Modular $A_4$ symmetry and light dark matter with gauged $U(1)_{B-L}$

We propose a radiative seesaw model with a light dark matter candidate (DM) under modular $A_4$ and gauged $U(1)_{B-L}$ symmetries, in which neutrino masses are generated via one-loop level, and we have a bosonic DM candidate $\sim 0.01-50$ GeV. DM is stabilized by nonzero modular weight and a remnant $Z_2$ symmetry of $U(1)_{B-L}$ symmetry. The naturalness of tiny DM mass is indirectly realized by the radiatively induced mass of neutral fermions that interact with our DM candidate. Thus, DM mass does not have to exceed the neutral fermion masses, whose scale is assumed to be 50 GeV. Taking several benchmark points of DM mass in DM analysis, we show the numerical analysis for the neutrino oscillation data in the normal and inverted hierarchy cases and find several features for both cases.