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.
Categories: hep-ph hep-ex