Dark matter for Majorana neutrinos in a $\mathbb{Z}_4$ symmetry

In the minimal Scotogenic model with fermionic dark matter, the neutrino Yukawa matrix is fine-tuned to satisfy current bounds on $\mu \rightarrow e \gamma$ while producing dark matter through thermal freeze-out. One way to avoid this bound is to open a new annihilation channel for dark matter (DM) with a scalar field that breaks the lepton number spontaneously. This introduces a physical Goldstone boson $a$, which can constitute a light relic and opens up the lepton flavor violating channel $\mu\rightarrow e a$. In this work, we introduce a model that contains the new annihilation channel for fermionic dark matter but avoids the introduction of the Goldstone boson by using a discrete $\mathbb{Z}_4$ instead of a continuous symmetry. In this way, cosmological constraints on the light relic and experimental limits on lepton flavor violating decays to the Goldstone boson are avoided. We analyze the experimental viability of the model, identifying the region of parameter space where lepton flavor violating constraints are not saturated, DM has the correct relic density, and DM direct detection experiments are sensitive to the DM candidate.