Sivers Asymmetry in Photoproduction of $J/\psi$ and Jet at the EIC

TL;DR

This paper predicts a sizable Sivers asymmetry in $J/$-jet photoproduction at the future EIC, using the GPM and NRQCD frameworks, to help determine the gluon Sivers function.

Contribution

It introduces a detailed calculation of the Sivers asymmetry in $J/$-jet production, including color states and TMD evolution effects, for the first time.

Findings

Sizable Sivers asymmetry predicted for EIC conditions.

Both color singlet and octet states significantly contribute.

TMD evolution impacts the magnitude of the asymmetry.

Abstract

We calculate the Sivers asymmetry in the photoproduction of almost back-to-back $J/\psi$-jet pair in the process $ep^\uparrow \to J/\psi+\mathrm{jet}+X$, which will be possible at the future planned electron-ion collider (EIC). We use the framework of generalized parton model (GPM), and NRQCD for calculating the $J/\psi$ production rate. We include contributions from both color singlet and color octate states in the asymmetry. We obtain sizable Sivers asymmetry that can be promising to determine the gluon Sivers function. We also investigate the effect of TMD evolution on the asymmetry.

Tables

Table 1. Table 1: Best fit parameters of Sivers function.

Best fit parameters
Evolution	$a$	$N_a$	$\alpha$	$\beta$	$\rho$	$⟨q_{⟂}^2⟩$ GeV2	Notation
DGLAP	$g$ D’Alesio et al. (2015)	0.65	2.8	2.8	0.687	0.25	SIDIS1
	$g$ D’Alesio et al. (2015)	0.05	0.8	1.4	0.576	0.25	SIDIS2
	$g$ D’Alesio et al. (2019a)	0.25	0.6	0.6	0.1	1.0	SIDIS3
TMD	$u$ Echevarria et al. (2014)	0.106	1.051	4.857		0.38	TMD-a
	$d$ Echevarria et al. (2014)	-0.163	1.552	4.857		0.38	TMD-b

Equations

$$e(l)+p^{\uparrow }(P)\to J/\psi (P_{\psi })+\mathrm{jet}(P_j)+X,$$

$$E_{\Psi }{E}_j\frac{d\sigma }{d^3{\mathbf{P}}_{\Psi }{d}^3{\mathbf{P}}_j}=\frac{d\sigma }{d^2{\mathbf{P}}_{\Psi \perp }dz{d}^2{\mathbf{P}}_{j\perp }d{z}_1}=$$

$$\times {\delta }^4(q+p-P_{\Psi }-P_j)\frac{1}{2\hat{s}}|{\mathcal{M}}_{\gamma a\to J/\psi a}{|}^2,$$

$$f_{\gamma /e}(x_{\gamma })=\frac{\alpha }{2\pi }\left[2m_e^2{x}_{\gamma }\left(\frac{1}{Q_{min}^2}-\frac{1}{Q_{max}^2}\right)+\frac{1+(1-x_{\gamma }{)}^2}{x_{\gamma }}\ln \frac{Q_{max}^2}{Q_{min}^2}\right],$$

$${\delta }^4(q+p-P_{\Psi }-P_j)=\frac{2}{x_{\gamma }s}\delta \left(1-z-z_1\right)\delta \left(x_a-\frac{M^2+P_{\Psi \perp }^2}{z{x}_{\gamma }s}-\frac{P_{j\perp }^2}{z_1{x}_{\gamma }s}\right){\delta }^2({\mathbf{p}}_{a\perp }-{\mathbf{P}}_{\Psi \perp }-{\mathbf{P}}_{j\perp }),$$

$$\frac{d\sigma }{d^2{\mathbf{q}}_{\perp }dz{d}^2{\mathbf{K}}_{\perp }}=$$

$$\frac{d{\sigma }^{\uparrow (\downarrow )}}{d^2{\mathbf{q}}_{\perp }dz{d}^2{\mathbf{K}}_{\perp }}=$$

$$A_N^{W({\varphi }_q)}\equiv \frac{\int d{\varphi }_{q}W({\varphi }_q)(\mathrm{d}{\sigma }^{\uparrow }-\mathrm{d}{\sigma }^{\downarrow })}{\int d{\varphi }_q(\mathrm{d}{\sigma }^{\uparrow }+\mathrm{d}{\sigma }^{\downarrow })}\equiv \frac{\int d{\varphi }_{q}W({\varphi }_q)\mathrm{d}\Delta \sigma ({\varphi }_q)}{\int d{\varphi }_{q}2\mathrm{d}\sigma },$$

$$-\sin ({\varphi }_q)=\frac{(\mathbf{S}\times \hat{\mathbf{P}})\cdot {\mathbf{q}}_{\perp }}{|\mathbf{S}\times \hat{\mathbf{P}}||\hat{\mathbf{P}}\times {\mathbf{q}}_{\perp }|}.$$

$$\mathrm{d}\Delta \sigma \equiv$$

$$\Delta {\hat{f}}_{a/p^{\uparrow }}(x_a,{\mathbf{q}}_{\perp })$$

$$={\Delta }^N{f}_{a/p^{\uparrow }}(x_a,q_{\perp })\hat{\mathbf{S}}\cdot (\hat{P}\times {\hat{\mathbf{q}}}_{\perp })$$

$$=-{\Delta }^N{f}_{a/p^{\uparrow }}(x_a,q_{\perp })\sin ({\varphi }_q)$$

$$=-\frac{2}{M_p}{f}_{1T}^{\perp }(x_a,q_{\perp })\hat{\mathbf{S}}\cdot (\hat{P}\times {\mathbf{q}}_{\perp }).$$

$$|{\Delta }^N{f}_{a/p^{\uparrow }}(x_a,q_{\perp })|\le 2f_{a/p}(x_a,q_{\perp }),\mathrm{or}\frac{q_{\perp }}{M_p}|f_{1T}^{\perp }(x_a,q_{\perp })|\le f_{a/p}(x_a,q_{\perp }).$$

$${\Delta }^N{f}_{a/p^{\uparrow }}(x_a,q_{\perp })=\left(-2\frac{q_{\perp }}{M_p}\right)f_{1T}^{\perp }(x_a,q_{\perp })=2{\mathcal{N}}_a(x_a)f_{a/p}(x_a)h(q_{\perp })\frac{e^{-q_{\perp }^2/⟨q_{\perp }^2⟩}}{\pi ⟨q_{\perp }^2⟩},$$

$${\mathcal{N}}_a(x_a)=N_a{x}_a^{\alpha }(1-x_a{)}^{\beta }\frac{(\alpha +\beta {)}^{(\alpha +\beta )}}{{\alpha }^{\alpha }{\beta }^{\beta }},$$

$$h(q_{\perp })=\sqrt{2e}\frac{q_{\perp }}{M^{\prime }}{e}^{-q_{\perp }^2/M^{\prime 2}},$$

$$\rho =\frac{M^{\prime 2}}{⟨q_{\perp }^2⟩+M^{\prime 2}},$$

$${\Delta }^N{f}_{a/p^{\uparrow }}(x_a,q_{\perp })=2\frac{\sqrt{2e}}{\pi }{\mathcal{N}}_a(x_a)f_{a/p}(x_a)\sqrt{\frac{1-\rho }{\rho }}{q}_{\perp }\frac{e^{-q_{\perp }^2/\rho ⟨q_{\perp }^2⟩}}{⟨q_{\perp }^2{⟩}^{3/2}}.$$

$$f_{a/p}(x_a,q_{\perp })=\frac{1}{\pi ⟨q_{\perp }^2⟩}f(x_a)e^{-q_{\perp }^2/⟨q_{\perp }^2⟩}.$$

$$f(x_a,b_{\perp },\mu )=\int d^2{\mathbf{p}}_{a\perp }{e}^{-i{\mathbf{b}}_{\perp }\cdot {\mathbf{p}}_{a\perp }}f(x_a,p_{a\perp },\mu ),$$

$$f(x_a,p_{a\perp },\mu )=\frac{1}{(2\pi {)}^2}\int d^2{\mathbf{b}}_{\perp }{e}^{i{\mathbf{b}}_{\perp }\cdot {\mathbf{p}}_{a\perp }}f(x_a,b_{\perp },\mu ).$$

$$f(x_a,b_{\perp },Q_f,\zeta )=f(x_a,b_{\perp },Q_i)R_{pert}\left(Q_f,Q_i,b_{\ast }\right)R_{NP}\left(Q_f,b_{\perp }\right).$$

$$\displaystyle R_{pert}\left(Q_{f},b_{\ast}\right)=\mathrm{exp}\Big{\{}{-\int_{c/b_{\ast}}^{Q_{f}}\frac{d\mu}{\mu}\left(A\log\left(\frac{Q_{f}^{2}}{\mu^{2}}\right)+B\right)}\Big{\}},$$

$$\displaystyle R_{NP}(Q_{f},b_{\perp})=\mathrm{exp}\Bigg{\{}-\Big{[}g_{1}^{\mathrm{TMD}}+\frac{g_{2}}{2}\log\frac{Q_{f}}{Q_{0}}\Big{]}b_{\perp}^{2}\Bigg{\}},$$

$$f(x_a,b_{\perp },Q_i)=\sum _{i=g,q}{\int }_x^1\frac{d\hat{x}}{\hat{x}}{C}_{i/a}(x_a/\hat{x},b_{\perp },{\alpha }_s,Q_i)f_{i/p}(\hat{x},c/b_{\ast })+\mathcal{O}(b_{\perp }{\Lambda }_{QCD}),$$

$$f_{1T}^{\perp }(x_a,p_{a\perp },Q_f)=-\frac{1}{2\pi p_{a\perp }}{\int }_0^{\infty }d{b}_{\perp }{b}_{\perp }{J}_1(p_{a\perp }{b}_{\perp })f_{1T}^{\prime \perp }(x_a,b_{\perp },Q_f),$$

$$f_{a/p}(x_a,p_{a\perp },Q_f)=\frac{1}{2\pi }{\int }_0^{\infty }d{b}_{\perp }{b}_{\perp }{J}_0(p_{a\perp }{b}_{\perp })f_{a/p}(x_a,b_{\perp },Q_f),$$

$$f_{1T}^{\prime \perp }(x_a,b_{\perp },Q_i)\simeq \frac{M_p{b}_{\perp }}{2}{T}_{a,F}(x_a,x_a,Q_i),$$

$$T_{a,F}(x_a,x_a,Q_i)={\mathcal{N}}_a(x_a)f_{a/p}(x_a,Q_i),$$

$$(a){\mathcal{N}}_g(x_g)$$

Full text

I Introduction

Single spin asymmetries and transverse momentum dependent parton distributions (TMD pdfs) are objects of a lot of interest in recent days in hadron physics. Among the TMD pdfs Sivers function Sivers (1990) is of particular interest. This gives the distribution of unpolarized quarks/gluons in a transversely polarized nucleon, which is not left-right symmetric with respect to the plane formed by the transverse momentum and spin of the nucleon. In some model calculations Sivers function is shown to be related to the quark orbital angular momentum through the GPD $E_{q}$ Burkardt (2004a). Sivers function introduces an asymmetry, for example, in the azimuthal angle of the observed final state hadron in semi-inclusive deep inelastic scattering (SIDIS) and in the azimuthal angle correlations of the lepton pair in Drell-Yan process or back-to-back jets in $pp$ collision, called the Sivers asymmetry. The first transverse moment of the Sivers function is related to the twist-three Qiu-Sterman function Qiu and Sterman (1991). First experimental information on non-zero Sivers function for quarks was obtained from HERMES Airapetian et al. (2005) and COMPASS Adolph et al. (2012) results. Since then, quite a lot of advances have been made, both in theory and experiment. Parametrization of quark Sivers function has been obtained in Anselmino et al. (2017) and for gluons in D’Alesio et al. (2015); D’Alesio et al. (2019a) by fitting data from RHIC within the DGLAP evolution approach. TMDs evolve with scale in a different way compared to the collinear pdfs. Much progress have been made in the past few years to understand the TMD evolution Aybat et al. (2012a); Aybat and Rogers (2011); Collins and Rogers (2017), and unpolarized distributions and fragmentation function have been calculated at NNLO Echevarria et al. (2016). A parametrization of the Sivers function incorporating the TMD evolution has been obtained in Echevarria et al. (2014), however, gluon Sivers function (GSF) is not yet known. Therefore, compared to the quark TMDs, gluon TMDs are much less known, and these will also be investigated at the future electron-ion collider (EIC) Accardi et al. (2016) and the future fixed target plans at the LHC Brodsky et al. (2013); Kikoła et al. (2017); Trzeciak et al. (2017). Gluon TMDs satisfy the positivity bounds first derived in Mulders and Rodrigues (2001). A phenomenological bound on GSF was obtained Burkardt (2004b), commonly known as Burkardt sum rule, from the requirement that the net transverse momentum of all partons (quarks and gluons) in a transversely polarized nucleon should vanish. In Anselmino et al. (2009), a fit to the data from SIDIS at low scale indicates that this sum rule is almost saturated by contribution from u and d quarks, however, there may still be about $30\%$ contribution from GSF.

Sivers function is a T-odd object and initial and final state interactions play an important role in Sivers asymmetry Brodsky et al. (2002). They are resummed into the gauge link or Wilson line in the operator definition of the Sivers function that is needed for color gauge invariance Boer et al. (2003). Gluon TMDs have two gauge links, in contrast to quark TMDs, that have only one. This introduces process dependence in them. The Sivers function in SIDIS is expected to be equal in magnitude but opposite in sign compared to the Sivers function appearing in the Drell-Yan process Brodsky et al. (2002). Recent data from RHIC Adamczyk et al. (2016) as well as COMPASS Aghasyan et al. (2017) seem to favour the sign change, however more data are needed Anselmino et al. (2017) . The GSF for any process, in general can be written in terms of two independent functions, one of them has an operator structure that is C-even, and the other, C-odd. In the literature, the former is called a f-type Sivers function and the latter, d-type Bomhof and Mulders (2007); Buffing et al. (2013). In fact, one of these two (d-type) is not constrained by the Burkardt sum rule. More experimental data are needed to precisely determine the GSF. As is well known, $J/\psi$ production in $ep$ and $pp$ collision is an effective method to probe the gluon TMDs including the GSF Kishore and Mukherjee (2019); Lansberg et al. (2017); Rajesh et al. (2018); Mukherjee and Rajesh (2017a); D’Alesio et al. (2017a); Mukherjee and Rajesh (2016); Mukherjee and Rajesh (2017b), as contribution to the Sivers asymmetry comes already at leading order (LO) through the virtual photon-gluon or gluon-gluon fusion processes, respectively. Data on Sivers asymmetry in $J/\psi$ production are available from COMPASS collaboration Matoušek (2016), although with large error bars, it can be qualitatively explained by a LO calculation in NRQCD based color octet model Mukherjee and Rajesh (2017a). In a recent work D’Alesio et al. (2019b), maximal values of the azimuthal asymmetries in back-to-back electroproduction of $J/\psi$ and a jet is estimated within the TMD factorization framework by neglecting the intrinsic transverse momentum of the initial parton in the hard part. Another interesting process to probe the GSF is quasi-real photoproduction of a hadron D’Alesio et al. (2017b) or $J/\psi$ Godbole et al. (2012, 2013); Rajesh et al. (2018). Contribution to the single spin asymmetry (SSA) comes from $J/\psi$ observed in the forward region that is when the transverse momentum ($p_{T}$) of $J/\psi$ is small. In this work, we investigate the possibility to probe the GSF in quasi-real photoproduction of back-to-back $J/\psi$ and jet by employing the generalized parton model (GPM) wherein the intrinsic transverse momentum of the initial parton is considered in the hard part, which will be possible in the future EIC. This will be sensitive to the GSF in a different kinematical region, and $J/\psi$ observed need not be in the forward region.

As mentioned above, initial and final state interactions play a very important role in the SSAs. TMD factorization has been proven only for certain processes. The current status of the TMD factorization for heavy quarkonium production in $pp$ collisions can be found in Echevarria (2019). The most widely used approach to calculate amplitudes for $J/\psi$ production is based on non-relativistic QCD (NRQCD). In this approach, the amplitude is factorized into a soft non-perturbative part and a hard part Carlson and Suaya (1976); Berger and Jones (1981); Baier and Ruckl (1981, 1982); Braaten and Fleming (1995); Cho and Leibovich (1996a, b). The heavy quark pair is produced in color singlet (CS) or color octet (CO) states in hard interaction. This is calculated in the perturbation theory. Then this heavy quark pair hadronizes to a quarkonium by emitting soft gluons. The hadronization process is described in terms of long distance matrix elements (LDMEs), that are obtained by fitting experimental data. The LDMEs have definite scaling properties with respect to the velocity parameter $v$, which is assumed to be small $v<<1$ Lepage et al. (1992). The theoretical estimates are arranged in a double expansion in powers of $v$ and the strong coupling, $\alpha_{s}$. NRQCD has been successful in explaining hadroproduction data from TEVATRON Abe et al. (1997); Acosta et al. (2005) and also $J/\psi$ photoproduction data from HERA Adloff et al. (2002); Aaron et al. (2010); Chekanov et al. (2003); Abramowicz et al. (2013). Both CS and CO contributions are needed to explain the HERA data Rajesh et al. (2018). In this work we calculate the weighted Sivers asymmetry, $A_{N}^{\sin(\phi_{q})}$, in photoproduction of back-to-back $J/\psi$ and jet at EIC in NRQCD by incorporating both CS and CO states. The plan of the paper is as follows. In section II we give the analytic expressions of the asymmetry. In section III we present the TMD evolution framework. Numerical results and conclusion are given in section IV and V respectively.

II Sivers Asymmetry

We consider the photoproduction process

[TABLE]

where the arrow in the superscript indicates the polarization of the proton. The letters in the round brackets represent the four momentum of the corresponding particles. We consider the proton-electron center of mass (C.M) frame wherein the proton and electron move along the +$z$ and -$z$ direction. The transverse plane, defined in Figure 1, is orthogonal to the momentum of proton direction. The initial scattering electron radiates the virtual photon that will interact with the proton. The four momentum square of the virtual photon is $q^{2}\approx-2E_{e}E_{e}^{\prime}(1-\cos\theta)$ with $E_{e}$ and $E_{e}^{\prime}$ are energies of the initial and final scattered electron respectively. In the forward scattering limit, photoproduction, the four momentum of the virtual photon $q^{2}=-Q^{2}\rightarrow 0$ as a result the virtual photon becomes the real photon. The dominant subprocess for $J/\psi$ production is the $\gamma(q)+g(p)\rightarrow J/\psi(P_{\psi})+g(P_{j})$ at next-to-leading order (NLO) in $\alpha_{s}$. The quark (antiquark) initiated subprocess can also contribute $\gamma+q(\bar{q})\rightarrow J/\psi+q(\bar{q})$. The unpolarized differential cross section for $ep\rightarrow J/\psi+\mathrm{jet}+X$ process can be written as follows

[TABLE]

where $a=g,u,d,s,\bar{u},\bar{d},\bar{s}$. The $x_{\gamma}$ and $x_{a}$ are the light-cone momentum fractions of photon and partons respectively, and ${\bm{p}}_{a\perp}$ is the transverse momentum of the initial parton. We have assumed TMD factorization in the GPM model with the inclusion of intrinsic transverse momentum of the initial parton in the hard part. When the exchanged photon is quasi-real, the inreaction takes place through the Weizs$\ddot{a}$ker-Williams distribution function of the electron, $f_{\gamma/e}(x_{\gamma})$, this describes the density of photons inside the electron and is given by Frixione et al. (1993)

[TABLE]

where $\alpha$ is the electromagnetic coupling and $Q^{2}_{min}=m_{e}^{2}\frac{x_{\gamma}^{2}}{1-x_{\gamma}}$, $m_{e}$ being the electron mass. We have considered $Q^{2}_{max}=1$ GeV2 for estimating the Sivers asymmetry. The unpolarized TMD, $f_{a/p}$, represents the density of unpolarized partons inside an unpolarized proton with momentum fraction $x_{a}$ and transverse momentum $p_{a\perp}$. The $\hat{s}$, $\hat{t}$ and $\hat{u}$ are the Mandelstam variables at partonic level and their definitions are given in appendix A. $\mathcal{M}_{\gamma a\rightarrow J/\psi a}$ is the amplitude of gluon and quark (anti-quark) initiated subprocesses. The square of the amplitude is calculated using the NRQCD model, for more details Ref.Rajesh et al. (2018) is referred for gluon channel, and the quark (anti-quark) channel matrix elements are given in the appendix B. The CS and CO states i.e., $\leftidx{{}^{3}}{S}{{}_{1}}^{(1,8)}$, $\leftidx{{}^{1}}{S}{{}_{0}}^{(8)}$ and $\leftidx{{}^{3}}{P}{{}_{J}}^{(8)}$ are considered for $J/\psi$ production. The center of mass (C.M) energy of the proton-electron system is $s=(P+l)^{2}$. In Eq.(2), the inelastic variables $z=\frac{P\cdot P_{h}}{P\cdot q}$ and $z_{1}=\frac{P\cdot P_{j}}{P\cdot q}$ are the energy fractions transferred from photon to $J/\psi$ and jet respectively in the proton rest frame. In photoproduction, the inelastic variables $z$ and $z_{1}$ can be measured in experiments using the Jacquet-Blondel method Adloff et al. (2002); Aaron et al. (2010); Abramowicz et al. (2013). By using the definitions of four momenta as given in appendix A, the momentum conservation delta function can be decomposed as

[TABLE]

where ${\bm{P}}_{\Psi\perp}$ and ${\bm{P}}_{j\perp}$ are the transverse momentum of the $J/\psi$ and jet respectively and their azimuthal angles are represented with $\phi_{1}$ and $\phi_{2}$ such that $\delta\phi=\phi_{2}-\phi_{1}-\pi$ as shown in Figure 1.

Now, we define the sum and difference of transverse momenta of $J/\psi$ and jet as ${\bm{q}}_{\perp}={\bm{P}}_{\Psi\perp}+{\bm{P}}_{j\perp}$ and ${\bm{K}}_{\perp}=({\bm{P}}_{\Psi\perp}-{\bm{P}}_{j\perp})/2$. We are interested in the case where $|{\bm{q}}_{\perp}|\ll|{\bm{K}}_{\perp}|$ i.e., the $J/\psi$ and jet are almost back to back in the transverse plane as shown in Figure 1. The azimuthal angle of $\bm{q}_{\perp}$ is denoted with $\phi_{q}$. After integrating over $z_{1}$, $x_{a}$ and ${\bm{p}}_{a\perp}$, one obtain the following expression

[TABLE]

For a tranversly polarized proton, the differential cross section is given by

[TABLE]

The weighted Sivers asymmetry is defined as Boer et al. (2016)

[TABLE]

where $\mathrm{d}\sigma^{\uparrow(\downarrow)}$ indicates the polarized cross section in the process where one of the initial particle is transversely polarized with respect to its momentum direction. The azimuthal weight factor $W(\phi_{q})=-\sin(\phi_{q})$ which is given by Bacchetta et al. (2004, 2007)

[TABLE]

It has been advertised that the $J/\psi$ production probes the gluon TMDs and the quark contribution can be safely neglected in the kinematical region considered because of the insignificant contribution of quarks w.r.t gluon, which will be discussed in the results section. Hence the dominant contribution to the Sivers asymmetry comes from the gluon channel. The numerator of the asymmetry is sensitive to the Sivers function in $J/\psi$ production

[TABLE]

with $\Delta\hat{f}_{a/{p^{\uparrow}}}(x_{a},{\bm{q}}_{\perp})$ being the Sivers function, describes the number density of unpolarized partons in a transversely polarized proton with mass $M_{p}$. The analytic expressions for contributions from different states can be found in Rajesh et al. (2018). As only $J/\psi$ was observed there, we had integrated over the phase space of the final gluon $a=g$, whereas here, the final parton is producing the observed jet. Sivers function in Trento convention Bacchetta et al. (2004) is given by

[TABLE]

The Sivers function fulfills the following positivity bound

[TABLE]

Following Ref.D’Alesio et al. (2015); D’Alesio et al. (2019a), we adopt the Gaussian parametrization for Sivers function within the DGLAP evolution approach as given below

[TABLE]

where $f_{a/p}(x_{a})$ is the usual collinear parton distribution function (PDF) which follows the DGLAP evolution equation and

[TABLE]

with $|N_{a}|\leq 1$ and

[TABLE]

as a result the Sivers function satisfies the positivity bound for all values of $x_{a}$ and $q_{\perp}$. If we define the parameter

[TABLE]

such that $0<\rho<1$, then Eq. (12) becomes

[TABLE]

The unpolarized gluon TMD sitting in the denominator of the asymmetry is parametrized as Gaussian distribution

[TABLE]

The best fit parameters of Sivers function have been extracted for quarks Anselmino et al. (2017) and gluons D’Alesio et al. (2015); D’Alesio et al. (2019a) from SIDIS and RHIC data respectively. In Ref. D’Alesio et al. (2019a), new set of best fit parameters of GSF are extracted for $\langle q_{\perp}^{2}\rangle=1$ GeV2 and are tabulated in Table 1.

III TMD Evolution

In DGLAP evolution, collinear PDFs evolve with only the probing scale ($\mu$). However, in TMD evolution approach, TMDs evolve with both the intrinsic transverse momentum ($p_{a\perp}$) of the parton and the probing scale. The TMD evolution framework is derived in the impact parameter space ($b_{\perp}$) Aybat and Rogers (2011)

[TABLE]

and in the momentum space is given by

[TABLE]

Usually, TMDs depend on two scales that are renormalization scale ($\mu$) and auxiliary scale ($\zeta$) Collins (2013); Aybat et al. (2012b). In order to cure the light-cone (rapidity) divergences in TMD factorization, the scale $\zeta$ has been introduced. One can obtain the renormalization group and Collins-Soper equations by taking scale evolution with respect to $\mu$ and $\zeta$ respectively. The unpolarized TMD expression at a given final scale $Q_{f}=\sqrt{\zeta}=M$ is obtained by solving RG and CS equations Echevarria et al. (2014); Collins (2013); Aybat et al. (2012b) and is given below

[TABLE]

where $Q_{i}=c/b_{\ast}(b_{\perp})$ is the initial scale of the TMD with $c=2e^{-\gamma_{\epsilon}}$ and $\gamma_{\epsilon}\approx 0.577$. In line with Ref.Echevarria et al. (2014), we adopt the $b_{\ast}$ prescription to avoid hitting the Landau pole by freezing the scale $b_{\perp}$. Here, $b_{\ast}(b_{\perp})=\frac{b_{\perp}}{\sqrt{1+\left(\frac{b_{\perp}}{b_{\mathrm{max}}}\right)^{2}}}\approx b_{\mathrm{max}}$ when $b_{\perp}\rightarrow\infty$ and $b_{\ast}(b_{\perp})\approx b_{\perp}$ when $b_{\perp}\rightarrow 0$. The $R_{pert}$ and $R_{NP}$ are the perturbative and the nonperturbative parts of the TMD respectively, which are given below

[TABLE]

and

[TABLE]

where the anomalous dimensions are $A=\sum_{n=1}^{\infty}\left(\frac{\alpha_{s}(\mu)}{\pi}\right)^{n}A_{n}$ and $B=\sum_{n=1}^{\infty}\left(\frac{\alpha_{s}(\mu)}{\pi}\right)^{n}B_{n}$, and the coefficients for gluon case are $A_{1}=C_{A}$, $A_{2}=\frac{1}{2}C_{F}\left(C_{A}\left(\frac{67}{18}-\frac{\pi^{2}}{6}\right)-\frac{5}{9}C_{A}N_{f}\right)$ and $B_{1}=-\frac{1}{2}(\frac{11}{3}C_{A}-\frac{2}{3}N_{f})$, and for quark case are $A_{1}=C_{F}$, $A_{2}=\frac{1}{2}C_{F}\left(C_{A}\left(\frac{67}{18}-\frac{\pi^{2}}{6}\right)-\frac{5}{9}N_{f}\right)$ and $B_{1}=-\frac{3}{2}C_{F}$ Echevarria et al. (2014). The unpolarized TMD at the initial scale can be written as

[TABLE]

where $C_{i/a}$ is the perturbatively calculated process independent coefficient function, and is different for each type of TMD. The derivative of the Sivers function follows the same evolution equation as given in Eq.(20) but the $g_{1}^{\mathrm{TMD}}$ changes to $g_{1}^{\mathrm{Sivers}}$ in the $R_{NP}$ factor. The Sivers function $f_{1T}^{\perp}(x_{a},{p}_{a\perp},Q_{f})$ and it’s derivative are related by Fourier transformation as below Aybat et al. (2012b)

[TABLE]

and the unpolarized TMD is given by

[TABLE]

The derivative of the Sivers function at the initial scale $Q_{i}$ can be written in terms of Qiu-Sterman function as below Boer et al. (2003); Ji et al. (2006)

[TABLE]

where $T_{a,F}(x_{a},x_{a},Q_{i})$ is the Qiu-Sterman function, which is usually assumed to be proportional to collinear PDF Kouvaris et al. (2006); Echevarria et al. (2014)

[TABLE]

the definition of $\mathcal{N}_{a}(x_{a})$ is given in Eq.(13). Note that here we have used the fact that Sivers function in Drell-Yan process have opposite sign compared to the Sivers function in semi-inclusive DIS. So far the best fit parameters of GSF have not been extracted in the TMD evolution approach. However only the $u$ and $d$ quark Sivers function are known which were extracted in Echevarria et al. (2014) from SIDIS data within TMD evolution scheme, which are tabulated in Table 1. As so far a fit for the gluon Sivers function is not avavilable in the TMD evolution approach, in order to show the effect of this evolution on the asymmetry, we use an exploratory approach, namely, following Ref.Boer and Vogelsang (2004), we define the following two set of parametrizations for GSF by using the known $u$ and $d$ quark Sivers function parameters

[TABLE]

We denote the first parametrization as TMD-a and second one as TMD-b. As the sign of $N_{u}$ and $N_{d}$ are opposite, TMD-a gives a small GSF and TMD-b gives a large GSF. As mentioned in the introduction, Burkardt sum rule Burkardt (2004b) gives a constraint on the GSF. However, in order to implement the contstraint from the Burkardt sum rule contribution from all quark favours need to be included, and sea quark Sivers function is still not well constrained. Assuming contributions only from $u$ and $d$ quarks and gluons, we have checked that TMD-a parametrization satisfies the Burkardt sum rule, violation is about $1\%$; wheras the TMD-b parametrization violates the sum rule by about $19\%$. The numerical values of best fit parameters are estimated Echevarria et al. (2014) at $Q_{0}=\sqrt{2.4}~{}\mathrm{GeV}$, $b_{\mathrm{max}}=1.5\mathrm{~{}GeV^{-1}}$, $g_{2}=0.16\mathrm{~{}GeV^{2}}$ and $\langle p^{2}_{s\perp}\rangle=0.282\mathrm{~{}GeV^{2}}$ with $g_{1}^{\mathrm{pdf}}=\langle p^{2}_{a\perp}\rangle/4=\langle q^{2}_{\perp}\rangle/4$ and $g_{1}^{\mathrm{sivers}}=\langle p^{2}_{s\perp}\rangle/4$. The numerator and denominator parts of Eq.(7) in TMD evolution approach can be written as the following

[TABLE]

[TABLE]

IV Numerical Results

In this section, we discuss the numerical results of Sivers asymmetry in $ep^{\uparrow}\to J/\psi+\mathrm{jet}+X$ photoproduction process, where the proton is transversely polarized. We consider the situation $|{\bm{q}}_{\perp}|\ll|{\bm{K}}_{\perp}|$ i.e., the produced pair of $J/\psi$ and jet are almost back to back in the transverse plane as shown in Figure 1. This configuration is feasible in the future proposed Electron-Ion collider (EIC) with C.M energy from 20 to 150 GeV. The NLO photon-gluon fusion and quark (anti-quark) initiated subprocesses, $\gamma g\rightarrow J/\psi g$ and $\gamma q(\mathrm{~{}or~{}}\bar{q})\rightarrow J/\psi q(\mathrm{~{}or~{}}\bar{q})$, are considered. The NRQCD model is employed for $J/\psi$ production, and the CS and CO states are considered for both numerator and denominator parts of Eq.(7). If we do not detect the jet in the final state then CS state does not contribute to the asymmetry, because the initial and final state interactions between the final state parton and remnant of the proton get canceled with each other as discussed in Yuan (2008). The values of long distance matrix elements (LDMEs) are taken from Ref.Chao et al. (2012). There are different set of LDMEs in the literature and the asymmetry is found to be independent of the choice of LDME set.

There are two types of $J/\psi$ photoproductions that are resolved and direct photon contributions. The resolved photoproduction, the photon splits into partons which subsequently interact with the partons of proton, contributes to $J/\psi$ production in the low $z$ region ($z<0.3$). While in the direct photoproduction, the photon directly interacts electromagnetically with the partons from the proton. For direct inelastic $J/\psi$ photoproduction one has to consider $0.3<z<0.9$ as discussed in Ref.Rajesh et al. (2018). The fragmentation of gluon and heavy quark can also contribute to $J/\psi$ production at high transverse momentum of the $J/\psi$ Li and Liu (1998). The feed-down contribution from an excited state $\psi(2S)$ and the decay of $\chi_{c}$ states contribution to $J/\psi$ are $15\%$ Chekanov et al. (2003) and $1\%$ Butenschoen and Kniehl (2010); Artoisenet et al. (2009) respectively, are not considered in this work. The final state parton becomes soft at $z\rightarrow 1$ which leads to infrared singularity. Therefore, to calculate the asymmetry for direct inelastic $J/\psi$ photoproduction we consider $z=0.3$. The mass of $J/\psi$ is taken to be $M=3.1$ GeV. The cteq6l1 PDF sets are used for collinear PDFs Buckley et al. (2015).

The Sivers asymmetry is calculated at EIC for $\sqrt{s}=45$ and 100 GeV within the DGLAP and TMD evolution approaches. In DGLAP evolution approach, the GSF has been extracted in D’Alesio et al. (2015) from pion data at RHIC for fixed Gaussian width $\langle q_{\perp}^{2}\rangle=0.25$ GeV2. Recently refitted the RHIC data with new set of GSF parameters for $\langle q_{\perp}^{2}\rangle=1$ GeV2 D’Alesio et al. (2019a). However, GSF has not been extracted yet in TMD evolution approach. The $u$ and $d$ quark Sivers functions are extracted in Ref.Echevarria et al. (2014) using TMD evolution approach. For numerical estimation of Sivers asymmetry, the $u$ and $d$ quark Sivers function best fit parameters are used for GSF as defined in Eq.(III) wthin TMD evolution approach. The best fit parameters of GSF and quark Sivers function are tabulated in Table 1. The convention of figures as follows. The obtained Sivers asymmetry in DGLAP approach is represented with SIDIS1, SIDIS2 and SIDIS3. The TMD-a and TMD-b represent the Sivers asymmetry in TMD evolution approach.

In Figure 2 - 4, the asymmetry in DGLAP and TMD evolution approaches as a function of $q_{\perp}$ is shown at $\sqrt{s}=45$ and 100 GeV for $K_{\perp}=3$ GeV respectively at $z=0.3$. The asymmetry is shown in the range $0\leq q_{\perp}\leq 1$ GeV which is considered to satisfy the condition $|{\bm{q}}_{\perp}|\ll|{\bm{K}}_{\perp}|$. The value of $K_{\perp}=3$ GeV is chosen of the order of $J/\psi$ mass. For higher values of $K_{\perp}$ and $z$ the gluon channel contribution is suppressed because the momentum fraction of the parton, $x_{a}$, depends quadratically on $K_{\perp}$ which can be seen from Eq.(4).

The maximized Sivers asymmetry, $A_{N}^{Max}$, at $\sqrt{s}=45$ GeV is shown in Figure2. Here, we saturated the Sivers function bound by adopting $\mathcal{N}_{a}=1$ and $\rho=2/3$ D’Alesio et al. (2019a) in the parametrization of Sivers function which is given in Eq.(12). In the left panel of Figure2, the gluon and quark (antiquark) channels contribution to $A_{N}^{Max}$ is shown, and the quark (antiquark) channel contribution is insignificant compared to gluon channel. On this basis we can say that the pair of $J/\psi$+jet photoproduction process is an effective channel that probes the poorly known GSF in the kinematical region considered here. We have neglected the quark channel contribution to the numerator part of the asymmetry in Figure 3 - 4. In the right panel of Figure2, the individual CS and CO states contribution to $A_{N}^{Max}$ is shown. The $\leftidx{{}^{3}}{S}{{}_{1}}^{(1)}$ and $\leftidx{{}^{1}}{S}{{}_{0}}^{(8)}$ states contribute largely to $A_{N}^{Max}$ which is independent of $\sqrt{s}$.

In the left panel of Figure 3 , the weighted Sivers asymmetry, $A_{N}^{\sin(\phi_{q})}$, is estimated to be about 3%, 1% and 6% respectively for SIDIS1, SIDIS2 and SIDIS3 set of GSF parameters at $\sqrt{s}=45$ GeV. The $A_{N}^{\sin(\phi_{q})}$ is reduced about 2% for $\sqrt{s}=$100 GeV as shown in the right panel of Figure 3 . In Figure 4, negative Sivers asymmetry is shown in TMD evolution approach. The sign of the Sivers asymmetry depends on relative magnitude of $N_{u}$ and $N_{d}$ and these have opposite sign which can be observed in Table 1. For TMD-b parameter set, the $\mathcal{N}_{g}$ is assumed to be proportional to $\mathcal{N}_{d}$ of $d$ quark, see the Eq.(III). The values of $N_{d}$ is negative which leads to negative asymmetry. The average of $u$ and $d$ quarks $x_{a}$-dependent factor, $\mathcal{N}_{a}$, is defined for gluon as given in Eq.(III) for TMD-a parameter set. The magnitude of $\mathcal{N}_{d}$ is comparable but slightly dominant compared to $\mathcal{N}_{u}$ that leads to negative and small asymmetry for TMD-a parameter set. In Figure 4, $A_{N}^{\sin(\phi_{q})}$ is estimated maximum of 8% and 4% at $\sqrt{s}=45$ and 100 GeV for TMD-b parameter set.

V Conclusion

In this work, we gave an estimate of the Sivers asymmetry in almost back-to-back $J/\psi$ and jet photoproduction at the future EIC. We assumed TMD factorization for this process and used generalized parton model, incorporating the intrinsic transverse momenta. The quasi-real photoproduction takes place through the Weizs$\ddot{a}$ker-Williams photon distribution of the electron. We used NRQCD to calculate the $J/\psi$ production and incorporated both CS and CO contributions to the asymmetry. Major contribution comes from ${}^{3}S_{1}^{(1)}$ and ${}^{1}S_{0}^{(8)}$ states. We have also shown the effect of the TMD evolution on the asymmetry. In fact the Sivers asymmetry is positive without incorporating the TMD evolution, whereas it becomes negative when evolution is incorporated. We have obtained sizable Sivers asymmetry where the main contribution comes from the gluon Sivers function and the quark contribution is small. Therefore, back-to-back production of $J/\psi$ and jet at the future EIC is a promising tool to access the gluon Sivers function.

Acknowledgment

We would like to thank Cristian Pisano and Pieter Taels for useful discussions. The work of S.R. is supported by Fondazione Sardegna under the project “Quarkonium at LHC energies”, CUP F71I17000160002 (University of Cagliari). A.M. would like to thank University of Cagliari and INFN, Cagliari, Italy for hospitality where the final stage of this work was completed.

Appendices

Appendix A Kinematics

We consider the frame in which the proton and electron are moving along +$z$ and -$z$-axises respectively and their four momenta are given by

[TABLE]

The C.M energy of electron-proton system is $s=(P+l)^{2}$. The above four momenta in light-cone coordinate system can be written as

[TABLE]

where $n_{+}$ and $n_{-}$ are two light-like vectors with $n_{+}\cdot n_{-}=1$ and $n_{+}^{2}=n^{2}_{-}=0$.

[TABLE]

We assume that the quasi-real photon is collinear to the electron. The quasi-real photon and parton four momenta are given by

[TABLE]

[TABLE]

where $x_{\gamma}=\frac{q^{-}}{l^{-}}$ and $x_{a}=\frac{p^{+}}{P^{+}}$ are the light-cone momentum fractions. The four momentum of the $J/\psi$ and final parton are given by

[TABLE]

[TABLE]

The inelastic variables are defined as $z=\frac{P\cdot P_{\Psi}}{P\cdot q}=\frac{P_{\Psi}^{-}}{q^{-}}$ and $z_{1}=\frac{P\cdot P_{j}}{P\cdot q}=\frac{P_{j}^{-}}{q^{-}}$. By using the above relations, we can write down the expressions of Mandelstam variables as below

[TABLE]

[TABLE]

[TABLE]

Here $M$ being the mass of $J/\psi$.

Appendix B Matrix elements for $\gamma+q(\bar{q})\to J/\psi+q(\bar{q})$ subprocess

In this section the matrix elements for $\gamma+q(\bar{q})\to J/\psi+q(\bar{q})$ channel are presented:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

The matrix elements for $\gamma g\to J/\psi+g$ channel are given in our previous paper Rajesh et al. (2018).