PHYSICAL REVIEW D VOLUME 29, NUMBER 3 1 FEBRUARY 1984

Horizontal permutation symmetry, fermion masses, and pseudo-Goldstone bosons in SU(2), x U( 1)

R. Yahalom Department of Physics, Technion-Israel Institute of Technology, Haifa, Israel

(Received 28 June 1983)

We perform a systematic study of all possible SU(2IL XU(1) models with S j as a horizontal sym- metry in an attempt to find a realistic fermion mass matrix for six quarks. The appearance of ac- cidental symmetries in the Higgs potential leads to ambiguities in the tree-level vacuum expectation values and the minimalization of the effective potential is therefore carried out including one-loop corrections. A way of eliminating undesired effects of almost-zero-mass pseudo-Goldstone boson without explicit S3 breaking is presented. We suggest a specific s U ( 2 ) ~ XU( 1) X S3 model with four Higgs isodoublets. Taking the experimental value of sine, as an input, the other weak mixing an- gles are predicted. We find that sin02=0. 14, ~ inO~=10-~ , and S=O. The B-meson lifetime is found to be 4.7 X 10-l3 sec and is essentially independent of the t-quark mass. CP violation occurs only by spontaneous breaking. Other topics, such as flavor-changing neutral currents and the Higgs-boson masses, are discussed.

I. INTRODUCTION

In spite of its outstanding success to predict many features of the known weak interaction phenomenology, the standard SU(2)L XU(1) theory1 of the electroweak in- teraction fails to explain the existence of multiple families of fermions-the so-called "generation problem."2 The masses and mixing angles of the fermions are free parame- ters that must be introduced by hand. There has been a considerable effort to overcome this difficulty. It was suggested that one way to constrain the generation struc- ture, reduce the number of independent parameters, and predict some relations between the masses and mixing an- gles, is to enlarge the symmetry by the addition of "hor- izontal" symmetries that act upon generation space.3 The theory is then invariant under SU(2), X U ( l ) X G H , when GH can be either discrete4 or c o n t i n u ~ u s . ~ One of the successes of this approach is the relation tan2@, = md /m, .6

Imposing GH on the Lagrangian ensures that such rela- tions are natural and higher-order corrections, if any, are finite and calculable.' In this paper we use a discrete group as GH. This choice is the simplest in the sense that it avoids the problems of additional gauge bosons in the case of a continuous and local symmetry, or massless Goldstone bosons if it is a global symmetry.'

The origin of GH does not interest us here. It may come as a low-energy approximation of a grand unifica- tion theory, or as a consequence of some substructure of the quarks. Also we do not concern ourselves with the origin of the Higgs scalars, which can be either elementary or condensates of fermion-antifermion pairs if a dynami- cal breaking scheme is used. The purpose of this work is to present models based on a specific discrete group and investigate the structure of this theory with an emphasis on the Higgs sector. I t is clear that in order to get the desired results we must enlarge the number of scalars. The mass spectrum turns to be crucially dependent on the

representation content of these fields under G H . It also depends on the pattern of the spontaneous symmetry breaking and on the exact solutions to the minimum of the Higgs potential.

Here we consider models in which GH is S3, the permu- tation group of three indexes (see Appendix for a discus- sion of this symmetry). The flavor-symmetry group of the gauge sector with n generations is GF=SU(n) X SU(n) XU(1). The minimal SU(2), XU(1) theory is in- dependent of the choice of generation. It is therefore ob- vious that S3 as a subgroup of GF is a natural choice for GH in the case of three generations of fermions. This symmetry (sometimes combined with additional symme- try) has been used before within the SU(2)L XU(1) theory in specific models [see Yamanaka, Sugawara, and Pakva- ~ a , ~ Derman and ~ s a o , ~ Weldon and ~ e ~ e r s , ~ ( ~ ) and Refs. 9- 121. In this gaper, a general, detailed analysis of all possible models with S3 and three generations is carried out.

In the next section we investigate the true minimum of various Higgs polynomials invariant under S3, for dif- ferent assignment of representations for the scalar fields. The exact solutions for the vacuum expectation values (VEV's) are found. Since V ( 4 ) is at most a quartic poly- nomial, it acquires a continuous, global symmetry. This pseudosymmetry'3 is not shared by the rest of the La- grangian.

In Sec. 111 we construct all possible mass terms compa- tible with S3 invariance of the Yukawa interaction with three generations. Ambiguities in the zero-order calcula- tions, which are the result of the degeneracy of the VEV's under the pseudosymmetry are displayed. After perform- ing one-loop corrections one model survives as a candidate for further study. All the rest fail in some phenomenolog- ical tests. It should be stressed that no explicit ad hoc soft breaking terms of S3 are introduced and only natural VEV's are used by calculating the effective potential for

536 @ 1984 The American Physical Society

29 - HORIZONTAL PERMUTATION SYMMETRY, FERMION MASSES, . . . 537

each model separately. This is to be compared, for exam- ple, to Ref. 9.

In Sec. IV we discuss the issue of the pseudo-Goldstone boson (PGB) that appears in our models and show how it can be decoupled.

In Sec. V a specific model is examined. Because of the proliferation of the Higgs section there are flavor- changing neutral currents (FCNC). As was already no- ticed before, one should abandon the naturalness of Glashow, Iliopoulos, and Maiani for the sake of calculabil- ity.14 Experiments put constraints on the Higgs-boson masses. The ~ o b a ~ a s h i - ~ a s k a w a ' ~ matrix and other re- sults are discussed. We are able to construct a calculable mass spectrum and a mixing angles pattern without spoil- ing the S3 symmetry.

To establish our notation, we write the S U ( ~ ) L X U( 1) X S3 Lagrangian as

where 2Yd is the scalar part of 2 that will be elaborated upon in Sec. 11:

2, is the Yukawa interaction between the scalars and fer- mion fields to be studied in Sec. 111:

Here and after, 4, p, X , <, @, f1 stand for scalar fields; 111, f, F for fermions. a , /3, y are generation indexes and i,j,k are the number of scalar indexes.

11. THE POTENTIAL

Here we investigate the Higgs potential for several dif- ferent representation assignments of the scalar fields under S3. Some of the solutions that are found in this sec- tion have been discovered but as will be shown, they covered a narrow range of a much wider spectrum of solutions. A systematic study of the potential is carried out here. We assume that all the Higgs fields are SU(2)L doublets, p'=(pt,pq). Throughout this section we use the notation of basis 2 [see Appendix, Eq. (A2)]. Note that the form of the potential and the mass matrix is fixed by the choice of basis, but the mass spectrum and the Kobayashi-Maskawa (KM) matrix do not depend on it. At the end of this section we transform the VEV's to the notation of basis 1 [Appendix, Eq. (Al)] for later use.

The most general, renormalizable potential invariant under SU(2)L XU( 1) is

Imposing S3 on V(4), we reduce the number of indepen- dent parameters according to the representation of the field under S3.

A. Case a

Here we consider the minimal possibility of two iso- doublets, transforming as an S3 doublet:

We immediately note that Va is invariant under an extra global U'( 1 ) symmetry

We expect two phenomena to occur13: (1) The existence of a pseudo-Goldstone boson which is

massless in zero order but picks up a finite, small mass from higher-order corrections if the same U'( 1) symmetry is not shared by the rest of the ~ a ~ r a n ~ i a n . ' ~

(2) Infinitely many solutions at the minimum, which transform to each other by the action of the pseudosym- metry. We substitute the most general solution allowed by gauge invariance under SU(2)L x U( 1 ):

[ o 1, [ u sina siny 91 = , ,iS/z p2= , ,ins cosy , -i8/2

and demand charge-conserving solutions ( y = O ) . V , takes the following form:

The U'( 1) symmetry is now apparent. It is found that two different minima exist for two separate ranges of the pa- rameters of Va:

In the first solution (al), which has already been noted by the authors of Ref. 9, the U'( 1) is equivalent to the U(1) gauge transformation of SU(2) XU( 1 ). A PGB is not ex- pected; it is absorbed by the gauge boson. The second minimum represents an infinity of solutions and without performing higher-order calculations it is impossible to determine which one is the true VEV. As will be shown in the next section it is forbidden to choose one of the solutions ad hoc, as it leads to wrong results.

B. Case b

We add a third isodoublet po which transforms as a singlet of S3 and demand a reflection symmetry R; po-po (or, alternatively, cpo can transform as the antisymmetric { lA ) singlet of S3 without the requirement of the R sym- metry):

538 R. YAHALOM - 29

Vb= ~~+p~(p~0)+h~(p$p0)~+~(qd$0g)(p~q1+p:p2) C. Case c

In this case the third field, pO, is an S3 singlet 11, j and we do not impose a reflection symmetry:

Again note that Vb is invariant under the same U ( i ) sym- After substitution of pL, as in Eq. (8) we find metry defined by Eq. (6). Using the most general charge- conserving gauge-invariant solution one finds

0 (po)=vo , ( p ~ ) = v ~ o s a e ~ ( ~ + ~ ' / ~ , 0

(8) ( p 2 ) = U sina e " ~ - p ' / ~ , + sin(a)cos 1 . (13)

+(K + ~ ) u ~ ~ v ~ + 2 8 ~ ~ ~ ~ ~ s i n 2 c r cosy . Two different minima exist:

0 b!: ( p o ) = ~ o ,

v cod45 ~

v sin(45 -ao/2)e-i8"2

(9) Because of the appearance of the phase /3 it is clear that the potential is no longer invariant under U'( 1 ) and there will not be any PGB.

Although the last fact may look promising we found later on that the Yukawa couplings contain at least two additional parameters (because of pol. It turns out that the masses and mixing angles are not calculable. Der- man" has used an equivalent potential by assuming that the three Higgs isodoublets transform as the three-

(10) dimensional reducible representation of S3. We shall not deal with this case further.

As we are forced to return to solutions a and b we rnust devise some mechanism to solve the problem of the PGB. 'The most common strategy is to break GH softly by the addition of extra terms by hand. In our view such a pro- cess is unnatural and we search for other options. There is no special reason why there is only one S j singlet. In fact, as we can show, if there are two such fields the prob- lem of the PGB can be solved, under certain conditions.

D. Case d

p 2 , p 2 , 8 < 0 . There are four isodoublets of Higgs fields. Two of

One of the infinitely many solutions of case (bl), /3=90, was found and used by Pakvasa and Sugawara.1° The fact that this is only one of many other possibilities was al- ready noted by Goffin, Segr;, and eld don.^ It does not change the conclusions of Ref. 10 since this paper studied the case of two generations of quarks, and in that case the whole Lagrangian is invariant under U'( 1 ). Unfortunately it is clear that a true Goldstone boson rnust appear in this case. While the authors of Ref. 9 chose to overcome the problem of the PGB when dealing with three generations of quarks by adding a soft-breaking tern1 of S3 to V(d) we shall try to preserve the symmetry, as is described further on.

them transform as an S j doublet (pZ2) and the others are S3 singlets ( x l ,xz 1.

We demand that the fields ( x l ,x2 ) carry a similar gIo- bal phase symmetry as ip l ,pz ) , i.e., they are invariant under the transformation U"( 1 ):

We do not concern ourselves with the origin of this symmetry but in Sec. XV we show that under these as- sumptions the PGB can almost completely decouple from ordinary matter.

The most general potential invariant under S j XU1'( 1 ) is

29 HORIZONTAL PERMUTATION SYMMETRY, FERMION MASSES, . . . 539

The most general solution consistent with gauge free­dom and charge conservation is

[<<*>?> | \vcosaeHr+f3)/2~

[<<p°2>r [vsmaeHy-p)/2

kcos0e' (£- r ) /2 1

\v0sinde-He+r)n

<*?> <X°2)

(16)

By direct substitution we found many solutions to this potential for different ranges of the parameters. Among all these solutions we chose to look at the two that are similar to b l and b2:

d l : <<P.>

(<P°2>,

<X°2)

v cosa e ip/2

u s i n a e - 1 ^ 7 2

v0cos6el€/2

v0sm6e-i€/2

(17)

sin2a = cosaosin20; y==0; 8 < 0 ; (p—a)<0; cosa0—[8/(p—a)](v0/v)2>0, 6 is a complicated function of the parameters.

d2: [<*>?> • \<<P°2>]

'<*?> <x°2)

1 Vl

=

eiM |

e-ipn\

v0co$6eun

u0sin ee-un

(18)

a = 4 5 ° , 8 < 0 , y=0, (p—a)>0, 0 a function of the pa­rameters. We close this section by rewriting the various relevant solutions of the VEV's when basis 1 is used in­stead of basis 2 (see Appendix). We call these fields <j> and £ (instead of q> and X, respectively).

a l : <*?>=», <$) = ±iv. (19)

d l : <4>°2>

<S2>

-om-^v

-O(e') VI

eia-/2

c - / « 7 2

,iff/2

-iff/2

(20)

| a ' = 4 5 - a , | < 9 ' = 4 5 - 0 ,

d2:

O(0') =

<S?>1

cos/?' —sinjS'

sin^' cos/?'

= O(0')

= 0 ( e ' )

0 (21)

Solution b l (b2) is identical in its form to d l (d2) for the fields <f>i and 02-

Solutions d l and d2 (bl, and b2) have a global, continu­ous symmetry. In the present convention it is 0(2), in­stead of U'( 1). The only way to choose the correct VEV is by calculating the effective potential, including one-loop

corrections. As Keff depends on the structure of the mass matrix, these calculations will be performed after the Yu­kawa terms are investigated in the next section.

III. THE MASS MATRIX

The Yukawa term <if Y which is ordinarily constrained only by SU(2)LXU(1) invariance is now fixed by the re­quirement of invariance under S3. (In the rest of this sec­tion we write the mass matrix for the Q = — y quarks only; for the Q — + y quarks, substitute (<f)) —> {<f>* ):)

Jf Y = 4>L [ 2 r ^ / | ^ + H . C .

= 2 «Ar [ (^L)p®(^ )«®(^ ) r ] + H.C. (22) P$q,r,t

The coupling matrices Tt are fixed by this equation, when ((t/'L)p<S>(t/'i?)^<8>(^) j stands for the direct product of the left- and right-fermion and the Higgs-fields multi-plets belonging to representations p,q9r, respectively, under GH. The free parameters g ^ r are nonzero only if the direct product contains GH singlets, t is the multiplicity of the singlets in the direct product, and is equal to one if GH = S3. In Table I we describe all the possible mass ma­trices, M = y£tiri<f>i with three generations of fermions and three scalar isodoublets (case c in Sec. II). (^1,^2) a r e

to be understood as the VEV of the *S3 doublet; 0O is t n e

VEV of the S3 singlet. For the potentials of cases a,b>d,<f>0

should be erased from these matrices. The transformation properties of the fermions are given by the name of the representation: {ls} =s, {\A } — A, (2} = J D . In order to find M9 basis 1 is used together with the proper Clebsch-Gordan coefficients [see Appendix, Eqs. (Al) and (A5)]. The order of the fermion fields in the multiplet is not necessarily as the order of the generations. In Table I, rf)T~(rpx,xl)2^^)y where (^1^2)1. 1S a n S3 doublet and (^3)1, is an S3 singlet. The possibility that all the left-handgd fields are singlets under S3 has been checked and proved to be unsuccessful for all possible assignments of the right-handed fields.

When one tries to find an interesting model with solu­tion a l [Eq. (19)] it is found that the mixing angles are unacceptable (0, 45°, 90°) or that they cannot be expressed as mass ratios. These are typical cases of natural flavor conservation.14

When (f>0 is not decoupled [case c, Eq. (13)] we see from Table I that there are too many parameters and a detailed study showed that it is impossible to find any relations be­tween the masses and mixing angles (see also Derman11).

The promising cases are therefore solutions b and d [Eqs. (20) and (21)], but as was seen in the last section, there are infinite many solutions, characterized by a sym­metry parameter /?.

The diagonalization of the mass matrix is done by means of a biunitary transformation

M(dmgonal)==MD==WfMV; MDMl^w\MM^)W .

(23)

The mixing matrix (KM matrix) is defined as

540 R. YAHALOM - 29

TABLE I. Mass matrices for S3-invariant Lagrangian.

L Y R M

ize MM', and use the result to find the minima of [ - Tr( M41nM2)].

where W U diagonalizes the Q = + sector and wd the Q = - L sector.

If the mass spectrum (MD) depends on the free parame- ter /3 we expect the matrix UKM to be /3-dependent and the masses and mixing angles can be found only by consider- ing Vef f . It was shown by S. weinberg,16 and S. Coleman and E. weinberg'' that

with m, M, and B as the scalar-boson, fermion, and gauge-boson mass matrices, respectively.

It is dear that in our case m and B do not depend on /3 and only cause a small shift in the values of u and uo. To find /3, we substitute the VEV (with /3 as a free parameter) in each of the models in Table I, calculate MM', diagonal-

A. Solution d2 (b2, a2)

From Eq. (21) (we use /3 instead of /?' throughout this section)

The left-handed fermions are assigned as follows:

The right-handed fields transform according to the specific model from Table I. Table I1 summarizes M and MM' where the fermions are arranged in generation or- der. Only representative types of matrices are displayed. For example, entries 9- 12 in Table I lead to the same mass matrices in the sense that the mass spectrum is iden- tical in all of them.

The masses in models A1 and A2 are independent of /3. In such a case O(2) is a symmetry of the Lagrangian and one expects a true Goldstone boson, unless at least one of the charged sectors is chosen to be assigned to model A3.

For model A3 it was shown in a previous letterla that an arbitrary choice of fl is not allowed. If we take fl=n/4 (as was done by the author of Ref. 121, and assume b >>a >>c we get the following values for the KM angles from Eqs. (23) and (24)

If one chooses /3=9W with the same hierarchy of parame- ters then

7 1 sinel I emdms/mb2, e2= -e3= - - 2 '

(27)

We show now that neither of these relations is correct and that this model cannot give any mass relations. Assuming again b >>a >>c and diagonalizing M M t one finds

From the requirement of minimalization of Veff [Eq. (2511 it is found that /3=(7/3)n (Ref. 19), n =O,l, . . . , . With this value of /3 we get

29 - HORIZONTAL PERMUTATION SYMMETRY, FERMION MASSES, . . . 54 1

TABLE 11. Mass matrices for solution d2. cs=cos/3, sg=sinS.

Entries in

Table I Name

0 0 0 acg bcg ccg asp bsB C S ~

0 0 0 acg bcg csg asg bsg -ccg

0 ccg csg

bcg ass acg bsg acg -asg

From Table I1 we rewfite the mass matrix and its eigen- values:

As before only the d sector is treated, and for the u mass matrices replace ( 4 ) by ( 4 )*.

To find the masses we substitute (4, ) and ($2) from Eq. (20). The mass spectrum of B1 and B2 does not de- pend on fi:

B1:

m 1 2 = 0 ,

m 2 2 = 0 , (33)

MM+ is diagonalized by W:

The masses of model B3, on the other hand, do depend on fi and we must look at the effective potential. We assume again b >>a >>c and get the following masses:

, (30)

It is seen that, as claimed, the mixing angles cannot be ex- pressed as a function of the masses, in contradiction to the cases fi=45",90" that turned out to be wrong.

c2 0 ac

0 a 2 + b 2 0

ac 0 a 2

M =

B. Solution d l (bl) m 22--a 2v 2( 1 + sin2a')

+c2v2[1-g(ar)/(1+sin2a')] , m:e(b2+a2cosa')v2 ,

g (a' ) = [cos3a' s i n 2 ~ ( 4 cos22fi - 1

O c O b 0 a O a O

Table I11 summarizes the typical mass matrices for this solution. The fermion representations are

, MMt=

+3sin2a'+l] . After solving the minima of Veff it is found that

model B1: dR,sR,bR:( lSj ;

modelB2: dR,sR: ( I s ] ; bR: ( I A ] ;

model B3: dR: (1,); [bR,sR]: ( 2 ) .

542 R. YAHALOM 29 -

TABLE 111. Mass matrices for solution dl.

Name M ~ M M + / L I

0 0 0

131 ~ ( 4 ~ ) b ( & ) ~ ( 4 2 )

a ( $ , ) b ( 4 1 ) ~ ( 4 1 )

0 ~ ( 4 2 ) ~ ( 4 1 )

B3 b ( d 2 ) ~ ( 4 , ) b ( d l ) ~ ( 4 1 )

IV. PSEUDO-GOLDSTONE BOSON @=Ua4, R = U e c ,

0 0 0

0 a$(l--sin2/3cosa') a $ ( c & - ' " ' - ~ & + ' ~ ' )

a $ ( c & + i a ' - ~ & -la' ) ao2( 1 +sin2/3cosa1)

c 2 2ac sin2/3 cosa' 2ac cosp wsa'

2ac sin2/3cosa1 2a2+ b2( 1 -sin2pcosa') b2cos2/3cosa'+ 2i(a 2 - f b2)sina' 1

2ac cos2/3 cosa' b2cos2/3cosa'- 2i(a - $b 2)sina') 2a + b 2( 1 + sin2D cosa')

We saw in the last section that solutions bl and d l are the only possible choices, with /3=45" as is required from one-loop corrections. The mass of the pseudo-Goldstone boson, present in these models, is changed by these correc- tions from zero to some small value. This value cannot be made large enough to prevent FCNC from occurring at an unrealistic level. The experimental bounds on the branch- ing ratio of processes such as KL - + I l +&? and the KL-Ks mass difference can be obeyed only if the PGB is made to decouple from the ordinary fermions. In this section we use the procedure suggested by Haber, Kane, and Ster- lingz0 in order to find the PGB for solution d l of the po- tential and show how it is indeed decoupled.

Following Georgi and ~ a n o ~ o u l o s ~ ~ we first change the basis of the fields 4i and fi to Qi and Ri, respectively, in such a way that only Q1 and R1 have a nonzero VEV. From Eqs. (20) and (37) the VEV's of case d l are (€=/i=45O)

'

Because this value of /3 is forced even if we use for one of

ca = cos [ f 1, sa =sin [ f 1, etc. ,

of 0(2), i.e., Veff has the same solution for /i-t/3+60n. J

Define

the charged sectors model B2 (or B1) while the other is B3, we shall use it for all B-type models. We stress again that Veff fixes this value of /3 and as a result, also all the other physical consequences.

We also note that there is a discrete unbroken symmetry

so that

(@'?)=v, (fl'?)=uo,

(Q;>=(n; ) =o .

"0COS [p ] -ivosin [$

Call L, those generators of the symmetry group of the po- tential that are broken at the vacuum or belong to a pseu- dosymmetry (see Ref. 20 for more details). If ( 4 ) is the VEV "vector" in the real representation (when the real and imaginary parts of 4, are separated) then the zero- mass eigenstates of the scalars mass matrix can be found by projecting an arbitrary field in the "direction" of

29 - HORIZONTAL PERMUTATION SYMMETRY, FERMION MASSES, . . . 543

(La ( 4 ) ). We find that the Goldstone boson (GB) in our model is

+'=1m(4), +'=Re(+), etc. (40)

Expressing Eq. (40) in terms of @ and fl one finds

To find the PGB we use the generator of O(2) in the eight-dimensional real representation of the neutral scalar fields, (4, f

where I is a 2 x 2 unit matrix,

( ~ T , f T ) = ( ~ ~ a , ~ , ~ , - ~ ~ a , ~ o ~ e , O , O , - ~ o ~ e ) . (42)

The PGB is

This field is not orthogonal to the GB in Eq. (40). Pro- jecting out the part of Eq. (43) perpendicular to the GB (which is absorbed by the Z O and decouples) and using the fields @, ln we get

Only 4 is coupled to the fermions and if we choose vo >>v, we can make this coupling, u / ( u ~ ~ + u ~ ) ~ / ~ , as small as we like by decreasing the value of v. [(v; + v 2 ) is bounded from above by the gauge-boson mass.] The mass of 4; [the analog of the single scalar field in the minimal SU(2) X U(1) theory] is bounded between few GeV and 1 TeV, but the other Higgs-boson masses are not bounded and can be made as large as one wi~hes .~ '

It is interesting to note that in this scheme the vector- boson masses are set by the scale of vo, while the fermion masses are fixed by the scale of v. Similar suggestions have been previously made by several authors (see, i.e., Ref. 20).

V. A REALISTIC MODEL

We use the results of the previous sections to give an ex- ample of a possible realistic model.

Solution d l of the potential [Eq. (20)] is combined with mode B2 (Table 111) for the Q = + f quarks and with model B3 for the Q = - f quarks. The following S3 as- signments for the fermions are adopted:

Using the values that were found for the VEV's and fl [Eqs. (36) and (3711, and diagonalizing M~ and Mu, one finds WU, wd, and the quark masses [only terms of O(mi/mi + ) have been kept in W]:

WU=

The parameters a,, b,, etc., in Eq. (47) are free and the quarks masses cannot be predicted by the model. The mass of the u quarks turns to be zero at tree level. It is evident from the mass matrix of model B2 (Table 111) that this result is in- dependent of the specific VEV we use and arises directly from the representation content of the fermion and scalar fields. In such a case, there will be one-loop corrections to the mass matrix (as was shown by ~ e i n b e r ~ ' ~ , ~ ~ ) and we ex- pect mu to be nonzero after higher-order calculations are performed:

c 1 3 1 0

-SlCd C1Cd-k SdS3 -Sd+cds3

islsd -i(sdc1 -cds3) -i(cd+sds3)

uKM=( w")'( wd)=

1 0 0

O Cu S u

0 is, -icU , , Wd=

c 1 S1 0

- S ~ C Z c ~ c ~ + s ~ s ~ -s2+c2s3

-sls2 s2c1 -c2s3 c2 +s2S3

. (49)

544 R. YAHALOM 29 -

U K M is in the known K M form without a CP phase and with c 3 e 1 , s 1 s 3 e 0 where

O,, f (a ') , and s3 are defined by Eq. (48). The angle a' is a free parameter that can be chosen to have an arbitrary value for a large range of the parameters of the potential [Eqs. (17) and (2011. We fix a' by demanding that s l =sinOc=0.229 (Ref. 23) thereby we get a fully predict- ed K M matrix. We use for the effective (current) quark masses the values2 (in MeV)

From Eq. (49) one gets that a'=8.3", ~ ~ e 0 . 1 4 4 , s ~ ~ I o - ~ , SO that

These values are in accordance with the known limits on the mixing angles.24

The b quark decays mainly to c (we expect Uub to have some small value when UKM is calculated to higher or-

ders). The B lifetime is given to a good approximation by23,25

rB =$rp(mp/mb)5 I Ucb 1 - 2 = 4 . 7 ~ 10-l3 sec . (51)

This result is well within the best experimental limit of 1.4X 10-l2 ~ e c . ~ ~

As the B-meson lifetime is essentially determined by Ucb, its main contribution comes from sz which, as seen from Eqs. (48) and (491, is almost independent of the t quark mass for m, 2 20 GeV. In our model the CP phase 6 in UKM [Eq. (5011 is zero and therefore we do not get a strong dependence of TB on m,, as predicted by previous analysis where 6#0.~'

There are two Higgs doublets (q5,-) that couple to the fermions. Out of them, after spontaneous symmetry breaking of SU(21, XU( 11, we are left with two charged and three neutral scalars. The influence of such scalars has been investigated by quite a few authors (see, for ex- ample, Refs. 20, 28, and 29). It is important to note that in our model there are flavor-changing neutral currents (FCNC) (caused by neutral-Higgs-boson exchange) and spontaneous CP breaking. In Sec. IV it was shown that the PGB is coupled very weakly and by decreasing the ra- tio u / v o we can make its effect compatible with experi- ment. Using the rotated Higgs fields defined by Eq. (39) and the relation v - ' =23/4~F1/2, we write the couplings of the neutral fields Qi :

where M represents the diagonal mass matrices of the fer- mions, f the fermions mass eigenvectors, and

R = $ ( l + y 5 ) ; ~ = 3 ( 1 - ~ ~ ) .

The matrices H are found by writing Y y in the form

and transforming ( r 1 , r 2 ) to the basis in which (a1,@,) and f j are defined [see Eqs. (23) and (39)]. We assume that the charged leptons have the same Yukawa structure as the Q = - f quarks. It is found that

states, which are a linear combination of Q; and have approximately the same mass =mH) is imposed by the processes KL -I l & and the KL-Ks mass difference. In Refs. 28 and 30 a detailed study of Higgs-boson-mediated FCNC has been performed. Using the result of the calcu- lation of McWilliams and Li in Ref. 28 one finds that

From the known limit of 2X on the branching ratio of KL - tpe we get

mH 2 330 GeV . (56)

Note the absence of FCNC involving the u quark in this approximation. All FCNC are mediated by Q2 whose mass can be as large as we need. The coupling of Q2 to (ds) is proportional to m, and not mb, in spite of the fact that we study the three generations case. The strongest limits on mH (we assume that the two Higgs mass eigen-

A higher, but less accurate, bound on mH is found from AmK. Using the experimental value Amk =2.5 X 10-l5 GeV one gets the constraintz8

m H - ' ~ ~ . 9 ~ ( ~ d ) s d ( ~ ~ loF5 . (57)

We thus find that m~ > 5 TeV. The bound of few TeV is also needed because of CP-violation effects (see Ref. 29 and references therein for treatment of this subject).

29 - HORIZONTAL PERMUTATION SYMMETRY, FERMION MASSES, . . .

VI. SUMMARY

S3 has been suggested as a possible horizontal symmetry group. A systematic and general study of the Higgs po- tential and Yukawa interaction in the case of three genera- tions has led us to examine the influence of a pseudosym- metry on the mass matrix and mixing angles. Previously suggested solutions of the minimum of the potential were found and further analysis which included one-loop corrections to the effective potential has shown that these models are unacceptable. While other works that used S j with three generations have either postulated an ad hoc structure of VEV or broken the symmetry explicitly in or- der to avoid the problem of the PGB, we suggested a new model based on four Higgs doublets which preserve the S3 invariance of the potential, and calculated the true vacu- um of the theory. We showed that Sg can serve as a possi- ble explanation for some of the relations between the fer- mion masses and mixing parameters. We have to stress, nevertheless, that this approach is limited by our ig- norance about the origin of the Higgs scalars and the ad- ditional horizontal symmetries, and one must wait for a deeper level of understanding of these topics in the future.

ACKNOWLEDGMENT

I am grateful to Dr. M. Gronau for suggesting this study, for helpful discussions, and for useful remarks con- cerning this work.

APPENDIX

In this appendix we describe the relevant properties and representations of s ~ . ~ ~ S3 is a six-element group and has three irreducible representations; two singlets (one sym- metric ( 1, ) and the other antisymmetric ( lA ) ) and one doublet ( 2) . The three-dimensional representation is reducible: ( 3 ) -+ ( 2 ) + ( 1, ) . The direct product of two doublets is

In this paper we work with two different choices of basis for the doublet. If pa, p b , pc (or 4a, +b, 4c) transform as the three-dimensional representation, we de- fine

basis 1:

basis 2:

i 4 3 2 = - ( ~ a +u2pb + ~ q c

W,e2?ri/3, J=75 .

These fields transform as a singlet (+o;po) or doublets (4142);(p1,q2) under the permutation of (a , b , c ) . The transformation of the doublet under S3 can be summarized in basis 1 as

I ecosa sina ' 3 = -esina cosa

2 7r

I €=? I ; a = - n ,

3

and the second basis is connected to the first by the fol- lowing transformation:

In order to find the VEV we use basis 2. For the Yukawa interaction we use basis 1. The Clebsch-Gordan coeffi- cient for the reduction of the direct product can be ex- pressed in basis 1 by the following notation: If ma and cPb

are two S3 doublets, the result of the reduction of qemb then

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