Three-loop induced neutrino mass model in a non-invertible symmetry

We propose a new type of radiatively induced neutrino masses at three-loop level based on the Ma model, introducing a non-invertible symmetry in the class under a ${\mathbb Z_2}$ gauging of ${\mathbb Z_6}$ symmetry and adding three isospin doublet vector-like fermions $L'$ and singlet boson $S_0$. Under this symmetry, the Yukawa interactions directly related to the neutrino masses are not allowed at tree-level. However it is allowed at one-loop level due to $L'$ and $S_0$ as well as $\eta$, which is no longer invariant under this symmetry. Therefore, the symmetry is dynamically broken. Intriguingly, $\eta$ plays important roles in contributing to both the radiative matrices $y^\eta$ and $m_\nu$. After constructing our model, we show some numerical analyses to satisfy the lepton flavor violations, muon anomalous magnetic dipole moment, and a boson dark matter candidate $S_0$ or $\eta_R$ for the cases of normal hierarchy and inverted hierarchy. Then, we demonstrate allowed space for our input parameters.