Particle Swarm Optimization Based Analysis to Unlocking the Neutrino Mass Puzzle using $A_{4}\times Z_{3}\times Z_{10}$ Flavor Symmetry

Abstract: New research has highlighted a shortfall in the Standard Model (SM) because it predicts neutrinos to have zero mass. However, recent experiments on neutrino oscillation have revealed that the majority of neutrino parameters indeed indicate their significant mass. In response, scientists are increasingly incorporating discrete symmetries alongside continuous ones for better justification of observed patterns of neutrino mixing. In this study, we have examined a model within $A_4\times Z_3\times Z_{10}$ symmetry to estimate the neutrino masses using particle swarm optimization technique for both mass hierarchy of neutrino. This model employed a hybrid seesaw mechanism, a combination of seesaw mechanism of type-I and type-II, to establish the effective Majorana neutrino mass matrix. After calculating the mass eigenvalues and lepton mixing matrix upto second order perturbation theory in this framework, this study seeks to investigate the scalar potential for vacuum expectation values (VEVs), optimize the parameters, $U_{PMNS}$ matrix, neutrino masses: $|{m_{1}^{\prime}}^N|=0.0292794-0.0435082\ eV$, $|{m_{2}^{\prime}}^N|=1.78893\times 10^{-18}-0.0293509\ eV$, $|{m_{3}^{\prime}}^N|=0.0307414-0.0471467\ eV$, $|{m_{1}^{\prime}}^I|=0.00982013-0.0453623\ eV$, $|{m_{2}^{\prime}}^I_|=0.0379702-0.0471197\ eV$, and $|{m_{3}^{\prime}}^I|=0.0122063-0.027544\ eV$, effective neutrino mass parameters: $\langle {m_{ee}} \rangle^N=(0.170-3.93)\times10^{-2}\ eV$, $\langle {m_{\beta}} \rangle^N=(0.471-1.39)\times10^{-2}\ eV$, $\langle {m_{ee}} \rangle^I=(1.85-4.55)\times10^{-2}\ eV$ and $\langle {m_{\beta}} \rangle^I=(2.26-4.56)\times10^{-2}\ eV$, are predicted for both mass hierarchy through particle swarm optimization (PSO), showing strong agreement with recent experimental findings.