Broken Scaling Neutrino Mass Matrix and Leptogenesis based on A$_4$ Modular invariance

In this work, we have proposed a modular $A_4$ symmetric model of neutrino mass which, simultaneously, explains observed baryon asymmetry of the Universe(BAU). In minimal extension of the standard model(SM) with two right-handed neutrinos we work in a supersymmetric framework. At Type-I seesaw level, the model predicts scaling in the neutrino mass matrix. In order to have correct low energy phenomenology, we propose two possible scenarios of scale-breaking in the neutrino mass matrix emanating from Type-I seesaw. Scenario-1 is based on the dimension-5 Weinberg operator whereas scenario-2 implements Type-II seesaw via scalar triplet Higgs superfields($\Delta,\bar{\Delta}$). Interestingly, the breaking patterns in both, otherwise dynamically different scenarios, are similar which can be attributed to the same charge assignments of superfields($\Delta,\bar{\Delta}$) and the Higgs superfield $H_u$ under modular $A_4$ symmetry. The breaking is found to be proportional to the Yukawa coupling of modular weight 10($Y_{1,1'}^{10}$). We, further, investigates the predictions of the model under scenario-2 (Type-I+II) for neutrino mass, mixings and matter-antimatter asymmetry of the Universe. The model predicts normal hierarchical neutrino masses and provide a robust range ($0.05-0.08$)eV for sum of neutrino masses($\sum m_{i}$). Lepton number violating $0\nu\beta\beta$ decay amplitude($M_{ee}$) is obtained to lie in the range ($0.04-0.06$)eV. Future $0\nu\beta\beta$ decay experiments such as NEXT and nEXO shall pose crucial test for the model. Both $CP$ conserving and $CP$ violating solutions are allowed in the model. Interesting correlations are obtained, specially, between Yukawa couplings of modular weight 2 and complex modulus $\tau$. The model exhibit consistent explanation of BAU for right-handed Majorana neutrino mass scale in the range ($(1-5)\times10^{13}$)GeV.