Semi-analytical results for $e^+e^-\to J/\psi + X_{{\rm non\,}c\bar{c}}$ up to $\mathcal{O}(\alpha_s v^2)$ at B factories

Abstract

Within the NRQCD factorization framework, we investigate the color-singlet contribution to $e^+e^- \to J/\psi + X_{{\rm non\,}c\bar{c}}$ at B factories, computing the $\mathcal{O}(\alpha_s)$, $\mathcal{O}(v^2)$, and $\mathcal{O}(\alpha_s v^2)$ corrections to both the unpolarized cross section and the $J/\psi$ angular distribution. The $\mathcal{O}(\alpha_s v^2)$ correction is obtained for the first time, and the validity of NRQCD factorization at this order is explicitly verified. Using the differential equation method, the short-distance coefficients are obtained as asymptotic expansions in $r = m_c/\sqrt{s}$ up to $r^{40}$, which reproduce exact results with high precision at B factory energies, achieving relative errors around $10^{-14}$ for the cross section and around $10^{-7}$ for the angular distribution. Notably, with the same input parameters, our $\mathcal{O}(\alpha_s)$ and $\mathcal{O}(v^2)$ corrections are consistent with those reported in the literature. Phenomenologically, the $\mathcal{O}(\alpha_s)$ correction (with $\mu_R=\sqrt{s}/2$) reaches about $50\%$ of the leading-order cross section, while the $\mathcal{O}(v^2)$ and $\mathcal{O}(\alpha_s v^2)$ corrections are accidentally small. After including feeddown contributions from $\psi(2S)$, the predicted cross section $0.523_{-0.197}^{+0.285}$ pb agrees with the {\tt Belle} measurement within uncertainties. However, the predicted angular distribution parameter $0.120_{-0.036}^{+0.041}$ deviates from the experimental value $5.71\pm 2.51$ by more than $2\sigma$, calling for further experimental and theoretical investigations.