J/ψ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J/\psi $$\end{document} production in polarized and unpolarized ep collision and Sivers and cos2ϕ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\cos 2\phi $$\end{document} asymmetries

We calculate the Sivers and cos2ϕ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\cos 2\phi $$\end{document} azimuthal asymmetries in J/ψ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J/\psi $$\end{document} production in the polarized and unpolarized semi-inclusive ep collision, respectively, using the formalism based on the transverse momentum-dependent parton distributions (TMDs). The non-relativistic QCD-based color octet model is employed in calculating the J/ψ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J/\psi $$\end{document} production rate. The Sivers asymmetry in this process directly probes the gluon Sivers function. The estimated Sivers asymmetry at z=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z=1$$\end{document} is negative, which is in good agreement with the COMPASS data. The effect of TMD evolution on the Sivers asymmetry is also investigated. The cos2ϕ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\cos 2\phi $$\end{document} asymmetry is sizable and probes the linearly polarized gluon distribution in an unpolarized proton.


I. INTRODUCTION
Single spin asymmetry (SSA) has been playing a vital role in spin physics since the observation of large SSA in high energy pp collision experimentally [1,2,3,4,5].SSA arises in scattering process in which the target or one of the colliding proton is transversely polarized with respect to the scattering plane.In order to explain the SSA theoretically, it requires the nonperturbative quark or gluon correlators and there are two approaches for it.First one is based on generalized factorization [6], where one includes intrinsic transverse momentum in the parton distribution functions and fragmentation functions (TMDs).This approach is applicable when the process involves two scales, namely a hard and a soft scale.Example of such process is semi inclusive deep inelastic scattering ( SIDIS), where the hard scale is the virtuality of the gauge boson exchanged and the soft scale can be characterized by the transverse momentum of the observed hadron.Another such process is Drell-Yan (DY), where the hard scale is the same as SIDIS and the soft scale is the transverse momentum of the lepton pair produced.This approach is phenomenologically well studied [7,8,9,10,11,12,13,14,15]. The second approach describes the SSAs in terms of collinear higher twist quark-gluon correlators.This formalism uses collinear factorization and was originally proposed in [16,17,18,19,20] and further developed by [21,22,23].This is useful for processes having only one hard scale like SSA in pp collision.
Among the single spin asymmetries, the Sivers asymmetry is one of the most important and well studied asymmetry, both theoretically and experimentally.This asymmetry involves the Sivers function [24].The asymmetry arises because the distribution of quarks and gluons in a transversely polarized proton is not left-right symmetric with respect to the plane formed by its transverse momentum and spin direction.The Sivers effect leads to an asymmetry in the azimuthal angle of the hadron produced in SIDIS and has been observed in HERMES [25,26] and COMPASS experiments [27,28] for proton target and by JLab Hall-A collaboration for 3 He target [29].The Sivers function has been shown in a model dependent way to be related to the orbital angular momentum of the quarks and gluons [30,31].The first transverse moment of the Sivers function is related to the quark-gluon twist three Qiu-Sterman function [32].A detailed discussion of such relations can be found in [33].
Sivers function is a T-odd (time reversal odd) object .The operator definitions of the quark and gluon Sivers function need gauge links (one for quark Sivers function and two for gluon Sivers function) for color gauge invariance.As these gauge links or Wilson lines depend on the specific process under consideration, this introduces non-universality or process dependence in the Sivers function [32].For gluon Sivers function, there are two gauge links and the process dependence is more involved.However, the gluon Sivers function for any process can be written in terms of two "universal" gluon Sivers functions [34], one involving a C-even operator (f-type) , the other a C-odd operator (d-type).
Gluon Sivers function (GSF) plays an important role in understanding the SSAs observed in pp collision as well as those in SIDIS over a wide kinematical region.What is more interesting is that different experiments probe different gluon Sivers functions.Burkardt's sum rule [35] gives a bound on the GSF.This sum rule is derived from the fact that the total transverse momentum of all partons in a transversely polarized proton should vanish.Fits to SIDIS data at low scale have found that this sum rule is almost saturated by contribution from the u and d quark's Sivers function [36], however there is still room for about 30% contribution from GSF.Moreover, one of the gluon Sivers functions (d-type) is not constrained by the Burkardt's sum rule.Apart from SIDIS and DY [36,37,38], Sivers effect has been studied theoretically in several ep ↑ collision processes, among them photoproduction of J/ψ [39,40,41], heavy quark pair and dijet production in ep ↑ scattering [42].In SSA in proton-proton collision, the process dependent initial and final state interactions play a major role and usually need to be carefully taken into account [43].
J/ψ production in ep ↑ scattering provides direct access to the GSF (f-type) through the leading order (LO) subprocess.It has been shown that [44], due to the final state interactions in ep and pp scattering process, SSA in heavy quarkonium production is zero in ep scattering when the heavy quark pair is produced in a color singlet state, whereas for pp scattering the SSA is zero when the heavy quark pair is produced in color octet state.Quarkonium production has been studied in unpolarized pp scattering within TMD evolution formalism in [45,46].In Ref. [39,40,41], SSA in J/ψ production in ep ↑ collision using low virtuality electroproduction approximation (photoproduction) is studied in color evaporation model (CEM) and sizable asymmetries are reported.In this work, we investigate the Sivers asymmetry in the semi-inclusive process e + p ↑ → e + J/ψ + X and the cos 2φ azimuthal asymmetry in the unpolarized process e + p → e + J/ψ + X using non-relativistic Quantum Chromo Dynamics (NRQCD) based color octet model (COM) [47].In COM, the cc pair is produced in the color octet state that forms J/ψ by emitting soft gluons [48].The COM is based on a factorization formula in NRQCD.The cross section is described in terms of a product of a perturbative part, where the initial state partons form a cc pair having definite color and total angular momentum quantum numbers, and a non-perturbative matrix element through which the cc pair forms J/ψ.These matrix elements are obtained by fitting data and they are universal.We use a recent extraction [49] for the gluon Sivers function from the SSA data in pp collision at RHIC.The TMDs (unpolarized as well as the Sivers function) depend on the scale, as a result the SSA also depends on the scale [50].The scale dependence is given by the TMD evolution and is usually performed in the impact parameter or b ⊥ -space [51,52].There are different schemes of performing the TMD evolution, and an improved evolution scheme called CSS2 has been proposed.A detailed discussion of the evolution schemes and scheme transformation issues are discussed in the recent paper [53].The evolution in the renormalization scale and rapidity scales are performed using renormalization group and Collins-Soper (CS) equations.To incorporate the correct evolution at large b ⊥ value a nonperturbative Sudakov factor is included in the evolution which is usually obtained by fitting the data.We also study the effect of TMD evolution on the Sivers asymmetry in J/ψ production in COM.
The cos 2φ azimuthal asymmetry was observed experimentally long ago both in unpolarized SIDIS [54,55] and DY [56,57] processes.Recently, HERMES [58] and COMPASS [59] experiments reported sizable azimuthal asymmetries in low transverse momentum region.In [12] it was suggested that the cos 2φ asymmetry could be explained by the Boer-Mulders effect.
The cos 2φ asymmetry arises in the unpolarized cross section due to the correlation between the transverse spin and transverse momentum of the parton inside the nucleon.As a result, Boer-Mulders TMD function appears along with cos 2φ term in the unpolarized cross section.

Quark (anti-quark) version Boer-Mulders function, h ⊥q
1 (T-odd), represents the transversely polarized quark (anti-quark) distribution inside an unpolarized hadron.h ⊥q 1 has been extracted in [60,61,62] from cos 2φ asymmetry SIDIS data assuming a relation with Sivers function.However, gluon Boer-Mulders function, h ⊥g 1 (T-even), has not been extracted yet.h ⊥g 1 represents the linearly polarized gluon distribution inside an unpolarized hadron.cos 2φ asymmetry in the production of J/ψ in unpolarized semi-inclusive ep collision process directly allows us to probe h ⊥g 1 .The paper is organized as follows.Sivers asymmetry and TMD evolution are presented in Sec.II and Sec.III respectively.Sec.IV and Sec.V discuss the cos 2φ azimuthal asymmetry and numerical results respectively along with the conclusion in Sec.VI.

II. SIVERS ASYMMETRY
Single spin asymmetry for the semi-inclusive process A ↑ + B → C + X is defined as 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.We consider the process, where the electron scatters by the transversely polarized proton target.The letters within the brackets represent the four momentum of the corresponding particle.We follow the generalized factorization theorem where the intrinsic partonic transverse momentum is taken into account unlike the collinear factorization.The kinematics considered below are different from [39,40,41].
We consider the frame as shown in FIG. 1, in which the proton and virtual photon are moving along −z and +z axes respectively.The four momenta of target system P and virtual photon q = l − l are given by with Q 2 = −q 2 and Bjorken variable, x B = Q 2 2P.q (up to proton mass correction).Here, M p is mass of the proton.The leptonic four momenta are expanded in terms of n − = P and n + = n = (q + x B P )/P.q [63] as follows here, y = P.q P.l .The invariant mass of electron-target system is s = (P + l) 2 = 2P.l= 2P.qy and then we have The virtual photon-target invariant mass is defined as . Using Sudakov decomposition, the four momenta of the initial gluon k and the final hadron where, x = k.n is the longitudinal momentum fraction, z = P.P h /P.q and P 2 hT = −P 2 hT .Mass of the J/ψ is denoted with M .In line with Ref. [63] , we assume that generalized factorization theorem allows to factorize the unpolarized differential cross section as dσ = 1 2s The leptonic tensor is given by The gluon-gluon correlator, Φ µµ g (x, k ⊥ ), describes the hadron to parton transition which is parametrized in terms of eight TMDs at leading twist.The gluon correlator is defined for unpolarized and transversely polarized hadron respectively as below [64] where g µµ T = g µµ − P µ n µ /P.n − P µ n µ /P.n is the transverse metric tensor.Here we have kept only the part of the hadronic tensor for transverse polarization, that contributes to the Sivers asymmetry.f g 1 and h ⊥g 1 represent the unpolarized and linearly polarized gluon distribution functions inside the unpolarized hadron respectively.f ⊥g 1T , gluon Sivers function, describes the density of unpolarized gluons inside the transversely polarized hadron.The only LO subprocess for J/ψ production is γ * g → cc.In Eq.( 8), M γ * g→J/ψ is the amplitude of J/ψ production.J/ψ production mechanism, for instance, contains both perturbative and nonperturbative regimes which need to be separated out systematically.We employ the COM to calculate the amplitude of J/ψ bound state.The detailed calculation is discussed in the Appendix.In COM framework, initially heavy quark pair produced in a definite quantum state which can be calculated using perturbation theory up to a fixed order in α s .The long distance matrix element (LDME), 0 | O J/ψ n | 0 , contains the transition probability of J/ψ production from heavy quark pair.
The momentum conservation delta function can be decomposed as The phase space factors in Eq.( 8) can be written as follows The differential cross section can be expressed in terms of TMDs by substituting parameterization of gluon correlator, the leptonic tensor and Eq.(A.65)-(A.69) in Eq.( 8).Using Eq.( 9)-( 13) and after integrating with respect to x and z, one obtains The azimuthal angle of the initial gluon transverse momentum is denoted with φ.For obtaining Eq.( 14), φ = φ h is understood where φ h is the azimuthal angle of the J/ψ.In Eq.( 14), only the unpolarized gluon contribution is taken into consideration.The effect of linearly polarized gluon contribution will be discussed in the Sec.IV.We define A 0 and A 1 as with N = 2(4π) 2 α s αe 2 c .A 1 does not contribute to the Sivers asymmetry.The numerical values of the different states LDME are taken from Ref. [46], Set-I in Table-I.Following Ref. [65], the numerator term of the Sivers asymmetry is given below when the target proton is transversely polarized The gluon Sivers function as per Trento convention is given by [66] ∆ The scale dependency in the definition of TMD is suppressed in this section.The denominator term is given by where the GSF ∆ N f describes the probability of finding an unpolarized gluon inside a transversely polarized proton which is defined as

III. EVOLUTION OF TMDS
In this section the evolution of TMDs is studied.It is generally assumed that the unpolarized gluon TMDs obey the Gaussian distribution.The Gaussian parameterization of unpolarized TMD is given by Here, x and k ⊥ dependencies of the TMD are factorized.f g/p (x, µ) is the collinear PDF which is measured at the scale µ = M (mass of J/ψ).The collinear PDF obeys the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) scale evolution.We choose a frame where the polarized proton is moving along −z axis with momentum P and is transversely polarized with S = S T (cos φ s , sin φ s , 0).The transverse momentum of the J/ψ is P hT = P hT (cos φ h , sin φ h , 0) where, φ s and φ h are the azimuthal angles which are defined in FIG. 1.The parameterization of GSF is given by [49, 67] here h(k ⊥ ) is defined as follows Therefore, the k ⊥ dependent part of Sivers function can now be written as where we defined The GSF has been extracted first time in pion production at RHIC [68] by D'Alesio et al. [49].
In this analysis [49], the best fit parameter sets are denoted with SIDIS1 and SIDIS2.Recently, M. Anselmino et al. [67] have extracted the quark and anti-quark Sivers function from latest SIDIS data.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 [69] : We call the parameterization (a) and (b) as BV-a and BV-b respectively.The best fit parameters are tabulated in Finally, we are in position to write the final expressions of Eq.(1) within DGLAP evolution formalism.Using Eq.( 21)-( 28), the sin(φ h − φ s ) weighted numerator part of Eq.( 1) is given by and the denominator term as follows Now, we adopt the framework implemented in Ref. [70] to study the TMD evolution.In general, TMDs are defined in impact parameter (b ⊥ )-space as below and the inverse Fourier transformation is Generally, TMDs depend on both renormalization scale (µ) and auxiliary scale (ξ) which is introduced to regularize the light-cone divergences in TMD factorization formalism [6,51].
Taking the scale evolution with respect to µ and ξ the renormalization group (RG) and Collins-Soper (CS) equations are obtained.By solving these equations one obtains the TMD PDF expression which is evolved from the initial scale Here, R pert is the perturbative part.The nonperturbative part of the TMDs is denoted with where the anomalous dimensions are denoted with A an B respectively and these have perturbative expansion that can be written as : Here the anomalous dimension coefficients These coefficients are derived up to 3-loop level in Ref. [72].The nonperturbative part is given by It is known that [51] the derivative of Sivers function, f ⊥ (x, b ⊥ , Q f ), follow the same evolution as that of the unpolarized TMD.The TMD evolution equation of unpolarized gluon TMD PDF is and derivative of gluon Sivers function is The TMD density function at the initial scale, f g 1 (x, b ⊥ , Q i ), can be written as the convolution of coefficient function times the regular collinear PDF [51] where C i/g is the perturbatively calculated coefficient function which is process independent.
C i/g is different for each type of TMD PDF.The collinear PDF is probed at the scale c/b * rather than the scale µ in contrast to the DGLAP evolution.The unpolarized and Sivers function TMDs in terms of collinear PDF at leading order in α s are given by [51, 70] where where N g (x) definition is given in Eq. (24).The numerical values of the free parameters are estimated [70] by global fit of SSA in SIDIS process from pion, kaons and charged hadrons production at Jlab, HERMES and COMPASS, which are tabulated in TABLE I.However, only the u and d quark's free parameters are extracted and gluon parameters are not known yet.To estimate SSA we use two parameterizations as given in Eq.( 28).We call the parameterization and the unpolarized gluon TMD is given by Using above expressions, the Eq.( 17), including the weight factor sin(φ h − φ s ) and ( 19) in TMD evolution framework can be written as follows IV. cos 2φ AZIMUTHAL ASYMMETRY Now, let's consider the unpolarized process i.e., e(l)+p(P ) → e(l )+J/ψ(P h )+X.Taking into account the linearly polarized gluons along with the unpolarized gluons in the gluon correlator, the Eq.( 14) can be written as The definitions of A 0 and A 1 are given in Eq.( 15) and Eq.( 16) respectively.The B 0 and B 1 are defined as below The dependence of the cross section on azimuthal angle vanishes when intrinsic parton transverse momentum k ⊥ = 0.The cos 2φ asymmetry is defined as [58,59] To estimate the cos 2φ asymmetry, we need the parameterization of TMDs.For unpolarized TMD, we follow the Gaussian parameterization as defined in Eq.( 21).The widely used Gaussian parameterization for linearly polarized gluon distribution function is given by [73] where, r (0 < r < 1) is the parameter.The upper bound on We consider k 2 ⊥ = 0.25 GeV 2 [73] and r = 1 3 and 2 3 [73] for numerical estimation.

V. NUMERICAL RESULTS
We have estimated the Sivers and cos 2φ asymmetries respectively in polarized and unpolarized SIDIS processes using TMD factorization formalism at √ s = 4.7 GeV (JLab), √ s = 7.2 GeV (HERMES), √ s = 17.33 GeV (COMPASS) and √ s = 45.0GeV (EIC).In this work, NRQCD color octet model (COM) is used for J/ψ production.The color octet states 1 S 0 , 3 P 0 , 3 P 1 and 3 P 2 are taken into account for the LO subprocess γ * g → cc of charmonium production.M = 3.096 GeV and m c = 1.4 GeV are considered for J/ψ and charm quark mass respectively.
Recently extracted gluon Sivers function [49] from RHIC data and quark's Sivers function [67] from latest SIDIS data have been employed in DGLAP evolution approach.The SSA as a function of P hT is negative, and is decreasing as the center of mass energy of the experiment increasing, which is maximum around 30% at JLab energy.Moreover, Sivers asymmetry as a function of Bjorken variable (x B ) is negative and is maximum for SIDIS1 GSF parameters.
Echevarria et al. [70], have extracted u and d quark's Sivers function by fitting data from JLab, HERMES and COMPASS within TMD evolution formalism.We use best fit parameters of these for gluon Sivers function as defined in Eq.( 28) in CSS TMD evolution approach.
Sivers asymmetry with respect to P hT obtained from SIDIS1 parameters is more at JLab and HERMES whereas SSA obtained from BV-b set parameters is dominant at COMPASS and EIC experiments.Basically, SSA is proportional to gluon Sivers function which is considered as an average of u and d quark's x-dependent normalization N (x) in TMD-a parameterization.The sign of the asymmetry depends on relative magnitude of N u and N d and these have opposite sign which can be observed in TABLE I.Note that our kinematics is different from previous works in [39,40,41], which also affects the sign.The magnitude of N u (x) is comparable but slightly dominant compared to N d (x) at EIC √ s.Therefore, the estimated Sivers asymmetry as a function of P hT using TMD-a parameters for EIC experiment is almost zero and positive.For JLab experiment, the estimated Sivers asymmetry by all the parameterizations except SIDIS1 is almost close to zero.
The delta function in Eq.( 12) implies that z = 1 (LO).In FIG. 6, the obtained Sivers asym-metry at z = 1 is compared with COMPASS data [77].Interestingly, all the set of parameters give negative asymmetry.However, estimated SSA with BV-b set of parameters is within the error bar of the experiment.In Ref. [76], negative gluon Sivers asymmetry with more than two standard deviation, A Siv P GF = −0.23 ± 0.08, is reported in SIDIS process based on Monte carlo simulation analysis.As stated before, it is expected that the Sivers function has different sign in DY and SIDIS process, which comes from the gauge link.Sivers function in SIDIS has been extracted by COMPASS [76,77], HERMES [26] and JLab [28] collaboration.However, information about the DY Sivers function has not been explored, since polarized DY process has not been measured ever.Only very recently, data is available in DY process pp ↑ → W ± /Z + X [78].Anselmino et al. [67] have first time attempted to study the nonuniversality signature i.e., sign change of Sivers function, however, they could not draw a definite conclusion about it due to poor data, although data for W − production seem to favor the sign change.
The cos 2φ asymmetry is shown in FIG.7-10 as a function of x B and P hT for r = 1/3 and r = 2/3.To obtain cos 2φ asymmetry, the Gaussian parameterizations for unpolarized and linearly polarized gluon distribution functions are used, as defined in Eq.( 21) and (51).
Until now, experimental investigation has not been done to extract the unknown Boer-Mulders function, h ⊥g 1 .In Ref. [45,46], the effect of h ⊥g 1 on the unpolarized differential cross section of J/ψ production in pp collision is explored.The J/ψ production in unpolarized ep collision process is also a reliable channel to probe the h ⊥g 1 by measuring cos 2φ asymmetry.It is obvious from Eq.( 48) that the negative cos 2φ asymmetry as function of x B and P hT is obtained due to the dominant contribution of 1 S 0 state compared to the other states ( 3 P 0 , 3 P 1 and 3 P 2 ).cos 2φ asymmetry as a function of P hT is almost same for all the experiments, however, maximum value of < cos 2φ > decreases with √ s.The maximum of 26% cos 2φ asymmetry as a function of x B is observed at EIC experiment.The integration ranges are 0 < P hT < 0.64 GeV, 0.7 < y < 0.9 and 0.0001 < x B < 0.35.

VI. CONCLUSION
We have calculated the Sivers and cos 2φ asymmetries in the production J/ψ in polarized and unpolarized ep collision respectively.J/ψ production process gives direct access to the gluon Sivers function at leading order through the channel γ * g → cc.We used the NRQCD based color octet model and a formalism based on TMD factorization.Sizable negative Sivers asymmetry is observed in J/ψ production.The estimated SSA at z = 1 is compared with COMPASS data and is in considerable agreement.We investigated the effect of TMD evolution on the Sivers asymmetry.Moreover, Sizable cos 2φ asymmetry is obtained in unpolarized SIDIS process which allows to probe the Boer-Mulders function, h ⊥g 1 .Thus the asymmetries in the polarized and unpolarized SIDIS processes are important observables to give valuable information on the gluon Sivers function and linearly polarized gluon TMD respectively.Further work would involve taking into account higher order corrections to the asymmetry, where effect of the charmonium production mechanism is likely to play an important role.

ACKNOWLEDGMENT
We would like to thank Mauro Anselmino for fruitful discussion during his stay at IIT Bombay.Cristian Pisano is thanked for useful discussion.
As per Ref. [73,79], the amplitude of the quarkonium bound state can be written as bellow (1,8a) ] = where k is the relative momentum of the heavy quark in the quarkonium rest frame.The eigenfunction of the orbital angular momentum L is Ψ LLz (k ).We follow the similar calculation as reported in [73], hence only the important steps are presented below and for more details Ref. [73] is preferred.From FIG. 11, the amplitude of heavy quark pair is given by O µν (q, k, P h , k ) = ij 3i; 3j|8a g s (ee c ) γ ν / P h /2 + / k − / q + m c (P h /2 + k − q) 2 − m 2
The initial scale of the TMDs is Q i = c/b * (b ⊥ ), where c = 2e −γ with γ ≈ 0.577.The widely used b * prescription is adopted to avoid hitting the Landau pole by freezing the scale b ⊥ .Here, b * (b ⊥ ) = b ⊥ 1+ b ⊥ bmax 2 ≈ b max when b ⊥ → ∞ and b * (b ⊥ ) ≈ b ⊥ when b ⊥ → 0. The perturbative evolution kernel is given by

TABLE I .
Best fit parameters of Sivers function.
[70]se the nonuniversality property of Sivers function for only SIDIS1 and SIDIS2 parameters since these parameters are extracted in DY process[70]