Quark Masses and Mixing with A4 Family Symmetry

The successful A4 family symmetry for leptons is applied to quarks, motivated by the quark-lepton assignments of SU(5). Realistic quark masses and mixing angles are obtained, in good agreement with data. In particular, we find a strong correlation between |V_{ub}| and the CP phase beta, thus allowing for a decisive future test of this model.

Since the original papers [1,2] on the application of the non-Abelian discrete symmetry A 4 to quark and lepton families, much progress has been made in understanding the case of tribimaximal mixing [3] for neutrinos in a number of specific models [4]. As for quarks, the generic prediction [2] is that its mixing matrix is just the unit matrix, which can become realistic only if small mixing angles can be generated by interactions beyond those of the Standard Model, such as in supersymmetry [5]. Other ideas of quark mixing include the judicious addition of terms which break the A 4 (as well as the residual Z 3 ) symmetry explicitly [6]. In this paper, motivated by the quark-lepton assignments of SU(5), we study a new alternative scenario, where realistic quark masses and mixing angles are obtained, entirely within the A 4 context.
In SU(5) grand unification, the 5 * representation contains the lepton doublet (ν, l) and the quark singlet d c , whereas the 10 representation contains the lepton singlet l c and the quark doublet (u, d) and singlet u c . In the successful A 4 model for leptons, (ν i , l i ) transform as a 3 whereas l c i transform as 1, 1 ′ , and 1 ′′ . This means that we should choose Assuming as in the leptonic case three Higgs doublets Φ i = (φ + i , φ 0 i ) transforming as 3 under A 4 , the relevant Yukawa couplings linking d i with d c j are given by resulting in the 3 × 3 mass matrix: where ω = exp(2πi/3), h i are three independent Yukawa couplings, and v i are the vacuum expectation values of φ 0 i . (For details of the A 4 multiplication rules, see for example the original papers [1,2] or the more recent review [7].) On the other hand, the Higgs doublets linking u with u c must be different because of the latter's A 4 assignments. We choose here two Higgs doublets transforming as 1 ′ and 1 ′′ , then where m 2 , µ 3 come from 1 ′ and m 3 , µ 2 from 1 ′′ . This matrix is also symmetric because of the usually assumed SU(5) decomposition of 10 × 10 × 5 → 1.
In minimal SU(5), there is just one 5 representation of Higgs bosons, yielding thus only two invariants, i.e. 10 × 10 × 5 → 1 (for the uu c mass matrix) and 5 * × 10 × In the limit |µ 2 | << |m 2 | and |µ 3 | << |m 3 |, we obtain the three eigenvalues of M uu c as with mixing angles In the d sector, we note first that where Its eigenvalue equation is If |v 1 | = |v 2 | = |v 3 | = |v| as assumed in the original papers and all those of Ref. [4], then A = 3|v| 2 , B = 0, and the three eigenvalues are simply 3|h 1,2,3 | 2 |v| 2 . We choose them instead to be different, but we still assume |h 3 | 2 >> |h 2 | 2 >> |h 1 | 2 . In that case, we find and the mixing angles are given by thereby requiring the condition which has unity as an upper bound. Using current experimental values for the left-hand side, we see that quark mixing in the d sector alone cannot explain the observed V CKM . Taking We show in the following how all quark masses and mixing angles are realistically obtained in this model.
We note first that our quark mass matrices are restricted by our choice of A 4 representations to have only 5 independent parameters each. In the down sector, the Yukawa couplings h 1,2,3 can all be chosen real, A is just an overall scale, and B is complex. The 5 independent parameters can be chosen as the 3 quark masses, and 2 angles. In the up sector, we can choose µ 2 and m 2,3 to be real, with µ 3 complex. The 5 independent parameters can be chosen as the 3 quark masses, 1 angle, and 1 phase. Since we also have 10 observables (6 quark masses, 3 angles, and 1 phase), it may appear that a fit is not so remarkable. However, the forms of the 2 mass matrices are very restrictive, and it is by no means trivial to obtain a good fit. Indeed, we find that V ub is strongly correlated with the CP phase β. If we were to fit just the 6 masses and the 3 angles, the structure of our mass matrices would allow only a very narrow range of values for β at each value of |V ub |. This means that future more precise determinations of these two parameters will be a decisive test of this model.
Our quark mass matrices are given at the SU(5) unification scale in principle. However, the A 4 flavor symmetry is spontaneously broken at the electroweak scale. Therefore, the forms of our mass matrices are not changed except for the magnitudes of the Yukawa couplings between the unification and electroweak scales. Our numerical analyses are presented at the electroweak scale.
In order to fit the ten observables (six quark masses, three CKM mixing angles and one phase), 1,000,000 random numbers have been generated for the ten parameters of our model.
We then choose the parameter sets which are allowed by the experimental data. First we show the prediction of |V ub | versus β in Figure 1, with the following nine experimental inputs Here we use the tighter constraints on the mass ratios of light quarks, i.e. m u /m d and m s /m d , consistent with the well-known successful low-energy sum rules [12]. Clearly, future more precise determinations of |V ub | and β will be a sensitive test of our model.
We should also comment on the hierarchy of h i and v i . The order of h i are fixed by quark mixings. The ratio of h 2 /h 3 ≃ λ 2 is required by the V cb mixing (λ ≃ 0.22), on the other hand, h 1 /h 2 ≃ λ comes from V us . Once h i are fixed, quark masses determine the hierarchy of v i as follows: v 1 /v 3 ≃ λ 2 and v 2 /v 3 ≃ λ ∼ λ 1/2 . These hierarchies of h i and v i are also consistent with the magnitude of J CP , which is given by In summary, the A 4 family symmetry (which has been successful in understanding the mixing pattern of neutrinos) is applied successfully as well to quarks, motivated by the quark-lepton assignments of SU ( University for hospitality during a recent visit.