Search for Singly Charged Higgs in Vector Boson Scattering at the ep Colliders

We search for a five-plet singly charged Higgs $H_5^\pm$ in the Georgi-Machacek model at the ep colliders. The charged Higgs bosons are produced via the $ZW^{\pm}$ fusion process $p\, e^- \to j\, e^-\, H_5^\pm$, and decay as $H_5^\pm \to Z\, W^{\pm} \to (l^+ l^-)\, (jj)$. With a detector-level simulation at the FCC-eh and LHeC, a multi-variate analysis is performed to yield limits on the production cross section times branching ratio $\sigma (p\, e^- \to j e^- H_5^\pm) \times {\rm BR}(H_5^\pm \to Z\, W^\pm)$ and on the model parameter $\sin\theta_H$ for charged Higgs masses between 200 and 1000 GeV. The effects of electron beam polarization are also investigated.