Baryon Asymmetry and Corrections to Scaling Neutrino Mass Matrix in Type-I+II Seesaw Model Under A \(_4\) Modular Invariance

The finite modular groups are isomorphic to permutation groups e.g. \(\Gamma _{3}\simeq \) A \(_4\) . Apart from usual irreducible representations of the permutation groups, they have modular weights as new symmetry charges. The Yukawa couplings transforms as modular forms of complex modulus \(\tau \) acquiring suitable charges of the underlying symmetry. In this work, we propose a scenario implementing the correction to scaling neutrino mass matrix and investigate baryogenesis based on A \(_4\) modular symmetry within Type-I+II seesaw framework. In fact the scaling neutrino mass matrix results in vanishing reactor mixing angle ( \(\theta _{13}\) ), inverted ordering of neutrino masses with vanishing lowest neutrino mass eigenvalue ( \(m_{3}=0\) ). In the proposed model, field content comprises of the standard model particles, two chiral neutrino superfields ( \(N_{1}^{c}\) , \(N_{2}^{c}\) ) and scalar singlet weighton field ( \(\phi \) ) which results in scaling neutrino mass matrix through Type-I seesaw. The correction to scaling neutrino mass matrix is manifested through Type-II seesaw obtained by introducing a supersymmetric pair of scalar triplet fields ( \(\Delta \) , \(\bar{\Delta }\) ). In particular, correction to scaling neutrino mass matrix is found to be proportional to modular Yukawa couplings of weight 10 ( \(Y_{1,1'}^{10}\) ). The model satisfies the neutrino oscillation data and cosmological constraint on sum of neutrino masses ( \(\sum m_{i}\le 0.12\) eV). The modular Yukawa couplings of modular weight 2 are sensitive to the imaginary part of complex modulus \(\tau \) , only. Also, we have studied the implications of the model for neutrinoless double beta decay ( \(0\nu \beta \beta \) ). The effective Majorana mass parameter ( \(M_{ee}\) ) is found to be within ( \(0.04-0.06\) ) eV which is well within the sensitivity reach of \(0\nu \beta \beta \) decay experiments. Furthermore, there exist robust lower bound on sum of neutrino masses ( \(\sum m_{i}\ge 0.05\) eV). Also, in order to generate a consistent baryon asymmetry of the universe the right-handed neutrino mass is found to be in the range \(((1-5)\times 10^{13})\) GeV implying that the flavor effects are negligible.