Perturbative QCD for $J/\psi$ Inclusive Production Via Initial State Radiation at $e^+e^-$ collider

Up to the next-leading order (NLO) of quantum chromodynamics (QCD), the process $e^+e^-\to J/\psi +X$ with the center-of-mass (CM) energy range from 3.7 to 10.6 GeV is calculated. At 10.6 GeV, the results is consistent with the experiment results at the Belle. However, the predictions are much smaller than the measurement at BESIII at low CM energy range from 3.7 to 4.6 GeV. This indicates that the convergence of QCD perturbative expansion becomes worse as the CM energy becomes lower and closer to the inclusive $J/\psi$ production threshold. For a further study of the QCD mechanism on $J/\psi$ production at $e^+e^-$ collider with different CM energy, the initial state radiation effect of $e^+e^-\to J/\psi+gg$ and $e^+e^-\to J/\psi+c \bar{c}$ are calculated at the QCD NLO. The results are plotted and the numbers of events for different CM energy bins are provided for the designed SuperKEKB. This provides a method to precisely test the validity of perturbative prediction on $J/\psi$ production in future measurements.


I. INTRODUCTION
QCD is the theory of strong interaction between quarks and gluons in the Standard Model. It exhibits two main properties, color confinement and asymptotic freedom [1,2]. Therefore, there are both perturbative and non-perturbative parts of QCD in the calculation of processes involves hadrons. The study on J/ψ related processes provides a good method to probe both perturbative and non-perturbative aspects of QCD dynamics. On one side, J/ψ's leptonic decays make it easy to be measured in the experiments. On the other side, J/ψ is a bound state of cc pair where c quark is heavy so that J/ψ related processes can be well factorized into perturbative and non-perturbative parts in theoretical calculation. In 1995, in order to explain the experimental measurements on J/ψ and ψ production at the Tevatron [3], a nonrelativistic QCD (NRQCD) factorization formalism was proposed based on the color-octet (CO) mechanism [4]. It allows consistent theoretical prediction to be made and improved order by order in the strong coupling constant α s and the heavy quark relative velocity v, although the CO NRQCD long-distance matrix elements (LDMEs) which thought to be universal can only be obtained by fitting experimental data.
In the last twenty years, many important progresses have be achieved on both experimental and theoretical studies for J/ψ related processes at different colliders.
But things are quite different in e + e − colliders. The * Electronic address: twain@ihep.ac.cn † Electronic address: wangyudong@ihep.ac.cn ‡ Electronic address: jxwang@ihep.ac.cn signature for CO production of J/ψ in e + e − annihilation at B-factories was suggested in Ref. [19] and its contribution is at the endpoint of J/ψ momentum spectra due to the kinematics of the two-body final states. The cross sections for inclusive J/ψ production in e + e − annihilation were measured by BABAR and the Belle [20][21][22][23]. But this CO signal was not observed. In Ref. [24], the authors tried to spread the CO signal by resuming the CO contribution. There was a suggestion in Ref. [25] to observe the CO contribution at e + e − → J/ψ + γ + X where CO channels have larger contribution than the colorsinglet (CS) channels at different range in energy spectrum of the photon . The experimental results are several times larger than the leading-order CS prediction [26,27]. This large discrepancy was resolved by including the NLO QCD corrections, relativistic corrections, and feeddown contribution from higher excited states (see e.g. Refs. [28][29][30][31]). Therefore the contributions from CS channels can already explain the experimental data and almost no space is left for CO contribution [32].
In experimental measurements, as many exotic states will decay into J/ψ. J/ψ production is very important background in the search for those exotic states. The precise measurement of e + e − → J/ψπ + π − was performed by BESIII Collaboration [33]. Except the contributions from decay of exotic states, the contribution for J/ψ inclusive production from continuous background obtained in this measurement is much larger. It may provides space for CO contribution [34]. Thus a detailed study for inclusive J/ψ production in e + e − collider with the center-of-mass (CM) energy range from 3.7 to 4.6 GeV is interesting. But the energy range is near inclusive J/ψ production threshold, so the validity of perturbative QCD calculation is strongly doubted. Previous studies have already shown that the CS result at the NLO can describe the the experimental measurement on inclusive J/ψ production at B-factories energy (10.58 GeV). Therefore the question, from which CM energy can CS arXiv:1903.08161v1 [hep-ph] 19 Mar 2019 perturbative results describe inclusive J/ψ production at e + e − collider, is the point we will address in this work. This paper is organized as follows. we give a detailed study for inclusive J/ψ production in e + e − collider with CM energy range from 3.7 to 10.58 GeV in Sec. II. In Sec. III, we suggest to measure inclusive J/ψ production by using the initial state radiation (ISR) effect at the Bfactories and present a detailed calculation at the QCD NLO for it. The summary and conclusion are in Sec. IV.

II. THE CROSS SECTION AT THE NLO FROM
3.70 TO 10.58 GEV As the CM energy is not enough for J/ψ + cc production at the BESIII, the calculation is almost same as J/ψ + gg case [30] at B factories, but with much smaller √ s. To perform the calculation, FDC package [35] is used to generate quad-precision FORTRAN Codes, which is essential to deal with serious numerical unstable problem in the calculation of virtual corrections near threshold. A two-cutoff method [36] is used to treat the infrared divergences in real correction processes, and after the check of cutoff independence, δ s = 10 −3 and δ c = 2 × 10 −5 are chosen. More details about the calculations can be found in Refs. [30,31].
For numerical results, the approximation M J/ψ = 2m c is our default choice. m c is set to be 1.4, 1.5 and 1.6 GeV and the renormalization scale µ r is chosen as 2m c or √ s/2 in the calculations which give an uncertainty of the results. The radial wave function at the origin of J/ψ, |R s (0)| 2 , is set to 1.006 GeV 3 for m c = 1.5 GeV. m π is set to 139.6 MeV according to PDG [37]. For the value of strong coupling constant α s , two-loop formula is used. To produce two π in the final states, a cut, m 2 X ≥ (2m π ) 2 is introduced on the invariant mass of gg for e + e − → J/ψgg, ggg for e + e − → J/ψggg and gqq for e + e − → J/ψgqq.
In Table I, the total cross sections of inclusive J/ψ production at the NLO with different √ s are shown, as well as the ratio of NLO results to LO ones. Unlike the Bfactory case, the ratio here are much larger, ranging from 1.90 to 2.73. In some sense, this larger ratio indicates that the convergence of QCD perturbative expansion becomes bad here.
In Fig.1, total cross section for inclusive J/ψ production is shown. The scale dependence of total cross section with √ s = 4.0 GeV is presented in Fig. 2. It shows clearly that the renormalization scale dependence is not improved for the NLO results in comparison with LO ones, and it gives more evidences that the convergence of QCD perturbative expansion become bad in this case.
In Fig. 3, we compare the results with experimental measurements in Ref. [33]. It can be seen that both LO and the NLO theoretical predictions are far away from experimental data, even though there is a large K factor. It probably means that the perturbative calculation becomes very bad in this situation. As the CM energy  becomes lower and closer to the inclusive J/ψ production threshold, the theoretical result of perturbative calculation loses its convergence gradually. On the other side close to 10.58 GeV, the result is consistent with the measurements from Belle [38]. It is a interesting question to know the boundary where the QCD perturbative calculation is not suitable any more. Considering that the energy in the collider is limited and it is impossible to obtain an experimental curve like the one in Fig. 3 to compare with ours. However, there is another way in which we can do the comparison by utilizing the ISR effect of QED. The ISR effect of e + e − → J/ψ + X is discussed in next section.

III. THE ISR CONTRIBUTION AT B-FACTORIES
In B-factories, there were heavy quarkonium related experimental measurement via ISR effect [39,40], and there was also heavy quarkonium related theoretical study via ISR at LO [41].
In this section, the ISR effect in the processes e + e − → J/ψgg and e + e − → J/ψcc is numerically calculated at the SuperKEKB energy. With the ISR factorization formula from the factorization of the mass and the infrared singularities [42], the total cross sections can be represented as: where D e −(+) (x, s) is the distribution function of the probability to find an electron (positron) with a momentum fraction x within the original electron (positron). Without loss of generality, we use D e (x, s) instead in the following discussion. σ R (x 1 x 2 s) is the cross section with reduced CM energy s = x 1 x 2 s. D e (x, s) satisfies following evolution equation (see as Ref. [43][44][45]) and is the regularized splitting function. By defining x = x 1 x 2 , we have with On the other hand, if we define It can be obtained, with the substitution x 2 = t/x 1 , Therefore F (x, s) can be obtained through taking the derivative of R(x, s) (with negative sign). It follows from Eq.(2) that R(x, s) satisfies the equation Defining another function D(x, s) can be obtained through taking the derivative of G(x, s). Again from Eq.(2), a similar equation as Eq. (9) is found for G(x, s) This equation differs from Eq.(9) through the substitution α(Q 2 ) → α(Q 2 )/2. Following the procedures in Ref. [46], R(x, s) and G(x, s) is obtained as with Thus F (x, s) and D e (x, s) are obtained as In the SuperKEKB [47,48], it will collide electrons at 7 GeV with positrons at 4 GeV. The invariant variable √ s = √ 112 GeV. The half-crossing angle θ c in the detector is 41.5 mrad, within which particles can not be measured. The calculations are performed at the CM frame of partons, then the Lorentz boost is performed from this frame to the laboratory frame for all the involved particles. Here we use a cut to make sure that the angle between outgoing J/ψ and beam at the laboratory frame is larger than the cross angle, θ J/ψ > θ c . The same cut is also applied for the recoiling particle X (here the momentum of X is the sum of all other final state particles), namely θ X > θ c . In really experimental measurement, a hadron is reconstructed from its decay products. Thus even a hadron is inside the cross angle, it could be still observed if its decay products are not. The cut will work better when involves the related Monte Carlo simulation in data analysis.
In Fig. 4, the ISR results with reduced CM energy ranging from 3.8 to 10.58 GeV are presented at both LO and NLO with m c = 1.5 GeV and µ r = 2m c . At low reduced CM energy there is obvious effect of the angle cut on NLO results and the effect becomes smaller as the energy becomes larger. Meanwhile, LO results with and without cut are almost coincide, which means the effect of cut is negligible. Besides, in comparison with LO results, the NLO result shows larger contribution from NLO correction at lower reduced CM energy, and the correction becomes smaller as the reduced CM energy becomes closer to initial CM energy. This behavior of NLO correction consists with what we have seen in Fig. 3. The error caused by the uncertainty of charm mass and the renormalization scale is shown in Fig. 5. Each center curve which describes the behavior of m c = 1.5 GeV is followed by a band that the upper and lower boundaries are corresponding to m c = 1.4 and 1.6 GeV. From Fig. 3, we know clearly that the measurement for J/ψπ + π − by BESIII is a few times larger than the QCD NLO prediction for inclusive J/ψ + X production, therefore experimental measurement for J/ψπ + π − via ISR effect will certainly be larger than the curve in Fig. 5 for the reduced CM energy ranging from 3.70 to 4.60 GeV. It means that the curve of ISR effect measured in future experiment will break QCD perturbative prediction in the small reduced CM energy range but be in agreement with QCD perturbative prediction at large reduced CM energy range. So the comparison between experimental measurement and QCD perturbative prediction in the curve is expecting.
KEKB has a peak luminosity of 2.1083×10 34 cm −2 s −1 and the SuperKEKB project requires a peak luminosity of 8 × 10 35 cm −2 s −1 which is 40 times larger than KEKB. SuperKEKB is designed to operate for 11 years. The total integrated luminosity accumulated by the Belle detector have reached 1.04ab −1 [49] . The goal of the BelleII detector in SuperKEKB is to accumulate an integrated luminosity of 50ab −1 [47]. In Table II, with the parameter set by m c = 1.5 GeV and µ r = 2m c , the number of events are estimated roughly at different bins of reduced CM energy.
Similar calculations in e + e − → J/ψcc is also performed. The cut effect is presented in Fig. 6. It shows that the effect of the angle cut in the detector is less than 0.1% no matter at LO or NLO, and the two curves are almost overlap with each other. The same situation is also seen in Fig. 7. From Fig. 4 and Fig. 6, the results are consistent with the ones in Refs. [30,31] when √ s close to 10.58 GeV. And the numbers of events estimated for the SuperKEKB are shown in Table III.

IV. SUMMARY AND CONCLUSION
In summary, the NLO QCD corrections of inclusive J/ψ production in e + e − annihilation with √ s range from 3.7 to 10.58 GeV are calculated. And it is found that even the QCD NLO results of the CM energy from 3.7 to 4.6 GeV are still far away from the recent experimental measurements of the BESIII. The perturbative prediction becomes bad when the CM energy is closer to J/ψ production threshold. On the other side close to 10.58 GeV, the result is consistent with the measurements in the Belle [38]. It is interesting to study the comparison between experimental measurement and QCD perturbative prediction for the processes with the CM energy from 3.7 to 10.6 GeV. By utilizing the ISR effect of QED, the ISR effect of e + e − → J/ψ + gg and e + e − → J/ψ + cc are calculated at the QCD NLO with consideration of the uncertainty from the charm quark mass and renormal-ization scale. The results are plotted and the number of event for different reduced CM energy bin are provided for future SuperKEKB. It provides a way to precisely test the validity of perturbative prediction on J/ψ production at e + e − collider with different reduced CM energy. So we suggest to measure the ISR effect on J/ψ production in future experiments.