Radiative lepton seesaw model in a non-invertible fusion rule and gauged $B-L$ symmetry

We propose a radiative lepton seesaw model with the ${\mathbb Z}_3$ Tambara-Yamagami fusion rule and gauged $B-L$ symmetry. The heavier right-handed neutral fermions ($N_R$) simultaneously contribute to the masses of charged-leptons and neutrinos, and three families of $N_R$ are necessary in realizing realistic mass matrices. On the other hand, we introduce a gauged $B-L$ symmetry to preserve the lepton seesaw mechanism. As a result, this gauged symmetry not only assures the three families of $N_R$ from view point of chiral gauge anomaly cancellation, but also provides a dominant annihilation cross section via Higgs portal in order to explain the correct relic density of dark matter. Finally, we show several numerical results satisfying the neutrino oscillation data, lepton flavor violations, muon $g-2$, and relic density of dark matter.