A new type of lepton seesaw model in a modular $A_4$ symmetry

We propose a new type of lepton seesaw model introducing a modular $A_4$ flavor symmetry in which isospin doublet vector fermions play an important role in constructing seesaw mechanisms for both the charged-lepton mass matrix and the neutrino one. The charged-lepton mass matrix is induced via the Dirac seesaw with a two-by-two block mass matrix. On the other hand, the neutrino mass matrix is generated via a seesaw with four-by-four block mass matrix where the right-handed neutral fermions are also added. Remarkably the neutral fermion mass matrix provides us a new type of seesaw formula for active neutrino mass. Through our chi square analysis, we find some tendencies about observables focusing on two fixed points $\tau=i,\ \omega$.