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    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.


    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


    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 ,

  • 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


    The most general solution consistent with gauge free-dom and charge conservation is

    [ | \vcosaeHr+f3)/2~

    [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 f