Linear seesaw model with a modular $S_4$ flavor symmetry

We discuss a linear seesaw model with as minimum field content as possible, introducing a modular $S_4$ with the help of gauged $U(1)_{B-L}$ symmetries. Due to rank two neutrino mass matrix, we have a vanishing neutrino mass eigenvalue, and only the normal mass hierarchy of neutrinos is favored through the modular $S_4$ symmetry. In our numerical $\Delta \chi^2$ analysis, we especially find rather sharp prediction on sum of neutrino masses to be around $60$ meV in addition to the other predictions.