Unveiling the CP-odd Higgs in a Generalized 2HDM at a Muon Collider

We revisit the generalized two-Higgs-doublet model (2HDM) with a minimally perturbed lepton specific (Type-X) Yukawa sector in context of the muon collider. The model offers a large region of parameter space where it can provide a solution to the observed muon ($g-2$) excess. Specifically the low mass CP odd scalar of this model plays an important role to fit this anomaly. Interestingly, the coupling of the pseudo scalar with the muons is proportional to $\tan \beta$ (the ratio of VEVs of the Higgs doublets) which dominantly controls the one loop and two loop contributions of the muon magnetic moment, therefore it can be probed directly at a muon collider with greater sensitivity in comparison to Large Hadron Collider (LHC). Primarily focusing on a parameter space where muon $g-2$ anomaly is satisfied, we explore the available theoretical and experimental constraints arising from vacuum stability, unitarity, lepton falvour violation (LFV), electroweak precision constraint, B-meson decay and collider experiments on the model. In the parameter space where all these constraints are satisfied, we propose to study the production of the light pseudo-scalar $A$ in association with a photon at a $3$ TeV muon collider. Subsequent decay of $A$ into a $\tau$ pair will produce a striking $l^{+}{l}^{-}\gamma$ plus missing energy signature.