Quark and lepton model with flavor specific dark matter and muon $g-2$ in modular $A_4$ and hidden $U(1)$ symmetries

We propose a quark and lepton model explaining their masses, mixings, and CP violating phases, introducing modular $A_4$ and hidden gauged $U(1)$ symmetries. The hidden $U(1)$ brings us heavier Majorana fermions that are requested by chiral anomaly cancellations, and we work on a canonical seesaw scenario due to their neutral particles.In this framework, we search for favorite parameter space to satisfy both the experimental values and show predictions, applying the $\chi$ square analysis.Then, we discuss a bosonic dark matter candidate that only annihilates into muon state due to the modular $A_4$ flavor symmetry where we suppose the main interaction of dark matter to be Yukawa terms. And we study muon anomalous magnetic dipole moment where there are not any constraints of lepton flavor violations thanks to this flavor symmetry. Finally, we show the allowed space to satisfy the observed relic density of dark matter and the muon anomalous magnetic dipole moment.