Single Spin Asymmetry in Charmonium Production
We present estimates of Single Spin Asymmetry (SSA) in the electroproduction of taking into account the transverse momentum dependent (TMD) evolution of the gluon Sivers function and using Color Evaporation Model of charmonium production. We estimate SSA for JLab, HERMES, COMPASS and eRHIC energies using recent parameters for the quark Sivers functions which are fitted using an evolution kernel in which the perturbative part is resummed up to next-to-leading logarithms (NLL) accuracy. We find that these SSAs are much smaller as compared to our first estimates obtained using DGLAP evolution but are comparable to our estimates obtained using TMD evolution where we had used approximate analytical solution of the TMD evolution equation for the purpose.
Keywords:Charmonium, SSA, TMD evolution
Transverse Single Spin Asymmetries (SSA’s) arise in the scattering of a transversely polarized nucleon off an unpolarized nucleon (or virtual photon) target when the final observed hadrons have asymmetric distribution in the transverse plane perpendicular to the beam direction depending on the polarization vector of the scattering nucleon. One of the two major theoretical approaches to explain these asymmetries is the Transverse Momentum Dependent (TMD) approach[1; 2] which is based on a pQCD factorization scheme which includes the spin and TMD effects in the collinear factorization scheme. An important Transverse Momentum Dependent Distribution (TMD) is the Sivers function which is related to the density of unpolarized partons in a transversely polarized nucleon.
The number density of partons inside proton with transverse polarization , three momentum and intrinsic transverse momentum of partons, is expressed in terms of the Sivers function, , as
Heavy quark and quarkonium systems are natural probes to study gluon Sivers function as the production and the differential distributions of the produced charmonium are directly dependent on the intrinsic transverse momentum distribtuion of the gluon especially at low transverse momentum. We have proposed study of SSA in production as a possible probe of gluon Sivers function and have estimated SSA in photo production (i.e. low virtuality electro production) of in scattering of electrons off transversely polarized protons, using the color evaporation model (CEM) of charmonium production[4; 5]. Here, we present some preliminary results containing improved estimates of asymmetry taking into account the TMD evolution of the Sivers function up to next-to-leading logarithm (NLL) order. A more detailed analysis can be found in Ref. .
2 Transverse Single Spin Asymmetry in
In the process under consideration, at LO, there is contribution only from a single partonic subprocess and hence it provides a clean probe of gluon Sivers function. The CEM expression for electroproduction of can be generalized by taking into account the transverse momentum dependence of the William Weizsaker (WW) function and gluon distribution function and can be written as
We use the following model for gluon Sivers function proposed by Anselmino et al.
is the -dependent normalization for which we have used . Since there is no information on gluon Sivers function, one parameterizes the gluon Sivers function in terms of quark Sivers function. The -dependent normalization for quarks is 
where and are best fit parameters. The weighted Sivers asymmetry integrated over the azimuthal angle of  is given by
where and are the azimuthal angles of the and proton spin respectively. The weight factor is and .
3 QCD evolution of TMDs
Early phenomenological fits of Sivers function were performed either neglecting QCD evolution or applying DGLAP evolution only to the collinear part of TMD parametrization. Recently a TMD factorization formalism has been derived and implemented by Collins et al. . TMD evolution describes how the form of distribution changes and also how the width changes in momentum space. A strategy to extract Sivers function from SIDIS data taking into account the TMD evolution was proposed by Anselmino et al.. In our earlier work, we estimated SSA in electroproduction of using this strategy . The energy evolution of a general transverse momentum dependent distribution(TMD) is more naturally described in b-space. The TMDPDF in -space evolves according to
where is the perturbative part of the evolution kernel, is the non-perturbative part and . The perturbative part is given by
where . The non-perturbative exponential part contains a Q-dependent factor universal to all TMD’s (), and a factor which gives the gaussian width in -space of the particular TMD ().
The -dependent TMD’s in momentum space are obtained by Fourier transforming . The perturbative evolution kernel , which drives the -evolution of TMD’s, becomes independent of in the limit i.e. . The integration can then be performed analytically and dependent PDF’s can be obtained. In Ref. , we had used this analytical solution of approximated TMD evolution equations given by Anselmino et al.  to estimate the asymmetry.
Here, we present our improved estimates based on exact solution of evolution equations. Echevarria et al. have recently considered solution of TMD evolution equations up to NLL accuracy and have performed a global fit of all the experimental data on the Sivers asymmetry in SIDIS using this formalism. Since the derivative of -space Sivers function satisfies the same evolution equation as the unpolarized PDF, its evolution is given by
Here, is the twist three Qiu-Sterman quark gluon correlation function which is related to the first moment of quark Sivers function and can be expressed in terms of the unpolarized collinear PDFs [14; 11].
The expansion coefficients with the appropriate gluon anomalous dimensions at NLL are
Choosing the initial scale , the term vanishes at NLL. Taking Fourier transform of Eq. (6), one gets which is related to Sivers function through
4 Numerical Estimates
We will now present our estimates of SSA in photoproduction of for JLAB, HERMES, COMPASS and eRHIC energies. A detailed discussion of results can be found in Ref. . Figs. 1-5 show the y and distribution for different experiments with parameterizations TMD Exact-1, TMD Exact -2 and TMD as given in Table 1. TMD-e1 parameter set, extracted at , is for the exact solution of TMD evolution equations extracted in Ref. . TMD-a is the parameter set fitted to analytical approximated solution of the Sivers function extracted in Ref. . For estimates using NLL kernel, we have used the most recent parameters by Echevarria et al. 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 . Fig. 6 shows a comparison of asymmetries at all energies.
We have compared estimates of SSA in electroproduction of 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.
Acknowledgements.A.M. would like to thank the organizers of LC2014 and Department of Physics, NCSU for their warm hospitality and University of Mumbai, University Grants Commission, India and Indian National Science Academy for financial support. This work was done under Grant No 2010/37P/47/BRNS of Department of Atomic Energy, India. R.M.G. wishes to acknowledge support from the Department of Science and Technology, India under Grant No. SR/S2/JCB-64/2007 under the J.C. Bose Fellowship scheme.
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