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The heavy quarkonium inclusive decays using the principle of maximum conformality

The next-to-next-to-leading order (NNLO) pQCD correction to the inclusive decays of the heavy quarkonium $η_Q$ ($Q$ 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 $(^{+4\%}_{-34\%})$ for $η_c$ and $(^{+0.0}_{-9\%})$ for $η_b$, by varying the scale within the range of $[m_Q, 4m_Q]$. The principle of maximum conformality (PMC) provides a systematic way to fix the $α_s$-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 $η_Q$ inclusive decay width can be greatly improved. Taking the relativistic correction $\mathcal{O}(α_{s}v^2)$ into consideration, the ratios of the $η_{Q}$ decays to light hadrons or $γγ$ are: $R^{\rm NNLO}_{η_c}|_{\rm{PMC}}=(3.93^{+0.26}_{-0.24})\times10^3$ and $R^{\rm NNLO}_{η_b}|_{\rm{PMC}}=(22.85^{+0.90}_{-0.87})\times10^3$, respectively. Here the errors are for $Δα_s(M_Z) = \pm0.0011$. As a step forward, by applying the Pad$\acute{e}$ approximation approach (PAA) over the PMC conformal series, we obtain approximate NNNLO predictions for those two ratios, e.g. $R^{\rm NNNLO}_{η_c}|_{\rm{PAA+PMC}} =(5.66^{+0.65}_{-0.55})\times10^3$ and $R^{\rm NNNLO}_{η_b}|_{\rm{PAA+PMC}}=(26.02^{+1.24}_{-1.17})\times10^3$. The $R^{\rm NNNLO}_{η_c}|_{\rm{PAA+PMC}}$ ratio agrees with the latest PDG value $R_{η_c}^{\rm{exp}}=(5.3_{-1.4}^{+2.4})\times10^3$, indicating the necessity of a strict calculation of NNNLO terms.