An analysis of the heavy quarkonium inclusive decays up to NNNLO level using the principle of maximum conformality

Abstract

The next-to-next-to-leading order (NNLO) pQCD correction to the inclusive decays of the heavy quarkonium \(\eta_Q\) (\(Q\) 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 renormalization scale dependence due to its weaker pQCD convergence, e.g. about \(\pm11\%\) for \(\eta_c\) and \(\pm17\%\) for \(\eta_b\), 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 \(\alpha_s\)-running behavior of the process, which satisfies all the requirements of renormalization group invariance. We show that the resultant PMC conformal series is renormalization scale independent, thus the precision of the \(\eta_Q\) inclusive decay width can be greatly improved. Taking the relativistic correction \(\mathcal{O}(\alpha_{s}v^2)\) into consideration, the ratios of \(\eta_{Q}\) decays to light hadrons or \(\gamma\gamma\) are: \(R^{\rm NNLO}_{\eta_c}|_{\rm{PMC}}=(3.93^{+0.26}_{-0.24})\times10^3\) and \(R^{\rm NNLO}_{\eta_b}|_{\rm{PMC}}=(22.85^{+0.90}_{-0.87})\times10^3\), respectively. Here the errors are for \(\Delta\alpha_s(M_Z) = \pm0.0011\). As a step forward, by applying the Pad\(\acute{e}\) approximation approach (PAA) over the PMC conformal series, we give the NNNLO predictions for those two ratios, e.g. \(R^{\rm NNNLO}_{\eta_c}|_{\rm{PAA+PMC}} =(5.66^{+0.65}_{-0.55})\times10^3\) and \(R^{\rm NNNLO}_{\eta_b}|_{\rm{PAA+PMC}}=(26.02^{+1.24}_{-1.17})\times10^3\). The \(R^{\rm NNNLO}_{\eta_c}\) ratio agrees well with the latest PDG value \(R_{\eta_c}^{\rm{exp}}=(5.3_{-1.4}^{+2.4})\times10^3\), indicating the importance of a strict NNNLO calculation.