Neutrino phenomenology in a left-right $D_4$ symmetric model

We present a minimal left-right symmetric flavor model and analyze the predictions for the neutrino sector. In this scenario, the Yukawa sector is shaped by the dihedral $D_4$ symmetry which leads to correlations for the neutrino mixing parameters. We end up with four possible solutions within this model. We further analyzed the impact of the upcoming long-baseline neutrino oscillation experiment DUNE. Due to its high sensitivity, DUNE will be able to rule out two of the solutions. Finally, the prediction for the neutrinoless double beta decay for the model has also been examined.


I. INTRODUCTION
A number of phenomenal experimental evidences over the past two decades have established the fact that neutrinos oscillate through their propagation path [1][2][3][4], which implies non-zero neutrino masses and mixings. This fact provides an undoubtedly motivation for the existence of physics beyond the Standard Model (SM), as neutrinos are massless in the SM. Furthermore, the experimental efforts in understanding the neutrino properties have determined the two mass-squared differences and large lepton mixing angles. From global fits of neutrino oscillation data [5] (other global analysis can be found in [6,7]), the best fit values and the 1σ intervals for a normal neutrino mass ordering (NO) are given by 1 Moreover, the theory behind the dynamical origin of neutrino mass and their flavor mixing pattern and whether they are Majorana or Dirac fermions, are yet unanswered. The simplest idea behind these shortcomings relies on the assumption that neutrinos are Majorana particles and their tiny masses are generated through a seesaw mechanism [8][9][10][11][12][13]. Interesting extensions of the SM featured by the inherent new physics signatures are those that consider a left-right (LR) symmetric nature [14][15][16][17]. For instance, LR symmetric models have the virtue of accounting for the small neutrino masses from the contribution of two mechanisms, the type-I and type-II seesaw, which implies the existence of new particles.
If the LR breaking and the masses of the new scalar fields are of O(TeV), this minimal setup produces sizeable contributions to lepton flavor violating (LFV) decays, lepton number violation as well as CP violating processes [19][20][21][22][23][24][25]. Therefore, this scenario turns out to be appealing for experimental searches among the low-energy LFV processes [26].
Further constraints apply to this model from LHC searches of new physics [27][28][29][30][31]. On the other hand, the LR symmetry is also possible to be broken at higher energies, such as the grand unification theory (GUT) scale, leading to gauge coupling unification [32,33]. This makes LR models interesting frameworks from the perspective of GUTs like SO(10) [34,35].
On top of gauge symmetries, one can impose additional global symmetries that relate the flavor structure of the SM. In the last decade, there have been a tremendous amount of works in that direction, for reviews see [36,37]. Nevertheless, It is particularly interesting the interplay between the LR symmetry and a discrete flavor symmetry. This combination shapes and correlates the Yukawa sector, giving predictions for the flavor observables, i.e. masses and mixings [38][39][40][41][42]. In this work, we study the effects of combining a non-Abelian discrete flavor symmetry D 4 with LR symmetry. The D 4 flavor symmetry group has been explored in [43][44][45][46][47][48][49][50][51][52][53][54], not in combination with a LR symmetric model to the best of our knowledge.
Among many of the consequences of this model, is the appearance of two-texture zeros in the neutrino mass matrix, in a similar way to other discrete flavor symmetry models. Under the Glashow-Frampton-Marfatia classification [55] for the two-zero texture Majorana neutrino mass matrices, we get an A 2 texture zero matrix. This model also predicts a non trivial mass matrix for the charged leptons.
The outline of the paper is as follows: in Sec. II we present the model and charge assignments. There, we describe the lepton sector, that is, we give the invariant Lagrangian of the theory. We explain the procedure of our analysis in Sec. III as well as show our results for the neutrino predictions within the model. Our final comments and summary are given in Sec. IV.

II. LEFT-RIGHT D 4 SYMMETRIC MODEL
We consider an extension of the minimal left-right symmetric model by adding a D 4 flavor symmetry. Besides postulating a symmetry that shapes the Yukawa sector, we add two flavon fields, ξ and η transforming as a singlet and doublet under D 4 , respectively.
In Table I we provide the matter content and charge assignments of the model. In this framework, the symmetry breaking goes like where G F = D 4 ⊗ Z 2 and its breaking is associated to the non-zero vevs of the flavon fields ξ and η . We assume the following sequential symmetry breaking is the flavour breaking scale and Λ LR is the left-right symmetry breaking scale 2 .
Given the matter content shown in Table I, the Yukawa Lagrangian (up to dimension-5) for the leptons can be expressed as where the bi-doublet Φ can be read as 2 With this assumption the flavon fields decouple from the theory having only an impact on the Yukawa couplings. Then, in this energy regime the scalar potential is approximate to the minimal LRSM one [18]. Additionally, since Λ LR >> Λ EW , the new scalars do not have an important contribution to LFV processes [26].
Note that the Dirac neutrino mass matrices stem from the dimension-5 operators. Hence, from Eq. (4) after spontaneous symmetry breaking (SSB), one gets that the mass matrix for charged leptons as where with Assuming a vev alignment η ∼ (1, 0) T , the mass matrix for the charged leptons becomes The matrix M can be diagonalised by a bi-unitary transformation as and the neutrino mass matrix is given by where In this scenario, the Dirac neutrino mass matrix turns out to be After the SSB, light neutrino eigenstates acquire their masses through the type-I and type-II seesaw mechanism. Hence, since v R >> v L , v 1 , v 2 , light-neutrino masses are given by, The left-right symmetric nature of the theory demands a relation between the Yukawa couplings mediating the interaction of leptons with the scalar triplets, i.e. Y R = Y L .
The left-right exchange symmetry can be realized through either C or P transformations. Here we choose to use P-transformations, which demand the hermiticity of Dirac type fermion mass matrices, that is, III. NEUTRINO PHENOMENOLOGY From Eq. (9) one can notice that the mass matrix for charged leptons is non-diagonal.
The left-right symmetry gives further relations for leptonic Yukawas, as mentioned in the previous section. Using this fact, the mass matrix for charged leptons, as given by Eq. (9), can be recasted as The phases of this matrix can be absorbed in a pair of diagonal phase matrices (P and P ), which will define a real charged lepton matrix basis In this basis the neutrino mass matrix becomes where M ν is the neutrino mass matrix in the interaction basis, Eq. (14).
SinceM is symmetric, it is diagonalised as where O is an orthogonal matrix and one can easily get the expressions for a , b and c in terms of the charged fermion masses. This is done by computing the invariants of the charged leptons mass matrix, namely TrM , TrM 2 and detM . Then, the matrix elements in Eq. (16) as functions of the masses read as With this information one can compute the rotation matrix O and is given by Regarding the neutrino mass matrix, this is obtained using eqs. (12)(13)(14) and turns out to where a ν , b ν and c ν and d ν are complex numbers. Then, in this model the light-neutrino masses are computed through diagonalization ofM ν in Eq. (21). This mass matrix and the neutrino mass eigenstates are related as follows, where m ν i are the light neutrino masses and the unitary matrix U ν follows the PDG parameterization [56]. Therefore, in this model, the lepton mixing matrix (also known as Pontecorvo-Maki-Nakagawa-Sakata (PMNS) mixing matrix [57,58]) is defined by 3 , 3 Similar structure for charged leptons and neutrinos was obtained in the context of S 3 flavor symmetry [59,60].
where the angles θ ij correspond to the mixing angles determined by neutrino oscillation experiments, δ represents the Dirac type CP-violating phase and K is the Majorana phase diagonal matrix.
A. Results These contours are shown using the red, orange, and yellow colors, respectively. The bestfit value has been marked with a 'black-dot'. It can be seen from the left-panel that the solution A is ruled out by the present data at 5σ C. L., whereas the solution D is marginally allowed at 3σ C. L., but only for the CP-conserving values, namely around δ = 0, 2π. We also notice that the solutions B and C are allowed at 1σ C. L. Furthermore, it can be seen that among the four cases only the solution C is able to explain the latest best-fit value of neutrino oscillation data.
Similarly, in the right-panel of Fig. 1, we show the compatibility of the model by considering the simulated results of the next generation long baseline oscillation experiment, DUNE [61]. The allowed parameter space of DUNE in the (sin 2 θ 23 − δ) plane is found using the latest best-fit value of neutrino oscillation data. For the numerical simulation of DUNE, the GLoBES package was used [62,63] along with the auxiliary files in Ref. [64]. It was assumed a running time of 3.5 years in both neutrino and antineutrino modes for DUNE, i.e. DUNE [3.5 + 3.5]. The detailed numerical procedure that have been followed to simulate data coincides with the one performed in [65,66]. Notice from the right-panel that DUNE results would significantly improve the precision of both the parameters. It is observed that sin 2 θ 23 is constrained to values between (0.45, 0.58), whereas δ is restricted to the range (0.95, 1.88)π at 5σ C. L. after DUNE[3.5 + 3.5] running time. Therefore, one can infer that the precise measurement of both parameters (θ 23 and δ) by DUNE, the solution D will be ruled out at 5σ C. L., still allowed by the latest global-fit data. In this model we also have a prediction for lepton number violating processes such as the neutrinoless double beta decay (0νββ). Ongoing experiments that are looking for the signatures of 0νββ decays are namely, GERDA Phase-II [67], CUORE [68], SuperNEMO [69], KamLAND-Zen [70] and EXO [71]. The half-life of these processes can be expressed as [72,73], where G 0ν represents the two-body phase-space factor, M 0ν is the nuclear matrix element and | m ee | is the effective Majorana neutrino mass. The expression of | m ee | is given by, where V L stands for lepton mixing matrix as mentioned in Eq. (22). Fig. 2 shows the prediction for 0νββ decay. For comparison, we first show the allowed 3σ parameter space in (m light − | m ee |)-plane using the latest global analysis of neutrinos oscillation data [5], as shown by the gray color. We proceed to compute the effective Majorana neutrino mass in In addition, the light green-vertical band represents the bound on m light coming from the cosmological limit on the sum of neutrino masses provided by the Planck Collaboration, namely m ν < 0.12 eV at the 95% C.L. [74,75]. Furthermore, as pointed out before, from the left-panel Fig. (1) on can observe that DUNE can rule out solution D. This also has an impact for the prediction of 0νββ. As a final remark, notice that the allowed solutions are compatible only with normal neutrino mass ordering.    data. More importantly, due to the high potential of the DUNE experiment to improve the precision of θ 23 and to probe δ, it gives further restrictions to the parameter space as shown in the right-panel in Fig. (1). Using this, DUNE will be able to rule out two of the solutions, namely A and D. We also provided the prediction for neutrinoless double beta decay in terms of the lightest neutrino mass for a mass range of ∼ (10 −3 − 10 −2 ) eV, which we summarize in Fig. (2). where I is identity matrix.