Connecting Vector-like Muons, pNGB Dark Matter and Electroweak Phase Transition through Collider and Gravitational Wave
We study an extension of the Standard Model with two complex scalar singlets, $S_2$ and $S_3$, charged under $\mathbb{Z}_2$ and $\mathbb{Z}_3$, respectively, and a muon-philic vector-like lepton $ψ$ carrying the same $\mathbb{Z}_3$ charge as $S_3$. The pseudoscalar associated with the $\mathbb{Z}_2$ sector remains stable due to the CP symmetry of the potential and serves as the dark matter (DM) candidate, with its mass generated through the corresponding soft breaking of the global $U(1)$ symmetry. The vector-like muon couples to the $\mathbb{Z}_3$ scalar and renders the second pseudoscalar unstable, thereby realizing an effectively single-component pNGB DM scenario. The pNGB nature strongly suppresses the tree-level spin-independent direct-detection cross section, while viable parameter regions reproduce the observed relic abundance and satisfy LHC monojet constraints. We further compute the one-loop contribution to DM-nucleon scattering and find that the resulting cross section remains below current experimental limits while being potentially accessible to future direct-detection experiments. At a multi-TeV muon collider, scalar mediated $ t$-channel processes can significantly enhance vector-like-muon pair production. We perform a detailed multivariate analysis for a future muon collider at 3 TeV center-of-mass energy. We also identify viable benchmark points exhibiting strong first-order electroweak phase transitions (SFOEWPT) with successful bubble nucleation, which can generate potentially observable stochastic gravitational wave (GW) signals. These results highlight the complementarity of dark matter searches, muon-collider probes, and SFOEWPT, with the resulting GW signals providing an additional probe of the extended scalar sector.