Neutrino masses and lepton-number violation in the littlest Higgs scenario

PHYSICAL REVIEW D 72, 053007 (2005)

Neutrino masses and lepton-number violation in the littlest Higgs scenario

Tao Han and Heather E. Logan*Department of Physics, University of Wisconsin, Madison, Wisconsin 53706, USA

Biswarup Mukhopadhyaya and Raghavendra SrikanthHarish-Chandra Research Institute, Chhatnag Road, Jhusi, Allahabad - 211 019, India

(Received 15 June 2005; published 14 September 2005)

*Present ad1125 Colonel

1550-7998=20

We investigate the sources of neutrino mass generation in little Higgs theories, by confining ourselves tothe ‘‘littlest Higgs’’ scenario. Our conclusion is that the most satisfactory way of incorporating neutrinomasses is to include a lepton-number violating interaction between the scalar triplet and lepton doublets.The tree-level neutrino masses generated by the vacuum expectation value of the triplet are found todominate over contributions from dimension-five operators so long as no additional large lepton-numberviolating physics exists at the cutoff scale of the effective theory. We also calculate the various decaybranching ratios of the charged and neutral scalar triplet states, in regions of the parameter spaceconsistent with the observed neutrino masses, hoping to search for signals of lepton-number violatinginteractions in collider experiments.

DOI: 10.1103/PhysRevD.72.053007 PACS numbers: 14.60.Pq, 14.80.Cp

I. INTRODUCTION

Little Higgs theories [1–3] represent a new attempt toaddress the problem of quadratic divergence in the mass ofthe Higgs boson responsible for electroweak symmetrybreaking. This approach treats the Higgs boson as part ofan assortment of pseudo-Goldstone bosons, arising from aglobal symmetry spontaneously broken at an energy scale�, typically on the order of 10 TeV. There is also anexplicit breakdown of the overseeing global symmetryvia gauge and Yukawa interactions, thereby endowing theGoldstone bosons with a Coleman-Weinberg potential andmaking them massive. The Higgs mass is thus protected bythe global symmetries of the theory and only arises radia-tively due to the gauge and Yukawa interactions. As aneffective theory valid up to the scale �, the model is rathereconomical in terms of the new fields introduced in order tofulfill the necessary cancellation for the quadratic diver-gence at the one-loop level. The model requires, in additionto new gauge bosons and vectorlike fermions, the existenceof additional scalars belonging to certain representations ofthe standard model (SM) gauge group.

Aside from the crucial vectorlike T quark, the fermionicsector can essentially have the same appearance as in theSM. There is no attempt made to address the origin offermion masses and mixing. In fact, the theory wouldencounter extremely stringent constraints from the absenceof excessively large flavor-changing neutral currents andCP violation in the fermionic sector [4] if the scale re-sponsible for flavor physics is at the order of the cutoffscale �. Flavor issues are thus ostensibly left out as prob-lems awaiting the more fundamental theory at higher en-ergies, the so-called UV completion of the theory, that

dress: Department of Physics, Carleton University,By Drive, Ottawa, Ontario K1S 5B6, Canada.

05=72(5)=053007(13)$23.00 053007

would hopefully lead to the SM structure or similar as aneffective low-energy realization.

However, one may like to remember that the only areawhere experimental hints of new physics have been foundso far is the neutrino sector [5]. It is therefore both inter-esting and important to see if little Higgs theories canaccommodate neutrino masses and mixing as suggestedby the observed data. It is even not unreasonable to say thatit will be a vindication of little Higgs theories if they atleast suggest a mechanism for the generation of neutrinomasses. The present work aims to buttress this attempt. Arethe neutrinos acquiring their masses through interactionwith new particles already postulated in the theory? Whatcan be the detectable signatures of the model carryingimprints of the fact that its low-energy Lagrangian andparticle spectrum address the issue of neutrino masses? Weexamine these questions by adopting the ‘‘littlest Higgs’’(LtH) model [2], which has been extensively studied inrecent literature.

We explore the most economic extension of the basicmodel that is required to accommodate neutrino massesand is consistent with the demand that it does not affect thecancellation of quadratic divergences in the SM Higgsmass. In particular, we make use of the fact that the LtHscenario contains, in addition to the usual Higgs doublet,an additional set of scalars that form a complex triplet [6]under the SU(2) gauge group of the standard model withhypercharge Y � 1 �Q � I3 � Y�. This complex tripletforms part of the assortment of Goldstone bosons when aglobal SU(5) breaks down to SO(5) at the scale � in thismodel. There is an additional gauged SU�2� � U�1� be-yond that of the SM, which is also spontaneously broken atscale �; some of the aforementioned Goldstone bosons areabsorbed as longitudinal components of the extra gaugebosons. Ten scalar degrees of freedom remain after this,and are found to consist of a doublet (H) and a complex

-1 © 2005 The American Physical Society

http://dx.doi.org/10.1103/PhysRevD.72.053007

HAN, LOGAN, MUKHOPADHYAYA, AND SRIKANTH PHYSICAL REVIEW D 72, 053007 (2005)

triplet (�) under the electroweak SU(2). The complextriplet offers a chance to introduce lepton-number violatinginteractions into the theory. We find that the most satisfac-tory way of incorporating neutrino masses is to exploitsuch an interaction of the lepton doublets, leading to aMajorana mass for neutrinos and lepton-number violationby two units. Then we proceed to examine the parameterrange of this model consistent with the observed neutrinomasses, and look at the consequence it has on the phe-nomenology of the model. In particular, we focus on thedecays of the additional SU(2) triplet scalar states intro-duced in this scenario, which can have masses of order aTeV. We present calculations of the decay branching ratiosof the triplet states, discuss the complementary roles ofdifferent decay channels to test the scenario, and commenton their potential collider signatures within the region ofthe parameter space that is consistent with the observedneutrino masses.

Our paper is organized as follows. In Sec. II, the status ofneutrino mass generation with a heavy right-handed neu-trino is first briefly reviewed. We then take up the case ofneutrino masses without any right-handed neutrino, andshow that the LtH construction can accommodate theobserved neutrino mass and mixing patterns. In particular,with the help of the complex triplet, one obtainsdimension-four lepton-number violating operators (�L �2). The Majorana neutrino masses and their mixing can begenerated by these operators consistent with current ob-servations without necessarily pushing the couplings totiny values; instead, the smallness of the neutrino massescan be driven in part by a tiny triplet vacuum expectationvalue (vev). We also discuss the �L � 2 operators with thefull gauge symmetry of the model and find that in such ascenario the couplings would have to be of order 10�11 toaccommodate the observed neutrino masses. In Sec. III, westudy the decay channels of the triplets. These, we empha-size, constitute the characteristic signals of the triplet andallow a test of the mechanism of neutrino mass generation.We summarize and conclude in Sec. IV. The features of theLtH scenario and the interactions of the triplet that arerelevant for our phenomenological study are summarizedin Appendix A. The triplet decay partial widths are listed inAppendix B.

II. NEUTRINO MASSES

A. Neutrino masses with right-handed neutrinos

In the SM as well as the simplest little Higgs construc-tions, there are no right-handed neutrino states that aresinglets under SM gauge interactions. By introducingright-handed neutrinos (NR), one can obtain gauge-invariant Dirac mass terms from the SU(2) doublets ofthe leptons L and the Higgs H,

yDijLLiHyNRj � H:c:; (1)

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with i; j being generation indices, as well as Majoranamass terms

�MijNcRiNRj � H:c: � MijNT

RiC�1NRj � H:c:; (2)

where C is the charge-conjugation operator in the notationof, e.g., Ref. [7].

The Dirac terms alone lead to a contribution to theneutrino mass of the order m� � y

Dv. Since the neutrinomasses are known to be at most of order 0.3 eV [8], theYukawa couplings would have to be extremely small, yD &

10�12. While technically natural, such tiny Yukawa cou-plings are difficult to rationalize.

Including the Majorana terms, light neutrino masses aregenerated at the order �yDv�2=M [9] by virtue of the well-known seesaw mechanism [10]. If we assume that theYukawa couplings yDij are naturally of the order of unity,thenM * 1013 GeV in order to obtain a neutrino mass lessthan about 0.3 eV. The problem, however, is that if we takethe Majorana mass scale to be near the little Higgs cutoff� ’ 10 TeV, then all of the Yukawa couplings would haveto be quite small and all roughly equal, yDij & 10�5 for allthree generations. This is in contrast to the correspondingcharged leptons, for which the Yukawa couplings exhibit alarge hierarchy between generations. Of course, the right-handed neutrino mass that determines the seesaw scalecould be much higher than �, as in the usual seesawscenario within the standard model. However, in thiswork we wish to look for alternative explanations of theneutrino masses within the context of the LtH scenariowith observable signatures that do not rely upon physicsabove the cutoff scale �.

B. Neutrino masses in the absence of right-handedneutrinos

To us, the solution seems to be in avoiding the introduc-tion of massive right-handed neutrinos altogether in a littleHiggs scenario. One can still construct Majorana massterms with the help of the Higgs triplet in the LtH model,obtained from a dimension-four �L � 2 coupling,

L � iYijLTi �C

�1Lj � H:c: (3)

Note that the definition of � here includes ��i�, as evidentfrom Eq. (A3). With the vev of �0 being v0, the inducedneutrino masses are of the order of Yv0. With a sufficientlysmall v0, as preferred by the precision electroweak data[11], adequate neutrino masses may be generated. Theoccurrence of such Majorana masses has already beendiscussed in the context of general models with tripletscalars [6,12].

In the LtH model, however, some additional caution isnecessary, since here we have an effective theory witha rather low cutoff. It can be argued that, if there islepton-number violating physics at the scale �, then it ispractically impossible to prevent the appearance of

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NEUTRINO MASSES AND LEPTON-NUMBER VIOLATION . . . PHYSICAL REVIEW D 72, 053007 (2005)

dimension-five operators of the form

Y5�HL�2

�(4)

giving rise to neutrino masses on the order of Y5v2=�. Thiscontribution to the neutrino masses is inadmissibly large ifY5 is naturally of order unity. Of course, one may suppressthe neutrino mass by requiring that the seesaw scale cor-responding to lepton-number violation is not � ��10 TeV�but some higher scale, perhaps corresponding to a grandunification scenario. However, as we have mentionedabove, this solution is somewhat unsatisfying in the littleHiggs context, since the entire issue of grand unification isunclear in a UV-incomplete theory.

The way out of the difficulty is to postulate that there isno additional lepton-number violating physics at the scale�, and that the only �L � 2 effect comes from the cou-pling given by Eq. (3). Such a postulate is plausible in thesense that the operator of Eq. (3) is renormalizable andindependent of the cutoff. Such a postulate also keeps thescenario minimal in terms of particle content, since right-handed neutrinos, unlike the scalar triplets, do not arisefrom any intrinsic requirement of the model. The absenceof right-handed neutrinos at or below the scale � preventsthe potentially dominant dimension-five operators ofEq. (4). Such operators can then arise only through loopeffects involving the �L � 2 couplings of the �L to thescalar triplet. As we demonstrate below in Sec. II C, thestructure of the Coleman-Weinberg potential ensures thatthe contributions of these loop-induced dimension-fiveoperators to the neutrino masses are subleading comparedto the tree-level �L � 2 interaction given above.

Thus, neutrino masses are perhaps best implemented inthe LtH model solely in terms of the tree-level �L � 2interaction of the scalar triplet. So far there is no need toattribute the effect to a high scale, since lepton-numberconservation is not dictated by any underlying symmetry ofthe theory. The relevance of this term is further accentuatedby the fact that the triplet vev in any case has to be quitesmall compared to the electroweak scale, in order to beconsistent with the limits on the � parameter [11,13]. Thus,seeds of small neutrino masses can already be linked to theelectroweak precision constraints.

It should be noted that although the LL� interactionterm is invariant under the standard SU�2�L � U�1�Y sym-metry, it breaks the full �SU�2� � U�1�2 gauge invarianceof the LtH model. The LL� interaction term is invariantunder the two U(1) symmetries so long as the U(1) chargesof the lepton doublet are chosen to cancel anomalies in thefull theory (see Sec. II D for details). On the other hand,this term breaks the �SU�2�2 part of the full gauge sym-metry because the triplet � is a Goldstone boson of the fulltheory and transforms nonlinearly under the two SU(2)s,while L transforms as a doublet under only SU�2�1. Wenote however that the real motivation for this enlarged

053007

gauge symmetry is the cancellation of potentially largequadratically divergent contributions to the Higgs mass.Apart from that, there is no requirement that such aninvariance holds in all sectors of the theory.

It can be seen through explicit calculation that thecancellation of quadratic divergences is not affected solong as the noninvariance under �SU�2� � U�1�2 is con-fined only to the lepton-number violating interaction of thetriplet. In particular, the global symmetry structure in thegauge and top-quark sectors that protects the Higgs mass atone-loop level is not affected by the new LL� interaction.The only effect of this interaction on the Coleman-Weinberg potential [14] for the scalars is a contributionto the coefficient of the triplet mass, ��2 (see Appendix Afor details), for which the modified one-loop expression is

��2 �a2

�g2

s2c2 �g02

s02c02

�� 8a0�2

1 � a00 Tr�YYy�; (5)

where a00 is an arbitrary O�1� constant reflecting the UV-incomplete nature of the theory. The overall constraint tobe satisfied by the modified expression is that ��2 shouldremain positive, so that the triplet vev, purportedly small, isgenerated through doublet-triplet mixing only. If a00 ispositive, it results in a slight enhancement of the tripletscalar mass compared to that in the minimal LtH scenario.Thus the introduction of the LL� interaction seems to beconsistent with the fundamental spirit of the little Higgsapproach. We lay out the full interaction terms of Eq. (3) inAppendix A for future phenomenological considerations.

C. Constraints from neutrino masses

Since our first concern is to see the viability of thisproposal, we begin by assuming neutrino masses to be oforder 0.1 eV. The left-handed Majorana neutrino massmatrix resulting from Eq. (3) in this scenario is

M ij � Yijv0: (6)

We neglect CP-violating phases. Then Y is a (3� 3)symmetric matrix with six independent parameters. Thephysical neutrino masses are the product of v0 and theeigenvalues of Y. The elements of Y can in principle beas large as perturbation theory permits; we consider themto have a natural size of order unity. The triplet vev v0 isrestricted to be & 1 GeV from the constraints on the �parameter [11,13].

The smallness of the neutrino masses leaves us with twoextreme alternatives, as described below.

(1) T

-3

he elements of Y are very small, typically of theorder 10�10, and v0 � 1 GeV. This means that theLL� interaction term in Eq. (3) supplies the physicsresponsible for the smallness of neutrino masses.

(2) Y
’ 1 together with an extremely small v0, arisingfrom a tiny value of the induced doublet-tripletmixing coefficient �h�h in the Coleman-Weinbergpotential. In this case the Coleman-Weinberg


potential provides the physics behind the smallnessof neutrino masses, while the origin of bi-largemixing has to be sought in the relative values ofthe different Yij.

The first option leads to very small couplings, which couldbe argued to be unnatural. One needs to remember that thephysics linked with Yij is not only lepton-number violationbut also lepton-flavor violation. Therefore the coupling inEq. (3) must have its origin at a scale much higher than �,in order to avoid unacceptable flavor violation in the low-energy theory and the appearance of large dimension-fiveoperators. Thus the explanation for the smallness of theneutrino masses is pushed up to scales much higher than �.

The second scenario, on the other hand, has a certainadvantage. In addition to generating neutrino masses of theright order, one also has to explain the observed bi-largemixing pattern in the neutrino sector. A model-independentfit of such mixing requires one to fine-tune the elements ofY. Having all six elements on the order of 10�10 enhancesthe degree of fine-tuning even further. It may therefore be aslightly less disquieting prospect to envision the ‘‘fine-tuned’’ elements of Y as being close to unity, and have avery small vev for the triplet. The generation of such asmall vev must be accomplished by appropriate values ofthe parameters that determine �h�h at the scale �. As canbe seen from the detailed expressions listed in Appendix A1, a small triplet vev can arise, for example, from a can-cellation of the gauge and Yukawa contributions to theColeman-Weinberg potential. While a theoretical explana-tion has to await the UV completion of the scenario, this

νc l ν

φ− W−

W−φ0 h0

(a)

× ×

νc l

φ−

Gh0

×

FIG. 1. Representative one-loop diagrams giving ris

053007

situation is consistent with all other aspects of the model,and has distinctive phenomenological implications. Thuswe have chosen to explore such implications in detail,remembering all along that the final explanation for thesmallness of the neutrino masses is linked to the UVcompletion of the LtH model.

To summarize, we will concentrate only on the operatorof Eq. (3), with the requirement m� ’ Yv

0 ’ 10�10 GeV.Within this constraint, the �L � 2 coupling Y [which isactually a (3� 3) matrix] and the triplet vev v0 can varyover a wide range in our formulation. As we shall see in thenext section, the phenomenological consequences are es-pecially interesting in the parameter ranges

10�5 < Yij & 1; 0:1 MeV> v0 > 1 eV: (7)

It is important not to overlook other potentially signifi-cant contributions to the neutrino masses throughdimension-five operators induced at the one-loop level.Some representative diagrams leading to such operatorsare shown in Fig. 1, where we have worked in the ’t Hooft-Feynman gauge. All of these diagrams give neutrinomasses of the form Mij�TLiC

�1�Lj.The neutrino mass from Fig. 1(a) is

Mij � ivv0MWg3Yij��

2p

Z d4p

�2��41

�p2 �m2��p

2 �M2W�

2

�Yijv0�

g4v2

32��2p�2m2

�

: (8)

νc l ν

φ− W−

G−φ0

h0

h0

(b)

× × ×

ν

W−

− h0

(c)

×

e to neutrino masses via dimension-five operators.

-4

TABLE I. Charge assignments of the lepton and scalar fieldsand of the operator in Eq. (11) under the two U(1) gauge groupsand hypercharge, with �;� � 1; 2.

�� L LT��C�1L�

U�1�1 3=5 3=10� ye 6=5� 2yeU�1�2 2=5 �4=5� ye �6=5� 2yeHypercharge 1 �1=2 0


Clearly this is a subleading contribution compared to Yijv0,being suppressed by a loop factor times v2=m2

�. Similarly,the contribution from Fig. 1(b) is

M ij ��Yijv0�g2�h��hv2

32��2p�2m2

�

� �Yijv0�

g2v2

24��2p�2f2

; (9)

where we have used the relation �h��h � �4��2=3 �

�4m2�=3f2 (see Appendix A for details). This is again

suppressed by a loop factor times v2=f2.The contribution from Fig. 1(c) is

Mij � ig2

4

Yij��2p �h�hfv

2Z d4p

�2��41

�p2 �m2��p

2 �M2W�

2

�Yijv0�

g2

32��2p�2; (10)

where we have made use of the relation v0 ��h�hv2=2��2f from the minimization conditions of theColeman-Weinberg potential and ��2f2 ’ m2

� (seeAppendix A). Unlike the other diagrams, the neutrinomass contribution from this diagram is suppressed byonly a loop factor. This is because the size of the diagramin Fig. 1(c) is controlled by the doublet vev v. The specialrelationships among parameters in the Coleman-Weinbergpotential enables one to reexpress the contribution in termsof the triplet vev v0. The contribution is larger than thatfrom Figs. 1(a) and 1(b), although the loop factor ensuresthat it is subleading. Such a contribution can play a poten-tially important role in determining the precise values ofthe neutrino mixing angles.

There is also a diagram in which the vertical W� propa-gator in Fig. 1(a) is replaced by the charged GoldstoneG�.Since the coupling of G� to ��; ‘ is suppressed by m‘=MW ,this diagram gives only a small contribution. Similarly,diagrams involving a virtual Z boson and neutrinos arenegligible. Finally, the above expressions are subject toadditional corrections due to doublet-triplet mixing, whichare further suppressed by v0=v.

D. �L � 2 operators with larger symmetry

Thus far, our approach to constructing lepton-numberviolating operators has been guided by the SM gaugeinvariance and naturalness considerations, subject to theexperimental constraints on the neutrino masses and mix-ing patterns. The treatment of the scalar triplet separatefrom the doublet requires some mechanism to split theinteractions of these two components of the nonlinear �field, which is beyond the scope of our phenomenologicalconsiderations in the current work. Nevertheless, it istempting to ask if one can instead construct operatorsthat respect the full gauge symmetry of the LtH model,namely �SU�2� � U�1�2 gauge invariance.

Following the conventions of Refs. [2,15], in which thethird-generation quark doublet is extended to �T �

053007

�bLtLTL�, we write the lepton doublets as LT � �‘L�L�.We can then write down the following lepton-flavor violat-ing operator,

L LFV � �12Yijf�L

Ti ��C

�1�Lj�� H:c:; (11)

where i; j are generation indices and �;� � 1; 2 are SU(5)indices. This operator is gauge invariant under both theSU�2�1;2 gauge groups and under hypercharge. This opera-tor is also gauge invariant under both of the U�1�1;2 gaugegroups if the lepton charges under the two U(1) groups aregiven by Y1�L� � �3=10 and Y2�L� � �1=5. In the nota-tion of Ref. [15], this corresponds to ye � 3=5, as shown inTable I. This is the same condition that ensures anomalycancellation among the SM fermions. This can be under-stood as follows. The anomaly cancellation condition issatisfied when the U�1�1;2 charges of the fermions areproportional to their hypercharges. Since the operator inEq. (11) conserves hypercharge, the anomaly-free condi-tion is sufficient to ensure that this operator also conservesthe U�1�1;2 charges.

Expanding the upper two-by-two block of the matrix�� in terms of the scalar fields H and � (seeAppendix A), we have

�� 2

f

�� =��2p

��=��2p

�0

!�

1

f2

h�h� h�h0

h�h0 h0h0

!

� � � � : (12)

Inserting this into Eq. (11), we obtain

LLFV � Yij

��TLiC

�1�Lj

��0 �

1

2fh0h0

�

� ��TLiC�1‘Lj � ‘TLiC

�1�Lj��

1��2p ��

1

2fh�h0

�

� ‘TLiC�1‘Lj

��

1

2fh�h�

�� H:c: (13)

Clearly, the nonlinear sigma model has served to relate thedimension-four �ci �j�

0 coupling to the dimension-five�ci �jh

0h0 coupling. This gives rise to a mass matrix forthe neutrinos involving both v0 and v:

M ij � Yij

�v0 �

v2

4f

�: (14)

Equation (13) gives, to the leading order, the following

-5


dimension-four couplings of scalars to left-handed leptonpairs:

L dim -4LFV � Yij

�‘TLiC

�1‘Lj�� 1��2p ��TLiC

�1‘Lj

� ‘TLiC�1�Lj�� TLiC

�1�Lj�0

�

� Y�ij

�‘LiC‘Lj

T�� 1��2p ��LiC‘Lj

T

� ‘LiC�LjT�� LiC�LjT�0�

�; (15)

where in the last two lines we have explicitly written outthe Hermitian conjugate piece. Note that �0 is a complexfield containing real scalar and pseudoscalar degrees offreedom, �0 � ��s � i�p�=

��2p

.The expression (11) is invariant under the full �SU�2� �

U�1�2 gauge symmetry and preserves the nonlinear sigma-model form for the scalar interactions. However, the priceto pay in such an approach is that one has to includedimension-five terms proportional to H2 from the begin-ning, and thus have contributions to the neutrino massesproportional to v2=f. Unlike the dimension-five operatorsgenerated by the diagrams in Fig. 1, these contributions arenot proportional to Yijv0 times a loop suppression factorand cannot in general be made small, since f ’ TeV if wehave to stabilize the Higgs mass. As a result, this approachalmost invariably ends up requiring values

Yij � 10�11; (16)

for the �L � 2 couplings of all i; j. They are indeedunnaturally small. This implies the need for a more funda-mental explanation for neutrino masses beyond the effec-tive theory at the scale �.

On the other hand, in our approach of separating thelepton-number violating couplings of � and H, one canavoid extreme fine-tuning of the Yij couplings and at thesame time ensure neutrino masses of a size consistent withexperimental data. This is because our starting point is thedimension-four renormalizable operator of Eq. (3), as op-posed to the higher dimensional ones discussed in thealternative approach. Thus our formulation by keepingonly the L-violating terms of Eq. (3) is independent ofthe cutoff. It is admittedly a phenomenological approach,and assumes that, whatever be the mechanism responsiblefor the breakdown of �SU�2� � U�1�2 in the L-violatingsector, any additional induced term proportional to �v2=f�is suppressed. We nonetheless feel that this approach isquite general and model-independent, especially becausethe cancellation of quadratically divergent contributions tothe SM Higgs mass remains unaffected, as was discussedin Sec. II B.

We take this opportunity to note that an attempt has beenrecently made in Refs. [16,17] similar to the approach ofEq. (11). In Ref. [17], this operator was given in the form

053007

L LFV � zij��f�LTi ��C

�1�LTj �� H:c:; (17)

which is equivalent to our result if LT � ��‘� is used inEq. (17). The authors of Refs. [16,17] also found the sameconclusion as in Eq. (16), that Yij � 10�11.

III. DECAYS OF THE TRIPLET STATES

We now examine the observable consequences of thescalar triplet having a vev and lepton-number violatinginteractions compatible with the observed neutrino masses.In particular, we consider the decays of the scalar tripletinto various characteristic final states, and discuss theirobservable signals in future collider experiments.

First of all, we note that the mechanism of scalar massgeneration through the Coleman-Weinberg mechanism[14] in the LtH model implies that the members of thetriplet, ��, ��, �s, and �p (where �s and �p are thescalar and pseudoscalar components of�0), are degenerateat lowest order with a common mass m�. Their masses aresplit by electroweak symmetry breaking effects, leading tomasses m��1�O�v2=m2

��. The mass splittings are thusquite small for m� MW , and we will neglect them inwhat follows. The relevant interaction terms for the �L �2 processes are given in Table II in Appendix A. The other� couplings conserving the lepton number have been givenin Ref. [15]. For completeness, they are also tabulated inTable III in Appendix A. The possible decays of the tripletstates are

�� ! ‘�i ‘�j ; W�W�;

�� ! ‘�i ��‘j ; t �b; T �b; W�Z; W�h;

�s ! �i�j; ��i ��j; t�t; b �b; t �T � �tT; ZZ; hh;

�p ! �i�j; ��i ��j; t�t; b �b; t �T � �tT; Zh: (18)

The full set of partial decay widths is listed in Appendix B.To clearly see the interesting physics points, we discuss

the partial decay widths for the doubly charged Higgsboson for m� MW ,

�� ! ‘�i ‘�i � �

jYiij2m�

8�;

�� ! W�L W�L �

v02m3�

2�v4 ;

�� ! W�T W�T �

g4v02

4�m�;

where WL (WT) stands for the longitudinal (transverse)component of the W boson. We first point out that the�L � 2 processes, �� ! ‘�i ‘

�j , are all driven by the

lepton-number violating Yukawa coupling Yij. These de-cays to the lepton states will constitute the smoking gunsignatures of the scenario proposed by us. The decays intotwo gauge bosons, on the other hand, depend directly on v0,

-6

0

0.2

0.4

0.6

0.8

1

0.00010.00001 0.001

fo

oitar

gni

hc

nar

B φ

++

v’ (GeV)

mφ = 2 TeV

W+L W+

L

l+i l+

j

0

0.2

0.4

0.6

0.8

1

800 1000 1200 1400 1600 1800 2000 2200 2400

fo

oitar

gni

hc

nar

B φ

++

mφ (GeV)

W+L W+

L

l+i l+

j

v’ = 6×10-5 GeV

(a) (b)

FIG. 2 (color online). Branching ratios of �� (a) versus the triplet vev for Yv0 � 10�10 GeV and m� � 2 TeV and (b) versus m�for v0 � 6� 10�5 GeV.


the triplet vev. The m� factors in the numerator in thedecay to the longitudinally polarized gauge bosons comefrom the typical enhancement �m2

�=M2W�

2 over the decay tothe transversely polarized gauge bosons, governed by theGoldstone-boson equivalence theorem. The W�T W

�T mode

with a genuine gauge coupling thus becomes vanishinglysmall at higher m�.

The complementarity between the ‘�‘� and W�W�

channels for small and large values of v0 is clearly seenin Fig. 2: for m� � 2 TeV, the two channels are compa-rable when v0 6� 10�5. In the calculation of thebranching ratios of � decays, we sum over all six lepton-flavor combinations in a flavor-democratic way and weassume

Yv0 10�10 GeV � 0:1 eV; (19)

so that neutrino masses lie in the expected range. Note thatfor v0 6� 10�5 GeV, this implies that Y 1:6� 10�6.While these couplings are still very small, we consider thisparameter freedom to be a strength of our analysis: ourapproach allows Y �O�1� with a very small v0 but at thesame time includes the possibility of small Y as well,allowing a large region of parameter space with interestingphenomenology. We also present the branching ratio as afunction of the �� mass in Fig. 2(b) for v0 �6� 10�5 GeV. Here one can see the effect of the differentm� dependence of the ‘�‘� and W�W� final states.

It is interesting to note that the experimental data onneutrino mixing require that at least some of the off-diagonal terms in Y must be of the same order as thediagonal terms when written in the charged lepton massbasis. Although the details of the structure depend on theparticular neutrino mass matrix, one can, assuming some-thing like a flavor-democratic scenario, immediately envi-sion flavor violating decays such as �� ! e��; ��

of sizable strength. Such lepton-flavor violating decays area striking signal of this scenario, where events with two

053007

like-sign different-flavor leptons can be observed in adecay final state which reconstructs to an invariant masspeak at m�.

The branching ratios of �� and �0 receive additionalcontributions from decays to heavy quarks. Of course, anSU(2) triplet has no dimension-four couplings to quarks.However, in the LtH model such couplings arise from(i) mixing between the triplet and the SU(2) doubletHiggs at order v0=v, and (ii) a dimension-five operatorinvolving both H and � that arises from the expansion ofthe nonlinear sigma field in the top-quark YukawaLagrangian, Eq. (A8); inserting the H vev, this yieldscouplings of � to heavy quark pairs suppressed by v=f.Both of these contributions to the � couplings to heavyquarks are controlled by the relevant Yukawa couplings,mq=v. The two contributions, proportional to v0=v andv=f, respectively, can be seen in the couplings given inTable III.

We are interested in the parameter region v=f v0=v,in which case the couplings of �� and �0 to heavy quarksare dominated by the dimension-five nonlinear sigma-model operators, yielding an interesting signal of the littleHiggs structure in the top sector of the model. Neglectingfinal-state masses, the partial decay widths are

�� ! ‘�i ��j� �jYijj

2m�

8�;

�� ! t �b; �tb� ��s ! t�t�

��p ! t�t� Ncm

2t

16�f2 m�; (20)

where Nc � 3 is the number of colors. The triplet cou-plings to T �b and T �t also involve the top sector parameters�1 and �2 (see Appendix A for details) and the decaywidths are proportional to ��1=�2�

2. We illustrate ourresults for �1 � �2. Exact formulas for the partial widthsare given in Appendix B. Figures 3 and 4 show that the

-7

0

0.2

0.4

0.6

0.8

1

1x101x101x10 -10 -9 -8

fo

oitar

gni

hc

nar

B φ

+

v’ (GeV)

mφ = 2 TeV

f = 1 TeVMT = √2 TeV

λ1 = λ2 = 1

l+ −ν

t −b

T −b

0

0.2

0.4

0.6

0.8

1

800 1000 1200 1400 1600 1800 2000 2200 2400

fo

oitar

gni

hc

nar

B φ

+

mφ (GeV)

l+ −ν

t −b

T −b

v’ = 3×10-9 GeV

(a) (b)

FIG. 3 (color online). Branching ratio of �� (a) versus the triplet vev for Yv0 � 10�10 GeV and m� � 2 TeV and (b) versus m� forv0 � 3� 10�9 GeV.


decays of �� and �0 are dominated, approximately fromv0 � 2� 10�9 GeV upwards, by the heavy quark finalstates.

Note that we have treated the triplet mass as a freeparameter because of the arbitrary constants a and a0 inthe coefficient of the triplet mass-squared, as explained inAppendix A. On the other hand,MT is proportional to f forfixed �1; �2. Therefore a large value of f in our approach,while the free parameter m� is held fixed, will suppress thedecays into the T quark. Our results are presented forMT �

��2p

TeV.For ��, the most interesting parameter range is where

the elements of Y range between 0:1 and 1, or equivalentlyv0 lies between 10�9 and 10�10 GeV. In this case ��

decays mostly into SM leptons, with branching fractionscontrolled by the structure of the Yij matrix, whichof course directly controls the neutrino masses andmixings. The signatures of �� would then be quite

0

0.2

0.4

0.6

0.8

1

fo

oitar

gni

hc

nar

B φ

s

v’ (GeV)

mφ = 2 TeV

f = 1 TeVMT = √2 TeV

λ1 = λ2 = 1

ν ν

t −t

t −T+T −t

(a)

1x10 -10 1x10 -91x10 -8

FIG. 4 (color online). Branching ratios of �s (a) versus the triplet vv0 � 3� 10�9 GeV. The branching ratios of �p are virtually ident

053007

distinct from those of a charged scalar coming froma two-Higgs-doublet model, such as in supersymmetrictheories, in which the charged Higgs couplings to leptonsare directly proportional to the charged lepton masses.It should also be remembered that this region, withY �O�1�, corresponds to the least number of fine-tunedparameters in the theory. For larger values of v0, however,the decays of �� will be dominated by the heavy quarkfinal states t �b (and T �b, if kinematically allowed) whichare difficult to distinguish from the decays of the chargedHiggs of a two-Higgs-doublet model. For v0 below10�4 GeV, the most distinct signals of the triplet willbe the �� decays directly into like-sign dileptons. Itshould be noted that the �� does not have any hadronicdecay modes to compete with the �L � 2 decays in thisrange of parameters. For larger values of v0, the mostdistinct signals of the triplet will come from �� !W�W�, giving rise to like-sign dileptons from the W

0

0.2

0.4

0.6

0.8

1

800 1000 1200 1400 1600 1800 2000 2200 2400

fo

oitar

gni

hc

nar

B φ

s

mφ (GeV)

ν ν

t −t

t −T+T −t

v’ = 3×10-9 GeV

(b)

ev for Yv0 � 10�10 GeV and m� � 2 TeV and (b) versus m� forical for the parameter ranges shown.

-8


decays which can be identified with suitable event selec-tion criteria.

In the same spirit, the neutral triplet states �s;�p arecharacterized by their invisible decays into two neutrinosfor Y * 0:1, or equivalently v0 & 10�9, as shown in Fig. 4for�s. The branching ratios of�p are virtually identical inthis parameter range. This makes the neutral scalar �s andthe pseudoscalar �p quite different in appearance fromtheir counterparts in either the SM or a two-Higgs-doubletmodel. Such invisible decays can lead to a detection of theneutral triplet through missing energy signatures or theidentification of an invisible state recoiling against a Zboson at a high-energy linear e�e� collider.

For the ��, �s and �p, the additional decay modes�� ! W�h, �s ! hh, �p ! Zh are available with thesame strength as the W�Z and ZZ modes. However, allthese channels are suppressed by v0=v, and they do notstand a chance against either the heavy quark final states orthe �L � 2 modes. Therefore, the production of the SMHiggs from triplet decays will be unobservable in thisscenario.

It should be noted that the region of the parameter spacethat gives rise to these interesting signals involving leptonswill not be accessible in the scenario described in Sec. II Dand Refs. [16,17], in which the LL� operator is related tothe dimension-five �LH�2 operator through the nonlinearsigma-model field. Thus the decays of the triplet states canserve to distinguish between alternative scenarios for neu-trino mass generation in the LtH model.

A final comment about the decay length of the triplets isin order here. In the region where the ‘�‘� channeldominates, the lifetime of �� (with all flavors summedover) is given by

�8�9

v02

�Yijv0�2

�1 TeV

m�

�� 6:6� 10�28 sec : (21)

For Yij 1:6� 10�6 (or v0 6� 10�5 GeV), one finds ’ 2:2� 10�16 sec for m� � 2 TeV. This gives a decaylength ‘d & 0:1 �m, which is too short to show up as adisplaced vertex in the decay. Taking a larger value for v0

suppresses the partial width into like-sign lepton pairs, butthe WW mode then grows quickly and the decay lengthremains small.

IV. SUMMARY AND CONCLUSIONS

We have considered the simplest possible scenario forgenerating the neutrino masses within the context of thelittlest Higgs model by coupling the scalar triplet present inthe model to the leptons in a �L � 2 interaction. This termthen generates neutrino masses through the triplet vev.Although this term does not obey the overseeing �SU�2� �U�1�2 gauge invariance, it does not affect the cancellationof quadratic divergences in the Higgs mass. We alsoshowed that all contributions coming from dimension-

053007

five operators remain subdominant so long as one assumesthat there is no lepton-number violating new physics at thescale �. Following the phenomenological requirement ofkeeping the neutrino masses in the required range, we areled to a situation where either the lepton-number violatingYukawa coupling or the triplet vev has to be very small.The second possibility, presumably triggered by some yet-unknown feature of the Coleman-Weinberg effective po-tential, allows one to retain the lepton-number violatingcouplings to be O�1�, a situation that seems less fine-tunedfrom the viewpoint of allowing bi-large mixing in theneutrino sector.

We have also investigated the decays of the triplet scalarstates in this scenario and identified their characteristicfeatures associated with lepton-number violation. Themost striking signature comes from the doubly chargedscalar decays. The crucial test is the complementaritybetween the final states of W�W� and ‘�‘�: While thetriplet vev controls the W�W� mode and thus the final-state branching ratios over a large range, the region corre-sponding to Y 1 leads to significant �L � 2 modes,with possibly large lepton-flavor violation. Different com-plementarity exists for the other triplet scalar decays:between SM heavy quarks (independent of v0) and the�L � 2 modes. Moreover, the singly charged scalar maydecay to charged leptons with nearly universal couplings,unlike the charged Higgs in typical two-Higgs-doubletmodels. Another interesting consequence is the ‘‘invis-ible’’ decay of the neutral triplet state into two neutrinos.These decays would allow one to distinguish models oflepton-flavor violation within the littlest Higgs scenarioand directly constrain the elements of the �L � 2 couplingmatrix which controls the neutrino masses and mixings.

ACKNOWLEDGMENTS

We thank Bob McElrath and Liantao Wang for numer-ous discussions about the neutrino mass issues in the littleHiggs scenarios. B. M. thanks the hospitality of thePhenomenology Institute at the University of Wisconsin–Madison, where this work was initiated. T. H. would like tothank the CERN Theory Division for hospitality during thefinal stage of this work. T. H. and H. E. L. were supportedin part by the U.S. Department of Energy under GrantNo. DE-FG02-95ER40896 and in part by the WisconsinAlumni Research Foundation.

APPENDIX A: THE LITTLEST HIGGS MODEL

1. Brief summary of the LtH model

The little Higgs approach conceives the Higgs bosonas a member of a set of pseudo-Goldstone bosons. In theoriginal version of the littlest Higgs scenario [2] to bediscussed here, the pseudo-Goldstone bosons arise whena global SU(5) symmetry is broken down to SO(5) at a

-9


scale �� 4�f. These pseudo-Goldstone bosons are de-scribed by a nonlinear sigma model below the scale �.

The breakdown of the global symmetry is triggered by avacuum expectation value �0 of the sigma-model field,

� � ei�=f�0ei�T=f; (A1)

where � �Pa�aXa and Xa correspond to the 14 broken

SU(5) generators. Explicitly, we have

�0 �

12�2

112�2

0@ 1A;

� �

02�2Hy��

2p �y

H��2p 0 H��

2p

� HT��2p 02�2

0BBB@1CCCA;

(A2)

where we have suppressed the Goldstone modes that willlater be eaten by broken gauge generators, and we define

H � �h�; h0�; � � �i��

2p

��2p �0

0@ 1A: (A3)

An �SU�2� � U�1�2 subgroup of the global SU(5) isgauged. The �0 vev that is responsible for the breakdownof the global symmetry also breaks the gauged �SU�2� �U�1�2 down to the SM electroweak gauge symmetrySU�2�L � U�1�Y . Under the electroweak gauge group, Hand � transform as a complex doublet and a complextriplet, respectively.

The gauge interaction of the sigma field is encoded in itscovariant derivative:

L � �f2

8TrjD��j2; (A4)

where

D�� @�� iXj�1;2

�gjWaj��Q

aj�� QaT

j �

� g0jBj��Yj�� YTj �: (A5)

Here Qaj are the SU(2) generators and Yj are the U(1)

generators, which explicitly break the global SU(5) sym-metry:

Qa1 �

�a2

03�3

� �; Qa

2 �03�3

�a�2

� �; (A6)

Y1 � 110 diag��3;�3; 2; 2; 2�;

Y2 � 110 diag��2;�2;�2; 3; 3�:

(A7)

Notice that setting g1 � g01 � 0 leaves unbroken an SU(3)subgroup of the global SU(5) symmetry; we call thisremaining global symmetry SU�3�1. Similarly, settingg2 � g02 � 0 leaves unbroken a second SU(3) subgroupof the global SU(5) symmetry, which we call SU�3�2. The

053007

Higgs doublet H transforms nonlinearly under both ofthese global SU(3) symmetries, and thus remains an exactGoldstone boson so long as these global symmetries arenot explicitly broken. A Higgs mass term can thus begenerated only by interactions involving both g1 and g2

(or both g01 and g02); this serves to forbid the diagrams thatgenerate the quadratic divergence in the Higgs mass at oneloop. However, logarithmically divergent diagrams con-tributing to the Higgs mass at one loop involve both gaugecouplings g1 and g2 (or both g01 and g02) and thus break theglobal SU(3), thereby leading to contributions to the Higgsmass.

In order to cancel the quadratic divergence arisingthrough the top-quark Yukawa coupling, we have to in-troduce a heavy vectorlike quark pair �T; Tc�, where T isleft-handed and has charge�2=3. Including this vectorlikepair, the top Yukawa Lagrangian is

L t ��1

2fijkxy�i�jx�kyt

c � �2fTTc � H:c:; (A8)

where �T � �bL; tL; T� and tc is an SU(2) singlet. Theindices i; j; k take the values 1; 2; 3, whereas x; y take thevalues 4; 5. It should be noted here that the coupling �1

preserves the global SU�3�1 and breaks SU�3�2, while �2

preserves SU�3�2 and breaks SU�3�1. This ensures that theHiggs mass-squared is protected from quadratic divergen-ces involving the top-quark sector at one loop.Diagonalizing the mass matrix arising from Eq. (A8), wefind the physical top quark t and a heavy isospin-singlet‘‘top partner’’ T:

mt ’�1�2��2

1 � �22

q v; MT ’ f��2

1 � �22

q: (A9)

The gauge and top-quark interactions generate a Higgspotential at one loop via the Coleman-Weinberg mecha-nism [14], which is given by

VCW � ��2f2 Tr��y�� i�h�hf�H�yHT �H��Hy�

��2HHy � �h4�HHy�2 � �h��hH�y�Hy

� �h2�2HHy Tr��y�� 2�2�Tr��y��2

� ��4 Tr��y��y��; (A10)

with coefficients

��2 �a2

�g2

s2c2 �g02

s02c02

�� 8a0�2

1 (A11)

�h�h � �a4

�g2 �c

2 � s2�

s2c2 � g02�c02 � s02�

s02c02

�� 4a0�2

1

(A12)

�h4 � 14��2 ; �h��h � �4

3��2 ; ��2�2 � �16a0�21

(A13)

-10

TABLE III. Feynman rules for lepton-number conserving �couplings to SM particles, from Ref. [15]. All particles andmomenta are outgoing.

��W��W�� 2ig2v0g��

��W��Z� �i�g2=cW�v0g��

��W��h �ig�v0=v��ph � p�� bt ��i=

��2pv��mtPR �mbPL��v=f� � 4�v0=v�

�� bT ��imt=��2pv��v=f� � 4�v0=v��1=�2�PR

�sZ�Z� i��2p�g2=c2

W�v0g��

�shh i2��2pm2��v

0=v2�

�sW��W�� 0

�s �tt ��imt=��2pv��v=f� � 4�v0=v�

�s �bb ��imb=��2pv��v=f� � 4�v0=v�

�s �tT ��imt=��2pv��v=f� � 4�v0=v��1=�2�PR

�s �Tt ��imt=��2pv��v=f� � 4�v0=v��1=�2�PL

�pZ�h ��2p�g=cW��v0=v��ph � p��

�p �tt ��mt=��2pv��v=f� � 4�v0=v�5

�p �bb �mb=��2pv��v=f� � 4�v0=v�5

�p �tT �mt=��2pv��v=f� � 4�v0=v��1=�2�PR

�p �Tt �mt=��2pv��v=f� � 4�v0=v��1=�2�PL

. . . PHYSICAL REVIEW D 72, 053007 (2005)

��4 � �2a3

�g2

s2c2 �g02

s02c02

��

16a0

3�2

1; (A14)

where c and s (c0 and s0) are the gauge coupling mixingparameters for the SU(2) [U(1)] gauge groups, respectively[15]. Here a, a0 are parameters of O�1� that encapsulate thecutoff dependence of the gauge and top sectors, respec-tively, of the UV-incomplete theory. The parameters �2

and �h2�2 are generated through logarithmic contributions.Electroweak symmetry breaking is triggered if �2 > 0,whereby the scalar doublet acquires a vev. The triplet vevis kept small by keeping ��2 positive; it originates inmixing with the doublet via �h�h. The minimization con-ditions for VCW, in terms of hh0i � v=

��2p

, h�0i � v0, are

v2 ��2

�h4 ��2h�h

��2

; v0 ��h�hv2

2��2f: (A15)

Note that terms of the form H2�2; �4 give a subleadingcontribution to Eq. (A15) and have been neglected. In orderto ensure electroweak symmetry breaking, we should have�h4 � ��2

h�h=��2�> 0. The resulting masses for the tripletstates � and the physical Higgs boson h after electroweaksymmetry breaking are

m2� ’ ��2f2; m2

h ’ 2��h4 �

�2h�h

��2

�v2 ’ 2�2: (A16)

It should also be noted that ��2 , as expressed above, getsmodified by an additional term once �L � 2 interactionsare switched on, as has been shown in Sec. II B.

2. Lepton-number violation

When we introduce the �L � 2 interaction of Eq. (3) inorder to give rise to neutrino masses, one of its effects is toadd an extra term to the expression of Eq. (A11) for ��2 , asshown in Eq. (5). This contribution is typically small in theparameter ranges that we consider.

As for the �L � 2 interactions of the triplet �, expand-ing Eq. (3) explicitly one can obtain the full lepton-numberviolating interaction vertices. The dimension-four cou-plings are given in Eq. (15). The Feynman rules for the

NEUTRINO MASSES AND LEPTON-NUMBER VIOLATION

TABLE II. Feynman rules for �L � 2 couplings. All particlesand momenta are outgoing. C is the charge-conjugation operator.Since Yij is symmetric under �i; j� we have combined thesymmetric vertices involving ��, �s and �p and writtenthem only for i � j.

��‘�i ‘�j �i � j� 2iY�ijPRC

��‘�i ��j i��2pY�ijPRC

�s�i�j�i � j� i��2pYijC

�1PL�s ��i ��j�i � j� i

��2pY�ijPRC

�p�i�j�i � j� ��2pYijC

�1PL�p ��i ��j�i � j�

��2pY�ijPRC

053007

�L � 2 interactions are given in Table II. The relevantlepton-number conserving interactions between the tripletstate and SM particles [15] are given as Feynman rules inTable III. For the �shh coupling, we have included thesymmetry factor, Feynman rule � iL� 2, and used therelation in Eq. (A15) to write �h�h in terms of v0.

APPENDIX B: TRIPLET DECAY PARTIALWIDTHS

In this appendix we present the formulas for the tripletdecay partial widths. We define the standard kinematicfunction ��x; y; z� � x2 � y2 � z2 � 2xy� 2xz� 2yzand use the scaled mass variable ri � mi=m�. For thedoubly charged scalar ��, we have

�� ! ‘�i ‘�j � �

8<:

18� jYijj

2m�; �i � j�1

4� jYijj2m�; �i < j�

�� ! W�T W�T � �

1

4�g4v02

m�

�1=2�1; r2W; r

2W��

4r2W � ��1; r

2W; r

2W�

q

g4v02

4�m�;

�� ! W�L W�L � �

1

4�g4v02

2m�

�1=2�1; r2W; r

2W��

4r2W � ��1; r

2W; r

2W�

q��1� 4r2

W�2

4r4W

v02m3

�

2�v4 ; (B1)

where in the last two expressions we have shown theapproximate result neglecting final-state masses comparedto m�. We use the subscripts T and L to denote the

-11


transverse and longitudinal polarizations of the SM gauge bosons.For the singly charged scalar ��, we have

�� ! ‘�i ��j� �1

8�jYijj2m�;

�� ! W�T ZT� �1

4�g4v02

m�c2W

��1=2�1; r2

W; r2Z��

4r2W � ��1; r

2W; r

2Z�

q�

��4r2

Z � ��1; r2W; r

2Z�

q �

g4v02

8�m�c2W

;

�� ! W�L h� �1

4�g2v02

v2

m�

2r2W

��3=2�1; r2

h; r2W��

4r2h � ��1; r

2h; r

2W�

q�

��4r2

W � ��1; r2h; r

2W�

q �v02m3

�

4�v4 ;

�� ! W�L ZL� �1

4�g4v02

2m�c2W

��1=2�1; r2

W; r2Z��

4r2W � ��1; r

2W; r

2Z�

q�

��4r2

Z � ��1; r2W; r

2Z�

q ��1� r2

W � r2Z�

2

4r2Wr

2Z

v02m3

�

4�v4 ;

�� ! t �b� �Nc4�

r2t m

3�

4f2

��1=2�1; r2

t ; r2b��1� r

2t � r2

b��4r2

t � ��1; r2t ; r

2b�

q�

��4r2

b � ��1; r2t ; r

2b�

q �Ncm2

t m�

32�f2 ;

�� ! T �b� �Nc4�

r2t m

3�

4f2

��1=2�1; r2

T; r2b��1� r

2T � r

2b��

4r2T � ��1; r

2T; r

2b�

q�

��4r2

b � ��1; r2T; r

2b�

q ��1

�2

�2Ncm2

t m�

32�f2

��1

�2

�2�1� r2

T�2:

(B2)

For the neutral scalar �s, we have

��s ! �i�j � ��i ��j� �

8><>:1

8� jYijj2m�; �i � j�

14� jYijj

2m�; �i < j�

��s ! ZTZT� �1

4�g4v02

2m�c4W

�1=2�1; r2Z; r

2Z��

4r2Z � ��1; r

2Z; r

2Z�

q g4v02

8�m�c4W

;

��s ! hh� �1

4�

v02m3�

v4

�1=2�1; r2h; r

2h��

4r2h � ��1; r

2h; r

2h�

q v02m3

�

4�v4 ;

��s ! ZLZL� �1

4�g4v02

4m�c4W

�1=2�1; r2Z; r

2Z��

4r2Z � ��1; r

2Z; r

2Z�

q �1� 4r2Z�

2

4r4Z

v02m3

�

4�v4 ;

��s ! t�t� �Nc4�

r2t m

3�

4f2

�1=2�1; r2t ; r

2t ��

4r2t � ��1; r2

t ; r2t �

p �1� 4r2t �

Ncm2t m�

16�f2 ;

��s ! b �b� �Nc4�

r2bm

3�

4f2

�1=2�1; r2b; r

2b��

4r2b � ��1; r

2b; r

2b�

q �1� 4r2b�

Ncm2bm�

16�f2 ;

��s ! T �t� t �T� �Nc4�

r2t m

3�

2f2

��1=2�1; r2

T; r2t ��1� r

2T � r

2t ��

4r2T � ��1; r

2T; r

2t �

q�

��4r2

t � ��1; r2T; r

2t �

q ��1

�2

�2Ncm

2t m�

16�f2

��1

�2

�2�1� r2

T�2:

(B3)

Finally, for the neutral pseudoscalar �p, we have

053007-12


��p ! �i�j � ��i ��j� �� 1

8� jYijj2m�; �i � j�

14� jYijj

2m�; �i < j�

��p ! ZLh� �1

4�

g2v02m�

v2c2Wr

2Z

��3=2�1; r2

h; r2Z��

4r2h � ��1; r

2h; r

2Z�

q�

��4r2

Z � ��1; r2h; r

2Z�

q �v02m3

�

2�v4 ;

��p ! t�t� �Nc4�

r2t m

3�

4f2

�1=2�1; r2t ; r

2t ��

4r2t � ��1; r2

t ; r2t �

p Ncm

2t m�

16�f2 ;

��p ! b �b� �Nc4�

r2bm

3�

4f2

�1=2�1; r2b; r

2b��

4r2b � ��1; r

2b; r

2b�

q Ncm

2bm�

16�f2 ;

��p ! T �t� t �T� �Nc4�

r2t m

3�

2f2

��1=2�1; r2

T; r2t ��1� r

2T � r

2t ��

4r2T � ��1; r

2T; r

2t �

q�

��4r2

t � ��1; r2T; r

2t �

q ��1

�2

�2Ncm

2t m�

16�f2

��1

�2

�2�1� r2

T�2:

(B4)

In the ��, �s, �p couplings to quarks, we have neglected v0=v relative to v=f and included the color factor, Nc � 3.

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053007

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Neutrino masses and lepton-number violation in the littlest Higgs scenario