# A Short Theoretical Review of Charmonium Production

Authors: An-Ping Chen; Yan-Qing Ma; Hong Zhang
Journal: Advances in High Energy Physics
 (2022)

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## Abstract

In this paper, we review the current status of the phenomenological study of quarkonium production in high-energy collisions. After a brief introduction of several important models and effective field theories for quarkonium production, we discuss the comparisons between theoretical predictions and experimental measurements.


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## Body

## 1. Introduction

Since the discovery of theJ/ψ in 1974, heavy quarkonium has been on the focus of much experimental and theoretical attention. Heavy quarkonium is a bound state consisting of a heavy quark (Q) and its antiquark (Q¯). Depending on the flavor of the quark pair, there are charmonium and bottomonium. The production of a heavy quarkonium involves three different momentum scales: the heavy quark mass mQ (mc≈1.5GeV and mb≈4.5GeV in on-shell scheme), which governs the perturbative creation of the heavy quark pair (QQ¯); the heavy quark momentum mQv in the quarkonium rest frame; and the typical heavy quark kinetic energy mQv2, which governs the nonperturbative hadronization of the QQ¯ to physical quarkonium. Here, v is the typical heavy quark velocity in the quarkonium rest frame (v2≈0.3 for charmonium and v2≈0.1 for bottomonium). Due to the nonrelativistic nature of the bound state, heavy quarkonium production at high-energy collisions is a very important process to test our understanding of QCD.Experimentally, many quarkonium states are relatively simple to identify in different colliders because of their clean experimental signatures and reasonably high yields. By virtue of these advantages, the heavy quarkonium is considered a promising tool to study the inner parton structure of the initial-state hadrons, such as the parton distribution functions (PDFs) and the transverse momentum-dependent distributions (TMDs) of proton. In recent years, heavy quarkonium production is also studied in heavy ion collisions to probe the quark-gluon plasma (QGP). The heavy quark pair is first produced in hard scattering at the early stage of the collisions and then interacts with the QGP and hadronizes to the heavy quarkonium on its way out of the QGP. Therefore, a good understanding of the heavy quarkonium production mechanism could facilitate our understanding of all these QCD objects.A lot of data for the quarkonium production in different high-energy collisions have been collected. TakeJ/ψ as an example, the cross section of e+e−⟶J/ψ+X (e+e− collision) has been measured by the Belle and BaBar collaborations, the cross section of e+e−⟶e+e−+J/ψ+X (γγ collision) has been measured by DELPHI collaboration at LEP, the yield and polarization of J/ψ production in ep⟶J/ψ+X (photoproduction) have been measured by H1 and Zeus at HERA, and the yield and polarization of J/ψ in hadroproduction (pp or pp¯ collision) have been measured by CDF at Tevatron, by PHENIX and STAR at RHIC, and by CMS, ATLAS, ALICE, and LHCb experiments at the LHC. In the meantime, lots of theoretical efforts have been made to explain these experimental measurements. In this article, we will briefly review the current status of the phenomenological study of quarkonium production. We first overview theoretical frameworks for describing the inclusive quarkonium production. Then, we review the current status of the comparison between theoretical predictions and experimental measurements. We put stress on charmonium production at hadron colliders, with brief overview of the quarkonium production in e+e−, ep, and γγ collisions. We refer the readers to several relevant reviews [1–5] for detailed discussions of other topics in quarkonium physics.


## 2. Quarkonium Production Mechanism

Heavy quarkonium production is usually separated into two steps: (1) the production of aQQ¯ pair with definite spin and color state in a hard collision, which could be calculated perturbatively, and (2) hadronization of the QQ¯ pair into a physical heavy quarkonium at a momentum scale much less than the heavy quark mass mQ, which is in principle nonperturbative. Different treatments of the nonperturbative transition from QQ¯ pair to the physical quarkonium lead to different theoretical models. In the following, we briefly describe some of the most widely used ones: the color evaporation model (CEM) [6–8], the color-singlet model (CSM) [9–11], the nonrelativistic QCD (NRQCD) factorization theory [12], the fragmentation function approach [13–17], and the most recently proposed soft gluon factorization (SGF) approach [18, 19].


### 2.1. The Color Evaporation Model (CEM)

In the CEM [6–8], it is assumed that every produced QQ¯ pair evolves into a specific heavy quarkonium if its invariant mass is below the open-charm/bottom threshold. It is further assumed that the probability for the QQ¯ pair to evolve into a specific quarkonium state H is given by a constant FH which is independent of momentum and process. Mathematically, the production cross section of H is expressed in the CEM as
(1)σH=FH∫2mQ2mDdσQQ¯dmQQ¯dmQQ¯,where2mD is the open-charm/bottom threshold. For each heavy quarkonium state, the CEM in equation (1) has one free parameter FH. The CEM is intuitive, simple, and successful to explain J/ψ production data. However, it has a very strong prediction that the production rate of any two different charmonium states depends on neither the process nor kinematic variables, which contradicts data from many experiments. For example, the ratio of production cross section of ψ2S to that of J/ψ in pp collisions clearly depends on their transverse momentum [20, 21]. To overcome these obstacles, an improved version of the model, the ICEM, was proposed [22], in which the momentum of the QQ¯ pair is assumed to be larger than the momentum of quarkonium H by a factor of mQQ¯/mH. One consequence is that the lower limit of the above integral is replaced by mH. It was shown that the ICEM can describe the charmonium yields as well as the ratio of ψ2S over J/ψ [22]. The ICEM was also combined with kT-factorization to describe quarkonium polarization [23–28].


### 2.2. The Color-Singlet Model (CSM)

In the CSM, theQQ¯ pair that evolves into the quarkonium is assumed to have the same color, spin, and orbital-angular-momentum quantum numbers as the heavy quarkonium. Particularly, it must be in a color-singlet state. Under this assumption, the production cross section for each quarkonium state H is related to the wave function (or its derivatives) of H around the origin, which can be extracted from the decay process of H or calculated from the potential model or lattice QCD. Therefore, the CSM effectively has no free parameters. At relatively low energies, the LO CSM predictions for quarkonium production agree with the experimental data, while at high energies, the LO CSM predictions have been shown to underestimate the experimental data of direct J/ψ and ψ2S production at s=1.8 TeV pp collisions [29] by more than an order of magnitude, which is known as the ψ2S surplus puzzle. In the past decade, it was found that the NLO and NNLO corrections to the CSM are significantly larger than the LO contributions [30–32]. Including these corrections relieves the inconsistency between the LO CSM prediction and the data. However, a full description of data is still difficult. Besides, given the very large corrections at NLO and NNLO, it is not clear that the perturbative expansion in αs is convergent. Moreover, in the case of P-wave production and decay, the CSM is known to be incomplete because it suffers from uncanceled infrared divergences. The last point can be rigorously cured in a more general framework of NRQCD factorization theory which we will discuss below.


### 2.3. The NRQCD Factorization Approach

NRQCD is an effective theory of QCD and reproduces full QCD dynamics at momentum scales of ordermQv and smaller. In NRQCD, the production cross section of a heavy quarkonium H is given by the factorization formula [12]:
(2)σH=∑nσnμΛOnHμΛ.Here,μΛ is the NRQCD factorization scale, which is the ultraviolet (UV) cutoff of the NRQCD effective theory, σn is the short-distance coefficient (SDC) which describes the production of a QQ¯ pair with quantum number n in the hard scattering, and OnHμΛ is the NRQCD long-distance matrix element (LDME) that describes the hadronization of the QQ¯ pair in state n into the heavy quarkonium H. The LDME is defined as the vacuum expectation value of a four-fermion operator in NRQCD, and each LDME has a known scaling behavior in powers of v. Then, the sum over n can be organized in powers of v. Therefore, equation (2) is a double expansion of αs and v. In practice, for a certain accuracy, one truncates the summation and keeps only a few LDMEs for each H production. The predictive power of the NRQCD factorization approach relies on the convergence of this velocity expansion, as well as the universality of LDMEs.As shown in equation (2), the NRQCD factorization contains contributions from both the color-singlet (CS) and the color-octet (CO) channels. If one sets the CO contributions to zero, one could recover the CSM for S-wave heavy quarkonium production. Thanks to the CO contributions, NRQCD solves the infrared divergence problem encountered in the CSM [33]. Although there is no all-order proof of NRQCD factorization for quarkonium production yet, it is found that the factorization holds at least to next-to-next-to-leading order (NNLO) in αs if the LDMEs are modified to be gauge complete [34–38].


### 2.4. The Fragmentation Function Approach

The SDCs in the NRQCD factorization formula in equation (2) suffer from large high-order αs corrections for heavy quarkonium produced at large transverse momentum pT≫mQ, which is an important kinematic region in high-energy colliders. In this region, the high-order corrections of the SDCs receive huge power enhancement in terms of pT2/mQ2, as well as large logarithmic corrections in terms of lnpT2/mQ2. To overcome these problems, a new QCD factorization approach was proposed to describe the large pT heavy quarkonium production [13–17]. In this approach, the cross section is first expanded by powers of mQ2/pT2. Both the leading-power (LP) term and next-to-leading-power (NLP) term of the expansion could be factorized systematically into parton-production cross sections convoluted with several universal FFs, i.e.,
(3)dσA+B⟶H+XpT=∑fdσ^A+B⟶f+XpTz,μF⊗Df⟶Hz,mQ,μF+∑κdσ^A+B⟶QQ¯κ+XPQQ¯κ=pTz,μF⊗DQQ¯κ⟶Hz,mQ,μF+OmQ4/pT4,where the first term on the right side gives the contribution of LP inmQ2/pT2 and the second term gives the NLP contribution. The symbol ⊗ represents the convolution of the light-cone momentum fraction z. In the first term, dσ^A+B⟶f+X is the semi-inclusive cross section for initial hadrons A and B to produce an on-shell parton f . The FFs Df⟶H represents the possibility of finding H in the hadronization products of parton f. The NLP contribution is similar, with an intermediate heavy quark pair in state κ instead of a single parton f.Since the NLP term is suppressed bymQ2/pT2, it seems to be not important at large pT. However, it is natural to expect that a heavy quark pair is more likely to evolve into a heavy quarkonium H, comparing to a single quark or gluon. Consequently, the double-parton FFs are more important than the single-parton ones. This nonpertubative enhancement could balance the perturbative mQ2/pT2 suppression in the intermediate pT range. There are more NLP contributions from other intermediate double partons besides a heavy quark pair. They are not included in equation (3) because they do not have this nonperturbative enhancement.The factorization formula in equation (3) is a double expansion of mQ2/pT2 and αs. The factorization scale μF is chosen at the same order of pT so no large logarithm exists. The FFs with different μF are related by a closed set of evolution equations. To make a theoretical prediction, one still needs a set of input FFs at a certain scale μ0~2mQ. These input FFs are nonperturbative and, in principle, should be extracted from fitting experimental data. However, since μ0~2mQ≫ΛQCD, it is natural to use NRQCD factorization to further factorize these input FFs. By doing this, all unknown input FFs could be expressed in terms of NRQCD LDMEs with the perturbatively calculable coefficients
(4)Df⟶Hz,mQ,μ0=∑nd^f⟶QQ¯nz,mQ,μ0,μΛOnHμΛ,(5)DQQ¯κ⟶Hz,mQ,μ0=∑nd^QQ¯κ⟶QQ¯nz,mQ,μ0,μΛOnHμΛ,whered^f⟶QQ¯n and d^QQ¯κ⟶QQ¯n describe the perturbative evolution of a parton f and a QQ¯ pair with quantum number κ into a QQ¯ pair in the state n, respectively. Mathematically, the FF+NRQCD factorization formula in equations (3), (4) and (5) is a reorganization of terms in equation (2). Physically, the FF+NRQCD factorization method correctly includes the evolution of a heavy quark pair when the relative velocity in the quarkonium rest frame is not much smaller than 1 and the NRQCD.During the past two decades, the FFs in equation (4) and (5) have been widely studied. The coefficients for all double-parton FFs to both S-wave and P-wave states are calculated up to Oαs in Refs. [39–41]. The coefficients for all single-parton FFs are available up to Oαs2[42–52] (see [39–41] for a summary and comparison). At αs3 order, the coefficients of gluon FFs to QQ¯3S11, QQ¯1P11, QQ¯1S01,8, and QQ¯3PJ1,8 are calculated in Refs. [38, 44, 53–60]. Recently, the heavy quark FFs to 1S01,8 state are obtained in Refs. [61, 62].With the input FFs and the evolution equations, FF+NRQCD factorization formalism provides a systematic reorganization of the cross section in terms of powers ofmQ2/pT2 and a systematic method for resumming the potentially large lnpT2/mQ2-type logarithms. It is expected to have a better convergence in the αs expansion than NRQCD.


### 2.5. The Soft Gluon Factorization Approach

As we will discuss later, the NRQCD factorization still encounters some difficulties in describing inclusive quarkonium production data. It is known long time ago that NRQCD has bad convergence in velocity expansion [63], which may be responsible for the phenomenological difficulties. The aim of SGF is to provide a framework with better convergence [18].In the SGF approach, the differential cross section of the quarkoniumH production is factorized as
(6)2π32PH0dσHd3PH≈∑n∫d4P2π4HnPFn⟶HP,PH,whereP is the momentum of the intermediate QQ¯ pair, PH is the momentum of H, and Fn⟶HP,PH is soft gluon distribution function (SGD), which describes the hadronization of the QQ¯ pair into physical quarkonium H by emitting soft hadrons. To account for the effect of soft hadron emission, which are mainly soft gluons perturbatively, the momentum of the intermediate state P is kept different from the observed quarkonium momentum PH, which is different from the treatment in NRQCD. The SGD is defined by QCD fields in a small loop momentum region. With an explicit definition of the small region and taking advantage of equations of motion, the SGF is shown to be equivalent to the NRQCD factorization [19]. Nevertheless, compared with NRQCD, the SGF resums a series of relativistic corrections originating from kinematic effects, which results in a better convergence in velocity expansion. It is expected that the SGF approach may provide a better description of experimental data.The first phenomenological application of the SGF approach was carried out in Ref. [64] for exclusive quarkonium production processes. It was shown there that, for ηc+γ production at B factories, the SGF provides the best description of experimental data among all existing theoretical calculations. Recently, a quarkonium fragmentation function in SGF has been calculated to NLO in Ref. [65], which is the first step to apply SGF to inclusive quarkonium production processes. With explicit NLO calculation, it was demonstrated that the SGF is valid at the NLO level. Phenomenological application for inclusive quarkonium production in the SGF approach is still missing.


## 2.1. The Color Evaporation Model (CEM)

In the CEM [6–8], it is assumed that every produced QQ¯ pair evolves into a specific heavy quarkonium if its invariant mass is below the open-charm/bottom threshold. It is further assumed that the probability for the QQ¯ pair to evolve into a specific quarkonium state H is given by a constant FH which is independent of momentum and process. Mathematically, the production cross section of H is expressed in the CEM as
(1)σH=FH∫2mQ2mDdσQQ¯dmQQ¯dmQQ¯,where2mD is the open-charm/bottom threshold. For each heavy quarkonium state, the CEM in equation (1) has one free parameter FH. The CEM is intuitive, simple, and successful to explain J/ψ production data. However, it has a very strong prediction that the production rate of any two different charmonium states depends on neither the process nor kinematic variables, which contradicts data from many experiments. For example, the ratio of production cross section of ψ2S to that of J/ψ in pp collisions clearly depends on their transverse momentum [20, 21]. To overcome these obstacles, an improved version of the model, the ICEM, was proposed [22], in which the momentum of the QQ¯ pair is assumed to be larger than the momentum of quarkonium H by a factor of mQQ¯/mH. One consequence is that the lower limit of the above integral is replaced by mH. It was shown that the ICEM can describe the charmonium yields as well as the ratio of ψ2S over J/ψ [22]. The ICEM was also combined with kT-factorization to describe quarkonium polarization [23–28].


## 2.2. The Color-Singlet Model (CSM)

In the CSM, theQQ¯ pair that evolves into the quarkonium is assumed to have the same color, spin, and orbital-angular-momentum quantum numbers as the heavy quarkonium. Particularly, it must be in a color-singlet state. Under this assumption, the production cross section for each quarkonium state H is related to the wave function (or its derivatives) of H around the origin, which can be extracted from the decay process of H or calculated from the potential model or lattice QCD. Therefore, the CSM effectively has no free parameters. At relatively low energies, the LO CSM predictions for quarkonium production agree with the experimental data, while at high energies, the LO CSM predictions have been shown to underestimate the experimental data of direct J/ψ and ψ2S production at s=1.8 TeV pp collisions [29] by more than an order of magnitude, which is known as the ψ2S surplus puzzle. In the past decade, it was found that the NLO and NNLO corrections to the CSM are significantly larger than the LO contributions [30–32]. Including these corrections relieves the inconsistency between the LO CSM prediction and the data. However, a full description of data is still difficult. Besides, given the very large corrections at NLO and NNLO, it is not clear that the perturbative expansion in αs is convergent. Moreover, in the case of P-wave production and decay, the CSM is known to be incomplete because it suffers from uncanceled infrared divergences. The last point can be rigorously cured in a more general framework of NRQCD factorization theory which we will discuss below.


## 2.3. The NRQCD Factorization Approach

NRQCD is an effective theory of QCD and reproduces full QCD dynamics at momentum scales of ordermQv and smaller. In NRQCD, the production cross section of a heavy quarkonium H is given by the factorization formula [12]:
(2)σH=∑nσnμΛOnHμΛ.Here,μΛ is the NRQCD factorization scale, which is the ultraviolet (UV) cutoff of the NRQCD effective theory, σn is the short-distance coefficient (SDC) which describes the production of a QQ¯ pair with quantum number n in the hard scattering, and OnHμΛ is the NRQCD long-distance matrix element (LDME) that describes the hadronization of the QQ¯ pair in state n into the heavy quarkonium H. The LDME is defined as the vacuum expectation value of a four-fermion operator in NRQCD, and each LDME has a known scaling behavior in powers of v. Then, the sum over n can be organized in powers of v. Therefore, equation (2) is a double expansion of αs and v. In practice, for a certain accuracy, one truncates the summation and keeps only a few LDMEs for each H production. The predictive power of the NRQCD factorization approach relies on the convergence of this velocity expansion, as well as the universality of LDMEs.As shown in equation (2), the NRQCD factorization contains contributions from both the color-singlet (CS) and the color-octet (CO) channels. If one sets the CO contributions to zero, one could recover the CSM for S-wave heavy quarkonium production. Thanks to the CO contributions, NRQCD solves the infrared divergence problem encountered in the CSM [33]. Although there is no all-order proof of NRQCD factorization for quarkonium production yet, it is found that the factorization holds at least to next-to-next-to-leading order (NNLO) in αs if the LDMEs are modified to be gauge complete [34–38].


## 2.4. The Fragmentation Function Approach

The SDCs in the NRQCD factorization formula in equation (2) suffer from large high-order αs corrections for heavy quarkonium produced at large transverse momentum pT≫mQ, which is an important kinematic region in high-energy colliders. In this region, the high-order corrections of the SDCs receive huge power enhancement in terms of pT2/mQ2, as well as large logarithmic corrections in terms of lnpT2/mQ2. To overcome these problems, a new QCD factorization approach was proposed to describe the large pT heavy quarkonium production [13–17]. In this approach, the cross section is first expanded by powers of mQ2/pT2. Both the leading-power (LP) term and next-to-leading-power (NLP) term of the expansion could be factorized systematically into parton-production cross sections convoluted with several universal FFs, i.e.,
(3)dσA+B⟶H+XpT=∑fdσ^A+B⟶f+XpTz,μF⊗Df⟶Hz,mQ,μF+∑κdσ^A+B⟶QQ¯κ+XPQQ¯κ=pTz,μF⊗DQQ¯κ⟶Hz,mQ,μF+OmQ4/pT4,where the first term on the right side gives the contribution of LP inmQ2/pT2 and the second term gives the NLP contribution. The symbol ⊗ represents the convolution of the light-cone momentum fraction z. In the first term, dσ^A+B⟶f+X is the semi-inclusive cross section for initial hadrons A and B to produce an on-shell parton f . The FFs Df⟶H represents the possibility of finding H in the hadronization products of parton f. The NLP contribution is similar, with an intermediate heavy quark pair in state κ instead of a single parton f.Since the NLP term is suppressed bymQ2/pT2, it seems to be not important at large pT. However, it is natural to expect that a heavy quark pair is more likely to evolve into a heavy quarkonium H, comparing to a single quark or gluon. Consequently, the double-parton FFs are more important than the single-parton ones. This nonpertubative enhancement could balance the perturbative mQ2/pT2 suppression in the intermediate pT range. There are more NLP contributions from other intermediate double partons besides a heavy quark pair. They are not included in equation (3) because they do not have this nonperturbative enhancement.The factorization formula in equation (3) is a double expansion of mQ2/pT2 and αs. The factorization scale μF is chosen at the same order of pT so no large logarithm exists. The FFs with different μF are related by a closed set of evolution equations. To make a theoretical prediction, one still needs a set of input FFs at a certain scale μ0~2mQ. These input FFs are nonperturbative and, in principle, should be extracted from fitting experimental data. However, since μ0~2mQ≫ΛQCD, it is natural to use NRQCD factorization to further factorize these input FFs. By doing this, all unknown input FFs could be expressed in terms of NRQCD LDMEs with the perturbatively calculable coefficients
(4)Df⟶Hz,mQ,μ0=∑nd^f⟶QQ¯nz,mQ,μ0,μΛOnHμΛ,(5)DQQ¯κ⟶Hz,mQ,μ0=∑nd^QQ¯κ⟶QQ¯nz,mQ,μ0,μΛOnHμΛ,whered^f⟶QQ¯n and d^QQ¯κ⟶QQ¯n describe the perturbative evolution of a parton f and a QQ¯ pair with quantum number κ into a QQ¯ pair in the state n, respectively. Mathematically, the FF+NRQCD factorization formula in equations (3), (4) and (5) is a reorganization of terms in equation (2). Physically, the FF+NRQCD factorization method correctly includes the evolution of a heavy quark pair when the relative velocity in the quarkonium rest frame is not much smaller than 1 and the NRQCD.During the past two decades, the FFs in equation (4) and (5) have been widely studied. The coefficients for all double-parton FFs to both S-wave and P-wave states are calculated up to Oαs in Refs. [39–41]. The coefficients for all single-parton FFs are available up to Oαs2[42–52] (see [39–41] for a summary and comparison). At αs3 order, the coefficients of gluon FFs to QQ¯3S11, QQ¯1P11, QQ¯1S01,8, and QQ¯3PJ1,8 are calculated in Refs. [38, 44, 53–60]. Recently, the heavy quark FFs to 1S01,8 state are obtained in Refs. [61, 62].With the input FFs and the evolution equations, FF+NRQCD factorization formalism provides a systematic reorganization of the cross section in terms of powers ofmQ2/pT2 and a systematic method for resumming the potentially large lnpT2/mQ2-type logarithms. It is expected to have a better convergence in the αs expansion than NRQCD.


## 2.5. The Soft Gluon Factorization Approach

As we will discuss later, the NRQCD factorization still encounters some difficulties in describing inclusive quarkonium production data. It is known long time ago that NRQCD has bad convergence in velocity expansion [63], which may be responsible for the phenomenological difficulties. The aim of SGF is to provide a framework with better convergence [18].In the SGF approach, the differential cross section of the quarkoniumH production is factorized as
(6)2π32PH0dσHd3PH≈∑n∫d4P2π4HnPFn⟶HP,PH,whereP is the momentum of the intermediate QQ¯ pair, PH is the momentum of H, and Fn⟶HP,PH is soft gluon distribution function (SGD), which describes the hadronization of the QQ¯ pair into physical quarkonium H by emitting soft hadrons. To account for the effect of soft hadron emission, which are mainly soft gluons perturbatively, the momentum of the intermediate state P is kept different from the observed quarkonium momentum PH, which is different from the treatment in NRQCD. The SGD is defined by QCD fields in a small loop momentum region. With an explicit definition of the small region and taking advantage of equations of motion, the SGF is shown to be equivalent to the NRQCD factorization [19]. Nevertheless, compared with NRQCD, the SGF resums a series of relativistic corrections originating from kinematic effects, which results in a better convergence in velocity expansion. It is expected that the SGF approach may provide a better description of experimental data.The first phenomenological application of the SGF approach was carried out in Ref. [64] for exclusive quarkonium production processes. It was shown there that, for ηc+γ production at B factories, the SGF provides the best description of experimental data among all existing theoretical calculations. Recently, a quarkonium fragmentation function in SGF has been calculated to NLO in Ref. [65], which is the first step to apply SGF to inclusive quarkonium production processes. With explicit NLO calculation, it was demonstrated that the SGF is valid at the NLO level. Phenomenological application for inclusive quarkonium production in the SGF approach is still missing.


## 3. Quarkonium Production inpp Collisions

### 3.1. HighpT Heavy Quarkonium Production

Based on the NRQCD factorization framework, the heavy quarkonium production inpp collisions has been widely studied. In the large pT region, the differential cross section of the quarkonium H production can be factorized as
(7)dσA+B⟶H+X=∑ndσ^nOnH=∑i,j,n∫dx1dx2fi/Ax1,μFfj/Bx2,μFdσ^i+j⟶QQ¯n+XμF,μΛOnHμΛ,wheref’s are the parton distribution functions (PDFs) for the partons in the initial colliding protons, μF is the collinear factorization scale, and dσi+j⟶QQ¯n+X is the partonic differential cross section.At LO inαs, the partonic differential cross sections of n=3S18,1S08,3PJ8 are scaled as pT−4, pT−6, and pT−6, respectively, which are more important than the CS contribution n=3S11, scaled as pT−8. By including the CO contributions, the ψnS surplus puzzle in CSM is solved naturally [66]. But NRQCD factorization encounters difficulties with charmonium polarizations. Since the dominant contribution is from the transverse 3S18 channel, LO NRQCD predicts that ψnS and YnS produced at hadron colliders are mainly transversely polarized [42, 67, 68]. On the contrary, experimental measurements at Tevatron and LHC find that these states are almost produced unpolarized [69–71]. In addition, the LO calculation in NRQCD is difficult to explain the observed cross section ratio Rχc=σχc2/σχc1 of the P-wave charmonia χcJ at Tevatron [72], as the 3S18 channel dominance predicts the ratio Rχc to be 5/3 by spin counting [73, 74], which is much larger than the measured value of 0.75.In the past decade, the perturbative SDCs for quarkonium production cross sections and polarizations have been calculated to NLO by three groups [75–81]. At this order, the 1S08 and 3PJ8 channels are pT−4 scaling, so their contributions are also important at large pT. Even though the NLO SDCs obtained by the three groups agree, they give very different predictions for J/ψ polarization, due to the different methods used in fitting the CO LDMEs.In Refs. [75–77], it was found that at high pT, the SDC of the 3PJ8 channel can be decomposed into a linear combination of the other two CO channels:
(8)dσ^3PJ8=r0dσ^1S08+r1dσ^3S18,wherer0=3.9,r1=−0.56 for the Tevatron and r0=4.1,r1=−0.56 for the LHC. Therefore, two linearly combined NRQCD LDMEs are introduced to fit the data:
(9)M0,r0H=OH1S08+r0mc2OH3P08,M1,r1H=OH3S18+r1mc2OH3P08,whereH is J/ψ or ψ2S. Using the Tevatron data [82, 83] with pT>7 GeV, the LDMEs are extracted as
(10)M0,r0J/ψ=7.4±1.9×10−2Gev3,M1,r1J/ψ=0.05±0.02×10−2Gev3.With these fitted LDMEs, the theoretical predictions for promptJ/ψ production are generally consistent with experimental data from the Tevatron and the LHC [82–84] (see Figure 1). As shown in Figure 1(b), the J/ψ polarization is roughly consistent with the CDF data, in which the J/ψ is approximately unpolarized. Detailed analysis shows that the transversely polarized contributions from 3S18 and 3PJ8 channels cancel each other. Thus, a possible mechanism is that the unpolarized 1S08 channel dominates, which leads to an unpolarized production [77, 85, 86]. For the ψ2S production, such cancellation is weak [87], and its polarization is still hard to explain.Figure 1
Comparison of NLO NRQCD calculations with promptJ/ψ data. (a) The pT distributions of prompt J/ψ production at the Tevatron and the LHC. (b) NLO NRQCD prediction for the polarization of J/ψ production at the Tevatron. Figures taken from Refs. [75, 77].
(a)(b)In Ref. [79], the authors determine the CO LDMEs by a global fit of a number of measurements. Using prompt J/ψ production data in pp (pT>3GeV) [88–94], ep (pT>1 GeV) [95–97], γγ [98], and e+e− [99] collisions, they determine all three CO LDMEs. In particular, they obtained a negative value for OJ/ψ3P08, and thus, the contributions from 3S18 and 3PJ8 channels add and enhance the transverse polarization at large pT. With these inputs, it was found that the J/ψ is transversely polarized at large pT as shown in Figure 2.Figure 2
Comparisons of the NLO NRQCD calculations in Ref. [79] for the J/ψ polarization with the Tevatron data. Figure from Ref. [79].In Ref. [80], the LDME determination is based on the yield data from CDF [89] and LHCb [94]. Similar to the method in Refs. [75, 77, 87], the data with pT<7GeV are not considered in the fitting. It is found that both OJ/ψ3S18 and OJ/ψ3PJ8 are negative. As the two LDMEs have the same sign, the cancellation between transversely polarized contributions still occurs, which explains the unpolarized J/ψ produced at large pT.Due to the approximate heavy quark spin symmetry (HQSS) of NRQCD [12], the production of J/ψ is closely related to the production of ηc by the following relations:
(11)Oηc3S18=OJ/ψ1S08,Oηc1S08=13OJ/ψ3S18,Oηc1P18=32J+1OJ/ψ3PJ8.Then, the measurement ofηc can provide a further test of the J/ψ LDMEs. The first prompt ηc hadroproduction cross section was measured in 2014 by the LHCb collaboration with the centre-of-mass energies at s=7 TeV and 8 TeV [100]. With the three sets of the J/ψ LDMEs above, the authors of Ref. [101] found that theoretical calculations overshoot experimental measurement, as shown in Figure 3. Detailed analysis shows that the LHCb data are almost saturated by the contribution from the CS 1S01 channel. This gives a constraint on OJ/ψ1S08. By assuming that the data are completely contributed from the 3S18 channel, the authors of Ref. [102] obtain an upper bound for Oηc3S18:
(12)Oηc3S18<1.46×10−2GeV3.Figure 3
The LHCb [100] measurements for prompt ηc hadroproduction at s=7TeV are compared with the NLO NRQCD calculations evaluated with three different LDME sets from Refs. [77, 79, 80]. Figures taken from Ref. [101].This bound is consistent with a previous study by the same authors on theJ/ψ yield and polarization [87], so they argue that the prompt production of ηc and J/ψ can be understood in the same theoretical framework. The authors of Ref. [103] fit both the CO and CS LDMEs, and their predictions are also compatible with data. Recently, the LHCb collaboration reports their measurement of the ηc hadroproduction cross section at s=13 TeV [104], which is consistent with the previous data.Within the same framework as that for theJ/ψ production, the study of Y production at NLO was carried out by two groups [81, 105, 106]. Their results show a slightly transverse polarization at large pT, consistent with the measurements by CMS [107] within experimental uncertainties.ForP-wave quarkonia, the first complete NLO study of χcJ production was performed in Ref. [108] in 2010. At NLO, the CS 3PJ1 channels are scaled as pT−4, which give large contributions at high pT. They also find that 3P11 decreases slower than 3P21, so the measured ratio of Rχc at the Tevatron [72] can be naturally explained (see Figure 4). In 2016, the authors of Ref. [109] performed a global analysis of the existing data on χc hadroproduction from the Tevatron [72] and the LHC [110–113]. In the meantime, the polarization of the χc was also predicted [114–116], which indicated that the polarization parameter λθ of the χc1 should be positive while that of the χc2 should be negative. The experimental measurement was not available until 2019 by the CMS collaboration [117]. Current experimental data seem to be consistent with the NLO results.Figure 4
Transverse momentum distribution of ratioRχc at the Tevatron with cut ∣yχcJ∣<1. Figure taken from Ref. [108].As mentioned in the previous section, in the highpT region, the partonic differential cross sections in the NRQCD factorization formula in equation (7) contain large logarithms like lnpT2/mQ2, which could ruin the convergence of perturbative expansion. Thus, resummations of these large logarithmic terms are necessary. This can be done by using FF+NRQCD factorization approach. Applying the FF+NRQCD factorization approach with LP approximation was carried out in Refs. [85, 118] for the hadroproduction of J/ψ, χcJ, and ψ2S at high pT, and good agreement with the measurements is obtained. It was also found that contributions from 3S18 and 3PJ8 channels should almost cancel with each other, so the produced J/ψ is almost unpolarized, which confirms the conclusion in Refs. [77, 87]. This implies that the qualitative results in the NLO NRQCD calculations are not changed by LP resummation. It is an interesting question whether the NLP resummation could change this conclusion.


### 3.2. LowpT Heavy Quarkonium Production

At the lowpT≲M (M is the quarkonium mass) region, the collinear factorization formalism in equation (7) is no longer applicable. At this regime, large αsln1/x (small x, x~M/s, where s is the collider center of mass energy) contribution arises at higher orders that may not be fully accounted for in the collinear factorization framework. Another source of O1 contribution is from the higher-twist multiparton matrix elements that are large at low pT. Both of these contributions can be computed systematically in the Color Glass Condensate (CGC) effective field theory [124, 125]. By combining the CGC and NRQCD formalisms, a new factorization framework for quarkonium production was proposed [123, 126], in which the SDCs in equation (2) are given by
(13)dσ^nd2p⊥dy=COαsπRp22π7Nc2−1∫d2k1⊥d2k⊥φp,ypk1⊥k1⊥2NYk⊥NYp⊥−k1⊥−k⊥Γ8κ,for the color-octet channels and
(14)dσ^nd2p⊥dy=CSαsπRp22π9Nc2−1∫d2k1⊥d2k⊥d2k′⊥φp,ypk1⊥k1⊥2×NYk⊥NYk′⊥NYp⊥−k1⊥−k⊥−k′⊥G1κ,for the color-singlet channels. In these expressions,p⊥ (y) is the transverse momentum (rapidity) of the produced heavy quarkonium, yp≡ln1/xp (Y≡ln1/xA) is the rapidity of gluons coming from dilute proton (dense proton), NY denotes the fundamental dipole amplitude, Γ8κ and G1κ are the hard parts, φ is an unintegrated gluon distribution inside the proton, and πRp2 is the effective transverse area of the proton. Such a CGC+NRQCD framework provides a good description of the ψnS yield [123] and polarization at low pT [127] in pp collisions, shown in Figures 5 and 6. Interestingly, the CGC+NRQCD result at small pT merges smoothly with the NLO NRQCD result at intermediate and large pT, providing an unified description for quarkonium production in the full pT region. Note that this framework is even more useful for quarkonium production in proton-nucleus collisions [128–131].Figure 5
Theψ2S (top curve) and J/ψ (other four curves) differential cross section as functions of pT. Data from [92, 94, 119–122]. Figure taken from Ref. [123].Figure 6
The angular distribution coefficientsλθ in the Collins-Soper frame as functions of pT. Data from [132–134]. Figure taken from Ref. [127].


## 3.1. HighpT Heavy Quarkonium Production

Based on the NRQCD factorization framework, the heavy quarkonium production inpp collisions has been widely studied. In the large pT region, the differential cross section of the quarkonium H production can be factorized as
(7)dσA+B⟶H+X=∑ndσ^nOnH=∑i,j,n∫dx1dx2fi/Ax1,μFfj/Bx2,μFdσ^i+j⟶QQ¯n+XμF,μΛOnHμΛ,wheref’s are the parton distribution functions (PDFs) for the partons in the initial colliding protons, μF is the collinear factorization scale, and dσi+j⟶QQ¯n+X is the partonic differential cross section.At LO inαs, the partonic differential cross sections of n=3S18,1S08,3PJ8 are scaled as pT−4, pT−6, and pT−6, respectively, which are more important than the CS contribution n=3S11, scaled as pT−8. By including the CO contributions, the ψnS surplus puzzle in CSM is solved naturally [66]. But NRQCD factorization encounters difficulties with charmonium polarizations. Since the dominant contribution is from the transverse 3S18 channel, LO NRQCD predicts that ψnS and YnS produced at hadron colliders are mainly transversely polarized [42, 67, 68]. On the contrary, experimental measurements at Tevatron and LHC find that these states are almost produced unpolarized [69–71]. In addition, the LO calculation in NRQCD is difficult to explain the observed cross section ratio Rχc=σχc2/σχc1 of the P-wave charmonia χcJ at Tevatron [72], as the 3S18 channel dominance predicts the ratio Rχc to be 5/3 by spin counting [73, 74], which is much larger than the measured value of 0.75.In the past decade, the perturbative SDCs for quarkonium production cross sections and polarizations have been calculated to NLO by three groups [75–81]. At this order, the 1S08 and 3PJ8 channels are pT−4 scaling, so their contributions are also important at large pT. Even though the NLO SDCs obtained by the three groups agree, they give very different predictions for J/ψ polarization, due to the different methods used in fitting the CO LDMEs.In Refs. [75–77], it was found that at high pT, the SDC of the 3PJ8 channel can be decomposed into a linear combination of the other two CO channels:
(8)dσ^3PJ8=r0dσ^1S08+r1dσ^3S18,wherer0=3.9,r1=−0.56 for the Tevatron and r0=4.1,r1=−0.56 for the LHC. Therefore, two linearly combined NRQCD LDMEs are introduced to fit the data:
(9)M0,r0H=OH1S08+r0mc2OH3P08,M1,r1H=OH3S18+r1mc2OH3P08,whereH is J/ψ or ψ2S. Using the Tevatron data [82, 83] with pT>7 GeV, the LDMEs are extracted as
(10)M0,r0J/ψ=7.4±1.9×10−2Gev3,M1,r1J/ψ=0.05±0.02×10−2Gev3.With these fitted LDMEs, the theoretical predictions for promptJ/ψ production are generally consistent with experimental data from the Tevatron and the LHC [82–84] (see Figure 1). As shown in Figure 1(b), the J/ψ polarization is roughly consistent with the CDF data, in which the J/ψ is approximately unpolarized. Detailed analysis shows that the transversely polarized contributions from 3S18 and 3PJ8 channels cancel each other. Thus, a possible mechanism is that the unpolarized 1S08 channel dominates, which leads to an unpolarized production [77, 85, 86]. For the ψ2S production, such cancellation is weak [87], and its polarization is still hard to explain.Figure 1
Comparison of NLO NRQCD calculations with promptJ/ψ data. (a) The pT distributions of prompt J/ψ production at the Tevatron and the LHC. (b) NLO NRQCD prediction for the polarization of J/ψ production at the Tevatron. Figures taken from Refs. [75, 77].
(a)(b)In Ref. [79], the authors determine the CO LDMEs by a global fit of a number of measurements. Using prompt J/ψ production data in pp (pT>3GeV) [88–94], ep (pT>1 GeV) [95–97], γγ [98], and e+e− [99] collisions, they determine all three CO LDMEs. In particular, they obtained a negative value for OJ/ψ3P08, and thus, the contributions from 3S18 and 3PJ8 channels add and enhance the transverse polarization at large pT. With these inputs, it was found that the J/ψ is transversely polarized at large pT as shown in Figure 2.Figure 2
Comparisons of the NLO NRQCD calculations in Ref. [79] for the J/ψ polarization with the Tevatron data. Figure from Ref. [79].In Ref. [80], the LDME determination is based on the yield data from CDF [89] and LHCb [94]. Similar to the method in Refs. [75, 77, 87], the data with pT<7GeV are not considered in the fitting. It is found that both OJ/ψ3S18 and OJ/ψ3PJ8 are negative. As the two LDMEs have the same sign, the cancellation between transversely polarized contributions still occurs, which explains the unpolarized J/ψ produced at large pT.Due to the approximate heavy quark spin symmetry (HQSS) of NRQCD [12], the production of J/ψ is closely related to the production of ηc by the following relations:
(11)Oηc3S18=OJ/ψ1S08,Oηc1S08=13OJ/ψ3S18,Oηc1P18=32J+1OJ/ψ3PJ8.Then, the measurement ofηc can provide a further test of the J/ψ LDMEs. The first prompt ηc hadroproduction cross section was measured in 2014 by the LHCb collaboration with the centre-of-mass energies at s=7 TeV and 8 TeV [100]. With the three sets of the J/ψ LDMEs above, the authors of Ref. [101] found that theoretical calculations overshoot experimental measurement, as shown in Figure 3. Detailed analysis shows that the LHCb data are almost saturated by the contribution from the CS 1S01 channel. This gives a constraint on OJ/ψ1S08. By assuming that the data are completely contributed from the 3S18 channel, the authors of Ref. [102] obtain an upper bound for Oηc3S18:
(12)Oηc3S18<1.46×10−2GeV3.Figure 3
The LHCb [100] measurements for prompt ηc hadroproduction at s=7TeV are compared with the NLO NRQCD calculations evaluated with three different LDME sets from Refs. [77, 79, 80]. Figures taken from Ref. [101].This bound is consistent with a previous study by the same authors on theJ/ψ yield and polarization [87], so they argue that the prompt production of ηc and J/ψ can be understood in the same theoretical framework. The authors of Ref. [103] fit both the CO and CS LDMEs, and their predictions are also compatible with data. Recently, the LHCb collaboration reports their measurement of the ηc hadroproduction cross section at s=13 TeV [104], which is consistent with the previous data.Within the same framework as that for theJ/ψ production, the study of Y production at NLO was carried out by two groups [81, 105, 106]. Their results show a slightly transverse polarization at large pT, consistent with the measurements by CMS [107] within experimental uncertainties.ForP-wave quarkonia, the first complete NLO study of χcJ production was performed in Ref. [108] in 2010. At NLO, the CS 3PJ1 channels are scaled as pT−4, which give large contributions at high pT. They also find that 3P11 decreases slower than 3P21, so the measured ratio of Rχc at the Tevatron [72] can be naturally explained (see Figure 4). In 2016, the authors of Ref. [109] performed a global analysis of the existing data on χc hadroproduction from the Tevatron [72] and the LHC [110–113]. In the meantime, the polarization of the χc was also predicted [114–116], which indicated that the polarization parameter λθ of the χc1 should be positive while that of the χc2 should be negative. The experimental measurement was not available until 2019 by the CMS collaboration [117]. Current experimental data seem to be consistent with the NLO results.Figure 4
Transverse momentum distribution of ratioRχc at the Tevatron with cut ∣yχcJ∣<1. Figure taken from Ref. [108].As mentioned in the previous section, in the highpT region, the partonic differential cross sections in the NRQCD factorization formula in equation (7) contain large logarithms like lnpT2/mQ2, which could ruin the convergence of perturbative expansion. Thus, resummations of these large logarithmic terms are necessary. This can be done by using FF+NRQCD factorization approach. Applying the FF+NRQCD factorization approach with LP approximation was carried out in Refs. [85, 118] for the hadroproduction of J/ψ, χcJ, and ψ2S at high pT, and good agreement with the measurements is obtained. It was also found that contributions from 3S18 and 3PJ8 channels should almost cancel with each other, so the produced J/ψ is almost unpolarized, which confirms the conclusion in Refs. [77, 87]. This implies that the qualitative results in the NLO NRQCD calculations are not changed by LP resummation. It is an interesting question whether the NLP resummation could change this conclusion.


## 3.2. LowpT Heavy Quarkonium Production

At the lowpT≲M (M is the quarkonium mass) region, the collinear factorization formalism in equation (7) is no longer applicable. At this regime, large αsln1/x (small x, x~M/s, where s is the collider center of mass energy) contribution arises at higher orders that may not be fully accounted for in the collinear factorization framework. Another source of O1 contribution is from the higher-twist multiparton matrix elements that are large at low pT. Both of these contributions can be computed systematically in the Color Glass Condensate (CGC) effective field theory [124, 125]. By combining the CGC and NRQCD formalisms, a new factorization framework for quarkonium production was proposed [123, 126], in which the SDCs in equation (2) are given by
(13)dσ^nd2p⊥dy=COαsπRp22π7Nc2−1∫d2k1⊥d2k⊥φp,ypk1⊥k1⊥2NYk⊥NYp⊥−k1⊥−k⊥Γ8κ,for the color-octet channels and
(14)dσ^nd2p⊥dy=CSαsπRp22π9Nc2−1∫d2k1⊥d2k⊥d2k′⊥φp,ypk1⊥k1⊥2×NYk⊥NYk′⊥NYp⊥−k1⊥−k⊥−k′⊥G1κ,for the color-singlet channels. In these expressions,p⊥ (y) is the transverse momentum (rapidity) of the produced heavy quarkonium, yp≡ln1/xp (Y≡ln1/xA) is the rapidity of gluons coming from dilute proton (dense proton), NY denotes the fundamental dipole amplitude, Γ8κ and G1κ are the hard parts, φ is an unintegrated gluon distribution inside the proton, and πRp2 is the effective transverse area of the proton. Such a CGC+NRQCD framework provides a good description of the ψnS yield [123] and polarization at low pT [127] in pp collisions, shown in Figures 5 and 6. Interestingly, the CGC+NRQCD result at small pT merges smoothly with the NLO NRQCD result at intermediate and large pT, providing an unified description for quarkonium production in the full pT region. Note that this framework is even more useful for quarkonium production in proton-nucleus collisions [128–131].Figure 5
Theψ2S (top curve) and J/ψ (other four curves) differential cross section as functions of pT. Data from [92, 94, 119–122]. Figure taken from Ref. [123].Figure 6
The angular distribution coefficientsλθ in the Collins-Soper frame as functions of pT. Data from [132–134]. Figure taken from Ref. [127].


## 4. Quarkonium Production ine+e− Annihilation at B Factories

The quarkonium production ine+e− annihilation at B factories is another important process to test the NRQCD factorization. For exclusive double charmonium production such as e+e−⟶J/ψ+ηc, the first theoretical calculation at LO in both αs and v [135–137] gives a production cross section about 2.3~5.5 fb. It is much smaller than the measured value, which is σJ/ψ+ηc×Bηc≥2=25.6±2.8±3.4 fb by Belle [138] and σJ/ψ+ηc×Bηc≥2=17.6±2.8−2.1+1.5 fb by BaBar [139], where Bηc≥2 is the branching fraction for the ηc decaying into at least two charged tracks. Later, the authors of Refs. [140, 141] show that the NLO QCD correction can substantially enhance the cross section with a K factor (the ratio of NLO to LO) of about 1.8~2.1. Meanwhile, the relative Ov2 correction is also found to be significant [135, 142, 143]. Including both the αs and v2 corrections may resolve the large discrepancy between theory and experiment. Recently, the NNLO QCD correction to e+e−⟶J/ψ+ηc has been completed [144], which gives a state-of-the-art calculation consistent with the BaBar measurement.For inclusiveJ/ψ production, the first measurements of the cross section are released by the BaBar [145] and Belle [138, 146] collaborations. It was found by Belle [138] that the cross section σe+e−⟶J/ψ+cc¯=0.87−0.19+0.21±0.17 fb is about a factor of 5 larger than the LO NRQCD factorization predictions including both the CS [147–151] and CO [151] contributions. Later, the Belle collaboration reported an updated measurement: σe+e−⟶J/ψ+cc¯=0.74±0.08+0.08+0.09fb [99], which is smaller than the previous one. But it is still much larger than the LO NRQCD calculation. The large gap between experiment and theoretical calculations is reduced by including the NLO QCD correction, which substantially enhances the cross section with a K factor of about 1.8 [152].To explainσe+e−⟶J/ψ+Xnon−cc¯ measured by Belle [99], the NLO QCD correction to the CS channel e+e−⟶J/ψ+gg is calculated in Refs. [153, 154], which increases the LO result by about 20~30%. As shown in Figure 7, the NLO CS contribution saturates the Belle measurement. The Ov2 relativistic correction [155, 156] and the QED initial state radiation (ISR) effect [157] have also been considered, which enhance the LO cross section by a factor of 20~30% and 15~25%, respectively. Surprisingly, including all these corrections leads to the CS contribution somewhat above the measurement by Belle, leaving little or no room for the contribution of the CO channel e+e−⟶J/ψ3PJ8,1S08+g. This provides an upper limit of the CO LDMEs. By assuming a vanishing CS contribution and including the NLO QCD correction to σe+e−⟶J/ψ3PJ8,1S08+g, the authors of Ref. [158] obtain
(15)M0,4.0J/ψ<2.0±0.6×10−2GeV3,Figure 7
Prompt cross sections ofe+e−⟶J/ψ+gg as functions of the renormalization scale μ at LO and NLO. The upper curves correspond to m=1.4GeV, and the lower ones correspond to m=1.5GeV. Figure taken from Ref. [153].which is much smaller than the value of the CO matrix element extracted from hadron colliders, i.e.,M0,4.0J/ψ≈7.4×10−2GeV3 [75, 76, 87], bringing the universality of LDMEs into question.The universality problem and the polarization puzzle discussed in the previous section are two outstanding problems in NRQCD factorization. A possible solution is to resum high-order relativistic corrections, since for charmonium,v2~0.3 is not a small number. This is exactly the motivation of the SGF approach. A detailed and comprehensive phenomenological study for inclusive quarkonium production in the SGF framework could help to understand these problems. In the meantime, more experimental data with smaller uncertainties are also indispensable.


## 5. Quarkonium Production inep and γγ Collisions

In the photoproduction of charmonia inep collisions at HERA, a quasi-real photon γ emitted from the incoming electron e interacts with a parton i from the proton p and produces a cc¯ pair that evolves into a charmonium state. There are two types of processes contributing to the photoproduction cross sections. The first is the direct photoproduction, in which the virtual photon interacts with the parton i electromagnetically. The second is the resolved photoproduction, in which the virtual photon emits a parton, which then interacts with the parton i. Combining collinear factorization and NRQCD factorization, the inclusive J/ψ photoproduction cross section can be written in the form of [159, 160]
(16)dσep⟶J/ψ+X=∑i,j,n∫dx1dx2dyfγ/ex1fj/γx2fi/Aydσ^ij⟶cc¯n+XOnJ/ψ,wherefγ/ex1 is the photon flux function, fj/γx2 is either δjγδ1−x2 or the PDF of parton j in the resolved photon, fi/Ay is the PDF of the proton, and dσ^ij⟶cc¯n+X is the partonic cross section.The NLO QCD correction to the partonic cross sectionsdσ^iγ⟶cc¯n+X for the direct production was first performed by Krämer in 1995 [161]. In 2009, complete NLO calculation was obtained in Refs. [159, 162, 163]. As shown in Figure 8, the yield data from HERA are roughly consistent with the global fit at NLO performed in Refs. [78, 160].Figure 8
NLO NRQCD calculation compared to HERA data. Figures taken from Ref. [160].In addition to the unpolarizedJ/ψ yield, the polarization of J/ψ photoproduction has also been studied [162, 163, 165]. Figure 9 shows the LO and NLO NRQCD results of polarization observables as a function of pT by Ref. [165] comparing two HERA datasets. It is found that NRQCD predicts the J/ψ produced at large pT to be approximately unpolarized, both at LO and NLO, which is confirmed by the H1 data (Figure 9(a)). However, the ZEUS measurement (Figure 9(b)) exhibits a tendency towards transverse polarization. A possible explanation is that diffractively produced vector mesons prefer to be strongly transversely polarized in the endpoint region z≈1 [165].Figure 9
NLO NRQCD calculations for polarization observables as functions ofpT compared to H1 [97] and ZEUS [164] data. Figures taken from Ref. [2].The inclusive cross section forJ/ψ production in e+e− colliders via γγ fusion has also been measured by the DELPHI collaboration [98]. Similar to the photoproduction, both direct photon and resolved photon contribute to the cross section. The first complete NLO NRQCD computation including the resolved contributions was studied in Ref. [160]. However, by using the LDMEs obtained from the global fit, the predicted cross sections are several times below the DELPHI data, as shown in Figure 10. This is caused by a cancellation between the 1S08 and 3PJ8 contributions owing to the negative value of OJ/ψ3P08 obtained by the global fit [2]. Thus, a global analysis of the world J/ψ data is still challenging.Figure 10
NLO NRQCD calculation compared to DELPHI data. Figure taken from Ref. [160].


## 6. Summary

In this article, we have reviewed some theoretical methods to describe heavy quarkonium production, including the color evaporation model, color-singlet model, NRQCD factorization, fragmentation function approach, and soft gluon factorization. We then emphasize the current status of the phenomenological study of charmonium production, mainly in the NRQCD factorization framework. We concentrate on the comparison between theoretical predictions and experimental measurements for the charmonium production in four important processes:pp collision, e+e− annihilation, ep collision, and γγ collision. After the NLO contributions are taken into account, the NRQCD factorization can give a qualitatively correct description for quarkonium production. In particular, by combining NRQCD factorization with CGC effective theory, a unified description for quarkonium hadroproduction in the full pT region is obtained. However, LDMEs determined from different choices of datasets can disagree with one another, and none of them are able to give a global description of all important observables, such as the total yield, momentum differential yield, and polarization. The polarization puzzle and the universality problem are two outstanding problems in NRQCD factorization, which may be caused by the bad convergence of velocity expansion. Hopefully, these difficulties could be resolved or relieved in the SGF framework with well-controlled relativistic corrections.



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