Pseudoscalar Higgs Production at Muon Colliders: The Role of One-Loop Effective Vertices

We investigate the production of the pseudoscalar Higgs boson $A$ at muon colliders within the framework of Type-II and Type-X Two-Higgs-Doublet Model (2HDM) at the Next-to-Leading Order (NLO), utilizing an Effective Field Theory (EFT) approach. In particular, we analyze the level of enhancement to the cross section due to the inclusion of the one-loop corrections involving $\gamma$ and $Z$ boson fusion compared to the tree-level contribution. We find that for Type-II, including the effective vertices of $\gamma\gamma A$, $\gamma Z A$ and $ZZ A$, could lead to an enhancement of a factor of $\sim 2$ at low $m_A$ and low $\tan\beta$, whereas for Type-X, the enhancement could reach $\sim 10$ in the same regime. We also investigate the impact of the COM energy and $\tan \beta$ on the production cross section. We find that for the region of the parameter space not excluded by experiment, cross sections of $\gtrsim 1$ fb for Type-II, and $\gtrsim 5$ fb for Type-X, are possible, making the proposed muon collider a feasible alternative for probing the 2HDM extended Higgs sector.