The next-to-next-to-leading-order QCD corrections to $e^+e^-\to \eta_c/\chi_{cJ}+\gamma$ at B factories

Abstract

We investigate the processes $e^+e^-\to \eta_c+\gamma$ and $e^+e^-\to \chi_{cJ}+\gamma$ at B factories within the NRQCD factorization framework, computing the corresponding helicity amplitudes through $\mathcal{O}(\alpha_s^2)$. 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 $\chi_{c0}+\gamma$, moderate for $\chi_{c1}+\gamma$, and substantial for $\eta_c+\gamma$ and $\chi_{c2}+\gamma$. Theoretical prediction for $\chi_{c1}+\gamma$ is consistent with the {\tt Belle} measurement within $2\sigma$, showing good agreement between theory and experiment. We also predict the angular distribution parameters $\alpha^H_\theta$, 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.