The heavy quarkonium inclusive decays using the principle of maximum conformality

TL;DR

This paper applies the principle of maximum conformality (PMC) to improve the precision of heavy quarkonium decay width calculations by eliminating renormalization scale ambiguities, and extends predictions to approximate NNNLO accuracy.

Contribution

It introduces the use of PMC single-scale method and Padé approximation to refine NNLO and NNNLO predictions for heavy quarkonium decay ratios, reducing theoretical uncertainties.

Findings

PMC reduces renormalization scale dependence in decay width calculations.

Predicted decay ratios agree with experimental data within uncertainties.

Approximate NNNLO predictions improve the accuracy of decay ratio estimates.

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 conventional renormalization scale dependence due to its weaker pQCD convergence, e.g. about $(^{+4\%}_{-34\%})$ for $\eta_c$ and $(^{+0.0}_{-9\%})$ 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 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 $\eta_Q$ inclusive decay width can be greatly improved. Taking the relativistic correction $\mathcal{O}(\alpha_{s}v^2)$ into consideration, the ratios of the $\eta_{Q}$ 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 obtain approximate 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 $R^{\rm NNNLO}_{\eta_c}|_{\rm{PAA+PMC}}$ ratio agrees with the latest PDG value $R_{\eta_c}^{\rm{exp}}=(5.3_{-1.4}^{+2.4})\times10^3$, indicating the necessity of a strict calculation of NNNLO terms.