Volume 134B, number 6 PItYSICS LI.IFTERS 26 January 1984

FLAVOR SYMMETRY AND PROTON DECAY

Ann NELSON L yman Laboratory o f Physics, Harvard University, Cambridge, MA 02138, USA

Received 17 November 1983

We consider a class of SU(5) grand unified models wilh supcrheavy fermions as well as the three light families. Mixing of superheavy and light fcrmions can cause Cabibbo-like suppression of proton decay. In a model with an SO(3) flavor sym-

I metry, the minimal SU (5) prediction m r = "5" mb is preserved, while the predictions for m e and rnu are not. ('abibbo favored proton decay modes are p ~ nV. and p --* Ke, but not p ~ K#. The top quark mass is Oc(mb/m d) m c ~- 150 GeV.

1. In t roduc t ion . The minimal SU(5) model predicts that the proton decays into 7re with a partial lifetime of 5 × 1029+-2 yr [1], which conflicts with recent ex- perimental data [2]. If there are no new particles lighter than the GUT scale, then the only ways to suppress this mode are to have either (~)t. and u~, or u~, e~, (ev) L and d~. in different SU(5) nmltiplets. The simplest possibility is to have additional heavy particles in the theory which can mix with the light families. If these

I

are in real multiplets such as 51. @ 5 Land 10 L @ lOt, the survival hypothesis of Georgi [31 predicts that at energies '~ MGU T the fermion content of the theory is three light families.

In general this scenario permits arbitrary masses and mixing between light and superheavy particles. It would be very attractive if the non observation of p 7re could be related in some way to the flavor problem. If flavor is a good symmetry at the GUT scale, we may gain some predictive power. The mixing between light and superheavy families gives us a new and rather sim- ple way to break flavor symmetry and produce an in- teresting light mass spectrum. In the next section we give an example o f such a model which in its most gen- eral form allows us to predict m b and mt, and the pat- tern of proton decay.

2. A f lavor s y m m e t r i c model . We consider a SU(5) grand unified theory with an SO(3) flavor symmetry and some additional fermions. Under SU(5)Gaug e ® SO(3)Globa I the left-handed fermions transform as

(10,3) @ (5,3) ~ (10,1) @ (10,1) @ (.~,1) @ (5,1).

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The light lliggs doublet of the Weinberg-Salam model is in a (5, 1), alongwith a superheavy SU(3)c triplet. The Yukawa couplings are:

Xu(10,3)(10,3)(5, l ) t l~s ¢ Xd(10,3)(5,3)(5, l)l-*tigg s

+ h.c.

An additional global symmetry forbids Yukawa cou- plings between the Higgs and the SO(3) singlet fermions. The SU(5) and SO(3) symmetries are spontaneously broken at the GUT scale, for example by superheavy bosons in (24,3) and (1,3) representations. SO(3) may also be explicitly softly broken. After these symmetries are broken, all of the particles in the (i-0, 1 ) and (5,1) find partners in linear combinations of the (10.3), (10, 1), (5,3) and (5,1) and become superheavy.

We will use the following notation to analyze the resulting light mass matrix and pr_oton decay. The super- heavy directions in the 10's and 5"s are labelled by four- vectors with the first three components corresponding to the SO(3) three-vector and the fourth to the SO(3) singlet. For example, the superheavy linear combina- tion of particles transforming as a (3,2, 1/6) under SU(3)c× SU(2)w X U ( I ) y will be labelled by the four- vector qSIt. To find the light mass matrix, for each SU(3)× SU(2)X U(I )muhip le t we find three vectors which are mutually orthogonal and orthogonal to the

ci etc. superheavy directions. These we call q~, u u , Now we can write down an effective field theory

with only light particles. One light particle corresponds to each of the light vectors we have found.

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Volume 134B, number 6 PHYSICS LETTERS 26 January 1984

The couplings of the light particles to the Higgs doublet (0) are:

• t u i [~ , c ] - d iT c]~ i jq' . c u E @ + Xi/qt. CdI. 0 + X~~rCel : ! + h,c.,

where

.=XU 1~.2 i c/ xd ~ oi,lc/ Xi/ u = ,3 qu u~, ~d. = la = 1,2.3 TM TM '

. - ,u , Xd(MGuT) = Xe(MGuT)- (1) p=1.2,3

The renormalization group predicts ~'d (rob) ~ 3Xe(mr) just as in minimal SU(5) [41. The matrices

.u=1.2.3 ] ,u=l ,2 .3-~u-u] '

(u = 1~2.3 t? ~e~/) '

all have at least one eigenvalue of I, with the other eigenvalues smaller, and so the minimal SU(5) predic- tion for m b / m r is unchanged. To see this, note that the couplings of the SO(3) triplet fermions to the light Higgs doublet actually have a SU(3) vector symmetry, with quarks and leptons translbrming as 3's and anti- quarks and antileptons as Ys. We can use this symme- try to write for the superheavy lepton directions

Sit * e~Sll * I~, =(Cl.O.O. -Sl) =(C2S3.C2C3.0.--s2) ,

ci real, ]ci[ 2 + Isil 2 = 1.

Now the vectors corresponding to the light particles are

i = ~u O, 1, 0,

O . O. 1.

s2s3, s2c3, 0. ~2"]1

ci _ C3 ' * 0, e u - -S3, .

O. 0 1. Oj

Using (1) we can see that the mass matrix for the leptons is:

S lS2S $1c3 @

mg! = ~.c (~) [s2e 3 - s 3 . (2)

(0 0

We could also use SU(3)v to write the up or down quark mass matrices in a similar way.

Inspecting these mass matrices tells us some inter- esting things about where the light and heavy particles sit in the SU(5) multiplet.

Since we know that two families are much lighter than the third, we have

[ml/m3] 2 + [m2/m312 + 1 = Ira31- 2 t r m m + ,~ 2

~ [SlS2S312 + lSlC312 + ls2c312 + ls312 ~ l, (3)

In English this means that all of the superheavy vec- tors have small fourth components and that the super- heavy quark and lepton directions are nearly orthogonal to the antiquark and antilepton directions.

Note that one linear combination of the two lightest quarks and leptons and of the lightest antiquarks and antileptons is mainly SO(3) singlet. These linear com- binations cannot both be mostly second family. The mass matrix (2) tells us that then we would have

Isls2s31>> Isle 3 I, Is2c31, and Is 3 I. Either the lightest quarks and leptons or lightest antiquarks and leptons or both must have large SO(3) singlet components. This restriction has important consequences for proton decay.

3. The quark mass matrix and proton decay. The SU(3)v symmetry does not commute with the cou- plings to the X and Y gauge bosons which mediate proton decay. However we can still use the SO(3) flavor symmetry to write:

qSU=(c , s* 0, 3 " ) , - * - , C - C ,

u, ~ u =(s* + e 6 * , c - s 6 * , - e * , 3'*).

The requirement that two quarks be much lighter than the third tells us that 13l, 13'[, lel, 16['~ 1 and [cl 2 + Isl 2 "~ I. Now d~ sH must have the form

( s*+cS '* ,c s S ' * , - e ' * , - 7 ' * ) )8'1,1e'1,17'1'~1

or the tltird family mixing in the KM matrix will be too large.

The light directions are:

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Volume 134B, number 6 PItYSICS LETTERS 26 January 1984

. ~'fi'c, •-~s*,

q ~ - [ s , c,

O, O,

o, i] O,

1,

IC+SS*, -S --8C, ci ~ / 3 ' S * ' "/C, Uta

! [,eS*, ~C,

o,! O, ,

1,

d~ i = (8')'e -+ 8'3"e').

Using (1) we find

m~ ~" m t 3' ,

0

(4)

rn~( ~- (3'6e ~ {5 '7 'e', m t -}" n/b). (5)

Since 0 c is small both up and down quark mass ma- trices have their second largest elements in the same row. We cannot have/3 ~ 6 ,7 , 8' , 7 ' , or the third family quarks will primarily decay into the lightest family, which conflicts with experiment. If'}, >> 6, t3 and 7 ' ~ 6',/3, then 0 c < rod~ms, also in conflict with experiment. We are left with only three possibilities, each producing a different type of proton decay.

( I ) 8 ~ 3' and 6' ~ 7 ' . There can be no significant suppression of p -+ 7re in this case. The lightest quarks and antiquarks are mainly in the SO(3) singlet 10 ~ 5. We know from the form of the lepton mass matrix that either the lightest lepton or the lightest antilepton is also in the SO(3) singlet. So either the operator

(qL 3'~ u~_ ) (~L 3'u eE) (6)

or

(q-t. 3'UuD (~-tt. 3'u d D ' (7)

or both are Cabbibo favored to contribute to p 4 rre. (2) 3' ~ 8,/3, 8 ' ~ 3",/3. This would suppress both

operators (6) and (7) as qL and u~, would be in differ- ent SU(5) multiplets, liowever now we can see that the mass matrix (3) predicts mu/m t ~ ~, ms/m b -~ 8'

t ] ". and 0 c -~ t3/8 , so m t -~ 0 c (mu/ms) m b which is ex- perimentally ruled out.

(3) 7 ' ~ 8',/3, 6 ~ 7,/3. Now the lightest qL and u~ are in the (10, 1) but the lightest d~ is in one of the (5,3)'s. If the antilepton in the (I0, 1) is not the e~_ but the/.t~, then operator (6) will primarily contribute to p - 7r/a rather than 7re. The lightest lepton is in the (5, 1) along with the strange antiquark, and so opera- tor (7) will cause p -+ Ke. Also now m d ~- f3', mc /m t

s, and 0 c ~-/3'/6. Therefore m t ~- Oc(mb/md) rnc, which is still allowed experimentally. Since the light q and u c are mainly in the (10, 1), which does not cou- ple to the colored ltiggs triplet, the contribution of the Higgs triplet to proton decay will be negligible un- less its mass is somewhat less than the GUT scale.

4. Conclusions. We find it encouraging that such a shnple flavor symmetry breaking mechanism gave us fermion mass matrices with so much structure. We now have a natural explanation for why m s, m d ¢ 3m u, 3m e but m b = 3mu. There are two possibilities for proton decay. Either the proton decays to rre with about the rate predicted by minimal SU(5) or the proton decays to Ke and rrp and the relation rn t ~ m e × (mb/ms) 0 c holds. We are currently looking at other, similar flavor symmetric models in hope of gaining more information about the pattern of quark and lep- ton masses and mixings.

It is a pleasure to thank Sheldon Glashow and Alvaro de Rujula for useful conversations. Special thanks are due H. Georgi for suggesting this class of models. This work was supported in part by the National Science Foundation under Contract No. PHY-82-15249.

References

[I I tl. Georgi and S.L. Glashow, Phys. Rev. Lett. 32 (1974) 438; W.J. Marciano, in: Proc. Fourth Workshop on (;rand unification (University of Pennsylvania) ed. A. Weldom (Birkhauser, Basel. 1983), to be published.

[21 B.M. Bionta et al., Phys. Rev. Left. 51 (1983) 27. {31 tl. Georgi, Nucl. Phys. B156 (1979) 126. [41 A.J. Buras, J. Ellis, M.K. Gaillard and D.V. Nanopoulos,

Nucl. Phys. B135 (1978) 66.

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