Abstract
The next-to-next-to-leading order (NNLO) pQCD prediction for the $\gamma\gamma^* \to \eta_c$ form factor was evaluated in 2015 using nonrelativistic QCD (NRQCD). A strong discrepancy between the NRQCD prediction and the BaBar measurements was observed. Until now there has been no solution for this puzzle. In this paper, we present a NNLO analysis by applying the Principle of Maximum Conformality (PMC) to set the renormalization scale. By carefully dealing with the light-by-light diagrams at the NNLO level, the resulting high precision PMC prediction agrees with the BaBar measurements within errors, and the conventional renormalization scale uncertainty is eliminated. The PMC is consistent with all of the requirements of the renormalization group, including scheme-independence. The application of the PMC thus provides a rigorous solution for the $\gamma\gamma^* \to \eta_c$ form factor puzzle, emphasizing the importance of correct renormalization scale-setting. The results also support the applicability of NRQCD to hard exclusive processes involving charmonium.
Highlights
The next-to-next-to-leading-order (NNLO) pQCD prediction for the γγà → ηc form factor was evaluated in 2015 using nonrelativistic quantum chromodynamics (QCD) (NRQCD)
We present a NNLO analysis by applying the Principle of Maximum Conformality (PMC) to set the renormalization scale
By carefully dealing with the light-by-light diagrams at the NNLO level, the resulting high precision PMC prediction agrees with the BABAR measurements within errors, and the conventional renormalization scale uncertainty is eliminated
Summary
For setting the renormalization scale has a number of defects [7], such as (i) pQCD predictions based on a guessed scale are incorrect for the Abelian theory—quantum electrodynamics (QED). In the case of QED, the renormalization scale of the running coupling αðq2Þ reflects the virtuality q2 of the photon propagator It can be set unambiguously by using the Gell-Mann-Low procedure [8], which automatically sums all proper and improper vacuum polarization contributions to the photon propagators to all orders;. It is clearly important to reanalyze the NNLO prediction for the TFF FðQ2Þ to clarify whether one is confronting the applicability of NRQCD or whether the discrepancy of experiment with theory is due to an improper choice of renormalization scale. We shall adopt the PMC to set the optimal renormalization scale for the TFF FðQ2Þ by carefully dealing with the NNLO contributions. FConvð0Þjμr1⁄41 GeV 1⁄4 cð0Þð1 − 0.25 − 1.11Þ: FConvð0Þjμr1⁄4mc 1⁄4 cð0Þð1 − 0.18 − 0.60Þ: FConvð0Þjμr1⁄42mc 1⁄4 cð0Þð1 − 0.13 − 0.36Þ: ð10Þ ð11Þ ð12Þ
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