Proton decay and fermion masses in supersymmetric SO(10) model with unified Higgs sector

Yunfei Wu and Da-Xin Zhang*

School of Physics and State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing 100871, China(Received 7 July 2009; published 25 August 2009)

We make a detailed analysis of the proton decay in a supersymmetric SO(10) model proposed by K.

Babu, I. Gogoladze, P. Nath, and R. Syed. We introduce quark mixing, and find that this model can

generate fermion mass without breaking the experimental bound on proton decay. We also predict large

Cabibbo-Kobayashi-Maskawa (CKM) unitarity violations. The CKM matrix V in this paper is defined as

normal, i.e. d0i ¼ Vijdj, where ij run from 1 to 3. The primed field is the weak eigenstate and the unprimed

field is the mass eigenstate.

DOI: 10.1103/PhysRevD.80.035022 PACS numbers: 12.10.Dm, 12.60.Jv

I. INTRODUCTION

In grand unification models [1,2] quarks and leptons areusually contained in the same multiplets. Consequently,baryon and lepton numbers are not conserved, in general. Ifthe models are supersymmetric, dimension-five operatorsmediated by the color-triplet Higgs superfields are domi-nant in these baryon and lepton number nonconservationprocesses [3,4]. These dimension-five operators are alsorelated to the fermion masses, and thus are highly predict-able in most of the supersymmetric unification models.

To build up unification models that generate correctfermion masses and fulfill the stability of baryons, oneusually needs to add in more Higgs multiplets and/ormore fermion multiplets. In Ref. [5], a unified Higgs sector

of 144þ 144 is used in the supersymmetric SO(10) model,so that the model is rather simple. The fermion masses ofthe third generation are generated through cubic couplingsat the price of introducing extra heavy matter superfields of45 and 10 dimension multiplets. The fermion masses of thefirst two generations arise from the Planck scale effects.The dimension-five operators, which mediate proton de-cay, are not very tightly connected to the fermion masses.Consequently, both the fermion masses and the stablebaryons are achieved in the model.

In our work, we analyze the problems of fermion massesand proton decay in this new model. By introducing quarksmixing, we obtain the Cabibbo-Kobayashi-Maskawa(CKM) unitarity breaking effects and the diagonal quarkmasses. The general form of the dimension-five operatorsfor baryon decay is given. We pick out the most importantproton decay mode, p ! Kþ ���. We fit the quartic and

cubic Yukawa couplings of the second and third generationat the unification scale. By using these values, we get thecouplings of dimension-five operators to analyze the pro-ton lifetime.

This paper is organized in the following way: A shortreview of the model is presented in Sec. II. In Sec. III, wegive the quark mixing and mass generation. The CKM

unitarity breaking is analyzed. In Sec. IV, we present thegeneral dimension-five operators and low energyLagrangian for proton decay. In Sec. V, the numericalresults are discussed. We find that the model can survivein some parameter space. Finally, we summarize ourresults.

II. REVIEW OF THE MODEL

The model of Ref. [5] uses the following superpotential:

W ¼ Mð144H � 144HÞþ �451

M0 ð144H � 144HÞ451ð144H � 144HÞ451þ �452

M0 ð144H � 144HÞ452ð144H � 144HÞ452þ �210

M0 ð144H � 144HÞ210ð144H � 144HÞ210; (1)

where M0 is supposed to be at the Plank scale and the �’sare the couplings after integrating out the corresponding 45or 210 dimension component fields. M is the mass of 144Higgs. Equation (1) gives the one-step breaking of SO(10)to the supersymmetric standard model and the doublet-triplet splitting.The terms responsible for the symmetry breaking are

WSB ¼ MQijP

ji þ

1

M0

���451 þ

1

6�210

�Qi

jPjiQ

klP

lk

þ 1

M0

��4�451 �

1

2�452 � �210

�Qi

kPkjQ

jlP

li; (2)

where all the fields are chiral supermultiplets and indices i,j run from 1 to 5. The P’s and Q’s are 24 dimension Higgs

of SU(5) coming from 144 and 144, respectively. To get thefollowing vacuum expectation values and minimization ofWSB,

hQiji ¼ q diagð2; 2; 2;�3;�3Þ;

hPiji ¼ p diagð2; 2; 2;�3;�3Þ; (3)

we need*dxzhang@phy.pku.edu.cn

PHYSICAL REVIEW D 80, 035022 (2009)

1550-7998=2009=80(3)=035022(7) 035022-1 � 2009 The American Physical Society

MM0

qp¼ 116�451 þ 7�452 þ 4�210: (4)

The D-flat condition requires that q ¼ p. The vacuumexpectation values in Eq. (3) break the SO(10) down tothe standard model gauge group.

The further electroweak symmetry breaking requirestwo of the Higgs doublets to be light and all the Higgstriplets to be heavy. The superpotential governing thedoublet-triplet splitting is given in Refs. [5,6]. We onlygive mass splitting results. The Higgs doublet pairs andtriplet pairs before splitting are

D1: ðQa;PaÞ; T1: ðQ�;P�Þ;D2: ðQa;P

aÞ; T2: ðQ�;P�Þ;

D3: ð~Qa; ~PaÞ; T3: ð~Q�; ~P

�Þ; T4: ð~Q�; ~P�Þ; (5)

where �, � take 1, 2, and 3, while a, b take 4 and 5. Thediagonalization of the doublets’ and triplets’ mass matricesuses the rotations

ðQ0a;P

0aÞð ~Q0

a; ~P0aÞ

� �¼ cos�D sin�D

� sin�D cos�D

� � ðQa;PaÞ

ð~Qa; ~PaÞ

� �;

ðQ0�;P

0�Þð ~Q0

�; ~P0�Þ

� �¼ cos�T sin�T

� sin�T cos�T

� � ðQ�;P�Þ

ð~Q�; ~P�Þ

� �;

(6)

where

tan�D ¼ 1

d3

�d2 þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffid2

2 þ d32

q �;

tan�T ¼ 1

t3

�t2 þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffit2

2 þ t32

q �:

(7)

Here

d1 ¼ � 2

5Mþ qp

M0

�296

5�451 � 16�452 �

392

15�210

�;

d2 ¼ � 8

5Mþ qp

M0

�� 1036

5�451 þ

1

2�452 þ

427

15�210

�;

d3 ¼ 2

ffiffiffi3

5

sqp

M0

�10�451 þ

5

4�452 �

5

6�210

�: (8)

The mass eigenvalues are found to be

MD1¼ Mþ qp

M0 ð180�451 þ 9�452 � 10�210Þ;

MD2;D3¼ 1

2

�d1 �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffid2

2 þ d32

q �: (9)

We will set MD3to be small, and get

Hu ¼ ~P0a and Hd ¼ ~Q0

a: (10)

Moreover, the triplet eigenstates’ masses are

MT1¼ Mþ qp

M0 ð180�451 þ 4�452 � 10�210Þ;

MT4¼ �Mþ qp

M0 ð�84�451 � 4�452 þ 2�210Þ;

MT2;T3¼ 1

2

�t1 �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffit2

2 þ t32

q �;

(11)

where

t1 ¼ � 2

5Mþ qp

M0

�576

5�451 � 11�452 �

302

15�210

�;

t2 ¼ � 8

5Mþ qp

M0

�� 816

5�451 � 2�452 þ

212

15�210

�;

t3 ¼ffiffiffi5

p qp

M0

�8�451 þ �452 �

2

3�210

�: (12)

The authors of [6] introduce two kinds of couplings togain fermion masses. All three generations achieve massesfrom the quartic couplings [5,6]

f� ð10Þij gð16i � 16jÞ10ð144� 144Þ10;f�ð10Þ

ij gð16i � 16jÞ10ð144� 144Þ10;f%ð126Þ

ij gð16i � 16jÞ126ð144� 144Þ126;f�ð45Þ

ij gð16i � 144Þ45ð16j � 144Þ45;f� ð120Þij gð16i � 16jÞ120ð144� 144Þ120;f�ð120Þ

ij gð16i � 16jÞ120ð144� 144Þ120;f�ð54Þ

ij gð16i � 144Þ54ð16j � 144Þ54;f�ð10Þ

ij gð16i � 144Þ10ð16j � 144Þ10;

(13)

where i, j are the generation indices. The extra 10-plet and45-plet extra heavy matter couples only to the third gen-eration fermions through the following superpotentials [6]:

W16�144�45 ¼ 1

2!hð45Þh�̂�

ðþÞjB�½�j�̂ðþÞ��iF̂ð45Þ�� ;

Wð45Þmass ¼ mð45Þ

F F̂ð45Þ�� F̂

ð45Þ�� ; (14)

and

W16�144�10 ¼ hð10Þh�̂�ðþÞjBj�̂ð�Þ�iF̂ð10Þ

� ;

Wð10Þmass ¼ mð10Þ

F F̂ð10Þ� F̂ð10Þ

� ; (15)

where �̂ represents the 16-dimension fermion and �̂ rep-resents the 144-dimension Higgs. We follow the authors of[5], defining

fðÞ � ihðÞ; (16)

to get the real couplings. From the above coupling forms,we can get all the Yukawa couplings contributing to thefermion masses.

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III. QUARK MIXING AND MASS GENERATION

The model of Ref. [5] provides a mechanism of gener-ating fermion masses. From Eqs. (13)–(15), we can deduceall the couplings. They can be found in Appendix A.

We assume that the quartic parts of the mass matrix forup-type quarks are already diagonalized to reduce com-plexity. The main difference from Ref. [6] is that theauthors of [6] neglect the fact that the quartic couplingmatrices might induce mixing between light quarks and theextra 45- and 10-plets. In this new scenario the down quarkmass matrix can be written as follows:

Md ¼

dc

sc16bc10bc45bc

d s 16b 45b 10bm11 m12 m13 0 0m21 m22 m23 0 0

m31 m32 m33 m00b mð10Þ

D

0 0 m0b 0 �2mð10Þ

f

0 0 mð45ÞD �2mð45Þ

f 0

0BBBBBBB@

1CCCCCCCA

;

(17)

where

m0b ¼

1

2fð10Þ

�hQ5iffiffiffiffiffiffi10

p þ h ~Q5i2

ffiffiffi3

p�; m00

b ¼ �2ffiffiffi2

pfð45ÞhP5i;

mð45ÞD ¼ �2

ffiffiffi2

pfð45Þp; mð10Þ

D ¼ ffiffiffi2

pfð10Þq; (18)

and all the m’s in the upper-left block are from the quarticcouplings.Noting that the quartic components of the upper-left 3�

3 part of Md are extremely small, the cubic componentswill receive little effect from the quartic couplings whendiagonalizing. In this sense, we can diagonalize the cubicpart first by takingm33 � 0 as in [5], and then consider howthe quartic couplings take effect. The matricesUt;b and Vt;b

can be found in Appendix A, and are slightly different fromRef. [6]. After diagonalizing the cubic part, we get

UbMdVTb ¼

m11 m12 m13 cos�Vbm13 sin�Vb

0m21 m22 m23 cos�Vb

m23 sin�Vb0

m31 cos�Ubm32 cos�Ub

�1 0 0m31 sin�Ub

m32 sin�Ub0 0 �2

0 0 0 �3 0

0BBBBB@

1CCCCCA; (19)

where �2 and �3 are the eigenmasses of rotated extra heavyfermions.

When �2 and �3 are extremely large, we can diagonalizethe light down-type quark mass matrix,

mijd ¼

m11 m12 m13 cos�Vb

m21 m22 m23 cos�Vb

m31 cos�Ubm32 cos�Ub

�1

0B@

1CA: (20)

If we denote the Yukawa couplings mid ¼ mij

d V0jk

ki , in the

up quark diagonalized basis V 0jk is analogous to the CKM

matrix.The matrices diagonalizing Md are

VTd ¼VT

b �V 0ijþO

��1

�2

;�1

�3

�þ���

¼

V 0ud V0

us V 0ub 0 0

V 0cd V0

cs V 0cb 0 0

V 0tdcos�Vb

V0tscos�Vb

V 0tbcos�Vb

sin�Vb0

�V0td sin�Vb

�V0ts sin�Vb

�V 0tb sin�Vb

cos�Vb0

0 0 0 0 1

0BBBBBBBB@

1CCCCCCCCA

þO��1

�2

;�1

�3

�þ��� (21)

and

Ud ¼ Ub þO��1

�2

;�1

�3

�þ � � � : (22)

The upper-left 3� 3 part of VTd is just the transpose of the

CKM matrix. When taking the quartic coupling for the upquarks to be diagonalized, the up quark mass matrix can bediagonalized easily by

VTu ¼ VT

t and Uu ¼ Ut: (23)

The mass matrices for the charged leptons have the samestructure as the down quarks, for they share the sameYukawa couplings.Besides mass matrix diagonalization, the CKM unitarity

violation can also be derived from Eq. (21) easily.

jVudj2 þ jVcdj2 þ jVtdj2 ¼ 1� jVtdj2tan2�Vb;

jVusj2 þ jVcsj2 þ jVtsj2 ¼ 1� jVtsj2tan2�Vb;

jVubj2 þ jVcbj2 þ jVtbj2 ¼ 1� jVtbj2tan2�Vb;

jVtdj2 þ jVtsj2 þ jVtbj2 ¼ 1� sin2�Vb:

(24)

When Vtd and Vts are very small in the right-hand side ofthe first two equations, the last two equations give the mostimportant unitarity violation. We denote

b ¼ jVtbj2tan2�Vb; t ¼ sin2�Vb

: (25)

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We require that j tan�Vbj 1 by examining the unitarity

bound on the CKM matrix [7]. This is different fromRef. [8], where the authors took j tan�Vb

j 1. Under

this new condition, the b� unification fb ¼ f gives

tan�D ¼ 5ffiffiffiffiffiffi30

p83

: (26)

From Sec. V, we will see that t and b are given at thepercent level by fixing the mass Yukawa couplings.

The other CKM violations are

VudVus þ VcdVcs þ VtdVts ¼ �VtdVtstan2�Vb

;

VusVub þ VcsVcb þ VtsVtb ¼ �VtsVtbtan2�Vb

;

VudVub þ VcdVcb þ VtdVtb ¼ �VtdVtbtan2�Vb

:

(27)

They are related to the phenomena of flavor changingneutral currents, with which we are not presentlyconcerned.

IV. DIMENSION-FIVE OPERATORSAND DECAY RATES

In supersymmetric unification models, the dominantmechanism of inducing proton decay is through thecolor-triplet Higgsino mediation. The resultingdimension-five operators are of the type LLLL andRRRR. We will focus on LLLL-type only to simplify ourdiscussion, although the RRRR-type can also be important[9].

The Yukawa couplings of the Higgs to the matter mul-tiplets are as follows:

WY ¼ hiuuci QiHu � V�

ijfjdd

ci QjHd � fiee

ci LiHd

þ YiQfQiQiHcf þ Yij

LfQiLj�Hcf þ Yij

ecfuci e

cjHcf

þ Yijqcfu

ci d

cj�Hcf; (28)

where the hiu’s, fjd’s, and fie’s are the Yukawa couplings

that give masses and the Y’s are the Yukawa couplings withthe color-triplet Higgs. f denotes different color triplets.

The dimension-five operators that cause the nucleondecay can be written explicitly as

W5 ¼ 1

MTf

YiQfY

klLfðQiQiÞðQkLlÞ

þ 1

MTf

YijecfY

klcfðuci ecjÞðuckdcl Þ: (29)

The total antisymmetry in the color index requires i � k,which implies the dominant mode is p ! K �� [10].

Dressing of wino to dimension-five operators gives thetriangle diagram factor [10,11]

fðu; dÞ ¼ M2

m2~u �m2

~d

�m2

~u

m2~u �M2

2

lnm2

~u

M22

� m2~d

m2~d�M2

2

lnm2

~d

M22

�;

(30)

where M2 is the wino mass. The resulting four-fermionoperators can be written as

L ¼ YðijkÞASði; j; kÞAL����½ðu�i d0�i Þðd0�j �kÞðfðuj; ekÞþ fðui; d0iÞÞ þ ðd0�i u�i Þðu�j ekÞðfðui; diÞþ fðd0j; �kÞÞ þ ðd0�i �kÞðd0�i u�j Þðfðui; ekÞþ fðui; d0jÞÞ þ ðu�i d0�j Þðu�i ekÞðfðd0i; ujÞ þ fðd0i; �kÞÞ�;

(31)

where the coupling YðijkÞ is defined as follows:

YðijkÞ ¼�

1

MT1

YiQ1Y

jkL1 þ

V�jk

MT2

½YiQ2Y

kL2cos

2�T

þ YiQ3Y

kL3sin

2�T þ ðYiQ3Y

kL2 þ Yi

Q2YkL3Þ

� cos�T sin�T

�þ V�

jk

MT3

½YiQ2Y

kL2sin

2�T

þ YiQ3Y

kL3cos

2�T � ðYiQ3Y

kL2 þ Yi

Q2YkL3Þ

� cos�T sin�T���2

2 : (32)

In Eq. (31), the function AS refers to the short-rangerenormalization effect between the unification and thesupersymmetry breaking scale, and AL is the long-rangerenormalization effect between the supersymmetry scaleand 1 GeV. All of these have been investigated thoroughlyin [10,12].The relevant terms for p ! Kþ þ ��� from Eq. (31) are

L ¼ AL����ððd�u�Þðs���Þ þ ðs�u�Þðd���ÞÞ� ½ASðc; u; sÞYð2; 1; 2ÞVcsVcdðfðc;�Þ þ fðc; d0ÞÞþ ASðt; u; sÞYð3; 1; 2ÞVtsVtdðfðt; �Þ þ fðt; d0ÞÞ�:

(33)

We neglect the �e mode for the smallness of the first-generation Yukawa couplings. The direct coupling to � issuppressed by the CKM matrix element. Although thecoupling to 10� is order 1, 10� is heavy and its contributionto the rotated � is highly suppressed.

Noting that YijL1 has only diagonal elements, it will have

no contribution. So �Hc1 or the first term in Yði; j; kÞ doesnot contribute to Eq. (33).We can use the chiral Lagrangian technique [13,14] to

obtain hadronic-level matrix elements,

hKþjðu; dÞLsLjpi ¼ �

f

�1þ

�D

3þ F

�mN

mB

�; (34)

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hKþjðu; sÞLdLjpi ¼ �

f

2D

3

mN

mB

; (35)

in the limitmu;d;s mN;B. All the parameters can be found

in [10,15].

V. NUMERICAL RESULTS AND DISCUSSION

In this section, we present some numerical results. Therecent Super-Kamiokande bound on proton decay is [7]

p!Kþþ �� > 1:6� 1033 yrs: (36)

In the present model, the doublet and triplet massesconnect to the 144 Higgs mass. We keep a pair of Higgsdoublets light.

When we set M0 in Eq. (1) to be at the Planck scale1019 GeV, we can get the relation between light Higgsmasses and the 144 Higgs mass. This is a fine-tuningproblem relating to the doublet-triplet splitting. We chooseone of the three doublets to be light while leaving othersheavy. Then we automatically get the heavy triplet Higgsmasses.

From Eq. (A3), the conditions tan�Ub 1 and

tan�Vb 1 give

mð10ÞF p and mð10Þ

F q; (37)

if we take fð10Þ and fð45Þ to be of order 1.The unification scale can be

MGUT � 2� 1016 GeV: (38)

The supersymmetry breaks at about 1 TeV. All the sfer-mion masses used in Eq. (30) are taken to be 1 TeV. Thewino mass is taken as M2 ¼ 300 GeV.

We take p ¼ q ¼ 1016 GeV and mð10ÞF ¼ mð45Þ

F ¼1017 GeV. After fine-tuning MD3

to the order of

102 GeV, the other Higgs doublets and triplets are fixedat the order of 1016 GeV. Because the number of Yukawacouplings in this model is redundant, we can just choosesome of them to generate the light fermion masses. Here

we take �ð10Þij ¼ %ð126Þ

ij ¼ 0. At the unification scale we fit

�ð10Þ, �ð120Þ, � ð10Þ, � ð120Þ, fð10Þ, and fð45Þ to get the correctfermion masses. Besides fermion masses, this model canhave a long enough proton lifetime even without the can-cellation introduced in Ref. [8], which implies vanishingdown-type fermion masses of the second generation. Weget the longest proton lifetimes without affecting the fer-

mion masses by fine-tuning �ð45Þ and taking �ð54Þ ¼�7=27�ð45Þ. The results are given in Table I, while wetake all the parameters in Eqs. (34) and (35) as [15]

� ¼ 0:0118 GeV3; D ¼ 0:8;

F ¼ 0:47; f ¼ 0:131 GeV;

mN ¼ 0:94 GeV; mB ¼ 1:15: GeV: (39)

From Table I, the longest possible proton lifetime de-creases when tan� increases. The present model dilutes therelation between fermion masses and dimension-five op-erators. We can choose some parameters to suppress thedimension-five operators without affecting the fermionmasses. Even for very large tan�, this model could havea long enough proton lifetime.The unitarity breaking of the CKM matrix can be ob-

tained directly from Eq. (21). Unitarity is good for the first

and second columns of the CKM matrix up to Oð�1

�2; �1

�3Þ,

because �2 and �3 are extra fermion masses at about1016 GeV. For the first two equations of (24), unitarityviolations are about 10�6 due to the smallness of Vtd andVts, within experimental constraints [7]. This model pre-dicts relatively large CKM breaking for b and t. Theirvalues are at the percentage level.

VI. SUMMARY

In this work we analyze the supersymmetric SO(10)

model with a unified 144þ 144 Higgs [5,6]. We introducethe fermion mixing and find large unitarity violations of theCKM matrix. We find that the proton lifetime is in agree-ment with the experiment bound for a rather large value oftan�.

ACKNOWLEDGMENTS

This work was supported in part by the National NaturalScience Foundation of China (NSFC) under GrantNo. 10435040.

APPENDIX A: MASS MATRICES AND YUKAWACOUPLINGS

In this section we list the detailed results of the massmatrices and the diagonalization matrices. From Eqs. (13)–(15) we can get all the couplings for the component fields.We put all of them in Tables II and III. The baryon-leptonnumber violating terms can also be found in Ref. [8].

TABLE I. Proton lifetime and CKM unitarity violation. b isobtained with jVtbj ¼ 0:77 [7].

tan� 2 3 6 10 20

� ð10Þ=10�20 GeV�1 �0:10 �0:20�0:30 �0:40 �0:80� ð120Þ=10�20 GeV�1 1.5 2.5 4.5 7.2 14.5

�ð10Þ=10�20 GeV�1 �0:40 �0:10�0:10 �0:70 �0:90�ð120Þ=10�20 GeV�1 5.0 2.8 2.3 6.2 7.2

�ð45Þ=10�19 GeV�1 �8:48 �1:58�1:35 �7:82 �6:47�ð54Þ=10�19 GeV�1 22.0 4.10 3.52 20.3 16.8

fð45Þ 1.24 1.04 0.96 0.95 0.95

fð10Þ 0.960 1.14 1.59 2.05 2.85

p=ð1033 yrsÞ 1:4� 102 84 38 4.3 1.9

b=ð%Þ 2.4 1.7 1.4 1.3 1.3

t=ð%Þ 3.0 2.2 2.0 1.8 1.8

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From Tables II and III we can easily write out the massmatrix. The matrices that used to diagonalize the lower-right 3� 3 part of the mass matrix (18) are given by

Ubðt;Þ ¼

1 0 0 0 00 1 0 0 00 0 cos�Ubðt;Þ � sin�Ubðt;Þ 00 0 sin�Ubðt;Þ cos�Ubðt;Þ 00 0 0 0 1

0BBBBB@

1CCCCCA (A1)

and

Vbðt;Þ ¼

1 0 0 0 00 1 0 0 00 0 cos�Vbðt;Þ � sin�Vbðt;Þ 00 0 sin�Vbðt;Þ cos�Vbðt;Þ 00 0 0 0 1

0BBBBB@

1CCCCCA: (A2)

They are just the matrices used in Ref. [6], when eliminat-ing the upper-left 2� 2 identity matrices.

The matrices used for t and have the same form. Allthe matrix elements are given by [6]

tan�Ub¼ � fð10Þqffiffiffi

2p

mð10Þf

; tan�Vb¼

ffiffiffi2

pfð45Þpmð45Þ

f

; (A3)

tan�U¼ 6

ffiffiffi2

pfð45Þp

mð45Þf

; tan�V¼ 3

ffiffiffi2

pfð10Þq

4mð10Þf

; (A4)

and

tan�Ut¼ � 4

ffiffiffi2

pfð45Þq

mð45Þf

; tan�Vt¼

ffiffiffi2

pfð45Þpmð45Þ

f

: (A5)

When taking the quartic couplings for up-type quarks tobe diagonalized, we can easily get the Yukawa couplingconstants for the masses.

hu ¼�24pffiffiffi5

p �ð10Þ þ 6pffiffiffi5

p���ð45Þ þ 4

3�ð120Þ

��cos�D

� ffiffiffi6

pp

�20�ð10Þ � 7

12�ð45Þ � 9

4�ð54Þ þ 20

9�ð120Þ

�

� sin�D; (A6)

fd ¼ q cos�Dffiffiffi5

p�11

15%ð126Þ þ 8� ð10Þ � �ð10Þ þ 56

3� ð120Þ

�

þ 2ffiffiffi6

pq

�1

9%ð126Þ þ 8� ð10Þ � 32

9� ð120Þ

�sin�D;

(A7)

fe ¼ 3q cos�Dffiffiffi5

p�� 3

15%ð126Þ þ 8� ð10Þ � �ð10Þ � 8

3� ð120Þ

�

þ 2ffiffiffi6

pq

�1

3%ð126Þ � 8� ð10Þ � 4

3� ð120Þ

�sin�D; (A8)

TABLE III. All the cubic couplings that contribute to masses and baryon-lepton number violating terms.

Couplings Mass terms Baryon number violating terms

J1 M̂iP̂jF̂ij �2

ffiffiffi2

pfð45Þ ð16bc�45Q�a þ 16Lb

45c�baÞPa ð�����16bc�45tc� þ 16La

45Qa�ÞP�

J2 �ijklmM̂ijP̂kF̂lm 1ffiffiffiffi

10p fð45Þ 4ð16tc�45Q�b�ba þ 45tc�

16Q�b�baÞPa 4ð16tc�45c þ 16c45tc� � ����16Q�a45Q�b�abÞP�

J3 �ijklmM̂ijP̂kl

n F̂mn 1ffiffi

2p fð45Þ 8

3 ð16tc�45Q�b�ba þ 45tc�16Q�b�baÞ~Pa 4ð16c45tc� � 16tc�

45cÞ~P�

K2 M̂ijQ̂jF̂i � 1

2ffiffiffiffi10

p fð10Þ ð16Qa�10bc� � 16c10Lb�baÞQa ð����16tc�10bc� þ 16Qa�10LaÞQ�

K3 M̂ijQ̂kijF̂

k 12ffiffi2

p fð10Þ ð216c10Lb�ba � 2316Qa�10bc�Þ ~Qa ð����16tc�10bc� � 16Qa�10LaÞ~Q�

TABLE II. All the quartic couplings that contribute to masses and baryon-lepton number violating terms. Generation indices areneglected. All the couplings are matrices of the three generations.

Coupling constants Mass terms Baryon number violating terms

J1 M̂ijM̂jP̂ki P̂k 2ð��ð45Þ � �ð54Þ þ 8�ð10Þ þ 8

3�ð120ÞÞ �3pðQa�dc� þ �abecLbÞPa 2pð����uc�dc� þQ�bLbÞP�

M̂ijM̂kP̂kj P̂i 2ð��ð45Þ þ �ð54ÞÞ ð2pQa�dc� � 3p�abecLbÞPa ð2p����uc�dc� � 3pQ�bLbÞP�

J2 �ijklmM̂ijM̂klP̂m

n P̂n � 1ffiffi

5p �ð10Þ �24puc�Q

�b�baPa 16pðuc�eþ � ����Q

�Q�ÞP�

�ijklmM̂inM̂jkP̂l

nP̂m 1ffiffi

5p ð��ð45Þ þ 4

3�ð120ÞÞ 6puc�Q

�b�baPa 4p��Q�Q� � uc�e

þÞP�

J3 �ijklmM̂ijM̂klP̂p

n P̂nmp 2�ð10Þ 40puc�Q

�b�ba~Pa 20pðuc�eþ � ����Q

�Q�Þ~P�

�ijklmM̂inM̂jkP̂p

n P̂lmp � 1

2�ð45Þ � 1

2�ð54Þ þ 4

3�ð120Þ � 20

3 puc�Q

�b�ba~Pa 40pðuc�eþ � ����Q

�Q�Þ~P�

�ijklmM̂npM̂jkP̂k

pP̂lmn

12 ð�ð45Þ � �ð54ÞÞ 34

3 puc�Q

�b�ba~Pa �20pðuc�eþ þ ����Q

�Q�Þ~P�

K1 M̂ijM̂klQ̂mn Q̂

n 2ð 215%ð126Þ þ � ð10Þ þ 23 �

ð120ÞÞ �24quc�Q�b�baQ

a 16qðuc�eþ � ����Q�Q�ÞQ�

K2 M̂ijM̂jQ̂ki Q̂k

1ffiffi5

p ð 115%ð126Þ þ 8� ð10Þ þ 83 �

ð120ÞÞ �3qðQa�dc� þ �abeþLbÞQa 2qð�����uc�dc� þQ�bLbÞQ�

M̂ijM̂kQ̂ki Q̂j

1ffiffi5

p ð 415%ð126Þ � �ð10Þ þ 163 �

ð120ÞÞ ð�2qQa�dc� þ 3q�abeþLbÞQa ð2q����uc�dc� þ 3qQ�bLbÞQ�

K3 M̂ijM̂jQ̂kl Q̂

lik 2ð 115%ð126Þ þ 8� ð10Þ � 8

3 �ð120ÞÞ 4qðQa�dc� þ �abeþLbÞ ~Qa �5qðQ�bLb � ����uc�d

c�Þ~Q�

M̂ijM̂kQ̂kl Q̂

lij

43 ð� 1

5%ð126Þ þ 4� ð120ÞÞ ð12q�abeþLb � 4

3qQa�dc�Þ~Qa ð2q����uc�dc� � 3qQ�bLbÞ ~Q�

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ht ¼ 3 sin�Utcos�Vt

fð45Þ�cos�Dffiffiffiffiffiffi

10p þ sin�D

2ffiffiffi3

p�; (A9)

fb ¼ � 1

2sin�Ub

cos�Vbfð10Þ

�cos�Dffiffiffiffiffiffi

10p þ sin�D

2ffiffiffi3

p�; (A10)

and

f ¼ � 1

2sin�U

cos�Vfð10Þ

�� cos�Dffiffiffiffiffiffi

10p þ

ffiffiffi3

psin�D2

�:

(A11)

We have neglected the generation indices in hu, fd, and fe,which should be 3� 3 matrices. ht, fb, and f are theYukawa couplings after diagonalization of the cubic cou-pling matrices.

The couplings for baryon-lepton number violating termsQQ and QL used in Sec. IV are given by

YQ1 ¼ �32q

�2

15%ð126Þ þ � ð10Þ þ 2

3� ð120Þ

�; (A12)

YQ2 ¼ 16pffiffiffi5

p �ð10Þ þ 4pffiffiffi5

p���ð45Þ þ 4

3�ð120Þ

�; (A13)

YQ3 ¼ 5ffiffiffi2

pp

�3�ð54Þ þ �ð45Þ � 16

3�ð120Þ � 4�ð10Þ

�;

(A14)

YL1 ¼ 2p

���ð45Þ þ 5�ð54Þ � 16�ð10Þ � 16

3�ð120Þ

�;

(A15)

YL2 ¼ qffiffiffi5

p�� 14

15%ð126Þ þ 3�ð10Þ � 16� ð10Þ � 64

3� ð120Þ

�;

(A16)

YL3 ¼ffiffiffi2

pq

�40� ð10Þ � 1

15%ð126Þ � 16

3� ð120Þ

�; (A17)

and

Y33Q2 ¼

4fð45Þffiffiffiffiffiffi10

p cos�Vtsin�Vt

: (A18)

Except for the last one, all the equations carry generationindices and are matrices. It is important to note that aftertaking the quartic couplings for the up quark to be diago-nal, the YQ’s and YL1 have only diagonal elements and YL1

has no contribution to Eq. (33).

APPENDIX B: FIELD NORMALIZATION

In deducing Tables II and III, we have used

@P̂ijk @P̂

ijk ¼ @~Pa@~Pa þ @~P�@~P� þ � � � : (B1)

So the triplets and doublets are normalized according to

ð@ ~Qa; @~PaÞ !

ffiffiffi3

p

2ffiffiffi2

p ð@~Qa; @~PaÞ;

ð@ ~Q�; @~P�Þ !

ffiffiffi2

p2

ð@ ~Q�; @~P�Þ:

(B2)

The other fields’ normalizations can be found in [5,6].

[1] H. Georgi, in Proceedings of the Williamsburg Meeting ofthe Division of Particles and Fields of the AmericanPhysical Society, edited by C. E. Carlson (AIP, NewYork, 1975), p. 575.

[2] H. Fritzsch and P. Minkowski, Ann. Phys. (N.Y.) 93, 193(1975).

[3] N. Sakai and T. Yanagida, Nucl. Phys. B197, 533 (1982).[4] S. Weinberg, Phys. Rev. D 26, 287 (1982).[5] K. S. Babu, I. Gogoladze, P. Nath, and R.M. Syed, Phys.

Rev. D 72, 095011 (2005).[6] K. Babu, I. Gogoladze, P. Nath, and R. Syed, Phys. Rev. D

74, 075004 (2006).[7] C. Amsler et al. (Particle Data Group), Phys. Lett. B 667, 1

(2008).[8] P. Nath and R.M. Syed, Phys. Rev. D 77, 015015 (2008).[9] T. Goto and T. Nihei, Phys. Rev. D 59, 115009 (1999).[10] J. Hisano, H. Murayama, and T. Yanagida, Nucl. Phys.

B402, 46 (1993).[11] P. Nath and R. Arnowitt, Phys. Rev. D 38, 1479 (1988).[12] J. Ellis, D. V. Nanopoulos, and S. Rudaz, Nucl. Phys.

B202, 43 (1982).[13] M. Claudson, M.B. Wise, and L. J. Hall, Nucl. Phys.

B195, 297 (1982).[14] S. Chadha and M. Daniels, Nucl. Phys. B229, 105 (1983).[15] Y. Aoki, C. Dawson, J. Noaki, and A. Soni, Phys. Rev. D

75, 014507 (2007).

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