Axion Dark Matter and additional BSM aspects in an extended 2HDM setup

We illustrate and discuss the phenomenology of a model featuring a two-Higgs doublet sector augmented by two $SU(2)$ singlet scalars. The gauge symmetry group is extended as well with a $U(1)_{B_{i}-L_{i}}$ component whose spontaneous breaking leads to the gauge boson which has an important effect in the muon (g-2). A global PQ symmetry is introduced upon its breaking we have the axion particle which is also DM in our work. In particular, we have focussed on Type-X and Type-II 2HDM models and found out that (g-2) can not be explained only by the scalar sector for Type-II 2HDM mainly due to stringent constraint from $b \rightarrow s \gamma$ resulted in $M_{H^{\pm}} > 800$ GeV. For Type-II 2HDM, we can have axion coupling with the gluons which generates the axion potential and possible explanation for the strong CP problem. The proposed model accommodates neutrino masses via the Type-I see-saw mechanism with an upper bound on the right-handed neutrino mass 1 GeV (1 TeV) for Type-II (Type-X) 2HDM due to the presence of Planck scale suppressed operators. Moreover, we also have additional scalars which affect the oblique parameters and hence the W-boson mass which leads us to explain the W-boson mass observed at CDF-II detector. The most stringent constraints on the masses and quartic couplings come from the perturbativity and potential bound from below conditions which leads to fine-tuning among the parameters in part of the parameter space. Finally, we discuss the possible detection prospects of the axion DM and the additional gauge boson.