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.