Probing the Inert Scalar Sector of the Inert Doublet Model via Vector-Boson Fusion at a Muon Collider
We investigate the sensitivity of a future high-energy muon collider (MuCs) to the inert scalar sector of the Inert Doublet Model (IDM) through a comprehensive set of $2\to4$ electroweak gauge-boson fusion processes. The scalar spectrum consists of two CP-even neutral scalars: $h$ and $H$, one CP-odd neutral scalar $A$, and a pair of charged scalars $H^\pm$. We assume the neutral scalar $H$ to be the lightest inert state and the dark matter (DM) candidate, while the observed $125$~GeV Higgs boson is identified with the scalar of the Standard Model-like doublet. We consider charged Higgs pair production: $\mu^+\mu^-\to H^+H^-\nu\bar{\nu}$ induced by WW fusion, as well as the associated channels: $\mu^+\mu^-\to HH^\pm\mu^\mp\nu$ and $\mu^+\mu^-\to AH^\pm\mu^\mp\nu$, induced by $WZ$ and $W\gamma$ fusion. Owing to their Vector Boson Fusion (VBF) origin, the corresponding cross sections increase significantly with collider energy, making them particularly well suited for exploration at a high-energy MuCs. We first study the $2\to 2$ subprocesses $WW\to H^+ H^-$ and $WV\to H^+ S_i$, $V=Z, \gamma$ and $S_i=H$ or $A$ and establish the destructive interference among the contributing diagrams. After imposing theoretical consistency requirements and current experimental constraints, including those from DM and collider searches, we evaluate the corresponding signal rates at $\sqrt{s}=3$, $10$, and $20$~TeV. We further perform a detector-level Monte Carlo analysis combined with a cut-based signal-background selection. Our results demonstrate that the sensitivity to inert scalar production increases significantly with collision energy, establishing the 10 and 20~TeV MuCs stages as powerful probes of the IDM inert scalar sector.