We evaluate the azimuthal asymmetries for the distributions of leading pions
inside a jet, produced in high-energy proton-proton collisions, in kinematic
configurations currently under active investigation at RHIC. Adopting a
transverse momentum dependent approach, which assumes the validity of a
perturbative QCD factorization scheme and takes into account all the spin and
intrinsic parton motion effects, we show how the main mechanisms underlying
these asymmetries, namely the Sivers and the Collins effects, can be
disentangled. Furthermore, we consider the impact of color-gauge invariant
initial and final state interactions and suggest a method for testing the
universality properties of the Sivers function for quarks.Comment: 4 pages, 3 figures, to appear in the proceedings of the 3rd Workshop
  on the QCD structure of the nucleon (QCD-N'12), Bilbao (Spain), 22-26 October
  201

D'Alesio, Umberto

Murgia, Francesco

Pisano, Cristian

English

arXiv

We evaluate the azimuthal asymmetries for the distributions of leading pions inside a jet, produced in high-energy proton-proton collisions, in kinematic configurations currently under active investigation at RHIC. Adopting a transverse momentum dependent approach, which assumes the validity of a perturbative QCD factorization scheme and takes into account all the spin and intrinsic parton motion

effects, we show how the main mechanisms underlying these asymmetries, namely the Sivers and the Collins effects, can be disentangled. Furthermore, we consider the impact of color-gauge invariant initial and final state interactions and suggest a method for testing the universality properties of the Sivers function for quarks

D'Alesio, U.

Murgia, F.

Pisano, C.

Scientific Open-access Literature Archive and Repository

DOI 10.1393/ncc/i2013-11593-3Colloquia: QCDN12IL NUOVO CIMENTO Vol. 36 C, N. 5 Settembre-Ottobre 2013Azimuthal distributions of pions inside jets at RHICU. D’Alesio(1)(2), F. Murgia(2) and C. Pisano(1)(2)(∗)(1) Dipartimento di Fisica, Universita` di Cagliari - Cittadella UniversitariaMonserrato, Italy(2) INFN, Sezione di Cagliari - C.P. 170, Monserrato, Italyricevuto il 18 Aprile 2013Summary. — We evaluate the azimuthal asymmetries for the distributions of lead-ing pions inside a jet, produced in high-energy proton-proton collisions, in kinematicconfigurations currently under active investigation at RHIC. Adopting a transversemomentum dependent approach, which assumes the validity of a perturbative QCDfactorization scheme and takes into account all the spin and intrinsic parton motioneffects, we show how the main mechanisms underlying these asymmetries, namelythe Sivers and the Collins effects, can be disentangled. Furthermore, we considerthe impact of color-gauge invariant initial and final state interactions and suggest amethod for testing the universality properties of the Sivers function for quarks.PACS 13.85.Ni – Inclusive production with identified hadrons.PACS 13.88.+e – Polarization in interactions and scattering.1. – IntroductionWe study the process p↑p → jetπ + X, where one of the protons is transverselypolarized and the jet has a large transverse momentum. Specifically, we calculate theazimuthal asymmetries in the distribution of leading pions around the jet axis within theso-called transverse momentum dependent framework, which takes into account all thepossible spin and intrinsic parton motion effects, assuming the validity of perturbativeQCD factorization. Within this approach, the leading-twist azimuthal asymmetries areexpressed in terms of convolutions of different transverse momentum dependent distribu-tion and fragmentation functions (TMDs) [1]. We focus on the ones which are the mostrelevant from a phenomenological point of view, namely the Sivers distribution functionand the Collins fragmentation function. By defining appropriate moments of the asym-metries, it is possible to separate the effects due to these two different TMDs, in stronganalogy with the case of semi-inclusive deep inelastic scattering (SIDIS). This will alsohelp in clarifying the role played by the Sivers and the Collins mechanisms in the sizablesingle-spin asymmetries observed at RHIC for inclusive pion production, where these twoeffects cannot be isolated.(∗) E-mail: cristian.pisano@ca.infn.itc© Societa` Italiana di Fisica 221222 U. D’ALESIO, F. MURGIA and C. PISANO-0.04-0.02 0 0.02 0.04 2  2.5  3  3.5  4  4.5  5  5.5  6  6.5p              SIDIS 1(GeV)j TANsin (φS − φHπ )ηj = 3.3     ≈ 0.3← xFπ+π0π−-0.2-0.15-0.1-0.05 0 0.05 0.1 0.15 0.2 2  2.5  3  3.5  4  4.5  5  5.5  6  6.5p              SIDIS 2(GeV)j TANsin (φS − φHπ )ηj = 3.3     ≈ 0.3← xFπ+π0π−Fig. 1. – Estimate of the Collins asymmetry Asin(φS−φHπ )N in the GPM approach adopting thetwo sets of parameterizations SIDIS 1 (left panel) and SIDIS 2 (right panel), at√s = 200GeVand fixed jet rapidity ηj = 3.3, as a function of the transverse momentum of the jet pjT .2. – Theoretical frameworkIn the hadronic center-of-mass, or in any other frame connected to it by a boost alongthe direction of the two colliding protons, the single-transverse polarized cross sectionfor the process p↑p→ jetπ + X, at leading order in pQCD, can be written as [1]2dσ(φS , φHπ ) ∼ dσ0 + dΔσ0 sinφS + dσ1 cosφHπ + dσ2 cos 2φHπ(1)+dΔσ−1 sin(φS − φHπ ) + dΔσ+1 sin(φS + φHπ )+dΔσ−2 sin(φS − 2φHπ ) + dΔσ+2 sin(φS + 2φHπ ),with φS being the azimuthal angle of the proton spin vector S relative to the jet produc-tion plane and φHπ being the azimuthal angle of the three-momentum of the pion aroundthe jet axis, as measured in the helicity frame of the fragmenting quark or gluon [1].By means of the azimuthal moments(2) AW (φS ,φHπ )N = 2∫dφSdφHπ W (φS , φHπ ) [dσ(φS , φHπ )− dσ(φS + π, φHπ )]∫dφSdφHπ [dσ(φS , φHπ ) + dσ(φS + π, φHπ )],where W (φS , φHπ ) is one of the circular functions in (1), it is possible to single out thedifferent angular modulations of the cross section. An estimate of the upper bounds of allthe azimuthal moments has been given in [1] for the ongoing experiments at RHIC. In thefollowing we will focus only on those (sizable) asymmetries that receive contributions fromthe Sivers and the Collins functions. These TMDs are known and their parameterizationshave been determined from independent fits to e+e− and SIDIS data.3. – Collins and Sivers asymmetries in the generalized parton modelIn this section the Collins and Sivers asymmetries are evaluated at the RHIC energy√s = 200GeV and at forward jet rapidity, within the generalized parton model (GPM)approach, according to which TMDs are considered to be process independent [1]. In ouranalysis, two different sets of parameterizations for the TMDs have been used, namedSIDIS 1 and SIDIS 2 [1].AZIMUTHAL DISTRIBUTIONS OF PIONS INSIDE JETS AT RHIC 223-0.06-0.04-0.02 0 0.02 0.04 0.06 0.08 0.1 2  2.5  3  3.5  4  4.5  5  5.5  6  6.5p               π+ (GeV)j TANsinφS ηj = 3.3     ≈ 0.3← xFSIDIS 1SIDIS 2qgqg-0.06-0.04-0.02 0 0.02 0.04 0.06 0.08 0.1 2  2.5  3  3.5  4  4.5  5  5.5  6  6.5p               π0 (GeV)j TANsinφS ηj = 3.3     ≈ 0.3← xFSIDIS 1SIDIS 2qgqg-0.06-0.04-0.02 0 0.02 0.04 0.06 0.08 0.1 2  2.5  3  3.5  4  4.5  5  5.5  6  6.5p               π−(GeV)j TANsinφS ηj = 3.3     ≈ 0.3← xFSIDIS 1SIDIS 2qgqgFig. 2. – The quark and gluon contributions to the Sivers asymmetry Asin φSN obtained in theGPM approach adopting the two sets of parameterizations SIDIS 1 and SIDIS 2, at√s =200GeV and fixed jet rapidity ηj=3.3, as a function of the transverse momentum of the jet pjT .The Collins asymmetry Asin(φS−φHπ )N , presented in fig. 1, is mainly given by a convolu-tion of the Collins fragmentation function and the transversity distribution. We note thatour prediction of a negligible value of Asin(φS−φHπ )N for neutral pions has been confirmed bypreliminary RHIC data [2]. The Sivers asymmetry AsinφSN is shown in fig. 2. Its quark andgluon contributions are depicted separately, but cannot be disentangled experimentally.In order to provide an estimate of the unknown gluon Sivers function, we have taken itpositive, saturating an upper bound derived from the analysis of PHENIX data for cen-tral production of neutral pions [5]. Recently the STAR Collaboration at RHIC reportedpreliminary data on the Sivers asymmetry for neutral pions, which turns out to be largerthan zero [2] and compatible with our results obtained within the GPM framework.The vertical dotted lines in figs. 1 and 2 delimit the range xF ≤ 0.3 in which TMDsare presently constrained by SIDIS data. Their extrapolation beyond this region leads toresults plagued by large uncertainties. Hence a measurement of the proposed observableswould shed light on the large x behavior of the Sivers and the transversity distributionfunctions.-0.04-0.02 0 0.02 0.04 0.06 2  4  6  8  10  12  14P               π+ (GeV)jTANsinφS ηj = 3.3= 0.3← xFSIDIS 1SIDIS 2GPMCGIGPMCGI-0.04-0.02 0 0.02 0.04 0.06 2  4  6  8  10  12  14P                π0 (GeV)jTANsinφS ηj = 3.3= 0.3← xFSIDIS 1SIDIS 2GPMCGIGPMCGI-0.04-0.02 0 0.02 0.04 0.06 2  4  6  8  10  12  14P                π− (GeV)jTANsinφS ηj = 3.3= 0.3← xFSIDIS 1SIDIS 2GPMCGIGPMCGIFig. 3. – The quark contribution to the Sivers asymmetry Asin φSN calculated in the GPM andin the CGI GPM approaches adopting two sets of parameterizations, SIDIS 1 and SIDIS 2, at√s = 500GeV and jet rapidity ηj = 3.3, as a function of the jet transverse momentum pjT .224 U. D’ALESIO, F. MURGIA and C. PISANO4. – The Sivers asymmetry in the color-gauge invariant parton modelIn contrast to the GPM approach adopted in the previous section, in the color-gaugeinvariant (CGI) GPM [3,4] TMDs can be process dependent, due to the effects of initial(ISI) and final (FSI) state interactions. A fundamental example (still to be confirmedby experiments) is provided by the ISI in SIDIS and the FSI in the DY processes, whichlead to two quark Sivers functions with an opposite relative sign. For the reaction understudy the quark Sivers function has in general a more involved color structure, since bothISI and FSI contribute [4]. However, at forward rapidities only the qg → qg channel givesa dominant contribution. As a consequence, our predictions for the Sivers asymmetriesobtained with and without ISI and FSI are comparable in size but have opposite signs,as depicted in fig. 3 at the RHIC energy√s = 500GeV. Therefore the measurementof a sizable asymmetry would verify one of the two approaches and test the processdependence of the Sivers function.Finally, we have also studied single-spin asymmetries for inclusive jet production,which are described solely by the Sivers function [1, 4]. Our predictions for AsinφSN turnout to be very similar to the ones for jet-neutral pion production, shown in the centralpanel of fig. 3. According to preliminary data reported by the AnDY Collaboration atRHIC, which have been analyzed very recently in the different framework of the twist-3collinear formalism [6], the Sivers asymmetry for p↑p→ jet+X is small and positive [7].These results seem to agree with the GPM predictions only for xF ≥ 0.3 and suggest theneed for further studies along these lines, aiming to confirm or disprove the validity ofour factorization hypotesis and to test the universality properties of the different TMDs.Currently, a further comparison with experiments is ongoing [8], in which the imposedkinematic cuts are as close as possible to the ones used at RHIC in the analysis of theazimuthal asymmetries for the processes p↑p→ jetπ + X and p↑p→ jet + X.REFERENCES[1] D’Alesio U., Murgia F. and Pisano C., Phys. Rev. D, 83 (2011) 034021.[2] Poljak N. (STAR Collaboration), Nuovo Cimento C, 35 (2012) 193.[3] Gamberg L. and Kang Z. B., Phys. Lett. B, 696 (2011) 109.[4] D’Alesio U. et al., Phys. Lett. B, 704 (2011) 637.[5] Anselmino M., D’Alesio U., Melis S. and Murgia F., Phys. Rev. D, 74 (2006) 094011.[6] Gamberg L., Kang Z. B. and Prokudin A., Phys. Rev. Lett., 110 (2013) 232301.[7] Nogach L. (AnDY Collaboration), arXiv:1212.3437 [hep-ex].[8] D’Alesio U., Murgia F. and Pisano C., in preparation.

Azimuthal distributions of pions inside jets at RHIC

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