SSA in electroproduction of $J/\psi$ and QCD-evolved TMD's

We estimate Sivers asymmetry in low virtuality photoproduction of $J/\psi$ using color evaporation model and taking into account $Q^2$- evolution of transverse momentum dependent PDF's and Sivers function. There is a substantial reduction in asymmetry as compared to our previous analysis wherein the $Q^2$-dependance came only from DGLAP evolution of collinear part of TMDs. The estimates of asymmetry are comparable to our earlier estimates in which we had used analytical solution of only an approximated form of the evolution equations. We have also estimated asymmetry using the latest parametrization by Echevarria {\it et al.} which are based on an evolution kernel in which the perturbative part is resummed to NLL accuracy.


Introduction
The issue of quarkonium production mechanism is an open question as none of theoretical models which are used to describe the non-perturbative transforma-tion of the QQ pair into quarkonium i.e. the Color Singlet Modeal (CSM) 1-3 , Color Evaporation Model (CEM) 4 and Non-Relativistic Quantum Chromodynamics (NRQCD) 5 , is able to explain satisfactorily all the data on both production cross section and polarization measurements. Thus, independent tests other than polarization measurements are needed to compare the different production mechanisms. One such possible test is transverse single spin asymmetry (SSA) in charmonium production 6,7,8 since the asymmetry in heavy quarkonium production is very sensitive to the production mechanism 9 .
One of the theoretical approaches that has been used to explain these asymmetries in Semi-Inclusive Deep Inelastic Scattering (SIDIS) and Drell Yan (DY) processes is based on a transverse momentum dependent factorization scheme 10,11 which involves transverse momentum dependent parton densities and fragmentation functions collectively referred to as TMD's. One such TMD of great interest is Sivers function which gives the probability of finding an unpolarized parton inside a transversly polarized nucleon.
In an earlier work, we proposed that transverse SSA in charmonium production can be used to study Sivers effect 6 . We presented first estimates of SSA in photoproduction (i.e. low virtuality electroproduction) of charmonium in scattering of electrons off transversely polarized protons using CEM. In the process that we considered, at LO, there is contribution only from a single partonic subprocess and hence, it can be used as a clean probe of gluon Sivers function. Subsequently, we improved our estimates taking into account TMD evolution of TMD PDF's and the Sivers function 12 . In present work, we present further improved estimates calculated using the latest fits to TMD PDF's given by Echevarria et al. 13 and compare them with our earlier estimates.

Transverse Single Spin
Asymmetry in e + p ↑ → J/ψ + X We will use a generalization of CEM for our estimates of asymmetry in photoproduction (low virtuality electroproduction) of J/ψ by taking into account the transverse momentum dependence of the WW function and the gluon distribution function 6 where f γ/e (y, E) is the William Weizsacker function 14 which gives distribution function of the photon in the electron. We assume k ⊥ dependence of pdf's to be factorized in gaussian form 15 For the k ⊥ dependent WW function also, we use a Gaussian form. Expression for the numerator of the asymmetry is 6 where q = p c + pc and theσ γg→cc 0 (M 2 ) is the partonic cross section in lowest order (LO).
In Eq. (3), ∆ N f g/p ↑ (x, k ⊥ ) is the gluon Sivers function for which we use the following parametrization 16 Since there is not enough data available to parametrize the gluon Sivers function, we have expressed gluon Sivers function in terms of quark Sivers function. For quark Sivers functions, we use the following normalization where a f , b f and N f are best fit parameters.
Taking sin(φ q − φ S ) as a weight, the asymmetry integrated over the azimuthal angle of J/ψ is where φ q and φ k ⊥ are the azimuthal angles of the transverse momenta q T and k ⊥ and x g,γ = M √ s e ±y .

QCD Evolution of TMD PDF's
Initial phenomenological fits of the Sivers function and other TMD's used TMDs which do not evolve with the scale of the process 19,16 . Our initial estimates of Sivers asymmetry were based on these parameters and the TMDs used were evolved using DGLAP evolution wherein only the collinear part evolves and the Gaussian width of the transverse part is assumed to be fixed. In recent years, the TMD factorization has been derived and implemented 11,20,21 . A strategy to extract Sivers function from SIDIS data taking into account the TMD Q 2 evolution was proposed by Anselmino et al. 22 . We have also estimated SSA in electroproduction of J/ψ production based on this strategy 12 . In present work, we compare our earlier estimates with improved estimates obtained using exact solution of evolution equation. In addition, we have also estimated asymmetry using the latest parametrization by Echevarria et al. 13 which are based on an evolution kernel in which the perturbative part is resummed to NLL accuracy.
The energy evolution of a general TMD F (x, k ⊥ , Q) is more naturally described in b-space. The b-space TMD's evolves with Q according to where R pert is the perturbative part of the evolution kernel, R N P is the nonperturbative part and b The perturbative part is given by where dD d ln µ = Γ cusp The anomalous dimensions A and B are known up to three loop level 23 . The non-perturbative exponential part contains a Q-dependent factor universal to all TMDs and a factor which gives the gaussian width in b-space of the particular TMD The b * prescription stitches together the perturbative part(which is valid at low b) and non-perturbative part(which is valid at large b. Q 2 -dependent TMD's in momentum space are obtained by Fourier transforming F (x, b, Q f ).

Approximate Analytical versus Exact Solution of TMD Evolution Equations
In the analytical approach of Anselmino et al. 22 , one assumes that the kernel R(Q, Q 0 , b), which drives the Q 2 -evolution of TMD's, becomes independent of b in large b limit, i.e. as b → ∞ , R(Q, Q 0 , b) → R(Q, Q 0 ). b integration can then be performed analytically and Q 2 dependent PDF's can be obtained. In our earlier work, we used Q 2 -evolved TMD PDF's obtained using this "approximate, analytical" approach. We will now compare these results with estimates obtained using exact solution of TMD evolution equations which can be obtained by solving the TMD evolution equation numerically 22 .
Recently, Echevarria et al. 13 have considered solution of TMD evolution equations up to NLL accuracy and have performed a global fitting of all experimental data on the Sivers asymmetry in SIDIS using this formalism. Since the derivative of b-space Sivers function satisfies the same evolution equation as the unpolarized PDF 21 , its evolution is given by where T q,F (x, x, Q) is the twist three Qui-Sterman quark gluon correlation function which is related to the first k ⊥ moment of quark Sivers function 24 and can be expressed in terms of the unpolarized collinear PDFs 25,13 .
The expansion coefficients with the appropriate gluon anomalous dimensions at NLL, A (1) , A (2) , B (1) and D (1) , are known 13 . Choosing the initial scale Q i = c/b, the D term vanishes at NLL. Taking Fourier transform of Eq. (9), one gets f ⊥g 1T (x, k ⊥ ; Q f ) which is related to Sivers function through

Numerical Estimates of Asymmetry using analytical and exact formalisms
We will now present our estimates of SSA in photoproduction of J/ψ for JLAB, HERMES, COMPASS and eRHIC energies. A detailed discussion of results can be found in Ref. 26. Figs. 1-5 show the y and k T distribution for different experiments with parameterizations TMD Exact-1, TMD Exact -2 and TMD a given in Table  1. TMD-e1 parameter set, extracted at Q 0 = 1.0 GeV, is for the exact solution of TMD evolution equations extracted in Ref. 22. TMD-a is the parameter set extracted using an analytical approximated form of evolution equations given in Ref. 22. For estimates using NLL kernel we have used the most recent parameters by Echevarria et al. 13 obtained by performing a global fit of all experimental data on Sivers asymmetry in SIDIS from HERMES, COMPASS and JLAb. We call this set TMD-e2. This set was fitted at Q 0 = √ 2.4 GeV. Fig. 6 shows a comparison of asymmetries at all energies.

Summary
We have compared estimates of SSA in electroproduction of J/ψ using TMD's evolved via DGLAP evolution and TMD evolution schemes. For the latter, we have chosen three different parameter sets fitted using an approximate analytical solution, an exact solution at LL and an exact solution at NLL. We find that the estimates given by TMD evolved PDF's and Sivers function are all comparable but substantially small as compared to estimates calculated using DGLAP evolved TMD's.