Sivers effect in Inelastic $J/\psi$ Photoproduction in $ep^\uparrow$ Collision in Color Octet Model

The prediction of single-spin asymmetry in inelastic photoproduction of $J/\psi$ in $ep^\uparrow$ collision is presented. At next-to-leading order, the dominating process is photon-gluon fusion, $\gamma+g\rightarrow J/\psi+g$ for the production of $J/\psi$ in $e+p^\uparrow\rightarrow J/\psi+X$, which directly probes the gluon Sivers function. Using the non-relativistic QCD based color octet model, the color octet states $\leftidx{^{3}}{S}{_1}^{(8)}$, $\leftidx{^{1}}{S}{_0}^{(8)}$ and $\leftidx{^{3}}{P}{_{J(0,1,2)}}^{(8)}$ contribution to $J/\psi$ production is calculated. Sizable asymmetry is estimated as a function of transverse momentum $P_T$ and energy fraction $z$ of $J/\psi$ in the range $0<P_T\leq 1$ GeV and $0.3<z\leq 0.9$. The unpolarized differential cross section of inelastic $J/\psi$ photoproduction is found to be in good agreement with H1 and ZEUS data.

contribution to J/ψ production is calculated. Sizable asymmetry is estimated as a function of transverse momentum P T and energy fraction z of J/ψ in the range 0 < P T ≤ 1 GeV and 0.3 < z ≤ 0.9. The unpolarized differential cross section of inelastic J/ψ photoproduction is found to be in good agreement with H1 and ZEUS data.

I. INTRODUCTION
Among the transverse momentum dependent pdfs (TMDs), Sivers function has attracted considerable interest in the scientific community in recent days, largely because of a large amount of experimental results coming in. The Sivers function gives the asymmetric distribution of unpolarized quarks/gluons inside a transversely polarized nucleon. The non-zero Sivers function gives a coupling between the intrinsic transverse momentum of the parton (quark/gluon) and the transverse spin of the nucleon [1,2], this gives an azimuthal asymmetry in the distribution of the final state particle in ep ↑ and pp ↑ collision that has been measured at HERMES [3][4][5], COMPASS [6][7][8][9], JLAB [10,11] and RHIC [12,13] respectively. Sivers function is a time reversal odd (T-odd) object [14]. The initial and final state interactions (gauge links) play an important role in the Sivers asymmetry. This gives a dependence on the specific process in which the Sivers function is studied. For example, Sivers function probed in semi-inclusive deep inelastic scattering (SIDIS) is expected to be the same in magnitude but opposite in sign compared to the one probed in the Drell-Yan (DY) process. More complex processes have complex gauge links [15]. Experimental data on the Sivers asymmetry have now made it possible for the extraction of u and d quark Sivers function [16], but the gluon Sivers function (GSF) is still unknown. There is no constraint on GSF except a positivity bound [17]. The GSF contains two gauge links, and the process dependence is more involved. It has been shown [18] that the GSF in any process can be written in terms of two independent Sivers functions, an f-type GSF (this contains [++] gauge link and also called WW gluon distributions) and a d-type GSF (this contains [+−] gauge link and are called dipole distributions) [18]. The operator structures in these two Sivers function have different charge conjugation properties.
Heavy quarkonium production in ep [19][20][21][22][23] and pp [24,25] collision has been studied theoretically quite extensively for probing the gluon TMDs, in particular the GSF and linearly polarized gluon distribution [26,27]. This is because the heavy quarkonium is produced at leading order (LO) through photon-gluon fusion (ep) or two gluon fusion (pp) channel. Although the production mechanism of heavy quarkonium is still not well established, the most widely used theoretical approach is based on non-relativistic QCD (NRQCD) [28]. This gives systematic way to separate the high energy and low energy effects of the production mechanism.
In this approach, the heavy quark pair is produced at a short distance in color singlet (CS) [29][30][31][32] or in color octet (CO) [33][34][35] configuration and then they hadronize to form a quarkonium state of given quantum numbers through a soft process. The short distance coefficients are calculated perturbatively for each process and the long distance matrix elements (LDMEs) are extracted from the experimental data. The LDMEs are categorized by performing an expansion in terms of the relative velocity of the heavy quark v in the limit v << 1 [36]. The theoretical predictions are arranged as double expansions in terms of v as well as α s . The heavy quark pair may be produced in CO state which then form the CS quarkonium by emitting a soft gluon.
NRQCD has been successful to explain the J/ψ hadroproduction at Tevatron [37,38], also data from J/ψ photoproducton at HERA [39][40][41][42] suggests substantial contribution from CO states [43][44][45][46][47][48]. In the single-spin asymmetry (SSA) in ep collision, when the J/ψ is produced in the CS state, the two final state interactions with quark and anti-quark lines cancel each other, and the final state interaction with unobserved particles cancel between diagrams having different cuts. As a result, SSA in J/ψ production in ep collision is zero when the heavy quark pair is produced in the CS state, and non-zero asymmetry can be observed when the pair is produced in CO state [49]. The final state interactions are more involved for pp collision processes, and there, non-zero SSA is expected when the heavy quark pair is produced in a CS state. In the study of TMDs in SSA in heavy quarkonium production, one assumes that TMD factorization holds for such processes.
In our previous work [20], we calculated the Sivers asymmetry in J/ψ electroproduction at LO, which is a photon-gluon 2 → 1 process, in color octet model (COM). We showed that the calculated asymmetry at z = 1 agrees within the error bar of the recent COMPASS [50] measurement . Here we extend the analysis to estimate the SSA in photoproduction of J/ψ at next-to-leading order (NLO). This allows to calculate the asymmetry over a wider kinematical region accessible to the present experiments at COMPASS and at the planned EIC in the future.
We will use NRQCD based COM in our calculation for estimating the asymmetry.
The paper is organized into five sections including the introduction in Sec.I. The SSA and J/ψ production framework are presented in Sec.II and Sec.III respectively. Sec.IV discusses about the numerical results. The conclusion of the paper is given in Sec.V. A few details of calculation are given in the appendices.

II. SINGLE-SPIN ASYMMETRY
In general the transverse single-spin asymmetry (SSA) is defined as following where dσ ↑ and dσ ↓ are respectively the differential cross-sections measured when one of the particle is transversely polarized up (↑) and down (↓) with respect to the scattering plane.
Here ↑ (↓) direction is the proton polarization direction along the +y (-y) axis with momentum along -z axis and the final hadron is produced in the xz plane as shown in FIG.1. We consider the inclusive process e(l) + p ↑ (P ) → J/ψ(P h ) + X. The virtual photon radiated by the initial electron scattering will interact with the proton. The virtual photon carries the momentum q such that q 2 ≈ −2EE (1 − cos θ) with E and E are energies of the initial and final electron respectively. In the forward scattering limit, however, the four momentum of virtual photon q 2 = −Q 2 → 0 as a result the virtual photon becomes the real photon. The dominating subprocess at NLO for quarkonium production in ep collision is photon-gluon fusion process, i.e., γ(q) + g(k) → J/ψ(P h ) + g(p g ). The letters within the round brackets represent the four momentum of each particle. There are two types of J/ψ photoproductions. One is the direct photoproduction in which the photon electromagnetically interacts with the partons of the proton. The second, resolved photoproduction wherein the photon acts as a source of partons and then they strongly interact with partons of the proton.
In this paper we have not considered the resolved photoproduction channel which basically contributes at low z region (z ≤ 0.3) [51], where z = P.P h P.q is the energy fraction transferred from the photon to J/ψ in the proton rest frame. The LO photon-gluon fusion subprocess (γ + g → J/ψ) contributes to elastic photoproduction at z = 1 [20]. The process of a colorless exchanged particle between quasi-real photon and proton, diffractive process, contributes to J/ψ production in the elastic region, i.e., z ≈ 1 and P T ≈ 0 GeV [52,53]. P T is the transverse momentum of J/ψ. Moreover, gluon and heavy quark fragmentation also contribute for quarkonium production significantly at P T > 4 GeV [54], which are excluded by imposing P T cut. The feed-down contribution from an excited state ψ(2S) and the decay of χ c states contribution to J/ψ are 15% [41] and 1% [53,55] respectively, are not considered in this work. Therefore, we impose the following kinematical cut 0.3 < z ≤ 0.9 to account for inelastic photoproduction [55,56] events only. For true inelastic J/ψ production, one has to impose low P T cut as in [55,56], however, to validate asymmetry calculation in the TMD framework, we have considered 0 < P T ≤ 1 GeV and low P T cut is not imposed. The softening of final gluon, i.e., z → 1, leads to infrared singularity in the inelastic photoproduction as shown in Eq.(B10).
Hence, z ≤ 0.9 kinematical cut is motivated to keep the final gluon hard and the perturbative calculation is under good control. At the same order in α s , another channel γ + q → J/ψ + q also gives the CO contribution to J/ψ production. Since the process is initiated by light quarks, the contribution is expected to be negligible compared to the photon-gluon fusion process [57].
For the dominating channel of J/ψ production through γg fusion, the contribution to the numerator of A N comes mainly from the gluon Sivers distribution [58]. As the heavy quark pair in the final state is produced unpolarized, there is no contribution from Collins function [24].
Also the linearly polarized gluons do not contribute to the denominator as long as the lepton is unpolarized [58]. Within the generalized parton model formalism, the differential cross section for an unpolarized process is given by Here x γ and x g are the light-cone momentum fractions of photon and gluon respectively. The Weizsäker-Williams distribution function, f γ/e (x γ ), describes the density of photons inside the electron which is given by [59] where α is the electromagnetic coupling and Q 2 min = m 2 e x 2 γ 1−xγ , m e being the electron mass. We have considered Q 2 max = 1 GeV 2 for estimating the SSA. For photoproduction of J/ψ at HERA, we have taken two different values of Q 2 max = 2.5 GeV 2 and 1 GeV 2 in line with H1 [39,40] and ZEUS [41,42] data respectively. The unpolarized gluon TMD, f g/p , represents the density of gluons inside an unpolarized proton. Theŝ,t andû are the Mandelstam variables whose definitions are given in appendix B. M γ+g→J/ψ+g is the amplitude of photon-gluon fusion process which will be discussed in Sec.III and its square is given in appendix A. The mass of J/ψ is represented with M . Now, we are in a position to write down the expression of numerator and denominator terms of Eq.(1) when the target proton is polarized and are given by and where ∆ N f g/p ↑ (x g , k ⊥g ), GSF, describes the density of unpolarized gluons inside the transversely polarized proton and is defined as below For estimating the SSA numerically, we have to discuss about the parameterization of TMDs.
Generally, it is assumed that the unpolarized gluon TMDs follow the Gaussian distribution.
The Gaussian parameterization of unpolarized TMD is Here, x g and k ⊥g dependencies of the TMD are factorized. The collinear PDF is denoted The collinear PDF obeys the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) scale evolution. We choose a frame (shown in FIG.1) as discussed in appendix B wherein the polarized proton is moving along −z axis with momentum P , is transversely polarizedŜ = (cos φ s , sin φ s , 0). The transverse momentum of the initial gluon is k ⊥g = k ⊥g (cos φ, sin φ, 0), For numerical estimation we have taken φ s = π/2. The parameterization of GSF is given by [16,60] here f g/p (x g , µ) is the usual collinear gluon PDF and The definition of h(k ⊥g ) is given by The k ⊥g dependent part of Sivers function can be written as where we defined D'Alesio et al. [60] have extracted the GSF from pion production data at RHIC [61] first time and two sets of best fit parameters were presented which are denoted with SIDIS1 and SIDIS2.
Moreover, using the latest SIDIS data Anselmino et al. [16] have extracted the quark and antiquark Sivers function. However, GSF has not been extracted yet from SIDIS data. Therefore, in order to estimate the asymmetry, best fit parameters of Sivers function corresponding to u and d quark will be used in the following parameterizations [62] : We call the parameterization (a) and (b) as BV-a and BV-b respectively. The best fit parameters are tabulated in TABLE I. The final expressions of numerator and denominator terms of Eq.(1) within DGLAP evolution approach are given by and

III. J/ψ PRODUCTION IN COM FRAMEWORK
Let us consider the J/ψ production in e + p → J/ψ + X process. The NLO subprocess is γ + g → J/ψ + g and the related Feynman diagrams to this process are shown in FIG.2.
The amplitude expression for bound state production in NRQCD framework can be written as below [63,64] γ(q) where k is the relative momentum of the heavy quark in the quarkonium rest frame. In Eq. (17), represents the amplitude of QQ pair without considering the external heavy quark and anti-quark legs, which is given by  k → −k . The color factor of each diagram is given by here the summation is over the colors of the outgoing quark and anti-quark. The SU(3) Clebsch-Gordan coefficients for CS and CO states respectively are given by and they project out the color state of QQ pair either it is in CS or CO state, where N c is the number of colors. The generators of SU(3) group in fundamental representation is denoted by t a which fallows Tr(t a t b ) = δ ab /2 and Tr(t a t b t c ) = 1 4 (d abc + if abc ). Using Eq.(24), we have the following color factors for the production of initial QQ in CO state The excluded heavy quark and anti-quark spinors are absorbed in the definition of spin projection operator which is given by [63,64] P SSz (P h , k ) = bearing Π SSz = γ 5 for singlet (S = 0) state and Π SSz = / ε sz (P h ) for triplet (S = 1) state. Here spin polarization vector of the QQ system is denoted with ε sz (P h ). Since the relative momentum k is very small w.r.t P h , Taylor expansion can be performed around k = 0 in Eq. (17). The first term in the expansion gives the S-wave amplitude. Since the radial wavefunction R 1 (0) = 0 for P −wave (L = 1, J = 0, 1, 2), one has to consider the second term in the Taylor expansion to calculate P -wave amplitude. By following Ref. [64], one obtains the S and P state amplitude expressions which are given by The following shorthand notations are defined in the above expressions For P − wave amplitude calculation, we use the Clebsch-Gordan coefficients as defined in Ref. [65,66] LzSz Here ε α Jz (P h ) is the polarization vector of bound state with J = 1 and it obeys the following relations The ε αβ Jz (P h ) represents the polarization tensor for J = 2 bound state and obeys the below relation [65,66] The R 0 (0) and R 1 (0) are the radial wave function and its derivative at the origin, and have the following relation with LDME [57] 0 The numerical values of LDMEs are given in TABLE II. Now, let's discuss about the each CO state ( 3 S 1 , 1 S 0 3 P J ) amplitudes in detail.

A. 3 S 1 Amplitude
We have the following symmetry relations for 3 S 1 state Using Eq.(39), we can sum the color factors and we have The diagrams 4 and 8 do not contribute to 3 S 1 state as from Eq. (39). The final amplitude expression for 3 S 1 state can be obtained by using Eq. (27) and is given by The symmetry relations for 1 S 0 state are given by One can sum the color factors using Eq.(43) and we have the below relation Using Eq.(27) the final amplitude expression for 1 S 0 state is given by where O 1 (0), O 2 (0) and O 3 (0) are given in Eq. (42) and C.

P J Amplitude
The symmetry relations for P -state (J = 0, 1, 2) are given by From above equations, we get the color factors as given in Eq. (44). Using these color factors, the Eq.(28) can be further simplified as below In order to calculate the amplitude expression for J = 0, 1 and 2, we have used the Clebsch-Gordan coefficients as defined in Eq.(31), (32) and (33). After summing and averaging over the colors and spins, the amplitude square of each state is given in appendix A.

IV. NUMERICAL RESULTS
In this section, we discuss the numerical results of SSA and inelastic photoproduction of J/ψ in polarized and unpolarized ep collision respectively. For numerical estimation of SSA, best fit parameters of GSF from [60] and up and down quark Sivers function parameters from [16] are considered. MSTW2008 [67] is used for PDF which is probed at the scale µ = M 2 + P 2 T . Mass of J/ψ, M=3.096 GeV is taken. The NLO subprocess γ + g → J/ψ + g is considered for J/ψ production in ep ↑ → J/ψ + X process. The COM is employed for calculating production rate of J/ψ. The 3 S  states amplitudes are calculated using FORM package [68], and are given in appendix A. For comparison, we have considered three sets of LDMEs from the References [45][46][47], which are tabulated in TABLE II. The LDMEs for J = 1, 2 are obtained by using the relations O J/ψ . The transverse momentum of the initial gluon k ⊥g in Eq.(2) is integrated within the limits 0 < k ⊥g < 3 GeV. We have noticed that the higher values of k ⊥g max (upper limit of the k ⊥g integration) do not affect the SSA and unpolarized differential cross section.
Ref. [ We have estimated the SSA at √ s = 100, 45 GeV (EIC) and √ s = 17.2 GeV (COMPASS) energies using Eq.(1) by fixing the J/ψ production plane as discussed in [69]. The SSA as a function of P T and z is obtained by integrating 0.3 < z ≤ 0.9 and 0 < P T ≤ 1 GeV respectively, and is shown in FIG.3-5. The light-cone momentum fraction x γ of quasi-real photon is integrated over the range 0 < x γ < 1 in FIG.3-6. The upper bound on the virtuality of the photon in Eq.(3), Q 2 max = 1 GeV 2 is considered in FIG.3-6. The integration w.r.t the light-cone momentum fraction of initial gluon x g in Eq. (15) and (16)  The obtained asymmetry using D'Alesio et al. [60] fit parameters of GSF is represented by "SIDIS1"and "SIDIS2". The "BV-a"and "BV-b"curves are obtained by using Anselmino et al. [16] fit parameters as defined in Eq. (14). As aforementioned, due to the final state interactions the asymmetry is nonzero when the heavy quark pair is produced in the CO state in ep collision [49]. Therefore, we have considered the initial heavy quark pair production is to be only in the In FIG.6, the unpolarized differential cross section as a function of P T and z using the LDMEs from Ref. [47] at EIC and COMPASS energies is shown. The CS state, 3 S 1 , contribution to J/ψ production is considered along with CO states to obtain the FIG.6. The energy spectrum of J/ψ, right panel in FIG.6, is restricted to z ≤ 0.9 as we are interested in the inelastic J/ψ production. The Gaussian parametrization of gluon TMD as defined in Eq. (7) with Gaussian width k 2 ⊥g = 1 GeV 2 is considered. For lower values of TMD width, i.e., k 2 ⊥g = 0.5 GeV 2 , the cross section differential in z is increased by 10% at low z region. Whereas the differential cross section as a function of P T is increased by 4.5% in the low P T region. The 3 P (8a) J state contribution to J/ψ production is significantly large compared to 3 S  The obtained unpolarized differential cross section of J/ψ using the LDMEs of Ref. [47] is compared with H1 data [39,40] in FIG.7. The theoretical results are calculated within the same kinematical region of H1 data, i.e., √ s = 318 GeV, P 2 T > 1 GeV 2 , 60 < W < 240 GeV, 0.3 < z < 0.9 and Q 2 max = 2.5 GeV 2 . The C.M energy of the photon-proton system is W and W 2 = (P + q) 2 ≈ x γ s, where s = (P + l) 2 is the C.M energy square of the proton-lepton system. The P T and W spectra obtained by considering the J/ψ production in CS state along with the CO states are in good agreement with data. However, the CS contribution to the J/ψ production is below the data. In FIG.7, the dσ/dz distribution is not well described by both CS and CO contributions of J/ψ. From FIG.7, it is obvious that the CO states contribution is dominated for higher z values. The H1 data are compared with the theoretical results obtained by using the LDMEs of Ref.
[45] and [46], which are presented in FIG.8. The LDMEs of [45] over estimate the result as shown in the left panel of FIG.8. Whereas Ref. [46] LDMEs predict the result very close to the data, which is illustrated in the right panel of FIG.8. The same behavior is also noticed for z and W spectra which are not shown. To assess the agreement between the data and theoretical results, f is calculated for three sets of LDMEs from the P T spectrum of FIG.7, FIG.8 and FIG.9 at a fixed k 2 ⊥g = 1 GeV 2 , which is tabulated in TABLE III. The χ 2 /d.o.f for k 2 ⊥g = 1 GeV 2 is observed to be smaller than that of k 2 ⊥g = 0.5 GeV 2 and k 2 ⊥g = 0.25 GeV 2 for three sets of LDMEs. Therefore, we have considered the unpolarized TMD Gaussian width to be k 2 ⊥g = 1 GeV 2 in the analysis of J/ψ photoproduction. Since the χ 2 /d.o.f for LDMEs of [47] is 7.92 and 2.541 for H1 and ZEUS data respectively, only the LDMEs of Ref. [47] have been used in the FIG.9 and FIG.10. The ZEUS data [41] are compared with theoretical results within the kinematical region √ s = 300 GeV, 50 < W < 180 GeV, 0.4 < z < 0.9 and Q 2 max = 1 GeV 2 , and is shown in FIG.9. The W and z spectra are obtained by integrating the P T over the range 1 < P T < 5 GeV. In FIG.10, the P T spectrum for each z bin is compared with H1 [40] and ZEUS [42] data. The P T spectrum is away from the data in the 0.3 < z < 0.5, 0.45 < z < 0.6 and 0.75 < z < 0.9 bins. However, the theoretical result is in good agreement with the data for the bin 0.6 < z < 0.75.
The each curve is obtained by taking into account the color singlet and color octet states contribution to J/ψ production. The integration ranges are 0 < P T ≤ 3 GeV and 0.3 < z < 0.9.

V. CONCLUSION
We have calculated the single-spin asymmetry and unpolarized differential cross section in the inelastic photoproduction of J/ψ in polarized and unpolarized ep collision respectively, where the scattered electron with small angle produces low virtuality photons. The NLO subprocess for J/ψ production is the photon-gluon fusion process γ + g → J/ψ + g. Within the NRQCD based COM framework, the color octet states 3 S J(0,1,2) contribution to J/ψ production is calculated. Sizable asymmetry is obtained as a function of P T and z in the kinematical range 0 < P T ≤ 1 GeV and 0.3 < z ≤ 0.9 respectively. The infrared singularity at z = 1, arises when the final gluon becomes soft, is excluded by restricting the analysis in the region z ≤ 0.9. The resolved photoproduction contribution is removed by considering z > 0.3.
We also presented the unpolarized differential cross section of inelastic J/ψ photoproduction as a function of P T , z and W , and is found to be in good agreement with the H1 and ZEUS data.
The sizable asymmetry indicates that the inelastic photoproduction of J/ψ in ep ↑ collision is a useful process to probe the gluon Sivers function over a wide kinematical region accessible to the future electron-ion collider (EIC). ACKNOWLEDGMENT We would like to thank Jean-Philippe Lansberg for fruitful discussion during his stay at IIT Bombay.