Scale-Fixed Predictions for $\gamma + \eta_c$ production in electron-positron collisions at NNLO in perturbative QCD

TL;DR

This paper provides NNLO QCD predictions for $ ext{eta}_c + ext{gamma}$ production in electron-positron collisions, eliminating renormalization scale ambiguities using the Principle of Maximum Conformality, and compares predictions with experimental data.

Contribution

It introduces a scale-and-scheme independent prediction method for $ ext{eta}_c + ext{gamma}$ production at NNLO using PMC and Padé approximation, improving theoretical accuracy.

Findings

NNLO cross-section prediction: approximately 41.18 fb at 10.6 GeV.

N3LO estimate: approximately 21.36 fb, consistent with BELLE measurements.

PMC reduces renormalization scale ambiguities and ensures scheme invariance.

Abstract

In the paper, we present QCD predictions for $\eta_{c} + \gamma$ production at an electron-position collider up to next-to-next-to-leading order (NNLO) accuracy without renormalization scale ambiguities. The NNLO total cross-section for $e^{+}+e^{-}\to\gamma+\eta_{c}$ using the conventional scale-setting approach has large renormalization scale ambiguities, usually estimated by choosing the renormalization scale to be the $e^+ e^-$ center-of-mass collision energy $\sqrt{s}$. The Principle of Maximum Conformality (PMC) provides a systematic way to eliminate such renormalization scale ambiguities by summing the nonconformal $\beta$ contributions into the QCD coupling $\alpha_s(Q^2)$. The renormalization group equation then sets the value of $\alpha_s$ for the process. The PMC renormalization scale reflects the virtuality of the underlying process, and the resulting predictions satisfy all of the requirements of renormalization group invariance, including renormalization scheme invariance. After applying the PMC, we obtain a scale-and-scheme independent prediction, $\sigma|_{\rm NNLO, PMC}\simeq 41.18$ fb for $\sqrt{s}$=10.6 GeV. The resulting pQCD series matches the series for conformal theory and thus has no divergent renormalon contributions. The large $K$ factor which contributes to this process reinforces the importance of uncalculated NNNLO and higher-order terms. Using the PMC scale-and-scheme independent conformal series and the $\rm Pad\acute{e}$ approximation approach, we predict $\sigma|_{\rm NNNLO, PMC+Pade} \simeq 21.36$ fb, which is consistent with the recent BELLE measurement $\sigma^{\rm obs}$=$16.58^{+10.51}_{-9.93}$ fb at $\sqrt{s} \simeq 10.6$ GeV. This procedure also provides a first estimate of the NNNLO contribution.