The next-to-next-to-leading order (NNLO) pQCD correction to the inclusive decays of the heavy quarkonium ηQ\documentclass[12pt]{minimal}
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\begin{document}$$\eta _Q$$\end{document} (Q being c or b) has been done in the literature within the framework of nonrelativistic QCD. One may observe that the NNLO decay width still has large conventional renormalization scale dependence due to its weaker pQCD convergence, e.g. about -34%+4%\documentclass[12pt]{minimal}
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\begin{document}$$\left( ^{+4\%}_{-34\%}\right) $$\end{document} for ηc\documentclass[12pt]{minimal}
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\begin{document}$$\eta _c$$\end{document} and -9%+0.0\documentclass[12pt]{minimal}
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\begin{document}$$\left( ^{+0.0}_{-9\%}\right) $$\end{document} for ηb\documentclass[12pt]{minimal}
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\begin{document}$$\eta _b$$\end{document}, by varying the scale within the range of [mQ,4mQ]\documentclass[12pt]{minimal}
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\begin{document}$$[m_Q, 4m_Q]$$\end{document}. The principle of maximum conformality (PMC) provides a systematic way to fix the αs\documentclass[12pt]{minimal}
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\begin{document}$$\alpha _s$$\end{document}-running behavior of the process, which satisfies the requirements of renormalization group invariance and eliminates the conventional renormalization scheme and scale ambiguities. Using the PMC single-scale method, we show that the resultant PMC conformal series is renormalization scale independent, and the precision of the ηQ\documentclass[12pt]{minimal}
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\begin{document}$$\eta _Q$$\end{document} inclusive decay width can be greatly improved. Taking the relativistic correction O(αsv2)\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {O}}(\alpha _{s}v^2)$$\end{document} into consideration, the ratios of the ηQ\documentclass[12pt]{minimal}
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\begin{document}$$\eta _{Q}$$\end{document} decays to light hadrons or γγ\documentclass[12pt]{minimal}
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\begin{document}$$\gamma \gamma $$\end{document} are: RηcNNLO|PMC=(3.93-0.24+0.26)×103\documentclass[12pt]{minimal}
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\begin{document}$$R^\mathrm{NNLO}_{\eta _c}|_{\mathrm{PMC}}=(3.93^{+0.26}_{-0.24})\times 10^3$$\end{document} and RηbNNLO|PMC=(22.85-0.87+0.90)×103\documentclass[12pt]{minimal}
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\begin{document}$$R^\mathrm{NNLO}_{\eta _b}|_{\mathrm{PMC}}=(22.85^{+0.90}_{-0.87})\times 10^3$$\end{document}, respectively. Here the errors are for Δαs(MZ)=±0.0011\documentclass[12pt]{minimal}
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\begin{document}$$\Delta \alpha _s(M_Z) = \pm 0.0011$$\end{document}. As a step forward, by applying the Pade´\documentclass[12pt]{minimal}
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\begin{document}$$\acute{e}$$\end{document} approximation approach (PAA) over the PMC conformal series, we obtain approximate NNNLO predictions for those two ratios, e.g. RηcNNNLO|PAA+PMC=(5.66-0.55+0.65)×103\documentclass[12pt]{minimal}
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\begin{document}$$R^{\mathrm{NNNLO}}_{\eta _c}|_{\mathrm{PAA+PMC}} =(5.66^{+0.65}_{-0.55})\times 10^3$$\end{document} and RηbNNNLO|PAA+PMC=(26.02-1.17+1.24)×103\documentclass[12pt]{minimal}
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\begin{document}$$R^{\mathrm{NNNLO}}_{\eta _b}|_{\mathrm{PAA+PMC}}=(26.02^{+1.24}_{-1.17})\times 10^3$$\end{document}. The RηcNNNLO|PAA+PMC\documentclass[12pt]{minimal}
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\begin{document}$$R^{\mathrm{NNNLO}}_{\eta _c}|_{\mathrm{PAA+PMC}}$$\end{document} ratio agrees with the latest PDG value Rηcexp=(5.3-1.4+2.4)×103\documentclass[12pt]{minimal}
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\begin{document}$$R_{\eta _c}^\mathrm{{exp}}=(5.3_{-1.4}^{+2.4})\times 10^3$$\end{document}, indicating the necessity of a strict calculation of NNNLO terms.
