Next-to-leading-order electroweak correction to $H\to Z^0\gamma$

Inspired by the recent observation of the Higgs boson radiative decay into $Z^0$ by {\tt ATLAS} and {\tt CMS} Collaborations, we investigate the next-to-leading-order (NLO) electroweak correction to this rare decay process in Standard Model (SM). Implementing the on-shell renormalization scheme, we find that the magnitude of the NLO electroweak correction may reach $7\%$ of the leading order (LO) prediction, much more significant than that of the NLO QCD correction, which is merely about $0.3\%$. After incorporating the ${\cal O}(\alpha)$ correction, the predicted partial width from various $\alpha$ schemes tend to converge to each other. Including both NLO electroweak and QCD corrections, the SM prediction for the branching fraction shifts from the LO value of $(1.40-1.71)\times 10^{-3}$ to $(1.55\pm 0.06)\times 10^{-3}$, considerably lower than the measured value ${\cal B}_{\rm exp}[H\to Z^0\gamma]=(3.4\pm 1.1)\times 10^{-3}$. Resolving this alarming discrepancy clearly calls for further theoretical investigations, and, more importantly, experimental efforts from {\tt HL-LHC} and the prospective Higgs factories such as {\tt CEPC} and {\tt FCC-ee}.