A modular A4 symmetry realization of two-zero textures of the Majorana neutrino mass matrix

Abstract

We show how to realize two-zero textures of the Majorana neutrino mass matrix M ν based on modular A 4 invariant models without flavons. In these models, all matter fields are assigned to three inequivalent singlets, 1, 1 ′ and 1 ′′ , of the finite modular group Γ 3 ≃ A 4 . Considering tensor products of the A 4 group, it is easy to make the charged lepton mass matrix M ℓ diagonal. Since not all modular forms of a specific weight and level 3 can be arranged into three inequivalent singlets of A 4 simultaneously, we can always make some entries in M ν vanish by properly assigning the representations and modular weights for the matter fields. We consider two cases where neutrino masses originate from the Weinberg operator and the type-I seesaw mechanism, respectively. For the former case, all seven viable two-zero textures of M ν (A 1,2 , B 1,2,3,4 and C) can be realized successfully. For the latter case, only five of them (namely A 1,2 , B 3,4 and C) can be achieved due to the intrinsic structure of the right-handed neutrino mass matrix M R in our assumption for simplicity. * Email: zhangdi@ihep.ac.cn Compelling evidences obtained from solar, atmospheric, reactor and accelerator neutrino experiments in the last two decades [1] have proved the existence of neutrino oscillations [2][3][4], implying that neutrinos have nonzero and nondegenerate masses, and flavor mixing in the lepton sector exists. The standard model (SM) itself tells us nothing about the quantitative details of Yukawa interactions, thus it is rather challenging to explore the underlying flavor structures of charged fermions and massive neutrinos. It calls for new physics beyond the SM to control the Yukawa couplings of quarks and leptons and understand flavor mixing. In view of the fact that a convincing flavor theory is lacking, approaches such as flavor symmetries and texture zeros or their combinations have been widely explored to shed light on the flavor secrets of fermions.The flavor symmetry with a non-Abelian discrete group is a popular approach to explain lepton flavor mixing pattern (see reviews [5][6][7][8] and references therein). Recently, a new and attractive approach with the modular symmetry applied to the lepton flavor problem has been proposed in [9]. In such a modular invariant model, only a few flavons or even no flavons need to be introduced and the Yukawa couplings or the right-handed neutrino mass matrix in the type-I seesaw mechanism [10-14] are regarded as modular forms which are holomorphic functions of the modulus τ and transform non-trivially under the modular symmetry. In the model without flavons, the vacuum expectation value (VEV) of the modulus τ is the only source of symmetry breaking. In the light of these advantages, a lot of works have been done and successfully predicted neutrino masses and mixing parameters in terms of a few input parameters, such as the models with the modular Γ 2 ≃ S 3 [15,16] 34,36,37] symmetry. This approach has also been extended to the quark sector [38][39][40][41][42].Assuming that neutrin...

Modular invariant models of leptons at level 7

“…The finite modular groups Γ 2 ∼ = S 3 [3][4][5][6], Γ 3 ∼ = A 4 [1,3,4,[7][8][9][10][11][12][13][14], Γ 4 ∼ = S 4 [13,[15][16][17][18][19]] and Γ 5 ∼ = A 5 [18,20,21] have been considered. For example, simple A 4 modular models can reproduce the measured neutrino masses and mixing angles [1,8,12].…”

The minimal seesaw model with a modular S4 symmetry

“…In this case, the flavon fields are no longer needed. In the literature, there have been a great number of works on the model building based on the modular group Γ N , which for a given value of N is isomorphic to the well-known non-Abelian discrete symmetry groups, e.g., Γ 2 S 3 [26][27][28][29], Γ 3 A 4 [30][31][32][33][34][35][36][37][38][39][40], Γ 4 S 4 [41][42][43] and Γ 5 A 5 [44][45][46]. Moreover, other interesting aspects of modular symmetries have also been studied, such as the combination of modular symmetries and the generalized CP symmetry [47], multiple modular symmetries [48,49], the double covering of modular groups [50], the A 4 symmetry JHEP05(2020)017 from the modular S 4 symmetry [51,52], the modular residual symmetry [45,53] and the unification of quark and lepton flavors with modular invariance [55].…”

Explicit perturbations to the stabilizer τ = i of modular $$ {A}_5^{\prime } $$ symmetry and leptonic CP violation

“…Once the modulus τ acquires its vev, the Yukawa couplings are determined and thus leptonic flavor mixing pattern is obtained. Finite modular groups, such as Γ 2 S 3 [13][14][15], Γ 3 A 4 [12,[16][17][18][19][20][21][22][23][24][25][26][27], Γ 4 S 4 [28][29][30][31][32][33], Γ 5 A 5 [34][35][36], Γ 7…”