Loop induced $H^\pm W^\mp Z$ vertices in the general two Higgs doublet model with CP violation

Understanding symmetry of extended Higgs models helps to construct the theory beyond the standard model and gives important insights for testing these models by experiments. The custodial symmetry plays an important role as a leading principle to construct non-minimal Higgs sectors. On the other hand, new sources of CP violation in the Higgs sector are essentially important to explain the baryon asymmetry of the Universe. In the two Higgs doublet model, it has been known that the CP violation and the custodial symmetry are not compatible, and thus the CP violating Higgs potential in general violates the custodial symmetry in the Higgs sector. In this paper, we discuss the loop-induced $H^\pm W^\mp Z$ vertices in the two Higgs doublet model, which are induced due to the custodial symmetry violation of the sector of the particles in the loop, and study how the vertices are affected by the CP violation. We calculate the $H^\pm W^\mp Z$ vertices in the most general two Higgs doublet model with the CP violation at the one-loop level and evaluate the decay modes $H^\pm \to W^\pm Z$ caused by these vertices. We obtain new contributions to the $H^\pm W^\mp Z$ vertices from the CP violating part of the model, in addition to the known contributions from the CP conserving part. Moreover, we find that an asymmetry between the decays $H^+ \to W^+ Z$ and $H^- \to W^- Z$ is caused by the CP violating phases in the model, some of which are important for the electroweak baryogenesis. We also briefly mention testability of these vertices at future collider experiments.