Predictive Pati-Salam theory of fermion masses and mixing

We propose a Pati-Salam extension of the standard model incorporating a flavor symmetry based on the $\Delta \left( 27\right)$ group. The theory realizes a realistic Froggatt-Nielsen picture of quark mixing and a predictive pattern of neutrino oscillations. We find that, for normal neutrino mass ordering, the atmospheric angle must lie in the higher octant, CP must be violated in oscillations, and there is a lower bound for the $0\nu\beta\beta$ decay rate. For the case of inverted mass ordering, we find that the lower atmospheric octant is preferred, and that CP can be conserved in oscillations. Neutrino masses arise from a low-scale seesaw mechanism, whose messengers can be produced by a $Z^{\prime }$ portal at the LHC.


Introduction
Apart from the discovery of neutrino oscillations [1,2], no other laboratory evidence for physics beyond the standard model has been so far unambiguously confirmed. Both the origin of neutrino mass itself, as well as the understanding of the mixing pattern, require an explanation from first principles. Moreover, there is a variety of other motivations for having beyond the standard model physics [3]. One of these is the pursuit of a dynamical explanation for the origin of parity violation in the weak interaction, whose basic V-A nature is put in by hand in the formulation of the standard model. With this in mind here we propose a flavored [4][5][6] Pati-Salam [7] extension of the standard model, addressing both the dynamical origin of the V-A nature of the weak force, as well as the related origin of neutrino mass. In addition, as we will see, the model can shed light upon the flavor problem and make predictions. The main features of our model include: • adequate implementation of ∆ (27) flavor symmetry in the Pati-Salam framework and symmetry breaking • consistent low-scale left-right symmetric seesaw mechanism for neutrinos [8][9][10][11][12] • predictive pattern of neutrino mixing summarized in Figs. 1 and 2 • realistic pattern of quark mixing, yielding a Froggatt-Nielsen [13]-like picture of the CKM matrix • lower bound for the 0νββ decay rate in Fig. 3 We note also that the model has a low-scale Z portal through which the TeV scale messengers S a can be pair-produced in Drell-Yan collisions at the LHC [14][15][16]. In addition, our model realizes a universal seesaw mechanism [17] for the down type quarks as well as the charged leptons, mediated by TeV scale exotic fermions. The latter should potentially lead to other phenomenological effects in the quark sector as well as lepton flavour violation effects.

The model
The where the subscript refers to fermion families. More explicitly, the standard model fermions are written in component form as Notice that we have extended the fermion sector of the original Pati-Salam model [7] by introducing three fermion singlets S i , in order to implement inverse and/or linear seesaw mechanisms for the generation of light active neutrino masses [8][9][10][11][12]. In addition, we have introduced vector-like fermions Ψ iL and Ψ iR so as to generate the standard model down-type quark and charged lepton masses, from a universal seesaw mechanism.
The particle content and gauge symmetry assignments are summarized in table 1.  Field and the scalars χ L,R , Φ k and Σ develop vacuum expectation values (VEVs) of the form where we have set These flavor symmetry scalar transformation properties are summarized in table 3.
With the above particle content and transformation properties the following relevant Yukawa terms arise: where r, s are SU (4) indices and y i , α i , β i , κ i , a i (i = 1, 2, 3), γ m (m = 1, 2, · · · , 6) and λ j (j = 1, 2) are O(1) dimensionless couplings. For an explanation of the ∆(27) notation used in the α, β and λ-terms, see Appendix A. It is noteworthy that the lightest of the physical neutral scalar states of (Φ j ) 11 , (Φ j ) 22 , φ i should be interpreted as the SM-like 125 GeV Higgs recently found at the LHC. Furthermore, our model at low energies corresponds to a seven Higgs doublet model with five scalar singlets (these scalar singlets come from ϕ l ). As we will show in section 4, the top quark mass only arises from (Φ 1 ) 11 . Consequently, the dominant contribution to the SM-like 125 GeV Higgs mainly arises from (Φ 1 ) 11 . We note also that the scalar potential of our model has many free parameters, so that we can assume the remaining scalars to be heavy and outside the LHC reach. Moreover, one can suppress the loop effects of the heavy scalars contributing to precision observables, by making an appropriate choice of the free parameters in the scalar potential. These adjustments do not affect the physical observables in the quark and lepton sectors, which are determined mainly by the Yukawa couplings.
The full symmetry group G exhibits the following spontaneous breaking pattern: Here v = 246 GeV is the electroweak symmetry breaking scale, and we assume that the Pati-Salam gauge symmetry is broken at the scale Λ P S ∼ ∼ > 10 6 GeV. This restriction follows from the experimental limit on the branching ratio for the rare neutral meson decays, such as B 0 → l ± i l ∓ j , mediated by the vector leptoquarks, as discussed in Refs. [18,19]. Furthermore, it is worth mentioning that Pati-Salam models with a quark-lepton unification scale of about ∼ > 10 6 GeV can fulfill gauge coupling unification [20]. A comprehensive study of gauge coupling unification in models that include all possible chains of Pati-Salam symmetry breaking in both supersymmetric and non-supersymmetric scenarios has been given in Ref. [20].

Understanding the model setup
In this section we try to motivate in more detail our choice for the model content and the transformation properties. First note that the Pati-Salam gauge symmetry SU (4) C ⊗ SU (2) L ⊗SU (2) R breaks down to the conventional left-right symmetry SU (3) C ⊗SU (2) L ⊗ SU (2) R ⊗ U (1) B−L by the VEV of the scalar field Σ, at the scale Λ P S ∼ > 10 6 GeV. The next symmetry breaking step is triggered by χ R , whose VEV is assumed to be in the few TeV scale, playing an important role in implementing the low-scale seesaw neutrino mass generation [9,10]. The breaking of the electroweak gauge group SU (3) C ⊗ SU (2) L ⊗ U (1) Y is triggered by the scalar fields Φ j , which acquire vacuum expectation values at the electroweak symmetry breaking scale v = 246 GeV.
Besides, note that the presence of the scalar field Σ transforming as the adjoint representation of SU (4) is also crucial in the implementation of the Universal Seesaw mechanism for down-type quarks and charged leptons, mediated by exotic fermions. This scalar field Σ acquires a VEV at the scale Λ P S , so that an insertion σ 6 Λ 6 in its corresponding Yukawa term generates a TeV scale contribution to the exotic charged lepton and exotic down-type quark masses. The charge of Σ under the Z 16 discrete group ensures that its different contributions to the charged leptons and down type quark masses will be comparable to the ones arising from the A i Ψ iL Ψ iR (i = 1, 2, 3) mass terms (which contribute equally to the down type quark and charged lepton masses). It is worth mentioning that we are assuming A i ≈ O(1) TeV. Let us note that the inclusion of the scalar field Σ is necessary to guarantee that the resulting down-type quark masses are different from the charged lepton masses, as it will be shown in Section 4.
Notice that the scalars φ i and ϕ l are needed to generate the mixing between the standard model charged leptons and down-type quarks with their exotic siblings, so as to implement the Universal Seesaw mechanism that gives rise to realistic masses for the standard model charged fermions. The scalar fields χ R , χ L and ξ i have Yukawa terms necessary for the implementation of the inverse and linear seesaw mechanisms, so as to generate the light active neutrino masses. This requires also that VEVs of ξ i are much smaller than the electroweak symmetry breaking scale.
The scalar fields ρ i , η i and τ i are needed to generate the diagonal 3 × 3 blocks that include the mixing of the neutrino states contained in F iL and F iR with the singlet neutrinos S i (i = 1, 2, 3), thus avoiding the transmission of the strong hierarchy in the up mass matrix to the light active neutrino mass matrix. Furthermore, the scalar field σ, charged under the Z 16 discrete group is need to generate the observed SM charged fermion mass and quark mixing hierarchy. In order to relate the quark masses with the quark mixing parameters, we assume that the scalar field σ acquires a VEV equal to λΛ, where λ = 0.225 is one of the Wolfenstein parameters and Λ is the cutoff of our model. In summary, the set of VEVs of the scalar fields is assumed to satisfy the following hierarchy: where χ L,R = v L,R . We now comment on the possible VEV patterns for the ∆(27) scalar triplets ρ, η, τ and ξ. Since the VEVs of the ∆(27) scalar triplets satisfy the hierarchy v ξ v ρ = v η = v τ ∼ Λ P S , the mixing angles of ξ with ρ, η and τ are very tiny since they are suppressed by the ratios of their VEVs, and consequently the method of recursive expansion proposed in Ref. [21] can be used for the analysis of the model scalar potential. In this scenario, the scalar potential for the ∆(27) scalar triplet ξ can be treated independently from the scalar potential for the ∆(27) scalar triplets ρ, η, τ , as shown in detail in the appendices B and C. One can see that the following VEV patterns for the ∆(27) scalar triplets are consistent with the scalar potential minimization equations for a large region of parameter space We now turn our attention on the role of each discrete group factor of our model. As will be seen in Sects. 4 and 5 the ∆ (27) discrete group is crucial for the predictivity of our model, giving rise to viable textures for the fermion masses and mixings. Notice that the ∆(27) discrete group is a non trivial group of the type ∆(3n 2 ) for n = 3, isomorphic to the [4]. Recently, this group has been used in multi-Higgs doublet models [22], SO(10) models [23], warped extra dimensional models [24] and [25,26]. We introduce the Z 16 discrete group, since it is the smallest cyclic symmetry that allows the Yukawa operator. This leads to the required λ 8 suppression needed to naturally explain the smallness of the up quark mass. The Z 16 group has been recently shown to be useful for explaining the observed SM charged fermion mass and quark mixing hierarchy, in the framework of a SU (3) C ⊗ SU (3) L ⊗ U (1) X models based on the A 4 and S 3 family symmetries [27][28][29].
As we will see in the next section, in our model the CKM matrix arises from the downtype quark sector. In order to get the correct hierarchy in the entries of the quark mass matrices yielding a realistic pattern of quark masses and mixing angles, we use a Z 4 discrete symmetry and the scalar bidoublets Φ j (j = 1, 2), one neutral and the another charged under Z 4 . This group was previously used in some other flavor models and proved to be helpful, in particular, in the context of Grand Unification [30,31], models with extended [32] and warped extradimensional models [33].
The Z 4 is the smallest cyclic symmetry that guarantees that the renormalizable Yukawa terms for the fermion singlets S i (i = 1, 2, 3) only involve the scalar fields ξ i , assumed to acquire very small VEVs. This feature is crucial to obtain an inverse seesaw contribution to the light active neutrino mass matrix, instead of a double seesaw contribution, thus giving rise to heavy quasi-Dirac neutrinos within the LHC reach.
It is worth noting that the Yukawa Lagrangian (2.8) possesses accidental U 1 -symmetries located in the non-SM sector with the field charge (Q) assignments: These are spontaneously broken by the VEVs of the corresponding scalar fields in Eq. (3.2).
As a result there appear massless Goldstone bosons with interaction strengths determined by the VEVs shown in Eq. (3.2). This leads to the presence of invisible Higgs decays [34] which are restricted by LEP as well as LHC searches [35]. From Eq. 1 -Goldstone with the 125 GeV Higgs boson is suppressed by the ratios of their VEVs (c.f. Ref. [21]). Alternatively, these Goldstones may also be avoided by adding explicit breaking trilinear terms in the scalar potential. A detailed study is beyond the scope of the present paper.

Quark masses and mixings
From the first line in Eq. (2.8), the up-type quark mass matrix is given by where y i (i = 1, 2, 3) are O(1) parameters and we set v GeV the scale of electroweak symmetry breaking and v σ = λΛ, with λ = 0.225 being one of the Wolfenstein parameters.
For the sake of simplicity, we assume v (j) 2 = 0 (j = 1, 2) so that the standard model down-type quarks and charged leptons acquire their masses from a universal seesaw mechanism, mediated by the exotic down-type quarks D i and charged leptons L i present in Ψ iR and Ψ iL . In this case the down-type charged fermion mass matrices take the form and the further simplification v φ i = v φ and v ϕ l = v ϕ .
Taking the limit M a , M b A i , the standard model down-type quark and charged lepton mass matrices become Notice that in our model the CKM matrix arises only from the down-type quark sector.
In order to recover the low energy quark flavor data, we assume that all dimensionless parameters of the SM down-type quark mass matrix are real, excepting a 5D , taken to be complex. The physical quark mass spectrum [36,37] and mixing angles [38] can be perfectly reproduced in terms of natural parameters of order one, as shown in Table 4, starting from the following benchmark point:

Lepton masses, mixing and oscillations
Here is where the predictive power of our flavor symmetry model is mainly manifest. From the neutrino Yukawa terms, we obtain the following neutrino mass terms : where the neutrino mass matrix reads where for the sake of simplicity, we set α i = β i = y i (i = 1, 2, 3) and Then, ν corresponds to the physical light neutrino mass matrix whereas M ν and M (3) ν are the heavy quasi-Dirac neutrino mass entries. Note that the physical eigenstates include three active neutrinos and six heavy, mainly isosinglet, neutrinos. The heavy quasi-Dirac neutrinos have a small splitting µ.

(5.25)
Eliminating ω 12 in the above relations using eq.(5.11), the mixing parameters depend ultimately on three angles ω 23 , ω 13 , ψ up to three discrete variables k 1 , k 2 , k 3 . Furthermore, without loss of generality these angles are restricted to ω 23 ∈ [−π, π], ω 13 ∈ [−π/2, π/2] and ψ ∈ [−π/2, π/2]. Notice that the angle ψ in this framework is responsible for the CP violating phase in the lepton sector, since the first stage of the diagonalization process yields a real symmetric matrix In figures 1 and 2, we give the allowed values for sin 2 θ 23 together with the corresponding J CP predictions in both mass orderings. For our analysis, we randomly generated parameter configurations for ω 23 , ω 13 and ψ corresponding to 3(1)σ values for the solar and reactor angles in Eqs. (5.18), (5.19). One sees that for the Normal Hierarchy (NH) case the allowed region is severely restricted, with a fourfold degeneracy. In this case CP must necessarily be violated in oscillations, and the predicted atmospheric angle lies in the higher octant, inside its 1σ region. In contrast, for the case of Inverted Hierarchy (IH) one sees that CP can be conserved in neutrino oscillations. Moreover, if violated, it is unlikely for CP to be maximally violated.
The predicted atmospheric angle lies inside its 2σ region, preferably in the first octant.

Neutrinoless double beta decay
In this section we determine the effective Majorana neutrino mass parameter characterizing the neutrinoless double beta (0νββ) decay amplitude. It is given by: 23 sin θ 0 cos ω 13 + 2i cos θ 0 sin ω 13 2 , where U 2 ei and m ν i are the lepton mixing matrix elements and the light active neutrino masses, respectively. The light active neutrino masses can be written in terms of the parameters of the model as We show in Figure 3 the effective Majorana neutrino mass parameter |m ββ | versus the lightest active neutrino mass for the cases of normal and inverted neutrino mass hierarchies.
In order to determine the predicted ranges for |m ββ | in our model, we have randomly generated the angles ω 23 , ω 13 and ψ, as well as the light active neutrino mass scale m ν = λ 2 v ξ √ 2+r 2 in a range of values where the neutrino mass squared splittings and the leptonic mixing parameters are consistent with the observed neutrino oscillation data. Our predicted range of values for the effective Majorana neutrino mass parameter has a lower bound, even in the case of normal hierarchy, indicating that a complete destructive interference among the three light neutrinos is prevented by our symmetry and the current oscillation data.
The corresponding 0νββ decay rates are within the reach of the next-generation bolometric CUORE experiment [48] or, more realistically, of the next-to-next-generation ton-scale 0νββ-decay experiments. It is worth mentioning that the Majorana neutrino mass parameter has an upper bound on |m ββ | ≤ (61 − 165) meV at 90% C.L, as indicated by the KamLAND-Zen experiment from the limit on the 136 Xe 0νββ decay half-life T 0νββ 1/2 ( 136 Xe) ≥ 1.07×10 26 yr [49]. This bound will be improved within a not too far future. The GERDA "phase-II"experiment [50,51] is expected to reach T 0νββ 1/2 ( 76 Ge) ≥ 2 × 10 26 yr, corresponding to |m ββ | ≤ 100 meV. A bolometric CUORE experiment, using 130 Te [48], is currently under construction and its estimated sensitivity is close to about T 0νββ 1/2 ( 130 Te) ∼ 10 26 yr, implying |m ββ | ≤ 50 meV. Furthermore, there are plans for ton-scale next-to-next generation 0νββ experiments with 136 Xe [52,53] and 76 Ge [50,54], asserting sensitivities over T 0νββ 1/2 ∼ 10 27 yr, corresponding to |m ββ | ∼ 12 − 30 meV. and 2. We find that, for normal neutrino mass ordering, the atmospheric angle must lie in the higher octant and that CP must be violated in oscillations. In contrast, for inverse hierarchy, the lower octant is favored and the range of allowed Jarlskog invariant extends from zero up to a non-maximal value. Our results concerning 0νββ decay are summarized in Fig.3. They indicate the existence of a lower bound for the 0νββ decay rate, a feature also encountered in other flavor models [55][56][57][58][59]. As mentioned, neutrino masses arise from a low-scale seesaw mechanism, whose messengers may be produced at the LHC either through a charged or neutral gauge portal [14,16,60]. Admittedly, the model is rather complex, especially in its scalar sector. However, it serves as a "proof-of-concept" attempt, namely, to our knowledge this is the first time that a fully realistic and to some extent predictive flavor realization of a Pati-Salam scenario is given. two triplets, i.e., 3 [0] [1] (which we denote by 3) and its conjugate 3 [0] [2] (which we denote by 3) and 9 singlets, i.e., 1 k,l (k, l = 0, 1, 2), where k and l correspond to the Z 3 and Z 3 charges, respectively [4]. The ∆(27) discrete group is a simple group of the type ∆(3n 2 ) with n = 3 and is isomorphic to the semi-direct product group (Z 3 × Z 3 ) Z 3 [4]. Indeed, the simplest group of the type ∆(3n 2 ) is ∆(3) ≡ Z 3 . The next group is ∆ (12), which is isomorphic to A 4 . Thus, the ∆(27) discrete group is the next simplest nontrivial group of the type ∆(3n 2 ). It is worth mentioning that one can write any element of the ∆ (27) discrete group as b k a m a n , where b, a and a correspond to the generators of the Z 3 , Z 3

Acknowledgments
and Z 3 cyclic groups, respectively. These generators fulfill the relations: The characters of the ∆(27) discrete group are shown in Table 5. Here n is the number of elements, h is the order of each element, and ω = e 2 is the cube root of unity, which satisfies the relations 1 + ω + ω 2 = 0 and ω 3 = 1. The conjugacy classes of 3 ω 2r+sp 0 0 Table 5. Characters of ∆ (27) ∆(27) are given by: : {ba p , ba p−1 a p−2 a 2 }, h = 3.
In the above formulas 3 A and 3 S 1,2 are an antisymmetric and two variants of symmetric triplets. The multiplication rules between ∆(27) singlets and ∆ (27) triplets are given by [4]: x (0,1) . (A.6) The tensor products of ∆(27) singlets 1 k, and 1 k , take the form [4]: From the equation given above, we obtain explicitly the singlet multiplication rules of the ∆(27) group, which are given in Table 6.