Comparison of transverse single-spin asymmetries for forward π 0 production in polarized pp , pAl and pAu collisions at nucleon pair c.m. energy s NN − − − √ =200 GeV
STAR Collaboration, J. Adam, L. Adamczyk, J. R. Adams, J. K. Adkins, G. Agakishiev, M. M. Aggarwal, Z. Ahammed, I. Alekseev, D. M. Anderson, A. Aparin, E. C. Aschenauer, M. U. Ashraf, F. G. Atetalla, A. Attri, G. S. Averichev, V. Bairathi, K. Barish, A. Behera, R. Bellwied, A. Bhasin, J. Bielcik, J. Bielcikova, L. C. Bland, I. G. Bordyuzhin, J. D. Brandenburg, A. V. Brandin, J. Butterworth, H. Caines, M. Calderón de la Barca Sánchez, D. Cebra, I. Chakaberia, P. Chaloupka, B. K. Chan, F-H. Chang, Z. Chang, N. Chankova-Bunzarova, A. Chatterjee, D. Chen, J. Chen, J. H. Chen, X. Chen, Z. Chen, J. Cheng, M. Cherney, M. Chevalier, S. Choudhury, W. Christie, X. Chu, H. J. Crawford, M. Csanád, M. Daugherity, T. G. Dedovich, I. M. Deppner, A. A. Derevschikov, L. Didenko, C. Dilks, X. Dong, J. L. Drachenberg, J. C. Dunlop, T. Edmonds, N. Elsey, J. Engelage, G. Eppley, S. Esumi, O. Evdokimov, A. Ewigleben, O. Eyser, R. Fatemi, S. Fazio, P. Federic, J. Fedorisin, C. J. Feng, Y. Feng, P. Filip, E. Finch, Y. Fisyak, A. Francisco, L. Fulek, C. A. Gagliardi, T. Galatyuk, F. Geurts, A. Gibson, K. Gopal, X. Gou, D. Grosnick, W. Guryn, A. I. Hamad, A. Hamed, S. Harabasz, J. W. Harris, S. He, W. He, X. H. He, Y. He, S. Heppelmann, S. Heppelmann, N. Herrmann, E. Hoffman, L. Holub, et al. (272 additional authors not shown)
CComparison of transverse single-spin asymmetries for forward π production inpolarized pp , p Al and p Au collisions at nucleon pair c.m. energy √ s NN = 200 GeV
J. Adam , L. Adamczyk , J. R. Adams , J. K. Adkins , G. Agakishiev , M. M. Aggarwal , Z. Ahammed ,I. Alekseev , , D. M. Anderson , A. Aparin , E. C. Aschenauer , M. U. Ashraf , F. G. Atetalla , A. Attri ,G. S. Averichev , V. Bairathi , K. Barish , A. Behera , R. Bellwied , A. Bhasin , J. Bielcik , J. Bielcikova ,L. C. Bland , I. G. Bordyuzhin , J. D. Brandenburg , A. V. Brandin , J. Butterworth , H. Caines ,M. Calder´on de la Barca S´anchez , D. Cebra , I. Chakaberia , , P. Chaloupka , B. K. Chan , F-H. Chang ,Z. Chang , N. Chankova-Bunzarova , A. Chatterjee , D. Chen , J. Chen , J. H. Chen , X. Chen ,Z. Chen , J. Cheng , M. Cherney , M. Chevalier , S. Choudhury , W. Christie , X. Chu , H. J. Crawford ,M. Csan´ad , M. Daugherity , T. G. Dedovich , I. M. Deppner , A. A. Derevschikov , L. Didenko , C. Dilks ,X. Dong , J. L. Drachenberg , J. C. Dunlop , T. Edmonds , N. Elsey , J. Engelage , G. Eppley , S. Esumi ,O. Evdokimov , A. Ewigleben , O. Eyser , R. Fatemi , S. Fazio , P. Federic , J. Fedorisin , C. J. Feng ,Y. Feng , P. Filip , E. Finch , Y. Fisyak , A. Francisco , L. Fulek , C. A. Gagliardi , T. Galatyuk ,F. Geurts , A. Gibson , K. Gopal , X. Gou , D. Grosnick , W. Guryn , A. I. Hamad , A. Hamed ,S. Harabasz , J. W. Harris , S. He , W. He , X. H. He , Y. He , S. Heppelmann , S. Heppelmann ,N. Herrmann , E. Hoffman , L. Holub , Y. Hong , S. Horvat , Y. Hu , H. Z. Huang , S. L. Huang ,T. Huang , X. Huang , T. J. Humanic , P. Huo , G. Igo , D. Isenhower , W. W. Jacobs , C. Jena ,A. Jentsch , Y. Ji , J. Jia , , K. Jiang , S. Jowzaee , X. Ju , E. G. Judd , S. Kabana , M. L. Kabir ,S. Kagamaster , D. Kalinkin , K. Kang , D. Kapukchyan , K. Kauder , H. W. Ke , D. Keane ,A. Kechechyan , M. Kelsey , Y. V. Khyzhniak , D. P. Kiko(cid:32)la , C. Kim , B. Kimelman , D. Kincses ,T. A. Kinghorn , I. Kisel , A. Kiselev , M. Kocan , L. Kochenda , D. D. Koetke , L. K. Kosarzewski ,L. Kramarik , P. Kravtsov , K. Krueger , N. Kulathunga Mudiyanselage , L. Kumar , S. Kumar ,R. Kunnawalkam Elayavalli , J. H. Kwasizur , R. Lacey , S. Lan , J. M. Landgraf , J. Lauret , A. Lebedev ,R. Lednicky , J. H. Lee , Y. H. Leung , C. Li , C. Li , W. Li , W. Li , X. Li , Y. Li , Y. Liang ,R. Licenik , T. Lin , Y. Lin , M. A. Lisa , F. Liu , H. Liu , P. Liu , P. Liu , T. Liu , X. Liu , Y. Liu ,Z. Liu , T. Ljubicic , W. J. Llope , R. S. Longacre , N. S. Lukow , S. Luo , X. Luo , G. L. Ma , L. Ma ,R. Ma , Y. G. Ma , N. Magdy , R. Majka , D. Mallick , S. Margetis , C. Markert , H. S. Matis ,J. A. Mazer , N. G. Minaev , S. Mioduszewski , B. Mohanty , M. M. Mondal , I. Mooney ,Z. Moravcova , D. A. Morozov , M. Nagy , J. D. Nam , Md. Nasim , K. Nayak , D. Neff , J. M. Nelson ,D. B. Nemes , M. Nie , G. Nigmatkulov , T. Niida , L. V. Nogach , T. Nonaka , A. S. Nunes ,G. Odyniec , A. Ogawa , S. Oh , V. A. Okorokov , B. S. Page , R. Pak , A. Pandav , Y. Panebratsev ,B. Pawlik , D. Pawlowska , H. Pei , C. Perkins , L. Pinsky , R. L. Pint´er , J. Pluta , J. Porter ,M. Posik , N. K. Pruthi , M. Przybycien , J. Putschke , H. Qiu , A. Quintero , S. K. Radhakrishnan ,S. Ramachandran , R. L. Ray , R. Reed , H. G. Ritter , O. V. Rogachevskiy , J. L. Romero , L. Ruan ,J. Rusnak , N. R. Sahoo , H. Sako , S. Salur , J. Sandweiss , S. Sato , W. B. Schmidke , N. Schmitz ,B. R. Schweid , F. Seck , J. Seger , M. Sergeeva , R. Seto , P. Seyboth , N. Shah , E. Shahaliev ,P. V. Shanmuganathan , M. Shao , A. I. Sheikh , W. Q. Shen , S. S. Shi , Y. Shi , Q. Y. Shou ,E. P. Sichtermann , R. Sikora , M. Simko , J. Singh , S. Singha , N. Smirnov , W. Solyst , P. Sorensen ,H. M. Spinka , B. Srivastava , T. D. S. Stanislaus , M. Stefaniak , D. J. Stewart , M. Strikhanov ,B. Stringfellow , A. A. P. Suaide , M. Sumbera , B. Summa , X. M. Sun , X. Sun , Y. Sun , Y. Sun ,B. Surrow , D. N. Svirida , P. Szymanski , A. H. Tang , Z. Tang , A. Taranenko , T. Tarnowsky ,J. H. Thomas , A. R. Timmins , D. Tlusty , M. Tokarev , C. A. Tomkiel , S. Trentalange , R. E. Tribble ,P. Tribedy , S. K. Tripathy , O. D. Tsai , Z. Tu , T. Ullrich , D. G. Underwood , I. Upsal , , G. Van Buren ,J. Vanek , A. N. Vasiliev , I. Vassiliev , F. Videbæk , S. Vokal , S. A. Voloshin , F. Wang ,G. Wang , J. S. Wang , P. Wang , Y. Wang , Y. Wang , Z. Wang , J. C. Webb , P. C. Weidenkaff ,L. Wen , G. D. Westfall , H. Wieman , S. W. Wissink , R. Witt , Y. Wu , Z. G. Xiao , G. Xie ,W. Xie , H. Xu , N. Xu , Q. H. Xu , Y. F. Xu , Y. Xu , Z. Xu , Z. Xu , C. Yang , Q. Yang ,S. Yang , Y. Yang , Z. Yang , Z. Ye , Z. Ye , L. Yi , K. Yip , Y. Yu , H. Zbroszczyk , W. Zha ,C. Zhang , D. Zhang , S. Zhang , S. Zhang , X. P. Zhang , Y. Zhang , Y. Zhang , Z. J. Zhang ,Z. Zhang , Z. Zhang , J. Zhao , C. Zhong , C. Zhou , X. Zhu , Z. Zhu , M. Zurek , M. Zyzak Abilene Christian University, Abilene, Texas 79699 AGH University of Science and Technology, FPACS, Cracow 30-059, Poland Alikhanov Institute for Theoretical and Experimental Physics NRC ”Kurchatov Institute”, Moscow 117218, Russia a r X i v : . [ nu c l - e x ] F e b Argonne National Laboratory, Argonne, Illinois 60439 American University of Cairo, New Cairo 11835, New Cairo, Egypt Brookhaven National Laboratory, Upton, New York 11973 University of California, Berkeley, California 94720 University of California, Davis, California 95616 University of California, Los Angeles, California 90095 University of California, Riverside, California 92521 Central China Normal University, Wuhan, Hubei 430079 University of Illinois at Chicago, Chicago, Illinois 60607 Creighton University, Omaha, Nebraska 68178 Czech Technical University in Prague, FNSPE, Prague 115 19, Czech Republic Technische Universit¨at Darmstadt, Darmstadt 64289, Germany ELTE E¨otv¨os Lor´and University, Budapest, Hungary H-1117 Frankfurt Institute for Advanced Studies FIAS, Frankfurt 60438, Germany Fudan University, Shanghai, 200433 University of Heidelberg, Heidelberg 69120, Germany University of Houston, Houston, Texas 77204 Huzhou University, Huzhou, Zhejiang 313000 Indian Institute of Science Education and Research (IISER), Berhampur 760010 , India Indian Institute of Science Education and Research (IISER) Tirupati, Tirupati 517507, India Indian Institute Technology, Patna, Bihar 801106, India Indiana University, Bloomington, Indiana 47408 Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou, Gansu 730000 University of Jammu, Jammu 180001, India Joint Institute for Nuclear Research, Dubna 141 980, Russia Kent State University, Kent, Ohio 44242 University of Kentucky, Lexington, Kentucky 40506-0055 Lawrence Berkeley National Laboratory, Berkeley, California 94720 Lehigh University, Bethlehem, Pennsylvania 18015 Max-Planck-Institut f¨ur Physik, Munich 80805, Germany Michigan State University, East Lansing, Michigan 48824 National Research Nuclear University MEPhI, Moscow 115409, Russia National Institute of Science Education and Research, HBNI, Jatni 752050, India National Cheng Kung University, Tainan 70101 Nuclear Physics Institute of the CAS, Rez 250 68, Czech Republic Ohio State University, Columbus, Ohio 43210 Institute of Nuclear Physics PAN, Cracow 31-342, Poland Panjab University, Chandigarh 160014, India Pennsylvania State University, University Park, Pennsylvania 16802 NRC ”Kurchatov Institute”, Institute of High Energy Physics, Protvino 142281, Russia Purdue University, West Lafayette, Indiana 47907 Rice University, Houston, Texas 77251 Rutgers University, Piscataway, New Jersey 08854 Universidade de S˜ao Paulo, S˜ao Paulo, Brazil 05314-970 University of Science and Technology of China, Hefei, Anhui 230026 Shandong University, Qingdao, Shandong 266237 Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai 201800 Southern Connecticut State University, New Haven, Connecticut 06515 State University of New York, Stony Brook, New York 11794 Instituto de Alta Investigaci´on, Universidad de Tarapac´a, Arica 1000000, Chile Temple University, Philadelphia, Pennsylvania 19122 Texas A&M University, College Station, Texas 77843 University of Texas, Austin, Texas 78712 Tsinghua University, Beijing 100084 University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan United States Naval Academy, Annapolis, Maryland 21402 Valparaiso University, Valparaiso, Indiana 46383 Variable Energy Cyclotron Centre, Kolkata 700064, India Warsaw University of Technology, Warsaw 00-661, Poland Wayne State University, Detroit, Michigan 48201 Yale University, New Haven, Connecticut 06520 and Institute of Physics, Bhubaneswar 751005, India (STAR Collaboration)
The STAR Collaboration reports a measurement of the transverse single-spin asymmetries, A N ,for neutral pions produced in polarized proton collisions with protons ( pp ), with aluminum nuclei( p Al) and with gold nuclei ( p Au) at a nucleon-nucleon center-of-mass energy of 200 GeV. Neutralpions are observed in the forward direction relative to the transversely polarized proton beam, inthe pseudo-rapidity region 2 . < η < .
8. Results are presented for π s observed in the STARFMS electromagnetic calorimeter in narrow Feynman x ( x F ) and transverse momentum ( p T ) bins,spanning the range 0 . < x F < .
81 and 1 . < p T < . c . For fixed x F < .
47, theasymmetries are found to rise with increasing transverse momentum. For larger x F , the asymmetryflattens or falls as p T increases. Parametrizing the ratio r ( A ) ≡ A N ( pA ) /A N ( pp ) = A P over thekinematic range, the ratio r ( A ) is found to depend only weakly on A , with (cid:104) P (cid:105) = − . ± . P is observed between the low- p T region, p T < . c , where gluonsaturation effects may play a role, and the high- p T region, p T > . c . It is further observedthat the value of A N is significantly larger for events with a large- p T isolated π than for eventswith a non-isolated π accompanied by additional jet-like fragments. The nuclear dependence r ( A )is similar for isolated and non-isolated π events. I. INTRODUCTION
The measurements and the evolving interpretationsof transverse single-spin asymmetries for forward pionproduction in high energy pp collisions have a rich his-tory [1–7]. These measurements guided the develop-ment of Quantum Chromo-Dynamics (QCD) based mod-els that incorporated quark helicity conservation, QCDfactorization, the nature of initial state parton motionor angular momentum, and the dynamics of fragmenta-tion within the scattering processes for polarized protons.The new transverse asymmetry measurements, presentedhere, again challenge aspects of current models for theapplication of QCD to the spin dependence of cross sec-tions. The π single-spin asymmetry, A N , is measured asa function of pion kinematics for collisions between po-larized protons and protons ( pp ), aluminum nuclei ( p Al)and gold nuclei ( p Au). Because A N for this process isexpected to be very sensitive to the QCD fields in thevicinity of a struck quark, the nuclear dependence of A N should be sensitive to phenomena that modify the localfields, for example, gluon saturation effects.This analysis presents the dependence of A N in the for-ward π production process, p ↑ + p (or A) → π + X. It isuseful to first define a simple azimuthal angle-dependentasymmetry, a N ( x F , p T , φ ), as the ratio of the differencein cross section for the two proton transverse spin states, σ ↑ and σ ↓ , to the sum of those cross sections for a pionproduced at x F (Feynman X) and p T (transverse mo-mentum), a N ( x F , p T , φ ) = σ ↑ ( x F , p T , φ ) − σ ↓ ( x F , p T , φ ) σ ↑ ( x F , p T , φ ) + σ ↓ ( x F , p T , φ ) (1)= A N ( x F , p T ) cos φ. (2)The three components of pion momentum are specifiedwith coordinates x F , p T and φ . The dependence ofthe pion differential cross section on transverse spin, ex-pressed as the pion momentum dependent asymmetry a N ( x F , p T , φ ) and the transverse single-spin asymmetry, A N ( x F , p T ), are defined in terms of the simple asym-metry accordingly, (Eq. 2). Referring to a right-handedcoordinate system, an initial state polarized proton is re- ferred to as spin “up” if it has a positive spin projectionalong the y axis while proton momentum is along the z axis. This polarized proton collides with an unpolarizedproton or nucleus traveling along the − z axis. A forwardpion has a positive longitudinal component of momen-tum p πL , given by a positive fraction x F = 2 p πL √ s of thepolarized proton momentum. The angle φ is the pionazimuthal angle about the z axis measured from the x axis positive direction. Equation 2 defines A N ( x F , p T )in terms of cross sections, which are differential in x F , p T and φ , with superscript arrows indicating the spindirections up or down, respectively. Symmetry requiresthat the φ dependence be proportional to cos φ . II. THE RELATION BETWEEN SCATTERINGWITH LONGITUDINAL AND TRANSVERSEPOLARIZATION
The unique features of the spin dependence of scatter-ing from quarks or gluons in transversely polarized pro-tons are best understood when contrasted with scatteringof partons in longitudinally (helicity) polarized protons.For a longitudinally polarized Dirac fermion, the depen-dence of cross section on the initial state spin is connectedto helicity conservation. For a relativistic electron orquark, the absorption or emission of a virtual photon (orsimilarly a gluon) cannot flip the helicity of a relativisticfermion. However, in a one dimensional scattering ex-ample, where a virtual photon is in a particular helicitystate, and is absorbed by a free quark at rest, the longitu-dinal spin component of the quark must flip as one unitof photon spin is absorbed by the quark, changing thestruck quark spin by one unit. Such a photon can be ab-sorbed by only one of the two possible initial quark spinstates, so cross sections thus can depend on initial statequark spin component along the final state direction or onthe final state helicity. But with absorption from a trans-versely polarized quark, where transverse spin states arecomposed of equal magnitude combinations of the twohelicity states, the cross section is the same for eithertransverse spin state. For scattering between small masselectrons and quarks, this generalizes to a cross section,that depends on Dirac fermion helicities but not on theirtransverse spins. Any cross section dependence on trans-verse spin is associated with the negligibly small helicityflip amplitudes.In the original quark model, where the spin of a polar-ized proton was attributed to the polarized quarks, it wasclear that the longitudinal polarization of these quarkscould be observed by the double helicity measurements inscattering between protons and electrons. Because deepinelastic scattering cross sections were most sensitive tothe up quarks, due to their larger electric charge, it wasa very early prediction of the quark model that the lon-gitudinal polarization of up quarks within the polarizedproton could be observed by measuring the dependenceof the lepton-proton cross section on the proton and lep-ton longitudinal spins [8].The longitudinal double spin lepton-proton scatteringmeasurements provided the mechanism for the first mea-surements of quark momentum dependent longitudinallypolarized quark distributions in a longitudinally polar-ized proton [9, 10]. Similar longitudinal double spinproton-proton cross sections depended upon the longitu-dinal polarization of partons, including gluons. Measure-ments and analysis of longitudinally polarized protonsremain an important topic for the STAR experiment, toconstrain longitudinal polarization densities of partons inthe proton. Global analyses of many experiments [11–13]have integrated the experimental results.In a frame where the proton was highly relativistic,where each quark momentum was nearly parallel to theproton momentum, the cross section did depend directlyon the helicity of the struck quark. The cross sectionassociated with a longitudinally polarized, nearly free,quark was calculable from hard scattering in helicity con-serving perturbative processes. The longitudinal doublespin asymmetry was then sensitive to the longitudinallypolarized struck quark in the longitudinally polarizedproton. However, the scattering cross section for sucha quark did not depend on the components of its spinmeasured along a transverse axis. Such a dependencewould have been associated with the parton flipping he-licity as it interacted, by absorbing or emitting a photonor gluon. Because the quark helicity-flip amplitude wasvanishingly small at high energies, early predictions, thatthe transverse spin dependence of the quark scatteringprocess should vanish at high energy, implied that A N should be small for high energy collisions [14]. Trans-verse spin dependence of cross sections are known to befurther suppressed because such dependencies requiredan interference between helicity amplitudes with differ-ent phases. Such a phase-shifted amplitude is not presentin the hard scattering part of leading twist perturbativeQCD (pQCD) processes.From the above discussion, it is clear that the helicityconserving hard parton amplitudes, which are apparentlydominant in the unpolarized cross sections, imply calcu-lable sensitivity of the parton cross sections to partonhelicity. This leads to longitudinal asymmetries, reflect- ing the polarization of partons in the proton. In contrast,the corresponding hard isolated amplitudes are insensi-tive to the transverse spin of the partons. The largetransverse spin asymmetries, A N , in pp collisions revealphysics beyond that of hard isolated parton scattering. III. MECHANISMS FOR NON-ZEROTRANSVERSE ASYMMETRY
The measurement of transverse spin asymmetries issensitive to effects that are very different than the physicsresponsible for longitudinal asymmetries. The traditionalpQCD calculations for hard scattering from protons re-lied on collinear factorization [15], where all parton mo-menta were characterized as propagating parallel to theparent proton momentum. Within this framework, thetransverse spin dependence was limited by the suppres-sion of hard scattering helicity-flip amplitudes. But morenuanced pictures of scattering of quarks in a transverselypolarized proton have emerged, utilizing parton densitydistributions that characterize both transverse and lon-gitudinal components of parton momentum. With sucha parton density distribution, the initial state partonmotion need not be parallel to the proton momentum,meaning that a helicity frame for the proton may notcompletely align with the helicity frame of the quark.A transverse momentum offset of (cid:126)k T , representing theaverage transverse momentum of the initial/final statequark relative to the initial/final state parent hadron,respectively, is added to the transverse momentum (cid:126)P T from the hard scattering process to form the observedpion transverse momentum, p πT = | (cid:126)P T + (cid:126)k T | . So whilethe quark scattering cross section has little direct depen-dence on the transverse spin of the quark, the pion pro-duction cross section can depend on the transverse spinof the proton through initial and final state interactionsleading to non-zero (cid:126)k T . If this bias of (cid:126)k T is correlatedwith the transverse spin of the proton, then non-zero A N will result. This kind of proton spin dependence of theobserved pion cross section is amplified by the extreme p T dependence of the hard pion cross section.The general expectation that the pion A N should fallwith increasing p T for p T above a nominal QCD momen-tum scale can be demonstrated in a simple model. If oneassumes that the forward hard scattering cross section ofa quark, with momentum fraction x , falls with increas-ing transverse momentum, p T , by a power law form withpower N , then dσdp T ∝ p − NT , (3)where p T = | (cid:126)P T | . If the scattered quark acquires trans-verse momentum (cid:126)k T = ± k T ˆ x from initial or final stateinteractions that is correlated with the polarized protonspin in the ± ˆ y directions, then we see that A N will alsofall with increasing p T . Assuming the hard scatteringtransverse momentum is much greater than the initialstate or final state transverse momentum ( p T (cid:29) k T ),then the difference in cross section when p πT is measuredalong the ± ˆ x direction leads to A N as in Eq. 4. If we as-sume a cross section form for p T , as in Eq. 3, expressing A N as a left-right asymmetry, we have A N ( x F , p T ) = σ ↑ ( x F , p T , − σ ↑ ( x F , p T , π ) σ ↑ ( x F , p T ,
0) + σ ↑ ( x F , p T , π ) (cid:39) ( p πT − k T ) − N − ( p πT + k T ) − N ( p πT − k T ) − N + ( p πT + k T ) − N (cid:39) N k T p πT (4)for small k T /p πT . This demonstrates that if the k T shift isindependent of the hard scattering p T , it is very naturalto expect the magnitude of the asymmetry to fall withincreasing observed transverse momentum p T at large p T .In previous measurements [5] of the p T dependence for A N with charged pions, the asymmetry has been seen toincrease with p T up to about p T < c . In an earlierSTAR measurement [7], it was observed that there waslittle evidence for A N falling with p T up to at least 3GeV/ c . In this paper, the data are analyzed to separatethe independent effects of p T and x F .Two classes of models have been introduced for for-ward A N , both involved the hard scattering of a leadingmomentum quark in the polarized proton and both de-pended upon secondary interactions to generate a spindependent contribution (cid:126)k T to the pion final state trans-verse momentum. The Sivers effect [16] involved aninitial state interaction before the hard scattering of aquark in a polarized proton, leading to initial state par-ton transverse momentum that depended on the protontransverse polarization. The Collins effect [17] generateda transverse spin dependent component to the final statepion transverse momentum from the fragmentation pro-cess of the scattered quark, which retained its initial statetransverse polarization through the hard scattering pro-cess. Closely related to Collins and Sivers models was anapproach involving higher twist calculation, where thescattered quark was correlated with a soft gluon, whichalso lead to a significant transverse asymmetry [18].Many model calculations attempt to describe forwardpion transverse spin asymmetries using one of these ap-proaches. While for both types of models the basicmechanism involves the production of a final state pionfrom fragmentation of a hard scattered parton, only theCollins approach explains large A N arising from the frag-mentation process. In contrast to pion production, jetproduction does not involve fragmentation. The Collinseffect therefore does not contribute to that asymmetry.Jet A N measurements in this kinematic region have beenpublished and the values of A N were observed to besmaller than measured pion asymmetries [19].Both the Sivers and the Collins approaches introduceda parton transverse momentum relative to the initial orfinal state hadron momentum to generate a transverse asymmetry without violating helicity conservation. Inthe Sivers picture, transverse momentum of initial statequarks can be connected to the initial state orbital an-gular momentum of a struck quark along the polariza-tion axis. While an orbiting quark does not, on average,have transverse momentum, Sivers argued that absorp-tive effects could break the left-right symmetric parton k T distribution to generate the required non-vanishingaverage k T = (cid:68) (cid:126)k T · ˆ x (cid:69) . Even though absorption does in-troduce phase changes, the calculation of this phase inthe conventional perturbative calculation was not fullyappreciated until it was noted in [20] that the Wilsonline contribution, formally required in the pQCD calcu-lation, did provide exactly the needed phase change fora non-zero A N [21].The emerging physical picture is that unlike the casefor longitudinal spin dependence, the observed large val-ues of A N derive not from the spin dependence of thehard scattering process between the pair of partons, butfrom the interaction between the scattered quark and theother constituents or fragments of the polarized proton.While from symmetry, A N must vanish at p T = 0, theexample of Eq. 4 demonstrates that the asymmetry isexpected to fall with transverse momentum above somenominal scale, (cid:126)k T . In recent years, there have been manycalculations based on Collins, Sivers or twist-3 collinearmethods, with a goal to reproduce the basic nature of A N dependence on kinematics [22–26]. Within the Collins orSivers methods it was necessary to account for the longi-tudinal and transverse momentum distributions of par-ton momentum within hadrons while traditional collinearparton densities or fragmentation functions involved onlylongitudinal distributions. In the twist-3 approach, onestarted with those traditional collinear parton densitiesor fragmentation functions and dynamically generatedthe transverse motion from interactions with other fieldsin the nucleon. A twist-3 calculation [24], involving fitsto many parameters, resulted in calculations that were inagreement with single inclusive deep inelastic scatteringasymmetries and with the x F dependence of π A N in pp collisions. This calculation also resulted in a nearlyflat, or very slowly falling, p T dependence above about3 GeV/ c for the π A N in pp scattering. While not ris-ing with p T , as do the new A N pp data presented in thispaper, the nearly flat p T dependence from the twist-3 cal-culation is interesting. It shows that the intuitive pictureof A N falling with P T , based on the simple arguments ofEq. 4, can involve a surprisingly large k T scale, well abovethe nominal QCD scale. IV. MEASUREMENTS OF A N INPROTON-NUCLEUS COLLISIONS
If the observed transverse single-spin dependent ampli-tude for forward pion production arises completely fromthe localized quark-gluon hard scattering process, thenthe environment that provides the soft gluon in the sec-ond proton or nucleus would not likely impact A N . Butwe know that the important source of A N is not thehard quark-gluon scattering process itself but primarilyinvolves the additional interactions with other fields inthe nucleon or nucleus, perhaps manifested by the gen-eration of parton transverse momentum relative to theparent hadron momentum. Because the mechanism re-sponsible for transverse spin asymmetries is not a simplelocal leading twist interaction but depends on the envi-ronment in which a parton interaction occurs, it is clearthat A N could be different for pp , p Al and p Au collisions.Even the simple model of Eq. 4 reminds us that a changein the shape of the p T dependence for pion productiondue to either nuclear absorption, rescattering, or modifi-cation of the gluon distribution, could lead to dependenceof A N on nuclear size.The measurement of how A N changes when the beamremnant partons of the proton are replaced with specta-tor partons of a nucleus is a subject of this paper. It isclear that the phase from the Wilson line integral, a lineintegral of the gauge vector potential color field alongthe struck quark trajectory, can give rise to color forcesbetween the struck quark and the rest of the polarizedproton. If there are also important color forces betweenthe hard scattering constituents and the residual specta-tor nucleus, then nuclear dependence of A N in pA scat-tering could result. Studies of the spin dependence ofthe interaction between the interacting quark and theresidual spectator nucleus have predicted large nuclear A -dependent transverse spin effects but at a lower trans-verse momentum scale than that of this analysis [27].A more recent calculation was based on lensing forces,with specific reference to the kinematics of this experi-ment [28]. The model addressed the dependence of A N on nuclear saturation as well as the p T dependence of A N .One mechanism that could provide nuclear A depen-dence of A N relates to the increase in gluon density inthe soft gluon distribution probed in forward scatter-ing. It is predicted that at low gluon x , when the gluondensity becomes large, saturation effects begin to playan important role. For interactions between soft gluonsand hard partons producing scattered pions in the range1 . < p T < . c , saturation effects might modifythe interaction, creating significant differences betweenthe corresponding scattering process in pp and pA col-lisions. Specific saturation models, such as the ColorGlass Condensate [29], predict interactions of the scat-tered quark with a condensate of gluons rather than ahard scatter from a single gluon. Such saturation cal-culations predict a change in the p T distribution of thecross section in regions of p T near the saturation scale,with a suppression of the cross section that increases withnuclear size. In the p T ≈ c range and at moreforward pseudo-rapidity than this measurement ( η ≈ R d Au in d Au scattering to produce π mesons [30] is sig- nificantly less than unity, suggesting a difference in thescattering process as the size of the nucleus is varied.In the same p T and rapidity range presented here, mea-surements of the nuclear modification factors for chargedhadrons (mostly charged pions) [31] showed suppressionin R d Au . This paper addresses the nuclear dependence of A N , noting, in particular, the lower end of the p T rangewhere evidence for saturation effects has already beenseen in the corresponding dA cross sections [30]. V. PHOTON AND π DETECTION IN THE FMS
These data from the Solenoidal Tracker At RHIC(STAR) experiment at the Relativistic Heavy Ion Col-lider (RHIC) were collected during the 2015 RHIC run,involving collisions between nucleons at center-of-massenergy √ s NN = 200 GeV per nucleon pair. The pho-ton pair from the decay of the π was detected with theSTAR forward electromagnetic calorimeter, referred to asthe Forward Meson Spectrometer (FMS) [32]. To mea-sure A N for forward π production, the STAR detectorsused in this analysis were the FMS and the Beam-BeamCounters (BBC).The two RHIC beams (yellow and blue beams) arebunched with up to 120 bunches in each ring. The smallangle scattering from the blue beam is associated withpositive rapidity. Only 111 bunches in each beam arefilled and a contiguous set of 9 bunches (the abort gap)are unfilled. Bunch spacing is 106 ns and the transversepolarization pattern is chosen for each fill according to apredefined pattern (either alternating the polarization di-rection from bunch to bunch or for pairs of bunches). Theblue beam polarization ranged between 50% and 60%.The BBCs are located at a distance of ± . < η < . pA collisionswe remove events with small signals in the east BBC,on the opposite side to the FMS, to reduce single beambackground.The FMS is a Pb-glass electromagnetic calorimeterconsisting of 1264 rectangular lead glass blocks or cells,stacked in a wall with front surface transverse to theSTAR beam line as shown in Fig. 1. The FMS cov-ers the range of forward pseudo-rapidity, 2 . < η < . π s in the STAR FMS have beendiscussed elsewhere [32].The small and large FMS cells have Pb-glass with dif-ferent compositions. For small and large cells the ratioof cell sizes is chosen to be proportional to the ratio of FIG. 1. The layout for the FMS calorimeter around theRHIC beam-line located about seven meters west of the nom-inal STAR interaction point. The FMS consists of lead glassblocks with lengths corresponding to 18 radiation lengths.There are 788 outer blocks with front face dimensions of 5.8 × × Moli´ere radii (transverse electro-magnetic shower dimen-sion); therefore a photon in the large cells will deposit itsenergy into a similar number of cells as a photon of thesame energy in the small cells. For a 10 GeV photon, theshower distributes measurable energy into about 10 cells.For higher energy photons, the number of involved cellsincreases. For a 30 GeV photon from the nominal inter-action point, incident at the center of a cell, about 80%of the photon energy is deposited in that cell. Fitting thedistribution of energy in cells to an expected distributionfrom a known shower shape, the transverse coordinatesof the incident photon (at shower maximum depth) canbe obtained with a resolution of about 20% of the celldimension.In the kinematic range discussed in this paper, ob-served photons from π decays have a separation rangingfrom a few cells to less than one cell. For the highest en-ergy π s, above 60 GeV, the shower shape from the twophotons starts to overlap into a small cell single cluster.Therefore, to reconstruct the highest energy π s, the dis-tribution of deposited energies in cells is fitted to a twophoton hypothesis, with parameters that represent thetwo photon energies and transverse position coordinates.The quality of these fits begins to degrade when the pho-ton separation is on the order of a single cell width.In addition to photons from π decays, the FMS mea-sures electrons and positrons. It also has some sensi- tivity to charged hadrons, such as π ± . On average, acharged pion deposits about 1/3 of its energy in theFMS. If the π is from the fragmentation of a high p T jet, the FMS sees many of the associated hadronic frag-ments with degraded energy sensitivity. These chargedhadron showers are fit to the photon shower shape and ifthe deposited energy is greater than 1 GeV, they are in-cluded in the list of low energy photon candidates. TheFMS is triggered by high transverse momentum local-ized FMS signals. Because these cross sections have asevere transverse momentum dependence, the partiallymeasured charged hadronic background contributes lit-tle to the trigger rate but does contribute background to π photon pair signals at high p T .The events from the FMS where obtained from twotrigger methods. The first method is called the boardsum trigger, which demands transverse energy to be de-posited in localized overlapping rectangles of the 32 FMScells. The second method is called the jet trigger, which issatisfied by deposition of transverse energy, with a higherthreshold than that of the board sum triggers, measuredwithin overlapping azimuthal regions of angle ∆ φ = π/ π p T aboveone of three adjustable thresholds, typically 1.6, 1.9 and2.2 GeV/ c . Triggers were prescaled to conserve detectorreadout bandwidth while sampling the different p T re-gions with similar statistical uncertainties. The pp datasample presented in this paper corresponds to an inte-grated luminosity of 34 pb − using the highest thresh-old triggers, which are not prescaled. The correspondinganalyzed luminosity for proton-nucleus collisions is 905nb − = . − and 206 nb − = . − for p Al and p Au, respectively, where the numerators are provided fordirect comparison of proton-nucleon luminosities.For each event, photon candidates are sorted into “coneclusters.” Each cone cluster includes a subset of the pho-ton candidates for which the momentum direction iswithin an angular cone of 0.08 radians about the conemomentum direction of included photons. For each pho-ton in the p T sorted photon event list, the photon istested for inclusion in the cone cluster list, testing thelargest p T clusters first. If not included in an existingcluster, this becomes the seed of a new cluster. Usu-ally, only one of these cone clusters will be associatedwith the large p T trigger. For this analysis of triggeredevents, only the leading p T cone cluster is searched for π candidates. This 0.08 radian cone radius, with nomi-nal kinematic pair cuts and for the pion energies around40 GeV, restricts the selected diphoton mass of photonpairs within a cone cluster to typically less than about 1GeV/ c . Searching for π candidates within a cone clus-ter greatly reduces the combinatorial photon pair possi-bilities and reduces diphoton background.At higher energy, the separation between π photonsbecomes small, on the scale of the cell size. In this case,fits to a two photon hypothesis tend to overestimate theseparation between these photons. For large energy pi-ons, or equivalently large x F , as seen in Fig. 2, cal-culated masses are preferentially smeared to larger val-ues. The π mass resolution is broadened significantlyto higher mass for π energies E π >
35 GeV in thelarge cells (lower pseudo-rapidity region) and for ener-gies above E π >
50 GeV in the small cells and higherpseudo-rapidity region of the FMS.The leading energy pair of photons in the highest p T cluster was analyzed, with selection based on the de-cay distribution of that two-photon pair. The condition Z < . Z = | E − E E + E | and E and E are the energies of the two photons. This selection waspreferred over a less restrictive one because it decreasesbackground under the π mass peak. It is the accountingfor background under the π peak that represents the ma-jority of the systematic uncertainty for the measurementof A N .While it is the intention to measure A N for inclusive π production, the selection of the highest-energy twophotons for the π candidates does sacrifice 10-15% ofthe pions, depending on kinematics. In proton-nucleuscollisions ( p Al and p Au), we apply an additional selec-tion criterion in order to remove a specific RHIC back-ground which is seen in the “abort-gap” events, betweenbuckets where the nuclear beam is not present. Theseevents are referred to as single-beam events. For pA col-lisions, we require that the east BBC have a minimumsignal (caused by the breakup of the nuclei). This re-moves about 5% of the lowest activity including mostperipheral collisions from this analysis, but also removesnearly all of the single-beam background. The residualsingle-beam background contributes significantly to thesystematic error only for a few of the high- x F bins.The residual single-beam background fraction in eachkinematic bin is estimated from events seen in the abort-gap bunches. The ratio of asymmetry for the single-beambackground to the π asymmetry is to be defined as R NB ,so A NB = R NB A N , where A N is the π asymmetry in theparticular kinematic bin. Consistent with asymmetriesobserved in the small number of events in the abort gap,we conservatively assume that R NB = 0 . ± . VI. THE INCLUSIVE A N MEASUREMENTS
In this paper A N for forward π production is mea-sured for pp , p Al, and p Au collisions. The high trans-verse momentum forward π is detected with the FMScalorimenter, detecting pions with pion pseudo-rapidity2 . < η < .
8. Candidate photon pairs passing the selec-tion are independently analyzed within kinematic regionsof p T and x F . In Fig. 2, the diphoton mass, M γγ , dis-tributions are shown for two example kinematic regions,for pp , p Al and p Au collisions. The two-photon mass dis-tributions are initially fitted to a quadratic backgroundshape plus a Gaussian pion shape in the mass region be-low the η peak. The Gaussian only approximately repre-sents the shape of the pion peak and that Gaussian shape is only used to determine a mass range above the pionpeak. To finally determine the background fraction, thequadratic background shape is constrained to be zero ata mass of zero and is fit to the mass distribution in thelimited mass region above the pion peak. Examples ofthese background fits are shown in Fig. 2. The pion sig-nal is obtained by counting the events in the pion peak,0 . < M γγ < .
255 GeV/ c , and subtracting the fit-ted background contribution in that region. The typicalfraction f B of background under the pion peak rangesfrom about 20% at very low x F to a few percent whenthe pion energy is larger. We define A B = R B A N is theasymmetry of the background under the π peak where R B is the fraction of non-pion background and A N is the π asymmetry.The value R B = 0 . ± .
33 was conservatively de-termined based on the asymmetry in the mass region(0 . < M γγ < . c ) above the pion peak andbelow the η meson peak . This background asymmetrycannot be well measured with significance within a singlekinematic bin, but is estimated based on an average overmany kinematic bins. Uncertainty in this backgroundcorrection is the most important contribution to the sys-tematic uncertainty in the π measurement of A N . A N for a given bin in x F and p T is extracted from the fits tothe uncorrected asymmetries, a ( φ ), which is determinedin each φ bin from the number of pions ( N ↑ and N ↓ ) de-tected when the proton polarization is up ↑ /down ↓ (seeFig. 3). The uncorrected asymmetry is a ( φ ) = N ↑ ( φ ) − N ↓ ( φ ) N ↑ ( φ ) + N ↓ ( φ ) . (5)The azimuthal dependence of a ( φ ) is fit to the form a ( φ ) = p + p cos φ. (6)The parameter p is proportional to A N but must becorrected for the polarization of the proton beam P B anda factor K to account for background effects, A N = p KP B . (7)The beam polarization varied for different RHIC fills.The polarization and beam luminosity were largest atthe start of a fill and decayed during the fill. To max-imize the use of available data acquisition bandwidth,STAR adjusts the FMS trigger prescale factors duringthe fill, collecting a larger fraction of available low p T cross section when the luminosity is lower. The analysisof RHIC polarization has been described by the RHICPolarimetry group [35]. In this analysis, the average po-larization for each kinematic data point is calculated byfolding the run by run polarization with the trigger ratecontributing to each kinematic point. For a given beamfill, there is variation in the average polarization of 1-2%for different kinematic regions. The variation of A N fromthese different polarizations is small with respect to theoverall uncertainties. The uncertainty on polarization is ) c (GeV/ gg M E v e n t s · c <2.2 GeV/ T p <0.27 2.0< F x : 0.21< pp c <0.255 GeV/ gg M B f (0.0005) – – =0.009 N A ) c (GeV/ gg M E v e n t s · c <3.5 GeV/ T p <0.47 3.0< F x : 0.37< pp c <0.255 GeV/ gg M B f (0.0014) – – =0.044 N A ) c (GeV/ gg M E v e n t s · c <2.2 GeV/ T p <0.27 2.0< F x Al: 0.21< p c <0.255 GeV/ gg M B f (0.0008) – – =0.008 N A ) c (GeV/ gg M E v e n t s · c <3.5 GeV/ T p <0.47 3.0< F x Al: 0.37< p c <0.255 GeV/ gg M B f (0.0014) – – =0.035 N A ) c (GeV/ gg M E v e n t s · c <2.2 GeV/ T p <0.27 2.0< F x Au: 0.21< p c <0.255 GeV/ gg M B f (0.0011) – – =0.014 N A ) c (GeV/ gg M E v e n t s · c <3.5 GeV/ T p <0.47 3.0< F x Au: 0.37< p c <0.255 GeV/ gg M B f (0.0016) – – =0.042 N A FIG. 2. Example invariant mass spectra for diphoton pairs selected within two kinematic regions (two columns) and threecollision types (rows: pp , p Al, p Au). The asymmetries A N for pion peaks are obtained within the mass region 0 . < M γγ < .
255 Gev/ c . For the indicated fitted backgrounds under the peaks, the fraction of background events is f B . The measured A N for the π , with all corrections applied, is included within each panel with statistical uncertainty followed by systematicuncertainty in parentheses. divided between scale uncertainties common throughoutthe running period and non-scale uncertainties that varyfill by fill. The scale uncertainties, ∆ P/P , are 3%, 3.1%,and 3.2% for pp , p Au, and p Al, respectively, and are notincluded in the point-by-point polarization measurement.When ratios of asymmetries are taken, the dominant po-larization uncertainty, like many of the other systematicuncertainties, tends to cancel in the ratio.In Eq. 7, K represents a correction factor to the asym-metry based on the estimates of backgrounds in the massregion 0 . < M γγ < .
255 GeV/ c . The largest partof the correction K of Eq. 7 was obtained from thebackground fraction f B under the peak with asymme-try A B = R B A N . The fraction f NB represents a smalladditional background fraction (typically 1 to 3%) frominteractions that cannot be associated with polarized pp or pA collisions with asymmetry A NB = R NB A N . Thenthe factor K is K = (cid:20)
11 + f B ( R B − (cid:21) (cid:20)
11 + f NB ( R NB − (cid:21) . (8) The systematic uncertainties on A N come from threesources: polarization error (typically < . − − K is the largest source of systematicerror in our measurement of A N . These uncertainties arecalculated individually for each given kinematic bin.The various systematic contributions to our p T uncer-tainty have been discussed in detail in a previous anal-ysis [32]. The transverse momentum error analysis us-ing that data, collected in 2012 and 2013, is applicablefor these 2015 data. That analysis determined the final σ p T /p T to be approximately 5-6%, an estimate we willadopt here. In both analyses, the dominant contributionlies in the uncertainty on the energy calibration of thedetector ( σ C ≈ π mass for 20-30 GeV π photon pairs in the large cells and 40-50 GeV pairs in thesmall cells. We have conservatively set our final error intransverse momentum, σ p T /p T = 7%, allowing for minor0 p f - - - ) f ( a - - c <2.2 GeV/ T p <0.27 2.0< F x pp: 0.21< 0.0013 – =0.0038 p – = p p f - - - ) f ( a - - c <3.5 GeV/ T p <0.47 3.0< F x pp: 0.37< 0.0022 – =0.0230 p – = p p f - - - ) f ( a - - c <2.2 GeV/ T p <0.27 2.0< F x Al: 0.21< p – =0.0036 p – -0.0006 = p p f - - - ) f ( a - - c <3.5 GeV/ T p <0.47 3.0< F x Al: 0.37< p – =0.0181 p – = p p f - - - ) f ( a - - c <2.2 GeV/ T p <0.27 2.0< F x Au: 0.21< p – =0.0074 p – -0.0001 = p p f - - - ) f ( a - - c <3.5 GeV/ T p <0.47 3.0< F x Au: 0.37< p – =0.0231 p – -0.0014 = p FIG. 3. Uncorrected transverse spin asymmetries for the same 6 kinematic regions as in Fig. 2. The azimuthal φ π distributions of the uncorrected asymmetries, a ( φ ), are shown for events in the mass range 0 . < M γγ < .
255 GeV/c .Fits to the functional form from Eq. 6 are shown with fitted parameter values p and p . differences with this analysis and the previous analysis.The value of the parameter p from Eq. 6 indicates theasymmetry of relative integrated luminosity, as measuredin the given kinematic region. RHIC spin patterns arechanged for each fill so the integrated luminosities forspin up and spin down bunches are nearly equal. Thedistributions of parameters p for the three collision sys-tem data sets ( pp , p Al and p Au) have weighted means of(0 . ± . − . ± . . ± . p , have corresponding χ values of32, 57 and 45 for 40 kinematic regions (39 degrees of free-dom). While the extracted values for A N depend only onthe p parameter, it is seen from the above that the val-ues of p parameters are small and for each beam dataset, the measurements of p in different kinematic regionsare internally consistent within each set.An A N point is extracted from each of 110 kinematicand “collisions beam type” bins based on the value ofparameter p from the fit to Eq. 6. As shown for a fewexample kinematic regions and beam types in Fig. 3,each two-parameter fit to the 20 azimuthal points results in a χ value. Over this large ensemble of such fits, thedistribution of measured χ values is in good agreementwith the theoretical χ distribution. For the pp , p Al and p Au data sets, the average χ s for the fits to Eq. 6 are18.5, 18.1 and 18.4 for 18 degrees of freedom, respectively.The examples shown in Figs. 2 and 3 represent only sixkinematic regions of 110 kinematic points at which A N has been calculated. The transverse single-spin asymme-try for the full data set is shown in Fig. 4.Even though A N is observed to differ among differ-ent nuclear collisions systems by 10% to 20%, it is aninstructive exercise to combine the data sets from differ-ent collision systems. In Fig. 5, the data points fromall beams and all transverse momenta are combined ineach of the six x F bins shown in other figures, with cen-ters located at x F = { } .All data from pp , p Al and p Au collisions are combinedand show the x F dependence for several p T regions. For x F < . A N seems to depend only weakly on trans-verse momentum, with a gentle increase in asymmetry atlarger p T , but at larger x F > .
47, it appears that A N ) c (GeV/ T p N A STAR 200 GeV p pp Al p Au p <0.21 F x c (GeV/ T p N A <0.47 F x ) c (GeV/ T p N A <0.27 F x c (GeV/ T p N A <0.61 F x c (GeV/ T p N A <0.37 F x c (GeV/ T p N A <0.81 F FIG. 4. The transverse momentum p T dependence of A N for 6 bins in Feynman x F . The events contributing are inclusive π s with selection in the invariant mass window 0 . < M γγ < .
255 GeV/c . Results for the three collision systems areshown, black squares for pp , blue circles for p Al and red triangles for p Au collisions. The event selection criteria are given in thetext. The statistical uncertainties are shown with vertical error bars and the filled boxes indicate the horizontal and verticalsystematic uncertainties. F x N A Au +p Al pp+p STAR c <2.0GeV/ T P < c c <3.0GeV/ T P < c c <4.0GeV/ T P < c c <5.0GeV/ T P < c c <7.0GeV/ T P < c FIG. 5. The x F dependence of the π A N is shown with data from the combined pp , p Al, and p Au data points, collectingpoints within x F intervals for frames from Fig.4 and the indicated p T range. Data points are shown separately for five intervalsof transverse momentum indicated by different symbols and plotted horizontally at the average x F for each combined point.Vertical error bars represent statistical uncertainties and the systematic horizontal and vertical uncertainties are shown withfilled boxes. p T .For each x F region, the ratios of A N for p Au( p Al) to A N for pp scattering are shown as a function of p T in Fig.6(7). The p T dependences of these ratios are consistentwith a constant ratio. Nevertheless, the A N ratios shownin Fig. 6 and Fig. 7 were separately averaged for low p T (1 . c < p T < . c ) and high p T ( p T > . c ). The fitted average values of A N ratios for eachplot in Figs. 6 and 7, averaging over the full p T rangefor each x F , are plotted in Fig. 8 as a function of log A .The systematic uncertainties in Fig. 6 and Fig. 7 arereduced to account for the correlated background correc-tions between pp and pA distributions. The non-beambackgrounds thus contribute the most to these system-atic errors with statistical uncertainty dominating.We parameterize the dependence of A N on nuclear size A with a power law form A N ( pA ) = A N ( pp ) A P . (9)To determine the exponent P for each of the six x F bins, the weighted means shown in Fig. 8 are fitted tothe power law form, r ( pA ) = (cid:28) A N ( pA ) A N ( pp ) (cid:29) all p T = A P . (10)The ratios, r ( pA ), as defined in Eq. 10, represent the ra-tio of nuclear suppression of A N in pA to A N observed pp scattering, averaged over the full observed p T range.For each region of x F , we fit to a power law in nuclearsize A with a fitted exponent, P . Recognizing that theuncertainties in the ratio of pA to pp are correlated, thesimple χ fit in the figure can be biased in the determi-nation of the exponent, P . We refer to this simple fit,with correlated uncertainties in the ratios, as a “Type 1”determination of P .A second method for determining the exponent, P ,without correlated uncertainties is to fit each point in p T and x F to the two-parameter form of Eq. 9, with param-eters A N ( pp ) and P . These fits are two-parameter fits tothree measurements within each kinematic region. Thenwith a weighted mean over p T of the exponents from fits,an average P is obtained for each x F region. This isreferred to as the “Type 2” method, and the bands cor-responding to the one sigma uncertainties in this “Type2” fit are shown in Fig. 8 as the shaded regions.Fitting the exponent of the A dependence of the ratiosseparately for the low and high p T regions, the exponents P L and P H are obtained, r L ( pA ) = (cid:28) A N ( pA ) A N ( pp ) (cid:29) p T < . / c = A P L (11) r H ( pA ) = (cid:28) A N ( pA ) A N ( pp ) (cid:29) p T > . / c = A P H . (12)Calculations of A N ratios by Hatta et al. [36] iden-tify an amplitude that is thought to be dominant in the saturation region and would scale as A N ∝ A − in p ↑ + A → π X . These calculations could apply to ourpresent measurements of A N for pp , p Al and p Au in thetransverse momentum range 1 . < p T < . c .Comparing gold with A=197 and proton collisions withA=1, this implies a reduction of A N for p Au by morethan a factor of 5. Above the saturation region, theypredict the A N will scale as A , indicating that the trans-verse single-spin asymmetry at larger p T could be similarfor pp and p Au collisions. The fitted values of the expo-nents P L and P H as functions of x F are shown in Fig. 9.The exponents are generally within about 5% of zero inboth the low and high p T regions and significantly dif-ferent from the value of − that has been predicted toapply in the region below the saturation scale.Another approach [37], based on a geometrical scalingof gluon distributions and with Collins-type fragmenta-tion, has also been used to calculate the transverse single-spin asymmetry. They predicted that for pion transversemomentum below the saturation scale, p T << Q s , the A N ratio is A N ( pA ) /A N ( pp ) (cid:39) Q sp Q sA , where Q sA is thesquare of the saturation scale for a nucleus with A nucle-ons. For p T well above the saturation scale, the ratio wasexpected to be 1. Models, which suggest that at large p T the ratio should approach a form with exponent zero, arein good agreement with these data. VII. ISOLATED A N MEASUREMENTS
It is observed here that the presence of soft photons orhadronic fragments in the vicinity of the highest p T pioncan decrease the asymmetry significantly, cutting A N inhalf in most kinematic regions. For a subset of the eventsshown in Fig. 4, there are exactly two photons with en-ergy greater than 1 GeV in the 0.08 radian cone aroundthe π event. We refer to “isolated” events as those witha highest p T cone cluster with only a single pair of pho-ton candidates. “Non-isolated” events are more jet-like,having at least three photon candidates within the cone.For a large fraction of the covered kinematics, about 1/3of the inclusively selected π events contributing to Fig.4 have an isolated π . These more exclusive events havegenerally larger values of A N .It is seen from the comparison of Fig. 4 with Figs.10, 11, and 12 that A N for isolated π s is significantlygreater than for the complementary part of the inclusiveevent set with additional fragments observed.The electromagnetic calorimeter has limited sensitiv-ity to charged pions, so isolation does not guarantee theabsence of hadrons other than π s. However, this ob-servation hints at the possibility that the asymmetry forjets with a leading energy π is much less than the sin-gle π asymmetry in this forward kinematic region. Theenhanced A N for events with no observed jet fragmentmay indicate that these events are not related to jet pro-duction with fragmentation.4 ) c (GeV/ T p ( pp ) N ) / A A u ( p N A STAR 0.139 – T p All 0.139 – T p Low <0.21 F x ) c (GeV/ T p ( pp ) N ) / A A u ( p N A STAR 0.046 – T p All 0.303 – T p Low 0.047 – T p High <0.27 F x ) c (GeV/ T p ( pp ) N ) / A A u ( p N A STAR 0.034 – T p All 0.112 – T p Low 0.036 – T p High <0.37 F x ) c (GeV/ T p ( pp ) N ) / A A u ( p N A STAR 0.046 – T p All 0.085 – T p Low 0.055 – T p High <0.47 F x ) c (GeV/ T p ( pp ) N ) / A A u ( p N A STAR 0.073 – T p All 0.246 – T p Low 0.076 – T p High <0.61 F x ) c (GeV/ T p ( pp ) N ) / A A u ( p N A STAR 0.234 – T p All 0.234 – T p High <0.81 F x FIG. 6. The transverse momentum p T dependence of the ratio of A N for p Au scattering to that for pp for six Feynman x F ranges. This figure refers to the same data as is plotted in Fig. 4. The event selection criteria are given in the text. Thestatistical uncertainties are shown with vertical error bars, and the filled boxes indicate systematic uncertainties appropriatefor the ratio. Horizontal lines indicate the fit to the average ratio over the region 1 . < p T < . c , 2 . < p T < . c and for the combined p T range. The observation that isolated π events have larger A N does not appear to depend upon the nuclear size A in pA collisions. In Fig. 13 the determination of theexponent P in the A dependence, defined in Eq. 9, hasbeen analyzed separately for isolated and non-isolatedevents. The average exponents are similar for these twosubsets of the data.This dependence of the measured A N on event topol-ogy is further described in a jet analysis [39], with someof these same data. Although technical aspects of thatanalyses differ from this one, the results are consistent inthose cases where the same quantity is measured. VIII. CONCLUSIONS
This new measurement of A N for forward π produc-tion, in pp , p Al and p Au collisions, determines the depen-dence on x F and p T . It is observed that A N generallyincreases with increeasing p T at fixed x F (0 . < x F < . p T up to 5 GeV / c. In many calculations, ex-emplified by the simple model of Eq. 4, A N is expectedto fall with p T when p T is significantly larger than somenominal QCD scale k T , representing the spin dependentpart of the transverse momentum shift due to initial orfinal interactions. The persistent rise in A N for p T wellbeyond the 1 GeV / c scale, is unexpected.Furthermore, the asymmetry A N , for forward π pro-duction is significantly larger for events with an observedisolated π than for events that show evidence of addi-tional fragmentation products. It is interesting to com-pare this result to the published A N for jets, from [19],where the asymmetry was observed to be small comparedto this π measurement. The Sivers picture, where aproton spin dependent transverse momentum k T is ac-quired from initial state interactions, is not the naturalchoice for explaining the difference in A N for isolatedand non-isolated π s in the final state. But neither isthe enhancement of A N for isolated pions expected inthe Collins picture, where jet fragmentation into multi-5 ) c (GeV/ T p ( pp ) N ) / A A l ( p N A STAR 0.214 – T p All 0.214 – T p Low <0.21 F x ) c (GeV/ T p ( pp ) N ) / A A l ( p N A STAR 0.057 – T p All 0.548 – T p Low 0.057 – T p High <0.27 F x ) c (GeV/ T p ( pp ) N ) / A A l ( p N A STAR 0.041 – T p All 0.147 – T p Low 0.042 – T p High <0.37 F x ) c (GeV/ T p ( pp ) N ) / A A l ( p N A STAR 0.057 – T p All 0.141 – T p Low 0.062 – T p High <0.47 F x ) c (GeV/ T p ( pp ) N ) / A A l ( p N A STAR 0.081 – T p All 0.371 – T p Low 0.083 – T p High <0.61 F x ) c (GeV/ T p ( pp ) N ) / A A l ( p N A STAR 0.239 – T p All 0.239 – T p High <0.81 F x FIG. 7. Similar to Fig. 6 but the ratio of A N in p Al to that in pp . ple hadrons imparts a spin dependent momentum k T tothe observed pion, to generate pion asymmetry.The kinematic dependence of A N on x F and p T issimilar for the three collision systems. The suppressionof A N in collisions with nuclear beams is modest, withthe typical A N ratios between p Au and pp greater than80%. When the suppression of A N is fit to a power lawnuclear A dependence, A N ( A ) ∝ A P , the measured ex-ponents from Type 2 fits are in the range of − .
00. The weighted average exponent in Fig. 9 is (cid:104) P (cid:105) = − . ± . r (Au) = 0 . ± .
02. For the Type 1 fits in the low p T region, the weighted average is (cid:104) P L (cid:105) = − . ± . r L (Au) (cid:39) . ± .
06. In the high p T region,the weighted average is (cid:104) P H (cid:105) = − . ± . r H (Au) (cid:39) . ± .
02. There is no significant differencebetween the exponent P H in the higher p T region and P L in the low p T region, where gluon saturation effectscould be most relevant. The general agreement betweenType 1 and Type 2 fits helps to give confidence in thefitting methods.This nuclear suppression of π A N is much lessthan that reported by the PHENIX collaboration, forpositively charged hadrons at somewhat lower pseudo- rapidity or lower x F . The fits from the PHENIX mea-surement favored an exponent P = − .
37 [38]. Un-like the result of this paper, the PHENIX results arenominally consistent with the prediction of Hatta et al.,( P = − / x F coverage by thePHENIX measurement, 0 . < x F < .
2, is below therange presented here, shown in Fig. 9. The range ofgluon momentum fractions, x , probed within the unpo-larized beams in this measurement is x < . x range probed in the PHENIX measurement.The distribution of exponents shown in Fig. 9 indi-cates that the P exponents slowly increase with increas-ing x F . The Type 2 data points can be fit to the lin-ear form P ( x F ) = P + P (cid:48) x F , yielding fitted parameters P = − . ± .
02 and P (cid:48) = 0 . ± .
05. Linear extrapo-lation of these data into the center of the PHENIX accep-tance gives an exponent P ( x F = 0 .
15) = − . ± . χ = ± P range, of approx-imately − . < P < − .
25. Comparing this to theSTAR extrapolated value, the difference appears signifi-cant. From the χ plot in the PHENIX paper, the value, P = − .
06, corresponds to χ (cid:39)
13. Of course, the linear6
Log A ( pp ) N ( p A ) / A N A P STAR Fit to Ratio = A0.036 (Type1) – P= -0.075 0.042 (Type2) – P= -0.049 <0.21 F x Log A ( pp ) N ( p A ) / A N A P STAR Fit to Ratio = A0.010 (Type1) – P= -0.058 0.012 (Type2) – P= -0.054 <0.27 F x Log A ( pp ) N ( p A ) / A N A P STAR Fit to Ratio = A0.007 (Type1) – P= -0.037 0.008 (Type2) – P= -0.026 <0.37 F x Log A ( pp ) N ( p A ) / A N A P STAR Fit to Ratio = A0.009 (Type1) – P= -0.032 0.010 (Type2) – P= -0.020 <0.47 F x Log A ( pp ) N ( p A ) / A N A P STAR Fit to Ratio = A0.013 (Type1) – P= -0.019 0.014 (Type2) – P= -0.001 <0.61 F x Log A ( pp ) N ( p A ) / A N A <0.81 F x P STAR Fit to Ratio = A0.056 (Type1) – P= -0.075 0.047 (Type2) – P= 0.000 <0.81 F x FIG. 8. The ratio of A N for pA scattering to that for pp scattering is shown for six x F regions, averaging over the full range of p T dependence. The fitted form for these ratios as a function of A is obtained using Type 1 and Type 2 analyses as described inthe text. The dependence of A N as a function of log A is displayed with a filled error band, obtained from the Type 2 analysis,shown as the dashed line. extrapolation is just an assumption.Combining all beam types to maximize statistics for A N measurements, for Feynman x F < .
47 the asymme-try A N increases with x F and with p T . For x F > . A N on p T flattens or may begin to fall with p T over the measured p T range.These measurements of the dependence of A N , for for-ward π production, on kinematics and event topology,should provide new input for ongoing theoretical studiesof the underlying dynamics for these processes. In pA collisions, the dependence of A N on nuclear size A hasbeen measured and is small. ACKNOWLEDGMENTS
We thank the RHIC Operations Group and RCF atBNL, the NERSC Center at LBNL, and the Open Sci-ence Grid consortium for providing resources and sup-port. This work was supported in part by the Office of Nuclear Physics within the U.S. DOE Office of Sci-ence, the U.S. National Science Foundation, the Min-istry of Education and Science of the Russian Federa-tion, National Natural Science Foundation of China, Chi-nese Academy of Science, the Ministry of Science andTechnology of China and the Chinese Ministry of Educa-tion, the Higher Education Sprout Project by Ministryof Education at NCKU, the National Research Founda-tion of Korea, Czech Science Foundation and Ministryof Education, Youth and Sports of the Czech Republic,Hungarian National Research, Development and Innova-tion Office, New National Excellency Programme of theHungarian Ministry of Human Capacities, Departmentof Atomic Energy and Department of Science and Tech-nology of the Government of India, the National ScienceCentre of Poland, the Ministry of Science, Education andSports of the Republic of Croatia, RosAtom of Russia andGerman Bundesministerium fur Bildung, Wissenschaft,Forschung and Technologie (BMBF), Helmholtz Associ-ation, Ministry of Education, Culture, Sports, Science,and Technology (MEXT) and Japan Society for the Pro-7 F x P - - - - STAR T P from Type 2 Fit, Averaged over All p T from Type 1 Fit, Averaged over Low p L P T from Type 1 Fit, Averaged over High p H P STAR T P from Type 2 Fit, Averaged over All p T from Type 1 Fit, Averaged over Low p L P T from Type 1 Fit, Averaged over High p H P P (pp)=A N (pA)/A N A Dependence: A Fit to Power Law
FIG. 9. Analyzing separately the low p T (1 . < p T < . c ) data and the higher p T data, the exponent, P , for nuclear A dependence of the asymmetry ratio A N ( pA ) A N ( pp ) = A P is shown as a function of x F . Points are included, averaging over the low p T region and high p T regions ( p T > . c ) separately. Examples of one parameter power law fits for P are shown in Fig. 8,where power dependence exponent P is plotted as a function of Feynman x F . The uncertainties shown are from fits describedin the text and are dominated by statistical uncertainties. The systematic uncertainties are small, mostly cancelling, in ratiosbetween different nuclear A data sets and are not separately shown. motion of Science (JSPS).8 ) c (GeV/ T p N A - STAR 200 GeV p not isolated pp isolated pp <0.21 F x ) c (GeV/ T p N A <0.47 F x ) c (GeV/ T p N A - <0.27 F x ) c (GeV/ T p N A <0.61 F x ) c (GeV/ T p N A - <0.37 F x ) c (GeV/ T p N A <0.81 F FIG. 10. The transverse momentum p T dependence of A N for pion production in six ( x F ) regions for pp collisions. The datafrom Fig. 4 have been divided into two parts based on whether the π is produced with additional jet-like fragments of energymore than 1 GeV, shown with filled markers, or in isolation shown with open markers. The event selection criteria for isolatedand non-isolated events are given in the text. The statistical uncertainties are shown with vertical error bars. The filled boxesindicate horizontal and vertical systematic uncertainties. ) c (GeV/ T p N A STAR 200 GeV p Al not isolated p Al isolated p <0.21 F x ) c (GeV/ T p N A <0.47 F x ) c (GeV/ T p N A <0.27 F x ) c (GeV/ T p N A <0.61 F x ) c (GeV/ T p N A <0.37 F x ) c (GeV/ T p N A <0.81 F FIG. 11. This plot is similar to Fig. 10 but for p Al collisions. ) c (GeV/ T p N A - STAR 200 GeV p Au not isloated p Au isolated p <0.21 F x ) c (GeV/ T p N A <0.47 F x ) c (GeV/ T p N A - <0.27 F x ) c (GeV/ T p N A <0.61 F x ) c (GeV/ T p N A - <0.37 F x ) c (GeV/ T p N A <0.81 F FIG. 12. This plot is similar to Fig. 10 but for p Au collisions. F x P - - - - P (pp)=A N (pA)/A N A Dependence: A Fit to Power Law
STARIsolated Type 2 FitNon-Isolated Type 2 Fit
FIG. 13. Comparison of the nuclear A dependence of A N for events with isolated π s and events with non-isolated π s. Theexponent P of the nuclear A dependence is shown as a function of x F . The exponent is defined in Eq. 9. The points shown areaveraged over the full p T range with a Type 2 fit at each p T and x F similar to the points of Fig. 9. [1] R. D. Klem, J. E. Bowers, H. W. Courant, H. Kagan,M. L. Marshak, E. A. Peterson, K. Ruddick, W. H.Dragoset, and J. B. Roberts, Measurement of Asymme-tries of Inclusive Pion Production in Proton Proton In-teractions at 6-GeV/c and 11.8-GeV/c. Phys. Rev. Lett.
36, 929 (1976).[2] S. Saroff et al, Single Spin Asymmetry in Inclusive Re-actions Polarized P , P Goes to π + , π − , and P at High P (t) at 13.3-GeV/ c and 18.5-GeV/ c . Phys. Rev. Lett.
Phys. Lett.
B261, 201 (1991).[4] D.L. Adams et al, Analyzing power in inclusive pi+ andpi- production at high x(F) with a 200-GeV polarizedproton beam.
Phys.Lett.
B264, 462 (1991).[5] I. Arsene et al, Single Transverse Spin Asymmetriesof Identified Charged Hadrons in Polarized p+p Colli-sions at s**(1/2) = 62.4-GeV.
Phys.Rev.Lett. √ s =200 and 62.4 GeV. Phys. Rev.
D90, 012006 (2014)[7] B.I. Abelev et al, Forward Neutral Pion Transverse SingleSpin Asymmetries in p+p Collisions at s**(1/2) = 200-GeV.
Phys.Rev.Lett.
Phys. Rev.
D1, 1376 (1970).[9] M. J. Alguard et al, Deep Inelastic Scattering of Polar-ized Electrons by Polarized Protons.
Phys. Rev. Lett.
Phys. Lett.
B206, 364 (1988).[11] Daniel de Florian, Rodolfo Sassot, Marco Stratmann, andWerner Vogelsang, Extraction of Spin-Dependent Par-ton Densities and Their Uncertainties.
Phys. Rev.
D80,034030 (2009).[12] Emanuele R. Nocera, Richard D. Ball, Stefano Forte,Giovanni Ridolfi, and Juan Rojo, A first unbiased globaldetermination of polarized PDFs and their uncertainties.
Nucl. Phys.
B887, 276 (2014).[13] Daniel de Florian, Rodolfo Sassot, Marco Stratmann, andWerner Vogelsang, Evidence for polarization of gluons inthe proton.
Phys. Rev. Lett.
Phys.Rev.Lett.
41, 1689 (1978).[15] John C. Collins, Davison E. Soper, and George F. Ster-man,
Factorization of Hard Processes in QCD . Adv. Ser.Dir. High Energy Phys.
5, 1 (1989).[16] D. W. Sivers, Hard scattering scaling laws for single spinproduction asymmetries.
Phys. Rev.
D43, 261 (1991).[17] J. C. Collins, S. F. Heppelmann, and G. A. Ladin-sky, Measuring transversity densities in singly polarizedhadron hadron and lepton - hadron collisions.
Nucl. Phys.
B420, 565 (1994). [18] J. Qiu and G. Sterman, Single transverse-spin asym-metries in hadronic pion production.
Phys. Rev.
D59,014004 (1999).[19] L. C. Bland et al, Cross Sections and Transverse Single-Spin Asymmetries in Forward Jet Production from Pro-ton Collisions at √ s = 500 GeV. Phys. Lett.
B750, 660(2015).[20] Stanley J. Brodsky, Dae Sung Hwang, and Ivan Schmidt,Initial-state interactions and single-spin asymmetries inDrell-Yan processes.
Nucl. Phys.
B642, 344 (2002).[21] John C. Collins, Leading twist single transverse-spinasymmetries: Drell-Yan and deep inelastic scattering.
Phys. Lett.
B536, 43 (2002).[22] C. Kouvaris, J. Qiu, W. Vogelsang, and F. Yuan, Singletransverse-spin asymmetry in high transverse momentumpion production in p p collisions.
Phys. Rev.
D74, 114013(2006).[23] M. Anselmino, M. Boglione, U. D’Alesio, S. Melis,F. Murgia, and A. Prokudin, Sivers effect and the singlespin asymmetry A N in p ↑ p → hX processes. Phys. Rev.
D88, 054023 (2013).[24] Koichi Kanazawa, Yuji Koike, Andreas Metz, and DanielPitonyak. Towards an explanation of transverse single-spin asymmetries in proton-proton collisions: the role offragmentation in collinear factorization.
Phys. Rev.
D89,111501 (2014).[25] M. Anselmino, U. D’Alesio, and S. Melis, Transversesingle-spin asymmetries in proton-proton collisions atthe AFTER@LHC experiment in a TMD factorisationscheme.
Adv. High Energy Phys.
JHEP
01, 111 (2019).[27] Yuri V. Kovchegov and Matthew D. Sievert, A NewMechanism for Generating a Single Transverse SpinAsymmetry.
Phys. Rev.
D86, 034028 (2012).[28] Yuri V. Kovchegov and M. Gabriel Santiago, LensingMechanism Meets Small- x Physics: Single TransverseSpin Asymmetry in p ↑ + p and p ↑ + A Collisions.
Phys.Rev.
D102, 014022 (2020).[29] Edmond Iancu, Andrei Leonidov, and Larry D. McLer-ran, Nonlinear gluon evolution in the color glass conden-sate. 1.
Nucl. Phys.
A692, 583 (2001).[30] John Adams et al, Forward neutral pion productionin p+p and d+au collisions at s(nn)**(1/2) = 200-gev.
Phys. Rev. Lett.
97, 152302 (2006).[31] I. Arsene et al, On the evolution of the nuclear modi-fication factors with rapidity and centrality in d + Aucollisions at s(NN)**(1/2) = 200-GeV.
Phys. Rev. Lett.
93, 242303 (2004).[32] Jaroslav Adam et al, Longitudinal Double-Spin Asym-metries for π s in the Forward Direction for 510 GeVPolarized pp Collisions.
Phys. Rev.
D98, 032013 (2018).[33] J. Kiryluk, Local polarimetry for proton beams with theSTAR beam beam counters. In
Spin physics. Polarizedelectron sources and polarimeters. Proceedings, 16th In-ternational Symposium, SPIN 2004, Trieste, Italy, Oc-tober 10-16, 2004, and Workshop, PESP 2004, Mainz,Germany, October 7-9, 2004 , pages 718 (2005). [34] C. A. Whitten, The beam-beam counter: A local po-larimeter at STAR. AIP Conf. Proc. pA col-lisions II: Fragmentation contribution. Phys. Rev.
D95,014008 (2017). [37] Zhong-Bo Kang and Feng Yuan, Single Spin AsymmetryScaling in the Forward Rapidity Region at RHIC.
Phys.Rev.
D84, 034019 (2011).[38] C. Aidala et al, Nuclear Dependence of the TransverseSingle-Spin Asymmetry in the Production of ChargedHadrons at Forward Rapidity in Polarized p + p , p +Al,and p +Au Collisions at √ s NN = 200 GeV. Phys. Rev.Lett. π and electromagnetic jets at forward ra-pidity in 200 and 500 GeV transversely polarized proton-proton collisions. arXiv:2012.11428 [hep-ex]arXiv:2012.11428 [hep-ex]