Comparison between ${\mu }^{-}{\mu }^{+}$ and $e^{-}{e}^{+}$ colliders for charged Higgs production in 2HDM

We study the phenomenology of the charged Higgs boson at future muon colliders. We investigate both the pair production ${\mu }^{+}{\mu }^{-}\to H^{+}{H}^{-}$, the single production ${\mu }^{+}{\mu }^{-}\to W^{\pm }{H}^{\mp }$, as well as the Vector Boson Fusion (VBF) $\{e^{+}{e}^{-},{\mu }^{+}{\mu }^{-}\}\to \nu \bar{\nu }{H}^{+}{H}^{-}$. We show that the neutral Higgs exchange diagrams in the muon collider case can lead to a significant boost in the cross sections through their Yukawa couplings. Our results for the muon collider are systematically compared to the corresponding ones at $e^{+}{e}^{-}$ machines. It is demonstrated that the vector boson fusion (VBF) $e^{+}{e}^{-}\to \nu \bar{\nu }{H}^{+}{H}^{-}$ can compete with the mentioned $2\to 2$ processes. We select benchmark points and perform signal-background analyses, taking into account detector simulations. We demonstrate the discovery region at $5\sigma$ and the excluded region at $2\sigma$ levels at a 3 TeV muon collider.