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

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.