Role of λ_(hH^+H^-) in Higgs boson decays h to bs in the 2HDM

Within the Two Higgs Doublet Model (2HDM) with $\mathcal{CP}$ conservation and a softly broken $Z_2$ symmetry, we analyze the flavor changing Higgs decays $h\to bs$ ($bs$ refers jointly to the two decay channels $b\bar s$ and $\bar b s$), where $h$ is identified with the SM-like Higgs boson discovered at the LHC. We provide a comprehensive study of the decay width $\Gamma (h \to bs)$ with particular focus on the most relevant effects from the triple Higgs coupling $\lambda_{hH^+H^-}$. Furthermore, we consider all the relevant theoretical and experimental constraints to determine which predictions for the $\mathrm{BR}\left(h\to bs\right)$ are still allowed by the current data. We find that the predictions for $\mathrm{BR}\left(h\to bs\right)$ in types II and III can be several orders of magnitude smaller compared to the SM value. In contrast, in type I and IV we find that the predicted enhancements in the decay rates with respect to the SM of up to about 70% and 50%, respectively, are still allowed. We discuss how these deviations from the SM are caused by interference effects controlled by the coupling $\lambda_{hH^+H^-}$ which can be large for very heavy $H^\pm$. To better understand the role of $\lambda_{hH^+H^-}$ in the $h \to bs$ decay we derive and analyze here the analytical results for the $hbs$ one-loop effective vertex that is generated by integrating out the heavy $H^\pm$.