Transverse Single Spin Asymmetries and Charmonium Production
Rohini M. Godbole, Anuradha Misra, Asmita Mukherjee, Vaibhav S. Rawoot
NNuclear Physics B Proceedings Supplement 00 (2018) 1–5
Nuclear Physics BProceedingsSupplement
Transverse Single Spin Asymmetries and Charmonium Production
Rohini M. Godbole a , Anuradha Misra b, ∗ , Asmita Mukherjee c , Vaibhav S. Rawoot b a Centre for High Energy Physics, Indian Institute of Science, Bangalore, India-560012 b Department of Physics,University of Mumbai,Santa Cruz(E), Mumbai, India-400098 c Department of Physics,Indian Institute of Technology,Bombay, Mumbai, India-400076
Abstract
We estimate transverse spin single spin asymmetry(TSSA) in the process e + p ↑ → J /ψ + X using color evaporationmodel of charmonium production. We take into account transverse momentum dependent(TMD) evolution of Siversfunction and parton distribution function and show that the there is a reduction in the asymmetry as compared to ourearlier estimates wherein the Q - evolution was implemented only through DGLAP evolution of unpolarized gluondensities. Keywords:
Single Spin Asymmetry, Charmonium
1. Introduction
Single spin asymmetries(SSA’s) arise in the scatter-ing of transversely polarized nucleons o ff an unpolar-ized nucleon (or virtual photon) target, when the finalstate hadrons have asymmetric distribution in the trans-verse plane perpendicular to the beam direction. SSAfor inclusive process A ↑ + B → C + X depends on thepolarization vector of the scattering hadron A and is de-fined by A N = d σ ↑ − d σ ↓ d σ ↑ + d σ ↓ (1)Non - zero SSAs have been observed over the years-in pion production at Fermilab[1] and at RHIC[2] in pp ↑ collisions as well as in Semi-inclusive deep inelas-tic scattering (SIDIS) experiments at HERMES[3] andCOMPASS[4]. These results have generated a lot ofinterest amongst theoreticians to investigate the mecha-nism involved and to understand the underlying physics. ∗ Speaker of the talk
Email addresses: [email protected] (Rohini M.Godbole), [email protected] (Anuradha Misra), [email protected] (Asmita Mukherjee), [email protected] (Vaibhav S. Rawoot)
The initial attempts to provide theoretical predic-tions of asymmetry, based on collinear factorization ofpQCD, led to estimates which were too small as com-pared to the experimental results[5]. In collinear fac-torization formalism, the parton distribution functions(PDF’s) and fragmentation functions (FF’s) are inte-grated over intrinsic transverse momentum of the par-tons and hence depend only on longitudinal momentumfraction x. The observation that SSAs calculated withincollinear formalism were almost vanishing suggestedthat these asymmetries may be due to parton’s trans-verse motion and spin orbit correlation. A generaliza-tion of factorization theorem, in the form of transversemomentum dependent (TMD) factorization which in-cludes the transverse momentum dependence of PDF’sand FF’s, was proposed as a possible approach to ac-count for the asymmetries[6].One of the TMD PDF’s of interest is Sivers function,which gives the probability of finding an unpolarizedquark inside a transversely polarized proton. The Siversfunction, ∆ N f a / p ↑ ( x , k ⊥ a ), defined by ∆ N f a / p ↑ ( x , k ⊥ a ) ≡ ˆ f a / p ↑ ( x , k ⊥ a ) − ˆ f a / ↓ ( x , k ⊥ a ) = ˆ f a / ↑ ( x , k ⊥ a ) − ˆ f a / ↑ ( x , − k ⊥ a ) (2) a r X i v : . [ h e p - ph ] O c t Nuclear Physics B Proceedings Supplement 00 (2018) 1–5 is related to the number density of partons inside a pro-ton with transverse polarization S, three momentum p and intrinsic transverse momentum k ⊥ of partons, andits spin dependence is given by ∆ N f a / p ↑ ( x , k ⊥ a ) = ∆ N f a / p ↑ ( x , k ⊥ ) S · ( ˆp × ˆk ⊥ ) (3)Parametrizations of quark Sivers distributionshave been obtained from fits of SSA in SIDISexperiments[7]. However, not much information isavailable on gluon Sivers function. Processes that havebeen studied with the aim of getting information aboutthis TMD are back to back correlations in azimuthalangles of of jet produced in pp ↑ scattering[8] and Dmeson production in pp ↑ scattering[9]. Heavy quarkand quarkonium systems have also been proposed asnatural probes to study gluon Sivers function due to thefact that the production is sensitive to intrinsic trans-verse momentum especially at low momentum[10]. Ithas been suggested, in the context of deep inelasticscattering[11] that the initial and final state interactionsmay lead to non-vanishing SSAs. Single transversespin asymmetry in heavy quarkonium production inlepton-nucleon and nucleon-nucleon collisions hasbeen investigated by Yuan etal taking into account theinitial and final state interactions [10] and it has beenshown that the asymmetry is very sensitive to the pro-duction mechanism. The three main models of heavyquarkonium production, which have been proposedand tested in unpolarized scattering, are Color SingletModel[12], Color Evaporation Model (CEM)[13, 14]and the NRQCD factorization approach[15]. It wasargued in Ref.[10] that the asymmetry should benon-zero in ep collisions only in color-octet modeland in pp collisions only in color-singlet model. Thus,SSA in charmonium production can be used to throwsome light on the issue of production mechanism. Inthis work, we present estimates of SSA in the process e + p ↑ → J /ψ + X and compare the results obtainedusing TMD evolution of PDF’s with our earlier resultswhich were obtained using DGLAP evolution only.
2. Transverse Single Spin Asymmetry in e + p ↑ → J /ψ + X The first estimate of SSA in photoproduction (i.e. lowvirtuality electroproduction) of J /ψ in the scattering ofelectrons o ff transversely polarized protons were pro-vided by us in Ref.[16] using Color Evaporation Model.In the process under consideration, at LO, there is con-tribution only from a single partonic subprocess andtherefore, it can be used as a clean probe of gluon Siversfunction. Color Evaporation Model (CEM) was introduced in1977 by Fritsch and was revived in 1996 by Halzen[13].This model gives a good description of photopro-duction data after inclusion of higher order QCDcorrections[17] and also of the hadroproduction CDFdata [18] after inclusion of k T smearing. In CEM, thecross-section for a quarkonium state H is some fraction F H of the cross-section for producing Q ¯ Q pair with in-variant mass below the M ¯ M threshold, where M is thelowest mass meson containing the heavy quark Q: σ CEM [ h A h B → H + X ] = F H (cid:88) i , j (cid:90) m M m d ˆ s × (cid:90) dx dx f i ( x , µ ) f j ( x , µ ) ˆ σ i j ( ˆ s ) δ ( ˆ s − x x s ) (4)We have used a generalization of CEM expressionfor electroproduction of J /ψ by taking into account thetransverse momentum dependence of the gluon distribu-tion function and the William Weizsacker (WW) func-tion which gives the photon distribution of the electronin equivalent photon approximation[19]. The cross sec-tion for the process e + p ↑ → J /ψ + X is then givenby σ e + p ↑ → e + J /ψ + X = (cid:90) m D m c dM c ¯ c dx γ dx g [ d k ⊥ γ d k ⊥ g ] × f g / p ↑ ( x g , k ⊥ g ) f γ/ e ( x γ , k ⊥ γ ) d ˆ σ γ g → c ¯ c dM c ¯ c (5)where f γ/ e ( x γ , k ⊥ γ ) is the distribution function of thephoton in the electron. We assume a gaussian form forthe k ⊥ dependence of pdf’s [7], f ( x , k ⊥ ) = f ( x ) 1 π (cid:104) k ⊥ (cid:105) e − k ⊥ / (cid:104) k ⊥ (cid:105) (6)where (cid:104) k ⊥ (cid:105) = . GeV . f γ/ e ( x γ , k ⊥ γ ) is also assumedto have a similar k ⊥ dependence and is given by f γ/ e ( x γ , k ⊥ γ ) = f γ/ e ( x γ ) 1 π (cid:104) k ⊥ γ (cid:105) e − k ⊥ γ / (cid:104) k ⊥ γ (cid:105) . (7)where f γ/ e ( x γ ) is the William Weizsacker function givenby [20]: f γ/ e ( y , E ) = απ + (1 − y ) y (cid:32) ln Em − (cid:33) + y (cid:34) ln (cid:32) y − (cid:33) + (cid:35) + (2 − y ) y ln (cid:32) − y − y (cid:33) (8)y being the energy fraction of the electron carried by thephoton. Nuclear Physics B Proceedings Supplement 00 (2018) 1–5 Using Eq. 3, the expression for the numerator of theasymmetry reduces to [16] d σ ↑ dyd q T − d σ ↓ dyd q T = (cid:90) m D m c [ dM ] (cid:90) [ dx γ dx g ] × (cid:90) [ d k ⊥ γ d k ⊥ g ] ∆ N f g / p ↑ ( x g , k ⊥ g ) × f γ/ e ( x γ , k ⊥ γ ) δ ( p g + p γ − q ) ˆ σ γ g → c ¯ c ( M ) (9)where q = p c + p ¯ c and ˆ σ γ g → c ¯ c ( M ) is the partonic crosssection[21]:ˆ σ γ g → c ¯ c ( M ) = e c παα s M (cid:20) (1 + γ − γ ) ln 1 + (cid:112) − γ − (cid:112) − γ − (1 + γ ) (cid:112) − γ (cid:21) (10) γ = m c / M and q T and k ⊥ are the transverse momentaof the gluon and J /ψ respectively with azimuthal angles φ q and φ k ⊥ : q T = q T (cos φ q , sin φ q , ) k ⊥ = k ⊥ (cos φ k ⊥ , sin φ k ⊥ , ) (11)The mixed product S · ( ˆ p × ˆ k ⊥ ) in ∆ N f g / p ↑ ( x g , k ⊥ g ) givesan azimuthal dependence of the form, S · ( ˆ p × ˆ k ⊥ ) = cos φ k ⊥ (12)Taking sin ( φ q − φ S ) as a weight[22], the asymmetry in-tegrated over the azimuthal angle of J /ψ is given by A sin ( φ q − φ S ) N = (cid:82) d φ q [ d σ ↑ − d σ ↓ ] sin ( φ q − φ S ) (cid:82) d φ q [ d σ ↑ + d σ ↓ ] (13)which finally leads to A N = (cid:82) d φ q [ (cid:82) m D m c [ dM ] (cid:82) [ d k ⊥ g ] ∆ N f g / p ↑ ( x g , k ⊥ g ) f γ/ e ( x γ , q T − k ⊥ g ) ˆ σ ] sin ( φ q − φ S )2 (cid:82) d φ q [ (cid:82) m D m c [ dM ] (cid:82) [ d k ⊥ g ] f g / P ( x g , k ⊥ g ) f γ/ e ( x γ , q T − k ⊥ g ) ˆ σ ] (14)where d σ = d σ dy d q T , x g ,γ = M √ s e ± y
3. Models for Sivers function
In our analysis, we have used the following parame-terization for the gluon Sivers function[7] ∆ N f g / p ↑ ( x , k ⊥ ) = N g ( x ) h ( k ⊥ ) f g / p ( x ) × e − k ⊥ / (cid:104) k ⊥ (cid:105) π (cid:104) k ⊥ (cid:105) cos φ k ⊥ (15) where N g ( x ) is an x dependent normalization. We haveused two di ff erent models for the functional forms of h ( k ⊥ ): In Model(1)[23] h ( k ⊥ ) = √ e k ⊥ M e − k ⊥ / M (16)whereas in Model(2)[9] h ( k ⊥ ) = k ⊥ M k ⊥ + M (17)where M = (cid:113) (cid:104) k ⊥ (cid:105) and M are best fit parameters.Here, we will present the results for Model I only. Theresults for Model II and a comparison of the two modelscan be found in Ref.[16]. For N g ( x ) also, we have usedtwo kinds of parametrizations [8](a) N g ( x ) = ( N u ( x ) + N d ( x )) / N g ( x ) = N d ( x ) ,where N u ( x ) and N d ( x ) are the normalizations for uand d quarks given by[8] N f ( x ) = N f x a f (1 − x ) b f ( a f + b f ) ( a f + b f ) a f a f b f b f (18)Here, a f , b f and N f are best fit parameters fitted fromnew HERMES and COMPASS data[24] fitted at (cid:104) Q (cid:105) = . GeV as given below: N u = . , a u = . , b u = . N d = − . , a d = . , b d = . M = . . We have estimated SSA using both Model I andII and parameterizations (a) and (b). The detailedresults can be found in Ref. [16].
4. TMD Evolution of PDF’s and Sivers Function
Early phenomenological fits of Sivers function wereperformed using experimental data at fixed scales andestimates of asymmetry were also performed either ne-glecting QCD evolution of TMD PDF’s or by apply-ing DGLAP evolution only to the collinear part of TMDparametrization. In our earlier estimates of asymmetry
Nuclear Physics B Proceedings Supplement 00 (2018) 1–5 in Ref.[16] also, we have assumed the Q -dependenceof PDF’s and the Sivers function to be of the form, f g / p ( x , k ⊥ ; Q ) = f g / p ( x ; Q ) 1 π (cid:104) k ⊥ (cid:105) e − k ⊥ / (cid:104) k ⊥ (cid:105) (19)and ∆ N f g / p ↑ ( x , k ⊥ ; Q ) = N g ( x ) f g / p ( x ; Q ) × √ e k ⊥ M π (cid:104) k ⊥ (cid:105) e − k ⊥ / (cid:104) k ⊥ (cid:105) S (20)where (cid:104) k ⊥ (cid:105) = . GeV . Note that the Q depen-dence of PDF comes from collinear PDF f g / p ( x ; Q )only which have been evolved using DGLAP evolution.More recently, energy evolution of TMD’s has beenstudied by various authors and a TMD evolution for-malism has been developed and implemented [25, 26].TMD evolution is more complicated as compared tocollinear counterpart because unlike collinear distribu-tions TMDs have rapidity divergences in addition tocollinear singularities. Thus TMD evolution describeshow the form of distribution changes and also how thewidth changes in momentum space. A strategy to ex-tract Sivers function from SIDIS data taking into ac-count the TMD Q evolution has been proposed [27].We have estimated SSA in electroproduction of J /ψ tak-ing into account this strategy. In this formalism, the Q dependence of PDF’s is given by f q / p ( x , k ⊥ ; Q ) = f q / p ( x , Q ) R ( Q , Q ) e − k ⊥ / w π w , (21)where, f q / p ( x , Q ) is the usual integrated PDF evaluatedat the initial scale Q and w ≡ w ( Q , Q ) is the “evolv-ing” Gaussian width, defined as w ( Q , Q ) = (cid:104) k ⊥ (cid:105) + g ln QQ · . (22) R ( Q , Q ) is the limiting value of a function R ( Q , Q , b T )that drives the Q -evolution of TMD’s in coordinatespace and is driven by R ( Q , Q , b T ) ≡ exp { ln QQ (cid:90) µ b Q d µ (cid:48) µ (cid:48) γ K ( µ (cid:48) ) + (cid:90) QQ d µµ γ F (cid:32) µ, Q µ (cid:33) } · (23)where b T is the parton impact parameter, b ∗ ( b T ) ≡ b T (cid:113) + b T / b , µ b = C b ∗ ( b T ) (24)with C = e − γ E where γ E = . b ∗ → b max . γ F and γ K are anomalous dimensions which are givenat O ( α s ) by γ F ( µ ; Q µ ) = α s ( µ ) C F π (cid:32) − ln Q µ (cid:33) (25) γ K ( µ ) = α s ( µ ) 2 C F π · (26)In the limit b T → ∞ , R ( Q , Q , b T ) → R ( Q , Q ).
5. Numerical Estimates
We have estimated SSA in electroproduction of J /ψ for JLab, HERMES, COMPASS and eRHIC energies.Our earlier calculation of asymmetry[16] had taken intoaccount energy evolution of PDF’s and Sivers functionusing DGLAP evolution. The details can be found inRef.[16].In Figs.1-5, we have presented a comparison of SSA’scalculated using DGLAP evolution and TMD evolutionof TMD PDF’s at various energies for Model I withparametrization (a). For TMD evolved Sivers function,we have used the parameter set fitted at Q = N u = . , N d = − . , b = . , a u = . , a d = . , M = . GeV , g = . . (27)It is found that the asymmetry is substantially reducedin all cases when TMD evolution of PDF’s and Siversfunction is taken into account. Here, we have usedparametrization (a) for our estimates. A more detailedanalysis with parametrization (b) and a comparison ofthe two parametrizations as well as of various parame-ter sets can be found in Ref[28].
6. Summary
Transverse SSA in electroprduction of J /ψ has beencalculated using color evaporation model of charmo-nium production. A TMD factorization formalismhas been used first with DGLAP evolved PDF’s andthen with TMD evolved PDF’s and Sivers function.Sizable asymmetry is predicted at energies of JLab,HERMES, COMPASS and eRHIC experiments in bothcases. However, it is found that there is a substantialreduction in asymmetry when TMD evolution is takeninto account. Substantial magnitude of asymmetryindicate that it may be worthwhile to look at SSA’s incharmonium production both from the point of view ofcomparing di ff erent models of charmonium production Nuclear Physics B Proceedings Supplement 00 (2018) 1–5 as well as comparing the di ff erent models of gluonSivers function. It is also clear that TMD evolutione ff ects are substantial and one must take them intoaccount for accurate predictions. ACKNOWLEDGEMENTS
I would like to thank the organizers of LC2012 Delhi fortheir kind hospitality. I would also like to thank DAEBRNS, India for financial support during this projectunder the grant No. 2010 / / / BRNS.
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D 72 , 054028(2005); arXiv:hep-ph / , 054010(2009) [arXiv:0901.3078 [hep-ph]].[24] M. Anselmino, M. Boglione, U. D’Alesio, S. Melis,F. Murgia and A. Prokudin, [hep-ph / ,114042 (2011) [arXiv:1101.5057 [hep-ph]], S. M. Ay-bat, J. C. Collins, J. -W. Qiu and T. C. Rogers,arXiv:1110.6428 [hep-ph], S. M. Aybat, A. Prokudin andT. C. Rogers, arXiv:1112.4423 [hep-ph].[27] M. Anselmino, M. Boglione and S. Melis, Phys. Rev. D , 014028 (2012) [arXiv:1204.1239 [hep-ph]].[28] R. M. Godbole, A. Misra, A. Mukherjee and V. S. Ra-woot, Phys. Rev. D , 014029 (2013) [arXiv:1304.2584[hep-ph]]. A N s i n ( φ q T - φ s ) yJLab, √ s =4.7 GeV TMDDGLAP 0 0.02 0.04 0.06 0.08 0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 A N s i n ( φ q T - φ s ) q T JLab, √ s =4.7 GeV TMDDGLAP
Figure 1: The Sivers asymmetry A sin( φ qT − φ S ) N for e + p ↑ → e + J /ψ + X at JLab energy ( √ s = . q T (bottom panel) for parametrization (a). Nuclear Physics B Proceedings Supplement 00 (2018) 1–5 A N s i n ( φ q T - φ s ) yHEMRES, √ s =7.2 GeV TMDDGLAP 0 0.02 0.04 0.06 0.08 0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 A N s i n ( φ q T - φ s ) q T HEMRES, √ s =7.2 GeV TMDDGLAP
Figure 2: The Sivers asymmetry A sin( φ qT − φ S ) N for e + p ↑ → e + J /ψ + X at HEMRES energy ( √ s = . q T (bottom panel) for parametrization (a). A N s i n ( φ q T - φ s ) yCOMPASS, √ s =17.33 GeV TMDDGLAP 0 0.02 0.04 0.06 0.08 0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 A N s i n ( φ q T - φ s ) q T COMPASS, √ s =17.33 GeV TMDDGLAP
Figure 3: The Sivers asymmetry A sin( φ qT − φ S ) N for e + p ↑ → e + J /ψ + X at COMPASS energy ( √ s = .
33 GeV) as a function of y (top panel)and q T (bottom panel) for parametrization (a). A N s i n ( φ q T - φ s ) yeRHIC-1, √ s =31.6 GeV TMDDGLAP 0 0.02 0.04 0.06 0.08 0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 A N s i n ( φ q T - φ s ) q T eRHIC-1, √ s =31.6 GeV TMDDGLAP
Figure 4: The Sivers asymmetry A sin( φ qT − φ S ) N for e + p ↑ → e + J /ψ + X at eRHIC-1 energy ( √ s = . q T (bottom panel) for parametrization (a). A N s i n ( φ q T - φ s ) yeRHIC-2, √ s =158.1 GeV TMDDGLAP 0 0.02 0.04 0.06 0.08 0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 A N s i n ( φ q T - φ s ) q T eRHIC-2, √ s =158.1 GeV TMDDGLAP