Charmonium Production In E～+e～- Annihilation

Using the non-relativistic quantum chromodynamics (NRQCD) factorization formalism, we calculate the inclusive (including e⁺e^-→η_c c(c|-), e⁺e^-→J/ψc(c|-), e^e-→x_cJc(c|-) (J=0, 1, 2) and e⁺e^-→δ_J c(c|-) (J=1, 2)) and exclusive (e⁺e^-→J/ψη_c and e⁺e^-→J/ψx_cJ (J=0, 1, 2)) production for S-wave, P-wave, and D-wave char-monia via double cc in e⁺e^- annihilation at s^1/2 = 10.6 GeV.The NRQCD factorization approach has achieved a number of successes in the elimination of the infrared divergence of inclusive decay rates and the predictions of production cross sections for heavy quarkonium states, such as, the explanation of the charmonia surplus production at the Tevatron and the elimination of the infrared divergence in B meson decays to P-wave charmonium. However, the most recent measurement on the double cc production has been reported by Belle, and found a very large fraction of J/ψproduction:σ(e⁺e^-→J/ψc(c|-))/σ(e⁺e^-→J/ψX) = 0.59_-0.13^+0.15±0.12, which means that the cross section of J/ψc(c|-) production is about 0.9 pb. It is much larger than the NRQCD predictions 0.1-0.2 pb taking into account the differences in the values of the input parameters. The discrepancy between NRQCD prediction and experimental data of charmonium inclusive production via double cc production in e⁺e^- annihilation is a challenge to NRQCD. In this paper, we calculate a series of the double cc inclusive cross sections, and expect these calculations other than J/ψc(c|-) production will further clarify the problem. We find that in the S-wave charmonium inclusive production via double cc, the color-octet processes cannot give large contributions, in contrast to the case of p(p|-) collisions, where the color-octet contributions are dominant. For instance, the color-octet contribution to J/ψc(c|-) production is only about 7%. Besides the color-octet contribution, we also discuss the contribution through two virtual photons in e⁺e^- annihilation. In general, the contributions from the process involving two virtual photons are suppressed by a factor of a²/a_s² compared to the process in- volving one virtual photon. However, the contributions from the process involving two virtual photons is enhanced by powers of E_cm/2m_c due to the difference of the virtuality of the virtual photons between the two precesses. This enhancement can compensate for the suppression factor that is associated with the coupling constants. Unfortunately, the contribution from the process involving two virtual photons is only about 15% of the cross section of the process involving one virtual photon in e⁺e^- annihilation at 10.6 GeV. Moreover, we find that the most striking challenge to NRQCD comes from the exclusive process. In the process of e⁺e^-→J/ψη_c, the experimental data may be an order of magnitude larger than the NRQCD predictions (with uncertainties due to the unknown decay branching fractions into ≥ 4-charged particles for .η_c). We also calculate the cross sections for the process of e⁺e^-→J/ψx_cJ (J = 0,1, 2) and the ratio of exclusive to inclusive cross sections. These predictions can be tested by future experimental measurements. The puzzles of charmonium production via double c_c|- production in e⁺e^- annihilation are really a striking challenge to NRQCD. If the Belle results are further confirmed, then some other possibilities have to be considered, for instance, the non-pertrbative effects for the heavy quarkonium production, and the new processes that have not yet been recognized.