Higgs Boson Production in Association with a Single Top Quark as a Probe of the Top Yukawa Coupling

This paper provides a detailed analysis of the associated production of the Higgs boson with a single top quark ($tH$) in proton-proton collisions at $\sqrt{s} = 13~\mathrm{TeV}$ and $14~\mathrm{TeV}$. Based on the ATLAS search, we have employed innovative modeling approaches to improve the sensitivity to new physics and improve Standard Model constraints. The major goals are to optimize the data selection, statistical error estimation, determination of physical limits on the top quark Yukawa coupling ($κ_t$), and future experimental projections for HL-LHC. Simulations with MadGraph5aMC@NLO at LO+MLM give cross-sections of $σ_{tHq}$ and $σ_{tWh}$ in SM, scaled by K-factors to simulate NLO accuracy. For inverted $κ_t = -1$ (ITC), positive enhancements are seen, consistent with constructive interference. Kinematic distributions ($H_T$, $p_T(h)$, $η[j]$, $ΔR[t,h]$) are checked against ATLAS expectations, confirming the optimized approach to maximize signal extraction and suppress systematics.