We assume a connection between the forward limit of the GPD E and the first moment of the Sivers transverse-momentum distribution, inspired by Burkardt’s idea of chromodynamic lensing. Then, we show that it is possible to fit at the same time the values of the nucleon anomalous magnetic moments and the data for semi-inclusive single-spin asymmetries originating from the Sivers effect. This opens a plausible way to quantifying the contribution of the partonic angular momentum to the spin of the nucleon, according to Ji’s definition

Radici, M.

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DOI 10.1393/ncc/i2013-11594-2Colloquia: QCDN12IL NUOVO CIMENTO Vol. 36 C, N. 5 Settembre-Ottobre 2013Partonic angular momentum and the Sivers asymmetryM. RadiciINFN - Sezione di Pavia - Pavia, Italyricevuto il 18 Aprile 2013Summary. — We assume a connection between the forward limit of the GPD Eand the first moment of the Sivers transverse-momentum distribution, inspired byBurkardt’s idea of chromodynamic lensing. Then, we show that it is possible to fitat the same time the values of the nucleon anomalous magnetic moments and thedata for semi-inclusive single-spin asymmetries originating from the Sivers effect.This opens a plausible way to quantifying the contribution of the partonic angularmomentum to the spin of the nucleon, according to Ji’s definition.PACS 14.65.Bt – Light quarks.PACS 14.20.Dh – Protons and neutrons.PACS 13.60.-r – Photon and charged-lepton interactions with hadrons.1. – IntroductionThe total angular momentum of a parton a (with a = q, q¯) at some scale Q2 can becomputed as a specific moment of generalized parton distribution functions (GPD) [1]2Ja(Q2) =∫ 10dxx(Ha(x, 0, 0;Q2) + Ea(x, 0, 0;Q2)).(1)The GPD Ha(x, 0, 0;Q2) corresponds to the familiar collinear parton distribution func-tion (PDF) fa1 (x;Q2). The forward limit of the GPD Ea does not correspond to anycollinear PDF. It is not possible to probe the function Ea in experiments in the for-ward limit. Assumptions are eventually necessary to constrain Ea(x, 0, 0;Q2). The onlymodel-independent constraint is the scale-independent sum rule∑qeqv∫ 10dxEqv (x, 0, 0) = κ,(2)where Eqv = Eq − E q¯ and κ denotes the anomalous magnetic moment of the parentnucleon.c© Societa` Italiana di Fisica 201202 M. RADICIDenoting the Sivers function by f⊥a1T , we propose this simple relation at a scale QL,f⊥(0)a1T (x;Q2L) = −L(x)Ea(x, 0, 0;Q2L),(3)where f⊥(0)a1T is the integral of the Sivers function over transverse momentum. Thisassumption is inspired by theoretical considerations [2] and by results of the spectatormodel [3]. L(x) is a flavor-independent function, representing the effect of the QCDinteraction of the active quark with the rest of the spectators inside the nucleon.The advantage of adopting the Ansatz of eq. (3) is twofold: first, the value of theanomalous magnetic moment fixes the integral of the GPD E and allows us to constrainthe valence Sivers function also outside the region where SIDIS data are available; second,our Ansatz allows us to obtain flavor-decomposed information on the x-dependence ofthe GPD E [4] and, ultimately, on quark and antiquark total angular momentum.2. – Extraction of Sivers function from dataTo analyze SIDIS data, we use the same assumptions adopted in ref. [5]: we neglect theeffect of TMD evolution, which has been studied only recently [6]; we assume a flavor- andscale-independent Gaussian transverse-momentum distribution for all involved TMDs,and we include the effect of the standard DGLAP evolution only in the collinear part ofthe parametrizing functions.Neglecting c, b, t flavors, we parametrize the Sivers function in the following way:f⊥qv1T (x, k2⊥;Q20)=Cqv√2eMM1πM21 〈k2⊥〉(1− x)fqv1 (x;Q20) e−k2⊥/M21 e−k2⊥/〈k2⊥〉 1− x/αqv|αqv − 1| .(4)For q¯, we use a similar function, excluding the last term. We used 〈k2⊥〉 = 0.14GeV2.M1 is a free parameter that determines the transverse-momentum width. We imposedconstraints on the parameters Ca in order to respect the positivity bound for the Siversfunction [7]. We multiply the unpolarized PDF by (1−x) to respect the predicted high-xbehavior of the Sivers function [8]. We introduce the free parameter αqv to allow for thepresence of a node at x = αqv , as suggested in refs. [3, 9-11].For the lensing function we use the Ansatz L(x) = K/(1 − x)η. The choice of thisform is guided by model calculations [3, 9], by the large-x limit of the GPD E [8], andby the phenomenological analysis of the GPD E proposed in ref. [12]. We checked aposteriori that there is no violation of the positivity bound on the GPD Eqv .We performed a combined χ2 fit to the nucleon anomalous magnetic moments (forour present purposes, we take κp = 1.793 ± 0.001, κn = −1.913 ± 0.001) and the Siversasymmetry with identified hadrons from refs. [13-15].We set the gluon Sivers function to zero (its influence through evolution is anywaylimited) and we chose Q0 = QL = 1GeV. We fixed αdv,sv = 0 (no nodes in the valencedown and strange Sivers functions). We explored several scenarios characterized bydifferent choices of the parameters related to the strange quark. In all cases, we obtainedvery good values of χ2 per degree of freedom (χ2/dof), around 1.34.All fits lead to a negative Sivers function for uv and large and positive for dv, inagreement with previous studies (see ref. [5] and references therein). The data are com-patible with vanishing sea-quark contributions (with large uncertainties). However, inthe x range where data exist, large Sivers functions for u¯ and d¯ are excluded, as well aslarge and negative for s¯. The Sivers function for sv is essentially unconstrained. It turnsPARTONIC ANGULAR MOMENTUM AND THE SIVERS ASYMMETRY 203Fig. 1. – The moment xf⊥(0) a1T (x;Q2L) of the Sivers function at the scale Q2L = 1GeV2 fora = uv, dv (top panel) and u¯, d¯ (lower panel). The uncertainty bands are produced by statisticalerrors on the fit parameters.out αuv ≈ 0.78, so there is little room for a node in the up Sivers function, also becauseof the constraint imposed by the anomalous magnetic moments.The best fit, with a χ2/dof 1.32, is reached for vanishing strange contribution Csv =C s¯ = 0. Our results for the Sivers function are shown in fig. 1 and is comparable withother extractions(for example, see ref. [6]). The results for the forward limit of the GPDE also turn out qualitatively similar to available extractions [12,16].3. – Quark and gluon contributions to the nucleon spinUsing eq. (1), we can compute the total longitudinal angular momentum carried byeach flavor q and q¯ at our initial scale Q2L = 1GeV2. Using the standard evolutionequations for the angular momentum (at leading order, with 3 flavors only, and ΛQCD =257MeV), we obtain the following results at Q2 = 4GeV2:Ju = 0.229± 0.002+0.008−0.012, J u¯ = 0.015± 0.003+0.001−0.000,Jd = −0.007± 0.003+0.020−0.005, J d¯ = 0.022± 0.005+0.001−0.000,Js = 0.006+0.002−0.006, Js¯ = 0.006+0.000−0.005.The first symmetric error is statistical and related to the errors on the fit parameters,while the second asymmetric error is theoretical and reflects the uncertainty introducedby other possible scenarios. In the present approach, we cannot include the (probablylarge) systematic error due to the rigidity of the functional form in eq. (4). The bias204 M. RADICIinduced by the choice of the functional form may affect in particular the determinationof the sea quark angular momenta, since they are not directly constrained by the valuesof the nucleon anomalous magnetic moments. Our present estimates (at Q2 = 4GeV2)agree well with other estimates, particularly with those ones based on the extraction ofthe GPD E [12, 16, 17] and on lattice simulations [18, 19]. Our study indicates a totalcontribution to the nucleon spin from quarks and antiquarks of 0.271 ± 0.007+0.032−0.028, ofwhich 85% is carried by the up quark.Our approach can be used also to estimate the size of the total angular momentumcarried by the gluons. In this case, we expect the lensing function to be different fromthat of the quarks. However, our extraction leaves little room for a nonzero gluon Siversfunction, since the quark Sivers function already saturates the so-called Burkardt sumrule [20]. If the Sivers function of the gluons is zero, our reasoning allows us to concludethat Eg is also zero, independent of the details of the lensing function. This would leadto a value of Jg = 0.215 at 4GeV2, which agrees with the result in refs. [16, 17], andseems compatible within errors with our finding of Jq+q¯ = 0.271±0.007+0.032−0.028. However,these considerations are strongly affected by the uncertainties on the sea-quark Siversfunctions outside the x range where data exists. Direct measurements of the sea-quarkand gluon Sivers functions are therefore highly necessary.At this point, we add a remark on the effect of TMD evolution on the Sivers function.The discussion in ref. [6] suggests that the inclusion of TMD-evolution effects might leadto larger values of the Sivers function at the starting scale Q20. If this were the case, wewould need to compensate the effect by a smaller size of the lensing function in orderto have an agreement with the anomalous magnetic moments. However, this will have anegligible net effect on the results for Ja, also because TMD evolution mildly modifiesthe Sivers function for sea quarks [6].REFERENCES[1] Ji X., Phys. Rev. Lett., 78 (1997) 610.[2] Burkardt M., Phys. Rev. D, 66 (2002) 114005.[3] Bacchetta A., Conti F. and Radici M., Phys. Rev. D, 78 (2008) 074010.[4] Bacchetta A. and Radici M., Phys. Rev. Lett., 107 (2011) 212001.[5] Anselmino M. et al., arXiv:1107.4446 [hep-ph].[6] Anselmino M., Boglione M. and Melis S., Phys. Rev. D, 86 (2012) 014028.[7] Bacchetta A., Boglione M., Henneman A. and Mulders P. J., Phys. Rev. Lett., 85(2000) 712.[8] Brodsky S. J. and Yuan F., Phys. Rev. D, 74 (2006) 094018.[9] Bacchetta A., Radici M., Conti F. and Guagnelli M., Eur. Phys. J. A, 45 (2010)373.[10] Kang Z. B. and Prokudin A., Phys. Rev. D, 85 (2012) 074008.[11] Boer D., Phys. Lett. B, 702 (2011) 242.[12] Guidal M., Polyakov M. V., Radyushkin A. V. and Vanderhaeghen M., Phys. Rev.D, 72 (2005) 054013.[13] Airapetian A. et al. (HERMES Collaboration), Phys. Rev. Lett., 103 (2009) 152002.[14] Alekseev M. et al. (COMPASS Collaboration), Phys. Lett. B, 673 (2009) 127.[15] Qian X. et al., Phys. Rev. Lett., 107 (2011) 072003.[16] Goloskokov S. and Kroll P., Eur. Phys. J. C, 59 (2009) 809.[17] Diehl M. and Kroll P., Eur. Phys. J. C, 73 (2013) 2397.[18] Bratt J. D. et al. (LHPC Collaboration), Phys. Rev. D, 82 (2010) 094502.[19] Brommel D. et al. (QCDSF-UKQCD Collaboration), PoS, LAT2007 (2007) 158.[20] Burkardt M., Phys. Rev. D, 69 (2004) 091501.

Partonic angular momentum and the Sivers asymmetry

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Partonic angular momentum and the Sivers asymmetry

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