Investigation of Charged Higgs Bosons Production from Vector-Like T Quark Decays at eγ Collider

Within the extended framework of the Two-Higgs-Doublet Model Type II (2HDM-II), enhanced by a vector-like quark (VLQ) doublet $TB$, we present a comprehensive analysis of the process $e^{-}\gamma \rightarrow b\nu_{e}\bar{T}$ at future high-energy $e\gamma$ colliders, focusing on the decays $\bar{T} \rightarrow H^{-} \bar{b}$ and $H^{-} \rightarrow \bar{t}b$. Using current theoretical and experimental constraints, we calculate production cross sections for both unpolarized and polarized beams at center-of-mass energies of $\sqrt{s} = 2$ and 3 TeV, demonstrating that polarized beams significantly enhance detection prospects by increasing production rates. By analyzing kinematic distributions, we establish optimized selection criteria to effectively separate signal events from background. At $\sqrt{s} = 2$ TeV with an integrated luminosity of 1500 fb$^{-1}$, we find exclusion regions within $s_R^d\in[0.085,0.16]$ for $m_T\in [1000,1260]$ GeV and a discovery potential within $s_R^d\in[0.14,0.17]$ for $m_T\in[1000,1100]$ GeV, with these regions expanding to $s_R^d\in [0.05,0.15]$ for $m_T\in[1000,1340]$ GeV and $s_R^d \in [0.11, 0.17]$ for $m_T\in[1000,1160]$ GeV at 3000 fb$^{-1}$. At $\sqrt{s} = 3$ TeV and 1500 fb$^{-1}$, we identify exclusion regions of $s_R^d\in[0.055,0.135]$ for $m_T \in [1000, 1640]$ GeV and discovery regions of $s_R^d \in [0.09, 0.15]$ for $m_T \in [1000, 1400]$ GeV, which further expand to $s_R^d \in [0.028, 0.12]$ for $m_T\in[1000,1970]$ GeV and $s_R^d \in[0.04,0.122]$ for $m_T\in[1000,1760]$ GeV at 3000 fb$^{-1}$. Our findings emphasize the increased detection potential at higher center-of-mass energies, particularly at 3 TeV compared to 2 TeV, with notable improvements when polarized beams are utilized. We also account for the effects of initial state radiation, beamstrahlung, and systematic uncertainties, which influence both exclusion and discovery prospects.