Relativistic correction to $e^{+}e^{-}\to J/ψ+gg$ at $B$ factories and constraint on color-octet matrix elements

TL;DR

This work applies NRQCD factorization to calculate the O(v^2) relativistic correction to the color-singlet process e^+e^- → J/ψ+gg at B-factories, with long-distance matrix elements extracted from J/ψ decays. The relativistic correction enhances the cross section by about 20–30%, comparable to the NLO α_s correction, and the combined corrections bring the color-singlet contribution close to Belle’s measured J/ψ+X_non-c c̄ cross section. Consequently, color-octet contributions are tightly constrained, suggesting their matrix elements may be smaller than earlier estimates. The analysis also highlights remaining theoretical and experimental uncertainties, underscoring the need for higher-order calculations and cross-experimental comparisons. Overall, the paper strengthens the case that color-octet effects in this channel are limited under current precision, influencing interpretations of quarkonium production mechanisms.

Abstract

We calculate the relativistic correction to $J/ψ$ production in the color-singlet process $e^{+}e^{-}\to J/ψ+gg$ at B-factories. We employ the non-relativistic QCD (NRQCD) factorization approach, where the short-distance coefficients are calculated perturbatively and the long-distance matrix elements are extracted from the decays of $J/ψ$ into $e^{+}e^{-}$ and light hadrons. We find that the $O(v^2)$ relativistic correction can enhance the cross section by a factor of 20-30%, comparable to the enhancement due to the $O(α_s)$ radiative correction obtained earlier. Combining the relativistic correction with the QCD radiative correction, we find that the color-singlet contribution to $e^{+}e^{-}\to J/ψ+gg$ can saturate the latest observed cross section $σ(e^{+}e^{-}\to J/ψ+X_{\mathrm{non-c\bar{c}}})=0.43 \pm0.09\pm0.09$ pb by Belle, thus leaving little room to the color-octet contributions. This gives a very stringent constraint on the color-octet contribution, and may imply that the values of color-octet matrix elements are much smaller than expected earlier by using the naive velocity scaling rules or extracted from fitting experimental data with the leading-order calculations.

Figures (2)

• Figure 1: The typical two among six Feynman diagrams in the process $e^{+}e^{-}\to J/\psi+gg$ at leading-order in QCD.
• Figure 2: Relativistic corrections to $\sigma(e^{+}e^{-}\to J/\psi+gg)$ as functions of $m_c$ with long-distance matrix elements determined from Eq.(\ref{['eqM2']}) (left-panel) and Eq.(\ref{['eqM4']}) (right-panel). In each figure the lower curve is the LO result and the upper one is the result including $v_c^2$ corrections.