Relativistic corrections to $J/ψ$ exclusive and inclusive double charm production at B factories

TL;DR

The paper analyzes relativistic corrections at order $v^2$ within NRQCD for both exclusive ($e^{+}e^{-}\to J/\psi+\eta_{c}$) and inclusive ($e^{+}e^{-}\to J/\psi+c\bar c$) double charm production at B factories. By deriving short-distance coefficients perturbatively and extracting long-distance matrix elements from charmonium decays, it shows substantial relativistic enhancements for the exclusive channel, which, when combined with NLO QCD corrections, can largely resolve the experimental discrepancies. In contrast, the inclusive process exhibits only tiny relativistic corrections, indicating that fragmentation-dominated dynamics and higher-order QCD effects are more influential there. Overall, the work supports a significant role for relativistic effects in exclusive double-charm production and highlights the ongoing importance of NLO corrections and alternative mechanisms for the inclusive case.

Abstract

In order to clarify the puzzling problems in double charm production, relativistic corrections at order $v^{2}$ to the processes $e^{+}e^{-}\to J/ψ+η_{c}$ and $e^{+}e^{-}\to J/ψ+c\bar{c}$ at B factories are studied in non-relativistic quantum chromodynamics. The short-distance parts of production cross sections are calculated perturbatively, while the long-distance matrix elements are estimated from $J/ψ$ and $η_c$ decays up to errors of order $v^4$. Our results show that the relativistic correction to the exclusive process $e^{+}e^{-}\to J/ψ+η_{c}$ is significant, which, when combined together with the next-to-leading order $α_{s}$ corrections, could resolve the large discrepancy between theory and experiment; whereas for the inclusive process $e^{+}e^{-}\to J/ψ+c\bar{c}$ the relativistic correction is tiny and negligible. The physical reason for the above difference between exclusive and inclusive processes largely lies in the fact that in the exclusive process the relative momentum between quarks in charmonium substantially reduces the virtuality of the gluon that converts into a charm quark pair, but this is not the case for the inclusive process, in which the charm quark fragmentation $c\to J/ψ+c$ is significant, and QCD radiative corrections can be more essential.

Figures (5)

• Figure 1: $e^{+}e^{-}\rightarrow (c\overline{c})_{{}^{3}S_{1}}+(c\overline{c})_{{}^{1}S_{0}}$
• Figure 2: $e^{+}e^{-}\rightarrow J/\psi+\eta_{c}$ cross sections with relativistic corrections to long-distance matrix elements extracted from charmonium decays (without NLO QCD radiative corrections). The lower line represents the LO result in $v$, and the upper line represents the result with $v^2$ corrections to the short-distance coefficients. Here the coupling constant is fixed as $\alpha_{s}=0.26$.
• Figure 3: $e^{+}e^{-}\rightarrow J/\psi+\eta_{c}$ cross sections with relativistic corrections to long-distance matrix elements extracted from charmonium decays (with NLO QCD radiative corrections). The lower line represents the LO result in $v$, and the upper line represents the result with $v^2$ corrections to the short-distance coefficients. Here the coupling constant is fixed as $\alpha_{s}=0.26$. Note that the QCD radiative corrections to the short-distance coefficients (with K=1.8) are included for the upper line but not the lower line.
• Figure 4: Feynman diagrams for $e^+ + e^-\rightarrow\gamma^*\rightarrow$$J/\psi$ + $c\bar{c}$.
• Figure 5: $R_{v}$ as a function of $\sqrt{s}$. Here $R_{v}$ is the ratio of the correction at order $v^2$ to the leading order result for the cross section of $e^{+}+e^{-}\rightarrow J/\psi+c\overline{c}$.