⇦Chinese Physics Letters, 2021, Vol. 38, No. 4, Article code 041201 Exploring $J/\psi$ Production Mechanism at the Future Electron-Ion Collider Jian-Wei Qiu (邱建伟)1, Xiang-Peng Wang (王翔鹏)2, and Hongxi Xing (邢宏喜)3,4* Affiliations 1Theory Center, Jefferson Lab, 12000 Jefferson Avenue, Newport News, VA 23606, USA 2High Energy Physics Division, Argonne National Laboratory, Argonne, Illinois 60439, USA 3Guangdong Provincial Key Laboratory of Nuclear Science, Institute of Quantum Matter, South China Normal University, Guangzhou 510006, China 4Guangdong-Hong Kong Joint Laboratory of Quantum Matter, Southern Nuclear Science Computing Center, South China Normal University, Guangzhou 510006, China Received 7 December 2020; accepted 19 February 2021; published online 6 April 2021 H. X. is supported by Guangdong Major Project of Basic and Applied Basic Research (Grant No. 2020B0301030008), and the National Natural Science Foundation of China (Grant Nos. 12022512 and 12035007), as well as the Science and Technology Program of Guangzhou (Grant No. 2019050001). J.-W.Q. is supported by the U.S. Department of Energy (Grant No. DE-AC05-06OR23177), under which Jefferson Science Associates, LLC, manages and operates Jefferson Lab. X.-P. W. is supported by the U.S. Department of Energy, Division of High Energy Physics (Grant No. DE-AC02-06CH11357). This work is also supported within the framework of the TMD Topical Collaboration.
*Corresponding author. Email: hxing@m.scnu.edu.cn Citation Text: Qiu J W, Wang X P, and Xing H X 2021 Chin. Phys. Lett. 38 041201    Abstract We propose to use transverse momentum $p_{\rm T}$ distribution of $J/\psi$ production at the future Electron Ion Collider (EIC) to explore the production mechanism of heavy quarkonia in high energy collisions. We apply QCD and QED collinear factorization to the production of a $c\bar{c}$ pair at high $p_{\rm T}$, and non-relativistic QCD factorization to the hadronization of the pair to a $J/\psi$. We evaluate $J/\psi$ $p_{\rm T}$-distribution at both leading and next-to-leading order in strong coupling, and show that production rates for various color-spin channels of a $c\bar{c}$ pair in electron-hadron collisions are very different from that in hadron-hadron collisions, which provides a strong discriminative power to determine various transition rates for the pair to become a $J/\psi$. We predict that the $J/\psi$ produced in electron-hadron collisions is likely unpolarized, and the production is an ideal probe for gluon distribution of colliding hadron (or nucleus). We find that the $J/\psi$ production is dominated by the color-octet channel, providing an excellent probe to explore the gluon medium in large nuclei at the EIC. DOI:10.1088/0256-307X/38/4/041201 © 2021 Chinese Physics Society Article Text $J/\psi$ production has been a focus of theoretical and experimental interest since its discovery over 45 years ago.[1,2] In particular, understanding its production mechanism and using it as a probe to QCD matter are among the most active research subjects in particle and nuclear physics.[3,4] The challenge is to understand the nonperturbative transition from a produced $c\bar{c}$ pair to a physical $J/\psi$. The non-relativistic QCD (NRQCD) factorization approach[5] to describe $J/\psi$ production in hadronic collisions is by far the most phenomenologically successful one, although there are still challenges to understand the polarization of produced $J/\psi$ and to resolve the difference between various sets of non-perturbative long distance matrix elements (LDMEs) extracted from the world data.[6–10] These LDMEs characterize the transition rates for various color-spin states of a produced $c\bar{c}$ pair to become a $J/\psi$ and should be process-independent. The universality of these LDMEs, which has not been confirmed, not even the sign, is a critical test of the NRQCD factorization approach to the $J/\psi$ production.[11,12] The $J/\psi$ production has been studied in every possible high energy collisions from $e^+e^-$,[13,14] photon-hadron,[15–17] and hadron-hadron ($hh$) collisions.[18–24] New ideas have been proposed to study $J/\psi$ production more differentially, such as measuring a $J/\psi$ within a produced jet[25–28] as well as its polarization.[28] With the recent decision to build the Electron-Ion Collider (EIC)[29] in US and the proposal of an EIC in China (EicC),[30] in this Letter, we propose to use transverse momentum $p_{\rm T}$ distribution of single inclusive $J/\psi$ production at the EIC to explore and to test the $J/\psi$ production mechanism. Unlike the traditional semi-inclusive deep inelastic scattering (SIDIS), we do not require to measure the scattered electron and define the $p_{\rm T}$ of produced $J/\psi$ in the electron-hadron frame. With a single observed hard scale, $p_{\rm T}$, we apply QCD and QED collinear factorization to colliding hadron and electron, respectively, and NRQCD factorization to the hadronization of produced $c\bar{c}$ pair to a physical $J/\psi$. The inclusiveness from not measuring the scattered electron helps us to eliminate a major uncertainty of QED radiative corrections in SIDIS.[31] We also discuss the similarity and difference from tagging or not tagging the scattered electron. In this Letter, we perform explicit calculations of short-distance hard parts of $J/\psi$ production at both leading order (LO) and next-to-leading order (NLO) in strong coupling constant $\alpha_{\rm s}$. We use the recently developed dipole subtraction method[32] to deal with the QCD divergences for heavy quarkonium production. With its high design luminosity,[29] we find from our NLO results that $J/\psi$ can be produced at a wide range of $p_{\rm T}$ with a large number of events ($> $1000) at the EIC. Most importantly, we find that the single inclusive $J/\psi$ production at the EIC is very sensitive to the difference between all available sets of LDMEs, which could provide new insights into the $J/\psi$ production mechanism. In addition, we are confident to predict that the $J/\psi$ produced at the EIC is likely unpolarized, and the production is dominated by gluon initiated partonic subprocesses. With the dominance of producing color-octet $c\bar{c}$ pairs, the $p_{\rm T}$-distribution of $J/\psi$ production at the EIC could be an excellent probe to explore the dense gluon medium in large nuclei, which should further motivate the measurement of single inclusive $J/\psi$ production at the EIC. Factorization and NLO Calculation. We consider single inclusive $J/\psi$ production in electron-hadron ($eh$) collisions, $e + h\to J/\psi(p) + X$, where the scattered outgoing electron is unobserved/integrated over. Single inclusive jet production in $eh$ collisions without observing the outgoing electron was introduced in Ref. [33]. Additional studies of the same process were carried out at NLO[34–36] and even at NNLO[37] in recent years. A key difference between the jet production and $J/\psi$ production at high $p_{\rm T}$ is that a $J/\psi$ is identified in the final-state, and its non-perturbative hadronization process might be interfered by the presence of soft gluons from the beam jet of the colliding hadron. Following the arguments given for QCD collinear factorization at both leading power and next-to-leading power production of $J/\psi$ at high $p_{\rm T}$ in hadron-hadron collisions,[38–41] we are confident that the impact of such soft gluon interactions should be suppressed by powers of $\varLambda_{\rm QCD}/p_{\rm T}$. Although there is no all-order proof of NRQCD factorization to represent the hadronization of a $c\bar{c}$ pair to a $J/\psi$ by an expansion of local LDMEs organized by powers of $\alpha_{\rm s}$ and the pair's relative velocity, the factorization was found valid up to two-loop calculations.[38,39] In this Letter, as an Ansatz, we apply NRQCD factorization for the $J/\psi$ hadronization. With a single hard scale $p_{\rm T}$, the same factorization formalism for $J/\psi$ production in hadron-hadron collisions should also apply for electron-hadron collisions, $$\begin{alignat}{1} &\frac{d\sigma_{eh\to J/\psi(p)}}{dp_{\rm T}d\eta} =\sum_{a,b,n} \int\frac{dx_a}{x_a}\frac{dx_b}{x_b}f_{a/e}(x_a,\mu_{\rm f}^2) f_{b/h}(x_b,\mu_{\rm f}^2)\\ &
\times \frac{d\hat{\sigma}_{ab \to c\bar c[n] }(x_a,x_b,p_{\rm T},\eta,m_{\rm c},\mu_{\rm f}^2)}{dp_{\rm T}d\eta} \langle{\mathcal O}^{J/\psi}_{[n]}\rangle,~~ \tag {1} \end{alignat} $$ with $J/\psi$ pseudorapidity $\eta$ and factorization scale $\mu_{\rm f}$. In Eq. (1), $a=e,\gamma$ in our calculations and could include all parton flavors at higher orders; $b=q,\bar{q},g$; $n$ sums over all color-spin states of the $c\bar{c}$ pair; $f_{a/e}$ is a collinear distribution to find an electron, a photon or a parton from the colliding electron; $f_{b/h}$ is the parton distribution function (PDF) of the colliding hadron; $\hat{\sigma}$ is the perturbatively calculable short distance hard part; and $\langle{\mathcal O}^{J/\psi}_{[n]}\rangle$ is the local LDME representing the probability for the state $c\bar{c}[n]$ to become a $J/\psi$. With the available collision energies at the EIC, we did not include the fragmentation contribution from resummation of $\log(p_{\rm T}/m_{\rm c})$-type logarithmic contributions.[38–41]
cpl-38-4-041201-fig1.png
Fig. 1. Leading order Feynman diagrams for $J/\psi$ production in $eh$ collisions.
We calculate the $\hat{\sigma}$ in Eq. (1) at both LO and NLO in strong coupling $\alpha_{\rm s}$, together with leading contributions in electromagnetic coupling $\alpha$, such that $d\sigma_{eh\to J/\psi(p)} \approx d\sigma^{\rm LO} + d\sigma^{\rm NLO}$. At the LO, as shown in Fig. 1, only gluon channel contributes, $$\begin{align} \frac{d\sigma^{\rm LO}}{dp_{\rm T}d\eta} = \,&\sum_{n={^{1} {S}_{0}^{[8]}},{^{3}
{P}_{J}^{[8]}}} \int\frac{dx_b}{x_b}f_{g/h}(x_b,\mu_{\rm f}^2)\\ &\times\frac{d\hat{\sigma}^{(2,1)}_{e+g \to c\bar c[n]}}{dp_{\rm T}d\eta} \langle{\mathcal O}^{J/\psi}_{[n]}\rangle,~~ \tag {2} \end{align} $$ where leading order $f_{e/e}=\delta(1-x_a)$ was used, $\hat{\sigma}^{(2,1)}$ denotes partonic cross section evaluated at $\mathcal O(\alpha^2\alpha_{\rm s})$. At this order, only color octet ${^{1}
{S}_{0}^{[8]}}$ and ${^{3}
{P}_{J}^{[8]}} c\bar{c}$ pair can contribute to high $p_{\rm T} J/\psi$ production in $eh$ collisions. On the other hand, all four leading $c\bar{c}$ states (${^{3}
{S}_{1}^{[1]}}$, ${^{1}
{S}_{0}^{[8]}}$, ${^{3}
{S}_{1}^{[8]}}$, ${^{3}
{P}_{J}^{[8]}}$) contribute at the LO in $hh$ collisions. That is, it is advantageous for $eh$ collisions to get better information on $\langle{\mathcal O}^{{J/\psi}}({{^{1}
{S}_{0}^{[8]}}})\rangle$ and $\langle{\mathcal O}^{{J/\psi}}({{^{3}
{P}_{0}^{[8]}}})\rangle$ [For ${^{3}
{P}_{J}^{[8]}}$ channel, we converted all P-wave LDMEs according to $\langle{\mathcal O}^{{J/\psi}}({{^{3}
{P}_{J}^{[8]}}})\rangle \approx (2J+1)\langle{\mathcal O}^{{J/\psi}}({{^{3}
{P}_{0}^{[8]}}})\rangle$]. At the NLO, one has to include both real and virtual contributions. For real contribution, new light-flavor quark (antiquark) channels open up, in addition to the gluon channel, as shown in Fig. 2. Since we do not measure the outgoing electron, the real contribution has QED collinear (CO) divergence (if the electron mass $m_e\to 0$) when the outgoing electron is collinear to the incoming one. Like QCD factorization, this QED CO divergence should be absorbed into $f_{\gamma/e}$. We adopt the method presented in Ref. [42] for QED divergence. In addition, there are QCD divergences. The outgoing light parton in Fig. 2 could be either soft or collinear to the incoming parton. The CO divergence should be absorbed into the PDFs of colliding hadron and the infrared (IR) divergence should be either canceled by the virtual diagrams in Fig. 3 or by soft one-loop correction to LDMEs. In practice, we use the dipole subtraction method, recently developed specifically for heavy quarkonium production.[32] This method is based on the standard dipole subtraction techniques for light and heavy flavors[43,44] and has been successfully implemented into calculations for $hh$ collisions.[45]
cpl-38-4-041201-fig2.png
Fig. 2. NLO Feynman diagrams for real contribution to $J/\psi$ production in $eh$ collisions.
cpl-38-4-041201-fig3.png
Fig. 3. NLO Feynman diagrams for virtual contribution to $J/\psi$ production in $eh$ collisions.
For the virtual contribution, shown in Fig. 3, the ultraviolet (UV) and IR divergences can be extracted through evaluation of some standard one-loop scalar integrals. Combining real and virtual contributions together, we obtain the finite partonic hard parts at NLO, $$\begin{alignat}{1} \frac{d\sigma^{\rm NLO}}{dp_{\rm T}d\eta} ={}& \sum_{b,n} \int\frac{dx_a}{x_a}\frac{dx_a}{x_a} \Big[ f_{e/e}(x_a)\frac{d\hat{\sigma}^{(2,2)}_{eb \to c\bar c[n] }}{dp_{\rm T}d\eta}\\ &+ f_{\gamma/e}(x_a) \frac{d\hat{\sigma}^{(1,2)}_{\gamma b \to c\bar c[n] }}{dp_{\rm T}d\eta} \Big] f_{b/h}(x_b) \langle{\mathcal O}^{J/\psi}_{[n]}\rangle,~~ \tag {3} \end{alignat} $$ where $\sum_n$ runs over four $c\bar{c}$ states: ${^{3}
{S}_{1}^{[1]}}$, ${^{1}
{S}_{0}^{[8]}}$, ${^{3}
{S}_{1}^{[8]}}$, and ${^{3}
{P}_{J}^{[8]}}$. The $\hat\sigma^{(2,2)}$, $\hat\sigma^{(1,2)}$ represent the finite contributions at $\mathcal O(\alpha^2\alpha_{\rm s}^2)$ and $\mathcal O(\alpha\alpha_{\rm s}^2)$, respectively. Notice that the QED CO divergence has been removed in $\hat{\sigma}^{(2,2)}$ and thus no quasi-real photon contribution in this term, the $\hat\sigma^{(1,2)}$ in Eq. (3) corresponds to the region where the exchanged photon is quasi-real. In the leading logarithmic approximation, the leading order electron and photon distributions in an electron are given by $$\begin{alignat}{1} f_{e/e}(x,\mu_{\rm f}^2) &= \delta(1-x_a),~~ \tag {4} \end{alignat} $$ $$\begin{alignat}{1} f_{\gamma/e}(x,\mu_{\rm f}^2) &= \frac{\alpha}{2\pi} \frac{1+(1-x)^2}{x} \Big[\ln\frac{\mu_{\rm f}^2}{x^2m_e^2}-1\Big],~~ \tag {5} \end{alignat} $$ with factorization scale $\mu_{\rm f}$. An all order resummation of the logarithms could be carried out by solving a QED version of DGLAP evolution.[31] In this Letter, we skip all details of standard NLO calculation. Our partonic hard parts, $\hat{\sigma}^{(2,1)}$, $\hat{\sigma}^{(2,2)}$ and $\hat{\sigma}^{(1,2)}$ are used for the following numerical results. Notice that we neglect the resolved contributions as they begin to show up at NNLO in this factorization approach, which is beyond the scope of the present study. Phenomenology for EIC. We perform following numerical analysis for the EIC kinematics, and our evaluated cross section is fully differential, which gives us the flexibility to implement any kinematic cuts for final state particles including the outgoing electron. We choose a center-of-mass energy at $\sqrt{s} = 141.4$ GeV, $J/\psi$ pseudo-rapidity cut $|\eta| < 4$, and $J/\psi$ transverse momentum cut $3 < p_{\rm T} < 15$ GeV to be comfortable with QCD factorization and enough $J/\psi$ events. We use CT14-nlo[46] for unpolarized proton PDFs and consider three quark flavors. We choose the factorization and renormalization scales $\mu_{\rm r} = \mu_{\rm f} = \sqrt{p_{\rm T}^2+M^2}$ with the $J/\psi$ mass $M=2m_{\rm c}=3.1$ GeV and $m_{\rm c}$ is the charm quark mass. We include four leading $c\bar{c}$ states for $J/\psi$ production with corresponding LDMEs from four groups: Bodwin et al.,[6] Butenschoen et al.,[7] Chao et al.,[8] and Gong et al.,[9] which are referred as Bodwin, Butenschoen, Chao and Gong, respectively, and presented in Table 1. The numerical values for these LDMEs are very different, even different in sign in some cases, which is mainly due to the different choice of data sets in their global fittings. Notice that a large part of the data used for Butenschoen fit is HERA data, which leads to distinct results comparing to the others as will be shown below.
Table 1. $J/\psi$ NRQCD LDMEs from four different groups.
$\langle {\mathcal O}(^3S_1^{[1]})\rangle$ $\langle {\mathcal O}(^1S_0^{[8]})\rangle$ $\langle {\mathcal O}(^3S_1^{[8]})\rangle$ $\langle {\mathcal O}(^3P_0^{[8]})\rangle$
GeV$^3$ $10^{-2}$ GeV$^3$ $10^{-2}$ GeV$^3$ $10^{-2}$ GeV$^5$
Bodwin 0 9.9 1.1 1.1
Butenschoen 1.32 3.04 0.16 $-0.91$
Chao 1.16 8.9 0.30 1.26
Gong 1.16 9.7 $-0.46$ $-2.14$
In order to illustrate the sensitivity to the nonperturbative NRQCD LDMEs, we show in Fig. 4 the differential cross section of inclusive $J/\psi$ production, without tagging the outgoing electron, as a function of $p_{\rm T}$. It is clear that the production rate, evaluated with all four sets of LDMEs in Table 1, is large enough for producing sufficient $J/\psi$ events at the future EIC. It is important to notice that predictions using LDMEs from Bodwin and Chao are very similar, but significantly different from those using other two sets of LDMEs. In addition, there is almost an order of magnitude difference in production rate between Bodwin/Chao and Butenschoen. Such drastic differences between predictions using the four different sets of LDMEs clearly demonstrates the discriminative power of this new observable on $J/\psi$ production mechanism. The middle panel in Fig. 4 shows the fractional contribution from the channel with a quasi-real photon, $R^{\gamma}$, defined as a ratio of contribution from the quasi-real photon channel [$\hat\sigma^{(1,2)}$ term in Eq. (3)] and the total contribution. Clearly, the quasi-real photon channel is dominant for all four sets of LDMEs, especially at lower $p_{\rm T}$ due to the $1/x$-dependence of $f_{\gamma/e}$ in Eq. (5). Notice that we used the leading logarithmic approximation for the photon distribution in an electron as shown in Eq. (5), the 70$\%$ contribution from quasi-real photon channel should be reduced once we consider the resummed photon distribution in an electron, but not in a significant way. Detailed analysis considering QED evolution of photon distribution in an electron will be presented in a future publication. We also show in the lowest panel the $K$-factors, defined as the ratios of the NLO over LO cross sections, for the processes with (cyan) and without (black) the quasi-real photon contribution. The large $K$-factor at NLO comes from the dominated quasi-real photon channel as it first appears at NLO, such a large $K$-factor is expected to be reduced when NNLO contribution is included.[37]
cpl-38-4-041201-fig4.png
Fig. 4. The NLO differential cross section as a function of $p_{\rm T}$. Four histograms correspond to four different sets of LDMEs, the green band shows the uncertainty from scale variation $\sqrt{p_{\rm T}^2+M^2}/2 < \mu_{\rm f} < 2\sqrt{p_{\rm T}^2+M^2}$. The middle panel shows the fraction from the quasi-real photon channel, while the lowest panel shows the $K$-factors for the processes with (cyan) and without (black) the quasi-real photon contribution.
To further explore the sensitivity to each particular $\langle{\mathcal O}^{J/\psi}_{[n]}\rangle$, we studied the cross section more differentially in terms of four different $c\bar{c}$ states. In Fig. 5, we used the LDMEs from Chao as an example, and found that contribution to the $J/\psi$ production is dominated by the production of ${^{1}
{S}_{0}^{[8]}}$ state for whole $p_{\rm T}$ region. We checked that this feature also holds true for the other three sets of LDMEs. The fact that the ${^{1}
{S}_{0}^{[8]}}$ channel clearly dominates the production rate and Bodwin and Chao have a very close value for corresponding $\langle{\mathcal O}^{{J/\psi}}({{^{1}
{S}_{0}^{[8]}}})\rangle$ should be the reason for the very similar predictions from Bodwin and Chao as shown in Fig. 4. Notice that the extracted $\langle{\mathcal O}^{{J/\psi}}({{^{1}
{S}_{0}^{[8]}}})\rangle$ from Gong has even closer value comparing to Bodwin, however, the opposite sign for $\langle{\mathcal O}^{{J/\psi}}({{^{3}
{P}_{0}^{[8]}}})\rangle$ in these two sets eventually lead to big difference between the blue and purple histograms in Fig. 4. The detailed analysis shown in Fig. 5 indicates that $J/\psi$ production in $eh$ collisions can be served as a very sensitive observable to constrain the LDME $\langle{\mathcal O}^{{J/\psi}}({{^{1}
{S}_{0}^{[8]}}})\rangle$. Since ${^{1}
{S}_{0}^{[8]}}$ has no polarization preference, we predict that high-$p_{\rm T} J/\psi$ produced in inclusive $eh$ collisions will be likely unpolarized. The measurement of $J/\psi$ polarization at the future EIC will be a stringent test of NRQCD factorization.
cpl-38-4-041201-fig5.png
Fig. 5. The NLO differential cross section as a function of $p_{\rm T}$ for $J/\psi$ production in $eh$ collisions. The four histograms correspond to the contributions from four different $c\bar{c}$ states, respectively.
If we extend this study to electron-nucleus collisions at the EIC, the dominance of production rate for the ${^{1}
{S}_{0}^{[8]}} c\bar c$ state in initial hard production will provide us a unique channel to study how a color octet $c\bar{c}$ state interacts with nuclear medium when it propagates through a large nucleus. Since our result on $J/\psi$ production is fully differential in phase space for final state electron, this allows us to study the production of $J/\psi$ by exchanging a virtual photon. We impose a constraint on the momentum transfer $Q^2 > 1$ GeV$^2$ for the exchanged photon, so that we could neglect the contribution from quasi-real photon channel. In this case, the QED CO divergence will be regularized by $Q^2$ cut, and the QED dipole subtraction in $\hat\sigma^{(2,2)}$ has to be removed. In Fig. 6, we show the $J/\psi$ cross section with $Q^2$ cut. As expected, the production rate becomes smaller compared to the result without constraint on $Q^2$, but still different sets of LDMEs lead to very different production rate. This feature has also been realized in the photoproduction process.[47] Our proposed approach of leptoproduction process will provide a complementary channel and new opportunity at future EIC to help resolve the long standing problem of heavy quarkonium production mechanism.
cpl-38-4-041201-fig6.png
Fig. 6. Same as Fig. 4, but with $Q^2>1$ GeV$^2$. The middle panel shows the fraction from initial gluon channel, while the lowest panel shows the $K$-factor.
We further isolate the contributions from initial quark and gluon channels. In the middle panel of Fig. 6, we plot $R_g$ as the gluon initiated fraction of total differential cross section. The large value of $R_g$ in a wide $p_{\rm T}$ region indicates that $J/\psi$ production in $eh$ collisions is dominated by initial gluon channel. This feature makes $J/\psi$ production in $eh$ collisions a good observable to probe the initial gluon distribution, not only in the small-$x$ region as pointed out in Ref. [48], but also in a wide $x$ range in colliding proton or nucleus at the EIC. Understanding the glue is one of the main science goals for future EIC. We show in the lowest panel of Fig. 6 the $K$-factor for the process with $Q^2$ cut. As expected, the absence of quasi-real photon contribution significantly reduces the $K$-factor comparing to that without $Q^2$ cut as shown in the cyan histogram in Fig. 4. In summary, we have proposed to measure the $p_{\rm T}$ distribution of inclusive $J/\psi$ production in the electron-hadron frame at the future EIC without tagging the outgoing electron. We apply QCD and QED collinear factorization to the production of a $c\bar{c}$ pair at high $p_{\rm T}$, and non-relativistic QCD factorization to the hadronization of the pair to a $J/\psi$, and argue that such factorization should be consistent with the factorization formalism used for hadron-hadron collisions. Using the dipole subtraction method within the framework of NRQCD, we have performed explicit calculations at both LO and NLO in $\alpha_{\rm s}$. It is found that the existing four sets of NRQCD LDMEs give very different predictions for this new proposed observable, which clearly demonstrates the uniqueness of future EIC in studying the $J/\psi$ production mechanism. We also find that the dominance of producing the ${^{1}
{S}_{0}^{[8]}} c\bar{c}$ state in the total contribution provides a solid prediction that $J/\psi$ produced in $eh$ collisions will likely be unpolarized. This prediction can provide a stringent test of NRQCD factorization and shed a new light on the $J/\psi$ production mechanism. Without tagging the outgoing electron, this observable will not be sensitive to the major uncertainty from QED radiative corrections in the traditional SIDIS. In addition, this new observable at the EIC could provide even more opportunities from its sensitivity to initial-state gluon distribution in nucleon or nucleus and its dominance to produce a color octet $c\bar{c}$ state propagating through the nuclear medium in $eA$ collisions. All these unique features of this new observable will make the $p_{\rm T}$ distribution of single inclusive $J/\psi$ production in $eh$ collisions a new and very important channel to study at the future EIC. References Experimental Observation of a Heavy Particle J Discovery of a Narrow Resonance in e + e − AnnihilationHeavy quarkonium: progress, puzzles, and opportunitiesAn experimental review of open heavy flavor and quarkonium production at RHICRigorous QCD analysis of inclusive annihilation and production of heavy quarkoniumFragmentation Contributions to J / ψ Production at the Tevatron and the LHCWorld data of J / ψ production consolidate nonrelativistic QCD factorization at next-to-leading order J / ψ Polarization at Hadron Colliders in Nonrelativistic QCDPolarization for Prompt J / ψ and ψ ( 2 s ) Production at the Tevatron and LHCFragmentation contributions to hadroproduction of prompt J / ψ , χ c J , and ψ ( 2 S ) statesNew observables in inclusive production of quarkoniaReview of quarkonium production: status and prospectsMeasurement of the e + e − → J / ψ c c ¯ cross section at s ≈ 10.6     GeV Study of inclusive J/ψ production in two-photon collisions at LEP II with the DELPHI detectorMeasurements of inelastic $J/\psi$ and $\psi^{\prime}$ photoproduction at HERAInelastic photoproduction of $J/\psi$ mesons at HERAInelastic production of J/ψ mesons in photoproduction and deep inelastic scattering at HERAMeasurement of the J / ψ meson and b -hadron production cross sections in p p ¯ collisions at s = 1960     G e V Transverse momentum dependence of J / ψ polarization at midrapidity in p + p collisions at s = 200     GeV J / ψ production at high transverse momenta in p + p and Cu + Cu collisions at s NN = 200 GeVMeasurement of J / ψ and ψ ( 2 S ) Prompt Double-Differential Cross Sections in p p Collisions at s = 7     TeV Measurement of the differential cross-sections of prompt and non-prompt production of $$J/\psi $$ J / ψ and $$\psi (2\mathrm {S})$$ ψ ( 2 S ) in pp collisions at $$\sqrt{s} = 7$$ s = 7 and 8 TeV with the ATLAS detectorMeasurement of prompt J/ψ and beauty hadron production cross sections at mid-rapidity in pp collisions at $ \sqrt{s}=7 $ TeVProduction of J/ψ and $ \varUpsilon $ mesons in pp collisions at $ \sqrt{s}=8 $ TeVStudy of J / ψ Production in JetsStudy of J / ψ meson production inside jets in pp collisions at s = 8 TeV NRQCD Confronts LHCb Data on Quarkonium Production within Jets J / ψ Production and Polarization within a JetElectron-Ion Collider: The next QCD frontierElectron ion collider in ChinaElectron-Ion Collider in ChinaFactorized approach to radiative corrections for inelastic lepton-hadron collisionsDipole subtraction at next-to-leading order in nonrelativistic-QCD factorizationExploring the structure of the proton through polarization observables in l p → jet X Single-inclusive production of hadrons and jets in lepton-nucleon scattering at NLODouble-longitudinal spin asymmetry in single-inclusive lepton scattering at NLOInclusive jet production as a probe of polarized parton distribution functions at a future EICSingle-inclusive jet production in electron–nucleon collisions through next-to-next-to-leading order in perturbative QCDFragmentation, nonrelativistic QCD, and NNLO factorization analysis in heavy quarkonium productionNonrelativistic QCD factorization and the velocity dependence of NNLO poles in heavy quarkonium productionHeavy quarkonium production at collider energies: Factorization and evolutionHeavy quarkonium production at collider energies: Partonic cross section and polarizationPolarized QED splittings of massive fermions and dipole subtraction for non-collinear-safe observablesA general algorithm for calculating jet cross sections in NLO QCDDipole formalism with heavy fermionsDipole subtraction vs. phase space slicing in NLO NRQCD heavy-quarkonium production calculationsNew parton distribution functions from a global analysis of quantum chromodynamicsQCD and strongly coupled gauge theories: challenges and perspectivesComprehensive Description of J / ψ Production in Proton-Proton Collisions at Collider Energies
[1] Aubert J et al. (E598 collaboration) 1974 Phys. Rev. Lett. 33 1404
[2] Augustin J et al. (SLAC-SP-017 collaboration) 1974 Phys. Rev. Lett. 33 1406
[3] Brambilla N et al. 2011 Eur. Phys. J. C 71 1534
[4] Tang Z B, Zha W M and Zhang Y F 2020 Nucl. Sci. Tech. 31 81
[5] Bodwin G T, Braaten E and Lepage G G 1995 Phys. Rev. D 51 1125; [Erratum: 1997 Phys. Rev. D 55 5853]
[6] Bodwin G T, Chung H S, Kim U R and Lee J 2014 Phys. Rev. Lett. 113 022001
[7] Butenschoen M and Kniehl B A 2011 Phys. Rev. D 84 051501
[8] Chao K T, Ma Y Q, Shao H S, Wang K and Zhang Y J 2012 Phys. Rev. Lett. 108 242004
[9] Gong B, Wan L P, Wang J X and Zhang H F 2013 Phys. Rev. Lett. 110 042002
[10] Bodwin G T et al. 2016 Phys. Rev. D 93 034041
[11] Lansberg J P 2020 Phys. Rep. 889 1
[12] Chung H S 2018 Proc. Sci. 336 007
[13] Pakhlov P et al. (Belle collaboration) 2009 Phys. Rev. D 79 071101
[14] Abdallah J et al. (DELPHI collaboration) 2003 Phys. Lett. B 565 76
[15] Chekanov S et al. (ZEUS collaboration) 2003 Eur. Phys. J. C 27 173
[16] Adloff C et al. (H1 collaboration) 2002 Eur. Phys. J. C 25 25
[17] Aaron F et al. (H1 collaboration) 2010 Eur. Phys. J. C 68 401
[18] Acosta D et al. (CDF collaboration) 2005 Phys. Rev. D 71 032001
[19] Adare A et al. (PHENIX collaboration) 2010 Phys. Rev. D 82 012001
[20] Abelev B et al. (STAR collaboration) 2009 Phys. Rev. C 80 041902
[21] Khachatryan V et al. (CMS collaboration) 2015 Phys. Rev. Lett. 114 191802
[22] Aad G et al. (ATLAS collaboration) 2016 Eur. Phys. J. C 76 283
[23] Abelev B et al. (ALICE collaboration) 2012 J. High Energy Phys. 2012(11) 065
[24] Aaij R et al. (LHCb collaboration) 2013 J. High Energy Phys. 2013(06) 064
[25] Aaij R et al. (LHCb collaboration) 2017 Phys. Rev. Lett. 118 192001
[26] Sirunyan A M et al. (CMS collaboration) 2020 Phys. Lett. B 804 135409
[27] Bain R, Dai L, Leibovich A, Makris Y and Mehen T 2017 Phys. Rev. Lett. 119 032002
[28] Kang Z B, Qiu J W, Ringer F, Xing H and Zhang H 2017 Phys. Rev. Lett. 119 032001
[29] Accardi A et al. 2016 Eur. Phys. J. A 52 268
[30] Cao X, Chang L, Chang N B et al. 2020 Nucl. Tech. 43 20001
Anderle D P, Bertone V, Cao X, Chang L, Chang N, Chen G, Chen X, Chen Z, Cui Z, Dai L et al. 2021 arXiv:2102.09222 [nucl-ex]
[31] Liu T, Melnitchouk W, Qiu J W and Sato N 2020 arXiv:2008.02895 [hep-ph]
[32] Butenschoen M and Kniehl B A 2020 Nucl. Phys. B 950 114843
[33] Kang Z B, Metz A, Qiu J W and Zhou J 2011 Phys. Rev. D 84 034046
[34] Hinderer P, Schlegel M and Vogelsang W 2015 Phys. Rev. D 92 014001; [Erratum: 2016 Phys. Rev. D 93 119903]
[35] Hinderer P, Schlegel M and Vogelsang W 2017 Phys. Rev. D 96 014002
[36] Boughezal R, Petriello F and Xing H 2018 Phys. Rev. D 98 054031
[37] Abelof G, Boughezal R, Liu X and Petriello F 2016 Phys. Lett. B 763 52
[38] Nayak G C, Qiu J W and Sterman G F 2005 Phys. Rev. D 72 114012
[39] Nayak G C, Qiu J W and Sterman G F 2006 Phys. Rev. D 74 074007
[40] Kang Z B, Ma Y Q, Qiu J W and Sterman G 2014 Phys. Rev. D 90 034006
[41] Kang Z B, Ma Y Q, Qiu J W and Sterman G 2015 Phys. Rev. D 91 014030
[42] Dittmaier S, Kabelschacht A and Kasprzik T 2008 Nucl. Phys. B 800 146
[43] Catani S and Seymour M H 1997 Nucl. Phys. B 485 291; [Erratum: 1998 Nucl. Phys. B 510 503]
[44] Phaf L and Weinzierl S 2001 J. High Energy Phys. 2001(04) 006
[45] Butenschoen M and Kniehl B A 2020 Nucl. Phys. B 957 115056
[46] Dulat S et al. 2016 Phys. Rev. D 93 033006
[47] Brambilla N, Eidelman S, Foka P, Gardner S, Kronfeld A S, Alford M G, Alkofer R, Butenschoen M, Cohen T D, Erdmenger J et al. 2014 Eur. Phys. J. C 74 2981
[48] Ma Y Q and Venugopalan R 2014 Phys. Rev. Lett. 113 192301