CP Violating $hW^+W^-$ Coupling in the Standard Model and Beyond

Inspired by the recent development in determining the property of the observed Higgs boson, we explore the $CP$-violating (CPV) $- c_{\rm CPV} h W^{+\, \mu\nu}\tilde{W}^{-}_{\mu\nu}/v$ coupling in the Standard Model (SM) and beyond, where $W^{\pm \, \mu \nu}$ and $\tilde{W}^{\pm\,\mu\nu}$ denote the $W$-boson field strength and its dual. To begin with, we show that the leading-order SM contribution to this CPV vertex appears at two-loop level. By summing over the quark flavor indices in the two loop integrals analytically, we can estimate the order of the corresponding Wilson coefficient to be $c^{\rm SM}_{\rm CPV} \sim {\cal O}(10^{-23})$, which is obviously too small to be probed at the LHC and planned future colliders. Then we investigate this CPV $hW^+ W^-$ interaction in two Beyond the Standard Model benchmark models: the left-right model and the complex 2-Higgs doublet model (C2HDM). Unlike what happens for the SM, the dominant contributions in both models arise at the one-loop level, and the corresponding Wilson coefficient can be as large as of ${\cal O}(10^{-9})$ in the former model and of ${\cal O}(10^{-3})$ for the latter. In light of such a large CPV effect in the $hW^+W^-$ coupling, we also give the formulae for the leading one-loop contribution to the related CPV $hZZ$ effective operator in the C2HDM. The order of magnitude of the Wilson coefficients in the C2HDM may be within reach of the high-luminosity LHC or planned future colliders.