Particle swarm optimization based analysis to unlocking the neutrino mass puzzle using \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$SU(2)_L \times U(1)_Y \times A_{4}\times S_2\times Z_{10} \times Z_{3}$$\end{document} flavor symmetry

• Published in: Scientific reports 2025-02, Vol.15
• Main Authors: Aslam, Muhammad Waheed, Zafar, Abrar Ahmad, Aslam, Muhammad Naeem, Bhatti, Abdul Aziz, Hussain, Talab, Iqbal, Munawar, Abdo, Mohammed S.
• Format: Article
• Language: English
• ISSN: 2045-2322
• DOI: 10.1038/s41598-024-81791-3

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 the observed patterns of neutrino mixing. In this study, we have examined a model within \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$SU(2)_L \times U(1)_Y \times A_{4}\times S_2\times Z_{10} \times Z_{3}$$\end{document} 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, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_{PMNS}$$\end{document} matrix, neutrino masses: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${m_{1}^{\prime }}^{(N)}(upper)=4.0000 \times 10^{-2}\ eV,$$\end{document} \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${m_{2}^{\prime }}^{(N)}(upper)=4.0000 \times 10^{-2}\ eV,$$\end{document} \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${m_{3}^{\prime }}^{(N)}(upper)=4.0000 \times 10^{-2}\ eV,$$\end{document} \documentcla