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.

The rest paragraphs are organized as follows: In Sect. 2 we give a description on the calculation formalism. In Sect. 3, the phenomenological results and discussions are presented. Sect. 4 is reserved as a summary.

Fig. 1

Some sample QCD Feynman Diagrams for $e^+e^- \rightarrow J/\psi +\chi _c$. a (${\mathcal {M}}_{\alpha \alpha _s}$) is the QCD tree-level diagram. b–f (${\mathcal {M}}_{\alpha \alpha _s^2}$) are NLO QCD corrections to a

Fig. 2

Some sample QED Feynman Diagrams for $e^+e^- \rightarrow J/\psi +\chi _c$. a, b ($\mathcal M_{\alpha ^2}$) are the QED tree-level diagrams, in which a denotes the typical SPF diagram. c–k ($\mathcal M_{\alpha ^2\alpha _s}$) are NLO QCD corrections to a, b

2 Calculation formalism

Up to the ${\mathcal {O}} (\alpha ^3)$-order level, the squared matrix element of $e^{+}e^{-} \rightarrow J/\psi +\chi _c$ can be written as,

$$\begin{aligned}&|({\mathcal {M}}_{\alpha \alpha _s}+{\mathcal {M}}_{\alpha \alpha _s^2})+({\mathcal {M}}_{\alpha ^2}+{\mathcal {M}}_{\alpha ^2\alpha _s})|^2 \nonumber \\&\quad =|{\mathcal {M}}_{\alpha \alpha _s}|^2+2\text {Re}({\mathcal {M}}_{\alpha \alpha _s}{\mathcal {M}}_{\alpha \alpha _s^2}^{*}) +2\text {Re}({\mathcal {M}}_{\alpha \alpha _s}{\mathcal {M}}_{\alpha ^2}^{*}) \nonumber \\&\qquad +2\text {Re}({\mathcal {M}}_{\alpha \alpha _s}{\mathcal {M}}_{\alpha ^2\alpha _s}^{*}) +2\text {Re}({\mathcal {M}}_{\alpha \alpha _s^2}{\mathcal {M}}_{\alpha ^2}^{*})+\cdots \end{aligned}$$

(1)

There are in total 84 QCD diagrams (4 tree-level, 60 one-loop and 20 counter-terms) and 72 QED diagrams (6 tree-level, 42 one-loop and 24 counter-terms) for $e^+e^- \rightarrow J/\psi +\chi _c$. Some sample Feynman diagrams are illustrated in Figs. 1 and 2. In calculating ${\mathcal {M}}_{\alpha ^2\alpha _s}$, we will not need to carry out the NLO QED corrections to $\mathcal M_{\alpha \alpha _s}$, namely Fig. 1a, since these topologies are compensated by the initial-state radiation diagrams, which are irrelevant to the exclusive productions of $e^+e^- \rightarrow J/\psi +\chi _c$.

Table 1 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

Ignoring the higher-order terms in $\alpha$, we divide the differential cross section into the following four parts:

$$\begin{aligned} d\sigma = d\sigma _{2}^{(0)}+d\sigma _{2}^{(1)}+d\sigma _{3}^{(0)}+d\sigma _{3}^{(1)} \end{aligned}$$

(2)

with

$$\begin{aligned} d\sigma _{2}^{(0)}\propto & {} |{\mathcal {M}}_{\alpha \alpha _s}|^2, \end{aligned}$$

(3)

$$\begin{aligned} d\sigma _{2}^{(1)}\propto & {} 2\text {Re}({\mathcal {M}}_{\alpha \alpha _s}{\mathcal {M}}_{\alpha \alpha _s^2}^{*}), \end{aligned}$$

(4)

$$\begin{aligned} d\sigma _{3}^{(0)}\propto & {} 2\text {Re}({\mathcal {M}}_{\alpha \alpha _s}{\mathcal {M}}_{\alpha ^2}^{*}), \end{aligned}$$

(5)

$$\begin{aligned} d\sigma _{3}^{(1)}\propto & {} 2\text {Re}({\mathcal {M}}_{\alpha \alpha _s}{\mathcal {M}}_{\alpha ^2\alpha _s}^{*})+2\text {Re}({\mathcal {M}}_{\alpha ^2}{\mathcal {M}}_{\alpha \alpha _s^2}^{*}). \end{aligned}$$

(6)

The first two terms $d\sigma _{2}^{(0,1)}$ and the second two terms $d\sigma _{3}^{(0,1)}$ are the usual QCD contributions and the newly introduced interference terms up to NLO level in $\alpha _s$, respectively.

For the purpose of isolating the ultraviolet (UV) and infrared (IR) divergences, we will adopt the usual dimensional regularization procedure with $D=4-2\epsilon$. The on-mass-shell (OS) scheme is employed to set the renormalization constants of the charm-quark mass $Z_m$ and the filed $Z_2$, and the $\overline{\mathrm{MS}}$-scheme for the QCD gauge coupling $Z_g$ and the gluon field $Z_3$,

$$\begin{aligned} \delta Z_{m}^{\mathrm{OS}}= & {} -3 C_{F} \frac{\alpha _s N_{\epsilon }}{4\pi }\left[ \frac{1}{\epsilon _{\text {UV}}}-\gamma _{E}+\text {ln}\frac{4 \pi \mu _r^2}{m_c^2}+\frac{4}{3}+{\mathcal {O}}(\epsilon )\right] , \nonumber \\ \delta Z_{2}^{\mathrm{OS}}= & {} - C_{F} \frac{\alpha _s N_{\epsilon }}{4\pi }\left[ \frac{1}{\epsilon _{\text {UV}}}+\frac{2}{\epsilon _{\text {IR}}}-3 \gamma _{E}+3 \text {ln}\frac{4 \pi \mu _r^2}{m_c^2} \right. \nonumber \\&\left. +4+{\mathcal {O}}(\epsilon )\phantom {\frac{1}{\epsilon _{\text {UV}}}}\right] , \nonumber \\ \delta Z_{3}^{\overline{\mathrm{MS}}}= & {} \frac{\alpha _s N_{\epsilon }}{4\pi }(\beta _{0}-2 C_{A})\left[ \frac{1}{\epsilon _{\text {UV}}}-\gamma _{E}+\text {ln}(4\pi )+{\mathcal {O}}(\epsilon )\right] , \nonumber \\ \delta Z_{g}^{\overline{\mathrm{MS}}}= & {} -\frac{\beta _{0}}{2}\frac{\alpha _s N_{\epsilon }}{4\pi }\left[ \frac{1}{\epsilon _{\text {UV}}}-\gamma _{E}+\text {ln}(4\pi )+{\mathcal {O}}(\epsilon )\right] , \end{aligned}$$

(7)

where $\gamma _E$ is the Euler’s constant, $\beta _{0}=\frac{11}{3}C_A-\frac{4}{3}T_Fn_f$ is the one-loop coefficient of the $\beta$-function and $n_f$ is the active quark flavor numbers, $N_{\epsilon }= \Gamma [1-\epsilon ] /({4\pi \mu _r^2}/{(4m_c^2)})^{\epsilon }$. In $\mathrm{SU}(3)_c$, the color factors are given by $T_F=\frac{1}{2}$, $C_F=\frac{4}{3}$ and $C_A=3$.

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 $\sigma ^{(0)}_3$.

The $\mu _r$ dependence of the total cross sections for $e^+e^- \rightarrow J/\psi +\chi _{cJ}$ ($J=0,1,2$) are illustrated in Fig. 3. As shown in this figure, for the $\chi _{c1}$ and $\chi _{c2}$ cases, the ${\mathcal {O}} (\alpha ^3\alpha _s^2)$-order terms, $\sigma ^{(1)}_3$, can largely counteract the $\sigma ^{(0)}_3$ contributions, especially when $\mu _r$ is around 3 GeV, consequently leading to a significant effect on the cross sections of $\sigma ^{(0)}_2+\sigma ^{(1)}_2+\sigma ^{(0)}_3$.

Fig. 3

The renormalization scale dependence of the total cross sections for $e^+e^- \rightarrow J/\psi +\chi _{cJ}$ ($J=0,1,2$) at $\sqrt{s}=10.6$ GeV, $m_c=1.5$ GeV

To investigate the relative importance of the newly introduced interference terms at higher collision energy, we define the ratio of $r=\sigma _3/\sigma _2$, namely $(\sigma ^{(0)}_3+\sigma ^{(1)}_3)/(\sigma ^{(0)}_2+\sigma ^{(1)}_2)$, as a function of $\sqrt{s}$, which is illustrated in Fig. 4, with $\mu _r=\sqrt{s}/2$. As demonstrated in this figure, the ingredient of $\sigma ^{(0)}_3+\sigma ^{(1)}_3$ will play an more and more important role as the center-of-mass energy rises, especially for the $\chi _{c1}$ case. To be specific, when $\sqrt{s}=30$ GeV, the values of 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.

Fig. 4

$r=\sigma _3/\sigma _2$ as a function of $\sqrt{s}$

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.