Low scale type I seesaw model for lepton masses and mixings

In contrast to the original type I seesaw mechanism that requires right-handed Majorana neutrinos at energies much higher than the electroweak scale, the so-called low scale seesaw models allow lighter masses for the additional neutrinos. Here we propose an alternative low scale type I seesaw model, where neither linear nor inverse seesaw mechanisms take place, but the spontaneous breaking of a discrete symmetry at an energy scale much lower than the model cutoff is responsible for the smallness of the light active neutrino masses. In this scenario, the model is defined with minimal particle content, where the right-handed Majorana neutrinos can have masses at the $\sim 50\mbox{ GeV}$ scale. The model is predictive in the neutrino sector having only four effective parameters that allow to successfully reproduce the experimental values of the six low energy neutrino observables.


I. INTRODUCTION
After minimally extending the Standard Model (SM) to include massive neutrinos, the observed fermion mass hierarchy is extended over a range of 13 orders of magnitude, from the lightest active neutrino mass scale up to the top quark mass. In addition, the small quark mixing angles decrease from one generation to the next while in the lepton sector this hierarchy is not present since two of the mixing angles are large and the other one is small.
Neither of these features in the flavor sector is explained in the SM. This is the so-called SM flavor puzzle, which has motivated the construction of theories with extended scalar and/or fermion sectors with additional continuous or discrete groups. In particular, extensions of the SM with non-Abelian discrete flavor symmetries are very attractive since they successfully describe the observed pattern of fermion masses and mixings (for recent reviews on discrete flavor groups see Refs. [1][2][3][4][5][6]), while they can naturally appear from the breaking of continuous non-Abelian gauge symmetries or from compactified extra dimensions (see Ref. [7] and references therein). Several discrete groups have been employed in extensions of the SM. In particular, A 4 is the smallest discrete group with one three-dimensional and three distinct one-dimensional irreducible representations where the three families of fermions can be accommodated rather naturally. This group has been particularly promising in providing a predictive description of the current pattern of SM fermion masses and mixing angles [8][9][10][11][12][13][14][15]. Despite several models based on the A 4 discrete symmetry have been proposed, most of them have a nonminimal scalar sector, composed of several SU (2) Higgs doublets, even in their low energy limit, and have A 4 scalar triplets in the scalar spectrum whose vacuum expectation value (VEV) configurations in the A 4 direction are not the most natural solutions of the scalar potential minimization equations. Thus, it would be desirable to build an A 4 flavor model which at low energies reduces to the SM model and where the different gauge singlet scalars are accommodated into A 4 singlets and one A 4 triplet [with VEV pattern in the (1, 1, 1) A 4 direction] which satisfies the minimization condition of the scalar potential for the whole range of values of the parameter space. To this end, in this work we propose an extension of the SM based on the A 4 family symmetry, which is supplemented by a Z 4 auxiliary symmetry, whose spontaneous breaking at an energy scale (v S ) much lower than the model cutoff (Λ) produces the small light active neutrino mass scale m ν . As we will show in the next sections, in this scenario the masses for the active neutrinos are produced by a type I seesaw mechanism [48][49][50][51] mediated by three ∼ 50 GeV right-handed Majorana neutrinos, where m ν ∝ (v S /Λ) 2 . Given the low mass scale of the right-handed neutrinos, this model can be classified as a low scale type I seesaw, as it has been coined in the literature [52][53][54][55][56][57][58][59][60][61][62]. There are different realizations of low scale seesaw models, as for example inverse or linear [35,46,[63][64][65][66][67][68][69][70][71][72][73][74][75][76][77][78] , where an additional lepton number violating mass parameter is added. In these models, the smallness of m ν is related to the smallness of the additional parameter. In our case, however, no extra small mass parameter has been included, and the smallness of the light neutrino masses is explained through the spontaneous breaking of the auxiliary discrete groups, which leads to a suppression in the Dirac neutrino mass matrix.
From the point of view of the low energy neutrino observables, the model makes very particular predictions for δ CP and θ 23 , which are not aligned with the central values of current fits. Therefore, future improvements in the precision of neutrino measurements will provide an experimental test of the model. Processes like (i) charged lepton flavor violating decays ( → γ) [11,12], (ii) flavor changing neutral currents, and (iii) rare top quark decays such as t → hc, t → cZ [15], are strongly suppressed, in contrast to other A 4 flavor models (that usually have several Higgs doublets), where these processes can have rates that are at the reach of forthcoming experiments.
The paper is organized as follows. In Sec. II we describe the model. In Sec. III we present a discussion on lepton masses and mixings and give the corresponding results. We draw our conclusions in Sec. IV. The Appendix provides a concise description of the A 4 discrete group.

II. MODEL DESCRIPTION
We propose an extension of the SM where the scalar sector is augmented by the inclusion of four gauge-singlet scalar fields and the SM gauge symmetry is supplemented by the A 4 × Z 4 discrete group. The symmetry G features the following spontaneous symmetry breaking pattern: where the symmetry-breaking scales satisfy the hierarchy v S ∼ O(1)TeV > v, v S is the scale of spontaneous breaking of the A 4 × Z 4 discrete group, and v = 246 GeV is the electroweak symmetry breaking scale. As mentioned before, the scalar sector of the SM is augmented by the inclusion of four SM gauge singlet scalars. We add these extra scalar fields for the following reasons: (i) to build nonrenormalizable charged leptons and Dirac neutrino Yukawa terms invariant under the local and discrete groups, crucial to generate predictive textures for the lepton sector; (ii) to generate a renormalizable Yukawa term for the right-handed Majorana neutrinos, that can give rise to ∼ 50 GeV masses for these singlet fermions. As we will see below, the observed pattern of SM charged lepton masses and leptonic mixing angles will arise from the spontaneous breaking of the A 4 × Z 4 discrete group. In order to generate the masses for the light active neutrinos via a type-I seesaw mechanism, we extend the fermion sector by including three right-handed Majorana neutrinos, which are singlets under the SM group. The lepton assignments under the group Here we specify the dimensions of the We assume the following vacuum configuration for the A 4 -triplet gauge singlet scalar ξ: which satisfies the minimization condition of the scalar potential for the whole range of values of the parameter space, as shown in Ref. [31]. With the particle content previously specified, we have the following relevant Yukawa terms for the lepton sector, invariant under the symmetries of the model: where the dimensionless couplings in Eq.
In what follows, we describe the role of each discrete group factor of our A 4 flavor model.
The A 4 discrete group yields a reduction of the number of model parameters, giving rise to predictive textures for the lepton sector, which are consistent with the lepton mass and mixing pattern, as will be shown in Sec. III. On the other hand, the Z 4 discrete group is the smallest cyclic symmetry allowing a renormalizable Yukawa term for the right-handed Majorana neutrinos, giving rise to a diagonal Majorana neutrino mass matrix that yields degenerate Majorana neutrinos with electroweak scale masses. In addition, the spontaneous breaking of the A 4 × Z 4 discrete group at an energy scale much lower than the model cutoff is crucial to produce small mixing mass terms between the active and sterile neutrinos, allowing the implementation of a low scale type I seesaw mechanism. Finally, we assume that the VEVs of the gauge singlet scalar fields ξ, ρ i (i = 1, 2, 3) satisfy the relation where v ξ ∼ v ρ ∼ v S is the discrete symmetry breaking scale and Λ is the model cutoff.
It is worth mentioning that this model at low energies corresponds to a singlet-doublet model [79,80]. Consequently, from a detailed analysis of the low energy scalar potential (as done for example in Ref. [81]) one can show that the 125 GeV SM-like Higgs boson has couplings close to the SM expectation, with small deviations of order v 2 /v 2 S ∼ O(10 −2 ). The TeV-scale singlet s 0 (s 0 = ξ, ρ j ) will mix with the CP -even neutral component of the SM Higgs doublet, h 0 , with a mixing angle γ ∼ O(v/v S ). Thus, the couplings of the singlet scalars to the SM particles will be equal to the SM Higgs couplings times the s 0 − h 0 mixing angle γ. The collider phenomenology of this scenario is well studied [82][83][84][85][86]. For TeV-scale singlets, the most stringent limits at the 8TeV LHC come from indirect searches. A global fit to all SM signal strengths constrains sin 2 γ ≤ 0.23 at 95% C.L. [87,88], that assuming O(1) couplings in the scalar potential translates to v S 500 GeV. For a summary of the sensitivity of future colliders see for example Table 1 of Ref. [86]. As we will see in the next section, there is a broad range of values of v S that are consistent with the observed light neutrino masses and current limits on singlet scalars.

III. NEUTRINO MASSES AND MIXINGS
The lepton Yukawa terms in Eq. (5) imply that the mass matrix for charged leptons is given by where so, Regarding the neutrino sector, we find that the resulting Dirac neutrino mass matrix reads where ω = e   where we can read that the typical mass scale of the light active neutrinos is It is noteworthy that the smallness of the active neutrino masses is a consequence of their inverse scaling with the square of the model cutoff, which is much larger than the breaking scale (v S ) of the discrete symmetries. We can see from Eq. (13) [89,90].
the experimental values taken from Ref. [89]. To give an example, for each hierarchy we  Tables I and II. From Figure 1, we can see that for the normal hierarchy, ∆m 2 31 , ∆m 2 21 , sin 2 θ 12 , and sin 2 θ 13 are evenly distributed in the allowed range. On the other hand, for sin 2 θ 23 and δ CP , the model features more definite predictions. The same behavior is found for the inverted hierarchy. It is worth mentioning that in a generic scenario, the neutrino Yukawa couplings are complex, thus the light active neutrino sector has eight parameters. However, not all of them are physical. Considering the case of real VEVs for the gauge-singlet scalars ρ 1 , ρ 2 , and ξ, the phase redefinition of the leptonic fields L L and N R allows to rotate away the phase of one of the neutrino Yukawa couplings, leading to seven physical parameters. On the other hand, if we consider complex VEVs for the gauge-singlet scalars ρ 1 , ρ 2 , and ξ, we can use their phases to set three of the four neutrino Yukawa couplings real. Therefore, in this case we are left with five effective parameters in the neutrino sector. However, for the sake of simplicity, we are considering a particular benchmark scenario with real neutrino Yukawa couplings, i.e., four effective parameters. In this simplified benchmark scenario, the complex phase responsible for CP violation in neutrino oscillation arises from the spontaneous breaking of the A 4 discrete group. This mechanism for inducing CP violation in the fermion sector where U 2 ej and m ν j are the PMNS mixing matrix elements and the Majorana neutrino masses, respectively. As we can see from Figure which is below the sensitivity of present 0νββ-decay experiments. The current experimental sensitivity on the Majorana neutrino mass parameter is obtained from the KamLAND-Zen limit on the 136 Xe 0νββ decay half-life, T 0νββ 1/2 ( 136 Xe) ≥ 1.07 × 10 26 yr [98], which yields the corresponding upper limit on the Majorana mass, |m ββ | ≤ (61 − 165) meV at 90% C.L. For other 0νββ-decay experiments see Refs. [99][100][101][102][103][104]. The experimental sensitivity of neutrinoless double beta decay searches is expected to improve in the near future. Note that the model predicts a range of values for neutrinoless double beta decay rates that can be tested by the next-generation bolometric CUORE experiment [102], as well as the next-to-next-generation ton-scale 0νββ-decay experiments [98,101,105,106].