Z. Phys. C 74, 171—178 (1997)

A model for the three lepton decay mode of the proton

Biswajoy Brahmachari1, Patrick J. O’Donnell2, Utpal Sarkar3

1High Energy Section, International Center for Theoretical Physics, I-34100 Trieste, Italy2Physics Department, University of Toronto, Toronto, Ontario M5S 1A7, Canada3Theory Group, Physical Research Laboratory, Ahmedabad 380 009, India

Received: 14 May 1996

Abstract. An extension of the left—right symmetric modelhas been constructed which gives in a natural way thethree lepton decay modes of the proton which have beensuggested as an explanation for the atmospheric neutrinoanomaly. We write down the potential which after minim-ization gives the proper choice of the Higgs spectrum.With this Higgs spectrum we then study the evolution ofthe gauge coupling constants and point out that for con-sistency one has to include effects of gravity.

1 Introduction

One expects to see produced in the atmosphere twice asmany muon neutrinos as electron neutrinos since de-tectors cannot distinguish between neutrinos and anti-neutrinos. The two water-Cerenkov detectors give a resultwhich is a factor of two smaller for the ratio R"N (lk)/N(l

e), a ratio in which many systematic uncertainties are

expected to cancel. The results are

Robs

/RMC

"0.60$0.07$0.05

from the Kamiokande experiment [1] (based on 6.1 Ktonyear) and

Robs

/RMC

"0.54$0.05$0.12

from IMB [2] based on 7.7 Kton year). The experimentslook for ‘‘contained’’ single prong events which are causedby neutrinos with energies below 2 GeV. The ratio R isestimated from the relative rates of sharply defined singlerings (muon-like) and diffused single rings (electron-like).(Other reported values [3]for this ratio are: Frejus [4]0.87$0.21 (1.56 Kton year), NUSEX [5] 0.99$0.40 (0.4Kton year), SOUDAN II [6] 0.69$0.19 (1 Kton year).

Although this atmospheric neutrino anomaly hasa popular explanation within the neutrino oscillationframework [7] there is an alternative explanation basedon three lepton decays of the proton [8]. The single ringevents have been analysed within the proton decay inter-pretation where it is argued that, if the proton decays into

a positron and two neutrinos with a lifetime of q(PP

e`ll)\4]1031 years, then the excess observed electronevents could be due to proton decay events [8]. Thelifetime for this particular decay mode of the proton [9] isconsistent with the present limit [10] for the expecteddominant decay mode of the proton q (PPe`n°)'5]1032 yr. The possibility that this particular decaymode might dominate over other usual decay modes wasconsidered earlier on general grounds [11—13] where itwas pointed out that it is difficult to have light neutrinosin the final decay product. It now appears that recent datafrom Kamiokande [14] may have ruled out a full explana-tion of their results coming from the three lepton decaymode. This does not, in itself rule out the possibility of thethree lepton decay mode of the proton. Indeed, in theSudbury Neutrino Observatory, it is proposed to measurethe recoil of the nucleus to see such a decay mode.Although the three lepton decay mode for the proton hasbeen discussed before [15], the conclusion there was thatit would not be possible. It has been shown however [13],that with the inclusion of Higg’s scalars the three leptonmode is possible.

In most models the left-handed neutrinos are light andthe right handed-neutrinos are heavy. Thus the decaymodes are restricted to

PPe`lLlL

or PPe`lLcl

Lor PPe`l

Lcl

Lc. (1)

Recently [13] it has been pointed out that these decaymodes are allowed in the framework of certain left—rightsymmetric models. In this paper we construct an explicitmodel in which three lepton decay mode of the proton isthe dominant one. We write down the general form of thepotential and show that the minima of the potential areconsistent with the choice of Higgs structure. We thenstudy the evolution of the gauge coupling constants in-cluding non-renormalizable interactions which may arisefrom the Planck scale physics [16, 17].

2 A proton decay mechanism

We work in the framework of the left—right sym-metric model [11—18] and start with the symmetry

breaking chain

Sº(4)]Sº(2)L]Sº(2)

R[,G

PS]

MPS

&"Sº (3)c]Sº(2)

L]Sº(2)

R]º (1)

(B~L)[,G

LR]

MR

&"Sº (3)c]Sº(2)

L]º(1)

Y[,G

std]

MW

&"Sº (3)c]º(1)

em.

In the minimal left—right symmetric model [11, 12, 18] theHiggs scalars consist of the following fields. The groupG

PSis broken by the vacuum expectation value (vev) of the

field H which transforms as (15, 1, 1) under the group GPS

.The right handed group is broken by the vev of a righthanded triplet Higgs field *

R,(1, 1, 3,!2)L(10, 1, 3).

By left—right parity this will imply the existence of the lefthanded triplet field *

L,(1, 3, 1, !2)L(10, 3, 1), which

gives Majorana mass to the left-handed neutrinos andwhose vev should be 41 GeV. (Where there are fournumbers, the first three correspond to the representationsof Sº (3)

c]Sº(2)

L]Sº (2)

Rwhile the last shows the º(1)

quantum numbers).Finally the electroweak symmetry breaking takes

place through the vev of a doublet scalar field /,

(1, 2, 2, 0)L(1, 2, 2). This field / also gives masses to thefermions. However this does not reproduce the rightquark—lepton mass ratios. For the right magnitude of thequark—lepton mass ratios we require yet another field m,(1, 2, 2, 0)L(15, 2, 2) [11]. The Sº (3)

csinglet component

of this field m, which acquires vev, has differentClebsch—Gordon coefficients for the Sº(3)

cand the º(1)

part of Sº(4). Hence they contribute to the quark andlepton masses with different coefficients. As a result, suit-able combinations of / and m can reproduce the rightquark—lepton mass ratios.

It was pointed out in [13] that with this minimalscalar content it is possible to get the decay mode requiredto explain the atmospheric neutrino problem. For this weneed the Sº(3)

ccolor triplet components of the fields

*L

and m, which we represent by *3L

and m3 respectively.(*

Lis (10, 3, 1) under G

PSand the (1, 3, 1, !2) component

acquires a vev; the 10 representation of Sº(4) decomposesunder Sº (3)

cas 6#3#1. Similarly, the 15 representa-

tion of Sº(4) decomposes under Sº (3)cas 8#3#31 #1).

Then the Yukawa couplings,

LYuk

"fql(q

LclL)*3R

L#f

dl(d

RclLc)m3R (2)

and the quartic scalar coupling,

Ls"jpr*3

L*3Lm3m1 (3)

give the (B—¸) conserving proton decay PPeL`l

LlLc

through the diagram of Fig. 1. This diagram will also give,with equal probability, the decay mode, PPk

L`l

LlLc.

Such a proton decay mechanism will give equal num-ber of sharp single rings (muon—like events) and diffusesingle rings (electron-like events). Since the proton decayevents contribute to both electron- and muon-like eventsthis seems to imply that the reduction of the ratio R

Fig. 1 Diagram giving PPeL`l

LlLc

cannot be explained by proton decay events. However,the weighted average of the two processes with ratiosR(proton decay)"1 and R (atmospheric neutrino)"2 (thetheoretical expected ratio for the muon-to-electron eventsif they have their origin only from the atmospheric neu-trinos) can in fact contribute to the atmospheric neutrinoanomaly.

To see this we note the observed numbers of electron-like [muon-like] events n

e(obs) [nk (obs)] are the sum of

the numbers of electrons [muons] produced by the atmo-spheric electron [muon] neutrinos l

e[lk] through scatter-

ing inside the detector ne(atm) [nk (atm)] and from the

decays of the protons into e`lLlLc [k`l

LlLc] inside the

detectors ne(prot) [nk (prot)]. That is,

Robs

"

nk(obs)

ne(obs)

"

nk (atm)#nk(prot)

ne(atm)#n

e(prot)

\0.6RMC

\1.2.

In [8] it was assumed that the proton decays into e`lLlLc

(and not muons), i.e., nk(prot)"0. They found that, bydoing a Monte Carlo simulation to obtain the protonlifetime, the atmospheric neutrino anomaly could beachieved with a proton lifetime of q

p\4]1031 years. In

the above relation this corresponds to ne(prot)\

(2/3)ne(atm)\(1/3)nk(atm).

In the present scenario the proton decays into bothelectrons and muons so that n

e(prot)\nk(prot). Thus for

the explanation of the atmospheric neutrino anomaly werequire n

e(prot)\4n

e(atm)\2nk(atm). Since the number of

proton decays is increased to give the same Robs

there willbe a reduction in the proton lifetime by a factor of 6. Thusin this scenario we can explain at least part of the atmo-spheric neutrino anomaly with a proton lifetime q

p\

(2/3)]1031 years, which is still consistent with presentexperiments on proton decay.

The amplitude for the process is given by

A"

jpr f 2ql

fdlSm1T

m2mÊm4*Ê(4)

where, Sm1T"S/T"250 GeV, jpr is the strength of thequartic coupling defined in (3) and the f

ql, f

dlare the

Yukawa coupling constants.

172

Then, taking reasonable values for the quartic and thequadratic Yukawa coupling parameters, jpr\10~2 andf\10~3, say, requires mmÊ and m*Ê to be relatively light.For the proton decay mode PPe

L`l

LlLc to be 1031 years

to explain the atmospheric neutrino anomaly, it has beenargued [11, 12] that the mass mmÊ can be as light as abouta TeV, which requires m*Ê\ few TeV. This can also beachieved naturally [13].

3 A general left—right symmetric potential

We now concentrate on the masses mmÊ and m*Ê. In earlierreferences [11—13] two new mechanisms were proposedwhich could give rise to appropriate masses for thesefields. Here we check the consistency of these two differentmechanisms when the complete potential with all thescalar fields is written and minimized. First we describethe two mechanisms which keep these two fields mmÊ andm*Ê light.

For light mmÊ it was argued [11, 12] that if there existsa field m@,(15, 2, 2) which can mix with the field m, thenfine tuning can give a large mass to one combination ofthe fields m and m@ and keep the other combination witha light mass. However, this will also keep the masses of thecolor octet and the color singlet light, which is undesirablefrom the point of view of evolution of the gauge couplingconstants. This problem is avoided [12] if instead of m@ weintroduce a field s,(6, 2, 2) under G

PS. If the symmetry

group GPS

is embedded in the unified group SO(10) thenthis field is contained in a 54-plet of SO (10), which isrequired to break the symmetry of the large group. Themixing of the field s with m can then give a mass matrixwhich may be fine tuned to give only a light color tripletfield. We shall discuss the details of this mechanism ata later stage. In the rest of the article we shall use the fields and not m@.

For the field *3 to remain light we have to alter theway in which the left-right symmetric model gets broken,although, as we shall see later, this particular method willmake the mechanism in the last paragraph consistent withthe minimization of the general potential. For this pur-pose we introduce a singlet field g,(1, 1, 1, 0)L(1, 1, 1),to break the left-right parity (usually this is referred to asD-parity) at a different scale from the left—right symmetrybreaking scale M

R[19]. This field transforms under D as

gP!g. The scalar and the fermionic fields transformunder D-parity as *

L,RP*

R,Land t

L,RPt

R,L, while

/ and m stay the same. Then with the field g we can addnew terms to the lagrangian,

Lg*"!Mgg (*sR*R!*s

L*

L)!jgg2 (*s

L*L#*s

R*R). (5)

In theories where the triplet Higgs breaks D-parity alongwith Sº(2)

Rwe have m*

Lm*

R. The masses of the fields

*Land *

Rare not be the same when D-parity is broken by

the vev of the field g. When g gets a non-zero vacuumexpectation value, the masses are given by,

m2*L"m2*!MgSgT#jgSgT2 and

m2*R"m2*#MgSgT#jgSgT2

where m* is the mass at which the left—right symmetrybreaking of Sº(2)

Ris broken spontaneously. So, in the

absence of the g field, both these * fields will have mass\m*. With SgT present the parameters in the three termscan be tuned to make m*L

vanish. The field *L

will thenacquire mass of the order of a TeV from radiative correc-tions. The same sets of parameter will also makem*R

heavy and lead to a solution

SgT\S*RT<S*

LT and MgBm*R

BS*RTBm*

and m*L;S*

RT (6)

Thus we can have m*Ê\m*L\ few TeV even when m*R

\

S*RT is as large as 1010 GeV.We now write the most general potential with all the

fields present in the minimal left—right symmetric modelwith the additional fields m, s and the D-parity odd- singletfield g. We then show the two mechanisms required tokeep the color triplet fields light are consistent with theminima of the potential. To simplify the expression wedefine,

/1,/; /

2,q

2/*1q2; m

1,m ; m

2,q

2m*1q2.

The most general potential with all the fields is,

»(/1, /

2, *

L, *

R, m

1, m

2, g, s)

"»”#»*#»g#»m#»s#»g”#»g*#»*”

#»”m#»*m#»gm#»sm (7)

where the different terms in this expression are given by,

»”"!+

i, j

k2ijtr (/s

i/

j)# +

i, j,k, l

jijkl

tr(/si/j)tr(/s

k/l)

# +i, j,k, l

j@ijkl

tr(/si/j/sk/l)

»*"!k2 [tr (*sL*L)#tr (*s

R*R)]#o

1[tr(*s

L*L)2

#tr(*sR*R)2]#o

2[tr(*s

L*L*sL*L)#tr(*s

R*R*sR*R)]

#o3

tr(*sL*L*s

R*R)

»g"!k2g g2#bg g4

»m"+i, j

m2ijtr(ms

imj)# +

i,j,k, l

nijkl

tr(msimjmskml)

# +i,j,k, l

pijkl

tr(msimj) tr (ms

kml)

»s"M2s tr(ss s)#js1[tr (ss s)]2#js

1tr(ss sss s)

»*”"#+i,j

aij[tr (*s

L*L)#tr(*s

R*R)] tr(/s

i/j)

#+i,j

bij[tr(*s

L*L/

i/sj)

#tr(*sR*

R/si/j)]#+

i,j

cijtr (*s

L/

i*R/s

j)

»g*"!Mg g [tr(*sL*L)!tr(*s

R*

R)]

#jg g2 [tr(*sL*L)#tr(*s

R*R)]

173

»g”"+i, j

dijg2 tr (/s

i/

j)

»”m" +

i,j,k, l

uijkl

tr(/si/

jmskml)# +

i,j,k, l

vijkl

tr(/si/

j) tr(ms

kml)

»*m"#+i,j

aij[tr(*s

L*L)#tr(*s

R*R)] tr(ms

imj)

#+i,j

bij[tr(*s

L*Lmimsj)#tr(*s

R*Rmsimj)]

#+i,j

cijtr(*s

Lmi*

Rmsj)#+

i, j

jprij

[tr (*L*Lmimj)

#tr(*R*Rmimj)#tr(*

L*Rmimj)]

»gm"+i,j

dijg2 tr(ms

imj)

»sm"Pg[tr(ms*R)!tr(ms*

L)]#M[tr(ms*

R)#tr(ms*

L)].

We have not written the Sº(4) indices explicitly for sim-plicity. For example, if we include the Sº(4) index, theterm o

2tr(*s

L*L*sL*

L) in our notation will actually mean

two terms, oa2tr(*sa

L*

La*sbL

*Lb) and ob

2tr(*sa

L*

Lb*sbL

*La).However, as far as the minimization and the consistency

of the model is concerned, we only have to replace o2

by(oa

2#ob

2). Otherwise the rest of the analysis will be unal-

tered. A more detailed analysis with explicit Sº(4) indiceswill not constrain or relax any of the constraints in thismodel.

The vacuum expectation values (vev) of the fields havethe following form:

S/T"Ak

0

0

k@B ; S*LT"A

0

vL

0

0B ;

S/I T"Ak@0

0

kB ; S*RT"A

0

vR

0

0B ;

SmT"AkI0

0

k3 @B ; Sm3 T"Ak3 @0

0

k3 B ;

SgT"g0; SsT"0.

The notation needs some clarification. For the fields/ and m we have used the representation in which rowscorrespond to the Sº (2)

Lquantum numbers (#1

2, !1

2)

and columns correspond to the Sº(2)R

quantum numbers(!1

2, #1

2). The field / is a singlet under the group Sº(4)

and hence it has a Sº (4) matrix representationdiag(1, 1, 1, 1). On the other hand the field m transformsunder Sº(4) as a 15 representation. Under the Sº(3)subgroup of Sº(4) the 15 decomposes as 8#3#31 #1.The Sº (4) matrix representation of the singlet is a trace-less diagonal matrix which is a unit matrix in the Sº(3)space. Hence the Sº(4) matrix representation of the com-ponent of m, which is a singlet under both the Sº(3)

Cand

the º(1) subgroups of Sº (4) and which acquires a vev, isdiag(1, 1, 1, !3). The Sº (2) representations are as above.Thus these fields / and m contribute differently to thequarks and leptons masses, and hence a proper combina-tion of the two fields give the correct mass relationsbetween the quarks and leptons [11]. For the fields

*L

and *R

we used the 2]2 triplet representations ofSº(2). Thus the components q1!iq2, which has the isos-pin #1 and hence charge neutral, acquire vev. (The elec-tric charge is ¹

3L#¹

3R#(B!¸)/2. For these repres-

entations B!¸"!2. So the charge neutral componentshould have ¹

3Lor ¹

3R"#1, meaning they should

contract with q1!iq2).

4 Minimization of the potential

It is almost impossible to minimize the potential withrespect to all the fields and then find the absolute minima.For this purpose one needs to simplify the problem con-siderably. As a first approximation one can extremize thepotential with respect to all the fields and then substitu-ting the vevs of the different fields to check if there is anyinconsistency. In the minimal left-right symmetric poten-tial, i.e., without the field g, there are no linear terms in anyfield so the usual practice is to replace the various fields bytheir vevs then extremize it with respect to these vevs andfinally check for consistency. We shall also follow thesame procedure, but we need to take care of the extralinear terms present in the potential. For these linearterms, we shall afterwards minimize the potential withrespect to those fields which do not acquire any vev. Thevanishing of these derivatives after substituting for thevevs will then impose new constraints which also have tobe satisfied.

After the spontaneous symmetry breaking, when thefields acquire a vev, the potential contains terms with k, k@,k3 , k3 @, v

Land v

R. We need only terms involving v

Land v

R.

These are given by,

»"!k2 (v2L#v2

R)#

o4(v4

L#v4

R)#

o@2

(v2Lv2R)

#2vLvR[(c

11#c

22) kk@#c

12(k2#k@2)]

#(v2L#v2

R) [(a

11#a

22#b

11) k2

#(a11#a

22#b

22) k@2#(4a

12#2b

12) kk@]

!Mg g0(v2

L!v2

R)#jg g20(v2

L#v2

R)

#(v2L#v2

R)[(a

11#a

22#b

11#jpr

11)k3 2

#(a11#a

22#b

22#jpr

22) k3 @2

#(4a12#b

12#jpr

12) k3 k3 @]

#2vLvR[(c

11#c

22#jpr

11#jpr

22) k3 k@

#(c12

#jpr12

) (k3 2#k3 @2)] (8)

where we have defined the new parameters o"4(o1#o

2)

and o@"2o3.

The minimization of this potential gives a constrainton v

Land v

R. Instead of minimizing this potential with

respect to the fields vL

and vR

separately, we considera combination, (L»/Lv

L) v

R!(L»/Lv

R) v

L"0, which gives

a relation among the fields vL

and vR

vLvR"

b1k2#b

2k3 2

Co!o@!4Mgg0

(v2L!v2

R)D

. (9)

174

where, b1"2c

12; and b

2"2(c

12#jpr

12) and we assumed

k@;k and k3 @;k3 . This allows us to have a very tiny vev forthe left-handed triplet field *

Lwhile keeping the vev of the

right-handed triplet field *R

very large. This is in agree-ment with what we required in (6).

Now consider the linear terms involving s given by»sm. These terms allow for the correct mixing between thecolor triplet components of the fields m and s. The massmatrix for m3 and s3 is now given by

M"Aa

c

b

dB (10)

where, a"m2, d"M2s and b"c"(Pg0#M) v

R, and

where we assumed all mijs are equal to m. If we now fine

tune parameters to make det M"0, i.e.,

(mMs)2"(Pg0#M)2v2

R

then one of the mass eigenvalues is zero. This fine tuningrequires that M must be negative and (Pg

0#M) be very

small and negative. In fact, DPg0#M D has to be of the

order of \M2W

/MR. This massless field will get a mass of

the order of \TeV after radiative corrections during theelectroweak symmetry breaking are included.

The terms linear in s required for this mechanism haveanother effect which was not transparent when we did theminimization of the potential with respect to the vevs ofthe various fields. If we first minimize with respect to thefield s and then substitute for the vevs of the various fields(which is not usually done since that complicates thecalculation), then there is an additional constraint,

vL"

Pg0#M

Pg0!M

vR. (11)

This means that to satisfy Eq. (9), we require DPg0#MD;

DPg0!MD. In other words, for v

L\M2

W/M

R, we need

D(Pg0#M)/(Pg

0!M) D\M2

W/M2

R. This is consistent with

the fine tuning used to keep the color triplet component ofm light, for which DPg

0#MD\M2

W/M

Rand DPg

0!

MD\MR. This would not have been possible if the field

g were not present. For example, in the original paper [12]where the field s was introduced and the field g was notrequired, this method would have led to an inconsistency.In the absence of the field g minimization of the potentialwith respect to the field s would have given a constraintvL"!v

R, which is inconsistent with the LEP data. Here

we need the field g to keep the left-handed color tripletlight and the fine tuning makes it consistent.

We now turn to the question of light *3. As mentionedearlier, to have M*\ few TeV, we require the coupling ofg and * as in (5). Also, in the most general potential theonly term which contributes at the level of g

0\M*R

is »g*,which is exactly the same as in (5). Thus the field *3\*Lcan be massless at that level. Then during the electroweak

phase transition this will again acquire mass throughradiative corrections of the order of a few TeV. This ismore natural in supersymmetric theories where the radi-ative corrections induce mass of the order of supersym-metry breaking scale, which are usually of the order ofa few TeV.

The mechanism just mentioned to make *3 light hasone drawback. It makes the other components of *

Lalso

very light. For example, the *6 can now mediate n!nNoscillation, which has to be suppressed. This problem issimilar to the doublet-triplet splitting of any other grandunified models. In the present scenario we assume thatalthough this field is light the coupling of *6 is very small,which can make the model safe. However, this is not thebest choice and if one can find some good solution to thedoublet-triplet splitting in other GUTs, then one has toincorporate the same mechanism here in future. With ourpresent assumption that the Yukawa coupling of *6 isvery small, we now have to check the consequences ofthese fields in the evolution of the gauge coupling con-stants.

Finally, we point out that the quartic coupling (givenby »*m) required for generating the required diagram isalso present in the general potential.

5 Evolution of the coupling constants

There have to be many light scalars for the present scen-ario to work. These scalars may destabilize the unificationof the gauge coupling constant at the unification scale. Inthe evolution of the gauge coupling constant with theseHiggs scalars included and with the mass scales as above itis impossible to have unification of the gauge couplingconstants using the LEP constraints [20]on sin h

wand a

s.

However higher dimensional operators, which might orig-inate from Planck scale physics such as quantum gravityor compactification of Kaluza-Klein theories or Super-string theories, can save the situation [16, 17]. Thus if thethree lepton decay mode of the proton survives all of theexperimental tests, then we may have an indication thatPlanck scale physics is actually modifying the boundaryconditions of the gauge coupling constants near the unifi-cation scale.

In our analysis we include the effect of the non-renor-malizable terms arising from Planck scale physics fromthe beginning using the notation and method of [17]. Wewrite down the generalized renormalization group equa-tions in which the Planck scale effects are parametrized interms of four extra parameters. We recover the usualrelations between the coupling constants in the absence ofgravity by setting the extra parameters to zero. It is whenwe do this we obtain a contradiction and the equationsfail to provide unification.

The evolution of gauge coupling constants with theenergy scale is governed by renormalization group equa-tions (RGE). Here we consider the RGE in one loopapproximation i.e. the gauge fields fermionic fields and thescalar fields contribute to the evolution of the gaugecouplings via one loop graphs only [21]. In this approxi-mation the renormalization group equation takes theform,

kda

i(k)

dk"2b

ia2i(k)

where, ai"g2

i/4n and the beta function is given in the

following generic form.

b"1

4n[!11

3N#4

3nf#1

6¹

s]. (12)

175

Table 1. Higgs scalars at varioussymmetry breaking scales. The º(1)quantum numbers are normalizedfrom their embedding in SO(10)

MUPM

cM

cPM

RM

RPM

W

(1,1,1)(15,1,1)(10,1,3) (1,1,3,J3

2)

(10,3,1) (6,3,1,!J32)#(3,3,1,J1

6)#(1,3,1,J3

2) (6,3,!J3

5)#(3,3,1

3,J3

5)#(1,3,J3

5)

(1,2,2) (1,2,2,0) (1,2, 12J3

5)#(1,2,~1

2J3

5)

(15,2,2) (3,2,2,J23)#(31 ,2,2,!J2

3)#(1,2,2,0) (3,2,1

6J3

5)#(3,2,7

6J3

5)#(1,2, 1

2J3

5)

(31 ,2,~16

J35)#(31 ,2~7

6J3

5)#(1,2~1

2J3

5)

(6,2,2)

Here N"1, 2, 3 or 4, the number of neutrinos is always3 and the scalars take on the values discussed below. Sincethere are a large number of scalar fields present in ourmodel their contribution will be substantial despite thesuppression by a factor of 6 in the beta function of thescalar term (¹

s). We list the scalar fields that contribute to

the RGE at different energy scales in Table. 1. For simpli-fication of notation we write M

cfor M

PSin this section.

We embed the Pati-Salam group GPS

into a largerGUT group SO(10). The SO (10) symmetry is broken bya 54-plet of Higgs field & at the scale M

U. The & is

a traceless symmetric field of the SO(10) and the vevs of& which mediates this symmetry breaking are given by,

S&T"1

J30&0

diag(1, 1, 1, 1, 1, 1, !32, !3

2, !3

2, !3

2).

(13)

where, &0"J6/5na

GM

Uand a

G"g2

0/4n is the GUT

coupling. The vev of a 45-plet field H breaks the symmetrygroup G

PSto G

LR,

SHT"1

J12 iH

0 A033

133

034

!133

033

034

043

043

044B (14)

where, 0mn

is a m]n null matrix and 1mm

is a m]m unitmatrix.

The (1, 1, 1) component of the 54-plet field & breaksthe SO(10) group at the scale M

Uand hence does not affect

the RGE. The (15, 1, 1) component of the 45-plet fieldH breaks G

PSand so contributes to the RGE between the

scale Mcand M

U. The color singlet part of the (10, 1, 3)

component of the 126-plet field *R

breaks the GLR

sym-metry. Then by extended survival hypothesis the colorsinglet component contributes to the RGE between thescale M

Rto M

cand all the components (10, 1, 3) contrib-

ute between the scales Mc

and MU

[21]. On the otherhand our proposed mechanism, allows all of the compo-nents of the field *

L,(10, 3, 1) to remain light and con-

tribute to the RGE at all energies between MW

and MU.

The bidoublet (1, 2, 2) field breaks the electroweak sym-metry group and contributes at all energies to the RGE.For the correct quark-lepton mass relation we also re-quire the bidoublet color singlet component of (15, 2, 2) toacquire a vev at the electroweak scale. For the potential wehave, the (6, 2, 2) field will mix with the color triplet and

Table 2. The modified beta functions (b3N"4nb

N) for the various

groups at different energy scales. In the table we use the notationbNxy, where N represents the group (N for Sº(N) and 1S for º (1)

S)

and xy means the beta functions within the scales Mx

and My

MUPM

cM

cPM

RM

RPM

z

b4uc

"!11/3 b3cr

"!29/6 b3crw

"!29/6b2Luc

"11/3 b2Lcr

"4/3 b2Lrw

"4/3b2Ruc

"11/3 b2Rcr

"3/2 b1Yrw

"49/6b1B~Lcr

"13

anti-triplet components of the (15, 2, 2) field and onecombination of these color triplet and anti-triplet fieldswill remain light. The color singlet and one combinationof the triplet and anti-triplet component will then con-tribute to the RGE at all energies, while the other combi-nation of the color triplet and anti-triplet will becomeheavy and contribute only between the energies M

cand MU.

The normalization of the U(1) quantum numbers atthe right handed breaking scale is fixed by the relation,

½"S3

5¹ 3

R#S

2

5½

B~L.

Combining results of (12) and Table 1 it is easy to writedown the following explicit form of the beta functions thatregulate the evolutionary behaviour of the gauge coup-lings at various energy scales. We assume that the numberof fermion generations is three.

We consider both symmetry breaking scales, MU

andM

c, to be very large so that Planck scale effects are not

negligible. We start with the renormalizable SO (10) invari-ant lagrangian,

¸"!12

Tr (FklFkl) (15)

and then include the non-renormalizable higher dimen-sional terms which have their origin in Planck scale phys-ics. We consider only terms of dimension 5 and 6, given by

¸(5)"!

1

2

g(1)M

Pl

Tr (Fkl/Fkl) (16)

¸(6)"!

1

2

1

M2Pl

[g(2)a

MTr(Fkl/2Fkl)#Tr (Fkl/Fkl/)N

#g(2)b

Tr(/2) Tr(FklFkl)#g(2)c

Tr(Fkl/) Tr(Fkl/)]

(17)

176

where g(n) are dimensional couplings of the higher dimen-sional operators. When any Higgs scalar / acquires vev/0, these operators induce effective dimension 4 terms

modifying the boundary conditions at the scale /0.

The symmetry breaking at MU

shifts the boundarycondition of the Sº (4) coupling constant with respect tothe Sº (2) couplings whereas the vevs of the 45-pletfield H contribute to the relative couplings of the Sº(3)and the º(1) constants. The G

PSinvariant effective lagran-

gian, modified by these higher dimensional operators, isgiven by,

!12

(1#e4) Tr(F(4)kl F(4)kl)!1

2(1#e

2) Tr(F(2L)kl F(2L)kl)

!12

(1#e2) Tr(F(2R)kl F(2R)kl) (18)

where,

e4"e(1)#e(2)

a#1

2e(2)b

e2"!3

2e(1)#9

4e(2)a#1

2e(2)b

.

and

e(n)i"CG

1

25naGH12 M

UM

PlDng(n)i

.

Then the usual GPS

lagrangian can be recovered with themodified coupling constants,

g24(M

U)"g6 2

4(M

U) (1#e

4)~1

g22L

(MU)"g6 2

2L(M

U) (1#e

2)~1

g22R

(MU)"g6 2

2R(M

U) (1#e

2)~1 (19)

where the g6iare the coupling constants in the absence of

the nonrenormalizable terms and giare the physical coup-

ling constants that evolve below MU. The modified

boundary condition then reads,

g24(M

U) (1#e

4)"g2

2L(M

U) (1#e

2)

"g22R

(MU) (1#e

2)"g2

0. (20)

At Mc

the symmetry group Sº (4)c

breaks down toSº(3)

c]º(1)

B~Lwhen the (15, 1, 1) component of the

45-plet of Higgs field H acquires a vev. The Sº (3)c]

º(1)B~L

invariant lagrangian is given by,

!12

(1#e@3) Tr(F(3)klF(3)kl)!1

2(1#e@

1) Tr(F(1)klF(1)kl)

where,

e@3"e@(2)

a#12e@(2)

b

e@1"7e@(2)

a#12e@(2)

b#12e@(2)

c.

and

e@(2)i

"

g@(2)i

/20

24M2Pl

"C1

20na4C

MI

MPlD2

D g@(2)i

where, i"a, b, c. Then the boundary condition at Mcbecomes,

g21(B~L)

(Mc) (1#e@

1)"g2

3c(M

c) (1#e@

3)"g2

4(M

c).

The matching conditions at the scale MR

are not modifiedby the Planck scale effects and are given by,

g~21Y

(MR)"3

5g~22R

(MR)#2

5g~21(B~L)

(MR)

g~22L

(MR)"g~2

2R(M

R). (21)

Using the above boundary conditions and the one looprenormalization group equation the unification couplingaU

can be related to the three couplings at the ¼ massscale (M

W) through the following relations [17] (we have

defined mij"ln M

i/M

j),

a~1y

(MW

)"a~1G

(1#3/5e2#2/5e

4#2/5(e@(1#e

4) ))

#(6/5 b2ruc

#4/5(1#e@1) b

4uc) m

uc

#(6/5 b2rcr

#4/5 b1blcr

)mcr#2b

1yrwm

rw

a~12

(MW

)"a~1G

(1#e2)#2 b

2lucm

uc#2 b

2lcrm

cr

#2b2lrw

mrw

a~13

(MW

)"a~1U

(1#e4)#2 b

4ucm

uc#2 b

3crm

cr

#2 b3crw

mrw

.

We define,

A"a~1Y

(MW)!a~1(M

Z)

and

B"a~12L

(MW

)#53

a~1Y

(MZ)!8

3a~13c

(MW

)

and relate them to the experimental numbers through thefollowing equations,

sin2hW"3

8!5

8aA

1!3

8

aas

"aB. (22)

If we now take all the es to be zero, then we have

mrw"!36.7#5.7m

uc

mcr"147.3!11.2m

uc.

There is no solution with positive mrw

and mcr

for anyvalue of M

ucwith the constraints, sin2h

W"0.2334,

as"0.12, and with the unification scale below the Planck

scale . In other words this means that if we do not considerthe effect of gravity, then in the presence of so many lightHiggs scalars it is not possible to have unification of thegauge coupling constants. Thus if the three lepton decaymode of the proton is the explanation of the atmosphericneutrino problem, gravity effects modify the low energypredictions of the grand unified theories.

We now consider the effects of the Planck scale. Forseveral choices of the es it is possible to have a mass scalesolution which may explain the atmospheric neutrinoproblem. To demonstrate this we present a few represen-tative solutions in Table 2. The light scalars contributefrom the scale M

z\100 GeV in the RG equation. The

unification scale MU

and the GPS

breaking scale Mcare

very close to each other (i.e., muc"ln (M

U/M

c) is very

small) and is considered here to be around MU\M

c\

1018 GeV (i.e., mrw#m

cr#m

uc"39); the right handed

breaking scale is around MR\1013 GeV (i.e., m

rw\28).

177

Table 3. Allowed ranges of parameters for unification

e@1

e2

e@3

e4

a~1G

mrw

mcr

muc

!1 !0.75 1 !1 55 28.32 9.73 0.93!1 !0.75 1 !1 57 28.04 10.34 0.61!1 !0.75 1 !1 59 27.75 10.95 0.28!0.99 !0.75 1 !1 59 27.76 10.95 0.28!0.90 !0.75 1 !1 59 27.76 10.94 0.28!0.80 !0.75 1 !1 59 27.77 10.93 0.28

With these values of the mass scales the three lepton decaymode of the proton would be the most dominant decaymode (with q (PPe`ll)\1031 yrs). Since the unificationscale is quite high now, conventional proton decay modesare very much suppressed (with q(PPe`n°)\1039 yrs.).Thus even if the three lepton decay mode of the proton isfound there is no conflict with the non-observation ofproton decay in other experiments.

6 Conclusions

We have presented an extension of the left—right symmet-ric model where the most dominant proton decay mode isthrough three leptons. The lifetime for this decay mode islarge enough to explain part of the atmospheric neutrinoanomaly, but apparently not the full anomaly. We haveminimized the complete potential to check the consistencyof the model. In the end we have carried out a renormaliz-ation group analysis to estimate the mass scales of themodel. We have shown that when the gravity inducedeffects coming from the Plank scale physics are included inthe renormalization group analysis, the couplings unify;an estimation of the several mass scales of the model thenbecomes possible.

Acknowledgement. We would like to thank Drs. DebajyotiChoudhuri and N.K. Mandal for many critical comments on themanuscript and helpful discussions. The work of US was supportedby a fellowship from the Alexander von Humboldt Foundation andthat of PJO’D by the Natural Sciences and Engineering Council ofCanada.

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