Transition magnetic moments of Majorana neutrinos in supersymmetry withoutR-parity in lightof neutrino oscillations

Marek Gozdz* and Wiesław A. Kaminski†

Department of Informatics, Maria Curie-Skłodowska University, pl. Marii Curie-Skłodowskiej 5, 20-031 Lublin, Poland

Fedor Simkovic‡ and Amand Faesslerx

Institute fur Theoretische Physik der Universitat Tubingen, D-72076 Tubingen, Germany(Received 4 July 2006; published 25 September 2006)

The transition magnetic moments of Majorana neutrinos ��ij (ij � e�, e�, ��) are calculated in grandunified theory (GUT) constrained Minimal Supersymmetric Standard Model (MSSM) with explicitR-parity violation. It is assumed that neutrinos acquire masses via one-loop (quark-squark and lepton-slepton) radiative corrections. The mixing of squarks, sleptons, and quarks is considered explicitly. Theconnection between ��ij and the entries of neutrino mass matrix is studied. The current upper limits on��ij are deduced from the elements of phenomenological neutrino mass matrix, which is reconstructedusing the neutrino oscillation data and the lower bound on the neutrinoless double beta decay half-life.Further, the results for ��e� , ��e� and ���� are presented for the cases of inverted and normal hierarchy ofneutrino masses and different SUSY scenarios. The largest values are of the order of 10�17 in units ofBohr magneton.

DOI: 10.1103/PhysRevD.74.055007 PACS numbers: 12.60.Jv, 11.30.Er, 11.30.Fs, 23.40.Bw

I. INTRODUCTION

Supersymmetric extensions of the standard model (SM)give natural framework for solving the hierarchy problem.They lead also to unification of interactions at energies ofthe order of mGUT � 1016 GeV. Supersymmetry (SUSY)itself is required by string theory, the best candidate by nowwhich resolves most of the SM problems.

The Minimal Supersymmetric Standard Model (MSSM)is the minimal in the interaction and particle content con-sistent extension of the SM. It provides rich phenomenol-ogy by the introduction of twice as many particles as weknow from the SM, the so-called superpartners of usualparticles. It possesses also an accidental symmetry calledR-parity, defined by means of the baryon (B), lepton (L),and spin (S) numbers as R � ��1�3B�L�2S. Conservationof R-parity implies that SUSY particles cannot decay intonon-SUSY ones. As a consequence the lightest SUSYparticle must be stable and may be considered as a goodcandidate for dark matter. Theoretically, however, nothingmotivates R-parity conservation and many models whichallow for breaking of this symmetry have been considered.Among them one can distinguish three groups. In the firstone R-parity violation (RpV) is introduced as a spontane-ous process triggered by a nonzero vacuum expectationvalue of some scalar field [1]. Other possibilities include

the introduction of additional bi- [2] or trilinear [3] RpVterms in the superpotential.

The most pressing motivation for looking on physicsbeyond the SM comes from the discovery of neutrinooscillations [4–7]. The experimental evidence for thisprocess clearly indicates that neutrinos do have nonzeromasses. What is more, the flavor and mass eigenstates arenot the same, which leads to mixing between them. Inexperiments focused on neutrino oscillations one can mea-sure the mixing angles as well as the differences of massessquared. For absolute values of masses one has to study thelarge scale structures of the Universe, the endpoint of theelectron spectrum in beta decay of Tritium [8], or searchfor signals of the hypothetical neutrinoless double betadecay (0���). The latter is also the only known processwhich distinguishes between Majorana and Dirac-like neu-trinos. Its observation will prove the Majorana nature ofthese particles.

The importance of theoretical studies of various proper-ties of neutrinos is obvious. In the present paper we focuson the transition magnetic moment, which is generated byinteractions of quark-squark and lepton-slepton self-energy loops with an external photon. This work is anatural continuation and extension of calculations pre-sented in Refs. [9–13].

In the following section we define the model, describe indetail our method of obtaining the supersymmetric particlespectrum, and the way in which unification (GUT con-straints) is imposed. In Sec. III we construct the neutrinomass matrices in loop mechanism, as well as calculate thetransition magnetic moments. In our present approach thesquark and quark mixings are taken into account. We

*Electronic address: mgozdz@kft.umcs.lublin.pl†Electronic address: kaminski@neuron.umcs.lublin.pl‡Electronic address: fedor.simkovic@fmph.uniba.sk; On leave

of absence from Department of Nuclear Physics, ComeniusUniversity, Mlynska Dolina F1, SK–842 15 Bratislava, Slovakia.xElectronic address: amand.faessler@uni-tuebingen.de

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present our results and state the conclusions in the last partof the paper.

II. GUT CONSTRAINTS AND PARTICLESPECTRUM

Let us fix our framework to be the MinimalSupersymmetric Standard Model with supersymmetrybreaking mediated by the gravity force (SUGRAMSSM). We follow the conventions and notation fromRef. [14]. We allow for R-parity violation by consideringexplicitly bilinear and trilinear RpV terms in the super-potential. We focus, however, on the trilinear part, as theeffects we want to study come from 1-loop processes,while the bilinear terms induce neutrino masses at treelevel.

The R-parity conserving part of the superpotential ofMSSM has the form

WMSSM � �ab��YE�ijLai Hb1

�Ej � �YD�ijQaixH

b1

�Dxj

� �YU�ijQaixH

b2

�Uxj ��H

a1H

b2 �; (1)

while its RpV part reads

WRpV � �ab

�1

2�ijkLai L

bj

�Ek � �0ijkLai Q

bjx

�Dxk

�

�1

2�xyz�00ijk �Ux

i�Dyj

�Dzk � �ab�

iLai Hb2 : (2)

The Y’s are 3 3 Yukawa matrices. L andQ are the SU�2�left-handed doublets while �E, �U and �D denote the right-handed lepton, up-quark and down-quark SU�2� singlets,respectively. H1 and H2 mean two Higgs doublets. Wehave introduced color indices x; y; z � 1; 2; 3, generationindices i; j; k � 1; 2; 3 and the SU�2� spinor indicesa; b; c � 1; 2. The tilde sign in ~�0, ~�00 denotes that all thedefinitions are given in the gauge basis for the quark fields.Later on we will change to the mass basis and drop thetilde.

The introduction of RpV terms implies the existence oflepton or baryon number violating processes, like the un-observed proton decay. Fortunately one may keep only onetype of terms and it is not necessary to have both at onetime. In order to get rid of too rapid proton decay and toallow for lepton number violating processes, like the neu-trinoless double beta decay, it is customary to set �00 � 0.

We supply the model with scalar mass term

L mass � m2H1hy1h1 �m2

H2hy2h2 � qym2

Qq� lym2

Ll

� um2Uuy � dm2

Ddy � em2

Eey; (3)

soft gauginos mass term (� � 1; . . . ; 8 for gluinos)

L gaug: �1

2�M1

~By ~B�M2~Wyi ~Wi �M3 ~gy�~g� � H:c:�;

(4)

as well as the supergravity mechanism of supersymmetry

breaking, by introducing the Lagrangian

L soft � �ab��AE�ijlai h

b1 �ej � �AD�ijq

axi h

b1

�djx

� �AU�ijqaxi hb2 �ujx � B�ha1h

b2 � B2�ilai h

b2�; (5)

where lowercase letters stand for scalar components ofrespective chiral superfields, and 3 3 matrices A aswell as B� and B2 are the soft breaking coupling constants.

The procedure of finding a GUT constrained low energyspectrum of the model consists of a few steps.

We take into account mass thresholds where SUSYparticles start to contribute [15] and use the SM 1-looprenormalization group equations (RGE) [16] below appro-priate threshold and 1-loop MSSM RGE [17] above it. The2-loop corrections as well as corrections coming from thepresence of RpV couplings are small and do not change theresults by more than few percent [18].

Initially all the thresholds are set to 1 TeV and aredynamically modified during the running of mass parame-ters. The values of Yukawa couplings Y at mZ are given bylepton and quark mass matrices M [19]

MU � vuSURYTUSyUL; MD � vdSDR

YTDSyDL

;

ME � vdSERYTESyEL;

(6)

where vd � hH01i and vu � hH

02i are the neutral Higgs

vacuum expectation values. Their ratio defines the angle� through the relation vu=vd � tan�. S matrices performdiagonalization so that one obtains eigenstates in the massrepresentation.

After evolving the dimensionless couplings from theelectroweak scale mZ up to mGUT � 2 1016 GeV, weunify the masses of gauginos, sfermions and squarks tobe equal to a common mass parameter m0. We set also themasses of all fermions to a common value m1=2. Thetrilinear soft couplings at mGUT are set according to theformula

A i � A0Yi; (7)

with A0 being another input parameter. We construct thesquark, slepton, chargino and neutralino mass matrices andin the next step evolve all the quantities down to mZ.During that running the tree-level Higgs potential is mini-mized, ie. the following set of equations is solved:

��B�2 > �j�j2 �m2H1��j�j2 �m2

H2�;

2B�< 2j�j2 �m2H1�m2

H2: (8)

In fact one should minimize the full 1-loop Higgs potential,but as the first approximation we consider only the tree-level. We compensate this by adding radiative corrections,which contain functions of particle mass eigenstates gen-erated by electroweak symmetry breaking (EWSB) mix-ing. In that way a proper EWSB mechanism is included inour procedure.

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To obtain the physical masses of supersymmetric particles one has to diagonalize proper mass matrices. For the downsquarks we have

m2~dk�

m2~dkL�m2

dk� 1

6 �2m2W �m

2Z� cos2� �mdk��AD�kk �� tan��

�mdk��AD�kk �� tan�� m2~dkR�m2

dk� 1

3 �m2W �m

2Z� cos2�

0@

1A; (9)

and for sleptons:

m2~ek�

m2~ekL�m2

ek� 1

2 �2m2W �m

2Z� cos2� �mek��AE�kk �� tan��

�mek��AE�kk �� tan�� m2~ekR�m2

ek� �m2

W �m2Z� cos2�

0@

1A; (10)

where m2W � m2

Zcos2W , and the Weinberg weak mixingangle is sin2W � 0:22. We have denoted by mdk and mek ,k � 1, 2, 3 the masses of down quarks and charged leptons,respectively. The L-handed elements are the eigenvalues ofthe running mass parameters m2

Q and m2L whereas the

R-handed ones are the eigenvalues of the singlet parame-ters m2

D and m2E. They are obtained from the RGE proce-

dure after diagonalization. The diagonalization procedureinvolves multiplication by orthogonal matrices which inthe standard trigonometric parameterization introducemixing angles between the weak and mass eigenstates inthe following way:

~d L � ~d1 cos� ~d2 sin; ~dR � �~d1 sin� ~d2 cos;

(11)

~eL � ~e1 cos� ~e2 sin; ~eR � �~e1 sin� ~e2 cos:

(12)

For completeness we list below the explicit expressions forthe mixing angles and mass eigenstates. For the d-typesquarks we finish with

sin2k � �2mdk��AD�kk �� tan��

��m2~dkL�m2

~dkR� 0:34m2

Z cos2��2

� 4m2dk��AD�kk �� tan��2��1=2; (13)

�AD�kk being the k-th eigenvalue of AD, and

m2~dk1;2�

1

2�m2

~dkL�m2

~dkR� �

1

4m2Z cos2�

�mdk

�mdk

�AD�kk �� tan�

sin2k

�: (14)

For sleptons the analogous expressions read:

sin2k � �2mek��AE�kk �� tan��

��m2~ekL�m2

~ekR� 0:04m2

Z cos2��2

� 4m2ek��AE�kk �� tan��2��1=2; (15)

m2~ek1;2�

1

2�m2

~ekL�m2

~ekR� �

1

4m2Z cos2�

�mek

�mek

�AE�kk �� tan�

sin2k

�: (16)

Each obtained solution is checked against various con-ditions. These are (i) finite values of Yukawa couplings atthe GUT scale; (ii) requirement of physically acceptablemass eigenvalues at low energies; (iii) FCNC phenome-nology (b! s� processes).

The model we have described has only five free parame-ters: three GUT parameters A0,m0,m1=2, the sign of� andtan�.

III. NEUTRINO MASSES AND TRANSITIONMAGNETIC MOMENTS

Many different proposals of generating neutrino massmatrix may be found in the literature [20]. In the presentpaper we use the loop mechanism of generating Majorananeutrino masses, possible in RpV models [10–13]. It is alsowell known that by considering photon interactions withthe particles in the loop a transition magnetic moment isgenerated.

A. The neutrino masses in R-parity breaking MSSM

Let us first recall the known results. Within the R-paritybreaking MSSM neutrinos are, in general, massive.

In the lowest order, the contribution to the neutrino massmatrix reads [10]

M treeii0 ��i�i0g2

2

M1�M2tan2W

4��m2W�M1�M2tan2W� sin2��M1M2�2�

;

(17)

where �i � �h~�ii � vd�i, and h~�ii are the vacuum expec-tation values of the sneutrino fields.

Going beyond the tree level one may consider diagramsdepicted on Fig. 1. The down-squark mixing is realizedthrough the angle . The quark mixing may be taken intoaccount in two ways: as mixing of down-type quarks and asmixing of up-type quarks (see discussion in Ref. [21]).Since in our case the d-quarks enter the loops, only their

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mixing will influence the results. The case of u-quarksmixing is equivalent to switching the mixing off and hasno impact on the outcome of our calculations. We denoteby V the standard Cabibbo-Kobayashi-Maskawa (CKM)quark mixing matrix.

Altogether the contribution to the neutrino mass matrixcoming from the squark-quark loop may be expressed as

M qii0 �

3

16�2

Xjkl

���0ijk�

0i0kl

Xa

VjaVlavqakmda

�

�

��0ijk�

0i0lj

Xa

VkaVlavqajmda

��: (18)

The �0-couplings in Eq. (26) are defined in the quark masseigenstate basis. They are related to the couplings inEq. (2) via

�0ijk � ~�0imn�VuL��jm�V

dR�kn: (19)

Here, the matrices VL;R rotate the quark gauge eigenstatesqgauge to the quarks mass eigenstates qmass as qmass

L;R �

VqL;Rqgauge. The CKM matrix is defined in the standard

way as V � VuLVdyL . The factor 3 in Eq. (18) comes from

the summation over three quark colors. The loop integral is

vqjk �sin2k

2

�lnxjk2

1� xjk2�

lnxjk11� xjk1

�: (20)

Here, we introduced two dimensionless quantities, xjk1 �

m2dj=m

2~dk1

and xjk2 � m2dj=m

2~dk2

.

The situation is simpler for the slepton-lepton loop(Fig. 2), as the lepton mixing is much weaker and wefeel justified to neglect it. By analogy with Eq. (18) weobtain

M ‘ii0 �

1

16�2

Xjk

�ijk�i0kj�v‘jkmej � v

‘kjmek�; (21)

with the loop integral taking the form

v‘jk �sin2k

2

�lnyjk2

1� yjk2�

lnyjk11� yjk1

�; (22)

where now yjk1 � m2ej=m

2~ek1

and yjk2 � m2ej=m

2~ek2

.

The functions vqjk and v‘jk, which are free of R-paritybreaking SUSY parameters, can be calculated by runningthe MSSM RGE, in which the whole low energy particle/sparticle spectrum is generated. In our calculation we haveused the following values for the quark sector: mu �5 MeV, md � 9 MeV, ms � 175 MeV, mc � 1:5 GeV,mb � 5 GeV, mt � 174 GeV, as well as the CKM matrixin the form

V �0:9755 0:2195 0:0031�0:2201 0:9746 0:0390�0:0055 �0:0387 0:9989

0@

1A: (23)

This corresponds to the sines of quark mixing angles s12 �0:2195, s23 � 0:039, and s13 � 0:0031 [21] in the standardtrigonometric parameterization. We have neglected thepossible appearance of the CP violating phase in the quarksector.

The values of the calculated loop integrals have beenarranged for clarity in the form of matrices. For the inputparameters A0 � 100 GeV, m0 � m1=2 � 150 GeV,tan� � 19, �> 0 we get:

vq �2:361 10�4 4:593 10�3 1:687 10�1

2:361 10�4 4:593 10�3 1:687 10�1

2:358 10�4 4:587 10�3 1:684 10�1

0B@

1CA;

(24)

v‘ �6:547 10�5 1:310 10�2 2:528 10�1

6:547 10�5 1:310 10�2 2:528 10�1

6:541 10�5 1:308 10�2 2:525 10�1

0B@

1CA:

(25)

For the second set of parameters A0 � 500 GeV, m0 �

m1=2 � 1000 GeV, tan� � 19, �> 0 we find:

vq �3:396 10�5 6:606 10�4 2:419 10�2

3:396 10�5 6:606 10�4 2:419 10�2

3:396 10�5 6:605 10�4 2:419 10�2

0B@

1CA;

(26)

v‘ �1:023 10�5 2:046 10�3 3:879 10�2

1:023 10�5 2:046 10�3 3:879 10�2

1:023 10�5 2:046 10�3 3:879 10�2

0B@

1CA:

(27)

B. The phenomenological neutrino mass matrix

The constraints on the various products of couplingconstants �i�i0 , �ijk�i0kj, and �0ijk�

0i0kj or their values can

λ ijk

ekLekR

ejL ejR

λ i kj

νiL νi L

λ ijk

ejRejL

ekR ekL

λ i kj

νiL νi L

FIG. 2. Feynman diagrams representing the slepton-leptonloop contribution to the Majorana neutrino mass.

λ ijk

dkLdkR

djL djR

λ i kj

νiL νi L

λ ijk

djRdjL

dkR dkL

λ i kj

νiL νi L

FIG. 1. Feynman diagrams representing the squark-quark loopcontribution to the Majorana neutrino mass.

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be find by a comparison of theoretical neutrino mass matrixcalculated within RpV MSSM with the phenomenologicalneutrino mass matrix, derived from analysis of neutrinodata by making viable assumptions. It is usually done byassuming that different contributions to theoretical neu-trino mass matrix do not significantly compensate eachother, i.e., it is possible to extract limits on individualcontributions without knowing the others.

The neutrino phenomenological mass matrix in flavorbasis, Mph, can be written as

M ph � U�MdiagUy; (28)

where

Mdiag � diag�m1; m2; m3�: (29)

Here mi are the moduli of neutrino mass eigenvalues. Thethree neutrino mixing scheme is assumed, namely

��L � U�i�iL; � � e;�; �; i � 1; 2; 3: (30)

��L are flavor neutrino states and �i are the masseigenstates.

For massive Majorana neutrinos the unitary Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix U in the standardparameterization has the form

U �c12c13 s12c13 s13e

�i

�s12c23 � c12s23s13ei c12c23 � s12s23s13ei s23c13

s12s23 � c12c23s13ei �c12s23 � s12c23s13e

i c23c13

0B@

1CA 1 0 0

0 ei2 00 0 ei3

0@

1A; (31)

where sij � sinij, cij � cosij. Three mixing angles ij(i < j) vary between 0 and �=2. The is the CP violatingDirac phase and2,3 are CP violating Majorana phases.Their values vary between 0 and 2�. The expression forMph

�� in terms of mi, ij, , 2, 3 is given in theAppendix.

The analysis of the Super-Kamiokande atmosphericneutrino data [4] and the global analysis of the data ofthe solar neutrino experiments and KamLAND experiment[6] yield the following best fit values of the relevantneutrino oscillation parameters:

j�m223j � 2:1 10�3 eV2; sin2223 � 1:00;

�m212 � 7:1 10�5 eV2; tan212 � 0:40:

(32)

The neutrino mass-squared difference is determined as�m2

ik � m2k �m

2i . For the angle 13 only the upper bound

is known. From exclusion plot obtained from the data ofthe reactor experiment CHOOZ [22] we have

sin 213 5 10�2 �90% c:l:�: (33)

The CP violating phases , 2 and 3, as well as theabsolute scale for the masses, remain undetermined.

Neutrino oscillation data are insensitive to the absolutescale of neutrino masses. The values of neutrino massesdepend on the lightest neutrino mass, on the neutrino massspectrum and the neutrino mass-squared differences �m2

12and �m2

23. The neutrino oscillation data are compatiblewith two types of neutrino mass spectra:

(i) The normal hierarchy (NH) mass spectrum, whichcorresponds to the case

m1 � m2 � m3: (34)

(ii) The inverted hierarchy (IH) of neutrino masses. It isgiven by the condition

m3 � m1 <m2: (35)

We note that it is common to label neutrino masses differ-ently in the case of the normal and the inverted spectra. Forboth spectra we have m2 >m1. But in the case of thenormal spectrum m3 is the mass of the heaviest neutrinoand in the case of the inverted hierarchy m3 is the mass ofthe lightest neutrino. This convention allows to keep thesame notation of the mixing angles for both spectra.Existing oscillation data are compatible both with normaland the inverted spectra.

The 0���-decay is one of the most sensitive knownways to probe the absolute values of neutrino masses andthe type of the spectrum. The most stringent lower boundon the half-life of 0���-decay were obtained in theHeidelberg-Moscow 76Ge experiment [23] (T0��exp

1=2 �

1:9 1025 yr). By assuming the nuclear matrix elementof Ref. [24] we end up with jm��j � U2

e1m1 �U2e2m2 �

U2e3m3 0:55 eV, where U is the neutrino mixing matrix

Eq. (31). With this input limit we can find the maximalallowed values for the matrix elements Mph

ij of the neu-trino mass matrix, which are as follows:

jMph�HMj

0:55 1:29 1:291:29 1:35 1:041:29 1:04 1:35

0@

1A eV: (36)

The first element of the matrix Mph�HM is fixed by theresults from the Heidelberg-Moscow experiment. In thecalculation of the remaining elements we used best fitvalues of mass-squared differences �m2

12, �m223 and mix-

ing angles 12, 23 from neutrino oscillation experiments[4,6]: sin213 � �0; 5 10�2�, ;2; 3 � �0; 2��; thewhole allowed mass parameter space of neutrinos wastaken into account.

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The phenomenological neutrino mass matrix can be constructed from the assumption of normal or inverted hierarchy ofneutrino masses. If mass-squared differences and mixing angles of (32) and (33) are considered we obtain

jMph�NHj � 10�4 eV�0:53; 45:2� �26:9; 98:9� �26:9; 98:9��26:9; 98:9� �173 254� �182 254��26:9; 98:9� �182 254� �173 254�

0@

1A; (37)

jMph�IHj � 10�2 eV�1:80; 4:52� �0:025; 3:12� �0:025; 3:12��0:025; 3:12� �0:028; 2:40� �0:945; 2:39��0:025; 3:12� �0:945; 2:39� �0:028; 2:40�

0@

1A: (38)

Here, we assumed that the mass of the lightest neutrino isnegligible. The allowed ranges have origin in the uncer-tainty coming from the 13 parameter and the CP violatingphases.

C. The magnetic moments of neutrinos

Once neutrinos are massive, they can have magneticmoments. We distinguish magnetic moments of Diracand Majorana neutrinos:

(i) The Dirac magnetic moment, which connects left-handed electroweak doublet neutrino �iL to a right-handed electroweak singlet neutrino �jR (i; j �e;�; �). One can express the effective Hamiltonian,which conserves the total lepton number, as

HDeff �

1

2�D�ij ��iL����jRF�� � H:c:; (39)

where ��Dijis Dirac diagonal (i � j) or transition

(i � j) magnetic moment between states �iL and�jR.A minimal extension of the SM with a right-handedsinglet neutrino yields a diagonal neutrino magneticmoment of [25]

�� �3

4�2

GFmem����2p �B: (40)

Here,�B � e=�2me� is the Bohr magneton andm� isthe neutrino mass. By using the neutrino oscillationdata one finds �� � 4 10�20 �B. It is believedthat larger values of magnetic moment are possiblein extensions of the SM, e.g., in models with largeextra dimensions [26]. A neutrino magnetic moment�� of the order of 10�11 �B might be an explanationof solar neutrino problem by flipping �e to sterileneutrinos [27–32].

(ii) The Majorana magnetic moment acts between �iLand �cjL chiral components of Majorana neutrinos,assuming gauge theory with only left-handed neu-trinos. As a consequence it violates the total leptonnumber by two units (�L � 2). The effectiveHamiltonian takes the form

HMeff �

1

2�Mij ��iL�

���cjLF�� � H:c: (41)

Majorana neutrinos cannot possess a flavor diagonal(i � j) magnetic moment due to the CPT theorem.They can have only transition (i � j) magnetic mo-ment [33]. From the definition one sees that ��ij

vanishes for i � j and also that magnetic moment isantisymmetric in indices i, j: ��ij � ���ji .

The limits on neutrino transition magnetic momentsarise from laboratory measurements of the �e � e scatter-ing cross section using solar, atmospheric and terrestrialneutrinos [34]. These experiments have put upper boundson �D

�ij , which are as follows:

�D�ej 0:9 10�10 �B; �D

��j 6:8 10�10 �B;

�D��j 3:9 10�7 �B: (42)

The most restrictive limits remains those from the astro-physical analysis of the energy-loss rate of stars due to the�! � �� process, which are [35]

�D�ij 3 10�12 �B: (43)

The assumption is that this energy-loss mechanism of giantred stars cannot exceed energy loss via weak processes. ForMajorana transition magnetic moments �M

�ij the abovelimits are twice stronger [36].

The cases of Dirac and Majorana magnetic momentshave fundamentally different physical applications andneed to be considered separately. In this paper we assumemassive neutrinos to be Majorana particles. In addition, weshall take advantage of the fact that neutrino magneticmoments have an intimate connection to their masses,which is model dependent. Recently, the contribution oftransition magnetic moments to the neutrino mass matrixwas estimated in [36]. It was found that if neutrino tran-sition magnetic moments are of the order of their currentupper bound, their contribution to Mij can exceed the

experimental value of Mphij . The purpose of this paper is

to discuss the impact of the Mphij on the transition magnetic

moments of Majorana neutrinos within the GUT con-strained MSSM with explicit R-parity violation.

In order to generate the magnetic moments within theR-parity breaking SUSY one needs to attach a photon toone of the internal lines in the loops from Figs. 1 and 2. Foreach Feynman diagram we have two possibilities. One can

GOZDZ, KAMINSKI, SIMKOVIC, AND FAESSLER PHYSICAL REVIEW D 74, 055007 (2006)

055007-6

attach the photon to quark (lepton) or squark (slepton)internal line of the loop. However, the contributions com-ing from photon-sparticle interactions are suppressed bybig masses of the latter and we may safely ignore them.

The contribution coming from the squark-quark looptakes again into account squark and quark mixing. Weend up with

�q�ii0 � �1� ii0 �

12Qdme

16�2

Xjkl

(�0ijk�

0i0kl

Xa

VjaVlawqakmda

� �0ijk�0i0lj

Xa

VkaVlawqajmda

)�B; (44)

where now the loop integral takes a slightly more compli-cated form

wqjk �sin2k

2

�xjk2 lnxjk2 � x

jk2 � 1

�1� xjk2 �2

� �x2 ! x1�

�: (45)

Here Qd � 1=3 is the d-quark charge in units of e, and medenotes the electron mass. We note that the resultingformula is antisymmetric as expected.

The contribution from the slepton-lepton loop reads

�‘�ii0� �1� ii0 �

4Qeme

16�2

Xjk

�ijk�i0kj

�w‘jkmej�w‘kjmek

��B;

(46)

where the loop integral is as previously equal to

w‘jk �sin2k

2

�yjk2 lnyjk2 � y

jk2 � 1

�1� yjk2 �2

� �y2 ! y1�

�: (47)

It is worthwhile to notice that if quark mixing is ne-glected, sin2k=2 � mdkm~dk=�m

2~dk1�m2

~dk2� and the masses

of squarks m~dk1, m~dk2

are replaced with their average valuem~dk , the expression for�q

�ii0 of Ref. [11] is reproduced. Thesame procedure allows to obtain the result of [11] also for�‘�ii0

. In this case sin2k=2 � mekm~ek=�m2~ek1�m2

~ek2� and

m~ek1, m~ek2

� m~ek (the average mass of sleptons) have to beconsidered.

The matrices wqjk and w‘jk in flavor space are evaluatedfor two representative sets of input parameters A0, m0,m1=2, tan� and�> 0. By assuming A0 � 100 GeV,m0 �

m1=2 � 150 GeV, tan� � 19, �> 0 we end up with

wq �2:582 10�12 5:024 10�11 2:332 10�9

6:759 10�10 1:315 10�8 6:068 10�7

2:750 10�7 5:350 10�6 2:421 10�4

0B@

1CA;

(48)

w‘ �1:223 10�14 2:446 10�12 5:163 10�11

2:691 10�10 5:384 10�8 1:132 10�6

4:832 10�8 9:668 10�6 2:019 10�4

0B@

1CA:

(49)

If larger SUSY mass scale is assumed with A0 � 500 GeV,m0 � m1=2 � 1000 GeV, tan� � 19, �> 0, we obtain

wq �9:895 10�15 1:925 10�13 8:809 10�12

2:780 10�12 5:408 10�11 2:466 10�9

1:383 10�9 2:690 10�8 1:216 10�6

0B@

1CA;

(50)

w‘ �5:316 10�17 1:064 10�14 2:126 10�13

1:301 10�12 2:602 10�10 5:195 10�9

2:755 10�10 5:511 10�8 1:098 10�6

0B@

1CA:

(51)

IV. RESULTS

The main purpose of our present work is to calculate thetransition magnetic moments for Majorana neutrinos. Toachieve this goal one proceeds as follows. First one findsthe values of the loop integrals v‘, vq, w‘ and wq withinsome GUT scenario. We have chosen two sets of parame-ters: �A0 � 100 GeV; m0 � m1=2 � 150 GeV� and �A0 �

500 GeV; m0 � m1=2 � 1000 GeV�, in both of them keep-ing tan� � 19 and positive �. This allows us to constructthe theoretical mass matrices, Eqs. (18) and (21), andconfront them with phenomenological ones. Next we cal-culate the contributions from the ‘-loop and the q-loop.The crucial point is that we consider each mechanismseparately, which means that only one element from thesums in Eqs. (18) and (21) is picked up at a time. Suchapproach is justified by the usual assumption that there isno fine-tuning between different contributions, whichtherefore may by analyzed separately. Explicitly, the sim-plified expressions read:

�q�ii0 ’ �1� ii0 �

4

3�Bme1Mq

ii0

�Pa VjaVlaw

qak=mqaP

a VjaVlavqakmqa

�max

� �1� ii0 �Mqii0f

qSUSY; (52)

�‘�ii0’ �1� ii0 �4�Bme1M‘

ii0

�w‘jk=mej

v‘jkmej

�max

� �1� ii0 �Mqii0f

‘SUSY; (53)

where the �. . .�max symbol denotes that we have checked allthe combinations of indices fj; k; lg in the first case andfj; kg in the second one, and picked up the dominantmechanism.

The coefficients f‘;qSUSY convert the neutrino masses intomagnetic moments. We list their values, obtained for four

TRANSITION MAGNETIC MOMENTS OF MAJORANA . . . PHYSICAL REVIEW D 74, 055007 (2006)

055007-7

different sets of GUT parameters, in Table I. The fq�CKMSUSY

parameter takes into account quark mixing, which is absentin fqSUSY. This additional mechanism slightly lowers themaximal allowed value of the magnetic moment One seesalso that the dependence on the tan� parameter is veryweak, therefore in the following we will discuss only thecase of large tan� � 19. For completeness we include alsoin the table the lower bounds on the f coefficients, whichwere obtained by finding the smallest of the expressions insquare brackets in Eqs. (52) and (53).

The resulting values of ��ii0for six different scenarios

are presented in Table II. Because of antisymmetricity thediagonal elements��ii are zero. The ranges of values in thefirst column are the highest upper limit and lowest upperlimit on the transition magnetic moment, coming fromdifferent mechanisms (different combination of couplingconstants �� or �0�0). They were calculated using theMph�HM mass matrix. One clearly sees that in this case

the different mechanisms give comparable results.However, the imposed GUT constraints have a muchbigger impact and introduce 2 orders of magnitudedifferences. The values from the first column for the firstset of GUT parameters (A0 � 100 GeV, m0 �m1=2 � 150 GeV) may be roughly compared with thepredictions published in Ref. [11] showing that for thisspecial case we our results are compatible with the pre-viously published.

The two remaining columns present the ranges of al-lowed values of the magnetic moments if one assumesnormal or inverted mass hierarchy of the neutrinos.These results take into account various mechanisms ofgenerating ��ii0

as well as the experimental uncertaintiesshown in mass matrices Mph�IH and Mph�NH. In this casethe ranges span roughly over 1–2 orders of magnitude. It isvisible that, regardless of the GUT parameters, one shouldnot expect the magnetic moment to be greater than

TABLE II. The Majorana neutrino transition magnetic moment ��ij (ij � e�, e�, ��) derived from phenomenological massmatrices (see text for details). The upper bounds on magnetic moments are obtained by using the current neutrino oscillation and the0���-decay data. The predictions for ��ij were calculated by assuming the inverted hierarchy or the normal hierarchy of neutrino

masses and that the lightest neutrino is massless. The ranges of allowed values correspond to uncertainty in SUSY coefficient fq;lSUSY

and in CP-violating violating phases. Two different GUT scenarios are considered. The remaining SUGRA parameters are tan� � 19and �> 0. �B is the Bohr magneton.

SUSY input in GeV Transition magnetic moment ��ij in �B

A0 m0 m1=2 ij 0��� constraints inverted hierarchy normal hierarchy

lepton-slepton loop mechanism100 150 150 e�; e� �0:62; 2:10� 10�15 �0:01; 5:00� 10�17 �0:13; 1:60� 10�17

�� �0:50; 1:70� 10�15 �0:45; 3:80� 10�17 �0:87; 4:10� 10�17

500 1000 1000 e�; e� �2:20; 5:50� 10�17 �0:04; 13:0� 10�19 �0:47; 4:20� 10�19

�� �1:80; 4:50� 10�17 �0:16; 1:00� 10�18 �0:32; 1:10� 10�18

quark-squark loop mechanism (without d-quarks mixing)100 150 150 e�; e� �0:41; 1:50� 10�16 �0:08; 36:0� 10�19 �0:08; 1:10� 10�18

�� �0:33; 1:20� 10�16 �0:30; 2:80� 10�18 �0:58; 3:00� 10�18

500 1000 1000 e�; e� �1:40; 4:00� 10�18 �0:03; 9:60� 10�20 �0:30; 3:00� 10�20

�� �1:20; 3:20� 10�18 �1:00; 7:30� 10�20 �2:00; 7:80� 10�20

quark-squark loop mechanism (with d-quarks mixing)100 150 150 e�; e� �0:36; 1:30� 10�16 �0:07; 31:0� 10�19 �0:76; 9:70� 10�19

�� �0:29; 1:00� 10�16 �0:27; 2:30� 10�18 �0:51; 2:50� 10�18

500 1000 1000 e�; e� �1:30; 3:40� 10�18 �0:03; 8:30� 10�20 �0:28; 2:60� 10�20

�� �1:10; 2:80� 10�18 �0:97; 6:40� 10�20 �1:90; 6:80� 10�20

TABLE I. The SUSY coefficients converting elements of neutrino mass mass matrix to transition magnetic moments of Majorananeutrinos (see Eqs. (52) and (53) for definition). It is assumed �> 0.

SUSY input The SUSY conversion coefficientA0 [GeV] m0 [GeV] m1=2 [GeV] tan� fqSUSY [eV�1] fq�CKM

SUSY [eV�1] f‘SUSY [eV�1]

100 150 150 5 �0:3; 1:0� 10�16 �2:8; 8:8� 10�17 �0:5; 1:5� 10�15

19 �0:3; 1:2� 10�16 �2:8; 9:8� 10�17 �0:5; 1:6� 10�15

500 1000 1000 5 �1:1; 2:8� 10�18 �1:0; 2:4� 10�18 �1:7; 4:1� 10�17

19 �1:1; 3:1� 10�18 �1:0; 2:7� 10�18 �1:7; 4:3� 10�17

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055007-8

10�17 �B, which is not possible to detect experimentallynowadays.

It is interesting to investigate the influence of quarkmixing on the results. When quark mixing is switchedoff, the maximal allowed �q

� values drop down approxi-mately by 10 to 20%.

In the case when the lightest neutrino mass is not set tozero, one may express all the remaining mass eigenstates interms of m1�3� and appropriate differences of massessquared, which are known from the results of the neutrinooscillation experiments. On Fig. 3 we present the allowedvalues for different elements of the neutrino mass matrix asthe function of m1�3� in the case of NH and IH. The shadedregions correspond to all possible combinations of the twoMajorana �12 and �13, and one Dirac phase , forsin213 � 0 (dark shading) and sin213 0:05 (lightershading). One sees clearly, that the phase factors as wellas possible nonzero mass of lightest neutrino modify theresults in a nontrivial fashion.

V. SUMMARY

Since the neutrinos have no electric charge they can beMajorana fermions. A Majorana neutrino is more naturalthan a Dirac neutrino as it allows to explain the smallnessof their masses in theories with grand unification.Majorana neutrinos have zero diagonal magnetic momentswhile they may in general have transition magnetic mo-ments ��ij (i � j). The question is, how large they are? Inthis paper we studied this issue within the GUT constrainedMSSM with explicit R-parity violation.

The transition magnetic moments were induced by at-taching photons to the internal lines of lepton-slepton andquark-squarks loop diagrams that generate neutrinomasses. ��ij has been written as a product of SUSYconversion coefficients fSUSY, which reflects the depen-dence on SUSY parameter space and quark mixing, and theelement of Majorana neutrino mass matrix in flavor basis.

Two possible GUT scenarios were considered: A0 �100 GeV, m0 � m1=2 � 150 GeV, and A0 � 500 GeV,m0 � m1=2 � 1000 GeV, in both of them having tan� �19 and positive � parameter. Our study showed that fSUSY

is nearly insensitive on the value of tan� and that itsdependence on the scale of GUT common mass and cou-pling constants parameters m0, m1=2, and A0 dominates.The difference in fSUSY for these two scenarios reacharound 2 orders of magnitude. It is worth mentioning thatcontributions to ��ij from lepton-slepton mechanismsdominates over contributions from quark-squark diagrams.

The transition magnetic moments were calculated byconsidering three differently constructed phenomenologi-cal neutrino mass matrices. The elements of the first one(Mph�HM) represent the maximal allowed values, if neu-trino oscillation data [4,6] and the lower bound on the0���-decay half-life of 76Ge [23] are taken into account.The corresponding upper limits on ��e� , ��e� and ����

have been found to be of the order of 10�15 �B and10�17 �B for the assumed two SUSY scenarios.

Two phenomenological neutrino mass matrices weregenerated by considering normal and inverted hierarchyof neutrino masses and that the mass of the lightest neu-trino is zero. This allowed to make predictions for transi-tion magnetic moments of Majorana neutrinos. For low(large) SUSY parameter scale we found the maximal value5 10�17 �B (1:3 10�18 �B). The dependence of theresults on the neutrino mass pattern is weak.

The magnetic moments ��e� , ��e� and ���� were calcu-lated also as a function of the lightest neutrino mass. Thebest fit values of neutrino oscillation parameters wereconsidered. Only for the mixing angle 13 it was assumedsin213 � 0 and sin213 � 5 10�2. It is worth mention-ing that the maximal mixing of atmospheric neutrinos(23 � �=4) implies that the allowed ranges of ��e� and��e� are the same, if the Dirac and Majorana CP violatingphases vary from 0 to�=2. The results show that especially

m3 [eV]10

-19

10-18

10-17

10-16

10-15

µ ν ij[µ

B]

m3 [eV]

10-4

10-3

10-2

10-1

100

m1 [eV]

10-19

10-18

10-17

10-16

10-15

µ ν ij[µ

B]

10-4

10-3

10-2

10-1

100

m1 [eV]

IH, ij=eµ, eτ IH, ij=µτ

NH, ij=eµ, eτ NH, ij=µτ

FIG. 3 (color online). Transition magnetic moments ofMajorana neutrinos ��e� , ��e� and ���� as function of thelightest neutrino mass m3 (the inverted hierarchy of neutrinomasses, upper panels) and m1 (the normal hierarchy of neutrinomasses, lower panels). The results are presented for sin213 � 0(the region with solid line boundaries) and sin213 � 5 10�2

(the region with dashed line boundaries). In the calculationthe best fit values of mass-squared differences �m2

12, �m223

and mixing angles 12, 23 were considered [4,6]. ForCP-violating phase we assumed ;2; 3 � �0; 2��. The largestvalue of the SUSY coefficient fq;lSUSY for the set of SUSYparameters �A0 � 100 GeV; m0 � m1=2 � 150 GeV�, tan� �19 and �> 0 was taken into account. IH and NH denote theinverted hierarchy and the normal hierarchy of neutrino masses,respectively. �B is the Bohr magneton.

TRANSITION MAGNETIC MOMENTS OF MAJORANA . . . PHYSICAL REVIEW D 74, 055007 (2006)

055007-9

��e�;e� (inverted hierarchy of neutrino masses) dependsstrongly on the values of CP phases.

The obtained results show that the values of theMajorana transition magnetic moments might be signifi-cantly above the scale of the Dirac-type magnetic momentin minimal extension of the SM with right-handed neutri-nos. However, the maximal values are still too small to betested in the present laboratory experiments or to haveastrophysical consequences.

ACKNOWLEDGMENTS

Thanks are due to S. G. Kovalenko for useful discussionon the R-parity violating MSSM. This work was supportedby the VEGA Grant agency of the Slovak Republic undercontract No. 1/3039/06, by the EU ILIAS project under

contract No. RII3-CT-2004-506222, by the DFG projectNo. 436 SLK 17/298 and the Polish State Committee forScientific Research under grants no. 2P03B 071 25 and1P03B 098 27. One of us (M. G.) greatly acknowledges thefinancial support from the Foundation for Polish Science.

APPENDIX

In this appendix, for convenience of the reader wepresent explicit expressions for the neutrino mass matrixin flavor basis, M�� (�;� � e;�; �), as functions of themoduli of neutrino mass eigenvalues mi, of mixing anglesij and of the CP violating phases , 2, 3 [37]. Thematrix element M is symmetric, i.e., it is defined by sixelements:

Mee � c213c

212m1� c2

13s212m2e�i22 � s2

13e2i m3e�i23 ;

Me� ��c12c13�c23s12� c12s23s13e�i �m1� c13s12�c23c12� s23s12s13e�i �m2e�i22 � c13s23s13ei m3e�i23 ;

Me� ��c12c13��s23s12� c23c12s13e�i �m1� c13s12�c12s23� c23s12s13e�i �m2e�i22 � c23c13s13ei m3e�i23 ;

M�� � �c223s

212� 2c23c12s23s12s13e�i � c2

12s223s

213e�2 �m1��c2

23c212� 2c23c12s23s12s13e�i � s2

23s212s

213e�2 �m2e�i22

� c213s

223m3e�i23 ;

M�� ���c23s23s212� c

223c12s12s13e

�i � c12s223s12s13e

�i � c23c212s23s

213e�2i �m1��c23c

212s23� c

223c12s12s13e

�i

� c12s223s12s13e

�i � c23s23s212s

213e�2i �m2e

�i22 � c23c213s23m3e

�i23 ;

M�� � �s223s

212� 2c23c12s23s12s13e

�i � c223c

212s

213e�2i �m1��c

212s

223� 2c23c12s23s12s13e

�i � c223s

212s

213e�2i �m2e

�i22

� c223c

213m3e

�i23 : (A1)

The maximal mixing of atmospheric neutrinos (23 � �=4) implies symmetry among some of elements of the neutrinomass matrix M��. If CP-violating phases are considered as free parameters the allowed ranges of jMe�j and jMe�j

(jM��j and jM��j) coincide each with other. If additional restriction is taken into account, namely sin2�13� � 0, weobtain

Mee � c212m1 � s2

12m2e�i22 Me� �1���2p c12s12��m1 �m2e�i22�; Me� � �

1���2p c12s12��m1 �m2e�i22�;

M�� �1

2�s2

12m1 � c212m2e

�i22 �m3e�i23�; M�� � �

1

2�s2

12m1 � c212m2e

�i22 �m3e�i23�;

M�� �1

2�s2

12m1 � c212m2e�i22 �m3e�i23�:

(A2)

In this case jMe�j � jMe�j and M�� �M��. In addition, for , 2, 3 between 0 and 2� the maximal and minimalabsolute values of all three elements M��, M�� and M�� are the same.

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