Probing CP-violating Higgs-gauge Boson Couplings at Future Muon Collider

We explore the sensitivity of future muon colliders to CP-violating interactions in the Higgs sector, specifically focusing on the process $\mu^- \mu^+ \to h \bar{\nu_{l}} \nu_{l} \to b\bar{b} \bar{\nu_{l}}\nu_{l}$. Using a model-independent approach within the framework of the Standard Model Effective Field Theory (SMEFT), we analyze the contribution of dimension-six operators to Higgs-gauge boson couplings, emphasizing CP-violating effects. To simulate the process, all signal and background events are generated through MadGraph. The analysis provides 95\% confidence level limits on the relevant Wilson coefficients $\tilde{c}_{HB}$, $\tilde{c}_{HW}$, $\tilde{c}_{\gamma}$, with a comparative discussion of existing experimental and phenomenological constraints. Our best constraints on the $\tilde{c}_{HB}$, $\tilde{c}_{HW}$, $\tilde{c}_{\gamma}$ with an integrated luminosity of 10 ab$^{-1}$ are $[-0.017428;0.018991]$, $[-0.002880;0.002586]$ and $[-0.010784;0.011381]$, respectively. In this context, this study highlights the capability of future muon collider experiments to probe new physics in the Higgs sector, potentially offering tighter constraints on CP-violating Higgs-gauge boson interactions than those provided by current colliders.