Gluon Sivers Function and Single Spin Asymmetry in e+ p ↑ →e+J/ψ+X
Rohini M. Godbole, Anuradha Misra, Asmita Mukherjee, Vaibhav S. Rawoot
aa r X i v : . [ h e p - ph ] O c t Gluon Sivers Function and Single Spin Asymmetryin e + p ↑ → e + J / y + X Rohini M. Godbole
Centre for High Energy Physics, Indian Institute of Science, Bangalore, India.E-mail: [email protected]
Anuradha Misra
Department of Physics, University of Mumbai, Mumbai, India.E-mail: [email protected]
Asmita Mukherjee
Department of Physics, Indian Institute of Technology Bombay, Mumbai, India.E-mail: [email protected]
Vaibhav S. Rawoot ∗ Department of Physics, University of Mumbai, Mumbai, India.E-mail: [email protected]
We propose measurement of transverse single spin asymmetry (SSA) in charmonium productionas a probe of gluon Sivers function. We estimate SSA in low virtuality electroproduction of J / y using color evaporation model of charmonium production and existing models of the gluon Siversfunction and find sizable asymmetry at JLab, HERMES, COMPASS and eRHIC energies. Sixth International Conference on Quarks and Nuclear PhysicsApril 16-20, 2012Ecole Polytechnique, Palaiseau, Paris ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/ ingle spin asymmetry in e + p ↑ → e + J / y + X Vaibhav S. Rawoot
1. INTRODUCTION
Transverse Single Spin Asymmetry (SSA) arises in the scattering of a transversely polarizedproton off an unpolarised hadron or nucleon if the scattering cross section depends on the directionof polarization. The Single Spin Asymmetry for inclusive process A ↑ + B → C + X is defined as A N = d s ↑ − d s ↓ d s ↑ + d s ↓ (1.1)where d s ↑ ( ↓ ) denotes the cross section for scattering of a transversely polarized hadron A offan unpolarized hadron B, with A upwards (downwards) transversely polarized w.r.t. the produc-tion plane. SSA’s significantly different from zero have been observed over last 35 years startingwith pion production in scattering of polarized protons off unpolarised proton target [1]. LargeSSA’s have been measured in pion production at Fermilab [2] as well as at BNL-RHIC in pp ↑ collisions [3]. SSA’s have also been observed by the HERMES [4] and COMPASS [5] collabora-tions, in polarized semi-inclusive deep inelastic scattering (SIDIS). The magnitude of the observedasymmetries has been found to be larger than what is predicted by perturbative quantum chromo-dynamics (pQCD) [6].Theoretically there are two major approaches to explain the SSA’s. One is the twist threeapproach and other is the transverse momentum dependent (TMD) approach which we have usedin the present work. The TMD approach is based on a pQCD factorization scheme in which spinand intrinsic transverse momentum effects are included in parton distribution functions (pdf’s) andfragmentation functions (ff’s). One of the difficulties in getting information about the spin andtransverse momentum dependent pdf’s and ff’s is that very often two or more of these functionscontribute to the same physical observable making it difficult to estimate each single one separately.The study of spin asymmetries requires extension of TMD factorization scheme to polarizedcase. Sivers in early 90’s proposed that there exists a correlation between the azimuthal distributionof an unpolarized parton and spin of its parent hadron [7]. Number density of partons inside protonwith transverse polarization S, three momentum p and intrinsic transverse momentum k ⊥ of partonsis expressed in terms of Sivers function D N f a / p ↑ ( x , k ⊥ ) ˆ f a / p ↑ ( x , k ⊥ , S ) = ˆ f a / p ( x , k ⊥ ) + D N f a / p ↑ ( x , k ⊥ ) S · ( ˆ p × ˆ k ⊥ ) (1.2) S · ( ˆ p × ˆ k ⊥ ) gives the correlation between the spin of the proton and intrinsic transverse momentumof the unpolarised quarks and gluons. There have been studies on the quark Sivers function inSIDIS and the gluon Sivers function in the process p ↑ p → DX [8, 9]. In this work, we proposecharmonium electroduction as another probe of the gluon Sivers function.
2. FORMALISM FOR ASYMMETRY IN J / y PRODUCTION
We have estimated SSA in photoproduction of charmonium in the process e + p ↑ → e + J / y + X . At leading order (LO), there is contribution only from a subprocess g g → c ¯ c . In addition, sincewe are using color evaporation model (CEM) for charmonium production, only one pdf is involved.Thus the process under consideration can be used as a clean probe of the gluon Sivers function.2 ingle spin asymmetry in e + p ↑ → e + J / y + X Vaibhav S. Rawoot
Also since the charmonium production mechanism can have implications for this SSA, its studycan help throw some light on the production mechanism of charmonium as well.Charmonium production process can be understood in terms of two distinct steps- productionof a c ¯ c pair (a short distance process) and a subsequent binding of this pair in to charmonium (along distance process). Various methods to describe this non-perturbative evolution of the c ¯ c pairinto charmonium lead to different models of charmonium production. As a first step in our inves-tigations of SSA in charmonium production, we have used the Color Evaporation model (CEM) ofcharmonium production. According to CEM, the cross section for charmonium production is pro-portional to the rate of production of c ¯ c pair integrated over the mass range 2 m c to 2 m D [10, 11, 12] s = Z m D m c dM c ¯ c d s c ¯ c dM c ¯ c (2.1)where m c is the charm quark mass and 2 m D is the D ¯ D threshold.The cross section for the low virtuality electroduction within CEM is s ep → e + J / y + X = Z m D m c dM c ¯ c Z dy dx f g / e ( y ) f g / p ( x ) d ˆ s g g → c ¯ c dM c ¯ c (2.2)where f g / e ( y ) is the distribution function of the photon in the electron which, in the WeizsakerWilliam approximation [13], is given by f g / e ( y , E ) = ap { + ( − y ) y (cid:18) ln Em − (cid:19) + y (cid:20) ln (cid:18) y − (cid:19) + (cid:21) + ( − y ) y ln (cid:18) − y − y (cid:19) } . (2.3)To calculate SSA in scattering of electrons off a polarized proton target, we assume general-ization of this CEM expression for low virtuality electroproduction of J / y by taking into accountthe transverse momentum dependence of the Weizsacker-Williams (WW) function and the gluondistribution function: s e + p ↑ → e + J / y + X = Z m D m c dM c ¯ c dx g dx g [ d k ⊥ g d k ⊥ g ] f g / p ↑ ( x g , k ⊥ g ) × f g / e ( x g , k ⊥ g ) d ˆ s g g → c ¯ c dM c ¯ c . (2.4)We assume k ⊥ dependence of pdf’s and WW function to be factorized in Gaussian form [8] f ( x , k ⊥ ) = f ( x ) p h k ⊥ i e − k ⊥ / h k ⊥ i (2.5)with h k ⊥ i = . GeV . The expression for the numerator of the asymmetry is d s ↑ dy d q T − d s ↓ dy d q T = Z m D m c dM Z [ dx g dx g d k ⊥ g d k ⊥ g ] D N f g / p ↑ ( x g , k ⊥ g ) × f g / e ( x g , k ⊥ g ) d ( p g + p g − q ) ˆ s g g → c ¯ c ( M ) (2.6)3 ingle spin asymmetry in e + p ↑ → e + J / y + X Vaibhav S. Rawoot where q = p c + p ¯ c , D N f g / p ↑ ( x g , k ⊥ g ) is the gluon Sivers function and M is invariant mass of the c ¯ c pair. The partonic cross section is [14]ˆ s g g → c ¯ c ( M ) = e c paa s M [( + g − g ) ln 1 + √ − g − √ − g − ( + g ) p − g ] (2.7)where g = m c / M .Sivers asymmetry integrated over the azimuthal angle of J / y with a weight factor sin ( f q T − f S ) is defined as A N = R d f q T [ R m D m c [ dM ] R [ d k ⊥ g ] D N f g / p ↑ ( x g , k ⊥ g ) f g / e ( x g , q T − k ⊥ g ) ˆ s ] sin ( f q T − f S ) R d f q T [ R m D m c [ dM ] R [ d k ⊥ g ] f g / P ( x g , k ⊥ g ) f g / e ( x g , q T − k ⊥ g ) ˆ s ] (2.8)where f q T and f S are azimuthal angles of J / y and proton spin respectively and x g , g = M √ s e ± y .
3. MODELS FOR SIVERS FUNCTION
We have used the following parameterization for the gluon Sivers function [8] D N f g / p ↑ ( x , k ⊥ ) = N g ( x ) h ( k ⊥ ) f g / p ( x ) e − k ⊥ / h k ⊥ i p h k ⊥ i cos f k ⊥ . (3.1)There is no information available about the gluon Sivers function from experimental data. Thevalance and sea quark Sivers distribution functions used are the ones extracted from the HERMESand COMPASS experimental data in SIDIS processes [15].The x dependent normalization for u and d quarks is given by, N f ( x ) = N f x a f ( − x ) b f ( a f + b f ) ( a f + b f ) a f a f b f b f (3.2)where a f , b f , N f and M are best fit parameters obtained by fitting SIDIS, HERMES and COM-PASS data [8].For N g ( x ) , we have used two choices [16](a) N g ( x ) = ( N u ( x ) + N d ( x )) / N g ( x ) = N d ( x ) .For h ( k ⊥ ) , we have used following two choices proposed by Anselmino etal [8, 9]: • Model I h ( k ⊥ ) = √ e k ⊥ M e − k ⊥ / M , (3.3) • Model II h ( k ⊥ ) = k ⊥ M k ⊥ + M , (3.4)where M = q h k ⊥ i and M are best fit parameters.4 ingle spin asymmetry in e + p ↑ → e + J / y + X Vaibhav S. Rawoot its A N s i n ( f q T - f s ) yI (a) -JLabI (a) -HERMESI (a) -COMPASSI (a) -eRHIC-1I (a) -eRHIC-2 A N s i n ( f q T - f s ) q T (GeV)I (a) -JLabI(a) -HERMESI(a) -COMPASSI(a) -eRHIC-1I(a) -eRHIC-2 Figure 1: (Color online) The single spin asymmetry A sin ( f qT − f S ) N for the e + p ↑ → e + J / y + X as a functionof y (left panel) and q T (right panel). The plots are for model I with parameterization (a) compared forJLab ( √ s = . √ s = . √ s = .
33 GeV) [dotted blue line], eRHIC-1 ( √ s = . √ s = .
4. NUMERICAL ESTIMATES
We have used the following best fit parameters from the recent HERMES and COMPASSdata [17] N u = . , a u = . , b u = . , N d = − . , a d = . , b d = . , M = . GeV . (4.1)In figure 1 we have shown the comparison of y and q T distribution of estimated SSA at JLab,HERMES, COMPASS and eRHIC for model I and parameterization (a) of the gluon Sivers func-tion. The estimates are obtained using GRV98LO for gluon distribution function and Weizsaker-Williams function for photon distribution. The results for model II and parametrization (b) aregiven in reference[18]. The hard scale involved in the calculation for all experiments is between4 m c and 4 m D as we are using color evaporation model. Hence the scale evolution of TMD’s is notexpected to affect much our estimates for the experiments at higher energies.According to our estimates sizable asymmetry is expected at various experiments coveringdifferent kinematical regions. Hence it is worthwhile to look at SSA’s in charmonium productionboth from the point of view of comparing different models of charmonium production as well ascomparing the different models of gluon Sivers function.
5. ACKNOWLEDGEMENT
V.S.R. would like to thank organizers of Sixth International Conference on Quarks and Nu-clear Physics 2012 (QNP 2012) for partial financial support and Department of Science and Tech-nology, India for travel support under the Grant No. SR/ITS/0041/2012-2013. R.M.G. wishesto acknowledge support from the Department of Science and Technology, India under Grant No.5 ingle spin asymmetry in e + p ↑ → e + J / y + X Vaibhav S. Rawoot
SR/S2/JCB-64/2007. A. Misra and V.S.R. would like to thank Department of Science and Tech-nology, India for financial support under the Grant No. SR/S2/HEP-17/2006 and the Departmentof Atomic Energy-BRNS, India under the Grant No. 2010/37P/47/BRNS.
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