A highly predictive and testable $A_{4}$ flavor model within type-I+II seesaw framework and associated phenomenology

We investigate neutrino mass model based on $A_4$ discrete flavor symmetry in type-I+II seesaw framework. The model has imperative predictions for neutrino masses, mixing and $CP$ violation testable in the current and upcoming neutrino oscillation experiments. The important predictions of the model are: normal hierarchy for neutrino masses, a higher octant for atmospheric angle ($\theta_{23}>45^{o}$) and near-maximal Dirac-type $CP$ phase ($\delta\approx\pi/2$ or $3\pi/2$) at $3\sigma$ C. L.. These predictions are in consonance with the latest global-fit and results from Super-Kamiokande(SK), NO$\nu$A and T2K. Also, one of the important feature of the model is the existence of a lower bound on effective Majorana mass, $|M_{ee}|\geq 0.047$eV(at 3$\sigma$) which corresponds to the lower part of the degenerate spectrum and is within the sensitivity reach of the neutrinoless double beta decay(0$\nu\beta\beta$) experiments.

Understanding this emerged picture of neutrino masses and mixing, which is at odds with that characterizing the quark sector, is one of the biggest challenge in elementary particle physics. The Yukawa couplings are undetermined in the gauge theories. To understand the origin of neutrino mass and mixing one way is to employ phenomenological approaches such as texture zeros [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23], hybrid textures [24][25][26][27][28], scaling [29][30][31][32][33][34][35][36], vanishing minor [37][38][39] etc. irrespective of details of the underlying theory. These different ansatze are quite predictive as they decrease the number of free parameters in neutrino mass matrix. The second way, which is more theoretically motivated, is to apply yet-to-be-determined non-Abelian flavor symmetry. In this approach a flavor symmetry group is employed in addition to the gauge group to restrict the Yukawa structure culminating in definitive predictions for values and/or correlations amongst low energy neutrino mixing parameters.
Another predictive ansatz is hybrid texture structure with one equality amongst elements and one texture zero in neutrino mass matrix. The hybrid texture of the neutrino mass matrix has been realized under S 3 ⊗Z 3 symmetry with in type-II seesaw framework assuming five scalar triplets with different charge assignments under S 3 and Z 3 [27]. Also, some of these hybrid textures have been realized under Quaternion family symmetry Q 8 [62]. In this work, we present a simple minimal model based on group A 4 with two right-handed neutrinos in type-I+II seesaw mechanism leading to hybrid texture structure for neutrino mass matrix. The same Higgs doublet is responsible for the masses of charged leptons and neutrinos [15]. In addition, one scalar singlet Higgs field χ and two scalar triplets ∆ i (i = 1, 2) are required to write A 4 invariant Lagrangian.
In Sec. II, we systematically discuss the model based on group A 4 and resulting effective Majorana neutrino mass matrix. Sec. III is devoted to study phenomenological consequences of the model. In this section we, also, study the implication to neutrinoless double beta decay (0νββ) process. Finally, in Sec. IV, we summarize the predictions of the model and their testability in current and upcoming neutrino oscillation/0νββ experiments.

II. THE A4 MODEL
The group A 4 is a non-Abelian discrete group of even permutations of four objects. It has four conjugacy classes, thus, have four irreducible representations(IRs), viz.: 1, 1 ′ , 1 ′′ and 3. The multiplication rules of the IRs are: ω ≡ e 2πi/3 and (a 1 , a 2 , a 3 ), (b 1 , b 2 , b 3 ) are basis vectors of the two triplets. Here, we present an A 4 model within type-I+II seesaw framework of neutrino mass generation. In this model, we employed one SU (2) L Higgs doublet Φ, one SU (2) L singlet Higgs χ and two SU (2) L triplet Higgs fields(∆ 1 , ∆ 2 ). The transformation properties of different fields under SU (2) L and A 4 are given in Table  I. These field assignments under SU (2) L and A 4 leads to the following Yukawa Lagrangian where,φ = iτ 2 φ * and y i (i = e, µ, τ, 1, 2, ∆ 1 , ∆ 2 ) are Yukawa coupling constants. The above Lagrangian leads to charged lepton mass matrix m l , right handed Majorana mass matrix m R and Dirac mass matrix m D given by after spontaneous symmetry breaking with vacuum expectation values(VEVs) as Φ 0 = ϑ √ 3 (1, 1, 1) T and χ 0 = ε for Higgs doublet and scalar singlet, respectively. Here U L is which diagonalizes m l , N = h χ ε, x = ϑy 1 and y = ϑy 2 . The type-I seesaw contribution to effective Majorana neutrino mass matrix is Using Eqns. (3) and (4) we get Assuming VEVs υ j (j = 1, 2) for scalar triplets ∆ 1 , ∆ 2 , respectively, the type-II seesaw contribution to effective Majorana mass matrix is where c = y ∆1 υ 1 and d = y ∆2 υ 2 . So, effective Majorana mass matrix is given as The charge lepton mass matrix m l can be diagonalized by the transformation where U R is unit matrix corresponding to right handed charged lepton singlet fields. In charged lepton basis the effective Majorana mass matrix is given by which symbolically can be written as where ∆ denotes the equality between elements and X denotes arbitrary non-zero elements. In literature, such type of neutrino mass matrix structure is referred as hybrid textures [24][25][26]. On changing the assignments of the fields we can have two more hybrid textures. For example, if we assign Similarly, the field assignments In the next section, we study the phenomenological con-sequences of these neutrino mass matrices.

III. PHENOMENOLOGICAL CONSEQUENCES OF THE MODEL
In charged lepton basis, the effective Majorana neutrino mass matrix, M ν is given by where V = U.P and U is Pontecorvo-Maki-Nakagawa-Sakata(PMNS) matrix and in standard PDG representation is given by where s ij = sin θ ij and c ij = cos θ ij . The phase matrix, P is where α, β are Majorana phases and δ is Dirac-type CP violating phase. The neutrino mass model described by Eqn.(10) imposes two conditions on the neutrino mass matrix M ν , viz.: where a = u = 1, b = m = n = 3 and v = 2 for neutrino mass matrix in Eqn. (9). It leads to two complex equations amongst nine parameters, viz.: three neutrino masses(m 1 , m 2 , m 3 ), three mixing angles(θ 12 , θ 23 , θ 13 ) and three CP violating phases(δ, α, β) and We solve Eqn. (16) and (17) for mass ratios m1 m3 and m2 m3 where the ratios R 13 ≡ m1 m3 and R 23 ≡ m2 m3 . The ratio R 23 can be obtained from R 13 using the transformation θ 12 → π 2 − θ 12 . The mass ratios R 13 and R 23 along with measured neutrino mass-squared differences provide two values of m 3 , viz.: m a 3 and m b 3 , respectively and is given by These two values of m 3 must be consistent with each other, which results in The ratios m1 m3 e −2i(β+δ) and m2 m3 e 2i(α−β−δ) , to first order in s 13 , is given by Using these approximated mass ratios we find  Using the experimental data shown in Table II With the help of constraints derived for δ, θ 23 for NH as well as IH, it is straightforward to show that R ν (Eqn. (22) Thus, the requirement that model-prediction for R ν must lie within its experimentally allowed range hints towards normal mass hierarchy. These approximated analytical results will be extremely helpful to comprehend the phenomenological predictions obtained from the numerical analysis which is based on the exact constraining Eqns. (18) and (19).
In numerical analysis, we have used Eqn.(22) as our constraining equation to obtain the allowed parameter space of the model i.e. R ν must lie within its 3σ experimental range. We have used latest global-fit data shown in Table II. The experimentally known parameters such as mass-squared differences and mixing angles are randomly generated with Gaussian distribution whereas CP violating phase δ is allowed to vary in full range (0 o − 360 o ) with uniform distribution(≈ 10 7 points). The mass ratios R 13 and R 23 depend on θ 12 , θ 23 , θ 13 and δ. Using experimental data shown in Table II, we first calculate the prediction of the model for R ν with normal as well as inverted hierarchy. It is evident from Fig. 1 that R ν is O(10 −1 ) for IH i.e. outside the experimental 3σ range of R ν which is, also, in consonance with above  In Fig. 2(a), we have depicted correlation between δ and θ 23 at 3σ. θ 23 = 45 o is not allowed because 1 − R 2 23 must be less than 1. Also, the point (θ 23 = 45 o , δ = 90 o or 270 o ) is not allowed otherwise R ν < 0. θ 23 is found to be above maximality and Dirac-type CP violating phase δ is constrained to a very narrow region in I st and IV th quadrant. In Fig. 2(b), 2(c) and 2(d), we have shown the normalized probability distributions of θ 23 and δ. The 3σ ranges of these parameters are given in Table III. One of the desirable feature of a neutrino mass model is its prediction of the observable(s) which can be probed outside the neutrino sector. One such process is 0νββ decay, the amplitude of which is proportional to effective Majorana neutrino mass |M ee | given by |M ee | = |m 1 c 2 12 c 2 13 + m 2 s 2 12 c 2 13 e 2iα + m 3 s 2 13 e 2iβ |. (29) In Fig. 3, we have shown sin 2 θ 23 − |M ee | correlation plot at 3σ. The important feature of the present model is the existence of lower bound |M ee | > 0.047eV(at 3σ) which is within the sensitivity reach of 0νββ decay experiments like SuperNEMO [63], KamLAND-Zen [64], NEXT [65,66], nEXO [67].
A similar analysis of neutrino mass matrices shown in Eqns. (11) and (12) reveals that these textures are not compatible with present global-fit data on neutrino masses and mixings including latest hints of normal hierarchical neutrino masses, higher octant of θ 23 and near maximal Dirac-type CP violating phase δ [7,20,68,69]. Furthermore, being a minimal model, it will have interesting implications for leptogenesis which will be discussed elsewhere. FIG. 2: δ − θ 23 correlation plot at 3σ (Fig. 2(a)) and probability distribution plots for θ 23 (Fig. 2(b)) and δ (Fig. 2(c) and 2(d)).

IV. CONCLUSIONS
In conclusion, we have presented a neutrino mass model based on A 4 flavor symmetry for leptons within type-I+II seesaw framework. The model is economical in terms of extended scalar sector and is highly predictive. The field content assumed in this work predicts three textures for M ν based on the charge assignments under SU (2) L and A 4 . However, only one(Eqn. (10)) is found to be compatible with experimental data on neutrino masses and mixing angles. We have studied the phenomenological implications of this texture in detail. The solar mass hierarchy i.e. R 2 23 − R 2 13 > 0 constrains Diractype CP violating δ to narrow ranges 87.50 o −89.70 o and 270.20 o − 272.50 o at 3σ. The sharp correlation between Dirac-type CP violating phase δ and atmospheric mixing angle θ 23 demonstrates the true predictive power of the model( Fig. 2(a)). The predictions for these less precisely known oscillation parameters(δ and θ 23 ) are remarkable which can be tested in neutrino oscillation experiments like T2K, NOνA, SK and DUNE to name a few. We have, also, calculated effective Majorana neutrino mass |M ee |. The important feature of the model is existence of lower bound on |M ee | which can be probed in 0νββ decay experiments like SuperNEMO, KamLAND-Zen, NEXT and nEXO. The main predictions of the model are: i. normal hierarchical neutrino masses.
A precise measurements of Dirac-type CP violating phase δ, neutrino mass hierarchy and θ 23 is important to confirm the viability of the model presented in this work.