. Aid et al. (H1 Collaboration), Z. Phys. C 71, 211 (1996).arXiv:hep-ex/9604006

789. C. Adloff et al. (H1 Collaboration), Eur. Phys. J. C 20, 639(2001). arXiv:hep-ex/0102050

790. W. Beenakker, R. Hopker, M. Spira, P.M. Zerwas, Phys. Rev.Lett. 74, 2905 (1995). arXiv:hep-ph/9412272

791. J. Kalinowski, R. Ruckl, H. Spiesberger, P.M. Zerwas, Phys.Lett. B 414, 297 (1997). arXiv:hep-ph/9708272

792. G. Moreau, E. Perez, G. Polesello, Nucl. Phys. B 604, 3 (2001).arXiv:hep-ph/0003012

793. G. Moreau, M. Chemtob, F. Deliot, C. Royon, E. Perez, Phys.Lett. B 475, 184 (2000). arXiv:hep-ph/9910341

794. S. Dimopoulos, L.J. Hall, Phys. Lett. B 207, 210 (1988)795. H.K. Dreiner, P. Richardson, M.H. Seymour, Phys. Rev. D 63,

055008 (2001). arXiv:hep-ph/0007228796. F. Deliot, G. Moreau, C. Royon, Eur. Phys. J. C 19, 155 (2001).

arXiv:hep-ph/0007288797. M. Chaichian, A. Datta, K. Huitu, S. Roy, Z.h. Yu, Phys. Lett.

B 594, 355 (2004). arXiv:hep-ph/0311327

798. S. Dimopoulos, R. Esmailzadeh, L.J. Hall, G.D. Starkman,Phys. Rev. D 41, 2099 (1990)

799. B. Allanach et al. (R Parity Working Group Collaboration),arXiv:hep-ph/9906224

800. E.L. Berger, B.W. Harris, Z. Sullivan, Phys. Rev. Lett. 83, 4472(1999). arXiv:hep-ph/9903549

801. M. Chaichian, K. Huitu, Z.H. Yu, Phys. Lett. B 490, 87 (2000).arXiv:hep-ph/0007220

802. E.D. Richter-Was, L. Poggioli, ATLAS note ATLAS-PHYS-98-131 (1998)

803. W.W. Armstrong et al., ATLAS technical proposal CERN-LHCC-94-43, 1994

804. A. Belyaev, M.H. Genest, C. Leroy, R.R. Mehdiyev, J. HighEnergy Phys. 0409, 012 (2004). arXiv:hep-ph/0401065

805. K.S. Babu, C.F. Kolda, Phys. Rev. Lett. 84, 228 (2000).arXiv:hep-ph/9909476

806. K.S. Babu, C. Kolda, Phys. Rev. Lett. 89, 241802 (2002).arXiv:hep-ph/0206310

807. A. Dedes, J.R. Ellis, M. Raidal, Phys. Lett. B 549, 159 (2002).arXiv:hep-ph/0209207

808. D. Chang, W.S. Hou, W.Y. Keung, Phys. Rev. D 48, 217 (1993).arXiv:hep-ph/9302267

809. M. Sher, Y. Yuan, Phys. Rev. D 44, 1461 (1991)810. M. Sher, Phys. Rev. D 66, 057301 (2002). arXiv:hep-ph/

0207136811. R. Kitano, M. Koike, S. Komine, Y. Okada, Phys. Lett. B 575,

300 (2003). arXiv:hep-ph/0308021812. S. Kanemura, Y. Kuno, M. Kuze, T. Ota, Phys. Lett. B 607, 165

(2005). arXiv:hep-ph/0410044813. A. Broncano, M.B. Gavela, E. Jenkins, Phys. Lett. B 552, 177

(2003) [Erratum: Phys. Lett. B 636, 330 (2006)]. arXiv:hep-ph/0210271

814. M.C. Gonzalez-Garcia, J.W.F. Valle, Phys. Lett. B 216, 360(1989)

815. S. Antusch, C. Biggio, E. Fernandez-Martinez, M.B. Gavela,J. Lopez-Pavon, J. High Energy Phys. 0610, 084 (2006).arXiv:hep-ph/0607020

816. M. Apollonio et al., Eur. Phys. J. C 27, 331 (2003). hep-ex/0301017

817. T. Araki et al. (KamLAND Collaboration), Phys. Rev. Lett. 94,081801 (2005). hep-ex/0406035

818. B. Aharmim et al. (SNO Collaboration), Phys. Rev. C 72,055502 (2005). nucl-ex/0502021

819. M.H. Ahn et al. (K2K Collaboration), Phys. Rev. Lett. 90,041801 (2003). hep-ex/0212007

820. Y. Ashie et al. (Super-Kamiokande Collaboration), Phys. Rev.D 71, 112005 (2005). hep-ex/0501064

821. P. Astier et al. (NOMAD Collaboration), Nucl. Phys. B 611, 3(2001). hep-ex/0106102

822. K. Eitel (KARMEN Collaboration), Nucl. Phys. Proc. Suppl.91, 191 (2001). hep-ex/0008002

823. Y. Declais et al., Nucl. Phys. B 434, 503 (1995)824. D.A. Petyt (MINOS Collaboration) First MINOS results from

the NuMI beam. http://www-numi.fnal.gov/825. M.C. Gonzalez-Garcia, Phys. Scr. T 121, 72 (2005).

hep-ph/0410030826. V. Barger, S. Geer, K. Whisnant, New J. Phys. 6, 135 (2004).

arXiv:hep-ph/0407140827. D.S. Ayres et al. (NOvA Collaboration), hep-ex/0503053828. J.J. Gomez-Cadenas et al. (CERN Working Group on Super

Beams Collaboration), hep-ph/0105297829. J.E. Campagne, A. Cazes, Eur. Phys. J. C 45, 643 (2006).

hep-ex/0411062830. P. Zucchelli, Phys. Lett. B 532, 166 (2002)831. S. Geer, Phys. Rev. D 57, 6989 (1998) hep-ph/9712290832. A. De Rujula, M.B. Gavela, P. Hernandez, Nucl. Phys. B 547,

21 (1999) hep-ph/9811390

Eur. Phys. J. C (2008) 57: 13–182 179

833. W.J. Marciano, A. Sirlin, Phys. Rev. Lett. 71, 3629 (1993)834. S. Eidelman et al. (Particle Data Group), Phys. Lett. B 592, 1

(2004)835. L. Fiorini (for the NA48/2 Collaboration), at ICHEP 2005836. V.D. Barger, G.F. Giudice, T. Han, Phys. Rev. D 40, 2987

(1989)837. P.H. Chankowski, A. Dabelstein, W. Hollik, W.M. Mosle,

S. Pokorski, J. Rosiek, Nucl. Phys. B 417, 101 (1994)838. J.F. Gunion, H.E. Haber, G.L. Kane, S. Dawson, The Higgs

hunter Guide (Addison–Wesley, Reading, 1990)839. A. Masiero, P. Paradisi, R. Petronzio, Phys. Rev. D 74, 011701

(2006). arXiv:hep-ph/0511289840. M. Krawczyk, D. Temes, Eur. Phys. J. C 44, 435 (2005).

arXiv:hep-ph/0410248841. P. Krawczyk, S. Pokorski, Phys. Rev. Lett. 60, 182 (1988)842. J.R. Ellis, J. Hisano, M. Raidal, Y. Shimizu, Phys. Lett. B 528,

86 (2002). arXiv:hep-ph/0111324843. I. Masina, Nucl. Phys. B 671, 432 (2003). arXiv:hep-ph/

0304299844. Y. Farzan, M.E. Peskin, Phys. Rev. D 70, 095001 (2004).

arXiv:hep-ph/0405214845. I. Masina, C. Savoy, Phys. Lett. B 579, 99 (2004). arXiv:

hep-ph/0309067846. K.S. Babu, B. Dutta, R.N. Mohapatra, Phys. Rev. Lett. 85, 5064

(2000). arXiv:hep-ph/0006329847. S. Abel, O. Lebedev, J. High Energy Phys. 0601, 133 (2006)848. T. Ibrahim, P. Nath, Phys. Rev. D 58, 111301 (1998) [Erratum:

Phys. Rev. D 60, 099902 (1999)]. arXiv:hep-ph/9807501849. J. Hisano, Y. Shimizu, Phys. Rev. D 70, 093001 (2004).

arXiv:hep-ph/0406091850. M. Brhlik, L.L. Everett, G.L. Kane, J.D. Lykken, Phys. Rev.

Lett. 83, 2124 (1999). arXiv:hep-ph/9905215851. S. Abel, S. Khalil, O. Lebedev, Phys. Rev. Lett. 86, 5850 (2001).

arXiv:hep-ph/0103031852. J.R. Ellis, S. Ferrara, D.V. Nanopoulos, Phys. Lett. B 114, 231

(1982)853. S. Abel, S. Khalil, O. Lebedev, Nucl. Phys. B 606, 151 (2001).

hep-ph/0103320854. D.A. Demir, O. Lebedev, K.A. Olive, M. Pospelov, A. Ritz,

Nucl. Phys. B 680, 339 (2004). hep-ph/0311314855. K.A. Olive, M. Pospelov, A. Ritz, Y. Santoso, Phys. Rev. D 72,

075001 (2005). hep-ph/0506106856. N. Arkani-Hamed, S. Dimopoulos, G.F. Giudice, A. Romanino,

Nucl. Phys. B 709, 3 (2005). hep-ph/0409232857. G.F. Giudice, A. Romanino, Phys. Lett. B 634, 307 (2006).

hep-ph/0510197858. D. Chang, W.-F. Chang, W.-Y. Keung, Phys. Rev. D 71, 076006

(2005). hep-ph/0503055859. A.D. Linde, Phys. Lett. B 129, 177 (1983)860. S. Weinberg, Phys. Rev. Lett. 59, 2607 (1987)861. V. Agrawal, S.M. Barr, J.F. Donoghue, D. Seckel, Phys. Rev. D

57, 5480 (1998). hep-ph/9707380862. R. Bousso, J. Polchinski, J. High Energy Phys. 06, 006 (2000).

hep-th/0004134863. N. Arkani-Hamed, S. Dimopoulos, S. Kachru, hep-th/0501082864. R. Arnowitt, J.L. Lopez, D.V. Nanopoulos, Phys. Rev. D 42,

2423 (1990)865. G. Degrassi, E. Franco, S. Marchetti, L. Silvestrini, J. High En-

ergy Phys. 11, 044 (2005). hep-ph/0510137866. M. Pospelov, A. Ritz, Phys. Rev. Lett. 83, 2526 (1999).

hep-ph/9904483867. M. Pospelov, A. Ritz, Phys. Rev. D 63, 073015 (2001).

hep-ph/0010037868. A. Arvanitaki, C. Davis, P.W. Graham, J.G. Wacker, Phys. Rev.

D 70, 117703 (2004). hep-ph/0406034869. C.A. Baker et al., Phys. Rev. Lett. 97, 131801 (2006).

hep-ex/0602020

870. O. Lebedev, W. Loinaz, T. Takeuchi, Phys. Rev. D 61, 115005(2000)

871. M.J. Ramsey-Musolf, Phys. Rev. D 62, 056009 (2000).arXiv:hep-ph/0004062

872. O. Lebedev, W. Loinaz, T. Takeuchi, Phys. Rev. D 62, 055014(2000). arXiv:hep-ph/0002106

873. J.H. Park, J. High Energy Phys. 0610, 077 (2006)874. M.J. Ramsey-Musolf, S. Su, arXiv:hep-ph/0612057875. M.A. Sanchis-Lozano, Contributed paper to the Workshop on

B-Factories and New Measurements, KEK, 13–14 September2006. arXiv:hep-ph/0610046

876. B.A. Campbell, D.W. Maybury, Nucl. Phys B 709, 419 (2005)877. W. Loinaz, N. Okamura, S. Rayyan, T. Takeuchi, L.C.R. Wije-

wardhana, Phys. Rev. D 70, 113004 (2004)878. M. Finkemeier, Phys. Lett. B 387, 391 (1996). arXiv:hep-ph/

9505434879. G. Czapek et al., Phys. Rev. Lett. 70, 17 (1993)880. D.I. Britton et al., Phys. Rev. Lett. 68, 3000 (1992)881. A.Y. Smirnov, R. Zukanovich Funchal, Phys. Rev. D 74, 013001

(2006). arXiv:hep-ph/0603009882. D. Pocanic, A. van der Schaaf (PEN Collaboration), PSI exper-

iment R-05-01883. M.A. Bychkov, Ph.D. thesis, University of Virginia (2005),

unpublished. http://pibeta.phys.virginia.edu/docs/publications/max_thesis/thesis.pdf

884. B.A. VanDevender, Ph.D. thesis, University of Virginia,2005, unpublished. http://pibeta.phys.virginia.edu/docs/publications/brent_diss.pdf

885. D. Bryman and T. Numao (PIENU Collaboration), TRIUMFexperiment 1072

886. NA48/2 Collaboration, Report CERN/SPSC/2000-003, 16 Jan-uary 2000

887. L. Fiorini, PhD thesis, Scuola Normale Superi-ore, Pisa, Italy, 2005, unpublished. Available fromhttp://lfiorini.home.cern.ch/lfiorini/

888. L. Fiorini (for the NA48/2 Collaboration), in Proc. HEP2005Europhysics Conference, Lisbon, 21–27 July 2005, PoS (HEP2005) 288

889. W.J. Marciano, A. Sirlin, Phys. Rev. Lett. 61, 1815 (1988)890. J.Z. Bai et al.(BES Collaboration), Phys. Rev. D 53, 20 (1996)891. V.V. Anashin et al. (KEDR Collaboration), Nucl. Phys. Proc.

Suppl. 169, 125 (2007). arXiv:hep-ex/0611046892. A. Lusiani (BaBar Collaboration), Nucl. Phys. Proc. Suppl. 144,

105 (2005)893. K. Abe et al. (Belle Collaboration), arXiv:hep-ex/0608046894. R. Decker, M. Finkemeier, Phys. Lett. B 334, 199 (1994)895. W. Buchmüller, R.D. Peccei, T. Yanagida, Annu. Rev. Nucl.

Part. Sci. 55, 311 (2005)896. W. Bernreuther, private communication897. L. Michel, Proc. Phys. Soc. A, 63, 514 (1950)898. W. Fetscher, H.-J. Gerber, K.F. Johnson, Phys. Lett. B 173, 102

(1986)899. P. Langacker, Commun. Nucl. Part. Phys. 19 (1989)900. A. Aktas et al. (H1 Collaboration), Phys. Lett. B 634, 173

(2006). arXiv:hep-ex/0512060901. W. Fetscher, H.-J. Gerber, Eur. Phys. J. C 15, 316 (2000)902. F. Scheck, Electroweak and Strong Interactions (Springer,

Berlin, 1996)903. J. Kirkby et al., PSI proposal R-99-06, 1999904. A. Barczyk et al. (FAST Collaboration), arXiv:0707.3904 [hep-

ex]905. W. Marciano, Phys. Rev. D 60, 093006 (1999)906. I.C. Barnett et al., Nucl. Instrum. Methods A 455, 329 (2000)907. Y.S. Tsai, in Stanford Tau Charm (1989), pp. 0387-0393908. Y.S. Tsai, Phys. Rev. D 51, 3172 (1995)909. J.H. Kuhn, E. Mirkes, Z. Phys. C 56, 661 (1992) [Erratum:

Z. Phys. C 67, 364 (1995)]

180 Eur. Phys. J. C (2008) 57: 13–182

910. J.H. Kühn, E. Mirkes, Phys. Lett. B 398, 407 (1997)911. I.I. Bigi, A.I. Sanda, Phys. Lett. B 625, 47 (2005)912. A. Datta et al., hep-ph/0610162913. J. Bernabeu et al., Nucl. Phys. B 763, 283 (2007)914. D. Delepine et al., Phys. Rev. D 74, 056004 (2006)915. A. Pich, Int. J. Mod. Phys. A 21, 5652 (2006)916. S. Anderson et al. (CLEO Collaboration),

Phys. Rev. Lett. 81, 3823 (1998)917. J.J. Sakurai, Phys. Rev. 109, 980 (1958)918. A.R. Zhitnitkii, Sov. J. Nucl. Phys. 31, 529 (1980)919. V.P. Efrosinin, I.B. Khriplovich, G.G. Kirilin, Yu.G. Kudenko,

Phys. Lett. B 493, 293 (2000). arXiv:hep-ph/0008199920. I.I. Bigi, A.I. Sanda, CP Violation (Cambridge University Press,

Cambridge, 2000)921. A.I. Sanda, Phys. Rev. D 23, 2647 (1981)922. N.G. Deshpande, Phys. Rev. D 23, 2654 (1981)923. H.Y. Cheng, Phys. Rev. D 26, 143 (1982)924. H.Y. Cheng, Phys. Rev. D 34, 1397 (1986)925. I.I.Y. Bigi, A.I. Sanda, Phys. Rev. Lett. 58, 1604 (1987)926. M. Leurer, Phys. Rev. Lett. 62, 1967 (1989)927. H.Y. Cheng, Phys. Rev. D 42, 2329 (1990)928. G. Belanger, C.Q. Geng, Phys. Rev. D 44, 2789 (1991)929. R. Garisto, G.L. Kane, Phys. Rev. D 44, 2038 (1991)930. M. Fabbrichesi, F. Vissani, Phys. Rev. D 55, 5334 (1997).

arXiv:hep-ph/9611237931. G.H. Wu, J.N. Ng, Phys. Lett. B 392, 93 (1997). arXiv:

hep-ph/9609314932. W.F. Chang, J.N. Ng, arXiv:hep-ph/0512334933. J.A. Macdonald et al., Nucl. Instrum. Methods Phys. Res. Sect.

A 506, 60 (2003)934. M. Abe et al., Phys. Rev. D 73, 072005 (2006)935. J-PARC experiment proposal P06. http://j-parc.jp/NuclPart/

Proposal_0606_e.html936. W. Bernreuther, U. Low, J.P. Ma, O. Nachtmann, Z. Phys. C 41,

143 (1988)937. M. Skalsey, J. Van House, Phys. Rev. Lett. 67, 1993 (1991)938. M. Felcini, Int. J. Mod. Phys. A 19, 3853 (2004).

arXiv:hep-ex/0404041939. O. Jinnouchi, S. Asai, T. Kobayashi, Phys. Lett. B 572, 117

(2003). arXiv:hep-ex/0308030940. M. Ahmed et al. (MEGA Collaboration), Phys. Rev. D 65,

112002 (2002). arXiv:hep-ex/0111030941. L. Willmann et al., Phys. Rev. Lett. 82, 49 (1999).

arXiv:hep-ex/9807011942. P. Wintz, in ICHEP 98, Vancouver, Canada, 23–29 July 1998943. J. Kaulard et al. (SINDRUM II Collaboration), Phys. Lett. B

422, 334 (1998)944. W. Honecker et al. (SINDRUM II Collaboration), Phys. Rev.

Lett. 76, 200 (1996)945. W. Bertl et al. (SINDRUM II Collaboration), Eur. Phys. J. C 47,

337 (2006)946. T. Mori et al., PSI proposal R-99-5, May 1999, available as UT-

ICEPP 00-02947. A. Baldini et al., Research Proposal to INFN, Septembet 2002.

http://meg.web.psi.ch/docs/948. S. Mihara et al., Cryogenics 44, 223 (2004)949. W. Bertl et al. (SINDRUM Collaboration), Nucl. Phys. B 260,

1 (1985)950. A. Czarnecki, W.J. Marciano, K. Melnikov, AIP Conf. Proc.

549, 938 (2002)951. T.S. Kosmas, I.E. Lagaris, J. Phys. G 28, 2907 (2002)952. R. Kitano, M. Koike, Y. Okada, Phys. Rev. D 66, 096002

(2002). arXiv:hep-ph/0203110953. T.S. Kosmas, J.D. Vergados, O. Civitarese, A. Faessler, Nucl.

Phys. A 570, 637 (1994)954. R.M. Dzhilkibaev, V.M. Lobashev, Sov. J. Nucl. Phys. 49, 384

(1989) [Yad. Fiz. 49, 622 (1989)]

955. M. Bachman et al., MECO, BNL proposal E 940, 1997956. Letter of Intent to J-PARC, L24: The PRISM Project, a Muon

Source of the World-Highest Brightness by Phase Rotation,2003

957. Letter of Intent to J-PARC, L26: Request for a Pulsed ProtonBeam Facility at J-PARC

958. G. Cvetic, C. Dib, C.S. Kim, J.D. Kim, Phys. Rev. D 66,034008 (2002) [Erratum: Phys. Rev. D 68 (2003) 059901].arXiv:hep-ph/0202212

959. J.R. Ellis, M.E. Gomez, G.K. Leontaris, S. Lola, D.V. Nanopou-los, Eur. Phys. J. C 14, 319 (2000). arXiv:hep-ph/9911459

960. T. Fukuyama, T. Kikuchi, N. Okada, Phys. Rev. D 68, 033012(2003). arXiv:hep-ph/0304190

961. C.X. Yuex, Y.M. Zhang, L.J. Liu, Phys. Lett. B 547, 252 (2002).arXiv:hep-ph/0209291

962. A. Ilakovac, Phys. Rev. D 62, 036010 (2000). arXiv:hep-ph/9910213

963. W.J. Li, Y.D. Yang, X.D. Zhang, Phys. Rev. D 73, 073005(2006). arXiv:hep-ph/0511273

964. D. Black, T. Han, H.J. He, M. Sher, Phys. Rev. D 66, 053002(2002). arXiv:hep-ph/0206056

965. R.D. Cousins, V.L. Highland, Nucl. Instrum, Methods Phys.Res. Sect. A 320, 331 (1992)

966. R. Barlow, Comput. Phys. Commun. 149, 97 (2002)967. K. Abe et al. (Belle Collaboration), arXiv:0708.3272 [hep-ex]968. Y. Miyazaki et al. (Belle Collaboration), Phys. Lett. B 648, 341

(2007). arXiv:hep-ex/0703009969. N.G. Unel, arXiv:hep-ex/0505030970. M. Della Negra et al., CMS Physics Technical Design Report,

vol. 1, 2006971. T. Speer et al., CMS Note 2006/032, 2006972. E. James et al., CMS Note 2006/010, 2006973. R. Santinelli, M. Biasini, CMS Note 2002/037, 2002974. R. Santinelli, Nucl. Phys. Proc. Suppl. 123, 234 (2003)975. M.C. Chang et al. (Belle Collaboration), Phys. Rev. D 68,

111101 (2003). arXiv:hep-ex/0309069976. F. Abe et al. (CDF Collaboration), Phys. Rev. Lett. 81, 5742

(1998)977. D. Ambrose et al. (BNL Collaboration), Phys. Rev. Lett. 81,

5734 (1998). arXiv:hep-ex/9811038978. W. Bonivento, N. Serra, LHCb Note 2007-028, 2007979. S.N. Gninenko, M.M. Kirsanov, N.V. Krasnikov, V.A. Matveev,

P. Nedelec, D. Sillou, M. Sher, CERN-SPSC-2004-016980. M. Sher, I. Turan, Phys. Rev. D 69, 017302 (2004). arXiv:

hep-ph/0309183981. D.E. Groom et al. (Particle Data Group), Eur. Phys. J. C 15, 1

(2000)982. B. Aubert et al. (BaBar Collaboration), Phys. Rev. Lett. 95,

191801 (2005). arXiv:hep-ex/0506066983. Y. Yusa et al. (Belle Collaboration), Phys. Lett. B 640, 138

(2006). arXiv:hep-ex/0603036984. G. Ingelman, A. Edin, J. Rathsman, Comput. Phys. Commun.

101, 108 (1997). arXiv:hep-ph/9605286985. M.V. Romalis, W.C. Griffith, E.N. Fortson, Phys. Rev. Lett. 86,

2505 (2001). arXiv:hep-ex/0012001986. R.D. Peccei, H.R. Quinn, Phys. Rev. Lett. 38, 1440 (1977)987. J.S.M. Ginges, V.V. Flambaum, Phys. Rep. 397, 63 (2004).

arXiv:physics/0309054988. I.B. Khriplovich, S.K. Lamoreaux, CP Violation Without

Strangeness: Electric Dipole Moments of Particles, Atoms, andMolecules (Springer, Berlin, 1997)

989. R.J. Crewther, P. Di Vecchia, G. Veneziano, E. Witten, Phys.Lett. B 88, 123 (1979) [Erratum: Phys. Lett. B 91, 487 (1980)]

990. K. Kawarabayashi, N. Ohta, Prog. Theor. Phys. 66, 1789 (1981)991. D. Chang, W.Y. Keung, A. Pilaftsis, Phys. Rev. Lett. 82, 900

(1999)

Eur. Phys. J. C (2008) 57: 13–182 181

992. O. Lebedev, M. Pospelov, Phys. Rev. Lett. 89, 101801 (2002).arXiv:hep-ph/0204359

993. M. Pospelov, A. Ritz, Y. Santoso, Phys. Rev. Lett. 96, 091801(2006). arXiv:hep-ph/0510254

994. M. Pospelov, A. Ritz, Y. Santoso, Phys. Rev. D 74, 075006(2006). arXiv:hep-ph/0608269

995. C. Grojean, G. Servant, J.D. Wells, Phys. Rev. D 71, 036001(2005). arXiv:hep-ph/0407019

996. D. Bodeker, L. Fromme, S.J. Huber, M. Seniuch, J. High EnergyPhys. 0502, 026 (2005). arXiv:hep-ph/0412366

997. S.J. Huber, M. Pospelov, A. Ritz, Phys. Rev. D 75, 036006(2007). arXiv:hep-ph/0610003

998. P.G. Harris et al., Phys. Rev. Lett. 82, 904 (1999)999. K.F. Smith et al., Phys. Lett. B 234, 191 (1990)

1000. I.S. Altarev et al., Phys. At. Nucl. 59, 1152 (1996) [Yad. Fiz.59(N7) 1204 (1996)]

1001. K. Green et al., Nucl. Instrum. Methods A 404, 381 (1998)1002. J.M. Pendlebury et al., Phys. Rev. A 70, 032102 (2004)1003. P.G. Harris, J.M. Pendlebury, Phys. Rev. A 73, 014101 (2006)1004. R. Golub, J.M. Pendlebury, Phys. Lett. A 62, 337 (1977)1005. C.A. Baker et al., Phys. Lett. A 308-1, 67 (2003)1006. C.A. Baker et al., Nucl. Instrum. Methods A 501, 517 (2003)1007. A. Steyerl et al., Phys. Lett. A 116, 347 (1986)1008. http://ucn.web.psi.ch1009. http://nedm.web.psi.ch1010. S. Groeger et al., J. Res. NIST 110, 179 (2005)1011. S. Groeger et al., Appl. Phys. B 80, 645 (2005)1012. K. Green et al., Nucl. Instrum. Methods A 404, 381 (1998)1013. http://p25ext.lanl.gov/edm/edm.html1014. R. Golub, K. Lamoreaux, Phys. Rep. 237, 1 (1994)1015. C.P. Liu, R.G.E. Timmermans, Phys. Rev. C 70, 055501 (2004).

arXiv:nucl-th/04080601016. O. Lebedev, K.A. Olive, M. Pospelov, A. Ritz, Phys. Rev. D 70,

016003 (2004). arXiv:hep-ph/04020231017. R. Stutz, E. Cornell, Bull. Am. Soc. Phys. 89, 76 (2004)1018. J.D. Jackson, Classical Electrodynamics, Wiley, New York

(1975)1019. A.J. Silenko, arXiv:hep-ph/06040951020. J. Bailey et al. (CERN Muon Storage Ring Collaboration),

J. Phys. G 4, 345 (1978)1021. F.J.M. Farley et al., Phys. Rev. Lett. 93, 052001 (2004).

arXiv:hep-ex/03070061022. Y.K. Semertzidis et al., arXiv:hep-ph/00120871023. J.P. Miller et al. (EDM Collaboration), AIP Conf. Proc. 698,

196 (2004)1024. A. Adelmann, K. Kirch, arXiv:hep-ex/06060341025. Y.K. Semertzidis et al. (EDM Collaboration), AIP Conf. Proc.

698 200 (2004). arXiv:hep-ex/03080631026. I.B. Khriplovich, Phys. Lett. B 444, 98 (1998). arXiv:

hep-ph/98093361027. I.I. Rabi, J.R. Zacharias, S. Millman, P. Kusch, Phys. Rev. 53,

318 (1938)1028. Y.F. Orlov, W.M. Morse, Y.K. Semertzidis, Phys. Rev. Lett. 96,

214802 (2006). arXiv:hep-ex/06050221029. E.D. Commins, J.D. Jackson, D.P. DeMille, Am. J. Phys. 75,

532 (2007)1030. E.D. Commins, Am. J. Phys. 59, 1077 (1991)1031. M.G. Kozlov, A.V. Titov, N.S. Mosyagin, P.V. Souchko, Phys.

Rev. A 56, R3326 (1997)1032. M.K. Nayak, R.K. Chaudhuri, B.P. Das, Phys. Rev. A 75,

022510 (2007)1033. N. Mosyagin, M. Kozlov, A. Titov, J. Phys. B 31, L763 (1998)1034. M.G. Kozlov, L. Labzowski, J. Phys. B 28, 1933 (1995)1035. A.N. Petrov et al., Phys. Rev. A 72, 022505 (2005)1036. T.A. Isaev et al., Phys. Rev. Lett. 95, 163004 (2005)1037. E.R. Meyer, J.L. Bohn, M.P. Deskevitch, Phys. Rev. A 73,

062108 (2006)

1038. A.N. Petrov, N.S. Mosyagin, T.A. Isaev, A.V. Titov, Phys. Rev.A 76, 030501(R) (2007)

1039. M.A. Player, P.G.H. Sandars, J. Phys. B 3, 1620 (1970)1040. Z.W. Liu, H.P. Kelly, Phys. Rev. A 45, R4210 (1992)1041. W.R. Johnson, D.S. Guo, M. Idrees, J. Sapirstein, Phys. Rev. A

34, 1043 (1986)1042. S.A. Murthy, D. Krause, Z.L. Li, L. Hunter, Phys. Rev. Lett. 63,

965 (1989)1043. M.V. Romalis, private communication1044. T.W. Kornack, M.V. Romalis, Phys. Rev. Lett. 89, 253002

(2002)1045. T.W. Kornack, R.K. Ghosh, M.V. Romalis, Phys. Rev. Lett. 95,

230801 (2005)1046. M.V. Romalis, E.N. Fortson, Phys. Rev. A 59, 4547 (1999)1047. C. Chin, V. Leiber, V. Vuletic, A.J. Kerman, S. Chu, Phys. Rev.

A 63, 033401 (2001)1048. D. Heinzen, private communication1049. D.S. Weiss, F. Fang, J. Chen, Bull. Am, Phys. Soc. APR03

J1.008 (2003)1050. J. Amini, H. Gould, arxiv.org/abs/physics/0602011 (2006)1051. Y. Sakemi, private communication1052. J.J. Hudson, B.E. Sauer, M.R. Tarbutt, E.A. Hinds, Phys. Rev.

Lett. 89, 023003 (2002)1053. M.R. Tarbutt et al., J. Phys. B 35, 5013 (2002)1054. B.E. Sauer, J. Wang, E. Hinds, J. Chem. Phys. 105, 7412 (1996)1055. D. DeMille et al., Phys. Rev. A 61, 052507 (2000)1056. L.R. Hunter et al., Phys. Rev. A 65, 030501(R) (2002)1057. D. Kawall, F. Bay, S. Bickman, Y. Jiang, D. DeMille, Phys. Rev.

Lett. 92, 133007 (2004)1058. E. Cornell, co-workers, private communication1059. N.E. Shafer-Ray, Phys. Rev. A 73, 34102 (2006);1060. N.E. Shafer-Ray, private communication1061. S.K. Lamoreaux, Phys. Rev. A 66, 022109 (2002)1062. F.L. Shapiro, Sov. Phys. Usp. 11, 345 (1968)1063. V.A. Dzuba, O.P. Sushkov, W.R. Johnson, U.I. Safronova, Phys.

Rev. A 66, 032105 (2002)1064. B.J. Heidenreich et al., Phys. Rev. Lett. 95, 253004 (2005)1065. T.N. Mukhamedjanov, V.A. Dzuba, O.P. Sushkov, Phys. Rev. A

68, 042103 (2003)1066. S. Lamoreaux, arXiv:physics/0701198 (2007)1067. C.-Y. Liu, S.K. Lamoreaux, Mod. Phys. A 19, 1235 (2004)1068. C.-Y. Liu, private communication1069. M. Mercier, Magnetism 6, 77 (1974)1070. J.L. Feng, K.T. Matchev, Y. Shadmi, Nucl. Phys. B 613, 366

(2001). arXiv:hep-ph/01071821071. A. Romanino, A. Strumia, Nucl. Phys. B 622, 73 (2002).

arXiv:hep-ph/01082751072. A. Bartl, W. Majerotto, W. Porod, D. Wyler, Phys. Rev. D 68,

053005 (2003). arXiv:hep-ph/03060501073. M. Aoki et al., J-Parc Letter of Intent, 20031074. A. Adelmann, K. Kirch, C.J.G. Onderwater, T. Schietinger,

A. Streun, to be published1075. P.A.M. Dirac, Proc. R. Soc. (London) A 117, 610 (1928),1076. P.A.M. Dirac, Proc. R. Soc. (London) A 118, 351 (1928).1077. P.A.M. Dirac, The Principles of Quantum Mechanics, 4th edn.

(Oxford University Press, London, 1958). Eq. 9.28 uses Dirac’soriginal notation

1078. B.C. Odom, D. Hanneke, B. D’Urso, G. Gabrielse, Phys. Rev.Lett. 97, 030801 (2006) [Erratum: Phys. Rev. Lett. 99, 039902(2007)]

1079. T. Kinoshita, M. Nio, Phys. Rev. D 73, 053007 (2006).arXiv:hep-ph/0512330

1080. J. Schwinger, Phys. Rev. 73 416L (1948)1081. J. Schwinger, Phys. Rev. 76, 790 (1949). The former paper con-

tains a misprint in the expression for ae that is corrected in thelonger paper

182 Eur. Phys. J. C (2008) 57: 13–182

1082. T. Blum, Phys. Rev. Lett. 91, 052001 (2003)1083. C. Aubin, T. Blum, PoS LAT2005:089 (2005). hep-lat/

0509064,1084. M. Hayakawa et al., PoS LAT2005:353 (2005). hep-lat/

05090161085. J.P. Miller, E. de Rafael, B.L. Roberts, Rep. Prog. Phys. 70, 795

(2007). arXiv:hep-ph/07030491086. It has been proposed that the hadronic contributions could also

be determined from hadronic τ decay, plus the conserved vector

current hypothesis, but this prescriptions seems to have internalconsistency issues which are still under study. See Ref. [1082]and references therein

1087. R.L. Garwin, L.M. Lederman, M. Weinrich, Phys. Rev. 105,1415 (1957)

1088. J. Bailey et al. (CERN-Mainz-Daresbury Collaboration), Nucl.Phys. B 150, 1 (1979)

1089. F.J.M. Farley, Nucl. Instrum. Methods A 523, 251 (2004).arXiv:hep-ex/0307024