Neutrino masses in R -parity violating supersymmetric models

PHYSICAL REVIEW D 69, 093002 ~2004!

Neutrino masses inR-parity violating supersymmetric models

Yuval Grossman*Stanford Linear Accelerator Center, Stanford University, Stanford, California 94309, USA,

Santa Cruz Institute for Particle Physics, University of California, Santa Cruz, California 95064, USA,and Department of Physics, Technion–Israel Institute of Technology, Technion City, 32000 Haifa, Israel

Subhendu Rakshit†

Department of Physics, Technion–Israel Institute of Technology, Technion City, 32000 Haifa, Israel~Received 4 December 2003; published 7 May 2004!

We study neutrino masses and mixing inR-parity violating supersymmetric models with generic soft super-symmetry breaking terms. Neutrinos acquire masses from various sources: tree level neutrino–neutralinomixing and loop effects proportional to bilinear and/or trilinearR-parity violating parameters. Each of thesecontributions is controlled by different parameters and has different suppression or enhancement factors thatwe identify. Within an Abelian horizontal symmetry framework these factors are related and specific predic-tions can be made. We find that the main contributions to the neutrino masses are from the tree level and thebilinear loops and that the observed neutrino data can be accommodated once mild fine-tuning is allowed.

DOI: 10.1103/PhysRevD.69.093002 PACS number~s!: 14.60.Pq

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I. INTRODUCTION

Neutrino oscillation experiments indicate that the neunos are massive@1#. The data are best explained with thfollowing set of parameters@2#:

Dm232 52.031023 eV2, Dm12

2 57.231025 eV2,

sin2u2350.5, sin2u1250.3, sin2u13,0.074, ~1.1!

where Dmi j2 [mi

22mj2 and u i j are the leptonic mixing

angles. Equation~1.1! tells us that the neutrino masses ehibit a mild hierarchy and that there is one somewhat smmixing angle (u13) and two large mixing angles (u12 andu23).

Any theory beyond the standard model~SM! needs toexplain this neutrino mass structure. One of the challengeto generate large mixing angles with hierarchical massGenerally, small mixing angles are associated with masserarchies and vice versa. This situation is avoided whendeterminant of the mass matrix is much smaller than its nral value, namely, when there are cancellations betweenferent terms in the determinant. Such cancellations can anaturally in models where different neutrinos acquire masfrom different sources. One such framework isR-parity vio-lating ~RPV! supersymmetry@3#, where generically a singleneutrino acquires a mass at the tree level via mixing withneutralinos while the other two neutrinos become massiveone-loop effects.

Neutrino masses in the framework of RPV supersymmtry have been widely studied@4#. In the earlier works, theonly loop contributions that were considered are fromloops that depend on trilinear RPV couplings. Later, it wrealized that the effect of sneutrino-antisneutrino mixi

*Electronic address: [email protected]†Electronic address: [email protected]

0556-2821/2004/69~9!/093002~10!/$22.50 69 0930

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@5,6# can be very important since it is related to loops thcontribute to the neutrino masses and depend on bilinRPV parameters@7,8#. In Ref. @9# many more loop contribu-tions besides the ‘‘traditional’’ trilinear ones were identifieThese loops were also studied in Refs.@10–13#.

In generic RPV models there are too many free paraeters and no specific predictions for the neutrino spectrcan be made. In general it is even not possible to identifyimportant contributions to the neutrino masses. In this pawe discuss the various contributions to neutrino massesidentify different suppression and enhancement factorseach of them. We also study one specific framework, thaAbelian horizontal symmetry, where specific predictions cbe made. We found that the main contributions to the ntrino masses are from the tree level and the sneutrneutralino loops and that the model can accommodateobserved data once mild fine-tuning is allowed.

II. THE MODEL

We start by describing the RPV framework. We follohere the notation of Ref.@7# where the model is described imore detail. In order to avoid the bounds from proton stabity, we consider the most general low-energy minimal supsymmetric standard model~MSSM! that conserves aZ3baryon triality @14#. Such a theory possesses RPV interations that violate lepton number. OnceR-parity is violated,there is no conserved quantum number that distinguishestween the lepton supermultipletsLm (m51,2,3) and thedown-type Higgs supermultipletHD . It is therefore conve-nient to denote the four supermultiplets by one symLa (a50,1,2,3), with L0[HD . We use Greek indices toindicate the four-dimensional extended lepton flavor spaand Latin ones for the usual three-dimensional flavor spa

The most general renormalizable superpotential is giby

©2004 The American Physical Society02-1

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Y. GROSSMAN AND S. RAKSHIT PHYSICAL REVIEW D69, 093002 ~2004!

W5e i j F2maLai HU

j 11

2labmLa

i Lbj Em1lanm8 La

i Qnj Dm

2hnmHUi Qn

j UmG , ~2.1!

where HU is the up-type Higgs supermultiplet, theQn aredoublet quark supermultiplets,Um (Dm) are singlet up-type~down-type! quark supermultiplets, andEm are the singletcharged lepton supermultiplets. The coefficientslabm areantisymmetric under the interchange of the indicesa andb.Note that them term of the MSSM@which corresponds tom0in Eq. ~2.1!# is now extended to a four-component vectma , and that the Yukawa matrices of the MSSM@whichcorrespond tol0i j8 and l0i j in Eq. ~2.1!# are now extendedinto rank-three tensors,lanm8 andlabm .

Next we consider the most general set of renormalizasoft supersymmetry breaking terms. In addition to the ussoft supersymmetry breaking terms of theR-parity conserv-ing ~RPC! MSSM, one must also add newA and B termscorresponding to the RPV terms of the superpotential.addition, new RPV scalar squared-mass terms also exisabove, we extend the definitions of the RPC terms to alindices of typea. Explicitly, the relevant terms are

Vsoft5~ML2!abLa

i* Lbi 2~e i j BaLa

i HUj 1H.c.!

1e i j F1

2AabmLa

i Lbj Em1Aanm8 La

i Qnj Dm1H.c.G ,

~2.2!

and we do not present the terms that are unchanged fromRPC~that can be found, for example, in Ref.@7#!. Note thatthe single B term of the MSSM is extended to a foucomponent vector,Ba , and that the single squared-materm for the down-type Higgs boson and the 333 leptonscalar squared-mass matrix are now part of a 434 matrix,(ML

2)ab . We further define

umu2[(a

umau2, ^HU&[1

A2vu ,

^na&[1

A2va , vd[uvau, ~2.3!

with

v[~ uvuu21uvdu2!1/252mW

g5246 GeV, tanb[

vu

vd.

~2.4!

These vacuum expectation values are determined viaminimum equations@7#.

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From now on we will work in a specific basis in the spaspanned byLa such thatvm50 andv05vd . The down-typequark and lepton mass matrices in this basis arise fromYukawa couplings toHD , namely,

~md!nm51

A2vdl0nm8 , ~m,!nm5

1

A2vdl0nm . ~2.5!

Note that due to the small RPV admixture with the chargHiggsinos, (m,)nm is not precisely the charged lepton mamatrix @11#. This small effect is not important for our analysis.

In the literature one often finds other basis choices. Tmost common is the one wherem05m and mm50. Ofcourse, the results for physical observables are indepenof the basis choice.~For basis independent parametrizatioof R-parity violation see@9,15–17#.!

III. NEUTRINO MASSES

The neutrino mass matrix receives contributions boththe tree level and from loops. In the following we review thvarious kinds of contributions and identify the leading fators that govern their magnitudes.

A. Tree level „µµ… masses

At tree level the neutrino mass matrix receives contribtions from RPV mixing between the neutrinos and the ntralinos~see Fig. 1!. The masses are calculated from the netral fermion ~neutralinos and neutrinos! mass matrix. Wework perturbatively, and thus at leading order we do ndistinguish betweenm and m0 . Then, the tree level neutrafermion mass matrix, with rows and columns correspondto $B,W3,HU ,nb%, is given by@7,18,19#

FIG. 2. Trilinear loop contribution to the neutrino mass matrThe blob on the scalar line indicates mixing between the lehanded and the right-handed squarks. A mass insertion on the inal quark propagator is denoted by the cross.

FIG. 1. Tree level neutrino mass in the mass insertion apprmation. A blob represents mixing between the neutrino andup-type Higgsino. The cross on the neutralino propagator signifiMajorana mass term for the neutralino.

2-2

S M1 0 mZsWvu /v 2mZsWvd /v 0 0 0

0 M2 2mZcWvu /v mZcWvd /v 0 0 0

mZsWvu /v 2mZcWvu /v 0 m m1 m2 m3

2mZsWvd /v mZcWvd /v m 0 0 0 0

0 0 m1 0 0 0 0 D , ~3.1!

NEUTRINO MASSES INR-PARITY VIOLATING . . . PHYSICAL REVIEW D 69, 093002 ~2004!

0 0 m2 0 0 0 0

0 0 m3 0 0 0 0

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where M1 is the B-ino mass,M2 is the W-ino mass,cW[cosuW andsW[sinuW. Integrating out the four neutralinowe get the neutrino mass matrix

@mn# i j(mm)5XTm im j , ~3.2!

where

XT5mZ

2mgcos2b

m~mZ2mgsin 2b2M1M2m!

;cos2b

m, ~3.3!

such thatmg[cW2 M11sW

2 M2 and in the last step we assumthat all the relevant masses are at the electroweak~or super-symmetry breaking! scale,m. The tree level neutrino masseare the eigenvalues of@mn# i j

(mm)

m3(T)5XT~m1

21m221m3

2!, m1(T)5m2

(T)50. ~3.4!

Here, and in what follows, we usem3>m2>m1 .We see that at the tree level only one neutrino is mass

Its mass is proportional to the RPV parameter(m i2 and to

cos2b. For large tanb the latter is a suppression factor. Awe discuss later, this suppression factor can be importan

B. Trilinear „l8l8 and ll… loops

The neutrino mass matrix receives contributions froloops that are proportional to trilinear RPV couplings~seeFig. 2!. These kinds of loops received much attention inliterature. Here we only present approximated expressthat are sufficient for our study. Full results can be found,example, in Ref.@7#.

Neglecting quark flavor mixing, the contribution of thl8l8 loops is proportional to the internal fermion mass ato the mixing between left and right sfermions. Explicitly,

@mn# i j(l8l8)'(

l ,k

3

8p2l i lk8 l jkl8

mdlDmdk

2

mdk

2

;(l ,k

3

8p2l i lk8 l jkl8

mdlmdk

m, ~3.5!

where mdkis the averagekth sfermion mass,Dmdk

2 is the

squared mass splitting between the twokth sfermions, and inthe last step we usedDmdk

2'mdk

m and mdk;m. There are

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similar contributions from loops with intermediate chargleptons wherel8 is replaced byl and there is no color factoin the numerator.

We see that the trilinear loop-generated masses arepressed by the RPV couplingsl82 (l2), by a loop factor,and by two down-type quark~charged lepton! masses. Thelatter factor, which is absent in other types of loops, maketrilinear contribution irrelevant in most cases.

C. Bilinear „BB… loop induced masses

Neutrinos acquire masses from loops that are proportioto bilinear RPV couplings. Here we discuss the contributiothat are proportional to two insertions of RPVBi parameters~see Fig. 3!. We also refer to the masses induced by thediagrams as the sneutrino splitting induced masses. Theson is that the twoB insertions in the scalar line also geneate splitting between the two sneutrino mass eigenstates.contribution of the BB loop diagram is related to thissneutrino mass splitting@5,9#. In particular, if the sneutrinosplitting vanishes the neutrino is massless.

The one-loop contribution to the neutrino mass matfrom theBB loop is given by@9#

@mn# i j(BB)5(

a

g2BiBj

4 cos2b~Za22Za1g8/g!2mxa

3$I 4~mh ,mn i,mn j

,mxa!cos2~a2b!

1I 4~mH ,mn i,mn j

,mxa!sin2~a2b!

2I 4~mA ,mn i,mn j

,mxa!%, ~3.6!

FIG. 3. TheBB loop-generated neutrino mass. Here the blodenote mixing of the sneutrinos with the neutral Higgs bosons.cross on the internal neutralino propagator denotes a Majoranafor the neutralino.

2-3

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where Zab is the neutralino mixing matrix witha,b51, . . . ,4 and

I 4~m1 ,m2 ,m3 ,m4!51

m122m2

2 @ I 3~m1 ,m3 ,m4!

2I 3~m2 ,m3 ,m4!#,

I 3~m1 ,m2 ,m3!51

m122m2

2 @ I 2~m1 ,m3!2I 2~m2 ,m3!#,

~3.7!

I 2~m1 ,m2!521

16p2

m12

m122m2

2 lnm1

2

m22

.

Assuming that all the masses in the right-hand side~RHS! ofEq. ~3.6! are of the order of the weak scale, we estimate

@mn# i j(BB);

g2

64p2cos2b

BiBj

m3. ~3.8!

In the above estimation, no cancellation between the difent Higgs loops were assumed. We do, however, expechave some degree of cancellation between these loopssee it note that if the threeI 4 functions in~3.6! were equal,@mn# i j

(BB) would vanish. The remnant of this effect is a partcancellation that becomes stronger in the decoupling limThen cos2(a2b)→0 andmH→mA and from Eq.~3.6! we seethat @mn# i j

(BB)→0. We discuss this cancellation in AppendA.

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lt.

Next we study theBB loop effect on the neutrino masseFor this we rewrite~3.6! as

@mn# i j(BB)5Ci j BiBj . ~3.9!

If all the elements of the matrixCi j were identical,@mn# i j(BB)

would have only one nonvanishing eigenvalue. Using~3.6!we see that such a situation arises when the sneutrinosdegenerate. More generally, we conclude that the contrtion to the light neutrinos receive potentially additional supression by a factor proportional to the nondegeneracy insneutrino sector.

In general we expect thatBa is not proportional toma .Then, one neutrino mass eigenstate acquires mass atlevel, and the other two from bilinear loops where the maof the lightest neutrino is proportional to the amount of nodegeneracy of the sneutrinos. We elaborate more on thisfect in Appendix B.

We conclude that theBB loop-generated masses are supressed by the RPV couplingsBB, by a loop factor, and by apossible effect due to the cancellation between the thHiggs loops. For large tanb the BB loop is enhanced bytan2b. The third neutrino mass may get an extra suppressproportional to the nondegeneracy among the sneutrinos

D. µB loops

Another type of diagrams that induce neutrino masfrom mixing between the sneutrinos and the neutral Higbosons is given in Fig. 4. The contribution from this diagrato the neutrino mass matrix is given by@9#1

ly by

@mn# i j(mB)5(

a,b

g2

4 cosbm iBj

mxb

mxa

Za3~Zb22Zb1g8/g!$2@Za4~Zb22Zb1g8/g!sina1~Za22Za1g8/g!Zb3cosa

1~Za22Za1g8/g!Zb4sina#cos~a2b!I 3~mh ,mxb,mn j

!1@Za4~Zb22Zb1g8/g!cosa

2~Za22Za1g8/g!Zb3sina1~Za22Za1g8/g!Zb4cosa#sin~a2b!I 3~mH ,mxb,mn j

!

1@Za4~Zb22Zb1g8/g!sinb1~Za22Za1g8/g!Zb3cosb1~Za22Za1g8/g!Zb4sinb#I 3~mA ,mxb,mn j

!%

1~ i↔ j !. ~3.10!

Assuming that all the masses are at the weak scale, this contribution to the neutrino mass matrix is given approximate@9#

1Note that we disagree with Ref.@9# on the sign of the term that originate from theA loop.

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@mn# i j(mB);

g2

64p2cosb

m iBj1m jBi

m2. ~3.11!

In the flavor basis these diagrams are expected to yield slar contributions to theBB loops. Yet, as pointed out in Re@13#, due to the dependence onm i , themB loop contributionto the neutrino masses is sub-leading. See Appendix Cdetails.

Similar to theBB loops, also in themB loop there is apartial cancellation between the different Higgs loops. Indecoupling limit this can be seen from Eq.~3.10!. Since, aswe just mentioned, the effect of themB loop is sub-leading,we do not elaborate on the decoupling effect.

We conclude that themB loop-generated masses are supressed by the RPV couplingsmB, by a loop factor, and bya possible effect due to Higgs decoupling. In the case whthe tree level contribution is dominant, their effect on tneutrino masses is second order in these small paramet

E. Other loops

There are many other loops that contribute to the neutmasses@9#. Almost all of them are suppressed by at least tYukawa interactions and are therefore likely to be negligib

There is only one contribution to the neutrino massesdepends on both bilinear and trilinear couplings and is spressed by only one Yukawa coupling. This diagramshown in Fig. 5. Neglecting squark flavor mixing, theml8contribution to the neutrino mass matrix is@9#

@mn# i j(ml8)'(

k

3

16p2gmdk

m il jkk8 1m jl ikk8

m. ~3.12!

There are similar contributions from diagrams withl insteadof l8 couplings, where leptons and sleptons are runningthe loop.

We see that theml8 diagrams are suppressed by the RPcouplingsml8, by a loop factor, and by one Yukawa copling. Also here, similar to the case of themB loops, oncethe tree level effect is taken into account, theml8 and mlcontributions to the light mass eigenstates are second oin the above-mentioned suppression factors.

FIG. 4. Neutrino Majorana mass generated by themB loop. Theblob on the external fermion line signifies mixing between a ntrino and a neutralino. The blob on the internal scalar line standsmixing between the sneutrinos and the neutral Higgs bosons.cross on the internal neutralino line denotes, as before, a Majomass term for the neutralino. There exists other diagrams withi↔ j .

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F. Model-independent considerations

As we discussed, there are many contributions to the ntrino masses that are suppressed by different small pareters. In general, the leading effects are model dependNevertheless, here we make some general remarks.

One neutrino is massive at the tree level and unless tabis very large, this is the dominant contribution tom3 . Theother neutrinos get masses at the loop level. Despite thetial cancellation between different Higgs loops, we expthe BB loops to be the dominant one. All the other contribtions are generically suppressed compared to it due tofollowing reasons:

~a! ThemB loop contribution to the light neutrino masseis second order in the small ratio between the loop-indumass and the tree level one.

~b! The l8l8 andll diagrams are doubly Yukawa suppressed.

~c! The ml8 and ml diagrams are singly Yukawa suppressed, and similar to the situation with themB loops, theircontributions to the light neutrino masses are second ordethe suppression factor.

Therefore, while there are several caveats as explaabove, the situation is likely to be as follows: The heavineutrino mass,m3 , arises at the tree level. The major contbution to m2 is from the BB loops. For nondegeneratsneutrinos,m1 is also generated by theBB loops. For degen-erate sneutrinos, however,m1 is very small and the majocontribution to it can arise from any of the other sourcNote that since neutrino oscillation data are not sensitivethe lightest neutrino mass, our ignorance of the mechanthat generatem1 is not problematic. In the following weconsider a specific model where we can explicitly checkrelevance of the different contributions.

IV. HORIZONTAL SYMMETRY

We work in the Abelian horizontal symmetry framewo@20#. The horizontal symmetry,H, is explicitly broken by asmall parameterl to which we attribute charge21. This canbe viewed as the effective low energy theory that comfrom a supersymmetric extension of the Froggatt-Nielsmechanism at a high scale@21#. Then, the following selec-tion rules apply:~a! Terms in the superpotential that carchargen>0 underH are suppressed byO(ln), while thosewith n,0 are forbidden by holomorphy;~b! soft supersym-

FIG. 5. Neutrino Majorana mass generated by theml8 loop.The blob on the external fermion line signifies a mixing betweeneutrino and an up-type Higgsino, which is then converted tgaugino. The cross on the internal fermion line stands for a Dmass insertion. There exists another diagram withi↔ j .

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2-5

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ibu-

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metry breaking terms that carry chargen underH are sup-pressed byO(l unu). For simplicity, in the following we as-sume that the horizontal charges of all the MSSM superfieare non-negative.

We identify the down-type Higgs doublet with the doubsuperfield that carries the smallest charge, which we choto be L0[Hd . ~To simplify the notation what we denotbefore asL0 is now L0 .) We order the remaining doubletaccording to their charges:

H~L1!>H~L2!>H~L3!>H~Hd!>0. ~4.1!

Similar ordering is made for the three generations ofcharged leptons and quarks.

Our methods of analyzing lepton and neutralino masstrices are described in detail in Refs.@18# and @22#, respec-tively. Specifically, we use the above-mentioned selectrules to estimate the magnitude of the relevant paramete

ma;mlH(La)1H(Hu),

Ba;m2lH(La)1H(Hu),

~ML2!ab;m2l uH(Lb)2H(La)u, ~4.2!

la jk8 ;lH(La)1H(Qj )1H(dk),

labk;lH(La)1H(Lb)1H( ,k).

Here m is the natural scale for them terms. We assume tham5O(m) and sincem0 is phenomenologically required tbe also ofO(m), we takeH(Hd)5H(Hu)50. Then we get

m i

m0;

Bi

B0;

~ML2!0i

~ML2!aa

;vl i jk

~m,! jk;

vl i jk8

~md! jk;lH(Li ). ~4.3!

We see that all the RPV parameters are suppressedcommon factor compared to their corresponding RPC pareters. In particular, this implies that the RPV trilinear coplings are very small since they are related to the small RYukawa couplings.

Several other parameters are expected to be ofO(1) inthe horizontal symmetry framework: In particular,

cosb, eH , eD , ~4.4!

such thateH is the suppression due to the effect of Higdecoupling@defined in Eq.~A4!# and eD is the suppressiondue to sneutrino degeneracy@defined in Eq.~B6!#. Yet, in thefollowing we keep them in order to understand what paraeters are needed to be fine-tuned in order to get a vimodel.

Now we can estimate the order of magnitude of the dferent contributions to the neutrino mass matrix

@mn# i j(mm);m0cos2blH(Li )1H(L j ),

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a-

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-le

-

@mn# i j(BB);

m0

cos2beLeHlH(Li )1H(L j ),

@mn# i j(mB);

m0

cosbeLeHlH(Li )1H(L j ), ~4.5!

@mn# i j(l8l8);m0eLS mb

v D 4

lH(Li )1H(L j ),

@mn# i j(ml8);m0eLS mb

v D 2

lH(Li )1H(L j ),

whereeL;1022 is the loop suppression factor. WhileeH andeL are in general different for the different contributions, wexpect them to be of the same order and thus we omit tidentification indices. In our simple model the overall scam0;O(m). It can be much smaller if there is a mechanisthat generates a common suppression factor for all the Rparameters.

In order to get the relative importance of the differecontributions to the neutrino masses we note the followpoints:

~1! There is a common factor,m0lH(Li )1H(L j ), to all thecontributions.

~2! Within the horizontal symmetry framework@mn# i j(BB)

;@mn# i j(mB) . As explained above, as long as the tree con

bution is dominant, this implies that the effect of@mn# i j(mB) on

the neutrino masses is negligible.

~3! Due to the extra Yukawa suppressions@mn# i j(ml8)

.@mn# i j(l8l8) . However, @mn# i j

(ml8) contributes to the neutrino mass at second order in the suppression factors. UEq. ~4.5! we get the ratio of these two contributions to thlightest neutrino mass

m1(ml8)

m1(l8l8)

;eL

cos2b. ~4.6!

We conclude that unless cosb is very small, the contributionof the l8l8 loops to the neutrino masses is more importathan that of theml8 ones.

We see that there are three possible important contrtions to the neutrino masses,@mn# i j

(mm) , @mn# i j(BB) , and

@mn# i j(l8l8) . Their relative effects are controlled by cosb,

eH , andeL @see Eq.~4.5!#. We assume that cosb, eH , andeD are not very small. Then, thel8l8 loops can be neglecteand we get the order of magnitude of the neutrino masse

m3;m0cos2bl2H(L3),

m2;m0

cos2beLeHl2H(L2), ~4.7!

m1;m0

cos2beLeHeDl2H(L1).

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If eD is very small then the lightest neutrino mass is domnated by thel8l8 loop contribution,

m1;m0l2H(L1)eLS mb

v D 4

. ~4.8!

Next we check whether the neutrino data can be explaiin our model. The mixing angles are given by@22#

sinu i j ;l uH(Li )2H(L j )u. ~4.9!

The requirement thatu23 andu12 are large@2# implies that

H~L3!5H~L2!5H~L1!. ~4.10!

A potential problem is that this choice of horizontal chargalso predicts largeu13. In order to generate a viable neutrinmass spectrum we require thatm3;1021 eV and m2;1022 eV. This is the case when

m0l2H(L3)cos2b;1021 eV ~4.11!

and

cos4b

eH;1021, ~4.12!

where we usedeL;1022.The requirement in Eq.~4.11! can be met once appropr

ate input parameters~or charges! are chosen. The ratio in~4.12! as well as the requirement of smallu13, however,required mild fine-tuning. Within our model both are prdicted to be of orderO(1) while the data suggest that theareO(1021).

Here we consider only a simple model base on aU(1)Hsymmetry. In more elaborate models, like those with mcomplicated symmetry group, e.g.,U(1)3U(1) or discretesymmetry group@23#, one may be able to achieve a viabmodel with less fine-tuning.

V. CONCLUSIONS

RPV supersymmetric models provide an alternative toseesaw mechanism. One virtue of RPV models is that tnaturally provide a mechanism for large mixing with hierachy, as indicated by the data.

We study the magnitudes of various sources of neutrmasses in RPV models. There are several parametersdetermine them and therefore there are several supprefactors in each of them. Thus, their relative importancemodel dependent. Due to the Yukawa suppression of thelinear loops, it is generally likely that the tree level and tBB loops are the dominant contributions.

We study one specific model with an Abelian horizonsymmetry. In this model indeed the tree level and theBBone-loop contributions are the dominant ones. We find tsuch a model can describe the neutrino data as long asfine-tuning is permitted.

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s

e

ey

ohationsri-

l

atild

ACKNOWLEDGMENTS

We thank Howie Haber, Sourov Roy and Jure Zupanhelpful discussions. S.R. thanks the Fermilab and SLtheory groups for hospitality while parts of this work wecompleted. He also acknowledges financial support fromLady Davis Trust. The work of Y.G. is supported by thDepartment of Energy, contract DE-AC03-76SF00515 aby the Department of Energy under grant No. DE-FG092ER40689.

APPENDIX A: HIGGS CANCELLATION IN THE BBLOOPS

Here we study the cancellation between the threeBBloops of Fig. 3. The weighted sum of the three Higgs progators, before integrating over the internal momentak, is

PS51

k22mh2

cos2~a2b!11

k22mH2

sin2~a2b!21

k22mA2

.

~A1!

For simplicity we use the tree level relations@24,25#

cos2~a2b!5mh

2~mZ22mh

2!

mA2~mH

2 2mh2!

, mZ22mh

25mH2 2mA

2 ,

~A2!

and we obtain

PS52k2~mZ

22mh2!~mA

22mh2!

mA2~k22mH

2 !~k22mA2 !~k22mh

2!. ~A3!

Consider the decoupling limit wheremH;mA@mh;mZ . Inthat limit theH andA propagators scale like one over theheavy mass squared. From Eq.~A3! we see that the weightesum scales like one over the heavy mass to the fourth po

While the partial cancellation is more severe in the decpling limit, it also occurs far away from that limit. The reason is that theI 4 function, defined in Eq.~3.7!, is not verysensitive to variation in one of its arguments as long as inot the largest one.

We have checked the effect of the summation overdifferent Higgs mediated diagrams numerically. We defithe following measure of the suppression factor

eH[U I ~mh!cos2~a2b!1I ~mH!sin2~a2b!2I ~mA!

uI ~mh!ucos2~a2b!1uI ~mH!usin2~a2b!1uI ~mA!uU ,

~A4!

where I (x)[I 4(x,mn i,mn j

,mxa) @see Eq. ~3.6!#, and the

i , j ,a indices ofeH are implicit. While the tree level relationis a good approximation of the effect, in the numerical cculation we use the two-loop spectrum for the Higgs bosmasses and mixing angles@26#. Some representative numbers are presented in Table I.

2-7

fkea-fos

otv

n

-

the

up-

t

ntpsatthedis-

ofwe

nf

is

the


We also checked the effect of tanb. We found thateHdecreases as tanb increases. Thus, the sensitivity o@mn# i j

(BB) to tanb is reduced. On one hand, it scales li1/cos2b @see Eq.~3.6!#, and on the other hand the cancelltion between the different Higgs loops becomes strongerlarge tanb. In fact, using the tree level Higgs mass relationwe found that asymptoticallyeH}cos2b. Thus, at the treelevel in the tanb→` limit, @mn# i j

(BB) is independent oftanb.

APPENDIX B: THE SUPPRESSION DUE TO SNEUTRINODEGENERACY

Here we study the effect of the sneutrino degeneracythe light mass eigenstate from theBB loops. We assume thathe heaviest neutrino acquires large mass at the tree leThen, for simplicity, we deal only with the loop contributioto the first two generations.

We define the mass squares of the two sneutrinos as

~mn2!1,2[mn

2~16D!. ~B1!

Computing theBB one-loop contributions up to orderD2, weget a mass matrix of the following form:

f 1S B1B1 B1B2

B2B1 B2B2D 1D f 2S B1B1 0

0 2B2B2D

1D2f 3S 3B1B1 B1B2

B2B1 3B2B2D 1O~D3! ~B2!

where

f 15mDeguD→0 , f 25]mDeg

]D UD→0

, f 351

2

]2mDeg

]D2 UD→0

,

~B3!

and

mDeg~D!5(a

g21

4 cos2b~Za22Za1g8/g!2mxa

3@ I 4~mh ,m,m,mxa!cos2~a2b!

1I 4~mH ,m,m,mxa!sin2~a2b!

2I 4~mA ,m,m,mxa!#, ~B4!

TABLE I. Numerical values of the suppression factor due tocancellation between different Higgs contributions in theBB loops.We usedmn1

5100 GeV,mn25200 GeV andmxa

5300 GeV.

tanb mh mA mH eH3103

4 92 184 190 9.12 81 426 430 4.720 106 285 284 1.114 106 294 294 0.3

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n

el.

wherem25mn2(11D). After diagonalization, we get the fol

lowing masses:

m25~B121B2

2! f 11O~D!,

m15B1

2B22

B121B2

2 ~4 f 1f 32 f 22!D21O~D3!. ~B5!

We see that the dominant contribution tom2 is the same asthat in the degenerate case. The leading contribution tom1 ,on the other hand, is proportional to the square ofsneutrino mass splitting.

We define the following measure of the degeneracy spression

eD[m1

m2, ~B6!

which is given by

eD5 f c

B12B2

2

~B121B2

2!2D2, f c5

4 f 1f 32 f 22

f 1

. ~B7!

We have checked numerically and found that typicallyf c;0.1. Thus, in addition to theD2 suppression, the lightesneutrino mass is also suppressed byf c .

APPENDIX C: µi DEPENDENT ONE-LOOPCONTRIBUTIONS

Here we explain why when the tree level is the dominacontribution to the neutrino mass matrix, the effect of loothat have onem i insertion are small. They appear onlysecond order in the ratio between the loop contribution tomass matrix and the tree level one. This effect was alsocussed in Ref.@13#.

We consider a two generation case with only one typeone-loop contribution at a time. First we assume thathave the following two contributions:

@mn# i j(mm)5Cm im j , @mn# i j

(Vm)5C«L~m iVj1m jVi !,~C1!

whereC is a constant andV is a normalized general vector iflavor space such thatuVu5umu. For example, in the case othemB diagram,Vi corresponds to the product ofBi with theloop function. The fact that the tree level is dominantencoded by the choice«L!1. The mass matrix is then

2-8


mn5CS m1212«LV1m1 m1m21«L~V1m21V2m1!

m1m21«L~V1m21V2m1! m2212«LV2m2

D . ~C2!

te

We see that the ratio of the two mass eigenstates is

m1

m2;O~«L

2!. ~C3!

Next consider a case where the loop effect is generawithout anym i insertion. For example

@mn# i j(mm)5Cm im j , @mn# i j

(VU)5Ci jVU«L~ViU j1VjUi !,

~C4!

Y

nt

e,

nd.Bv.

tt.er

ys.,

09300

d

whereU is another normalized vector andC;Ci jVU for any i

and j. For mn5@mn# (mm)1@mn# (VU) we generally get

m1

m2;O~«L!. ~C5!

Note that the above holds also forU5V. Comparing Eqs.~C5! and~C3! we see that diagrams with onem i insertion areunlikely to affect the neutrino masses significantly.

le,

ia,

o,

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Neutrino masses in R -parity violating supersymmetric models