PHYSICAL REVIEW D VOLUME 34, NUMBER I 1 JULY 1986

Z4 symmetry and the fourth generation

P. K. Mohapatra and R. N. Mohapatra Department of Physics and Astronomy, University of Maryland, College Park, Maryland 20742

(Received 12 December 1985)

The Z4 horizontal symmetry of weak interactions is used to study the masses and mixings of a possible fourth generation of quarks ( t l , b ' ) . We obtain the relation ( m , / m , ) = ( m b + m b t ) / ( m , + m , , ) which implies m b ~ / m , ~ << 1. Coupled with the known experimental constraints on the p parameter, this leads to m; 5 300 GeV and mb8 < 58 GeV. We also predict the mixing of the fourth-generation quarks to those of the known quarks.

The recent discovery of the t quark' at the CERN col- lider by the UAl group with m, in the range 30-50 GeV completes the third family of quarks and leptons. The next obvious question is whether there is a fourth genera- tion of quarks and leptons as well, and, in recent days, there has been a great deal of interest in this ~ubjec t .~ From a theoretical point of view, except for the fact that considerations such as cosmological He abundance, asymptotic freedom, etc., allow a fourth generation offer- mions, perhaps the only serious reason for such considera- tion is that a fourth heavy generation of fermions makes it easier to understand the estimated magnitudes of the present baryon-to-entropy ratio of the ~ n i v e r s e . ~ Recent- ly it has been shown that if superstrings are interpreted as preons instead of quarks there will be four generations.4

Two important questions concerning the fourth genera- tion of quarks are the masses of the t' and b' quarks and their mixing with the three lighter generations. If one as- sumes that m,, >>mb#, the one-loop corrections5 to the electroweak parameter p, coupled with the experimental constraint that p = 1.02 50.02, imply that m,, 5 300 GeV. To predict the mixing angles, one has to make additional assumptions concerning the structure of the quark mass matrices.

In this paper, we assume the existence of a Z4- horizontal symmetry among the four quark genera- tions6-* and attempt to predict the m,, and mbr as well as their mixing to lighter-quark generations. This symmetry has been used previously to predict successfully the weak mixings in the case of three quark generations. We show in this paper that applying it to models with four genera- tions leads to the following result:

Equation (1) obviously implies that m,, >>mb, leading to the prediction that m,, 5 300 GeV and mb, 5 58 GeV. We also predict the mixing matrix and weak-interaction prop- erties of t' and b' quarks.

THE MODEL

We consider the electroweak gauge group to be the left-right-symmetric group9 SU(2)L x SU( 2)R x U( 1 )B -L

with quark doublets Qa [a = 1, . . . , 4 for generations;

Qa =(u,,da ) ] assigned to the gauge group in a left-right symmetric manner:

We choose the minimal Higgs system,'' with 4(2,2,O) and hL(3,1,+2)+hR(1,3,+2) so as to understand the small- ness of the neutrino masses using the "see-saw" mecha- nism. The most general form of Yukawa coupling of 4 in this model can be written as

Parity and CP invariance of Y y implies that hub and Kab are real and satisfy the property that hab=hh and Lab =Kb. The Yukawa couplings h and hS lead to quark mass matrices once ( 4 )#O. To restrict the form of the Yukawa couplings and hence quark mass matrices, we propose that the model be invariant under a discrete Z4 symmetry under which quarks and the Higgs multiplet 4 transform as

This leads to h and 6 matrices of the form

34 231 - 01986 The American Physical Society

232 P. K. MOHAPATRA AND R. N. MOHAPATRA

By an appropriate change of basis, we can make h diago- nal (i.e., h 13 = h24-=0), while at the same time maintain- ing the form of h as in Eq. (6). At this stage, we have only eight arbitrary parameters.

The vacuum expectation value of $ is in general of the form

where a leads to CP violation. We will assume that a is very small, so that observed CP violation can be account- ed for by the right-handed currents as has been argued." From this point on, we will set a=O. We then obtain (where w =K'/K)

Equation (8) leads to the prediction thatI2

or since md, m,, and mc << mb, mb., m,, m,, we find

A priori, this allows for o as big as 1 or even bigger. However, we will subsequently see that a consistent pic- ture requires w << 1. Since all mixings are due to z and the mixings are known to be small, we will assume the various gij to be smaller than hi,. In that case, in the up- quark mass matrix, xw << h and we can safely ignore the off-diagonal elements in our analysis. We then obtain m , = h l l ~ , m c = h 2 2 ~ , m f = h 3 3 ~ , and mt,=hUK. Wefur- ther assume that ha, and jab satisfy the following hierarchical pattern:

We then apply the method of successive approximation to diagonalize Md .

First we diagonalize the matrix

The orthogonal matrix that diagonalizes this matrix is denoted by v"' and is parametrized by an angle 8,. We obtain the mixing angle 034 between the third and fourth generation to be

2(m,m,~02-mbmb~)1'2 tan203, =

m, - m,, (12)

We then write the 3 X 3 matrix M::; keeping k23#0 and diagonalize it by means of the matrix v'~'v(", where

where X1,X2 << 1. The important point is that this succes- sive diagonalization keeps M $ ~ in diagonal form to lead- ing order. We find

- - X1%-h23/mb and X2=-hz3/rni (14)

and

From now on, we set K = 1 and express all quark masses in units of K, which is the electroweak breaking scale. It is easy to convince oneself that XI is the mixing angle Vcb. Experimentally, I Vcb 1 =0.06+0.01 leading to x23.u0.31 10.05 GeV. Equation (15) then implies that

We thus see that consistency of the theory requires that w--0.08-0.175 which is consistent with our assumption that o << 1. Since experimentally mb /m, '~0.10-0.18, we expect that

This kind of relation between quark masses has also been obtained before in other m ~ d e l s . ' ~ We note, however, that if the last equation in Eq. (17) holds exactly, then Eq. (12) would imply 63,=0 and h34 =O. We, therefore, use the weaker relation that follows from Eqs. (10) and (16), i.e.,

We also find from Eq. (18) and the experimental lower bound on mbr 2 22 GeV that K34 < 16 GeV, which is in rough agreement with our Eq. (10').

We now proceed to the final stage of our diagonaliza- tion, which is achieved by the same procedure of succes- sive approximation by using matrix v ' ~ ' where

Again, we find that, within the framework of our ap- proximation,

and -

a3?-h 14/mb* . (20)

Using the fact that a, is the dominant contribution to the

34 - Z4 SYMMETRY A N D THE FOURTH GENERATION 233

Cabibbo angle, we can determine g12 = (mdm, ) ' I 2 TABLE I . The experimental observables. A M K ~ - K ~ , MBo-g O , d d

-40-50 MeV. Note that the bound on I Vub 1 50.008 rBo, , rBp-e? R 2 , , and RL for the two cases of the mass implies that ~ 4 0 MeV barring cancellations; i.e., h l 2 d d

and F l 4 are of the same order of magnitude. However, we also consider cases with larger K I 4 . We, thus, see that all - the parameters of our model except hM, K34 , and h 14 are determined to within a range and experimental data sets upper limits on AM, K34 , and K I 4 . TO get an idea about the kind of four-generation Kobayashi-Maskawa (KM) matrix VKM predicted by our model, we present two typi- cal cases.

Case (a): We choose w =O. 15, m, =40 GeV, m,,=300 GeV; z 1 2 ~ e 4 5 MeV, KI4~=250 MeV; K 2 3 ~ = 3 5 ~ MeV; Kj4~=4 GeV. In this case, we find

As a second example, we consider the following. Case (b): 0=0.19, mt=50 GeV, mt.=300 GeV;

z l 2 K = 5 5 MeV; L l 4 ~ = 5 0 MeV; KZ3~=300 MeV; z 3 4 ~ = 15 GeV. In this case, we obtain

PHENOMENOLOGICAL IMPLICATIONS

We now discuss some of the phenomenological implica- tions of our model. The first point we focus on is the life- time of the b ' quark. It is clear from the two examples given above that b' dominantly couples to t. Therefore, its lifetime ( r b , ) is vastly different in the two cases, when mb, > m, or mb. < m, . In general, rb8 of course depends on mba and m, via the mixing angles in VKM; however, the Vtb,, Vcb,, Vubf do not change very much in their relative magnitudes; therefore, simply taking into account the phase-space effects, we find

case ( a ) rb8= 1 . 6 ~ 1 0 - l ~ sec, mb,> m, ,

case ( b ) rb,=4.3X sec, mbz>m, ,

rb1=4.2X 10-l5 sec, mb, < m, .

matrix studied in the text.

Experimental observables Case (a) Case (b)

WK, -K, 1.25 x lo-'5 GeV 1.22 x lo-'' GeV

M a g o 5.3 X lo- '4 GeV 6.2 X GeV Ed- d

ME,"-8 ; 4.9 x lo-'' GeV 6.8 X lo-'' GeV

r B ~ d - B O d -9.4 x lo -15 GeV -3.2 x lo-'' GeV

r ~ ~ - ~ : - 9 . 7 7 ~ 10-l4 GeV -3.6 x 10-l4 GeV

R odd 1 0.021

R $d 0.57 0.72

Next, we discuss the effect of the fourth generation with typical masses mb,=45 and 58 GeV and m,,=300 GeV on the KL - Ks mass difference, B:-B and B:-B mix- ings, as well as the parameters rI2. We summarize our findings in Table I. (The parameter Rodd measures the ra- tio of like-sign dimuons to unlike-sign ones in B' produc- tion, and we have used the bag parameters B = 1 for all observables listed in the table.)

In conclusion, we have studied the implications of Z4- horizontal symmetry on the fourth-generation quarks. Our main prediction is that m,, 5 300 GeV and mbf 5 5 8 GeV. Such predictions have appeared in recent literature but we believe that since our work relies on the assump- tion of a discrete symmetry in a gauge model, it is stable under radiative corrections. We then predict the mixing angles of the b ' and t' quark to the lighter generations and study their implications for the experimentally ob- servable quantities in B O - B O mixing. It is interesting to note that, if we further require the existence of a grand- unified model that generates these mass matrices [such as some version of SOilO)], then we would expect the mass of the fourth-generation lepton r' to be related to mb,, i.e., mbg=2. 9m ,, . This would considerably sharpen the pre- dictions of our quark masses since experimentally m+ 2 22 GeV. We would expect, m,, ~ 2 2 - 2 4 GeV, mb,=63-69 GeV, and m , , ~ 3 0 0 GeV.

ACKNOWLEDGMENTS

We wish to thank C. Y. Chang for some discussions. This work was supported by a grant from the National Science Foundation.

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