Probing Single Production of Vector-Like $T$ Quarks through the $T \to H^+ b$ Decay Mode with $H^+ \to t b$ at the HL-LHC

We study the single production and decay of a vector-like top quark ($T$) (VLT) with charge $+2/3$ in the context of the Two Higgs Doublet Model Type II (2HDM-II), extended by a vector-like quark (VLQ) doublet ($TB$). Our focus is on the decay mode where the $T$ quark decays into a charged Higgs boson ($H^+$) and a bottom quark, followed by the subsequent decay of $H^+$ into a top and bottom quark. We specifically examine the collider process $pp \to qg \to T^{+}\bar{b}j \to H^{+}b\bar{b}j \to W^{+}b\bar{b}j \to 1\ell + 4b + 1j + \slashed {E}_T$, resulting in final states with multiple bottom quarks, a single jet, one lepton, and missing transverse energy. Our results demonstrate that this channel offers a promising opportunity for discovering both the VLT and the charged Higgs boson, providing strong detection potential across a broad parameter space.