Updated S3 Model of Quarks

A model proposed in 2004 using the non-Abelian discrete symmetry S3 for understanding the flavor structure of quarks and leptons is updated, with special focus on the quark and scalar sectors. We show how the approximate residual symmetries of this model explain both the pattern of the quark mixing matrix and why the recently observed particle of 126 GeV at the Large Hadron Collider is so much like the one Higgs boson of the Standard Model. We identify the strongest phenomenological bounds on the scalar masses of this model, and predict a possibly observable decay b ->s tau- mu+, but not b ->s tau+ mu-.


Introduction
With the discovery [1,2] of a particle of 126 GeV at the Large Hadron Collider (LHC) and no evidence for any others in a wide range of masses, extensions of the standard model are now severely constrained. In particular, if we want to understand the pattern of quark and lepton masses and their mixing in terms of a flavor symmetry, we are faced with a new theoretical challenge. In order to carry the flavor symmetry in the context of a renormalizable theory, new scalar multiplets are required. We now must have a good reason within the flavor model as to why the one observed light Higgs boson is so much like that of the Standard Model (SM). Of course we need also to understand within the same context of why the quark mixing matrix is nearly diagonal, whereas the neutrino (lepton) mixing matrix is close to tribimaximal.
In 2004, the family structure of quarks and leptons was explained in a model [3] using the non-Abelian discrete symmetry S 3 . It has a symmetry breaking pattern designed to allow for small 2 − 3 mixing in the quark sector and near-maximal 2 − 3 mixing in the neutrino sector. Whereas all the basic details were described for both quarks and leptons, that paper dealt mainly with neutrino mixing, with the specific assumption of negligible e−µ mixing in the charged-lepton mass matrix, although this 1 − 2 mixing is generally allowed by the S 3 symmetry and is unavoidable also in the quark sector. As a result of that arbitrary assumption, the neutrino mixing angle θ 13 was predicted to be very small: 0.02±0.01. Given that it has now been measured [4,5] at about 0.16, this prediction based on that arbitrary assumption is certainly ruled out, and the observed value of θ 13 should be attributed to e − µ mixing within the context of this model. The Higgs sector of this leptonic model has also been studied [6,7] for its collider signatures.
In this paper we study the quark sector itself in detail and show how the Higgs sector is constrained by present data. In particular, we identify two approximate residual discrete Z 2 symmetries, which allow the lightest scalar particle to be the observed 126 GeV particle, with the property that it is naturally very close to that of the SM.
In Sec. 2 we present the S 3 model of Ref. [3], writing down specifically all the quark and Higgs representations. In Sec. 3 we discuss the c − t and s − b quark sectors and show how they align in a symmetry limit, and the resulting phenomenological constraint from these two sectors. In Sec. 4 we add the u and d quarks and discuss the full 3 × 3 quark mixing matrix.
In Sec. 5 we consider the full scalar sector consisting of three Higgs doublets, and show how the one light neutral scalar of this sector resembles that of the standard model to a very good approximation, not being a result of fine tuning but based on symmetry. In Sec. 6 we derive the phenomenological constraint on the third Higgs doublet which is responsible for mixing the first family with the other two. In Sec. 7 we make a specific verifiable prediction of this model, i.e. b → sτ − µ + could have a branching fraction as large as 10 −7 , whereas b → sτ + µ − would be suppressed by a relative factor of m 2 µ /m 2 τ . In Sec. 8 we conclude.

The S 3 Model
The smallest non-Abelian discrete symmetry is the group S 3 of the permutation of three objects. It has six elements, and is isomorphic to the symmetry group of the equilateral triangle (identity, rotations by ±2π/3, and three reflections). It has three irreducible representations 1, 1 , 2, with the multiplication rules: The specific choice of 2 × 2 matrices for the 2 representation is only unique up to a unitary transformation. A particular practical and elegant one appeared in Ref. [8] which was followed in Ref. [3]. For a review, see Ref. [9]. In this representation, if so that and The consequence of this representation is the elegant result that the trilinear combination Consider all quarks as left-handed fields, so that the usual right-handed ones are repre- In analogy to the three quark families, there are also three Higgs with assignments:

The c − t and s − b Quark Sectors
In this sector, only Φ 1,2 are involved in the Yukawa interactions, i.e.
Consider now the Higgs potential of Φ 1,2 . In addition to the S 3 symmetrical term, we add a soft term which breaks S 3 , but preserves the discrete Z 2 symmetry Φ 1 ↔ Φ 2 . Hence, This has a minimum with v As a result, the normalized physical scalar bosons and their masses are given by Note that without the µ 2 2 term which breaks S 3 , A would be massless.
Consider now the generic structure of the c − t and s − b mass matrices. They are of the form given by Eq. (12) in Ref. [3], i.e.
Since they are both diagonalized by the same 2 × 2 unitary matrix, there is no mismatch, and the quark mixing matrix is diagonal, explaining to a good first approximation what is observed. Note that this result is based on the symmetry breaking pattern S 3 → Z 2 . Now h 0 may be identified with the corresponding SM Higgs boson which couples to (m s /2v)ss and (m b /2v)bb, etc.
As for H ± , H 0 , A, their Yukawa couplings are given by Hence H 0 and A will contribute significantly to B s −B s mixing, with resulting bounds on their masses. The tree-level effective four-fermion interaction is given by The matrix elements of the operators are calculated at the m H,A mass scale and evolved to the hadronic scale by using the anomalous dimension matrices given in Ref. [11,12,13]. The mass difference in the B s −B s system is then given by where For the bag model parameters we use the results from Ref. [14] estimated in the quenched approximation on the lattice (µ b = m RI−MOM b = 4.6 GeV):

The 3 × 3 Quark Mixing Matrix
The introduction of the third Higgs doublet Φ 3 will allow the u and d quarks to obtain mass, as well as mixing with the other quarks. Since these are all small, it is natural to assume that v 3 is small. (This also means that the Φ 3 mass may be naturally large, as shown later.) In that case, the h 0 of Eq. (12) is still a very good approximation to the one physical Higgs boson h 0 SM of the SM, i.e. we have an explanation of why our specific three-Higgs doublet model has a mass eigenstate h 0 which is very close to h 0 SM .

The 3 × 3 quark mass matrices are given by [3]
and (26) where s /c = v 2 /v 1 , and M u by We have rephased d R , s R , b R , u R , c R .t R as well as (u, d) L , (c, s) L so that only one complex phase δ remains. Hence with the CP violating parameter

The Complete Higgs Sector
To allow for v 1 = v 2 , the symmetry Φ 1 ↔ Φ 2 must be broken. This may be accomplished by adding to V 12 of Eq. (10) the term µ 2 . The coefficient µ 2 3 may be chosen naturally small, because µ 2 3 = 0 results in the extra Z 2 symmetry already discussed. In addition, we impose a new Z 2 symmetry so that Φ 3 and (u, d) L are odd and all other fields are even on Z 2 . The purpose of this symmetry is to forbid the quartic term Φ † which is allowed by S 3 (the reason for this will become clear later). However, we also allow this new Z 2 symmetry to be broken softly by the bilinear term µ 2 4 Φ † 3 (Φ 1 + Φ 2 ) + h.c., which preserves the Φ 1 ↔ Φ 2 interchange symmetry of V 12 . The complete scalar potential of this model is then Consider first v 1 = v 2 , this results in a deviation of h 0 from h 0 SM given by where (v 2 1 − v 2 2 )/4v 2 = 0.0207 from s = 0.04135 of Eq. (23).
and the mixing of The h 0 of our model is thus naturally equal to h 0 6 Constraint on Φ 3 The exchange of φ 0 3 directly contributes to ∆M K . The relevant effective interaction is given by This operator is again O 4 from Eq. (20), with the appropriate exchange of the quarks and the contribution to ∆m K is analogous to Eq. (23), i.e.
So far, we have been able to show that h 0 is very close to h 0 SM , and that the other physical scalars are of order 1 to 10 TeV, from B s −B s and K 0 −K 0 mixing respectively. All other effective flavor-changing neutral-current interactions in the quark sector such as b → sγ are suppressed.

Specific Prediction
Since the scalars of this model also have leptonic interactions, there are some lepton flavor violating processes in this model which are negligible in the SM. The corresponding Lagrangian to Eq. (17) for the µ − τ sector is given by Hence b → sτ − µ + proceeds again through the exchange of Experimentally the most interesting decay would be B s → τ + µ − . The branching ratio for this decay can be written as

Conclusion
In the post-Higgs era, any extension of the SM has to face the question of why it contains a light neutral scalar boson h 0 so much like to h 0 SM , in addition to being consistent with a myriad of phenomenological precision measurements. We show how this is possible in a model [3] proposed in 2004 based on the non-Abelian discrete symmetry S 3 . It has three Higgs doublets, and yet one becomes almost exactly that of the SM because of two ap-proximate residual Z 2 symmetries. It also explains why the quark mixing matrix is nearly diagonal, with just enough parameters to fit the data precisely. The model is constrained principally by B s −B s and K 0 −K 0 mixing, and has the unique prediction of b → sτ − µ + with a branching fraction for B s → τ + µ − decay as large as 10 −7 , but a strong suppression by the factor m 2 µ /m 2 τ for b → sτ + µ − decay.