Charged Higgs scalar production in single-top mode (and other) at future $ep$ colliders in the Minimal Supersymmetric Standard Model

We study charged Higgs boson production at future electron-proton colliders in the framework of the Minimal Supersymmetric Standard Model. We focus our attention to the case of single-top production and decay through the channel $t\to b{H}^{\pm }$ and of vector-scalar fusion via $W^{\pm \ast }{\mathrm{\Phi }}^{\ast }\to H^{\pm }$ (where $\mathrm{\Phi }=H,h$ and $A$). We consider the signature $H^{\pm }\to \tau {\nu }_{\tau }$ and compare it to the irreducible background from Standard Model interactions. For $M_{H^{\pm }}<m_t$, the $H^{\pm }$ signal is accessible through lepton universality breaking if $M_A<100-120$ GeV at both low and large values of $\tan \beta$. Furthermore, although the bulk of the production cross section comes from single-top events, a sizable contribution due to vector-scalar-scalar interactions should be observable at large $\tan \beta$, this possibly offering some insights into the structure of the scalar sector of the theory. The possibility of the CERN collider running in the LEPXLHC mode is considered in detail