A4 family symmetry and quark-lepton unification

We present a model of quark and lepton masses and mixings based on A4 family symmetry, a discrete subgroup of an SO(3) flavour symmetry, together with Pati-Salam unification. It accommodates tri-bimaximal neutrino mixing via constrained sequential dominance with a particularly simple vacuum alignment mechanism emerging through the effective D-term contributions to the scalar potential.


Introduction
There has recently been considerable interest in the use of the discrete group A 4 as a family symmetry .A particularly attractive feature of A 4 is the possibility of obtaining non-trivial vacuum alignment in a simpler way than for continuos family symmetries [18][19][20].Such non-trivial vacuum alignments are of interest since they can lead to tribimaximal neutrino mixing [26].In particular, in the framework of the see-saw mechanism with sequential dominance (SD) [27][28][29][30], such non-trivial vacuum alignment can lead to constrained sequential dominance (CSD) [31] in which tri-bimaximal neutrino mixing arises from simple relations between Yukawa couplings involving the dominant and leading subdominant right-handed neutrinos.
Despite the great interest in A 4 as a family symmetry, there does not yet exist in the literature a model in which quarks and leptons are unified.Part of the reason for this is that the left and right handed chiral components of the quarks and leptons are usually required to transform differently under the A 4 family symmetry [1-3, 7, 9, 15-22].If both helicity components transform in the same way then the A 4 family symmetry does not prevent trivial invariant operators which give a mass matrix contribution proportional to the unit matrix [32], rather than the desired hierarchical form.The situation is rather similar to the case of SO(3) family symmetry since A 4 may be regarded as a discrete subgroup of SO(3).In the case of SO(3) the solution to this problem is to accept the left-right asymmetry, and to construct partially unified models based on Pati-Salam gauge group [31].Such models can in principle be embedded directly into string theory, and may be consistent with SO (10) in a 5D framework [33], without the need for an explicit 4D SO (10) GUT, which in any case suffers from the doublet-triplet splitting problem.However, to best of our knowledge, no such Pati-Salam unified model with A 4 family symmetry exists in the literature.
In this paper we present a realistic model of quark and lepton masses and mixings based on A 4 family symmetry and Pati-Salam unification.The model goes along the lines of the SO(3) and Pati-Salam model discussed in detail in [33], and shares many of the desirable features of that model, in particular the flavons entered at the lowest possible order, which allowed the messenger sector to be explicitly specified.Also, as in [33], tribimaximal neutrino mixing emerges from the see-saw mechanism with CSD arising from vacuum alignment.However, whereas the vacuum alignment in SO(3) [31], assumed in [33], was rather involved, here, with the discrete subgroup A 4 , it will become remarkably simple.Here we will use the discrete radiative vacuum alignment mechanism proposed in [34] for the ∆(27) discrete symmetry model, based on discrete D-terms rather than the F-term mechanism discussed in [20] for discrete subgroups of SO(3) and SU (3).In fact the A 4 model presented here as a discrete version of the SO(3) models discussed in [31,33], mirrors the ∆(27) model discussed in [34] which is a discrete version of the SU (3) models discussed in [35][36][37].

The model
The model is based on a high-energy Pati-Salam SU (4) C ⊗ SU (2) L ⊗ SU (2) R supersymmetric model with Yukawa sector driven by a discrete subgroup of SO(3), the A 4 flavour symmetry and a pair of extra symmetry factors U (1) ⊗ Z 2 to forbid some unwanted operators.The construction goes along similar lines as in the case of a fully SO(3) invariant model studied in [33].However, sticking to a discrete subgroup of a Lie-group brings in several qualitative changes that require a separate treatment.In particular, it provides for a very effective tool to address the vacuum alignment issues that often make the SUSY models based on continuous flavour symmetries rather cumbersome due a proliferation of extra degrees of freedom.

The field content and symmetry breaking
The full set of the effective theory matter, Higgs and flavon fields and their transformation properties are given in Table 1.We embed the left-handed Standard Model matter fields into a triplet of A 4 while keeping the right-handed matter transform as the SO(3)-like A 4 singlet1 .Apart of the pair of MSSM light Higgs doublets h (arranged into the traditional Pati-Salam bidoublet) driving the electroweak symmetry breakdown we use a pair of heavy Higgs bosons H and H ′ to break the Pati-Salam gauge symmetry at a high scale and provide the Majorana mass terms for the right-handed neutrinos.

The Yukawa sector
In what follows we shall use upper indices for A 4 triplet components while the lower indices stand for the various species of structures in the game.The symmetries defined above allow field The basic Higgs, matter and flavon content of the model.
for the following contributions to the Yukawa superpotential: where x × y is the standard SO(3) cross-product, (x * y) i = s ijk x j y k (with s ijk being +1 for each permutation of {i, j, k} ∈ {1, 2, 3}) corresponds to the extra (symmetric) vector product in A 4 while I i (x, y, u, v) denotes the available independent quartic A 4 invariants, as discussed in Appendix A. Note that M is a generic symbol for the mass of the relevant messenger sector fields giving rise to the desired effective vertices in eq.(2.1).For sake of conciseness, we shall not discuss the messenger sector here and defer an interested reader to the study [33] for an example of such analysis.
After the spontaneous breakdown of the flavour symmetry (for details see Section 2.5 and Table 2) the Yukawa matrices generated from this superpotential piece read: like (15, 1, 3) under the Pati-Salam symmetry) responsible for the distinct charged sector hierarchies à la Georgi and Jarlskog [38] and σ denotes the VEV of the Georgi-Jarlskog field σ ≡ Σ /M f .The effective couplings y 23 and y 13 stem from the multiple contributions to the 13 and 23 elements of Y f LR due to the higher number of relevant cubic and quartic A 4 invariants.

The Majorana sector
The Majorana mass matrix is obtained from the superpotential of the form where as before I i stand for the various A 4 quartic invariants and the ellipsis denotes the higher order terms.It is easy to verify that the Majorana mass matrix emerging from here reads where only the relevant terms are displayed because the mixing in the right-handed neutrino sector due to the off-diagonal terms is negligible.

The generic results
In order to achieve a good fit to all the quark and lepton masses and mixing parameters one has to assume a hierarchy among the flavon VEV parameters ε f x .Since the relevant VEV scales emerge from a radiative symmetry breaking mechanism, as discussed in Section 2.5, it is completely natural to expect a certain hierarchy among them that in turn propagates to the order of magnitude differences in ε f x .The only extra assumption concerns the magnitude of the VEV of H ′ entering the Majorana sector analysis H ′ ≡ δ H H with δ H ≪ 1.However, a similar radiative mechanism like in the flavon case can play a role here thus making such an assumption as natural as the previous ones.
As it was shown previously in the context of an SO(3) model [33], the structures under consideration lead to a good fit of all the quark and lepton mass and mixing data provided • The naturalness of the hierarchy among the third and second generation Yukawa couplings as well as a moderate suppression of the V cb CKM mixing parameter are traced back to the higher-order origin of the relevant (Georgi-Jarlskog and 2-3 entry) operators.
• The first generation masses as well as the smallness of the V ub CKM mixing descend from the hierarchy of the relevant flavon VEVs as discussed in the next section.
• The neutrino sector conforms to the CSD conditions [31].The particular structure of the neutrino Yukawa matrix together with the hierarchy of the charged lepton Yukawa couplings leads to approximate tri-bimaximal mixing in the neutrino sector [26] characterized by the approximately maximal atmospheric mixing tan θ 23 ≈ 1, large solar mixing angle obeying sin θ 12 ≈ 1/ √ 3 and a small reactor angle θ 13 ≈ 0, in good agreement with the latest neutrino data, see e.g.[39,40] and references therein.
• Concerning the light neutrino mass spectrum, the large hierarchy in Y ν LR is effectively undone in the seesaw formula by the particular form of the Majorana mass matrix (2.3).
Thus, the model provides a very good description of all the known quark and lepton masses and mixing parameters.The only missing ingredient is the mechanism leading to the desired correlations among the VEVs of the various triplet flavon components shown in Table 2.

The vacuum alignment mechanism
The discrete nature of the flavour symmetry leads to a particularly simple option to achieve all the desired vacuum structures displayed in Table 2.As discussed in [34], in such a class of models the supergravity (SUGRA) induced D-terms can naturally lead to a set of extra quartic terms in the effective scalar potential.Such a set of terms, however, lead to a lift of the would-be degenerate vacua potentially emerging in a continuous case and thus makes the vacuum alignment mechanism straightforward.To force the system to depart from the symmetric state we shall assume a variant of a radiative symmetry breaking mechanism, as we now discuss.
Let us first consider the case of a single triplet φ.Apart from the obvious SO(3) invariant (φ † φ) 2 the discrete A 4 symmetry admits for instance a contraction like that breaks the rotational degeneracy of the would-be SO(3) symmetric vacua.Assuming that the scalar potential is governed by the terms4 it is easy to verify that the only vacuum structures that can arise in such a case (i.e. when all the mixing terms are negligible) are |φ| ∝ (1, 1, 1) and/or |φ| ∝ (1, 0, 0), (0, 1, 0), (0, 0, 1) ( where only the magnitudes of the components are so far specified.What matters is the sign of the SO(3)-breaking term λI 0 (φ † , φ, φ † , φ): if λ > 0 the "isotropic" option |φ| ∝ (1, 1, 1) is picked up while the VEV is maximally "anisotropic" (i.e. with just one nonzero entry in φ ) if λ < 0. Let us stress that the configurations (2.6) correspond to the case of an entirely hermitian field φ.Since both the I 0 and I 1 invariants (c.f.Appendix A) dominating the scalar potential (2.5) are phase-blind, the current mechanism does not specify the phase of any of the triplet components if φ = φ † .However, as it was shown in [31], what matters in achieving tri-bimaximal mixing via CSD is not the absolute phases of the flavon components but the equality of their magnitudes, and their complex orthogonality, which we shall shortly discuss.All this leads us to the following realization of the vacuum alignment mechanism: suppose each of the fields φ 123 , φ 1 and φ 3 has a potential of the form in Eq.(2.5), simply repeated for each field.Suppose each of the fields develop negative mass-squares through radiative effects around scales M 123 , M 1 and M 3 and let us arrange the "λ-terms" in the leading piece of the scalar potential in Eq.(2.5) so that they pick up VEVs in the directions allowed by eq.(2.6), in particular 5 : The stability of such a setup requires that the mixing arising from the "inhomogeneous" terms like 6 I i (φ † 123 , φ 123 , φ † 1,3 , φ 1,3 ) , i ∈ {0, 1, 3} should be suppressed with respect to the "pure" ones I i (φ † 123 , φ 123 , φ † 123 , φ 123 ) and ). Subsequently, φ23 and φ 23 can be generated if the interactions with the first stage fields φ 123 and φ 1,3 are dominated by the terms .8) where the ellipsis stands for SO(3) (and thus also A 4 ) invariant terms of the form Λ φ (φ † .φ) 2 necessary to lift the flat directions.If λ 123 and λ123 are positive, the VEVs of φ 23 and φ23 driven to the directions orthogonal to φ 123 while λ 1 , λ1 > 0 make their first component vanish and thus |φ 23 | , | φ23 | ∝ (0, 1, 1).Concerning the above mentioned ambiguity in fixing the phases of the vacuum alignment emerging from the simple potential (2.5), 5 The alignment of |φ1| ∝ (1, 0, 0) and |φ3| ∝ (0, 0, 1) is a just a choice of basis that we are free to make as long as there are no interactions binding the VEVs of φ1 and φ3 together.On the other hand, to make sure φ1 does not coincide with φ3 spontaneously a mixing term like |φ † 1 .φ3| 2 with a positive coefficient can be exploited. 6Here we again suppress all the triplet indices so that the generic symbols Ii(φ † A , φB, φ † C , φD) account for the various linearly independent A4 contractions.There are only 4 such independent structures for A = C and B = D, three if A = B = C = D and 2 if on top of that φ = φ † (i.e.only if φ is strictly neutral), for more details see Appendix A.
flavon VEV VEV direction VEV normalization (scale) Table 2: The vacuum alignment pattern generated by the mechanism specified in the text.The mass scales M i and the relevant quartic couplings λ i and Λ i are defined in Section 2.5, Eqs.(2.5) and (2.8).In the "VEV direction" column only the magnitudes of the relevant (in general complex) flavon VEVs are displayed.The minus sign in the case of φ 23 and φ23 illustrates the important π-difference of the 2nd and 3rd component VEV phases of φ 23 and φ 123 .
in particular φ 123 , the orthogonality condition φ 123 † .φ 23 = 0 together with φ 1 23 = 0 following from the minimisation of (2.8) is just enough to generate θ ν 13 close to zero [30] and tan θ ν 12 ∼ 1/ √ 2 regardless any particular arrangement of the φ 123 phases.The minus signs in Table 2 are for illustrative purposes and simply denote the π-shift in the relative phases of the components of φ 123 and φ 23 (or φ23 ) arising from the relevant orthogonality conditions.
At this point it is perhaps worth mentioning that the positivity of the λ-couplings above also ensures a better control over the magnitudes of the corresponding VEVs unlike the case of having an interaction with a negative coupling constant when a potentially large negative correction must be compensated by the explicit mass entry from the F-terms.This means that one can handle easily all the relevant scales without a need of an extra tuning of parameters in the would-be "effective wrong-sign masses" that might otherwise arise.
Concerning the alignment of φ 2 , a particular shape of its VEV is immaterial as long as it admits a nonzero projection to the second SO(3) coordinate.A particularly elegant setup can be obtained if for instance |φ 2 | ∝ (0, 1, 0) is generated via the same mechanism like φ 1,3 , repeating just as before the form of potential (2.5) with λ 2 > 0. To make sure the (0, 1, 0) option is picked up one can employ the orthogonality of all the φ 1 , φ 2 and φ 3 VEVs via the mixing terms of the form |φ † i .φj | 2 with positive coefficients so that the complete basis of the triplet space is spanned.
The results of our vacuum alignment mechanism are sumarized in Table 2.It is easy to see that all the mass scales in Table 2 are essentially free: since the potential (2.8) is fully SO(3) invariant, the anisotropy enters only through the A 4 terms driving the VEVs of φ 123 and φ 1,2,3 while φ23 , φ 23 can rotate freely to follow the constraints imposed through the interactions with φ 123,1 .Thus, even a small push in any particular direction is enough to imprint the desired alignment to all the relevant VEVs and we are free to choose M 23 and M 123 so that the desired VEV hierarchy is achieved.

Conclusions
We have constructed the first complete model of flavour based on A 4 family symmetry together with the SU (4) C ⊗SU (2) L ⊗SU (2) R Pati-Salam gauge symmetry.A 4 corresponds to the symmetry of the tetrahedron, and is a discrete subgroup of SO (3).Assuming the simple extra symmetry factors U (1) ⊗ Z 2 , we have performed an operator analysis of the model, and shown that the resulting effective Yukawa and Majorana couplings have a similar form to those discussed in [33], and when the messenger sector is completed, the resulting structures provide a good description of the fermion mass and mixing spectrum.In particular, the constrained sequential dominance is realized and tri-bimaximal neutrino mixing results, with calculable deviations expressed in terms of neutrino sum rules [31,41].The main simplification afforded by the discrete symmetry is in the vacuum alignment sector.Due to the discrete nature of the flavour symmetry a particularly simple vacuum mechanism emerges through the SUGRA induced D-term contributions to the effective scalar potential that lift the SO(3) vacuum degeneracy.We have shown that this discrete version of the radiative symmetry breaking mechanism may be achieved with a minimal number of fields that do not participate directly in the Yukawa sector, and that a realistic model can be constructed which incorporates all these features simultaneously.The A 4 model presented here may be regarded as being on the same footing as the ∆( 27) model presented in [34] At the quartic level one can easily check that the basic structures are invariant with respect to the action (A.1).However, this is not the end of the story yet as one must consider also the other permutations of the set of parameters {x, y, u, v}.
The symmetries of I 0,1,2 are such that all these expressions are actually invariant with respect to permutations of the first and second pair of arguments (and in case of I 0 even all of them), and thus what matters is just the pairings of {x, y, u, v}.In short, the independent structures emerging from eq. (A.2) correspond to I 0 , I 1 and I 2 with arguments (x, y; u, v), (x, u; y, v) and (x, v; y, u) only.Due to the maximal symmetry of I 0 one gets only 4 additional relevant structures from I 1 and I 2 , namely To demonstrate the completeness of such a "naively" constructed set of invariants it is sufficient to find a mapping of I 0,..,6 onto the set of "group-theoretical" purely triplet quartic invariants of the form (x.y)(u.v)= (3 ⊗ 3) 1 ⊗ (3 ⊗ 3) 1 , (x, y) 1 ′ (u, v) It is then easy to see that whenever x = u and y = v only 4 of these structures remain independent (for instance I 0 , I 3 , I 4 and one from the equal I 1,2,5,6 ).If on top of that y = x † then I 3 = I † 4 that allows for only three independent terms in a hermitean scalar potential and, finally, if all the arguments coincide there is only 2 such terms like for instance I 0 and one from I 1,..,6 left.