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<ul><li><p>PHYSICAL REVIEW D VOLUME 29, NUMBER 3 1 FEBRUARY 1984 </p><p>Horizontal permutation symmetry, fermion masses, and pseudo-Goldstone bosons in SU(2), x U( 1) </p><p>R. Yahalom Department of Physics, Technion-Israel Institute of Technology, Haifa, Israel </p><p>(Received 28 June 1983) </p><p>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. </p><p>I. INTRODUCTION </p><p>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.' </p><p>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 </p><p>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. </p><p>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. </p><p>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. </p><p>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 </p><p>536 @ 1984 The American Physical Society </p></li><li><p>29 - HORIZONTAL PERMUTATION SYMMETRY, FERMION MASSES, . . . 537 </p><p>each model separately. This is to be compared, for exam- ple, to Ref. 9. </p><p>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. </p><p>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. </p><p>To establish our notation, we write the S U ( ~ ) L X U( 1) X S3 Lagrangian as </p><p>where 2Yd is the scalar part of 2 that will be elaborated upon in Sec. 11: </p><p>2, is the Yukawa interaction between the scalars and fer- mion fields to be studied in Sec. 111: </p><p>Here and after, 4, p, X , </p></li><li><p>538 R. YAHALOM - 29 </p><p>Vb= ~~+p~(p~0)+h~(p$p0)~+~(qd$0g)(p~q1+p:p2) C. Case c </p><p>In this case the third field, pO, is an S3 singlet 11, j and we do not impose a reflection symmetry: </p><p>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 </p><p>0 (po)=vo , ( p ~ ) = v ~ o s a e ~ ( ~ + ~ ' / ~ , 0 </p><p>(8) ( p 2 ) = U sina e " ~ - p ' / ~ , + sin(a)cos 1 . (13) </p><p>+(K + ~ ) u ~ ~ v ~ + 2 8 ~ ~ ~ ~ ~ s i n 2 c r cosy . Two different minima exist: </p><p>0 b!: ( p o ) = ~ o , </p><p>v cod45 ~ </p><p>v sin(45 -ao/2)e-i8"2 </p><p>(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. </p><p>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- </p><p>(10) dimensional reducible representation of S3. We shall not deal with this case further. </p><p>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. </p><p>D. Case d </p><p>p 2 , p 2 , 8 < 0 . There are four isodoublets of Higgs fields. Two of </p><p>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. </p><p>them transform as an S j doublet (pZ2) and the others are S3 singlets ( x l ,xz 1. </p><p>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 ): </p><p>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. </p><p>The most general potential invariant under S j XU1'( 1 ) is </p></li><li><p>29 HORIZONTAL PERMUTATION SYMMETRY, FERMION MASSES, . . . 539 </p><p>The most general solution consistent with gauge free-dom and charge conservation is </p><p>[ | \vcosaeHr+f3)/2~ </p><p>[d,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.</p><p> 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. </p><p>When one tries to find an interesting model with solu-tion 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 </p><p>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 be-tween the masses and mixing angles (see also Derman11). </p><p>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 sym-metry parameter /?. </p><p>The diagonalization of the mass matrix is done by means of a biunitary transformation </p><p>M(dmgonal)==MD==WfMV; MDMl^w\MM^)W . </p><p>(23) </p><p>The mixing matrix (KM matrix) is defined as </p></li><li><p>540 R. YAHALOM - 29 </p><p>TABLE I. Mass matrices for S3-invariant Lagrangian. </p><p>L Y R M </p><p>ize MM', and use the result to find the minima of [ - Tr( M41nM2)]. </p><p>where W U diagonalizes the Q = + sector and wd the Q = - L sector. </p><p>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 </p><p>with m, M, and B as the scalar-boson, fermion, and gauge-boson mass matrices, respectively. </p><p>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- </p><p>A. Solution d2 (b2, a2) </p><p>From Eq. (21) (we use /3 instead of /?' throughout this section) </p><p>The left-handed fermions are assigned as follows: </p><p>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. </p><p>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. </p><p>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) </p><p>If one chooses /3=9W with the same hierarchy of parame- ters then </p><p>7 1 sinel I emdms/mb2, e2= -e3= - - 2 ' </p><p>(27) </p><p>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 </p><p>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 </p></li><li><p>29 - HORIZONTAL PERMUTATION SYMMETRY, FERMION MASSES, . . . 54 1 </p><p>TABLE 11. Mass matrices for solution d2. cs=cos/3, sg=sinS. </p><p>Entries in </p><p>Table I Name </p><p>0 0 0 acg bcg ccg asp bsB C S ~ </p><p>0 0 0 acg bcg csg asg bsg -ccg </p><p>0 ccg csg </p><p>bcg ass acg bsg acg -asg </p><p>From Table I1 we rewfite the mass matrix and its eigen- values: </p><p>As before only the d sector is treated, and for the u mass matrices replace ( 4 ) by ( 4 )*. </p><p>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: </p><p>B1: </p><p>m 1 2 = 0 , </p><p>m 2 2 = 0 , (33) </p><p>MM+ is diagonalized by W: </p><p>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: </p><p>, (30) </p><p>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. </p><p>c2 0 ac </p><p>0 a 2 + b 2 0 </p><p>ac 0 a 2 </p><p>M = </p><p>B. Solution d l (bl) m 22--a 2v 2( 1 + sin2a') </p><p>+c2v2[1-g(ar)/(1+sin2a')] , m:e(b2+a2cosa')v2 , </p><p>g (a' ) = [cos3a' s i n 2 ~ ( 4 cos22fi - 1 </p><p>O c O b 0 a O a O </p><p>Table I11 summarizes the typical mass matrices for this solution. The fermion representations are </p><p>, MMt= </p><p>+3sin2a'+l] . After solving the minima of Veff it is found that </p><p>model B1: dR,sR,bR:( lSj ; </p><p>modelB2: dR,sR: ( I s ] ; bR: ( I A ] ; </p><p>model B3: dR: (1,); [bR,sR]: ( 2 ) . </p></li><li><p>542 R. YAHALOM 29 - </p><p>TABLE 111. Mass matrices for solution dl. </p><p>Name M ~ M M + / L I </p><p>0 0 0 </p><p>131 ~ ( 4 ~ ) b ( & ) ~ ( 4 2 ) a ( $ , ) b ( 4 1 ) ~ ( 4 1 ) </p><p>0 ~ ( 4 2 ) ~ ( 4 1 ) </p><p>B3 b ( d 2 ) ~ ( 4 , ) b ( d l ) ~ ( 4 1 ) </p><p>IV. PSEUDO-GOLDSTONE BOSON @=Ua4, R = U e c , </p><p>0 0 0 </p><p>0 a$(l--sin2/3cosa') a $ ( c & - ' " ' - ~ & + ' ~...</p></li></ul>