Prospects for discovering a light charged Higgs boson within the NMSSM at the FCC-eh collider

We analyze the prospects of observing relatively light charged Higgs bosons ($h^{\pm}$) in their decays via $h^{-} \to s \bar c + s \bar u$ at the upcoming Future Circular Collider in hadron-electron mode (FCC-eh) with $\sqrt s \approx 3.5$ TeV. Assuming that the intermediate Higgs boson ($h_2$) is Standard Model (SM)-like, we study the production of $e^- b \to \nu_e h^{-} b$ (also $b$ could be $\bar b$ in both initial and final states) in the framework of the Next-to-Minimal Supersymmetric Standard Model (NMSSM). We consider constraints from Dark Matter (DM), super-particle and the Higgs boson data. The charged Higgs boson decays into light flavors leads to a three-jets with missing transverse energy signal with one $b$-tagged jet. Our results show that light charged Higgs bosons with mass close to, e.g., 114(121) GeV have the maximal significance of 3.2(1.8)$\sigma$, upon using normal cut based selections and after 1 ab$^{-1}$ of luminosity. However, we further adopt an optimization technique to enhance the latter to 4.4 (2.2)$\sigma$, respectively.