One-loop induced contributions to the rare decay of $A_0\to h_0{h}_0\gamma$ in Two Higgs Doublet Models

The analytic expressions for one-loop contributions to the rare decay process $A_0\to h_0{h}_0\gamma$ within the CP-conserving of Two Higgs Doublet Models are first reported in this paper. Analytic results are presented in term of scalar one-loop Passarino-Veltman functions following the standard output of the packages~{\tt LoopTools} and {\tt Collier}. In this context, physical results for the computed process are easily generated by using one of these packages. The numerical checks are proposed to verify for the analytic results in this paper. The checks rely on the renormalization conditions that the decay amplitude must be the ultraviolet finiteness and infrared finiteness. The amplitude consisting of an external photon always obeys the Ward identity. This will be confirmed numerically in this article. In phenomenological results, the decay rates of $A_0\to h_0{h}_0\gamma$ are evaluated at several points in the allowed regions of the parameter space. Furthermore, the differential decay widths with respect to the invariant mass of Higgs-pair in final states are studied.