Neutrino mass model at a three-loop level from a non-holomorphic modular $A_4$ symmetry

We study a three-loop induced neutrino mass scenario from a non-holomorphic modular $A_4$ flavor symmetry and reach the minimum scenario leading predictions of the lepton masses, mixing angles, and Dirac and Majorana phases, which are shown through the chi square analyses. In addition, we discuss the lepton flavor violations, muon anomalous magnetic moment, lepton universality, and relic density of the dark matter candidate. And, we find that we need to extend our model if we satisfy the observed relic density of dark matter within the limit of perturbation where it can be done by adding one singlet scalar boson without changing predictions in neutrino sector.