# Further studies on the exclusive productions of \(J/\psi +\chi _{cJ}\) (\(J=0,1,2\)) via \(e^+e^-\) annihilation at the *B* factories

## Abstract

By including the interference effect between the QCD and the QED diagrams, we carry out a complete analysis on the exclusive productions of \(e^+e^- \rightarrow J/\psi +\chi _{cJ}\) (\(J=0,1,2\)) at the *B* factories with \(\sqrt{s}=10.6\) GeV at the next-to-leading-order (NLO) level in \(\alpha _s\), within the nonrelativistic QCD framework. It is found that the \({\mathcal {O}} (\alpha ^3\alpha _s)\)-order terms that represent the tree-level interference are comparable with the usual NLO QCD corrections, especially for the \(\chi _{c1}\) and \(\chi _{c2}\) cases. To explore the effect of the higher-order terms, namely \({\mathcal {O}} (\alpha ^3\alpha _s^2)\), we perform the QCD corrections to these \({\mathcal {O}} (\alpha ^3\alpha _s)\)-order terms for the first time, which are found to be able to significantly influence the \({\mathcal {O}} (\alpha ^3\alpha _s)\)-order results. In particular, in the case of \(\chi _{c1}\) and \(\chi _{c2}\), the newly calculated \({\mathcal {O}} (\alpha ^3\alpha _s^2)\)-order terms can to a large extent counteract the \({\mathcal {O}} (\alpha ^3\alpha _s)\) contributions, evidently indicating the indispensability of the corrections. In addition, we find that, as the collision energy rises, the percentage of the interference effect in the total cross section will increase rapidly, especially for the \(\chi _{c1}\) case.

## 1 Introduction

The exclusive production of double charmonia via the \(e^{+}e^{-}\) annihilation at the *B* factories is an ideal laboratory for the study of heavy quarkonium. In the first place, the process is “clean”. To be specific, the color-octet effect is negligible and the contributions of the color-singlet channels are dominant, which is beneficial to draw a definite conclusion. On the experiment side, the measurements on the total cross sections of \(\sigma [e^+e^-\rightarrow J/\psi +\eta _c]\) and \(\sigma [e^+e^-\rightarrow J/\psi +\chi _{c0}]\) [1, 2, 3, 4] both significantly overshoot the leading-order (LO) QCD predictions [5, 6, 7, 8] based on the nonrelativistic QCD framework [9]. In order to deal with the large discrepancy between theory and data, a great amount of attempts have been tried [10, 11, 12, 13, 14, 15, 16, 17]. Among them, the next-to-leading-order (NLO) QCD correction [15, 16, 17] to the process is regarded as a breakthrough, significantly alleviating the tension between the theoretical predictions and the measured cross sections.

As pointed out in [5, 6], for \(e^+e^-\rightarrow J/\psi +\eta _c\), in addition to the mentioned above essential NLO QCD corrections, the interference between the QCD and QED tree-level diagrams, namely the \({\mathcal {O}} (\alpha ^3\alpha _s)\)-order terms, can also provide significant contributions, which can be ascribed to the large kinematic enhancements caused by the single-photon-fragmentation (SPF) topologies of the QED diagrams. Moreover, recently Sun et al. [18] find that the NLO QCD corrections to these \({\mathcal {O}} (\alpha ^3\alpha _s)\)-order terms can significantly further strengthen the effect of the interference terms.

Considering that the SPF topologies also exist in the process of \(e^+e^-\rightarrow J/\psi +\chi _{c}\), the cross terms between the QCD and QED diagrams probably can as well have a significant effect on the total cross section, deserving a separate investigation. For this purpose, by introducing the interference terms up to the \({\mathcal {O}} (\alpha ^3\alpha _s^2)\) order, we will carry out a further study on the exclusive production of \(J/\psi +\chi _{c}\) via \(e^+e^-\) annihilation at the *B* factories, providing a complete comparison between the interference effects and the usual QCD contributions, at the QCD NLO level, for the first time.

## 2 Calculation formalism

Total cross sections for \(e^{+}e^{-} \rightarrow J/\psi +\chi _{cJ}\) (in unit: fb), with \(J=0,1,2\). \(m_c=1.5\) GeV. \(\sqrt{s}=10.6\) GeV

\(J/\psi +\chi _{cJ}\) | \(\mu _r\) | \(\sigma _{2}^{(0)}\) | \(\sigma _{2}^{(1)}\) | \(\sigma _{3}^{(0)}\) | \(\sigma _{3}^{(1)}\) | \(\left| \frac{\sigma _{3}^{(0)}}{\sigma _{2}^{(1)}}\right| \) | \(\left| \frac{\sigma _{2}^{(1)}}{\sigma _{2}^{(0)}}\right| \) | \(\left| \frac{\sigma _{3}^{(1)}}{\sigma _{3}^{(0)}}\right| \) |
---|---|---|---|---|---|---|---|---|

\(J=0\) | \(2m_c\) | 6.62 | 3.79 | 0.24 | \(-0.01\) | 0.06 | 0.57 | 0.04 |

\(\sqrt{s}/2\) | 4.48 | 3.54 | 0.19 | 0.03 | 0.05 | 0.79 | 0.16 | |

\(\sqrt{s}\) | 3.14 | 3.11 | 0.16 | 0.05 | 0.05 | 0.99 | 0.31 | |

\(J=1\) | \(2m_c\) | 1.12 | \(-0.10\) | \(-0.08\) | 0.09 | 0.80 | 0.09 | 1.13 |

\(\sqrt{s}/2\) | 0.75 | 0.19 | \(-0.06\) | 0.05 | 0.32 | 0.25 | 0.83 | |

\(\sqrt{s}\) | 0.53 | 0.28 | \(-0.05\) | 0.03 | 0.18 | 0.53 | 0.60 | |

\(J=2\) | \(2m_c\) | 1.61 | \(-0.35\) | 0.13 | \(-0.10\) | 0.37 | 0.22 | 0.77 |

\(\sqrt{s}/2\) | 1.09 | 0.16 | 0.11 | \(-0.05\) | 0.69 | 0.15 | 0.45 | |

\(\sqrt{s}\) | 0.76 | 0.34 | 0.09 | \(-0.02\) | 0.26 | 0.45 | 0.22 |

## 3 Phenomenological results and discussions

Before presenting the phenomenological results, we first demonstrate the choices of the parameters in our calculations. The \(e^+e^-\) collision energy is assumed to be \(\sqrt{s}=10.6\) GeV. To keep the gauge invariance, both the masses of \(J/\psi \) and \(\chi _c\) are set to be \(2m_c\), with \(m_c=1.5\) GeV. \(\alpha =1/137\). For the NLO calculations, we employ the two-loop \(\alpha _s\) running, and one-loop \(\alpha _s\) running for LO. The values of \(|R_s(0)|^2\) and \(|R^{'}_p(0)|^2\) are taken as \(|R_s(0)|^2=0.81~\text {GeV}^3\) and \(|R^{'}_p(0)|^2=0.075~\text {GeV}^5\), respectively [19].

As a cross check for our calculations, with the same choices of the input parameters, we have obtained the same NLO QCD predictions, namely \(\sigma ^{(1)}_{2}\), as those of Refs. [17, 20].

The total cross sections for \(e^+e^- \rightarrow J/\psi +\chi _{cJ}\) (\(J=0,1,2\)) are presented in Table 1. One can see that, for the production of \(J/\psi +\chi _{c0}\), the contributions of the \({\mathcal {O}} (\alpha ^3\alpha _s)\)-order terms representing the interference effect between the QCD and QED tree-level diagrams, namely \(\sigma ^{(0)}_3\), is about \(6\%\) of the well-known positive and large NLO QCD corrections, \(\sigma ^{(1)}_2\). By calculating the NLO QCD corrections to \(\sigma ^{(0)}_3\), it is found that the newly obtained higher order terms, \(\sigma ^{(1)}_3\), can enhance \(\sigma ^{(0)}_3\) by about 4–\(31\%\). In the case of \(\chi _{c1}\), when \(\mu _r=3\) GeV, the \(\sigma ^{(0)}_3\) is important, which is almost identical to \(\sigma ^{(1)}_2\), further reducing the LO QCD cross section, \(\sigma ^{(0)}_2\). To our astonishment, the newly calculated \(\sigma ^{(1)}_3\) can even reach up to \(-113\%\) of \(\sigma ^{(0)}_3\), greatly compensating for the “reduction” effect caused by \(\sigma ^{(0)}_3\). With regard to the production of \(J/\psi \) in association with \(\chi _{c2}\), \(\sigma ^{(0)}_3\) can also provide a sizeable contribution comparing to the usual NLO QCD corrections \(\sigma ^{(1)}_2\). And, similar to the \(\chi _{c1}\) case, this significant \(\sigma ^{(0)}_3\) contribution will be still counteracted by \(\sigma ^{(1)}_3\) to a large extent. Therefore, to achieve a more precise prediction on the total cross sections for \(e^+e^- \rightarrow J/\psi +\chi _{cJ}\), it is definitely indispensable to incorporate the new \(\sigma ^{(1)}_3\) ingredient by calculating the NLO QCD corrections to \(\sigma ^{(0)}_3\).

*r*can reach up to \(15\%\), \(20\%\) and \(26\%\), corresponding to \(\chi _{c0}\), \(\chi _{c1}\) and \(\chi _{c2}\), respectively. Therefore, at the future \(e^+e^-\) collider with much higher collision energy, such as the ILC (International Linear Collider) and the \(\text {Super}-Z\) factory, for the exclusive production of \(e^+e^- \rightarrow J/\psi +\chi _c\), the interference effect may be fundamental, or even dominant in comparison with the usual QCD contributions.

## 4 Summary

In this paper, by introducing the cross terms between the QCD and QED diagrams, we carry out a further study on the exclusive productions of \(e^+e^- \rightarrow J/\psi +\chi _{cJ}\) (\(J=0,1,2\)) at the *B* factories, based on the NRQCD framework, providing a complete comparison between the interference effects and the usual QCD contributions, at the QCD NLO level, for the first time. It is found that the \({\mathcal {O}} (\alpha ^3\alpha _s)\)-order terms representing the interference effect between the born-level QCD and QED diagrams can provide nonnegligible contributions, which are comparable with the usual NLO QCD corrections, especially for the \(\chi _{c1}\) and \(\chi _{c2}\) cases. By calculating the QCD corrections to these \({\mathcal {O}} (\alpha ^3\alpha _s)\)-order terms for the first time, we find that the higher order terms, namely \({\mathcal {O}} (\alpha ^3\alpha _s^2)\), will lead to a significant effect on the \({\mathcal {O}} (\alpha ^3\alpha _s)\) results. Especially, in the case of \(\chi _{c1}\) and \(\chi _{c2}\), the newly calculated \({\mathcal {O}} (\alpha ^3\alpha _s^2)\)-order terms can largely counteract the \({\mathcal {O}} (\alpha ^3\alpha _s)\) contributions. Therefore, to achieve a sound estimate on the total cross sections for \(e^+e^- \rightarrow J/\psi +\chi _{cJ}\), it is indispensable to include the new \(\mathcal O (\alpha ^3\alpha _s^2)\)-order ingredient. In addition, it is found that, as the collision energy rises, the ratio taken by the interference effect between the QCD and the QED diagrams to the usual QCD cross section will increase rapidly, especially for the \(\chi _{c1}\) case.

## Notes

### Acknowledgements

This work is supported in part by the Natural Science Foundation of China under the Grant no. 11705034, by the Project for Young Talents Growth of Guizhou Provincial Department of Education under Grant no. KY[2017]135, and the Key Project for Innovation Research Groups of Guizhou Provincial Department of Education under Grant no. KY[2016]028.

## References

- 1.K. Abe et al. [Belle Collaboration], Observation of double c anti-c production in e\(+\) e\(-\) annihilation at s**(1/2) approximately 10.6-GeV. Phys. Rev. Lett.
**89**, 142001 (2002)Google Scholar - 2.K. Abe et al. [Belle Collaboration], Study of double charmonium production in e\(+\) e\(-\) annihilation at s**(1/2) 10.6-GeV. Phys. Rev. D
**70**, 071102 (2004)Google Scholar - 3.B. Aubert et al. [BaBar Collaboration], Measurement of double charmonium production in \(e^+e^-\) annihilations at \(\sqrt{s}=10.6\) GeV. Phys. Rev. D
**72**, 031101 (2005)Google Scholar - 4.T.V. Uglov, Double production at Belle. Eur. Phys. J. C
**33**, S235 (2004). https://doi.org/10.1140/epjcd/s2003-03-119-0 ADSCrossRefGoogle Scholar - 5.E. Braaten, J. Lee, Exclusive double charmonium production from e\(+\) e\(-\) annihilation into a virtual photon. Phys. Rev. D
**67**, 054007 (2003)ADSCrossRefGoogle Scholar - 6.K.Y. Liu, Z.G. He, K.T. Chao, Search for excited charmonium states in e\(+\) e\(-\) annihilation at s**(1/2) \(=\) 10.6-GeV. Phys. Rev. D
**77**, 014002 (2008)ADSCrossRefGoogle Scholar - 7.K.Y. Liu, Z.G. He, K.T. Chao, Problems of double charm production in e\(+\) e\(-\) annihilation at s**(1/2)\( = \)10.6-GeV. Phys. Lett. B
**557**, 45 (2003)ADSCrossRefGoogle Scholar - 8.K. Hagiwara, E. Kou, C.F. Qiao, Exclusive \(J/\psi \) productions at \(e^{+} e^{-}\) colliders. Phys. Lett. B
**570**, 39 (2003)ADSCrossRefGoogle Scholar - 9.G.T. Bodwin, E. Braaten, G.P. Lepage, Rigorous QCD analysis of inclusive annihilation and production of heavy quarkonium. Phys. Rev. D
**51**, 1125 (1995)ADSCrossRefGoogle Scholar - 10.J.P. Ma, Z.G. Si, Predictions for e\(+\) e\(-\) \(\rightarrow \) J/psi eta(c) with light-cone wave-functions. Phys. Rev. D
**70**, 074007 (2004)ADSCrossRefGoogle Scholar - 11.A.E. Bondar, V.L. Chernyak, Is the BELLE result for the cross section sigma(e\(+\) e\(-\) \(\rightarrow \) J / psi \(+\) eta(c)) a real difficulty for QCD? Phys. Lett. B
**612**, 215 (2005)ADSCrossRefGoogle Scholar - 12.V.V. Braguta, A.K. Likhoded, A.V. Luchinsky, Excited charmonium mesons production in e\(+\) e\(-\) annihilation at s**1/2 \(=\) 10.6-GeV. Phys. Rev. D
**72**, 074019 (2005)ADSCrossRefGoogle Scholar - 13.G.T. Bodwin, D. Kang, J. Lee, Reconciling the light-cone and NRQCD approaches to calculating e\(+\) e\(-\) \(\rightarrow \) J/psi \(+\) eta(c). Phys. Rev. D
**74**, 114028 (2006)ADSCrossRefGoogle Scholar - 14.Y.J. Sun, X.G. Wu, F. Zuo, T. Huang, The cross section of the process e\(+ +\) e\(-\)–> J/psi\(+\) eta(c) within the QCD light-cone sum rules. Eur. Phys. J. C
**67**, 117 (2010)ADSCrossRefGoogle Scholar - 15.Y. J. Zhang, Y. J. Gao , K. T. Chao, Next-to-leading order QCD correction to e\(+\) e\(-\) \(\rightarrow \) J / psi \(+\) eta(c) at s**(1/2) \(=\) 10.6-GeV. Phys. Rev. Lett.
**96**, 092001 (2006)Google Scholar - 16.B. Gong, J.X. Wang, QCD corrections to \(J/\psi \) plus \(\eta _c\) production in \(e^{+} e^{-}\) annihilation at \(S^{(1/2)}\) = 10.6-GeV. Phys. Rev. D
**77**, 054028 (2008)ADSCrossRefGoogle Scholar - 17.Y.J. Zhang, Y.Q. Ma, K.T. Chao, Factorization and NLO QCD correction in \(e^+e^- \rightarrow J/\psi (\psi (2S))+\chi _{c0}\) at B Factories. Phys. Rev. D
**78**, 054006 (2008). https://doi.org/10.1103/PhysRevD.78.054006 ADSCrossRefGoogle Scholar - 18.Z. Sun, X. G. Wu, Y. Ma , S. J. Brodsky, Exclusive production of \(J/\psi +\eta _c\) at the \(B\) factories Belle and Babar using the principle of maximum conformality. arXiv:1807.04503 [hep-ph]
- 19.E.J. Eichten, C. Quigg, Quarkonium wave functions at the origin. Phys. Rev. D
**52**, 1726 (1995). https://doi.org/10.1103/PhysRevD.52.1726 ADSCrossRefGoogle Scholar - 20.H. R. Dong, F. Feng , Y. Jia, \(O(\alpha _s)\) corrections to \(J/\psi +\chi _{cJ}\) production at \(B\) factories. JHEP
**1110**, 141 (2011) https://doi.org/10.1007/JHEP10(2011)141 (erratum: JHEP**1302**, 089, 2013, https://doi.org/10.1007/JHEP02(2013)089)

## Copyright information

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Funded by SCOAP^{3}.