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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 masseigenvalue, and only the normal mass hierarchy of neutrinos is favored throughthe modular $S_4$ symmetry. In our numerical $\Delta \chi^2$ analysis, weespecially find rather sharp prediction on sum of neutrino masses to be around$60$ meV in addition to the other predictions.

Linear seesaw model with a modular $S_4$ flavor symmetry