Mixed Electroweak-QCD Corrections to $H\to \gamma\gamma$

We present for the first time the complete three-loop mixed electroweak-QCD ($\mathcal{O}(\alpha\alpha_s)$) corrections for the decay channel $H \to \gamma\gamma$, by implementing three different on-shell $\alpha$ schemes in computing the electroweak correction. Our studies indicate that the $\mathcal{O}(\alpha_s)$ correction amounts to approximately $1.7\%$ of the leading-order prediction for the diphoton width, while the $\mathcal{O}(\alpha)$ correction varies from $-4.8\%$ to $1.4\%$ depending on the specific $\alpha$ scheme. The three-loop mixed electroweak-QCD correction may reach $0.6\%$, $0.5\%$, and $0.2\%$ of the LO diphoton width in $\alpha(0)$, $\alpha(M_Z)$, and $G_\mu$ schemes, respectively, which is much more significant than the less-than-$0.1\%$ contribution from the three-loop QCD correction. It is also worth noting that the inclusion of the ${\cal O}(\alpha\alpha_s)$ correction significantly reduces the scheme dependence of the partial width from $0.6$ keV at leading order down to $0.03$ keV. The state-of-the-art Standard Model predictions are $\Gamma[H \to \gamma\gamma] = 9.389\div 9.420$ keV, providing a valuable theoretical benchmark for future Higgs factory collider program.