Initial state radiation effects in inclusive $J/\psi$ production at B factories

Based on Monte Carlo techniques, we analyze the initial state radiation (ISR) effects in prompt $\jpsi$ inclusive production at B factories. ISR enhances cross section $\sigma(e^-e^+\to\jpsi+gg+X)$ by about $15-25\%$, which is almost the same size as the QCD and relativistic correction. Moreover, ISR slightly changes $\sigma(e^-e^+\to\jpsi+c\bar{c}+X)$. The $\jpsi$ momentum spectrum in $e^-e^+\to\jpsi+gg+X$ and in $e^-e^+\to\jpsi+c\bar{c}+X$ is softer after the photon showering from the initial $e^{\pm}$ beam radiation. After combining the QCD,relativistic, and ISR corrections,a more precise theoretical result is obtained. The new result provides a more stringent constraint of the color-octet contribution to $\sigma(e^-e^+\to\jpsi+X_{\rm{non-}c\bar{c}})$.


Introduction
A good way of studying the interplay between perturbative QCD and non-perturbative QCD is investigating the quarkonium production at various colliders. Prompt J/ψ production in hadronic collisions indicates that color-octet mechanism should play a crucial role in reducing the discrepancies between theory and experiments [1,2,3,4,5,6,7]. Although color-singlet contribution may receive large QCD corrections in several examples [8,9,10,11,12,13], it encounters the difficulty in explaining the yields and polarisation of prompt J/ψ simultaneously in hadronic collisions. In other words, a significant color octet component is essential to understand the large transverse momentum hadronic data in the current framework. On the contrast, in a relative small scale physics regime, it seems that color singlet one is already sufficient to explain the experimental data both at hadron colliders [14,15] and at B factories [16,17,18,19,20,21,22]. Hence, it is necessary to study the physics at these two scales further.
We will study the initial state radiation (ISR) effect to prompt J/ψ inclusive production at B factories in the article. Initial state radiation is a very important  ingredient that should be understood in investigating physics in electron-positron collisions. After including ISR effect in inclusive J/ψ production at B factories, we will obtain a more precise theoretical result. In general, the detailed studies of photon radiative corrections from initial e ± beams require Monte Carlo generators [39,40,41,42,43,44,45,46,47]. We interfaced a general-purposed matrix element and events generator HELAC-Onia [48,49,50,51,52] to the general photon shower program QEDPS [45,46,47] to include the initial state radiation in various e − e + anihilation processes. In the above two interested processes e − e + → J/ψ+cc+X and e − e + → J/ψ + gg + X, two representative Feynman diagrams with ISR are shown in Fig.1. After photon shower, the annihilating e − and e + no more make the head-on collision, as they might deviate from the beam axis by the radiation. The center-ofmass energy after showering √ŝ will be smaller than √ s = 10.6 GeV. The LO cross section for e − e + → J/ψ + gg + X, as shown in Fig.2, increases as √ŝ decreases near 10.6 GeV, while that for e − e + → J/ψ + cc + X changes very little as √ŝ decreases near 10.6 GeV. Therefore, we would expect that ISR effect should be siginificant in J/ψ + gg + X , while it might be not so important in J/ψ + cc + X. Moreover, since the factorizable of ISR and J/ψ production, ISR should not change QCD correction and relativistic correction unless the K factors of these two corrections are varying a lot with √ŝ near 10.6 GeV. Next, we will study the ISR effect in detail by including QCD correction and relativistic correction. The organization of the following article is: we will study the ISR effects in e − e + → J/ψ + cc + X in section 2 and in e − e + → J/ψ + gg + X in section 3, and draw our conclusion in section 4.

ISR in J/ψcc + X
The inclusive double charm production at B factories is one of the most interesting processes to probe the heavy quarkonium physics. The most precise measurement of its cross section by BELLE collaboration is [37] σ prompt (e − e + → J/ψ + cc + X) = 0.74 ± 0.08 +0.09 −0.08 pb. The QED and double photon contributions will enhance the cross section by 8+29 fb [22]. The feeddown contribution from ψ(2S) will enlarge the cross section by a factor of 1.355, while that from χ cJ is 21 fb [22,32]. The small color-octet contribution is 11 fb [32]. After combining all of these contributions, the prompt cross section will become 0.51(0.71) pb [22]. The relativistic correction was done in Ref. [53], and the authors found it was negligible.
However, the physical cross section should always include ISR at e − e + annihilation. In Fig.3, we show the cross sections with number of radiated photon for e − e + → J/ψ +cc+X. There are substaintial probability to radiate at least a photon. The averaged number of ISR photon in each event is about 0.88, which can be imagined since though there is a α supperation for radiating a photon, there is an extra log(m e / √ s) enhancement. In order to have a look at how the center-of-mass energy √ŝ changes after showering, we also plot the √ŝ / √ s distribution in Fig.3. The averaged value for √ŝ / √ s is about 0.98. It is quite close to 1, and it also indicates ISR correction should be small. Fig.4 shows the ISR effect in J/ψ momentum spectrum. We obtained the curves for NLO and NLO + ISR by normalizing the corresponding LO and LO + ISR 2 results by a NLO K factor from Ref. [22], because the K factor changes mildly in p *  specturm and with √ŝ [17]. Here, we take m c = 1.4 GeV, µ = 2m c , |R(0)| 2 = 1.01 GeV 3 . As expected, ISR correctes the momentum spectrum only a little. To make things clear, we also present the K ISR = σ LO+ISR /σ LO as a function of p * J/ψ in the right panel of Fig.4. ISR makes the J/ψ momentum spectrum a little softer. We also compared the BELLE measurement [37] with the theoretical prompt results in Fig.5. Because of the uncertainties in input parameters like m c , µ, we compared the experimental result with the theoretical ones in different parameter sets. It is shown that m c = 1.4 GeV, µ = 2m c is the closest set to the BELLE data [37], though there are still large uncertainties in experimental data. Finally, the prompt total cross sections are summarized in Tab.1. ISR decreases the cross section by a very little amount.

ISR in J/ψgg + X
In this section, we will study e − e + → J/ψ + gg + X. BELLE collaboration has measured the cross section for J/ψ + X non−cc in Ref. [37] as σ prompt (e + e − → J/ψ + X non−cc ) = 0.43 ± 0.09 ± 0.09 pb. (3.1) On the theoretical side, the NLO color-singlet cross section for prompt J/ψ + gg + X is 0.67(0.53) pb when m c = 1.4 GeV, µ = 2.8(5.3) GeV, |R(0)| 2 = 1.01 GeV 3 [18,19], which enhances LO cross secion by about 20 − 30%. Later, it was found in Ref. [20] that the relativistic correction also contribute a factor of 20 − 30% to the colorsinglet σ(e + e − → J/ψ + gg + X), which is comparable to that arises from QCD correction. Relativistic correction was also confirmed by other author [21]. From their calculations, the color-singlet result is already saturating the experimental data if we think the whole partonic cross section contributes to J/ψ + X non−cc final states.
As we discussed in the first section, ISR should also change the cross section siginificantly. We ploted the number of ISRphoton distribution and √ŝ / √ s in Fig.6. Compared with J/ψ + cc + X, there is a little larger probability to radiate a photon in J/ψ + gg + X. The reason is mainly relying in the fact that the cross section σ(e − e + → J/ψ + gg + X) will increase as √ŝ becomes smaller. The averaged number of photon in per event is enlarging to 1.04. Meanwhile, the averaged √ŝ / √ s is 0.93.
In other words, ISR effect is much more important in e − e + → J/ψ + gg + X.
Next, we are intending to include QCD correction,relativistic correction and ISR in our color-singlet results. Unlike the case in J/ψ +cc, the QCD correction to J/ψ + gg will make the J/ψ momentum spectrum much softer than the LO one [18], since at the endpoint, the LO result will suffer from large logarithms log(1 − E J/ψ /E max J/ψ ) due to kinematic reasons. The LO spectrum will change significantly at the endpoint with resummation of such logarithms, while resummation affects NLO spectrum in a very limited amount [18] 4 . On the other hand, the relativistic correction should not change the LO spectrum but only enhance it by a simple K factor. We use the formula to get the fixed-order result by taking account QCD and relativistic corrections in. Similar formula can be applied to the result with ISR dσ NLO+ISR dp * It is justified by the fact that ISR and QCD/relativistic correction can be factorized out. Moreover, the K factor of QCD/relativistic correction changes mildly with √ŝ [19,20]. The result is shown in Fig.7. ISR makes the J/ψ momentum spectrum softer, which is much clear from K ISR = σ LO+ISR /σ LO that is shown in the right panel of Fig.7. Another interesting thing is we want to see whether the K factors are sensitive to m c values. We established two plots in Fig.8. From it, we see both of K ISR = σ LO+ISR /σ LO and K NLO = K NLO(αs) + K NLO(v 2 ) − 1 are insensitive to m c .    To compare with BELLE measurement, we take the same bin size as theirs. The J/ψ momentum spectrum is shown in Fig.9. We take four different input parameter sets. The color-singlet result is already saturating the experiment data. With all of the three corrections (i.e. QCD correction,relativistic correction and ISR correction), there is more stringent room left for color-octet contribution in J/ψ + X non−cc . The total theoretical cross sections for e − e + → J/ψ +gg +X in various parameter sets are summarized in Tab.2. ISR enlarges the cross section about 15 − 25%. Although the cross sections are a little bit larger than the experimental data [37],considering large theoretical uncertainties, there are still many rooms to make the theoretical result lower. For example, one can take a lower value of |R(0)| 2 as done in Ref. [19]   potential model estimation [54]. In principle, the ratio R cc should be independent of the value of |R(0)| 2 in color-singlet case. We presented the theoretical R cc in Tab.3. We take the same parameter set in J/ψ + cc + X and J/ψ + gg + X and assume σ(e − e + → J/ψ + gg + X) = σ(e − e + → J/ψ + X non−cc ). It seems that the theoretical result is a little bit lower than BELLE measurement, but it is still within 2 standard deviation. Therefore, we expect that a more precise measurement will make the situation clear.Finally, we also listed the total cross sections σ(e − e + → J/ψ + X) = σ(e − e + → J/ψ + cc + X) + σ(e − e + → J/ψ + gg + X) in Tab.4. It is compatible with the experiment [37] value σ prompt (e − e + → J/ψ + X) = 1.17 ± 0.02 ± 0.07 pb.

Summary
The different conclusions drawn from large p T hadronic data and small scale e ± data motivate us to reconsider the theoretical results again. In this article, we keep our eyes on the issue about prompt J/ψ inclusive production at B factories. Since the cross sections in e − e + annihilation usually gain large corrections from ISR. We use Monte Carlo techniques to take ISR effect into our theoretical results. We found the effect of ISR photon shower to e − e + → J/ψ + cc + X is samll but it is large to e − e + → J/ψ + gg + X. ISR enhances the total cross section of e − e + → J/ψ + gg + X by a factor of 15 − 25% depending on the input values of parameters. Moreover, ISR makes the J/ψ momentum spectrum to be softer both in e − e + → J/ψ + cc + X and in e − e + → J/ψ + gg + X. It is important because it is thought as a good way to have a look at the color-octet contribution. Combining with QCD, relativistic and ISR corrections, we presented the theoretical results for these two processes.
To compare with experiment, feeddown contributions (mainly from ψ(2S)) are also included. Total cross sections for J/ψ + cc + X and J/ψ + gg + X are presented in Tab.1 and Tab.2 respectivley, while the J/ψ momentum spectra in BELLE bin size [37] are shown in Fig.5 and Fig.9. Due to large experimental and theoretical uncertainties, we are still unable to draw strong conclusions, but the corrections in J/ψ + gg + X really constraint the color-octet a lot. Finally, we also presented R cc with assumption σ(e − e + → J/ψ + gg + X) = σ(e − e + → J/ψ + X non−cc ), which should be more precise than the cross section alone. The theoretical result is lower than BELLE measurement [37], but is still in 2σ. More careful analysis both from theoretical and experimental sides is necessary in the future.