Inclusive Charmonium Production via Double $c \bar c$ in $e^+e^-$ Annihilation

TL;DR

The study addresses the challenge of double charm production in $e^+e^-$ annihilation by computing complete $O(α_s^2)$ inclusive cross sections for $η_c$, $J/ψ$, and $χ_{cJ}$ with an accompanying $c\bar c$ pair under NRQCD. It analyzes both color-singlet and color-octet contributions, comparing exact color-singlet results to charm-quark fragmentation, and estimates octet effects using NRQCD long-distance matrix elements. At the Belle/BaBar energy of $√s=10.6$ GeV, fragmentation substantially overestimates the cross sections for $χ_{c1}$ and $χ_{c2}$, while the octet component is small for $J/ψ$ but sizable for $χ_{c1}$ and $χ_{c2}$. These results provide NRQCD-based predictions to confront experimental data and clarify the energy regimes where fragmentation is a good approximation for charmonium production in association with charm quarks.

Abstract

Motivated by the recent observation of double charm quark pair production by the Belle Collaboration, we calculate the complete ${\cal O}(α_{s}^{2})$ inclusive production cross sections for $η_{c}$, $J/ψ$, and $χ_{cJ}$(J=0, 1, 2) plus $c\bar{c}$ in $e^+ e^-$ annihilation through a virtual photon. We consider both color-singlet and color-octet contributions, and give the analytical expressions for these cross sections. The complete color-singlet calculations are compared with the approximate fragmentation calculations as functions of the center-of-mass energy $\sqrt{s}$. We find that most of the fragmentation results substantially overestimate the cross sections (e.g. by a factor of $\sim$4 for $χ_{c1}$ and $χ_{c2}$) at the Belle and BaBar energy $\sqrt{s}=10.6$GeV. The fragmentation results become a good approximation only when $\sqrt{s}$ is higher than about 100GeV. We further calculate the color-octet contributions to these cross sections with analytical expressions. We find that while the color-octet contribution to $J/ψ$ inclusive production via double charm is negligible (only about 3%), the color-octet contributions to $χ_{c1}$ and $χ_{c2}$ can be significant.

Figures (9)

• Figure 1: Feynman diagrams for $e^+ + e^-\rightarrow\gamma^*\rightarrow$ Charmonium + $c\bar{c}$.
• Figure 2: Feynman diagrams for $e^+ + e^-\rightarrow\gamma^*\rightarrow$$c\bar{c}(^{2S+1}L_J^{(8)})$ + $c\bar{c}$.
• Figure 3: Cross sections for $e^+ e^-\rightarrow \eta_{c} + c\bar{c}$ plotted against the center-of-mass energy. Dotted line illustrates the fragmentation calculation and solid line illustrates the complete calculation. The cross sections are in units of $\sigma_{cc}= \sigma(e^{+}+e^{-}\rightarrow\gamma^{*}\rightarrow c\bar{c})$ times $10^{-4}$.
• Figure 4: Cross sections for $e^+ e^-\rightarrow J/\psi + c\bar{c}$ plotted against the center-of-mass energy. Dotted line illustrates the fragmentation calculation and solid line illustrates the complete calculation.
• Figure 5: Cross sections for $e^+ e^-\rightarrow \chi_{c0} + c\bar{c}$ plotted against the center-of-mass energy. Dotted line illustrates the fragmentation calculation and solid line illustrates the complete calculation.