Neutrino predictions from a left-right symmetric flavored extension of the standard model

We propose a left-right symmetric electroweak extension of the Standard Model based on the $\Delta \left( 27\right)$ family symmetry. The masses of all electrically charged Standard Model fermions lighter than the top quark are induced by a Universal Seesaw mechanism mediated by exotic fermions. The top quark is the only Standard Model fermion to get mass directly from a tree level renormalizable Yukawa interaction, while neutrinos are unique in that they get calculable radiative masses through a low-scale seesaw mechanism. The scheme has generalized $\mu-\tau$ symmetry and leads to a restricted range of neutrino oscillations parameters, with a nonzero neutrinoless double beta decay amplitude lying at the upper ranges generically associated to normal and inverted neutrino mass ordering.


I. INTRODUCTION
The SU(3) C ⊗ SU(2) L ⊗ U(1) Y gauge theory provides a remarkable description of the interactions of quarks and leptons as mediated by intermediate vector bosons associated to the Standard Model gauge structure. However, it is well-known to suffer from a number of drawbacks. Most noticeably, it fails to account for neutrino masses, needed to describe the current oscillation data [1]. Likewise, it does not provide a dynamical understanding of the origin of parity violation in the weak interaction. Last, but not least, it also fails in providing an understanding of charged lepton and quark mass hierarchies and mixing angles from first principles. Left-right symmetric electroweak extensions of the Weinberg-Salam theory address the origin of parity violation [2,3], while models based on non-Abelian flavor symmetries [4] address the flavor issues [5,6]. Combining these features is a desirable step forward. Indeed, a predictive Pati-Salam theory of fermion masses and mixing combining both approaches has been suggested recently [7].
In this paper we propose a less restrictive left-right symmetric electroweak extension of the Standard Model based on the SU(3) ⊗ SU(2) L ⊗ SU(2) R ⊗ U(1) B−L gauge group and the ∆ (27) family symmetry. Of the Standard Model fermions only the top quark acquires mass through a tree level renormalizable Yukawa interaction. Exotic charged fermions acquire mass from their corresponding tree level mass terms, while gauge singlet fermions can also have gauge-invariant tree level Majorana mass terms. The masses for the other electrically charged Standard Model fermions, namely quarks lighter than the top, as well as charged leptons, are all induced by a Universal Seesaw mechanism mediated by the exotic fermions. The mass hierarchies as well as the quark mixing angles arise from the spontaneous breaking of the ∆ (27) ⊗ Z 6 ⊗ Z 12 discrete family group, and the radiative nature of the inverse seesaw mechanism is guaranteed by spontaneously broken Z 4 and Z 12 symmetries, with Z 12 spontaneously broken down to a preserved Z 2 symmetry. The Cabibbo mixing arises from the up-type quark sector, whereas the down-type quark sector contributes to the remaining CKM mixing angles. On the other hand, the lepton mixing matrix receives its main contributions from the light active neutrino mass matrix, while the Standard Model charged lepton mass matrix provides Cabibbo-like corrections to these parameters. Finally, the masses for the light active neutrinos emerge from a low-scale inverse/linear seesaw mechanism [8][9][10][11][12] with one-loop-induced seed mass parameters [13,14].

II. THE MODEL
The model is based on the left-right gauge symmetry SU(3) ⊗ SU(2) L ⊗ SU(2) R ⊗ U(1) B−L supplemented by the discrete ∆ (27) ⊗ Z 4 ⊗ Z 6 ⊗ Z 12 family group. The particle content and gauge quantum numbers are summarized in table I, while the transformation properties of the fields under the discrete symmetries are presented in tables II, III and IV. Here ω = e Notice that the fermion sector of the original left-right symmetric model has been extended with two vectorlike up-type quarks T k , k = 1, 2, three vectorlike down type quarks B i , three vectorlike charged leptons E i and six neutral Majorana singlets S i , Ω i , with i = 1, 2, 3. The role of the new exotic vectorlike fermions is to generate the masses for Standard Model charged fermions from a Universal Seesaw mechanism. Neutrino masses are in turn produced by an inverse seesaw mechanism, triggered by a one loop generated mass scale [13,14] from the interplay of the gauge singlet fermions S i and Ω i .
Let us note that the neutrino Yukawa terms given in Eq. (6) have accidental U symmetries described in Table V. These are spontaneously broken by the VEVs of the scalar fields charged under these symmetries. As a result, massless Goldstone bosons are expected to arise from the spontaneous breakdown of these symmetries. However, these can acquire masses from scalar interactions like λρ 2 ξ 2 and M χ † 2L Φχ 2R , invariant under the symmetry group G of our model, but not under the accidental U symmetries, respectively.
We now explain the different group factors of the model. In the present model, the ∆ (27) group is responsible for the generation of a neutrino mass matrix texture compatible with the experimentally observed deviation of the tribimaximal mixing pattern. In addition it allows for renormalizable Yukawa terms only for the top quark, the gauge singlet Majorana fermions Ω i (i = 1, 2, 3) and tree level mass terms for the exotic charged fermions. This allows for their masses to appear at the tree level. Let us note that the ∆(27) discrete group is a non trivial group of the type ∆(3n 2 ), isomorphic to the semi-direct product group (Z 3 × Z 3 ) × Z 3 [4]. This group was proposed for the first time in Ref. [15] and it has been employed in order to construct the Pati-Salam electroweak extension proposed in [7]. This group has also been used in multiscalar singlet models [16], multi-Higgs doublet models [17,18], Higgs triplet models [19] SO(10) models [20][21][22], warped extra dimensional models [23], and models based on the SU (3) C ⊗SU (3) L ⊗U (1) X gauge symmetry [24][25][26].
The auxiliary Z 6 and Z 12 symmetries select the allowed entries of the charged fermion mass matrices and shape their hierarchical structure, so as to get realistic SM charged lepton masses as well as quark mixing out of order one parameters. We assume that the Z 12 symmetry is broken down to a preserved Z 2 symmetry, which allows the implementation of an inverse/linear seesaw mechanism [8][9][10][11][12] for the generation of the light active neutrino masses. This is triggered by one-loop-induced seed mass parameters, in a manner analogous to the models discussed in [13,14].
The spontaneously broken Z 4 symmetry also ensures the radiative nature of the inverse seessaw mechanism. This group was previously used in some other flavor models and proved to be helpful, in particular, in the context of Grand Unification [27][28][29], models with extended SU (3) C ⊗ SU (3) L ⊗ U (1) X gauge symmetry [30,31] and warped extradimensional models [32]. It is worth mentioning that one or both of the Z 6 and Z 12 discrete groups were previously used in some other flavor models and proved to be useful in describing the SM fermion mass and mixing pattern, in particular in the context of three Higgs doublet models [33], models with extended SU (3) C ⊗ SU (3) L ⊗ U (1) X gauge symmetry [14,30,34], Grand Unified theories [28] and models with strongly coupled heavy vector resonances [35].
Quark masses and mixing parameters are modeled with the help of the scalar singlets σ and η. We assume that these scalars acquire vacuum expectation values of order λΛ, where λ = 0.225 is the Cabibbo angle and Λ is the cutoff of our model. Consequently, we set the VEVs of the scalar fields to satisfy the following hierarchy: Here v = 246 GeV is the electroweak symmetry breaking scale and v kR ∼ few TeV (k = 1, 2) the scale of breaking of the left-right symmetry. The resulting mixing angles of ξ with ρ, η and τ are very tiny since they are suppressed by the ratios of their VEVs, which is a consequence of the method of recursive expansion proposed in Ref. [36]. Thus, the scalar potential for ξ can be studied independently from the corresponding one for ρ, η, τ . As shown in detail in Ref. [7], the following VEV alignments for the ∆(27) scalar triplets are consistent with the scalar potential minimization equations for a large region of parameter space: Summarizing, the full symmetry of the model exhibits the following spontaneous breaking pattern

A. Charged lepton sector
From the charged lepton Yukawa terms in Eq. (5), we find that the mass matrix containing the charged leptons in the basis (l 1L , l 2L , l 3L , E 1L , E 2L , E 3L ) versus (l 1R , l 2R , l 3R , E 1R , E 2R , E 3R ) takes the form: Given that the exotic charged lepton masses m Ei (i = 1, 2, 3) are much larger than v L and v R , it follows that the SM charged leptons get their masses from an Universal seesaw mechanism mediated by the three charged exotic leptons E i (i = 1, 2, 3). Then, the SM charged lepton mass matrix becomes where the effective Yukawas e ij are naturally expected to be of order one. The Standard Model charged lepton mass matrix is diagonalized by a unitary matrix through . In order to illustrate how the charged lepton mass spectrum arises from Eq. (10) B. Neutrino sector From the neutrino Yukawa interactions in Eq. (6), we obtain the following mass terms: where the neutrino mass matrix M ν is given in block form as: with m R = m Re ϕ and m I = m Im ϕ and the loop function [37]: The one-loop Feynman diagrams contributing to the entries of the Majorana neutrino mass submatrix µ are shown in Fig. 1. The splitting between the masses m Re ϕ and m Im ϕ arises from the term κ For the sake of simplicity, we assume that the singlet scalar field ϕ is heavier than the right-handed Majorana neutrinos Ω i (i = 1, 2, 3), so that we can restrict to the scenario and m 2 R − m 2 I m 2 R + m 2 I , for which the submatrix µ takes the form The structure of the resulting neutrino mass is a particular case of that in Ref. [38] (see below). Besides the three active Majorana neutrinos the physical states include the six heavy exotic neutrinos. After seesaw blockdiagonalization [39] we obtain where we have simplified our analysis setting γ i = κ i (i = 1, 2, 3). Here M (1) ν corresponds to the effective active neutrino mass matrix resulting from seesaw diagonalization, whereas M corresponds to the inverse seesaw piece [8,9], while the latter comes from the linear seesaw contribution [10][11][12].
Finally, the light active neutrino mass spectrum is in terms of a common mass scale m 0 C. Lepton mixing matrix The lepton mixing matrix is thus given by where we take V l to be approximated as with η 1 and η 2 of the same order as the Cabibbo parameter λ, as indicated by our estimate in Eq. (12). In the fully "symmetrical" presentation of the lepton mixing matrix [43,44] with c ij = cos θ ij and s ij = sin θ ij , we find the relation between mixing angles and the entries of U to be given as The Jarlskog invariant J CP = Im (U * 11 U * 23 U 13 U 21 ), takes the form [44] J CP = 1 8 sin 2θ 12 sin 2θ 23 sin 2θ 13 cos θ 13 sin(φ 13 This is the CP phase relevant for the description of neutrino oscillations. The two additional Majorana-type rephasing invariants I 1 = Im U 2 12 U * 2 11 and I 2 = Im U 2 13 U * 2 11 are given as I 1 = 1 4 sin 2 2θ 12 cos 4 θ 13 sin(−2φ 12 ) and I 2 = 1 4 sin 2 2θ 13 cos 2 θ 12 sin(−2φ 13 ) .
In Figure (2) we show the allowed values for the leptonic Dirac CP violating phase δ versus the atmospheric mixing parameter sin 2 θ 23 , for both normal and inverted neutrino mass orderings. These values were generated by randomly varying the model parameters ω 13 , ω 23 , ψ, |η 1 | and |η 2 | within a range that covers reactor and solar mixing angles inside the 2σ experimentally allowed range. In particular, we varied |η 1 | and |η 2 | in the range [0.5λ, 3λ]. Furthermore, the light active neutrino mass scale was randomly varied in the range 10 −4 eV < m 0 < 1eV, consistent with 2σ allowed values for the neutrino mass squared splittings.
To close this section we note that, in contrast to the Left-right symmetric model of Ref. [45], where the µ − τ symmetry is broken softly, our departure from µ − τ symmetry is induced by the mixing in the charged lepton sector, parameterized by the η 1 and η 2 angles, assumed to be of the same order as the Cabibbo angle λ.

IV. NEUTRINOLESS DOUBLE BETA DECAY
In this section we present the model predictions for neutrinoless double beta (0νββ) decay. The effective Majorana neutrino mass parameter is where m νi are the light active neutrino masses and U 2 ei are the squared lepton mixing matrix elements, respectively. The current experimental sensitivity on the Majorana neutrino mass parameter is illustrated by the horizontal band in Fig. (3) and comes from the KamLAND-Zen limit on the 136 Xe 0νββ decay half-life T 0νββ 1/2 ( 136 Xe) ≥ 1.07 × 10 26 yr [46], which translates into a corresponding upper bound on |m ββ | ≤ (61 − 165) meV at 90% C.L. as indicated by the horizontal band in Fig. (3). For those of other experiments see Ref. [47][48][49][50][51]. The "expected" regions for the effective Majorana neutrino mass parameter |m ββ | consistent with the constraints from the current neutrino oscillation data at the 2σ level are indicated by the other broad shaded bands. There are two cases, corresponding to normal and inverted neutrino mass orderings. These are generic, arising only by imposing current oscillation data. In contrast, the thinner (darker) bands include also the model predictions described in the previous section. These regions are obtained from our generated model points by imposing current neutrino oscillation constraints at the 2σ level.
One sees that our "predicted" ranges for the effective Majorana neutrino mass parameter have lower bounds in both cases, of normal and inverted mass orderings, indicating that a complete destructive interference amongst the three light neutrinos is always prevented in our model. These lower bounds for the 0νββ amplitude are general predictions of the present model, and can easily be understood.
In fact, as mentioned above, the structure of our µ − τ symmetric neutrino mass matrix is a particular case of that in Ref. [38]. Comparing with the results of Ref. [41] one sees that, indeed, the possible destructive interference amongst the three light neutrinos is prevented (as lim η 1 →0 I 1,2 = 0), thus explaining the absolute lower bound we obtain.
The experimental sensitivity of 0νββ searches is expected to improve in the near future. For our model, the predicted 0νββ decay rates may be tested by the next-generation bolometric CUORE experiment [52], as well as the next-to-next-generation ton-scale 0νββ-decay experiments [46,[53][54][55].

V. QUARK MASSES AND MIXINGS
In this section, we illustrate how the model is capable of reproducing the correct masses and mixings in the quark sector. From the quark Yukawa interactions, we find that the up-type mass matrix in the basis while the down type quark mass matrix is given as: Assuming the exotic quark masses to be sufficiently larger than v L and v R , it follows that the SM quarks lighter than the top quark all get their masses from a Universal seesaw mechanism mediated by the three exotic up-type and down-type quarks U i and D i (i = 1, 2, 3). It is worth mentioning that the top quark does not mix with the remaining up-type quarks. As a result, the SM quark mass matrices take the form: a22 + κλ 2 λ 6 a 12 λ 5 0 a 12 λ 5 y 2 11 + y 2 13 λ 2 λ 4 y 13 y 23 λ 5 y 13 y 33 λ 3 y 13 y 23 λ 5 m B 3 m B 2 y 2 22 + y 2 23 λ 2 λ 2 y 23 y 33 λ 2 y 13 y 33 λ 3 y 23 y 33 λ 2 y 2 where we have set Let us note that in our model, the dominant contribution to the Cabbibo mixing arises from the up-type quark sector, whereas the down-type quark sector contributes to the remaining CKM mixing angles. In order to recover the low energy quark flavor data, we assume that all dimensionless parameters of the SM quark mass matrices are real, except for b 13 , taken to be complex.
Starting from the following benchmark point:  invariant [58] are indeed consistent with the experimental data, as shown in Table VI. This establishes the viability of our model also for the quark sector. Note that the dimensionless parameters of the benchmark point (48) are all ∼ O(1) in absolute value. This means that our model reproduces the quark mass and mixing hierarchy by its symmetries resulting in certain distribution of the powers of λ among the entries of the mass matrices (46), (47).

VI. FEATURES OF THE MODEL
We now sum up the main theoretical features of our model. 2. The masses for the SM charged fermions lighter than the top quark arise from a Universal Seesaw mechanism mediated by charged exotic fermions. The quark mixing angles and the hierarchy between quark masses arise from the spontaneous breaking of the Z 6 ⊗ Z 12 discrete group.
3. The Cabibbo mixing arises from the up-type quark sector, whereas the down-type quark sector induces the remaining CKM mixing angles. On the other hand, the leptonic mixing parameters receive their dominant contributions from the light active neutrino mass matrix, whereas the charged lepton mass matrix provides Cabibbo-sized corrections to these parameters. 4. The masses for the light active neutrinos emerge from a one loop level inverse seesaw mechanism, whose radiative nature is guaranteed by the spontaneously broken Z 4 and Z 12 symmetries, with Z 12 spontaneously broken down to a preserved Z 2 symmetry.
5. The mass terms for the gauge singlet sterile neutrinos S i (i = 1, 2, 3) are generated from at one loop level, mediated by the real and imaginary components of the electrically neutral gauge singlet scalar ϕ as well as by the gauge singlet Majorana neutrinos Ω i (i = 1, 2, 3). These mass terms break lepton number by two units, triggering the one loop level inverse seesaw mechanism responsible for the light active neutrino masses.

VII. DISCUSSION AND CONCLUSIONS
In summary, we have built a viable extension of the left-right symmetric electroweak extension of the Standard Model capable of explaining the current pattern of SM fermion masses and mixings. Our model is based on the ∆(27) discrete symmetry, supplemented by the Z 4 ⊗ Z 6 ⊗ Z 12 discrete family group. In our model, the masses of the light active neutrinos emerge from a one loop level inverse seesaw mechanism, whereas the masses of the Standard Model charged fermions lighter than the top quark are produced by a Universal Seesaw mechanism. Of the Standard Model fermions only the top quark acquires mass through a tree level renormalizable Yukawa interaction. In our model the Cabibbo mixing arises from the up-type quark sector whereas the down-type quark sector contributes to the other CKM mixing angles. On the other hand, the leptonic mixing parameters receive their dominant contributions from the light active neutrino mass matrix, whereas the SM charged lepton mass matrix provide Cabibbo sized corrections. The observed hierarchy of SM charged fermion masses and mixing angles is caused by the spontaneous breaking of the ∆ (27) ⊗ Z 6 ⊗ Z 12 discrete flavor group, whereas the radiative nature of the inverse seessaw mechanism is guaranteed by spontaneously broken Z 4 and Z 12 symmetries, having Z 12 spontaneously broken down to a preserved Z 2 symmetry. Our model features a generalized µ−τ symmetry and predicts a restricted range of neutrino oscillations parameters, with the neutrinoless double beta decay amplitude lying at the upper ranges associated to normal and inverted neutrino mass ordering.
Notice also that our low-scale left-right symmetric radiative seesaw scheme not only accounts for the light neutrino masses and mixings that lead to oscillations and 0νββ-decay, but can also lead to signatures that can make it testable at collider experiments such as the LHC. For example, the heavy quasi Dirac neutrinos can be produced in pairs at the LHC, via a Drell-Yan mechanism mediated by a heavy non Standard Model neutral gauge boson Z . These heavy quasi Dirac neutrinos can decay into a Standard Model charged lepton and W gauge boson, due to their mixings with the light active neutrinos. Thus, the observation of an excess of events in the dilepton final states with respect to the SM background, would be a signal supporting this model at the LHC. Moreover, lepton flavor violation is expected in these decays, even if suppressed at low energies [59,60]. A detailed study of the collider phenomenology of this model is beyond the scope of the present paper and is left for future studies.