Decay of CP-even Higgs $H\rightarrow h \gamma \gamma$ in Two Higgs Doublet Model: one-loop analytic results, ward identity checks

The first analytic expressions for loop-induced contributions for the decay of CP-even Higgs $H\rightarrow h \gamma \gamma$ with $h$ being Standard-Model-like Higgs boson within the framework of Two Higgs Doublet Model are presented in this paper. The one-loop form factors for the decay processes are written in terms of the scalar one-loop Passarino-Veltman functions following the notations of the packages~{\tt LoopTools} and {\tt Collier}. Subsequently, physical results for the decay processes can be generated numerically by using one of the above-mentioned packages. The analytic expressions shown in this paper are verified by several numerical checks, for examples, the ultraviolet and infrared finiteness of one-loop amplitude. Furthermore, the amplitude satisfies the Ward-Takahashi identity due to on-shell photons in final states. The identity is also verified numerically in this work. In phenomenological studies, the differential decay rates as functions of the invariant mass of two photons in final states of $H\rightarrow h \gamma \gamma$ are first studied in parameter space of the Two Higgs Doublet Models.