# Gluon GPDs and exclusive photoproduction of quarkonium in forward region

## Abstract

Forward photoproduction of \(J/\psi \) can be used to extract generalized parton distributions (GPDs) of gluons. We analyze the process at twist-3 level and study relevant classifications of twist-3 gluon GPDs. At leading power or twist-2 level the produced \(J/\psi \) is transversely polarized. We find that at twist-3 the produced \(J/\psi \) is longitudinally polarized. Our study shows that in the high-energy limit the twist-3 amplitude is only suppressed by the inverse power of the heavy quark mass relatively to the twist-2 amplitude. This indicates that the power correction to the cross-section of unpolarized \(J/\psi \) can have a sizable effect.The poles in the hard factor convoluted with twist-3 gluon GPDs may concern the GPDs discontinuities, which will be briefly discussed in this work. We have also derived the amplitude of the production of \(h_c\) at twist-3, but the result contains end-point singularities. The production of other quarkonia has been briefly discussed.

## 1 Introduction

Properties of hadrons are described by their matrix elements of QCD operators. These matrix elements are nonperturbative. They can be calculated only with nonperturbative methods, or extracted from experimental data. A class of such matrix elements are Generalized Parton Distributions(GPDs) introduced in [1, 2, 3]. In [4, 5] it has been shown that one can obtain the quark or gluon contributions to the proton spin from quark or gluon twist-2 GPDs, respectively. Therefore, studies of GPDs and relevant processes both in theory and experiment will help to solve the so-called proton spin crisis and provide more information about the hadron’s inner structure.

There is a class of exclusive processes which can be used to extract GPDs. They can be extracted from deeply virtual Compton scattering (DVCS) in the forward limit, as suggested in [4, 5]. Besides DVCS, processes with a light hadron or a quarkonium instead of the real photon in the final state of DVCS can be described with GPDs. With the leading power approximation the amplitudes of the processes are given by convolutions of twist-2 GPDs with perturbative coefficient functions. The properties of twist-2 GPDs have been studied extensively (see the reviews in [6, 7, 8]). From twist-2 GPDs one can only obtain the sum of the spin part and the orbital-angular momentum part of quarks or gluons. Recently, it has been shown in [9, 10, 11, 12] that one can obtain each orbital-angular momentum part from twist-3 GPDs individually. Therefore, it is important to study twist-3 GPDs and how to extract them from experiment.

In general twist-3 GPDs only appear in power-suppressed contributions. In this work we study the power-suppressed contributions of photoproduction of \(J/\psi \) or \(\varUpsilon \). The amplitude at the leading power of \(J/\psi \)- or \(\varUpsilon \) photoproduction has been given in [15]. At the leading power, only a part of gluonic twist-2 GPDs is involved. The produced quarkonium at twist-2 level is only transversely polarized. At the next-to-leading power, the quarkonium is longitudinally polarized. If one can measure the polarization, then twist-3 GPDs can be accessed directly. It is interesting to note that the production rate of longitudinally polarized \(J/\psi \) or \(\varUpsilon \) can be large because the polarization vector is proportional to the energy in contrast to the transverse polarization vectors which are constant. This can enhance the power-suppressed contributions in the high energy limit. This motivates us to study the process at twist-3 level in this work. After we have completed the power-suppressed contributions of photoproduction of \(J/\varPsi \) in terms of twist-3 GPDs, we should take care of an important property of the GPDs to ensure that these calculations are safe. This property has been shown in a series of papers [16, 17, 18, 19, 20, 21, 22, 23], which tell that some twist-3 GPDs may be discontinuous. We will discuss this at the end of our calculation.

## 2 Definitions and properties of GPDs

*h*is a proton or a spin-1/2 hadron. The kinematical variables of the process are given by \(s=(p+q)^2\) and \(t= (p'-p)^2\). This process with \(J/\psi \) at leading twist has been studied extensively in, e.g., [13, 14, 15] and the references therein.

*Q*is a generic large scale of the process. It can be the heavy quark mass or \(\sqrt{s}\). In this kinematical region the momentum of \(J_Q\) is

For the process QCD factorization is expected in which the nonperturbative effect related to the quarkonium is included in matrix elements of non-relativistic QCD (NRQCD) and that of the initial hadron is included in various GPDs. The produced quarkonium mainly consists of a heavy quark *Q* and a heavy antiquark \({\bar{Q}}\). The relative velocity *v* between *Q* and \({\bar{Q}}\) inside the quarkonium is small. One can make a small velocity expansion and use NRQCD factorization in [24] for the quarkonium. In general, a quarkonium is a bound state of a heavy quark \(Q{\bar{Q}}\) pair combined with light partons, i.e., a quarkonium state is a superposition of a \(Q{\bar{Q}}\) state and states of \(Q{\bar{Q}}\) combined with gluons and light-quark pairs. We take the leading order of *v*. At this order quarkonium, considered in this work, can be taken only as a bound state of \(Q{\bar{Q}}\). The other components with light partons, like the \(Q{\bar{Q}} g\) component, contribute only at the higher orders in the small velocity expansion according to the power counting in [24].

For our calculation one can use a wave function \(\psi \) to project the relevant state of the produced \(Q{\bar{Q}}\) pair. The projection is standard. For \(J/\psi \) or \(\varUpsilon \) the derived amplitude will be proportional to the wave function at the origin. The amplitude of \(h_c\) is proportional to the derivative of the corresponding wave function at the origin. In this work we use the notation of wave functions. It should be noted that the wave function at the origin or the derivative are in fact the corresponding NRQCD matrix elements defined in [24]. The relations can be found in [24]. It will be important to express our results with these NRQCD matrix elements to perform NRQCD factorization beyond tree level. The Coulomb singularities appearing at higher orders of \(\alpha _s\) are factorized into NRQCD matrix elements as shown in [24]. In terms of wave functions, it is also noted that the effects of the detailed shapes of the wave functions are included in those NRQCD matrix elements at higher orders of the small velocity expansion. In this work we take the leading order of *v* and \(\alpha _s\).

*f*(

*x*) and

*g*(

*x*). The covariant derivative \(D^\mu \) is given by \(D^{\mu } (x) = \partial ^\mu + i g_s G^\mu (x)\). \({\tilde{D}}^\mu \) is defined with \(D^\mu \) in a similar way to \({\tilde{\partial }}^\mu \). All Lorentz indices \(\mu ,\nu \) and \(\rho \) are transverse. There is another twist-3 matrix element of gluonic GPDs, which is obtained by replacing \(f^{abc}\) in \(M_F^{\mu \nu \rho } (x_1,x_2,\varDelta )\) by \(d^{abc}\). This matrix element is irrelevant for the process in Eq. (1). But it will be relevant for the process when the produced quarkonium has the quantum number \(C=+\).

*F*one can derive the relation

*t*. There is a freedom by choosing different parameterizations. The parametrization in the above has the advantage that all spinor products \({\bar{u}}(p') \varGamma u(p)\) with \(\varGamma = \gamma ^+\), \(\sigma ^{+\mu }\) and \(\gamma ^+\gamma _5\) are at the order of \({{\mathcal {O}}}(\lambda ^0)\). There are constraints for these functions or GPDs from hermiticity, Bose symmetry and time-reversal symmetry. All \(G_i\) and \(F_i\) are real and have the properties

*P*standing for the principal-value prescription and by noting that \(F_{1,2,3,4}\) and \(G_{1,2,3,4}\) are real and \(H_{1,2,3,4}\) are complex, we obtain the relations among these GPDs from the relation in Eq. (10):

## 3 Calculations for \(J/\varPsi \) Production

*s*-channel helicity frame by introducing the 4-vector \(X^\mu = q+p\). The polarization vector \(\epsilon _L^\mu \) of this frame is given by

Before giving a discussion about our results, we notice that our amplitudes in Eqs. (29, 43) are \(U_{em}(1)\)-gauge invariant only at the leading order of \(\lambda \). Since in our results only the transverse part of the polarization vector of the initial photon is involved, the amplitudes are not exactly zero by replacing the polarization vector with the photon momentum \(q^\mu \). After replacement the amplitudes are in fact suppressed by one power of \(\lambda \) because \(q_\perp ^\mu \) is at order of \(\lambda \). Hence, the \(U_{em}(1)\)-gauge invariance of our result holds only at the leading order of \(\lambda \).

One may expect that the amplitude can be simplified in the high energy limit, i.e., \(s \gg m_Q^2\). In the process \(\xi \) is fixed as \(\xi =2m_Q^2/(s-2 m_Q^2) (1+ {{\mathcal {O}}}(\lambda )\). Formally, one sets \(\xi =0\) to significantly simplify the twist-3 amplitude in the limit \(s \gg m_Q^2\). But this will result in a divergent contribution. This can be seen from the contribution from the GPD \(H_g(x,\xi ,t)\) contained in \({{\mathcal {F}}}_g\). In the limit \(\varDelta ^\mu \rightarrow 0\) \(H_g\) is related to the gluon parton distribution *g*(*x*) as \( H_g (x,0,0)= x g(x)\). For \(x\rightarrow 0\) *g*(*x*) behaves as \(x^{-\alpha }\) with \(\alpha >0\). The pole contribution from \(H_g\) in the first line of Eq. (43) is expected for \(\xi \rightarrow 0\) as \(\xi ^{-\alpha }\), as discussed in [31]. The behavior of GPDs in the region \(\vert x\vert \sim \xi \) is crucial for the limit. It is clear that the integration regions with \(\vert x\vert \sim \xi \) and \(\vert x_{1,2}\vert \sim \xi \) are the most important ones. If one assumes that all GPDs have similar behaviors in these regions, one may only neglect the last two terms by power counting of \(\xi \). Thus, the behaviors of the GPDs may have a significant effect on our calculation. One behavior of the GPDs has been studied in a series of papers [16, 17, 18, 19, 20, 21, 22, 23], which show that some twist-3 GDPs may be discontinuous in some region. If this is true for the twist-3 GPDs in Eq. (43), we will not be allowed to do some of the integrations there. Thus, the power counting method cannot be applied to each twist-3 term of Eq. (43) individually. However, one has also shown [16] that some particular linear combinations of the twist-3 GPDs are free of discontinuity, and the amplitude may only depend on these linear combinations, although these GPDs may be discontinuous individually. The verification of a similar conclusion as regards the twist-3 GPDs here is another long work which is out of scope of this paper. Before this verification, one should always be careful when making use of the results here directly. Additionally, if a similar conclusion should be drawn here, it may be helpful to explain the poles of the hard factors which are convoluted to the twist-3 GPDs in Eq. (43). In general, it is not clear if the amplitude will be simplified in the limit. But the relative order of importance between twist-2- and twist-3 amplitude can be estimated, if we assume that the contributions from twist-3 GPDs can be transformed into terms that are free of discontinuity and, therefore, are finite.

By simply comparing the contribution from \(H_g\) in twist-2- or twist-3 contributions, one can already conclude that the twist-3 amplitude is suppressed only by a factor \(\varDelta _\perp /m_Q\). For \(J/\psi \) \(m_Q\) is \(m_c\); around 1 GeV. Therefore, the suppression is not strong. The weak suppression has a significant impact on extracting twist-2 gluon GPDs by using the leading amplitude given in Eq. (29), in which \(J/\psi \) is only transversely polarized. If the polarization of \(J/\psi \) is not measured or summed in experiment, then there will be a substantial contribution of longitudinally polarized \(J/\psi \) in the measured differential cross-sections. This can result in the extracted twist-2 GPDs by using the leading amplitude being not accurate and containing significant effects of higher-twists. This is an important observation of our work.

From the production of transversely polarized \(J/\psi \) or \(\varUpsilon \) one can only extract the gluon GPDs \(E_g\) and \(H_g\). It is possible to obtain information of other twist-2 GPDs through production of quarkonium with different quantum numbers. At leading power or twist-2 level, the exchanged two gluons in Fig. 1a can only be in a state with the quantum number \(C=+\). Therefore, \(h_c\) can be produced at twist-2. Other quarkonia, like \(\eta _c\) and \(\chi _{cJ}\), with \(J=0,1,2\), can only be produced through three-gluon exchange given in Fig. 1b, where the production is described with twist-3 GPDs obtained from \(M_F^{\mu \nu \rho }\) by replacing \(i f^{abc}\) in Eq. (6) by \(d^{abc}\).

## 4 Calculations for \(h_c\) Production

*x*-dependence of the GPD \(E_g\) and \(H_g\) contained in \({{\mathcal {F}}}_g\) can be found in [8, 36]:

*B*-meson decays. For processes factorized with GPDs, the existence of such divergences involving GPDs has been first observed in \(\rho \)-meson production in [31]. These divergences make QCD factorization impossible [32, 33, 34, 35]. To the best of our knowledge, there has so far appeared no rigorous way to deal with the singularities.

It should be emphasized that the existence of the end-point singularity in our twist-3 amplitude of \(h_c\) only means that we cannot evaluate it correctly in the framework of collinear factorization. But the contribution of the amplitude is power-suppressed. The amplitude of \(h_{c,b}\) at leading power or twist-2 has no such a singularity. If the power correction can be neglected, a physical prediction can still be made from Eq. (44) for \(h_{c,b}\) production. It is noted that in the twist-3 amplitude of \(J/\psi \) or \(\varUpsilon \) in Eq. (43) there is no evidence of existing end-point singularities. One can expect that the factorization holds in this case.

## 5 Summary

To summarize, we have studied at twist-3 level the forward photoproduction of a quarkonium, where the quarkonium can be \(J/\psi \) and \(h_c\). We have classified the relevant twist-3 gluon GPDs and studied their properties. The produced quarkonium at twist-2 is transversely polarized. At twist-3 it is longitudinally polarized and the amplitude is power-suppressed. Our result indicates that the twist-3 amplitude of \(h_c\) contains end-point singularities which spoil the factorization. We find that the twist-3 amplitude is only suppressed by the inverse of the heavy quark mass. This has an important implication for extracting twist-2 gluon GPDs, if the polarization of the produced quarkonium is not observed. In this case, the contribution of the longitudinal polarization can be significant and should be taken into account. Experimentally, the studied processes can be already studied at JLab, and in the future at EIC or After@LHC as proposed in [37] or [38], respectively.

## Notes

### Acknowledgements

The work is supported by National Nature Science Foundation of P.R. China (No. 11675241 and No. 11821505). The partial support from the CAS center for excellence in particle physics(CCEPP) is acknowledged.

## References

- 1.D. Mueller et al., Fortsch. Phys.
**42**, 101 (1994)ADSCrossRefGoogle Scholar - 2.
- 3.
- 4.
- 5.
- 6.
- 7.
- 8.A.V. Belitsky, A.V. Radyushkin, Phys. Rept.
**418**, 1 (2005). arXiv:hep-ph/0504030 ADSCrossRefGoogle Scholar - 9.X. Ji, X. Xiong, F. Yuan, Phys. Rev. Lett.
**109**, 152005 (2012). arXiv:1202.2847 [hep-ph]ADSCrossRefGoogle Scholar - 10.
- 11.
- 12.
- 13.DYu. Ivanov, A. Schafer, L. Szymanowski, G. Krasnikov, Eur. Phys. J. C
**34**, 297 (2004). arXiv:hep-ph/0401131 ADSCrossRefGoogle Scholar - 14.M. Vanttinen, L. Mankiewicz, Phys. Lett. B
**440**, 157 (1998). arXiv:hep-ph/9807287 ADSCrossRefGoogle Scholar - 15.J. Koempel, P. Kroll, A. Metz, J. Zhou, Phys. Rev. D
**85**, 051502 (2012). arXiv:1112.1334 ADSCrossRefGoogle Scholar - 16.Fatma Aslan, Matthias Burkardt, Phys. Rev. D
**98**, 014038 (2018)ADSCrossRefGoogle Scholar - 17.A.V. Belitsky, D. Mueller, Nucl. Phys. B
**589**, 611 (2000)ADSCrossRefGoogle Scholar - 18.N. Kivel, M.V. Polyakov, A. Schäfer, O.V. Teryaev, Phys. Lett. B
**497**, 73 (2001)ADSCrossRefGoogle Scholar - 19.A.V. Radyushkin, C. Weiss, Phys. Lett. B
**493**, 332 (2000)ADSCrossRefGoogle Scholar - 20.N. Kivel, M.V. Polyakov, Nucl. Phys. B
**600**, 334 (2001)ADSCrossRefGoogle Scholar - 21.A.V. Radyushkin, C. Weiss, Phys. Rev. D
**63**, 114012 (2001)ADSCrossRefGoogle Scholar - 22.A.V. Belitsky, D. Mueller, A. Kirchner, A. Schäfer, Phys. Rev. D
**64**, 116002 (2001)ADSCrossRefGoogle Scholar - 23.N. Kivel, M.V. Polyakov, M. Vanderhaeghen, Phys. Rev. D
**63**, 114014 (2001)ADSCrossRefGoogle Scholar - 24.G.T. Bodwin, E. Braaten and G. Peter Lepage, Phys.Rev.
**D51**1125, (1995) Erratum-ibid.D55 (1997) 5853, e-Print arXiv:hep-ph/9407339 - 25.A.V. Efremov, O.V. Teryaev, Sov. J. Nucl. Phys
**36**, 142 (1982)Google Scholar - 26.A.V. Efremov, O.V. Teryaev, Phys. Lett
**B150**, 383 (1985)ADSCrossRefGoogle Scholar - 27.J.W. Qiu, G. Sterman, Phys. Rev. Lett
**67**, 2264 (1991)ADSCrossRefGoogle Scholar - 28.J.W. Qiu, G. Sterman, Nucl. Phys. B
**378**, 52 (1992)ADSCrossRefGoogle Scholar - 29.J.W. Qiu, G. Sterman, Phys. Rev
**D59**, 014004 (1998)ADSGoogle Scholar - 30.R.L. Jaffe, A. Manohar, Nucl. Phys. B
**337**, 509 (1990)ADSCrossRefGoogle Scholar - 31.L. Mankiewicz, G. Piller, Phys. Rev. D
**61**, 074013 (2000). arXiv:hep-ph/9905287 ADSCrossRefGoogle Scholar - 32.J. Collins, L. Frankfurt, M. Strikman, Phys. Rev. D
**56**, 2982 (1997)ADSCrossRefGoogle Scholar - 33.A. Radyushkin, Phys. Lett. B
**380**, 417 (1996)ADSCrossRefGoogle Scholar - 34.A. Radyushkin, Phys. Lett. B
**385**, 333 (1996)ADSCrossRefGoogle Scholar - 35.A. Radyushkin, Phys. Rev. D
**56**, 5524 (1997)ADSCrossRefGoogle Scholar - 36.
- 37.A. Accardi et al., Eur. Phys. J.
**A52**(9), 268 (2016). arXiv:1212.1701 [nucl-ex]ADSCrossRefGoogle Scholar - 38.L. Massacrier, J.P. Lansberg, L. Szymanowski, J. Wagner, arXiv:1709.09044 [nucl-ex]

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