Scale-fixed predictions for γ + ηc production in electron-positron collisions at NNLO in perturbative QCD

In the paper, we present QCD predictions for γ + η_c production at an electron-positron collider up to next-to-next-to-leading order (NNLO) accuracy without renormalization scale ambiguities. The NNLO total cross-section for e⁺ + e⁻ → γ + η_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 $$ \sqrt{s} $$s . The Principle of Maximum Conformality (PMC) provides a systematic way to eliminate such renormalization scale ambiguities by summing the nonconformal β contributions into the QCD coupling α_s(Q²). The renormalization group equation then sets the value of α_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 renormalization scale-and-scheme independent prediction, σ|_NNLO,PMC ≃ 41.18 fb for $$ \sqrt{s} $$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 Padé approximation approach, we predict σ|_{NNNLO,PMC+Pade} ≃ 18.99 fb, which is consistent with the recent BELLE measurement $$ {\sigma}^{\mathrm{obs}}={16.58}_{-9.93}^{+10.51} $$σ obs = 16.58 − 9.93 + 10.51 fb at $$ \sqrt{s} $$s ≃ 10.6 GeV. This procedure also provides a first estimate of the NNNLO contribution.