The next-to-next-to-leading-order QCD corrections to $e^+e^-\to η_c/χ_{cJ}+γ$ at B factories

TL;DR

This work advances the QCD description of exclusive charmonium production in $e^+e^-$ annihilation by computing helicity amplitudes for $e^+e^-\to \eta_c+\gamma$ and $e^+e^-\to \chi_{cJ}+\gamma$ up to $\mathcal{O}(\alpha_s^2)$ within NRQCD. The authors construct composite asymptotic expressions for the short-distance coefficients by solving differential equations around four expansion points in $r=\frac{4m_c^2}{s}$, ensuring accurate coverage of the full physical range $0\le r\le 1$ with sub-percentage errors in most regions. They extract analytic leading and next-to-leading logarithms as $r\to 0$ and use the results to predict unpolarized cross sections and angular distributions, finding small to moderate perturbative corrections depending on the channel and substantial corrections for $\eta_c+\gamma$ and $\chi_{c2}+\gamma$. The theoretical predictions show good agreement with Belle data for $\chi_{c1}+\gamma$ within $2\sigma$, and the angular distribution parameters $\alpha^H_\theta$ are particularly stable and LDME‑independent, providing a clean test of NRQCD factorization at Belle II.

Abstract

We investigate the processes $e^+e^-\to η_c+γ$ and $e^+e^-\to χ_{cJ}+γ$ at B factories within the NRQCD factorization framework, computing the corresponding helicity amplitudes through $\mathcal{O}(α_s^2)$. The short-distance coefficients are obtained as series expansions in $r=\frac{4m_c^2}{s}$ around $r=0, 1/3, 2/3, 1$, using the method of differential equations. By combining the expansions from all four points, we construct composite asymptotic expressions that reproduce the exact results accurately over the full range $0 \leq r\leq 1$, with relative errors below $0.1\%$ over most of the domain and remaining under $1\%$ elsewhere. Analytic expressions for the leading and next-to-leading logarithmic terms are extracted in the limit $r\to 0$. Using these results, we compute the unpolarized cross sections and observe that the perturbative corrections are small for $χ_{c0}+γ$, moderate for $χ_{c1}+γ$, and substantial for $η_c+γ$ and $χ_{c2}+γ$. Theoretical prediction for $χ_{c1}+γ$ is consistent with the {\tt Belle} measurement within $2σ$, showing good agreement between theory and experiment. We also predict the angular distribution parameters $α^H_θ$, which are insensitive to NRQCD matrix elements and exhibit small theoretical uncertainties. These parameters further display good stability across different perturbative orders. With the high luminosity anticipated at {\tt Belle 2}, future experimental measurements will thus provide a clear test of NRQCD factorization.

Figures (7)

• Figure 1: Representative Feynman diagrams for the process $\gamma^* \to c\bar{c}+\gamma$ at various perturbative orders. (a) LO diagram; (b)–(c) NLO diagrams; (d)–(h) NNLO diagrams, where (h) corresponds to the light-by-light contribution. All diagrams were drawn using JaxoDrawBinosi:2008ig.
• Figure 2: Comparison between the asymptotic expansions around $r=0$ and the exact results for the real parts of the NLO coefficients $c^{H,(1)}_{\lambda_1,\lambda_2}$.
• Figure 3: Comparison between the asymptotic expansions around $r=0$ and the exact results for the imaginary parts of the NLO coefficients $c^{H,(1)}_{\lambda_1,\lambda_2}$.
• Figure 4: Comparison between the asymptotic expansions around $r=0$ and the exact results for the real parts of the NNLO coefficients $c^{H,(2)}_{\lambda_1,\lambda_2}$.
• Figure 5: Comparison between the asymptotic expansions around $r=0$ and the exact results for the imaginary parts of the NNLO coefficients $c^{H,(2)}_{\lambda_1,\lambda_2}$.