Probing the singlet-triplet scotogenic model with lepton flavor violation, gravitational waves and dark matter

We perform a comprehensive phenomenological analysis of neutrino masses, lepton-flavor violation, gravitational-wave signature induced by the first-order electroweak phase transition, and scalar dark matter within the framework of the singlet--triplet scotogenic model. We derive general analytical expressions for the one-loop Wilson coefficients of the penguin and box contributions to lepton-flavor-violating processes, considering generic scalar and fermion mediators. These general results are then applied to the singlet--triplet scotogenic model to calculate the branching ratios of $μ\to eγ$ and $μ\to e\overline{e}e$, as well as the coherent $μ$--$e$ conversion rate. We systematically investigate the lepton-flavor-violation phenomenology and the constraints from current and future experiments. It is found that the coupling $λ_5$ plays a key role in connecting neutrino masses, lepton flavor violation, and dark matter direct detection. We also study the finite-temperature effective potential and the gravitational-wave signatures of the resulting first-order phase transition. By analyzing the thermal history of the model, we identify the viable parameter space allowing for a first-order electroweak phase transition in the early Universe and find that the portal parameters $λ_1^Ω$ and $μ_1$ play a crucial role in determining the transition strength. The corresponding stochastic gravitational-wave spectra can be probed by future gravitational-wave observatories. As a representative example, we identify a benchmark point that simultaneously satisfies the current constraints from lepton-flavor-violation and dark-matter experiments and generates a stochastic gravitational-wave spectrum that may be probed by the Laser Interferometer Space Antenna, the Big Bang Observer, and the Ultimate Deci-hertz Interferometer Gravitational-wave Observatory.