An analysis of $H \to γγ$ up to three-loop QCD corrections

The principle of maximum conformality (PMC) provides a convenient way for setting the optimal renormalization scales for high-energy processes, which can eliminate the conventional renormalization scale error via an order-by-order manner. At present, we make a detailed PMC analysis on the Higgs decay $H\rightarrow γγ$ up to three-loop QCD corrections. As an important point of deriving reliable PMC estimation, it is noted that only those $\{β_i\}$-terms that rightly determine the running behavior of coupling constant via the renormalization group equation should be absorbed into the coupling constant, and those $\{β_i\}$-terms that pertain to the quark mass renormalization and etc. should be kept as a separate. To avoid confusion of separating and absorbing different types of $\{β_i\}$-terms into the coupling constant, we first transform the decay width in terms of top quark $\overline{\rm MS}$ mass into that of on-shell mass and then apply the PMC scale setting. After applying PMC scale setting, the final estimation is conformal and is scheme-independent and scale-independent. Up to three-loop QCD corrections, we obtain a PMC scale $μ^{\rm PMC}_{r}=242.3$ GeV $\sim 2M_H$, which is optimal and highly independent of any choice of initial scale. Thus, we obtain a more accurate scale-independent prediction by taking the Higgs mass as the same as that of ATLAS and CMS measurements, i.e., $Γ(H\rightarrow γγ)|_{\rm ATLAS}=9.504^{+0.226}_{-0.252}$ keV and $Γ(H\rightarrow γγ)|_{\rm CMS}=9.568^{+0.195}_{-0.191}$ keV, where the error is caused by the measured Higgs mass, i.e. the Higgs mass $M_{H}$ is taken as $125.5\pm0.2^{+0.5}_{-0.6}$ GeV for ATLAS and $125.7\pm0.3\pm0.3$ GeV for CMS, respectively.

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