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Quark-Lepton Mass Relation in a Realistic A4 Extension of the Standard Model

We propose a realistic A4 extension of the Standard Model involving a particular quark-lepton mass relation, namely that the ratio of the third family mass to the geometric mean of the first and second family masses are equal for down-type quarks and charged leptons. This relation, which is approximately renormalization group invariant, is usually regarded as arising from the Georgi-Jarlskog relations, but in the present model there is no unification group or supersymmetry. In the neutrino sector we propose a simple modification of the so called Zee-Wolfenstein mass matrix pattern which allows an acceptable reactor angle along with a deviation of the atmospheric and solar angles from their bi-maximal values. Quark masses, mixing angles and CP violation are well described by a numerical fit.


I. INTRODUCTION
Supersymmetric Grand Unified Theories (SUSY GUTs) are very attractive from the theoretical point of view as they allow to obtain the SM from a single unified gauge group [1,2]. Apart from predicting for instance the quantization of electric charge, they reduce the number of free parameters. For example, they give the right value for the electroweak mixing angle and may provide a good framework for the understanding of the flavour problem. Indeed, several GUT models have been studied in the literature, having as prediction a mass relation between down-quark masses and the charged leptons. For instance in the SU (5) unified framework Georgi and Jarlskog have found the mass relation [3] which is in good agreement with data to first approximation, assuming that holds at the GUT scale, and taking into account renormalization group running to low energies, with suitable SUSY threshold effects. Such mass relations are very welcome since, by itself, the Standard Model sheds no light on the flavour problem. However, the Large Hadron Collider has so far not found any evidence of physics beyond the Standard Model (SM). Indeed, the only major discovery to date has been that of a new boson which is entirely consistent with the properties of the SM Higgs boson, arising from a single Higgs doublet H. In particular the LHC has not so far found any evidence for Supersymmetry (SUSY) as indicated within the simplest Grand Unified Theories (GUTs), namely those which do not involve an intermediate scale such as SU (5).
Here we advocate an alternative TeV-scale approach to the flavour problem employing just the Standard Model gauge symmetry, supplemented only by a non-Abelian discrete flavour symmetry. For the latter we adopt A 4 , the discrete group of even permutations of four objects isomorphic to the group of symmetries of the tetrahedron. It is the smallest group containing triplet irreducible representations. Several A 4 -based flavour models have been suggested [4][5][6], for reviews see Refs. [7][8][9]. Recently three of us have proposed an SU(3) ⊗ SU(2) L ⊗ U(1) model [10] based on the discrete family symmetry A 4 leading to the quark-lepton mass relation: It is clear that Eq. (2) provides an interesting generalization of Eq. (1) which is found to be in very good agreement with data. Given that it is approximately renormalization group invariant, it holds at all mass scales [10]. In contrast arXiv:1301.7065v1 [hep-ph] 29 Jan 2013 to the Georgi-Jarlskog relation, Eq. (2) arises just from the flavour structure of the model, and the fact from the existence of two Higgs doublets selectively coupled to the up-and down-type fermions 1 . Both of these were assigned to be A 4 triplets, with one of them coupling to the down-type quarks and charged leptons. Note that the model in [10] employed an extended Higgs sector with three families of supersymmetric Higgs doublets H u and H d for which there is presently no evidence, with present data being consistent with a single Higgs doublet. Moreover, the model predicted V ub = 0 = V cb . While providing a good starting point for the CKM matrix, a derivation of the full quark mixing was lacking.
In the present paper we propose an alternative discrete family symmetry A 4 model, keeping the same motivation for introducing A 4 into the Standard Model [10], namely, to shed light on the flavour problem. Indeed the model presented here provides a fully realistic description of all quark and lepton masses and mixing angles, and in particular reproduces the successful quark-lepton mass relation in Eq. (2). In contrast to the previous construction in [10] its remains closer in spirit to the SM, since we do not assume supersymmetry nor unification, keeping a single Higgs doublet H instead of multi-Higgs doublets. In particular, we assign right-handed up quarks to singlet representation of A 4 instead of triplet as in the original model. The A 4 flavour symmetry is broken by SM singlet flavons which distinguish up-type quarks from down-type quarks and charged leptons, with additional flavons in the neutrino sector, where we require extra Abelian discrete groups to distinguish these sectors. Assuming full explicit breaking of A 4 through suitable scalar flavon multiplets we show that this simple modification of the model in [10] can describe all the CKM mixing parameters. Moreover, it is straightforward to obtain also the so called Zee-Wolfenstein mass matrix pattern in the neutrino sector using A 4 invariance [11]. However, since it predicts bi-maximal mixing, current neutrino oscillation data analysis [12] rule out such a pattern [13]. We propose a simple modification of the Zee-Wolfenstein model where all the mixing angle, as well as the reactor angle can be reproduced.
In the next section we introduce our model, in section III we obtain our quark-lepton mass relation, in section IV we give the fit for the quark mixing parameters, in section V we describe the neutrino mass generation mechanism and study its phenomenological implications, while in section VI we give our conclusions.

II. THE MODEL
The matter content and the flavour group assignment are given in Table I. Note that all the fermions, apart of the u R fields, are assigned to triplets of A 4 2 . In the scalar sector we have one SM Higgs doublet and four flavon fields. With respect to the model of Ref. [10] we have extra Abelian symmetries, namely Z u 2 , Z d 2 and Z 3 . The reason for imposing such a symmetries is because our present model is not supersymmetric. We are replacing the SUSY-Higgs doublets H u and H d (triplet of A 4 in Ref. [10]) with scalar SU L (2)-singlets (flavons) triplets of A 4 times the standard model Higgs doublet, namely It is clear that the Z u 2 and Z d 2 symmetries glue the ϕ u and ϕ d flavons fields to the up and down quark sectors, respectively, while the extra Z ν 3 symmetry is used to separate the charged and neutral fermion sectors. Table I: Matter contents of the model. 1 Such structure is required in supersymmetric models, but the mechanism proposed in Ref. [10] leading to Eq. (2) is more general, relying only of the two-doublet nature of the Higgs sector, as mentioned above. Here we abandon the use of supersymmetry. 2 Therefore the present model can not be embedded in any grand-unified framework.
The Lagrangian for quarks and charged leptons in our model is given by where α = 3 1 , 3 2 are the two triplet contractions of 3 × 3 = 1 + 1 + 1 + 3 + 3, which are the symmetric and the antisymmetric ones, while β = 1, 1 , 1 . Note that, while the A 4 flavour symmetry holds in the (non-renormalizable) Yukawa terms leading to charged fermion masses, we assume it to be completely broken in the scalar potential. Indeed, we assume that the scalar flavon multiplets get vacuum expectation values (vevs) in an arbitrary direction of A 4 , preserving none of its subgroups. This can be easily achieved by only including terms in the scalar potential which are SO(3) invariant as discussed in [14]. In this case the flavon scalar fields get vevs in arbitrary directions of A 4 , that is where To complete the model we need also to specify the mechanism of neutrino mass generation, see Sec. V, below.

III. THE CHARGED LEPTON-QUARK MASS RELATION
From the A 4 contraction rules (see appendix A) and the fact that the charged leptons and down-type quarks are in the same A 4 representations, one sees that the charged leptons and down-type quark mass matrices have the form where f = d, l. This special form is the same as obtained in Ref. [10,15]. With the redefinition of variables: the mass matrix for the mass matrix in Eq. (6) takes the form Let us now consider the system given by the following three invariants ( where S f = M f M † f . This system can be solved and we find in the limit r f α f , 1 and r f b f /a f . These equations are general in the sense that in the complex case, namely complex Yukawa couplings and vevs, the invariants, Eqs. (9)-(11) do not depend on the phases of the vevs v ui . Indeed, the only dependence on the relative phase of the Yukawa couplings, y f 1 and y 2 2 enters in the determinant, Eq. (9), and this is negligible in the above limit. From Eqs. (12), (13), (14), one finds simple relations for the second and third family masses, namely, from which we require a f b f in order to account for the second and third family mass hierarchy. Moreover, since the charged leptons and down-type quarks couple to the same Higgs and flavons, we have 3 r l = r d and α l = α d , so that, from Eq. (12), we obtain the mass relation [10] Now we turn to the up-type quark sector. From the Lagrangian in Eq. (4), the up quark mass matrix is given by In what follows we discuss the resulting structure of the quark mixing matrix.

IV. QUARK MIXING: THE CKM MATRIX
Recall that the down-type quark mass matrix takes the form in Eq. (6) while the up-type quark mass matrix has just been given in Eq. (18). Out of these matrices one finds the matrices M u · M u † and M d · M d † . Their diagonalization results in two unitary matrices V u,d for which one can obtain approximate analytical expressions. In the down sector, from M d · M d † , one finds, One sees that if α d ∼ O(1) the down sector gives about the Cabibbo angle in the 1 − 2 plane while the mixings in the 1 − 3 and 2 − 3 planes are negligible. On the other hand from the up quark sector one finds, approximately with coefficients of order one in front of the (i, j) if at least one of the Yukawa couplings is of order one 4 . Thus for the up quark matrix mixing factor V u we have that where we have assumed The overall quark mixing matrix is given by the product One sees how the Cabibbo angle arises from the down-type quark matrix mixing factor V d , while the V ub and V cb CKM mixing angles arise from the up quark matrix mixing factor V u . Taking λ ≈ 0.2 we obtain approximately the correct value for the mixing angle. However the order one parameters are relevant in order to exactly determinate the quark mixing angles. In order to obtain quantitative predictions for these we perform a global numerical fit. The experimental data used and the one σ error bars are given in the second column, taking the quark masses (at the scale of the M Z ) from [16] and the quark mixing angles from [17]. The third column displays the values predicted by our model when the values of its parameters are those in equations (27). The fourth column shows the number of standard deviations from the mean values, P i −Ō i /σ i , computed using the data from the second column. The value χ 2 = 0.39 is the sum of the squares of the numbers in the fourth column.
We note that the phases of the up couplings y u i can be reabsorbed by transforming the right-handed fields u Ri while in the down sector not all the phases can be removed. For simplicity we assume all the couplings to be real and we show how to make a fit of the quark mixing parameters (including the complex phase) and masses. We note that by taking ω to be the only phase in our parametrization one can fit for the CKM phase, namely the Jarlskog invariant 5 . In table II we compare our theoretical predictions with the current experimental values for the quark masses and CKM mixing parameters. The theoretical predictions are for the values: where λ = 0.2.

V. NEUTRINO MASSES AND MIXING
As in Ref. [10] here we consider an effective way to generate neutrino masses byà la Weinberg by upgrading the standard dimension-five operator to the flavon case, making it dimension-six, that is 6  Table II: Experimental and predicted quark masses and mixing parameters from our fit. Quark masses (at the scale of the MZ ) have been taken from [16], while quark mixing angles have been taken from [17]. The third column displays our predicted values from Eqs. (27). The fourth column shows the number of standard deviations from the mean values, Pi −Ōi /σi, computed using the data from the second column. The value χ 2 = 0.39 is the sum of the squares of the numbers in the fourth column.
After electroweak symmetry breaking it gives to the following Majorana neutrino mass matrix Note that, unlike the charged fermion case, only the symmetric contractions are allowed from the first operator in Eq. (28). We remark that these parameters in the neutrino sector are unrelated to those in the charged fermion sector.
Taking the limit where d = 0 the neutrino mass matrix has the well known Zee-Wolfenstein structure, which cannot reproduce the current neutrino oscillation data [12], since it gives to Bi-maximal mixing [13]. The addition of the unit matrix contribution proportional to d introduces deviations from maximal atmospheric mixing proportional to b and also introduces a non-zero reactor angle θ 13 ∼ (a − b)/(2d), while the solar angle is approximately given by tan 2θ 12 ∼ 2(a + b)/d, which reduces to maximal solar mixing in the Zee-Wolfenstein limit d → 0. One finds a strict correlation between the neutrinoless double beta decay rate and the magnitude of the parameter d. In fact, as seen in Fig. 1 we find a (weak) lower bound for the neutrinoless double beta decay rate, despite the fact that the model has a normal hierarchy neutrino spectrum, this follows from the presence of the flavour symmetry 7 . In our numerical scan we also obtain a restricted set of neutrino oscillation angles. For example the curved-shaped region in the left panel of Fig. 2 (orange in color version) corresponds to the "predicted" atmospheric angle consistent with the currently allowed values of the solar angle at 3 σ, from Ref. [12]. For comparison we display also the 1 σ bands for sin 2 θ 23 and sin 2 θ 13 . In the right panel in Fig. 2 we re-express our sin 2 θ 23 "prediction" in terms of the lightest neutrino mass m 1 , again keeping undisplayed oscillation parameters at 3 σ. The existence of the above restrictions reflects the fact that, as a result of the flavour symmetry, we have in total less parameters than observables to describe.

VI. CONCLUSIONS
We have proposed a realistic A 4 extension of the Standard Model leading to the quark-lepton mass relation given in Eq. (2). This successful and nearly renormalization invariant mass relation generalizes the Georgi-Jarlskog formula . Straight bands are the currently allowed 1 σ bands of the oscillation angles, taken from Ref. [12].
and arises outside the context of unification. Quark masses, mixing angles and CP violation are properly accounted for, while in the neutrino sector we obtain a generalized Zee-Wolfenstein mass matrix giving an acceptable reactor angle along with a deviation of the atmospheric and solar angles from their bi-maximal values. As seen in Fig. 1 the neutrinoless double beta decay rate correlates sharply with this deviation parameter, with a minimum allowed value despite the fact that the model has a normal hierarchy neutrino mass spectrum. Moreover we find that the atmospheric angle correlates with the lightest neutrino mass (right panel in Fig. 2) and with the reactor angle (left panel). From the theory point of view the model treats all fermion masses effectively, as arising from non-renormalizable Yukawa-like terms.