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

Abstract

Within the NRQCD factorization framework, we investigate the color-singlet contribution to $e^{+}{e}^{-}\to J/\psi +X_{\mathrm{non}c\bar{c}}$ 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 $0.530_{-0.113}^{+0.122}$ pb agrees with the {\tt Belle} measurement within uncertainties. However, the predicted angular distribution parameter $0.120_{-0.019}^{+0.027}$ deviates from the experimental value $5.71\pm 2.51$ by more than $2\sigma$, calling for further experimental and theoretical investigations.

Comments: 22 pages, 5 figures, 4 tables