application/xmlNeutrino masses induced by R-parity violation in a SUSY SU(5) model with additional [formula omitted]Yoshio KoidePhysics Letters B 595 (2004) 469-475. doi:10.1016/j.physletb.2004.05.072journalPhysics Letters BCopyright © 2004 Elsevier B.V. All rights reserved.Elsevier B.V.0370-26935951-412 August 20042004-08-12469-47546947510.1016/j.physletb.2004.05.072http://dx.doi.org/10.1016/j.physletb.2004.05.072doi:10.1016/j.physletb.2004.05.072http://vtw.elsevier.com/data/voc/oa/OpenAccessStatus#Full2014-01-01T00:14:32ZSCOAP3 - Sponsoring Consortium for Open Access Publishing in Particle Physicshttp://vtw.elsevier.com/data/voc/oa/SponsorType#FundingBodyhttp://creativecommons.org/licenses/by/3.0/

Journals

S300.3

PLB21129S0370-2693(04)00889-510.1016/j.physletb.2004.05.072Elsevier B.V.Phenomenology

Fig. 1

Radiative generation of neutrino Majorana mass.

Fig. 2

Effective 5̄qlLH̄d† term.

Neutrino masses induced by R-parity violation in a SUSY SU(5) model with additional 5̄′L+5′L

Yoshio

Koide

koide@u-shizuoka-ken.ac.jp

Department of Physics, University of Shizuoka, 52-1 Yada, Shizuoka 422-8526, Japan

Editor: T. Yanagida

Abstract

Within the framework of an SU(5) SUSY GUT model, a possible general form of the neutrino mass matrix induced by R-parity violation is investigated. The model has matter fields 5̄′L+5′L in addition to the ordinary matter fields 5̄L+10L and Higgs fields Hu+H̄d. The R-parity violating terms are given by 5̄L5̄L10L, while the Yukawa interactions are given by H̄d5̄′L10L. Since the matter fields 5̄′L and 5̄L are different from each other at the unification scale, the R-parity violation effects at a low energy scale appear only through the 5̄′L↔5̄L mixings. In order to make this R-parity violation effect harmless for proton decay, a discrete symmetry Z3 and a triplet–doublet splitting mechanism analogous to that in the 5-plet Higgs fields are assumed.

PACS

14.60.Pq12.60.Jv11.30.Hv11.30.Er

Introduction

As an origin of the neutrino masses, the idea of the radiative neutrino mass [1] is very interesting as well as the idea of the neutrino seesaw mechanism [2]. However, currently, the latter idea is influential, because it is hard to embed the former model into a grand unification theory (GUT). For example, a supersymmetric (SUSY) model with R-parity violation can provide radiative neutrino masses [3], but the model cannot be embedded into GUT, because the R-parity violating terms induce proton decay inevitably [4].Recently, the author [5] has proposed a model with R-parity violation within the framework of an SU(5) SUSY GUT: we have quark and lepton fields 5̄L+10L, which contribute to the Yukawa interactions as Hu10L10L and H̄d5̄L10L; we also have additional matter fields 5̄′L+5′L which contribute to the R-parity violating terms 5̄′L5̄′L10L. Since the two 5̄L and 5̄′L are different from each other, the R-parity violating interactions are usually invisible. The R-parity violating effects become visible only through 5̄L↔5̄′L mixings in low energy phenomena.In the previous model [5], a discrete symmetry Z3 has been assumed, and their quantum numbers have been assigned as 5̄L(−)+10L(+)+5̄′L(+)+5′L(+) and H̄d(0)+Hu(+), where we have denoted fields with the transformation properties Ψ→ω+1Ψ, Ψ→ω0Ψ and Ψ→ω−1Ψ (ω=ei2π/3) as Ψ(+), Ψ(0) and Ψ(−), respectively. Therefore, in the set 5̄L+10L, the fields 5̄L(−) and 10L(+) have different transformation properties each other. In contrast to the previous model, in the present Letter, we will propose a model with alternative assignments (1.1)(5̄L+10L)(+)+(5̄′L+5′L)(0)+H̄d(−)+Hu(+). Although the mechanism of the harmless R-parity violation is the same as the previous model, since the Z3 quantum number assignment is different from the previous one, the structure of the model is completely different from the previous one.In the present Letter, we will investigate not only the radiatively-induced neutrino masses, but also the contributions from the vacuum expectation values (VEV) of the sneutrinos, 〈ν̃〉, although in the previous paper the estimate of 〈ν̃〉 was merely based on an optimistic speculation.2

Harmless R-parity violation mechanism

Under the Z3 quantum number assignment (1.1), the Z3 invariant trilinear terms in the superpotential are only the following three terms: (2.1)Wtri=(Yu)ijHu(+)10L(+)i10L(+)j+(Yd)ijH̄d(−)5̄′L(0)i10L(+)j+λijk5̄L(+)i5̄L(+)j10L(+)k. Similarly, the Z3 invariant bi-linear terms are only two: H̄d(−)Hu(+) and 5̄L(0)HL(0). In order to give doublet–triplet splitting, we assume the following “effective” bi-linear terms (2.2)Wbi=H̄d(−)μ+gH〈Φ(0)〉Hu(+)+5̄′L(0)iM5−g5〈Φ(0)〉5′L(0)i+MiSB5̄L(+)i5′L(0)i, where Φ(0) is a 24-plet Higgs field with the VEV 〈Φ(0)〉=v24diag(2,2,2,−3,−3), so that, for example, the effective masses M(a) in the term 5̄L(0)′(a)5L(0)′(a) (5(2)L and 5(3)L denote doublet and triplet components of the fields 5L, respectively) are given by(2.3)M(2)=M5+3g5v24,M(3)=M5−2g5v24. The last term in Eq. (2.2) has been added in order to break the Z3 symmetry softly. We define the 5̄L↔5̄′L mixing as follows: (2.4)5̄′L(0)i=ci5̄qℓLi+si5̄heavyLi,5̄L(+)i=−si5̄qℓLi+ci5̄heavyLi, where si=sinθi and ci=cosθi. Then, we can rewrite the second and third terms in Eq. (2.2) as (2.5)∑a=2,3M(a)2+MiSB25̄heavyLi(a)5heavyLi(a), where 5heavyL=5′L(0) and (2.6)si(a)=M(a)M(a)2+MiSB2,ci(a)=MiSBM(a)2+MiSB2. The fields 5̄Liheavy(a) have masses (M(a))2+(MiSB)2, while 5̄Liqℓ(a) are massless. We regard 5̄qℓLi+10L(+)i as the observed quarks and leptons at low energy scale (μ<MGUT). (Hereafter, we will simply denote 5̄Liqℓ and 10L(+)i as 5̄Li and 10Li, respectively.)Then, the effective R-parity violating terms at μ<MGUT are given by (2.7)WR/eff=si(a)sj(b)λijk5̄(a)Li5̄(b)Lj10Lk. In order to suppress the unwelcome term dRcdRcuRc in the effective R-parity violating terms (2.7), we assume a fine tuning (2.8)M(2)∼MGUT,M(3)∼mSUSY,MiSB∼MGUT×10−1, where mSUSY denotes a SUSY breaking scale (mSUSY∼1 TeV), so that (2.9)si(2)=1−O10−2,ci(2)≃MiSBM(2)∼10−1,si(3)≃M(3)MiSB∼10−12,ci(3)=1−O10−24.Note that in the present model the observed down-quarks dRic=(5̄Liqℓ)(3) are given by (5̄Liqℓ)(3)≃(5̄′L(0)i)(3), while the observed lepton doublets (νLi,eLi)=(5̄Liqℓ)(2) are given by (5̄Liqℓ)(2)≃−(5̄L(+)i)(2).From Eq. (2.9), the R-parity violating terms dRcdRcuRc and dRc(eLuL−νLdL) are suppressed by s(3)s(3)∼10−24 and s(3)s(2)∼10−12, respectively. Thus, proton decay caused by terms dRcdRcuRc and dRc(eLuL−νLdL) is suppressed by a factor (s(3))3s(2)∼10−36. On the other hand, radiative neutrino masses are generated by the R-parity violating term (eLνL−νLeL)eRc with a factor s(2)s(2)≃1.The up-quark masses are generated by the Yukawa interactions (2.1), so that we obtain the up-quark mass matrix Mu as (Mu)ij=(Yu)ijvu, where vu=〈Hu(+)0〉. We also obtain the down-quark mass matrix Md and charged lepton mass matrix Me as (2.10)Md†=C(3)Ydvd,Me∗=C(2)Ydvd, where (2.11)C(a)=diagc1(a),c2(a),c3(a), so that (2.12)MdT=C(3)C(2)−1∗Me, where vd=〈H̄d(−)0〉. Note that MdT has a structure different from Me, because the values of ci(2) (i=1,2,3) can be different from each other. (The idea MdT≠Me based on a mixing between two 5̄L has been discussed, for example, by Bando and Kugo [6] in the context of an E6 model.)3

General form of the neutrino mass matrix

First, we investigate a possible form of the radiatively-induced neutrino mass matrix Mrad. In the present model, since we do not have a term which induces ê+R↔H̄+d(−) mixing, there is no Zee-type diagram [1], which is proportional to the Yukawa vertex (Yd)ij and R-parity violating vertex λijk.Only the radiative neutrino masses in the present scenario come from a charged-lepton loop diagram: the radiative diagram with (νL)j→(eR)l+(ẽLc)n and (eL)k+(ẽLc)m→(νLc)i. The contributions (Mrad)ij from the charged lepton loop are given, except for the common factors, as follows: (3.1)(Mrad)ij=sisjsksnλ∗ikmλ∗jnl(Me)∗klM̃eLR2T∗mn+(i↔j), where si=si(2), and Me and M̃eLR2 are charged-lepton and charged-slepton-LR mass matrices, respectively. (In the present Letter, we define the charged-lepton mass matrix Me and the neutrino mass matrix Mν as ēLMeeR and ν̄LMννLc, respectively, so that the complex conjugate quantities λ∗ijk and so on have appeared in the expression (3.1).) Since M̃eLR2 is proportional to Me, i.e., M̃2eLR=(A+μ(2)tanβ)Me (μ(2)=μ−3gHv24, and A is the coefficient of the soft SUSY breaking terms (Yd)ij(ν̃,ẽ)TLiẽLjcH̄d with A∼1 TeV), we obtain (3.2)(Mrad)ij=2A+μ(2)tanβsisjsksnλ∗ikmλ∗jnl(Me)∗kl(Me)∗nm. Since the coefficient λijk is antisymmetric in the permutation i↔j, it is useful to define (3.3)λijk=εijlLlk, and (3.4)K=SMeLT∗, where S=diag(s1,s2,s3). Then, the radiative neutrino mass matrix is given by (3.5)(Mrad)ij=m0−1sisjεikmεjlnKmlKnk. The coefficient m0−1 is calculated from one-loop diagram (Fig. 1

) as (3.6)m0−1=216π2A+μ(2)tanβFmẽR2,mẽL2, where (3.7)Fma2,mb2=1ma2−mb2lnma2mb2.

Next, let us investigate the contributions from the VEVs of sneutrinos 〈ν̃i〉. In general, the sneutrinos ν̃i can have VEVs vi≡〈ν̃i〉≠0[7], if there are one or more of the following terms: μi5̄LiHu in superpotential W, and Bi5̄LiHu+mHLi25̄LiH̄d† in the bilinear soft SUSY breaking terms Vsoft. In the present model, there is no such a term at tree level, because these terms are forbidden by the Z3 symmetry. However, only an effective mHLi2-term can appear via the loop diagram H̄d→(5̄qlL)c+(10L)c→5̄Lql (Fig. 2

). The contribution mHLi2 is proportional to (3.8)sisjλijk(Me)jk=siεijkK∗jk. On the other hand, the contribution MVEV from 〈ν̃i〉≠0 to the neutrino mass matrix is proportional to (3.9)v21v1v2v1v3v1v2v22v2v3v1v3v2v3v32, and vi≡〈ν̃i〉 are proportional to the values (mHLi2)∗, so that the mass matrix MVEV is given by (3.10)(MVEV)ij=ξm0−1sisjεiklεjmnKklKmn, where ξ is a relative ratio of MVEV to Mrad.

In conclusion, the neutrino mass matrix Mν in the present model is given by the form (3.11)(Mν)ij=m0−1sisjεiklεjmn(KknKml+ξKklKmn), i.e., (3.12)Mν=m0−1SK−KTK−KT−1TrKK−KKT(1+ξ)+K+KT−1TrKTrK−KK+KTKT+1Tr(KK)S, where 1 is a 3×3 unit matrix.4

General features of the neutrino mass matrix

In the present model, if the charged lepton mass matrix Me and the structure of λijk (i.e., Lij) are given, then we can obtain K=(SMeLT)∗, so that we can predict neutrino masses and mixings. However, at present, we have many unknown parameters, so that in order to give explicit predictions of the neutrino masses and mixings, we must put a further assumption on the parameters Kij. In the present section, we investigate general features of the neutrino mass matrix (3.11) (or (3.12)) without making any explicit assumptions about flavor symmetries.So far, the expression of Mν, (3.12), has been given in the initial flavor basis, where 5̄L(+)↔5̄′L(0) mixings have been taken place a diagonal form (4.1)S(a)=diags(a)1,s(a)2,s(a)3,C(a)=diagc(a)1,c(a)2,c(a)3, and the matrix K has been defined by Eq. (3.4), K=(SMeLT)∗. Since S, Me and L are transformed as (4.2)Me→M′e=U5†MeU∗10,L→L′=U5†LU10,S→S′=U5†SU5, under a rotation of the flavor basis (4.3)10L→10′L=U10†10L,5̄Lql→5̄Lqℓ′=U5†5̄Lqℓ, the matrix K transforms as (4.4)K→K′=U5TKU5. We have a great interest in the form of M′ν in the flavor basis with M′e=De≡diag(me,mμ,mτ). Hereafter, we denote the quantities M′ν, K′, and so on in the M′e=De basis as M̂ν, K̂ and so on, respectively. The matrix K̂ is expressed as (4.5)K̂=ŜDeL̂†≃DeURe†L†UeL, where U5=UeL and U10=UeR, and we have put Ŝ≃1 because of S≃1 as we have assumed in Eq. (2.9).Here, let us summarize general features of the present neutrino mass matrix (3.12).

(i)If the matrix K defined by Eq. (3.4) satisfies KT=K in the initial basis, the matrix K′ in the arbitrary basis also satisfies K′T=K′, so that the present model gives 〈ν̃i′〉=0 in the arbitrary basis. For such a case, the neutrino mass matrix is simply given by (4.6)Mν=−m−10S2KK−2KTrK−Tr(KK)+(TrK)2S.

(ii)When K is symmetric under the flavor 2↔3 permutation, the neutrino mass matrix Mν is also symmetric under the 2↔3 permutation. It is well known [8] that when the neutrino mass matrix M̂ν is symmetric under the 2↔3 permutation, the mass matrix M̂ν gives a nearly bimaximal mixing, i.e., sin22θ23=1 and |U13|2=0, which are favorable to the observed atmospheric [9], K2K [10] and CHOOZ [11] data. In the present model, the 2↔3 symmetry of M̂ν means that the parameters (4.7)K̂ij=KklUeLkiULelj, are symmetric under the 2↔3 permutation. In other words, the 2↔3 symmetry of M̂ν is due to special structures of UeL and K. For example, when K and UeL are given by the textures (4.8)K=K11000K22K230K32K33,(4.9)UeL=0−1212−s12c12cc12s12s, the matrix K̂ is 2↔3 symmetric: (4.10)K̂=faaa′gba′bg, so that the neutrino mass matrix M̂ν is also 2↔3 symmetric: (M̂ν)11=−2g2−b2m0−1,(M̂ν)12=(M̂ν)13=(M̂ν)21=(M̂ν)31=(a+a′)(g−b)m0−1,(Mν)22=(M̂ν)33=(a−a′)2(1+ξ)+2(aa′−fg)m0−1,(4.11)(M̂ν)23=(M̂ν)32=−(a−a′)2(1+ξ)+2(aa′−fb)m0−1, and K in the initial basis is given by (4.12)K11=g−b,K22=(g+b)c2−2(a+a′)cs+fs2,K33=(g+b)s2+2(a+a′)cs+fc2,K23=2−as2+a′c2+(g+b−f)cs,K32=2ac2−a′s2+(g+b−d)cs.

Finally, let us show a simple example which is suggested by above comments (i) and (ii). We assume that MeMe† on the initial basis is 2↔3 symmetric: (4.13)MeMe†=FAAAGBABG, so that ULe has a form of a nearly bimaximal mixing. For simplicity, we assume that ULe is given by the full bimaximal mixing form (4.14)UeL=UeLT=0−1212−121212121212, which demands the constraint F=B+G on the matrix (4.13). Then, the eigenvalues D2e=diag(m2e,m2μ,m2τ) are given by (4.15)m2e=G−B,m2μ=G+B−2A,m2τ=G+B+2A. On the other hand, we assume that K in the initial basis is given by the form (4.8) with K23=K32, so that we obtain a=a′ and (4.16)M̂ν=2m0−1−(g2−b2)a(g−b)a(g−b)a(g−b)a2−fg−(a2−fb)a(g−b)−(a2−fb)a2−fg. Note that the mass matrix (4.16) does not include the contributions (ξ-terms) from nonvanishing sneutrino VEVs because of KT=K. The mass matrix (4.16) gives the following eigenvalues and mixings: (4.17)mν1=(g−b)9(g+b)2+2f(g+b)+f2−(g+b+f)m−10,−mν2=−(g−b)9(g+b)2+2f(g+b)+f2+g+b+fm−10,mν3=−22a2−(g+b)fm−10,(4.18)Ûν=cνsν012sν−12cν−1212sν−12cν12,(4.19)sν=mν1mν1+mν2,cν=mν2mν1+mν2, so that we obtain (4.20)tan2θsolar=mν1mν2, together with sin22θatm=1 and |U13|2=0. For a further simple case with f=0, which demands (4.21)K23=K32=12(K33+K22), we obtain mν1=mν2/2=2(g2−b2)m0−1, so that (4.22)tan2θsolar=12,(4.23)R≡Δm221Δm232=34(g2−b2)2a4−(g2−b2)2, where we have considered (4.24)a2=18(K33−K22)2⪢g2−b2=K112(K33+K22)2. The result (4.22) is favorable to the recent solar [12] and KamLAND data [13]. By using the best fit values Δm2solar=7.2×10−5 eV2[12,13] and Δm2atm=2.4×10−3 eV2[9,10], we obtain (4.25)mν2mν3=|g2−b2|a2=4R3+4R=0.20, where R=Δm2solar/Δm2atm, and (4.26)mν1=0.0049eV,mν2=0.0098eV,mν3=0.050eV, where we have used the relation mν1/mν2=1/2 and Δm2atm=3m2ν2/4. Of course, this is only an example, and the result (4.22) is not a prediction which is inevitably driven from the general form of Mν.5

Summary

In conclusion, within the framework of a SUSY GUT model, we have proposed an R-parity violation mechanism which is harmless for proton decay and investigated a general form of the neutrino mass matrix Mν. As we have given in Eq. (3.12), the form of Mν is described in terms of the matrix K defined in Eq. (3.4). (i) If KT=K, the VEVs of sneutrinos are exactly zero, 〈ν̃i〉=0, in the arbitrary basis, so that Mν is given only by the radiative contributions. (ii) If K̂ is 2↔3 symmetric, then M̂ν is also 2↔3 symmetric, so that M̂ν can predict sin22θatm=1 and |U13|2=0.In order to demonstrate that the general form indeed has a phenomenologically favorable parameter range, we have given a simple example of K and MeMe† in the last part of the Section 4. Although such a simple form of K, (4.8), with the constraint (4.23) is likely, the investigation of the origin of the possible form K will be our next task. The purpose of the present Letter is not to give a special model for neutrino phenomenology, and it is to demonstrate that it is indeed possible to build a neutrino mass matrix model with R-parity violation, i.e., without a seesaw mechanism, even if the model is within a framework of GUT.The present model has assigned Z3 quantum numbers to the superfields differently from those in the previous model [5] with 5̄L↔5̄′L mixing: we have been able to assign the same Z3 quantum number to the matter fields 5̄L and 10L (and also to 5̄′L and 5′L). This reassignment will give fruitful potentiality for a further extension of the present model.

Acknowledgements

The author would like to thank J. Sato for helpful conversations. This work was supported by the Grant-in-Aid for Scientific Research, the Ministry of Education, Science and Culture, Japan (Grant Number 15540283).

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