Associated charged Higgs production within the 2HDM: $e^-e^+$ versus $\mu^-\mu^+$ colliders

Our goal is to investigate the charged Higgs phenomenology in the framework of 2HDM at the upcoming $e^+e^-$ and muon colliders. We are primarily concerned with the associated production processes with a fermion pair: $\ell^+ \ell^- \to \tau^+ \nu_{\tau }H^-$ and $\ell^+ \ell^- \to t \bar{b} H^-$, as well as with the W boson and a neutral Higgs boson: $\ell^+ \ell^- \to W^\pm H^\mp S$ ($S=h,\,H,\, A$) and $\ell^+ \ell^- \to W^\pm H^\mp Z$. We first update the results for $e^+e^- \to \{ \tau^+ \nu_{\tau }H^-\ , \ t \bar{b} H^-\}$ and then discuss our findings for $e^+e^- \to \{ W^\pm H^\mp S\}$ for various center of mass energies 500 GeV, 1 TeV, 1.5 TeV and 3 TeV. In the case of muon collider, we show that the new s-channel and t-channel diagrams can increase the cross sections by virtue of their Yukawa couplings. We systematically compare our results for the muon collider with those obtained at the International Linear (ILC) and Compact Linear (CLIC) colliders. We select benchmark points and conduct signal-background analyses, incorporating detector simulations. For a 3 TeV muon collider, our results show an exclusion region at the 2$\sigma$ level and a discovery region at the 5$\sigma$ level.