NNLO QCD Corrections to $D$-Wave Spin-Singlet Heavy Quarkonia Decay $\eta_{Q2}\to\gamma\gamma$ via the Principle of Maximum Conformality

In this paper, we perform a comprehensive study of the decay process $\eta_{Q2}\to\gamma\gamma$ for $D$-wave spin-singlet heavy quarkonia up to next-to-next-to-leading-order (NNLO) QCD corrections within the nonrelativistic QCD effective theory. Following its factorization formalism, the total decay width is decomposed into perturbatively calculable short-distance coefficients (SDCs) and nonperturbative $D$-wave long-distance matrix elements (LDMEs). The original NNLO series of SDCs suffers from sizable renormalization and factorization scale uncertainties. To eliminate such inherent scale ambiguities, we adopt the Principle of Maximum Conformality (PMC). We show that recursively applying the renormalization group equations for the running of $\alpha_s$ and $D$-wave LDMEs within the PMC framework yields an effective strong coupling $\alpha_s(Q_\ast)$ consistent with the expansion coefficients, resulting in a scale-invariant perturbative series. The determined PMC scales are $Q_\ast=1.483$ GeV for $\eta_{c2}$ and $Q_\ast=4.246$ GeV for $\eta_{b2}$. By removing divergent renormalon contributions, the PMC naturally improves the convergence of the perturbative series for SDCs. Our PMC predictions for the total decay widths are $\Gamma_{\eta_{c2}\to\gamma\gamma}^{\rm PMC} = 3.322^{+0.899}_{-0.828}\ \text{eV}$ and $\Gamma_{\eta_{b2}\to\gamma\gamma}^{\rm PMC} = 0.0188^{+0.0014}_{-0.0013}\ \text{eV}$. The uncertainties arise from variations of the charm and bottom quark masses $\Delta m_c=\pm 0.07$ GeV, $\Delta m_b=\pm 0.06$ GeV, as well as systematic errors from uncalculated higher-order corrections. The corresponding branching ratios are $\text{Br}(\eta_{c2}\to\gamma\gamma) = \big(7.463^{+2.020}_{-1.860}\big)\times 10^{-6}$ and $\text{Br}(\eta_{b2}\to\gamma\gamma) = \big(6.460^{+0.481}_{-0.447}\big)\times 10^{-7}$.