Self-consistent analysis for the $\eta_c\rightarrow \gamma\gamma$ process

TL;DR

This paper applies the Principle of Maximum Conformality to analyze the $ ext{eta}_c ightarrow ext{gamma} ext{gamma}$ process, improving theoretical predictions and supporting NRQCD's applicability to charmonium decays.

Contribution

It introduces the use of PMC to eliminate scale ambiguities in pQCD calculations of $ ext{eta}_c$ decay, aligning theory with experimental data.

Findings

PMC scale determined as 4.49 times the charm quark mass

Decay width prediction agrees with experimental value within uncertainties

Transition form factor matches precise experimental measurements

Abstract

The next-to-next-to-leading-order (NNLO) pQCD predictions for both the decay width and the transition form factor in the $\eta_c\rightarrow \gamma\gamma$ process, based on nonrelativistic QCD (NRQCD), deviate from precise experimental measurements. These significant discrepancies have cast doubt on the applicability of NRQCD to charmonium processes. In this paper, we analyze the $\eta_c\rightarrow \gamma\gamma$ process by applying the Principle of Maximum Conformality (PMC), a systematic method for eliminating renormalization scheme and scale ambiguities. The PMC renormalization scales are determined by absorbing the non-conformal $\beta$ terms which govern the behavior of the QCD running coupling via the Renormalization Group Equation. We obtain the PMC scale $Q_\star=4.49\,m_c$ for the $\eta_c\rightarrow \gamma\gamma$ decay width. Even after using the PMC method, the convergence of the pQCD series is still poor, which indicates the importance of uncalculated NNNLO and higher-order terms. The resulting value for $\Gamma_{\eta_c\rightarrow \gamma\gamma}$ is in agreement with the Particle Data Group's reported value of $\Gamma_{\eta_c\rightarrow \gamma\gamma}=5.1\pm0.4$ keV within the bounds of uncertainties. Moreover, the transition form factor obtained using the PMC is also in good agreement with precise experimental measurements. The application of the PMC suggests a potential resolution to $\eta_c\rightarrow \gamma\gamma$ puzzle and supports the applicability of NRQCD to charmonium processes.