Spin Structure of the Proton from Polarized Inclusive Deep-Inelastic Muon-Proton Scattering

PHYSICAL REVIEW D VOLUME 56, NUMBER 9 1 NOVEMBER 1997

Spin structure of the proton from polarized inclusive deep-inelastic muon-proton scattering

D. Adams,17 B. Adeva,19 E. Arik,2 A. Arvidson,22,a B. Badelek,22,24 M. K. Ballintijn,14 G. Bardin,18,b G. Baum,1 P. Berglund,7 L. Betev,12 I. G. Bird,18,c R. Birsa,21 P. Bjo¨rkholm,22,d B. E. Bonner,17 N. de Botton,18 M. Boutemeur,25,e F. Bradamante,21 A. Bravar,10 A. Bressan,21 S. Bu¨ltmann,1,f E. Burtin,18 C. Cavata,18 D. Crabb,23 J. Cranshaw,17,g T. Ç uhadar,2 S. Dalla Torre,21 R. van Dantzig,14 B. Derro,3 A. Deshpande,25 S. Dhawan,25 C. Dulya,3,h A. Dyring,22 S. Eichblatt,17,i J. C. Faivre,18 D. Fasching,16,j F. Feinstein,18 C. Fernandez,19,8 B. Frois,18 A. Gallas,19 J. A. Garzon,19,8 T. Gaussiran,17,k M. Giorgi,21 E. von Goeler,15 G. Gracia,19,h N. de Groot,14,l M. Grosse Perdekamp,3,m E. Gu¨lmez,2 D. von Harrach,10 T. Hasegawa,13,n P. Hautle,4,o N. Hayashi,13,p C. A. Heusch,4,q N. Horikawa,13 V. W. Hughes,25 G. Igo,3 S. Ishimoto,13,r T. Iwata,13 E. M. Kabuß,10 A. Karev,9 H. J. Kessler,5 T. J. Ketel,14 A. Kishi,13 Yu. Kisselev,9 L. Klostermann,14,s D. Kra¨mer,1 V. Krivokhijine,9 W. Kro¨ger,4,q K. Kurek,24 J. Kyyna¨raïnen,4,7,t M. Lamanna,21 U. Landgraf,5 T. Layda,4 J. M. Le Goff,18 F. Lehar,18 A. de Lesquen,18 J. Lichtenstadt,20 T. Lindqvist,22 M. Litmaath,14,u M. Lowe,17,j A. Magnon,18 G. K. Mallot,10 F. Marie,18 A. Martin,21 J. Martino,18 T. Matsuda,13,n B. Mayes,8 J. S. McCarthy,23 K. Medved,9 G. van Middelkoop,14 D. Miller,16 K. Mori,13 J. Moromisato,15 A. Nagaitsev,9 J. Nassalski,24 L. Naumann,4,b T. O. Niinikoski,4 J. E. J. Oberski,14 A. Ogawa,13 C. Ozben,2 D. P. Parks,8 A. Penzo,21 F. Perrot-Kunne,18 D. Peshekhonov,9 R. Piegaia,25,v L. Pinsky,8 S. Platchkov,18 M. Plo,19 D. Pose,9 H. Postma,14 J. Pretz,10 T. Pussieux,18 J. Pyrlik,8 I. Reyhancan,2 A. Rijllart,4 J. B. Roberts,17 S. Rock,4,w M. Rodriguez,22,v E. Rondio,24,u A. Rosado,12 I. Sabo,20 J. Saborido,19 A. Sandacz,24 I. Savin,9 P. Schiavon,21 K. P. Schu¨ler,25,x R. Segel,16 R. Seitz,10,y Y. Semertzidis,4,z F. Sever,14,aa P. Shanahan,16,i E. P. Sichtermann,14 F. Simeoni,21 G. I. Smirnov,9 A. Staude,12 A. Steinmetz,10 U. Stiegler,4 H. Stuhrmann,6 M. Szleper,24 K. M. Teichert,12 F. Tessarotto,21 W. Tlaczala,24 S. Trentalange,3 G. Unel,2 M. Velasco,16,u J. Vogt,12 R. Voss,4 R. Weinstein,8 C. Whitten,3 R. Windmolders,11 R. Willumeit,6 W. Wislicki,24 A. Witzmann,5,bb A. M. Zanetti,21 K. Zaremba,24 and J. Zhao6,cc ͑Spin Muon Collaboration͒ 1University of Bielefeld, Physics Department, 33501 Bielefeld, Germany 2Bogaziçi University and Istanbul Technical University, Istanbul, Turkey 3University of California, Department of Physics, Los Angeles, California 90024 4CERN, 1211 Geneva 23, Switzerland 5University of Freiburg, Physics Department, 79104 Freiburg, Germany 6GKSS, 21494 Geesthacht, Germany 7Helsinki University of Technology, Low Temperature Laboratory and Institute of Particle Physics Technology, Espoo, Finland 8University of Houston, Department of Physics, and Institute for Beam Particle Dynamics, Houston, Texas 77204 9JINR, Dubna, RU-141980 Dubna, Russia 10University of Mainz, Institute for Nuclear Physics, 55099 Mainz, Germany 11University of Mons, Faculty of Science, 7000 Mons, Belgium 12University of Munich, Physics Department, 80799 Munich, Germany 13Nagoya University, CIRSE and Department of Physics, Furo-Cho, Chikusa-Ku, 464 Nagoya, Japan 14NIKHEF, Delft University of Technology, FOM and Free University, 1009 AJ Amsterdam, The Netherlands 15Northeastern University, Department of Physics, Boston, Massachusetts 02115 16Northwestern University, Department of Physics, Evanston, Illinois 60208 17Rice University, Bonner Laboratory, Houston, Texas 77251-1892 18C.E.A. Saclay, DAPNIA, 91191 Gif-sur-Yvette, France 19University of Santiago, Department of Particle Physics, 15706 Santiago de Compostela, Spain 20Tel Aviv University, School of Physics, 69978 Tel Aviv, Israel 21INFN Trieste and University of Trieste, Department of Physics, 34127 Trieste, Italy 22Uppsala University, Department of Radiation Sciences, 75121 Uppsala, Sweden 23University of Virginia, Department of Physics, Charlottesville, Virginia 22901 24Soltan Institute for Nuclear Studies and Warsaw University, 00681 Warsaw, Poland 25Yale University, Department of Physics, New Haven, Connecticut 06511 ͑Received 11 February 1997͒

p We have measured the spin-dependent structure function g1 in inclusive deep-inelastic scattering of polarized muons off polarized protons, in the kinematic range 0.003ϽxϽ0.7 and 1 GeV2ϽQ2Ͻ60 GeV2. A next- p 2 2 p to-leading order QCD analysis is used to evolve the measured g1(x,Q ) to a fixed Q0. The first moment of g1 2 2 p at Q0ϭ10 GeV is ⌫1ϭ0.136Ϯ0.013 ͑stat͒ Ϯ0.009 ͑syst͒ Ϯ0.005 ͑evol͒. This result is below the prediction of the Ellis-Jaffe sum rule by more than two standard deviations. The singlet axial charge a0 is found to be 0.28Ϯ0.16. In the Adler-Bardeen factorization scheme, ⌬gӍ2 is required to bring ⌬⌺ in agreement with the quark-parton model. A combined analysis of all available proton, deuteron, and 3He data confirms the Bjorken sum rule. ͓S0556-2821͑97͒05521-5͔ PACS number͑s͒: 13.60.Hb, 13.88.ϩe

I. INTRODUCTION The first experiments on polarized electron-proton scattering were carried out by the E80 and E130 Collaborations at Deep-inelastic scattering of leptons from nucleons has re- SLAC ͓1͔. They measured significant spin-dependent asym- vealed much of what is known about quarks and gluons. The metries in deep-inelastic electron-proton scattering cross sec- scattering of high-energy charged polarized leptons on polar- tions, and their results were consistent with the Ellis-Jaffe ized nucleons provides insight into the spin structure of the and Bjorken sum rules with some plausible models of proton nucleon at the parton level. The spin-dependent nucleon spin structure. Subsequently, a similar experiment with a po- structure functions determined from these measurements are larized muon beam and polarized proton target was made by fundamental properties of the nucleon as are the spin- the European Muon Collaboration EMC at CERN 2 . With independent structure functions, and they provide crucial in- ͑ ͒ ͓ ͔ formation for the development and testing of perturbative a tenfold higher beam energy as compared to that at SLAC, and nonperturbative quantum chromodynamics ͑QCD͒. Ex- the EMC measurement covered a much larger kinematic amples are the QCD spin-dependent sum rules and calcula- range than the electron scattering experiments and found the tions by lattice gauge theory. violation of the Ellis-Jaffe sum rule ͓3͔. This implies, in the framework of the quark-parton model ͑QPM͒, that the total contribution of the quark spins to the proton spin is small. This result was a great surprise and posed a major prob- aNow at Gammadata, Uppsala, Sweden. lem for the QPM, particularly because of the success of the bDeceased. QPM in explaining the magnetic moments of hadrons in cNow at CEBAF, Newport News, VA 23606. terms of three valence quarks. It stimulated a new series of dNow at Ericsson Infocom AB, Sweden. polarized electron and muon nucleon scattering experiments eNow at the University of Montreal, H3C 3J7, Montreal, PQ, which by now have achieved the following: ͑1͒ inclusive Canada. f scattering measurements of the spin-dependent structure Now at the University of Virginia, Department of Physics, Char- p lottesville, VA 22901. function g1 of the proton with improved accuracy over an g enlarged kinematic range; ͑2͒ evaluation of the first moment Now at INFN Trieste, 34127 Trieste, Italy. p 1 p hNow at NIKHEF, 1009 AJ Amsterdam, The Netherlands. of the proton spin structure function, ⌫1ϭ͐0g1(x)dx, with iNow at Fermi National Accelerator Laboratory, Batavia, IL reduced statistical and systematic errors; ͑3͒ similar measure- 3 60510. ments with polarized deuteron and He targets, in order to jNow at the University of Wisconsin. measure the neutron spin structure function and test the fun- k p n Now at Applied Research Laboratories, University of Texas at damental Bjorken sum rule for ⌫1Ϫ⌫1 ͓4͔; ͑4͒ measure- Austin. ments of the spin-dependent structure function g2 for the lNow at SLAC, Stanford, CA 94309. proton and neutron; and ͑5͒ semi-inclusive measurements of mNow at Yale University, Department of Physics, New Haven, final states, which allow determination of the separate va- 06511. lence and sea quark contributions to the nucleon spin. nPermanent address: Miyazaki University, Faculty of Engineer- The recent measurements of polarized muon-nucleon ing, 889-21 Miyazaki-Shi, Japan. scattering have been done by the Spin Muon Collaboration oPermanent address: Paul Scherrer Institut, 5232 Villigen, Swit- ͑SMC͒ at CERN with polarized muon beams of 100 and 190 zerland. GeV obtained from the CERN Super Proton Synchrotron pPermanent address: The Institute of Physical and Chemical Re- ͑SPS͒ 450 GeV proton beam and with polarized proton and search ͑RIKEN͒, Wako 351-01, Japan. deuteron targets. Spin-dependent cross section asymmetries qPermanent address: University of California, Institute of Par- are measured over a wide kinematic range with relatively 2 ticle Physics, Santa Cruz, CA 95064. high Q and extending to small x values. The determination r 2 Permanent address: KEK, Tsukuba-Shi, 305 Ibaraki-Ken, Ja- of g1(x,Q ) for the proton and deuteron has been the prin- pan. cipal result of the SMC experiment, but g2 and semi- sNow at Ericsson Telecommunication, 5120 AA Rijen, The Neth- inclusive measurements have also been made. erlands. The recent measurements of polarized electron-nucleon tNow at the University of Bielefeld, Physics Department, 33501 scattering have been done principally at SLAC in experiment 3 Bielefeld, Germany. E142 ͓5͔͑beam energy Eeϳ19, 23, and 26 GeV, He tar- u Now at CERN, 1211 Geneva 23, Switzerland. get͒, E143 ͓6,7͔͑beam energy Eeϳ9, 16, and 29 GeV, H and v 3 Permanent address: University of Buenos Aires, Physics De- D targets͒, and E154 ͑Eeϳ48 GeV, He target͒. SLAC E155 partment, 1428 Buenos Aires, Argentina. with Eeϳ50 GeV and polarized proton and deuteron targets wPermanent address: The American University, Washington, will take data soon. The SLAC experiments provide inclu- D.C. 20016. sive measurements of g1 and g2 over a kinematic range of xNow at DESY. relatively low Q2 and do not extend to very small x values. yNow at Dresden Technical University, 01062 Dresden, Germany. However, the electron scattering experiments involve very zPermanent address: Brookhaven National Laboratory, Upton, high beam intensities and achieve excellent statistical accu- NY 11973. racies. Hence the electron and muon experiments are aaPresent address: ESFR, F-38043 Grenoble, France. complementary. Recently, the HERMES experiment has be- bbNow at F. Hoffmann-La Roche Ltd., CH-4070 Basel, Switzer- come operational and has reported preliminary results with a land. polarized 3He target ͓8͔. This experiment uses a polarized ccNow at Los Alamos National Laboratory, Los Alamos, NM electron beam of 27 GeV in the electron ring at the DESY 87545. ep collider HERA and an internal polarized gas 5332 D. ADAMS et al. 56 target. Both inclusive and semi-inclusive data were obtained, and polarized H and D targets will be used in the future. In this paper, we present SMC results on the spin- p p dependent structure functions g1 and g2 of the proton, obtained from data taken in 1993 with a polarized butanol target. First results from these measurements were published in Refs. ͓9, 10͔. We use here the same data sample, but present a more refined analysis; in particular, the influence of the radiative corrections on the statistical error on the asymmetry is now properly taken into account, resulting in an observable increase of this error at small x, and we allow for a Q2 p evolution of the g1 structure function as predicted by pertur- FIG. 1. Lepton and nucleon kinematic variables in polarized bative QCD. SMC has also published results on the deuteron lepton scattering on a fixed polarized nucleon target. d structure function g1 ͓11–13͔ and on a measurement of semi-inclusive cross section asymmetries ͓14͔. For a test of The spin-dependent part of the cross section can be writ- d ten in terms of two structure functions g and g which the Bjorken sum rule, we refer to our measurement of g1 . 1 2 The paper is organized as follows. In Sec. II we review describe the interaction of lepton and hadron currents. When the theoretical background. The experimental setup and the the lepton spin and the nucleon spin form an angle ␺,itcan data-taking procedure are described in Sec. III. In Sec. IV we be expressed as ͓16͔ discuss the analysis of cross section asymmetries, and in Sec. V we give the evaluation of the spin-dependent structure ⌬␴ϭcos ␺⌬␴ʈϩsin ␺ cos ␾⌬␴Ќ , ͑2.4͒ function g p and its first moment. The results for g p are dis- 1 2 where ␾ is the azimuthal angle between the scattering plane cussed in Sec. VI. In Sec. VII we combine proton and deu- n and the spin plane ͑Fig. 1͒. teron results to determine the structure function g1 of the The cross sections ⌬␴ʈ and ⌬␴Ќ refer to the two configu- neutron and to test the Bjorken sum rule. In Sec. VIII we rations where the nucleon spin is ͑anti͒parallel or orthogonal interpret our results in terms of the spin structure of the pro- to the lepton spin; ⌬␴ʈ is the difference between the cross ton. Finally, we present our conclusions in Sec. IX. sections for antiparallel and parallel spin orientations and ⌬␴ЌϭϪhl⌬␴T /cos ␾ the difference between the cross sec- II. THEORETICAL OVERVIEW tions at angles ␾ and ␾ϩ␲. The corresponding differential A. Cross sections for polarized lepton-nucleon scattering cross sections are given by 2 2 2 2 2 The polarized deep-inelastic lepton-nucleon inclusive d ⌬␴ʈ 16␲␣ y y ␥ y ␥ y ϭ 1Ϫ Ϫ g Ϫ g scattering cross section in the one-photon-exchange approxi- dxdQ2 Q4 ͫͩ 2 4 ͪ 1 2 2ͬ mation can be written as the sum of a spin-independent term ͑2.5͒ ¯␴ and a spin-dependent term ⌬␴ and involves the lepton helicity hlϭϮ1: and ϭ¯Ϫ 1 3 2 2 2 ␴ ␴ 2 hl␦␴. ͑2.1͒ d ⌬␴T 8␣ y ␥ y y ϭϪcos ␾ ␥ 1ϪyϪ g ϩg . dxdQ2d␾ Q4 ͱ 4 ͩ 2 1 2 ͪ For longitudinally polarized leptons the spin Sl is along the lepton momentum k. The spin-independent cross section for ͑2.6͒ parity-conserving interactions can be expressed in terms of For a high beam energy E, ␥ is small since either x is small two unpolarized structure functions F1 and F2 . These func- or Q2 high. The structure function g is therefore best mea- 2 1 tions depend on the four-momentum transfer squared Q and sured in the ͑anti͒parallel configuration where it dominates the scaling variable xϭQ2/2M␯, where ␯ is the energy of the spin-dependent cross section; g2 is best obtained from a the exchanged virtual photon and M is the nucleon mass. measurement in the orthogonal configuration, combined with The double-differential cross section can be written as a a measurement of g . In all formulas used in this article, we 2 1 function of x and Q ͓15͔: consider only the single-virtual-photon exchange. The interference effects between virtual Z0 and photon exchange in 2 2 2 d ¯␴ 4␲␣ 2ml deep-inelastic muon scattering have been measured ͓17͔ and ϭ xy2 1Ϫ F ͑x,Q2͒ dxdQ2 Q4x ͫ ͩ Q2 ͪ 1 found to be small and compatible with the standard model expectations. They can be neglected in the kinematic range 2 2 ␥ y of current experiments. ϩ 1ϪyϪ F ͑x,Q2͒ , ͑2.2͒ ͩ 4 ͪ 2 ͬ B. Cross section asymmetries where ml is the lepton mass, yϭ␯/E in the laboratory system, and The spin-dependent cross section terms, Eqs. ͑2.5͒ and ͑2.6͒, make only a small contribution to the total deep- inelastic scattering cross section and furthermore their con- 2Mx ͱQ2 ␥ϭ ϭ . ͑2.3͒ tribution is, in general, reduced by incomplete beam and tar- ͱQ2 ␯ get polarizations. Therefore they can best be determined 56 SPIN STRUCTURE OF THE PROTON FROM POLARIZED... 5333 from measurements of cross section asymmetries in which respectively, where F1 is usually expressed in terms of F2 the spin-independent contribution cancels. The relevant and R: asymmetries are 1ϩ␥2 ⌬␴ʈ ⌬␴Ќ A ϭ , A ϭ , ͑2.7͒ F1ϭ F2 . ͑2.16͒ ʈ 2¯␴ Ќ 2¯␴ 2x͑1ϩR͒ which are related to the virtual photon-proton asymmetries These relations are used in the present analysis for the evalu- A1 and A2 by 2 ation of g1 in bins of x and Q , starting from the asymmetries measured in the parallel spin configuration and using AʈϭD͑A1ϩ␩A2͒, AЌϭd͑A2Ϫ␰A1͒, ͑2.8͒ 2 2 parametrizations of F2(x,Q ) and R(x,Q ). where The virtual photon-proton asymmetry A2 is evaluated from the measured transverse and longitudinal asymmetries 2 ␴1/2Ϫ␴3/2 g1Ϫ␥ g2 Aʈ and AЌ : A1ϭ ϭ , ␴1/2ϩ␴3/2 F1

1 AЌ Aʈ TL A ϭ ϩ␰ . ͑2.17͒ 2␴ g1ϩg2 2 1ϩ␩␰ ͩ d D ͪ A2ϭ ϭ␥ . ͑2.9͒ ␴1/2ϩ␴3/2 F1 2 In Eqs. ͑2.8͒ and ͑2.9͒, D is the depolarization factor of the From Eqs. ͑2.3͒ and ͑2.9͒, A2 has an explicit 1/ͱQ depen- virtual photon defined below and d, ␩, and ␰ are the kine- dence and is therefore expected to be small at high energies. matic factors: The structure function g2 is obtained from the measured asymmetries using Eqs. ͑2.9͒ and ͑2.17͒. ͱ1ϪyϪ␥2y 2/4 dϭ D, ͑2.10͒ 1Ϫy/2 C. Spin-dependent structure function g1 The significance of the spin-dependent structure function ␥͑1ϪyϪ␥2y 2/4͒ ␩ϭ , ͑2.11͒ g1 can be understood from the virtual photon asymmetry A1 . ͑1Ϫy/2͒͑1ϩ␥2y/2͒ As shown in Eq. ͑2.9͒, A1Ӎg1 /F1 or ␴1/2Ϫ␴3/2ϰg1 . In order to conserve angular momentum, a virtual photon with ␥͑1Ϫy/2͒ helicity ϩ1orϪ1 can only be absorbed by a quark with a ␰ϭ 2 . ͑2.12͒ 1 1 1ϩ␥ y/2 spin projection of Ϫ 2 or ϩ 2 , respectively, if the quarks have no orbital angular momentum. Hence, g1 contains informa- The cross sections ␴1/2 and ␴3/2 refer to the absorption of a tion on the quark spin orientations with respect to the proton transversely polarized virtual photon by a polarized proton spin direction. for total photon-proton angular momentum component along In the simplest quark-parton model, the quark densities the virtual photon axis of 1/2 and 3/2, respectively; ␴TL is an depend only on the momentum fraction x carried by the interference cross section due to the helicity spin-flip ampli- quark, and g1 is given by tude in forward Compton scattering ͓18͔. The depolarization n f factor D depends on y and on the ratio Rϭ␴L /␴T of longi- 1 2 tudinal and transverse photoabsorption cross sections: g1͑x͒ϭ ei ⌬qi͑x͒, ͑2.18͒ 2 i͚ϭ1 y͑2Ϫy͒͑1ϩ␥2y/2͒ Dϭ 2 2 2 2 2 2 . where y ͑1ϩ␥ ͒͑1Ϫ2ml /Q ͒ϩ2͑1ϪyϪ␥ y /4͒͑1ϩR͒ ͑2.13͒ ϭ ϩ Ϫ Ϫ ϩ¯ ϩ Ϫ¯Ϫ From Eqs. ͑2.8͒ and ͑2.9͒, we can express the virtual ⌬qi͑x͒ qi ͑x͒ qi ͑x͒ q i ͑x͒ qi ͑x͒, ͑2.19͒ photon-proton asymmetry A1 in terms of g1 and A2 and find the following relation for the longitudinal asymmetry: ϩ ϩ Ϫ Ϫ qi (¯q i ) and qi (¯q i ) are the distribution functions of quarks ͑antiquarks͒ with spin parallel and antiparallel to the A g ʈ 2 1 nucleon spin, respectively, e is the electric charge of the ϭ͑1ϩ␥ ͒ ϩ͑␩Ϫ␥͒A2 . ͑2.14͒ i D F1 quarks of flavor i, and n f is the number of quark flavors involved. The virtual-photon asymmetries are bounded by positivity In QCD, quarks interact by gluon exchange, which gives 2 relations ͉A1͉р1 and ͉A2͉рͱR ͓19͔. When the term propor- rise to a weak Q dependence of the structure functions. The tional to A 2 is neglected in Eqs. ͑2.8͒ and ͑2.14͒, the longi- treatment of g1 in perturbative QCD follows closely that of tudinal asymmetry is related to A1 and g1 by unpolarized parton distributions and structure functions ͓20͔. 2 At a given scale Q , g1 is related to the polarized quark and A g 1 A ʈ 1 ʈ gluon distributions by coefficient functions Cq and Cg A1Ӎ , Ӎ 2 , ͑2.15͒ D F1 1ϩ␥ D through ͓20͔ 5334 D. ADAMS et al. 56

n 1 f e2 1 dy x x x x k S C0,S ,␣ ϭC0,NS ,␣ ϭ␦ 1Ϫ , g1͑x,t͒ϭ ͚ Cq ,␣s͑t͒ ⌬⌺͑y,t͒ q ͩ y s ͪ q ͩ y s ͪ ͩ y ͪ 2 kϭ1 n ͵x y ͫ ͩy ͪ f

x 0 x ϩ2n C ,␣ ͑t͒ ⌬g͑y,t͒ Cg ,␣s ϭ0. ͑2.26͒ f gͩy s ͪ ͩ y ͪ

x Note that g1 decouples from ⌬g in this scheme. NS NS ϩCq ,␣s͑t͒ ⌬q ͑y,t͒ . ͑2.20͒ Beyond leading order, the coefficient functions and the ͩ y ͪ ͬ splitting functions are not uniquely defined; they depend on the renormalization scheme. The complete set of coefficient 2 2 In this equation, tϭln(Q /⌳ ), ␣s is the strong coupling con- functions has been computed in the modified minimal sub- 2 stant, and ⌳ is the scale parameter of QCD. The superscripts traction (MS) renormalization scheme up to order ␣s ͓23͔. 2 ‘‘S’’ and ‘‘NS,’’ respectively, indicate flavor-singlet and The O(␣s ) corrections to the polarized splitting functions nonsinglet parton distributions and coefficient functions; Pqq and Pqg have been computed in Ref. ͓23͔ and those to ⌬g(x,t) is the polarized gluon distribution, and ⌬⌺ and Pgq and Pgg in ͓24,25͔. This formalism allows a complete ⌬qNS are the singlet and nonsinglet combinations of the po- next-to-leading order ͑NLO͒ QCD analysis of the scaling larized quark and antiquark distributions: violations of spin-dependent structure functions. 2 In QCD, the ratio g1 /F1 is Q dependent because the nf splitting functions, with the exception of Pqq , are different for polarized and unpolarized parton distributions. Both P ⌬⌺͑x,t͒ϭ͚ ⌬qi͑x,t͒, ͑2.21͒ gq iϭ1 and Pgg are different in the two cases because of a soft gluon singularity at xϭ0, which is only present in the unpolarized

n f n f n f case. However, in kinematic regions dominated by valence 1 1 2 NS 2 2 2 quarks, the Q dependence of g1 /F1 is expected to be small ⌬q ͑x,t͒ϭ ͚ ei Ϫ ͚ ek ͚ ek ⌬qi͑x,t͒. ͫ iϭ1 ͩ n f kϭ1 ͪ Ͳ n f kϭ1 ͬ ͓26͔. ͑2.22͒ D. Small-x behavior of g1 The t dependence of the polarized quark and gluon distribu- The most important theoretical predictions for polarized tions follows the Gribov-Lipatov-Altarelli-Parisi ͑GLAP͒ deep-inelastic scattering are the sum rules for the nucleon equations ͓21,22͔. As for the unpolarized distributions, the structure functions g1 . The evaluation of the ﬁrst moment of polarized singlet and gluon distributions are coupled by g1 ,

1 1 2 2 d ␣s͑t͒ dy x ⌫1͑Q ͒ϭ g1͑x,Q ͒dx, ͑2.27͒ S ͵0 ⌬⌺͑x,t͒ϭ Pqq ,␣s͑t͒ ⌬⌺͑y,t͒ dt 2␲ ͵x y ͫ ͩy ͪ requires knowledge of g1 over the entire x region. Since the x experimentally accessible x range is limited, extrapolations ϩ2n P ,␣ ͑t͒ ⌬g͑y,t͒ , ͑2.23͒ f qgͩy s ͪ ͬ to xϭ0 and xϭ1 are unavoidable. The latter is not critical because it is constrained by the bound ͉A1͉р1 and gives only a small contribution to the integral. However, the small d ␣ ͑t͒ 1 dy x s x behavior of g1(x) is theoretically not well established and ⌬g͑x,t͒ϭ Pgq ,␣s͑t͒ ⌬⌺͑y,t͒ dt 2␲ ͵x y ͫ ͩy ͪ evaluation of ⌫ depends critically on the assumption made 1 for this extrapolation. x From the Regge model it is expected that for Q2 ϩPgg ,␣s͑t͒ ⌬g͑y,t͒ , ͑2.24͒ p n p n Ϫ␣ ͩy ͪ ͬ Ӷ2M␯, i.e., x 0, g1ϩg1 and g1Ϫg1 behave like x ͓27͔, where ␣ is the→ intercept of the lowest contributing Regge trajectories. These trajectories are those of the pseudovector whereas the nonsinglet distribution evolves independently of p n the singlet and gluon distributions: mesons f 1 for the isosinglet combination, g1ϩg1 and of a1 p n for the isotriplet combination, g1Ϫg1 , respectively. Their intercepts are negative and assumed to be equal and in the d ␣ ͑t͒ 1 dy x NS s NS NS Ϫ Ͻ Ͻ ⌬q ͑x,t͒ϭ Pqq ,␣s͑t͒ ⌬q ͑y,t͒. range 0.5 ␣ 0. Such behavior has been assumed in most dt 2␲ ͵x y ͩ y ͪ analyses. ͑2.25͒ A flavor-singlet contribution to g1(x) that varies as ͓2 ln(1/x)Ϫ1͔ ͓28͔ was obtained from a model where an Here Pij are the QCD splitting functions for polarized parton exchange of two nonperturbative gluons is assumed. Even 2 Ϫ1 distributions. very divergent dependences like g1(x)ϰ(x ln x) were Expressions ͑2.20͒, ͑2.23͒, ͑2.24͒, and ͑2.25͒ are valid in considered ͓29͔. Such dependences are not necessarily con- all orders of perturbative QCD. The quark and gluon distri- sistent with the QCD evolution equations. butions, coefficient functions, and splitting functions depend Expectations based on QCD calculations for the behavior 2 on the mass factorization scale and on the renormalization at small x of g1(x,Q ) are twofold. scale; we adopt here the simplest choice, setting both scales Resummation of standard Altarelli-Parisi corrections equal to Q2. At leading order, the coefficient functions are gives ͓30–32͔ 56 SPIN STRUCTURE OF THE PROTON FROM POLARIZED... 5335

2 2 2 in Eq. ͑2.34͒ under isospin symmetry is equal to the neutron g1͑x,Q ͒ϳexp Aͱln͓␣x͑Q0͒/␣s͑Q ͔͒ln͑1/x͒, ͑2.28͒ ␤-decay constant gA /gV . If exact SU͑3͒ symmetry is assumed for the axial ﬂavor-octet current, the axial couplings for the nonsinglet and singlet parts of g1 . a3 and a8 in Eqs. ͑2.34͒ and ͑2.35͒ can be expressed in terms Resummation of leading powers of ln(1/x) gives of coupling constants F and D, obtained from neutron and hyperon ␤ decays ͓3͔,as NS 2 ϪwNS g1 ͑x,Q ͒ϳx , wNSϳ0.4, ͑2.29͒

S 2 ϪwS a3ϭFϩD, a8ϭ3FϪD. ͑2.37͒ g1͑x,Q ͒ϳx , wSϳ3wNS , ͑2.30͒ for the nonsinglet ͓33͔ and singlet ͓34͔ parts, respectively. The effects of a possible SU͑3͒ symmetry breaking will be discussed in Sec. VIII B. E. Sum-rule predictions The first moment of the polarized quark distribution for flavor qi , that is, ⌬qiϭ͐⌬qi(x)dx, is the contribution of 1. First moment of g1 and the operator product expansion flavor qi to the spin of the nucleon. In the QPM, ai is inter- A powerful tool to study moments of structure functions preted as ⌬qi and a0 as ⌬⌺ϭ⌬uϩ⌬dϩ⌬s. In this frame- is provided by the operator product expansion ͑OPE͒, where work, the moments of x⌬u, x⌬d, and x⌬s are bound by a the product of the leptonic and the hadronic tensors describ- positivity limit given by the corresponding moments of ing polarized deep-inelastic lepton-nucleon scattering re- xu,xd,xs,... obtained from unpolarized structure functions. duces to the expansion of the product of two electromagnetic In Sec. II F we will see that the U͑1͒ anomaly modifies this currents. At leading twist, the only gauge-invariant contribu- simple interpretation of the axial couplings. tions are due to the nonsinglet and singlet axial currents 2 2 When Q is above the charm threshold (2mc) , four fla- ͓35,36͔. If only the contributions from the three lightest vors must be considered and an additional proton matrix el- quark flavors are considered, the axial current operator Ak ement must be defined, can be expressed in terms of the SU͑3͒ flavor matrices ␭k (kϭ1,...,8)and␭0ϭ2I as ͓36͔ 15 s␮ s␮ ͗ps͉A ͉ps͘ϭ ͑auϩadϩasϪ3ac͒ϭ a15 , ␭k ␮ Ak ϭ¯␺ ␥ ␥ ␺, ͑2.31͒ 2ͱ6 2ͱ6 ␮ 2 5 ␮ ͑2.38͒ and the first moment of g1 is given by while the singlet matrix element becomes s␮(auϩadϩas S 2 NS 2 ϩa ). C1͑Q ͒ C1 ͑Q ͒ c s ⌫ p͑n͒͑Q2͒ϭ ͓͗ps͉A0 ͉ps͔͘ϩ ␮ 1 9 ␮ 6 2. Bjorken sum rule

3 1 8 The Bjorken sum rule ͓4͔ is an immediate consequence of ϫ ϩ͑Ϫ͒͗ps͉A␮͉ps͘ϩ ͗ps͉A␮͉ps͘ , NS ͫ ) ͬ Eqs. ͑2.32͒ and ͑2.34͒. In the QPM where C1 ϭ1, ͑2.32͒ p n 1 gA NS S ⌫1Ϫ⌫1ϭ . ͑2.39͒ where C1 and C1 are the nonsinglet and singlet coefficient 6 ͯg ͯ V functions, respectively. The proton matrix elements for momentum p and spin s, ͗ps͉Ai ͉ps͘, can be related to those of ␮ In this form, the sum rule was first derived by Bjorken from the neutron by assuming isospin symmetry. In terms of the current algebra and isospin symmetry, and has since been axial charge matrix element ͑axial coupling͒ for flavor q and i recognized as a cornerstone of the QPM. the covariant spin vector s , ␮ The Bjorken sum rule is a rigorous prediction of QCD in 2 the limit of infinite momentum transfer. It is subject to QCD s␮ai͑Q ͒ϭ͗ps͉¯qi␥5␥␮qi͉ps͘, ͑2.33͒ radiative corrections at finite values of Q2 ͓35,37͔. These 3 they can be written as QCD corrections have recently been computed up to O(␣s ) 4 ͓38͔ and the O(␣s ) correction has been estimated ͓39͔. Since 3 s␮ s␮ s␮ gA the Bjorken sum rule is a pure flavor-nonsinglet expression, ͗ps͉A ͉ps͘ϭ a ϭ ͑a Ϫa ͒ϭ , ͑2.34͒ ␮ 2 3 2 u d 2 ͯg ͯ these corrections are given by the nonsinglet coefficient V NS function C1 : 8 s␮ s␮ ͗ps͉A␮͉ps͘ϭ a8ϭ ͑auϩadϪ2as͒, ͑2.35͒ 2) 2) 1 gA ⌫ pϪ⌫nϭ CNS . ͑2.40͒ 1 1 6 ͯg ͯ 1 0 2 V ͗ps͉A␮͉ps͘ϭs␮a0ϭs␮͑auϩadϩas͒ϭs␮a0͑Q ͒, ͑2.36͒ NS Beyond leading order, C1 depends on the number of flavors 2 where the Q dependence of au , ad , and as is implied from and on the renormalization scheme. Table I shows the coef- NS now on and is discussed in Sec. II F. The matrix element a3 ficients ci of the expansion 5336 D. ADAMS et al. 56

NS S TABLE I. Higher-order coefficients of the nonsinglet and singlet coefficient functions C1 and C1 in the NS S S ϱ MS scheme. The coefficients c4 and c3 are estimates; c3 is unknown for n f ϭ4 flavors. The quantities a0 2 and a0(Q ) are discussed in Sec. II E 3.

ϱ 2 Nonsinglet Singlet (a0 ) Singlet ͓a0(Q )͔ NS NS NS NS S S S S S S n f c1 c2 c3 c4 c1 c2 c3 c1 c2 c3 3 1.0 3.5833 20.2153 130 0.3333 0.5496 2 1 1.0959 3.7 4 1.0 3.2500 13.8503 68 0.0400 Ϫ1.0815 1 Ϫ0.0666

2 2 2 2 1 S ␣s͑Q ͒ ␣s͑Q ͒ ϭa (Q ϭϱ) ͓36͔. The coefficients c in the third column of CNSϭ1ϪcNS ϪcNS 0 i 1 1 ͩ ␲ ͪ 2 ͩ ␲ ͪ Table I should be used to compute the coefficient function CS that appears in Eq. 2.44 . 2 3 2 4 1 ͑ ͒ ␣s͑Q ͒ ␣s͑Q ͒ ϪcNS ϪO͑cNS͒ , ͑2.41͒ 3 ͩ ␲ ͪ 4 ͩ ␲ ͪ 4. Higher-twist effects As for unpolarized structure functions, spin-dependent in the MS scheme. structure functions measured at small Q2 are subject to higher-twist ͑HT͒ effects due to nonperturbative contribu- 3. Ellis-Jaffe sum rules tions to the lepton-nucleon cross section. In the analysis of 2 In the QPM the coefficient functions are equal to unity, moments and for not too low Q , such effects are expressed 2 and assuming exact SU͑3͒ symmetry ͓Eq. ͑2.37͔͒ the expres- as a power series in 1/Q : sion ͑2.32͒ can be written 1 M 2 M 4 ͑0͒ ͑2͒ ͑2͒ ͑2͒ ⌫1ϭ a ϩ 2 ͑a ϩ4d ϩ4 f ͒ϩO 4 p͑n͒ 1 5 1 2 9Q ͩ Q ͪ ⌫1 ϭϩ͑Ϫ͒ 12 ͑FϩD͒ϩ 36 ͑3FϪD͒ϩ 3 as . ͑2.42͒ 1 ϭ a͑0͒ϩHT. ͑2.45͒ This relation was derived by Ellis and Jaffe ͓3͔. With the 2 additional assumption that asϭ0, which in the QPM means p n ⌬sϭ0, they obtained numerical predictions for ⌫1 and ⌫1 . (0,2) (2) (2) p Here a , d , and f are the reduced matrix elements of The EMC measurement ͓2͔ showed that ⌫1 is smaller than the twist-2, twist-3, and twist-4 components, respectively, their prediction, which in the QPM implied that ⌬⌺, the and M is the nucleon mass. The values of a(2) and d(2) for contribution of quark spins to the proton spin, is small. This proton and deuteron have recently been measured ͓41͔ from result is at the origin of the current interest in polarized deep- the second moment of g1 and g2 , and found to be consistent inelastic scattering. with zero. Several authors have estimated the HT effects for The moments of g and the Ellis-Jaffe predictions are also 1 ⌫1 ͓42–44͔ and for the Bjorken sum rule ͓45,46͔.Inthe subject to QCD radiative corrections. The coefficient func- literature, there is a consensus that such effects are probably NS tion C1 ͓Eq. ͑2.41͔͒ used for the Bjorken sum rule also negligible in the kinematic range of the data used to evaluate applies to the nonsinglet part. The additional coefficient ⌫ in this paper. S 1 function C1 for the singlet contribution in Eq. ͑2.32͒ has 2 3 been computed up to O(␣s ) ͓36͔ and the O(␣s ) term has F. Physical interpretation of aD and the U„1… anomaly also been estimated for n ϭ3 flavors ͓40͔: f 2 In the simplest approximation, the axial coupling a0(Q ) 2 2 2 2 3 is expected to be equal to ⌬⌺, the contribution of the quark S S ␣s͑Q ͒ S ␣s͑Q ͒ S ␣s͑Q ͒ C1ϭ1Ϫc1 Ϫc2 ϪO͑c3͒ , spin to the nucleon spin. However, in QCD the U͑1͒ ͩ ␲ ͪ ͩ ␲ ͪ ͩ ␲ ͪ 2 anomaly causes a gluon contribution to a0(Q ) ͓47–49͔ as ͑2.43͒ well, which makes ⌬⌺ dependent on the factorization scheme, while a0 is not. The total fraction of the nucleon and the coefficients cS are shown in Table I. The QCD- i spin carried by quarks is the sum of ⌬⌺ and Lq , where Lq is corrected Ellis-Jaffe predictions for asϭ0 become the contribution of quark orbital angular momentum to the nucleon spin. Recently, it was pointed out ͓50͔ that this sum

p͑n͒ NS 1 gA 1 is scheme independent because of an exact compensation ⌫1 ϭC1 ϩ͑Ϫ͒ ϩ ͑3FϪD͒ ͫ 12 ͯg ͯ 36 ͬ between the anomalous contribution to ⌬⌺ and to Lq . V The decomposition of a0 into ⌬⌺ and a gluon contribu- 1 tion is scheme dependent ͓51͔. In the Adler-Bardeen ͑AB͒ ϩ CS͑3FϪD͒. ͑2.44͒ 9 1 ͓52͔ factorization scheme ͓53͔

Since a0ϭa8ϩ3as , the assumption asϭ0 is equivalent to 1 ϱ 2 2 a0ϭa8ϭ3FϪD. The quantity 3FϪD is independent of In Ref. ͓36͔, a0 and a0(Q ) are referred to as ⌺inv and ⌺(Q ), 2 ϱ Q , and so the assumption a0ϭa8 should be made for a0 respectively. 56 SPIN STRUCTURE OF THE PROTON FROM POLARIZED... 5337

2 ␣s͑Q ͒ a ͑Q2͒ϭ⌬⌺Ϫn ⌬g͑Q2͒, ͑2.46͒ 0 f 2␲ where the last term was originally identiﬁed as the anomalous gluon contribution ͓47–49͔. In this scheme ⌬⌺ is independent of Q2; however, it cannot be obtained from the measured a0 without an input value for ⌬g. In other schemes ⌬⌺ 2 2 is equal to a0(Q ), but then it depends on Q ͓51͔. The differences between these two schemes do not vanish when 2 2 2 Q ϱ because ␣s(Q )⌬g remains ﬁnite when Q ϱ ͓47→͔. →

G. Spin-dependent structure function g2

Phenomenologically, the structure function g2 can be understood from the spin-flip amplitude that gives rise to the interference asymmetry A2ϰg1ϩg2 of Eq. ͑2.9͒, owing to the absorption of a longitudinally polarized photon by the nucleon. There are two mechanisms by which this can occur ͓54͔. In the first, allowed in perturbative QCD, the photon is absorbed by a quark, causing its helicity to flip, but since helicity is conserved for massless fermions, this process is strongly suppressed for small quark masses. In the second, which is of a nonperturbative nature, the photon is absorbed by coherent parton scattering where the final-state quark con- serves helicity by absorption of a helicity Ϫ1 gluon. Wandzura and Wilczek have shown ͓55͔ that g2 can be decomposed as

2 WW 2 2 g2͑x,Q ͒ϭg2 ͑x,Q ͒ϩ¯g2͑x,Q ͒. ͑2.47͒

WW The term g2 is a linear function of g1 ,

1 dt WW 2 2 2 g2 ͑x,Q ͒ϭϪg͑x,Q ͒ϩ g1͑t,Q ͒ . ͑2.48͒ ͵x t

The term ¯g2 is due to a twist-3 contribution in the OPE ͓16͔ and is a measure of quark-gluon correlations in the nucleon ͓56͔. FIG. 2. Schematic layout of the muon beam and forward spectrometer. The individual detectors are discussed in the text see In the simplest QPM, g2 vanishes because the masses and ͑ transverse momenta of quarks are neglected. The predictions Table III͒.In͑b͒, B11 is a compensating dipole that is used on of improved quark-parton models which take these aspects when taking data with transverse target polarization. In ͑c͒,B8is into account depend critically on the assumptions made for the forward spectrometer magnet and referred to as the FSM in the the quark masses and the nucleon wave function ͓56͔. text. A right-handed coordinate system is used with its origin at the The Burkhardt-Cottingham sum rule predicts that the ﬁrst center of B8. The x axis points along the beam direction, and the z axis points upwards ͓out of the page in ͑b͒ and ͑c͔͒. moment of g2 vanishes for both the proton and the neutron ͓57͔: dinally polarized muons from polarized protons in a solid butanol target ͑Fig. 2͒. The energy of the incoming positive 1 muons, 190 GeV, is measured with a magnetic spectrometer ⌫2ϵ g2͑x͒dxϭ0. ͑2.49͒ ͵0 in the beam momentum station ͑BMS͒. The scattered muons are detected in the forward spectrometer ͑FS͒. They are iden- This sum rule is derived in Regge theory and relies on as- tiﬁed by coincident hits in arrays of hodoscopes located up- sumptions that are not well established. Its validity has there- stream and downstream of a hadron absorber; their momenta fore been the subject of much debate in the recent theoretical are measured with a large-acceptance, high-resolution mag- literature ͓16,58,59͔. netic spectrometer. The beam polarization is measured with a polarimeter located downstream of the FS. The high energy III. EXPERIMENTAL METHOD of the beam provides a kinematic coverage down to x ϳ0.003 for Q2Ͼ1 GeV2 and a high average Q2. A small A. Overview data sample was collected with a beam energy of 100 GeV The experiment involves principally the measurement of and transverse target polarization for the measurement of the p cross section asymmetries for inclusive scattering of longitu- asymmetry A2 . 5338 D. ADAMS et al. 56

FIG. 3. Muon polarization P␮ as a function of muon beam energy E␮ ͓60͔ for a monochromatic pion beam of 205 GeV ͑solid line͓͒Eq. ͑3.1͔͒ and mean P␮ vs E␮ as calculated by beam transport simulations ͓60͔͑dashed line͒.

The counting-rate asymmetries measured in this experiment vary from 0.001 to 0.05 depending on the kinematic region. To assure that the asymmetries measured do not depend on the incident muon ﬂux, the polarized target is sub- divided into two cells which are polarized in opposite directions. Frequent reversals of the target spin directions in both cells strongly reduce systematic errors arising from time- dependent variations of the detector efﬁciencies. Such errors FIG. 4. Schematic layout of the beam polarimeter for the muon are further reduced by the high redundancy of detectors in decay measurement ͑a͒ and for the muon-electron scattering mea- the forward spectrometer. The muon beam polarization is not surement ͑b͒. The different components of the apparatus are dis- reversed in this experiment. cussed in the text. The lead glass electromagnetic calorimeter and The statistical errors of the counting-rate asymmetries are the shower veto detector are labeled as LGC and SVD, respectively. Ϫ1 Ϫ1/2 proportional to (P␮Pt) (N) , where P␮ and Pt are the beam and target polarizations, respectively, and N is the The beam is naturally polarized because of parity viola- number of events. Hence high values of P␮ and Pt as well as tion in the weak decays of the parent hadrons. For mono- high N are important. chromatic muon and hadron beams, the polarization is a function of the ratio of muon and hadron energies ͓61͔:

B. Muon beam 2 2 m␲,Kϩ͑1Ϫ2E␲,K /E␮͒m␮ P ϭϮ , ͑3.1͒ The SMC experiment ͑NA47͒ is installed in the upgraded ␮ m2 Ϫm2 muon beam M2 of the CERN SPS ͓60͔. A beryllium target is ␲,K ␮ bombarded with 450-GeV protons from the SPS, and sec- ondary pions and kaons are momentum selected and trans- where the Ϫ and ϩ signs refer to positive and negative ported through a 600-m-long decay channel where for 200 muons, respectively ͑Fig. 3͒. For a given pion energy, the GeV about 5% decay into muons and neutrinos. The remain- muon intensity depends on the ratio E␲,K /E␮ ; this ratio was ing hadrons are stopped in a 9.9-m-long beryllium absorber optimized using Monte Carlo simulations of the beam trans- for the 190-GeV muon beam. Downstream of the absorber, port ͓62,63͔ to obtain the best combination of beam polar- muons are momentum selected and transported into the ex- ization and intensity. perimental hall. The beam intensity was 4ϫ107 muons per SPS pulse; these pulses are 2.4 s long with a repetition period of 14.4 s. C. Measurement of the beam polarization The beam spot on the target was approximately circular with A polarimeter downstream of the muon spectrometer al- a rms radius of 1.6 cm and a rms momentum width of lows us to determine the beam polarization by two different Ϸ2.5%. The momentum of the incident muons is measured methods. The first involves measuring the energy spectrum ϩ ϩ for each trigger in the BMS located upstream of the experi- of positrons from muon decay in flight, ␮ e ¯␯␮␯e , mental hall ͑Fig. 2͒. The BMS employs a set of quadrupoles which depends on the parent-muon polarization→͓64͔. The (Q) and a dipole ͑B6͒ in the beam line, with a nominal second method involves measuring the spin-dependent cross vertical deflection of 33.7 mrad. Four planes of fast scintil- section asymmetry for elastic scattering of polarized muons lator arrays ͑HB͒ upstream and downstream of this magnet on polarized electrons ͓65͔. The two methods require differ- are used to measure the muon tracks. The resolution of the ent layouts for the polarimeter and thus cannot be run simul- momentum measurement is better than 0.5%. taneously. 56 SPIN STRUCTURE OF THE PROTON FROM POLARIZED... 5339

The measured positron spectrum is corrected for the over- all detector response. The response function is obtained from a Monte Carlo simulation that generates muons according to the measured beam phase space. The simulation accounts for radiative effects at the vertex and external bremsstrahlung, the geometry of the setup, and chamber efficiencies. The Monte Carlo events were processed using the same procedure as applied to the real data. The response function is obtained by dividing the Monte Carlo spectrum by the Michel spectrum of Eq. ͑3.2͒. The polarization P␮ can be determined by fitting Eq. ͑3.2͒ to the measured decay spectrum corrected for the detector response. Figure 5 shows the sensitivity of the Michel spectrum to the muon polarization. The systematic error in the P␮ determination is mainly due to uncertainties in the response function, the main contributions to which are uncer- FIG. 5. The Michel spectrum predictions for P␮ϭϪ1, 0, and tainties in the MWPC efficiencies and in the background ϩ1 are shown by the solid, dashed, and dotted lines, respectively. rejection. Background due to external ␥ conversion, ␮ϩ ␮ϩ␥ ␮ϩeϩeϪ, is measured using the charge- 1. Polarized-muon decay conjugate→ process→ with a ␮Ϫ beam and was found to be The energy spectrum of positrons from the decay negligible. Other contributions to the systematic error arise ϩ ϩ from uncertainties in y , in radiative effects at the vertex, ␮ e ␯e¯␯␮ ͓66͔ can be expressed in terms of the ratio of e → and in the alignment of the wire chambers. positron and muon energies, y eϭEe /E␮ , and of the muon polarization P ͓67,68͔: ␮ 2. Polarized-muon-electron scattering In QED at first order, the differential cross section for dN 5 2 4 3 1 2 8 3 ϭN0 Ϫ3yeϩ yeϪP␮ Ϫ3yeϩ ye , elastic scattering of longitudinally polarized muons off lon- dye ͫ3 3 ͩ3 3 ͪͬ gitudinally polarized electrons is ͓69͔ ͑3.2͒ 2 d␴ 2␲reme 1 1 1 where N is the number of muon decays. ϭ 2 Ϫ ϩ ͑1ϩPeP␮A␮e͒, 0 dy␮e E␮ ͩy y␮eY 2ͪ ␮e The polarimeter configuration for this measurement is ͑3.3͒ shown in Fig. 4͑a͒. It consists of a 30-m-long evacuated decay volume, followed by a magnetic spectrometer and an where me is the electron mass, re the classical electron ra- 2 Ϫ1 electromagnetic calorimeter to measure and identify the de- dius, y ␮eϭ1ϪE␮Ј /E␮ , and Yϭ(1ϩm␮/2meE␮) is the ki- cay positrons. The beginning of the decay path is defined by nematic upper limit of y ␮e . The cross section asymmetry the shower veto detector ͑SVD͒, which consists of a lead foil A␮e for antiparallel ͑ ͒ and parallel ͑ ͒ orientations of the followed by two scintillator hodoscopes. Along the decay incoming muon and target↑↓ electron spins↑↑ is path, tracks are measured with multiwire proportional chambers ͑MWPCs͒. The decay positrons are momentum ana- d␴↑↓Ϫd␴↑↑ 1Ϫy␮e /Yϩy␮e/2 lyzed in a magnetic spectrometer consisting of a 6-m-long A␮eϭ ϭy␮e 2 . ͑3.4͒ d␴↑↓ϩd␴↑↑ 1Ϫy ␮e /Yϩy ␮e/2 small-aperture dipole magnet followed by another set of

MWPCs. This spectrometer and the BMS, which measures The measured asymmetry Aexpt is related to A␮e by the parent muon momentum, were intercalibrated in dedi- cated runs to 0.2%. A lead glass calorimeter ͑LGC͒ is used to Aexpt͑y ␮e͒ϭPeP␮A␮e͑y ␮e͒, ͑3.5͒ identify the decay positrons. The trigger requires a hit in each SVD plane, in coinci- where Pe and P␮ are the electron and muon polarizations, dence with a signal from the LGC above a threshold of about respectively. The measured asymmetries range from about 15 GeV. Events with two or more hits in both planes within 0.01 at low y ␮e to 0.05 at high y ␮e . a 50-ns time window are rejected. This suppresses back- The experimental setup for the ␮-e scattering measure- ground from incident positrons originating upstream of the ment is shown schematically in Fig. 4͑b͒. The lead foil is polarimeter and rejects events with more than one muon. removed from the SVD, and only the hodoscopes of the SVD In the off-line analysis, events whose energy E␮ was mea- are used to tag the incident muon which is tracked in three sured in the BMS and experienced a large energy loss in the MWPCs installed upstream of the magnetized target. Be- SVD are rejected. A single track is required, both upstream tween the target and spectrometer magnet, three additional and downstream of the magnet. To reject muon decays inside chambers measure the tracks of the scattered muon and of the magnetic field volume, the upstream and downstream the knock-on electron. Downstream of the magnet, the muon tracks are required to intersect in the center of the magnet. and the electron are tracked in two wire-chamber telescopes Decay positrons are identified by requiring that the momen- sharing a large MWPC. The electron is identified in the tum measured by the polarimeter spectrometer matches the LGC, and the muon is detected in a scintillation-counter ho- energy deposition in the LGC. doscope located behind a 2-m-thick iron absorber. 5340 D. ADAMS et al. 56

The polarized electron target is a 2.7-mm-thick foil made of a ferromagnetic alloy consisting of 49% Fe, 49% Co, and 2% V. It is installed in the gap of a soft-iron flat-magnet circuit with two magnetizing solenoidal coils ͓70͔. The magnet circuit creates a saturated homogeneous field of 2.3 T along the plane of the target foil. In order to obtain a component of electron polarization parallel to the beam, the target foil was positioned at an angle of 25° to the beam axis. To determine the target polarization, the magnetic flux in the foil under reversal of the target-field orientation is measured with a pickup coil wound around the target. The mag- netization of the target was found to be constant along the foil to within 0.3%. The electron polarization is determined from the magnetomechanical ratio gЈ of the foil material. A measurement of gЈ for the alloy used does not exist; a value of gЈϭ1.916Ϯ0.002 has been reported for an alloy of 50% Fe and 50% Co ͓71͔. We assume that the addition of 2% V does not affect gЈ, but we enlarge the uncertainty to Ϯ0.02. The resulting polarization along the beam axis is ͉Pe͉ ϭ0.0756Ϯ0.0008. The loss of ␮-e events because of the internal motion of K-shell electrons ͓72͔ affects the asymmetry Aexpt by less than Ϫ0.001 and was therefore neglected. To measure the cross section asymmetry, the target-field orientation was changed between SPS pulses by reversing the current in the coil. The vertical component of the magnetizing field provides a bending power of 0.05 T m, which gives rise to a false asymmetry. This effect was compensated for by alternating the target angle every hour between 25° and Ϫ25° and averaging the asymmetries obtained with the two orientations. The trigger requires a coincidence between the two SVD FIG. 6. The QED radiative corrections to the asymmetry A␮e ͑a͒ without experimental cuts. b The correction to the asymmetry if hodoscope planes, an energy deposition of 15 GeV or more ͑ ͒ the following experimental cuts are included in the calculation: ͑i͒ in the LGC, and a signal in the muon hodoscope ͑MH͒. The recoil electron energy greater than 35 GeV, ͑ii͒ energy difference scattering vertex is reconstructed from the track upstream between initial and final states less than 40 GeV, and ͑iii͒ angular and the two tracks downstream of the magnetized target. The cuts on both outgoing muon and electron. The corrections ␦␮e are three tracks were required to be in the same plane to within given in percent. 20° and the reconstructed vertex to be within Ϯ50 cm of the target position. The two outgoing tracks were required to of Eq. ͑3.3͒ are evaluated using the program ␮ela ͓73͔. The 3 have an opening angle larger than 2 mrad and to satisfy the corrections are calculated up to O(␣QED) with finite muon two-body kinematics of elastic scattering to within 1 mrad. mass and found to be negligible once the experimental cuts Since the electron radiates in the target, we use the scattered are applied ͑Fig. 6͒. muon energy to calculate y ␮e . The polarization P␮ϭAexpt(y␮e)/A␮e(y␮e)Pe in bins of y ␮e Background originates from bremsstrahlung (␮ϩ ␮ϩ␥) is shown in Fig. 7. The main contributions to the systematic followed by conversion and pair production→ error are the uncertainty of the flux normalization, the false (␮ϩ ␮ϩeϩeϪ). It was determined experimentally by us- asymmetry, the uncertainty of the target polarization, and the ing a→␮Ϫ beam with a similar setup and triggering on ␮Ϫeϩ background subtraction. coincidences. Most of the background was eliminated by requiring that the energy conservation between the initial and final states be satisfied within 40 GeV. This requirement rejects very few good events. The background correction to the beam polarization is Ϫ0.012Ϯ0.004. The experimental asymmetry was obtained from data samples taken with the two different target field orientations. The data samples were normalized to the incident muon fluxes using a random trigger technique. A possible false asymmetry due to the target magnetic field was studied using both a Monte Carlo simulation of the apparatus and data taken with an unpolarized polystyrene target under the same experimental conditions. In both cases the resulting asymme- FIG. 7. Beam polarization vs the ratio of electron and muon try was found to be consistent with zero. The radiative cor- energies from polarized ␮-e scattering. The dashed line represents QED rections ␦␮eϭ(A␮e /A␮eϪ1) to the first-order cross section the average value. Only the statistical errors are shown. 56 SPIN STRUCTURE OF THE PROTON FROM POLARIZED... 5341

TABLE II. Quantities ͑in moles͒ of the various chemical elements in the target volume.

Element Quantity Element Quantity Element Quantity

1H 185.70 F 0.24 Cu 00.36 3He 6.00 Na 0.17 O 22.70 4He 23.00 Cr 0.17 C 71.80 Ni 0.14

The two target cells were each 60-cm-long, cylindrical, polyester-epoxy mesh cartridges of 5 cm diameter, separated by a 30-cm gap. The target consisted of 1.8-mm butanol glass beads. The total amount of target material was 1.42 kg, with a packing fraction of 0.62 and a density of 0.985 g/cm3 at 77 K. The concentration of paramagnetic electron spins in FIG. 8. Cross section of the SMC polarized target. the target material was 6.2ϫ1019 spins/ml. In addition to butanol, the target cells contained other material, mostly the 3. Beam polarization 3He-4He cooling liquid and the NMR coils for the polariza- The beam polarization obtained from the ␮-e scattering tion measurement ͑Table II͒. experiment in 1993 is ͓74,75͔ In the 2.5-T field and at a temperature below 1 K, the electron spins are nearly 100% polarized. When their reso- nance line is saturated at a frequency just above or below the P␮ϭϪ0.779Ϯ0.026͑stat͒Ϯ0.017͑syst͒ ͑3.6͒ absorption spectrum centered around the frequency of ␯e Ϸ69.3 GHz at 2.5 T, negative and positive proton polariza- for E␮ϭ188 GeV. The polarization measured by the muon tions are obtained. This technique was applied to polarize the decay method in 1993, P␮ϭϪ0.803Ϯ0.029 ͑stat͒ material in the two target cells in opposite directions. Modu- Ϯ0.020 ͑syst͒, has been published earlier ͓9͔. Both results lation of the microwave frequencies with a 30-MHz ampli- are compatible. An alternative analysis with a larger data tude and a 1-kHz rate increased the polarization buildup rate sample for the muon decay method is in progress, and the by 20% and resulted in a gain in maximum polarization of systematic uncertainties of our previous analysis are being 6%. This method was originally developed to improve the reevaluated. The result of the ␮-e scattering Eq. ͑3.6͒ is used polarization of a deuterated butanol target ͓80͔. in this paper. For E␮ϭ100 GeV a value of P␮ϭϪ0.82 The DR ͓81͔ cools the target material to a temperature Ϯ0.06 was used for the analysis of the A2 measurement. below 0.5 K, while absorbing the microwave power applied This is based on the measurement reported in Ref. ͓64͔. for DNP. Once a high polarization is reached, the micro- Monte Carlo simulations of the muon beam ͓60͔ are consis- waves are turned off and the target material is cooled to 50 tent with these measurements of P␮ for both beam energies. mK. At this temperature the proton spin-lattice relaxation We have evaluated the average polarization of our accepted time exceeds 1000 h at 0.5 T. Under these ‘‘frozen-spin’’ event sample taking into account the energy dependence of conditions, the polarization is preserved during field rotation the muon polarization. The polarization was calculated on an and during measurements with transverse spin. To avoid pos- event-by-event basis using Eq. ͑3.1͒ and assuming a mo- sible systematic errors, the proton polarizations were re- noenergetic pion beam ͑Fig. 5͒. versed by DNP once a week. The superconducting magnet system consists of a solenoid with a longitudinal field of 2.5 T aligned with the beam D. Polarized target axis and a dipole providing a perpendicular ‘‘holding’’ field The polarized proton target uses the method of dynamic of 0.5 T. The solenoid has a bore of 26.5 cm into which the nuclear polarization ͑DNP͓͒76͔ and contains two oppositely DR with the target cells is inserted; this diameter corre- polarized target cells exposed to the same muon beam ͑Fig. sponds to an opening angle of Ϯ65 mrad with respect to the 8͓͒2͔. The solid target material is butanol ͓CH3͑CH2͒3OH͔ upstream end of the target. Sixteen correction coils allow the plus 5% water doped with paramagnetic EHBA-Cr͑V͒ mol- field to be adjusted to a relative homogeneity of Ϯ3.5 ecules. A superconducting magnet system ͓77͔ and a ϫ10Ϫ5 over the target volume. In addition, the trim coils 3He-4He dilution refrigerator ͑DR͓͒78͔ provide the strong were used to suppress the super-radiance effect ͓82͔, which magnetic field and the low temperature required for high can cause losses of the negative proton polarization while the polarization, and allow for frequent inversion of the field and field is being changed. The spin directions were reversed thus of the polarization vectors. Additional subsystems in- every 5 h with relative polarization losses of less than 0.2%. clude a double microwave setup needed for the DNP and a This was accomplished by rotating the magnetic field vector ten-channel NMR system to measure the spin polarization of the superimposed solenoid and dipole fields, with a loss of ͓79͔. During data taking, the nuclear spin axis is aligned data-taking time of only 10 min per rotation ͓83͔. The dipole either along or perpendicular to the beam direction in order field was also used to hold the spin direction transverse to the to measure Aʈ or AЌ , respectively. beam for the measurement of AЌ . 5342 D. ADAMS et al. 56

TABLE III. Detectors of the muon spectrometer.

Modules Pitch Size Wire Modules Pitch Size Dead Hodoscope ϫplanes ͑cm͒ ͑cm͒ chamber ϫplanes ͑cm͒ ͑cm͒ zone ͑cm͒

BHA-B 2ϫ8 0.4 8ϫ8 P0A-E 5ϫ8 0.1 л14 V123 5ϫ1 various PV1 1ϫ4 0.2 150ϫ94 H1 2 7.0 250ϫ130 PV2 1ϫ6 0.2 154ϫ100 л8 H2 cal 4 28.0 560ϫ280 P123 3ϫ3 0.2 180ϫ80 л13 H3 2 15.0 750ϫ340 W12 2ϫ8 2.0 220ϫ120 л12 H4 1 15.0 996ϫ435 W45 6ϫ4 4.0 530ϫ260 л13–25 H1Ј,3Ј,4Ј 1 1.4 50ϫ50 P45 5ϫ2 0.2 л90 л12 S1,2,4 1 various ST67 4ϫ8 1.0 410ϫ410 л16 H5 1ϫ2 various 19ϫ20 P67 4ϫ2 0.2 л90 л12 H6 1ϫ2 various л14 DT67 3ϫ4 5.2 500ϫ420 83ϫ83

The proton polarization was measured with ten series- 1. Spectrometer layout tuned Q-meter circuits with five NMR coils in each target Three stages of the spectrometer can be distin- cell ͓84,85͔. The polarization is proportional to the integrated guished: tracking of the incident muon, tracking and mo- NMR absorption signal, which was determined from con- mentum measurement of the scattered muon, and muon iden- secutively measured response functions of the circuit with tification. The beam tracking section upstream of the target is and without the NMR signal. The latter was obtained by composed of two scintillator hodoscopes ͑BHA and BHB͒ increasing the magnetic field and thus shifting the proton and the P0B MWPC. A set of veto counters ͑V1.5, V3, V2.1, NMR spectrum outside the integration window. The calibra- and V2͒ defines the beam spot size. Beam tracks are recon- tion constant was obtained from a measurement of the ther- structed with an angular resolution of 0.1 mrad and an effi- mal equilibrium ͑TE͒ signals at 1 K, where the polarization ciency better than 90% for intensities up to 5ϫ107␮/spill. is known from the Curie law PTEϭtanh(h␯ p/2kT) The momentum of the scattered muon is measured with a Ӎ0.002 553; T is the lattice temperature, k the Boltzmann conventional large-aperture dipole magnet ͑FSM͒ and a system of more than 100 planes of MWPCs ͑Table III͒. The constant, and ␯ p is the proton Larmor frequency. The accuracy of the TE calibration signal contributed to the polariza- FSM is operated with bending powers of 2.3 and 4.4 T m at tion error by ⌬P/Pϭ1.1% ͓79͔. The NMR signals were 100 and 190 GeV beam energies, respectively, correspond- measured every minute during data taking. The polarizations ing to a horizontal beam deflection of 7 mrad. The angular measured with the individual coils were averaged for each resolution for scattered muons is 0.4 mrad. The large MW- PCs are complemented by smaller MWPCs with a smaller target cell and over the duration of one data taking run of wire pitch, to increase the redundancy and the resolution of typically 30 min. All measurements inside the same cell the spectrometer in the high-rate environment at small scat- agreed to better than 3%. To detect a possible radial inho- tering angles. mogeneity, two of the five coils in the upstream target cell Scattered muons are identified by the observation of a were at the same longitudinal position, but one was in the track behind a 2-m-thick iron absorber. The muon identifica- center and the other at a radius of 1 cm. No significant dif- tion system consists of streamer tubes, MWPC and drift ference was found between the polarizations measured by tubes. To cope with the high beam intensity, the streamer these two coils. tubes were operated with voltages at which their pulse The characteristic polarization buildup time was 2–3 h. However, the highest polarizations of ϩ0.93 and Ϫ0.94 were achieved only after several days of DNP. The average polarization during the data taking was 0.86, and the relative error in the average polarization of the target was estimated to be 3%.

E. Muon spectrometer and event reconstruction The spectrometer is similar to the setups used by the EMC ͓86͔ and the NMC ͓87͔͑Fig. 2͒. Aging chambers were re- placed and new ones added to improve the redundancy of the muon tracking and to extend the kinematic coverage to smaller x. A major new streamer tube detector ST67 was constructed to identify and measure scattered muon positions downstream of the absorber. Triggers were optimized for improved kinematic coverage, in particular in the region of FIG. 9. Kinematic ranges for triggers T1, T2, and T14 at 190 small x. GeV. 56 SPIN STRUCTURE OF THE PROTON FROM POLARIZED... 5343

TABLE IV. Kinematic cuts applied for the Aʈ and AЌ analysis.

Kinematic Aʈ analysis AЌ analysis variable E␮ϭ190 GeV E␮ϭ100 GeV ␯ у15 GeV у10 GeV y р0.9 р0.9

p␮Ј у19 GeV у15 GeV ␪ у9 mrad у13 mrad

Final data sample for Aʈ analysis Final data sample for AЌ analysis x range 0.003рxр0.7 0.0008рxр0.7 0.006рxр0.6 0.0035рxр0.6 Q2 range 1рQ2р90 0.2рQ2р90 1рQ2р30 0.5рQ2р30 Events 4.5ϫ106 6.0ϫ106 8.8ϫ105 9.6ϫ105 heights were close to the electronic threshold. Their efficien- The reconstruction of the scattered muon tracks starts in cies were thus very sensitive to the ambient pressure and the muon identification system behind the hadron absorber temperature, and a high-voltage feedback system was devel- ͑ST67, DT67, P67͒. Tracks found in this system are extrapo- oped to stabilize the average streamer pulse height within lated upstream and reconstructed in the MWPC and drift 1%. chambers between the absorber and the FSM ͑W45, P45, W12, P0E͒. The next step in the reconstruction is the track 2. Triggers finding in the FSM chambers ͑P123, P0D͒, starting with the vertical coordinates which are fitted by straight lines. Hori- The read-out of the detectors was triggered by predefined zontal coordinates matching the downstream tracks are coincidence patterns of hits in different planes of searched for on circular trajectories inside the FSM. Because scintillation-counter hodoscopes. Three physics triggers pro- 2 of the high track multiplicity in the FSM aperture, each ex- vide a coverage of different x and Q ranges ͑Fig. 9͒. All trapolation of a downstream track through the magnetic field triggers require that there is no hit in any of the beam- is tested with a spline fit and the best track is retained. In the defining veto counters. vertex chambers ͑PV12, P0C͒, hits are selected using the The large-angle trigger T1 requires a coincidence pattern extrapolated track reconstructed in the magnet and are fitted of the hodoscopes H1, H3, and H4. This trigger has a good by a straight line. It is verified that the reconstructed muon acceptance for scattering angles ␪ larger than 20 mrad. Tar- track satisfies the trigger conditions. get pointing of the scattered muon is also required. The ac- The vertex position in the target is computed as the point ceptance decreases for smaller angles, but extends to ␪ of closest distance of approach between the beam and the Ϸ3 mrad. The small-angle trigger T2 uses the smaller hodo- scattered-muon tracks. Tracks are propagated through the scopes H1Ј,H3Ј, and H4Ј. This trigger covers the range magnetic field in the target using a Runge-Kutta method, 5 mradр␪р15 mrad. It has a more limited x range than T1. taking into account energy loss and multiple scattering. In 2 However, at a given x, T2 selects events with lower Q than case of multiple beam tracks, the vertex with the best space- T1. A small-x trigger T14 is provided by the S1, S2, and S4 time correlation between the beam and the scattered-muon counters which are placed close to the beam to cover scat- track is chosen. The vertex is reconstructed with resolutions tering angles down to 3 mrad with good efficiency. The of better than 30 and 0.3 mm along and perpendicular to the counters for T2 and T14 were located downstream of the beam direction, respectively. spectrometer magnet where scattered muons of low scattering angle and low momenta are expected. The acceptance of F. Data taking the triggers T1 and T14 extends down to xӍ0.5ϫ10Ϫ3 and thus is sensitive to elastic scattering of muons from atomic The data presented in this paper were taken during 134 days of the 1993 CERN SPS fixed-target run. Most data were electrons, xϭme /m p ͑Fig. 9͒. The trigger rate per SPS spill was about 200 for T1, 50 for T2, and 100 for T14. taken with longitudinal target polarization, at a beam energy Other triggers include normalization and beam-halo trig- of 190 GeV. For 22 days, data were taken with the target gers, which were used for calibration, alignment, and effi- polarized transversely to the beam, at a beam energy of 100 ciency calculations. GeV. A total of 1.6ϫ107 deep-inelastic-scattering events were reconstructed from the data with a longitudinally polarized 3. Event reconstruction target, using the three physics triggers T1, T2, and T14. The The track finding starts with the beam-track reconstruc- integrated muon flux was 1.7ϫ1013. tion. The momentum of the incident muons is computed With transverse target polarization, only T1 was used and from the hit pattern in the BMS hodoscopes. The beam track 1.6ϫ106 events were reconstructed. The transverse target upstream of the target is found from the hits in the BHA and field was always in the same vertical direction, and the spin BHB hodoscopes and the P0B wire chamber. A coincidence direction was inverted by microwave reversal a total of 10 is required between the hits in the BMS and those in the times. The integrated muon flux at 100 GeV was 0.2 beam hodoscopes. ϫ1013. 5344 D. ADAMS et al. 56

target spin conﬁgurations are used. The quantity AT ϭAЌ cos ␾ is determined separately for the upstream and downstream target cells from the four counting rates into the upper and lower vertical halves of the spectrometer for the two transverse spin directions.

1. Aʈ analysis

The number of muons, Nu and Nd , scattered in the upstream and downstream target cells, respectively, is given by

Nuϭnu⌽au¯␴͑1Ϫ fP␮PuAʈ͒, ͑4.1͒

Ndϭnd⌽ad¯␴͑1Ϫ fP␮PdAʈ͒, ͑4.2͒

where ⌽ is the integrated beam flux, Pu and Pd are the polarizations in the two target cells, nu and nd are the area densities of the target nucleons, and au and ad are the corresponding spectrometer acceptances. The dilution factor f accounts for the fact that only a fraction of the target nucleons is polarized ͑Sec. IV C͒. The flux ⌽ and the spin- independent cross section ¯␴ cancel in the evaluation of the raw counting-rate asymmetries, Araw and ArawЈ , obtained be- FIG. 10. Vertex distributions of scattered muons after kinematic fore and after target polarization reversal: cuts: ͑a͒ along the beam direction and ͑b͒ in the plane perpendicular to the target axis, at the location of one of the NMR coils. In ͑a͒, NuϪNd NdЈϪNuЈ Arawϭ , ArawЈ ϭ . ͑4.3͒ the dashed lines indicate the fiducial cuts on the target volume NuϩNd NdЈϩNuЈ which coincide with the entry and exit windows of the target cells; most events outside the shaded region originate from interactions Provided that nu /nd is constant and that the ratio of ac- with the 3He-4He cooling liquid. The small peak at xϷϪ3.9 m ceptances is the same before and after polarization reversal arises from scattering in the exit window of the target cryostat. In and close to unity, i.e., au /adϭauЈ/adЈӍ1, then the acceptan- ͑b͒, the outer circle indicates the wall of the target cells and the ces a and the densities n cancel in the average of the raw inner circle shows the radial cut applied. Scattering from the tubular asymmetries, so that NMR coils is clearly visible.

1 ArawϩArawЈ G. Event selection A ϭϪ . ͑4.4͒ ʈ fP P ͫ 2 ͬ ␮ t Since the Aʈ and AЌ data were recorded at different beam energies, they cover different kinematic ranges and are sub- If au /adÞauЈ/adЈ , a ‘‘false’’ asymmetry ensues, ject to different kinematic cuts ͑Table IV͒. A cut at small ␯ rejects events with poor kinematic resolution, whereas a cut 1 rϪ1 rЈϪ1 at high y removes events with large radiative corrections. A A ϭϪ Ϫ . ͑4.5͒ false 2 fDP P ͫrϩ1 rЈϩ1ͬ cut on the momentum of the outgoing muon reduces the ␮ t contamination by muons from ␲ and K production in the Ϫ3 The virtual photon-proton asymmetry A1ӍAʈ /D ͓Eq. ͑2.15͔͒ target and subsequent decay to a few 10 . The cut on ␪ was is thus given by only applied for the analysis with Q2у1 GeV2. It rejects events with poor vertex resolution. 1 ArawϩArawЈ Cuts were also applied to the beam phase space to ensure A1ϭϪ ϪAfalse . ͑4.6͒ that the beam flux was the same for both target cells. Fidu- fDP␮Pt ͫ 2 ͬ cial cuts on the target volume reject events from material outside the target cells ͑Fig. 10͒. Less than 10% of the raw In these expressions, D is the depolarization factor ͓Eq. data were discarded because of instabilities in the beam in- ͑2.13͔͒, rϭnuau /ndad , rЈϭnuauЈ/ndadЈ , and Pt is the tensity, detector efficiencies, and target polarization. The size weighted average of the target cell polarizations, of the final data samples after all cuts is shown in Table IV. ⌺͉Pu͉Nuϩ⌺͉Pd͉Nd ⌺͉PuЈ͉NuЈϩ⌺͉PdЈ͉NdЈ 2Ptϭ ϩ . IV. DATA ANALYSIS ⌺Nuϩ⌺Nd ⌺NuЈϩ⌺NdЈ ͑4.7͒ A. Evaluation of cross section asymmetries

The two cross section asymmetries Aʈ and AЌ ͓Eq. ͑2.7͔͒ Equation ͑4.6͒ provides an unbiased estimate of the cross are evaluated from counting rate asymmetries. To determine section asymmetry for large numbers of events. To avoid

Aʈ the four measured counting rates from the upstream and possible biases for the number of events involved, a maxi- downstream target cells with the two possible antiparallel mum likelihood technique was developed which allows a 56 SPIN STRUCTURE OF THE PROTON FROM POLARIZED... 5345

p 2 2 2 2 TABLE V. The virtual photon-proton asymmetry A1 for Q Ͼ1 GeV ͑above separation line͒ and Q Ͼ0.2 GeV ͑below line͒. In the last column, the ﬁrst error is statistical and the second is systematic. ͗Araw͘ is the straight average of Araw and ArawЈ in Eq. ͑4.4͒. The values for p A1 have been corrected for radiative effects as described in Sec. IV B.

͗Q2͘ 2 rc p x range ͗x͘ (GeV ) ͗P␮͗͘y͗͘D͗͘f͗͘1/␳͗͘Araw͘ A1 A1 0.003–0.006 0.005 1.32 Ϫ0.79 0.791 0.80 0.070 1.50 0.004 0.007 0.083Ϯ0.041Ϯ0.006 0.006–0.010 0.008 2.07 Ϫ0.78 0.748 0.76 0.081 1.39 0.003 0.008 0.044Ϯ0.037Ϯ0.004 0.010–0.020 0.014 3.56 Ϫ0.78 0.704 0.72 0.090 1.30 0.003 0.010 0.061Ϯ0.032Ϯ0.004 0.020–0.030 0.025 5.73 Ϫ0.78 0.660 0.68 0.096 1.24 0.003 0.012 0.068Ϯ0.044Ϯ0.005 0.030–0.040 0.035 7.80 Ϫ0.78 0.634 0.66 0.099 1.21 0.002 0.015 0.041Ϯ0.052Ϯ0.003 0.040–0.060 0.049 10.44 Ϫ0.78 0.603 0.64 0.102 1.18 0.006 0.017 0.104Ϯ0.045Ϯ0.007 0.060–0.100 0.077 15.01 Ϫ0.78 0.551 0.60 0.106 1.14 0.009 0.020 0.180Ϯ0.045Ϯ0.013 0.100–0.150 0.122 21.41 Ϫ0.78 0.498 0.55 0.112 1.10 0.013 0.022 0.289Ϯ0.058Ϯ0.019 0.150–0.200 0.173 27.80 Ϫ0.79 0.456 0.51 0.118 1.08 0.012 0.022 0.276Ϯ0.080Ϯ0.019 0.200–0.300 0.242 35.54 Ϫ0.79 0.417 0.47 0.127 1.05 0.010 0.019 0.246Ϯ0.082Ϯ0.017 0.300–0.400 0.342 45.45 Ϫ0.78 0.377 0.43 0.139 1.02 0.021 0.010 0.499Ϯ0.132Ϯ0.036 0.400–0.700 0.482 57.09 Ϫ0.78 0.337 0.39 0.156 0.99 0.022 Ϫ0.006 0.527Ϯ0.174Ϯ0.041

0.0008–0.0012 0.001 0.28 Ϫ0.78 0.808 0.85 0.044 1.74 Ϫ0.001 0.002 Ϫ0.032Ϯ0.077Ϯ0.004 0.0012–0.002 0.002 0.44 Ϫ0.78 0.794 0.83 0.054 1.65 0.002 0.003 0.085Ϯ0.055Ϯ0.007 0.002–0.003 0.003 0.69 Ϫ0.78 0.781 0.80 0.062 1.56 0.001 0.004 0.031Ϯ0.054Ϯ0.004 0.003–0.006 0.004 1.19 Ϫ0.78 0.763 0.77 0.073 1.46 0.003 0.006 0.059Ϯ0.034Ϯ0.005 0.006–0.010 0.008 2.04 Ϫ0.78 0.738 0.75 0.082 1.38 0.003 0.008 0.050Ϯ0.036Ϯ0.004

T 1␥ common analysis of all events in each x bin. In this method, ¯␴ ϭv¯␴ ϩ¯␴tail , Aʈ /D is computed from the event weights wϭ fDP␮ using the expression T 1␥ ⌬␴ ϭv⌬␴ ϩ⌬␴tail , ͑4.10͒ 1 ⌺wuϪ⌺wd ⌺wdϪ⌺wu Ј A1ϭϪ 2 2 ϩ 2 2 ϪAfalse . 2Pt ͫͩ⌺wuϩ⌺wdͪ ͩ⌺wdϩ⌺wuͪ ͬ where ¯␴T is the total, i.e., measured, spin-independent cross ͑4.8͒ section, ¯␴1␥ is the corresponding one-photon-exchange cross section, and ¯␴ is the contribution to ¯␴T from the elastic tail As explained in Sec. IV C, in the actual analysis we use a tail and the inelastic continuum. The corresponding differences weight wϭ f DP . A Monte Carlo simulation conﬁrmed Ј ␮ of the cross sections for antiparallel and parallel orientations that this method does not introduce any biases. of lepton and target spins are denoted by ⌬␴. The factor v accounts for vacuum polarization and also includes contribu- 2. A analysis Ќ tions from the inelastic tail close in x. The decomposition in A similar formalism applies to the measurement of the Eq. ͑4.10͒ depends on the fraction of the inelastic tail in- transverse asymmetry AЌ , where the event yields are given cluded in v and is therefore to some extend ambiguous. As a by N(␾)ϭn⌽a¯␴(1ϪfP␮PT AЌ cos␾). Here AЌ is ob- result of a cancellation of the different contributions, v is tained for each target cell separately from ͓N(␾)ϪN(␾ close to unity. Using the program TERAD ͓89͔, we ﬁnd 0.98 Ϫ␲)͔/͓N(␾)ϩN(␾Ϫ␲)͔ and AЌ /d becomes ϽvϽ1.03 in the kinematic range of our data. For simplicity we set v to unity in our analysis and attribute all corrections AЌ Ϫ1 ⌺ fd cos ␾ ⌺ fd cos ␾ Ј ϭ ϩ to ␴tail ͓88͔. d 2P ͗P ͘ ͫͩ ⌺͑ fd cos ␾͒2 ͪ ͩ ⌺͑ fd cos ␾͒2 ͪ ͬ ␮ t

ϪAfalse , ͑4.9͒ where ͗Pt͘ is the average target polarization before and after reversal in absolute value. To obtain the same statistical accuracy for AЌ /d and Aʈ /D, more data are required for AЌ /d due to its dependence on cos ␾ and also, to a lesser extent, to the fact that dϽD.

B. Radiative corrections QED radiative corrections are applied to convert the measured asymmetries ͑4.8͒ and ͑4.9͒ to one-photon-exchange FIG. 11. The dilution factor f ͑solid line͒ and the effective di- asymmetries. These corrections are calculated using lution factor f Јϭ␳ f ͑dashed line͒ as a function of x. 5346 D. ADAMS et al. 56

p TABLE VII. The virtual photon-proton asymmetry A1 as a function of x and Q2. Only statistical errors are shown.

͗Q2͘ ͗Q2͘ 2 p 2 p ͗x͘ (GeV ) A1 ͗x͘ (GeV ) A1 0.0009 0.25 Ϫ0.122Ϯ0.110 0.0345 7.77 0.058Ϯ0.082 0.0010 0.30 0.033Ϯ0.137 0.0359 10.15 Ϫ0.012Ϯ0.095 0.0011 0.34 0.082Ϯ0.169 0.0474 2.94 Ϫ1.114Ϯ0.589 0.0014 0.38 0.209Ϯ0.081 0.0473 5.49 Ϫ0.117Ϯ0.142 0.0017 0.46 0.042Ϯ0.102 0.0478 7.83 0.241Ϯ0.094 0.0018 0.55 Ϫ0.086Ϯ0.109 0.0484 10.96 0.123Ϯ0.068 0.0023 0.58 0.114Ϯ0.085 0.0527 14.73 0.058Ϯ0.098 0.0025 0.70 Ϫ0.009Ϯ0.094 0.0738 5.33 0.359Ϯ0.239 0.0028 0.82 Ϫ0.025Ϯ0.102 0.0744 7.88 0.212Ϯ0.142 p FIG. 12. The virtual photon asymmetry A1 as a function of x. 0.0036 0.88 Ϫ0.006Ϯ0.065 0.0751 11.09 0.214Ϯ0.088 The error bars show statistical errors only; the systematic errors are 0.0043 1.14 0.089Ϯ0.054 0.0762 16.32 0.203Ϯ0.068 indicated by the shaded area. 0.0051 1.43 0.119Ϯ0.067 0.0855 23.04 0.066Ϯ0.105 0.0057 1.70 Ϫ0.033Ϯ0.118 0.1193 7.36 0.456Ϯ0.242 Neglecting A and thus implying A ϭ⌬␴/(2D␴), the 2 1 0.0070 1.42 0.037Ϯ0.094 0.1199 11.16 0.480Ϯ0.159 radiative corrections to the one-photon asymmetry A1␥ can 1 0.0072 1.76 0.014Ϯ0.073 0.1204 16.47 0.364Ϯ0.110 be written as 0.0077 2.04 Ϫ0.045Ϯ0.071 0.1208 24.84 0.199Ϯ0.098 T 1␥ rc 0.0085 2.34 0.166Ϯ0.085 0.1293 34.28 0.172Ϯ0.137 A1 ϭ␳͑A1 ϩA1 ͒, ͑4.11͒ 0.0092 2.72 0.145Ϯ0.093 0.1713 14.15 0.288Ϯ0.143 1␥ T rc 1␥ 0.0122 2.15 0.184Ϯ0.090 0.1717 24.92 0.349Ϯ0.156 with ␳ϭv¯␴ /¯␴ and A1 ϭ⌬␴tail/2vD¯␴ . 1␥ T rc 0.0125 2.82 0.020Ϯ0.067 0.1742 39.54 0.212Ϯ0.123 The ratio ¯␴ /¯␴ and the radiative correction A1 are evaluated using the program POLRAD ͓90,91͔. The asymme- 0.0141 3.52 0.066Ϯ0.053 0.2384 14.53 0.139Ϯ0.176 p 0.0165 4.43 0.085Ϯ0.069 0.2396 29.71 0.110Ϯ0.132 try A1(x) required as input is taken from Refs. ͓2, 9, 6͔, and p rc 0.0184 5.43 Ϫ0.042Ϯ0.113 0.2462 52.76 0.413Ϯ0.131 the contribution from A2 is neglected. The uncertainty in A1 p 0.0235 2.95 0.189Ϯ0.176 0.3392 15.29 0.644Ϯ0.354 is estimated by varying the input values of A1 within the rc 0.0236 4.38 Ϫ0.026Ϯ0.086 0.3408 29.82 0.814Ϯ0.241 errors. The factor ␳ and the additive correction A1 are shown in Table V at the average Q2 of each x bin. 0.0242 5.75 0.107Ϯ0.070 0.3432 61.49 0.333Ϯ0.179 We have incorporated ␳ into the evaluation of the dilution 0.0263 7.42 0.072Ϯ0.080 0.4747 26.74 0.541Ϯ0.306 factor, f Јϭ␳ f , on an event-by-event basis. Using the weight 0.0339 4.14 0.003Ϯ0.174 0.4858 71.58 0.518Ϯ0.213 T 0.0341 5.81 0.097Ϯ0.119 wϭ f ЈDP␮ , we directly obtain A1 /␳ on the left-hand side of 1␥ Eq. ͑4.8͒ and thus A1 ͓Eq. ͑4.11͔͒. T TABLE VI. Contributions to the systematic errors at the average The radiative corrections to the transverse asymmetry AЌ 2 WW Q of the x bin. are evaluated as above, however assuming that g2ϭg2 ͓55͔. The additive correction is much smaller than the statis- ͗x͘ ⌬Afalse ⌬Pt ⌬P␮ ⌬ f Ј ⌬rc ⌬A2 ⌬R tical error and has been neglected. 0.005 0.0021 0.0025 0.0033 0.0016 0.0012 0.0006 0.0027 0.008 0.0019 0.0013 0.0017 0.0008 0.0012 0.0007 0.0012 C. Dilution factor 0.014 0.0019 0.0018 0.0024 0.0011 0.0011 0.0009 0.0021 In addition to butanol, the target cells contain the NMR 0.025 0.0018 0.0020 0.0027 0.0013 0.0010 0.0002 0.0031 coils and the 3He–4He coolant mixture. The composition in 0.035 0.0018 0.0012 0.0016 0.0008 0.0010 0.0003 0.0016 terms of chemical elements is summarized in Table II. The 0.049 0.0018 0.0031 0.0041 0.0020 0.0009 0.0003 0.0040 dilution factor f can be expressed in terms of the number nA 0.077 0.0019 0.0054 0.0071 0.0035 0.0009 0.0004 0.0080 of nuclei with mass number A and the corresponding total T 0.122 0.0019 0.0087 0.0114 0.0058 0.0010 0.0005 0.0112 spin-independent cross sections ¯␴A per nucleon for all the 0.173 0.0020 0.0083 0.0109 0.0056 0.0010 0.0005 0.0110 elements involved: 0.242 0.0020 0.0074 0.0097 0.0051 0.0009 0.0022 0.0105 ¯T 0.342 0.0020 0.0150 0.0197 0.0107 0.0007 0.0025 0.0236 nH•␴H f ϭ . ͑4.12͒ ¯T 0.482 0.0020 0.0158 0.0208 0.0117 0.0008 0.0030 0.0293 ⌺AnA•A•␴A 0.0011 0.0032 0.0010 0.0013 0.0011 0.0009 0.0005 0.0017 ¯T ¯T 0.0016 0.0027 0.0025 0.0034 0.0026 0.0010 0.0008 0.0035 The total cross section ratios ␴A/␴H for D, He, C, and Ca are n p 0.0025 0.0024 0.0009 0.0012 0.0009 0.0011 0.0011 0.0012 obtained from the structure function ratios F2/F2 ͓87͔ and A d 0.0044 0.0021 0.0018 0.0024 0.0014 0.0012 0.0007 0.0023 F2 /F2 ͓92͔. The original procedure leading from the mea- T T 0.0078 0.0020 0.0015 0.0020 0.0010 0.0012 0.0008 0.0015 sured cross section ratios ¯␴A/¯␴H to the published structure function ratios was inverted step by step involving the iso- 56 SPIN STRUCTURE OF THE PROTON FROM POLARIZED... 5347

p FIG. 14. The virtual photon-proton asymmetry A1 as a function of x from this experiment, compared with data from the EMC and SLAC E80, E130, and E143 experiments. For E143, the structure p p p function ratio g1/F1 is shown instead of A1 . The errors are statistical only.

ence in the statistical error given in Table V and the one reported in Ref. ͓9͔ is due to the change in the beam polarization measurement ͑Sec. III C͒, but this is only a 2% effect. The dilution factor is shown in Fig. 11 where it is compared to the ‘‘naive’’ expectation for a mixture of 62% butanol ͓CH3͑CH2͒3OH͔ and 38% helium by volume, f Ӎ0.123. The rise of f at xϾ0.3 is due to the decrease of the n p ratio F2/F2 , whereas the drop in the small-x range is due to the larger contribution of radiative processes from elements with mass number much larger than hydrogen.

p D. Longitudinal cross section asymmetry FIG. 13. The virtual photon-proton asymmetry A1 as a function 2 of Q , for constant values of x. The solid circles are data from this 1. Results for Ap experiment. The data of the EMC and E143 experiments are also 1 p shown as open circles and squares, respectively. The virtual photon asymmetry A1 is calculated from Eqs. ͑4.8͒, ͑4.11͒, and ͑4.13͒ under the assumption that Afalseϭ0. scalarity corrections and radiative corrections ͑TERAD͒. For The uncertainty introduced by this assumption is estimated unmeasured nuclei the cross section ratios are obtained in the using Eq. ͑4.5͒. A d p 2 2 same way from a parametrization of F2 (x)/F2(x) as a func- The results for A1 for Q у1 GeV are shown in Table V tion of A ͓93–95͔. and in Fig. 12. The numbers for f , D, and kinematic quan- The dilution factor also accounts for the contamination tities given in Table V are mean values within the bins cal- 2 from material outside the finite target cells due to vertex culated with the weighting factor ( f ЈDP␮) . In addition to resolution. This correction is applied as a function of the the results given in Ref. ͓9͔, we include here data obtained scattering angle, and the largest contamination occurs for the with the T14 trigger ͑Sec. III E 2͒. In Table V and Fig. 12, angles between 2 and 9 mrad, which results in a reduction of we also show data in the kinematic range 0.2 GeV2рQ2 the dilution factor by about 6%. The correction needed be- р1 GeV2, 0.0008рxр0.003. These data are not used to p cause of the NMR coils ͑Fig. 10͒ is convoluted with the evaluate g1 or its first moment. distribution of the beam intensity profile. p In the actual evaluation of Eqs. ͑4.8͒ and ͑4.9͒ we use an TABLE VIII. Results on the asymmetry A2 . Only statistical p effective dilution factor f Ј ͑Fig. 11͒: errors are given. The A2 values are the average values from the two target cells. f Јϭ␳ f , ͑4.13͒ x range ͗x͗͘Q2͘(GeV2) A p as discussed in Sec. IV B. The present procedure guarantees 2 a proper calculation of the statistical error in the asymmetry, 0.006–0.015 0.010 1.4 0.002Ϯ0.109 in contrast to our previous analysis ͓9–12͔ where all radia- 0.015–0.050 0.026 2.7 0.041Ϯ0.076 tive effects were included as an additive radiative correction. 0.050–0.150 0.080 5.8 0.017Ϯ0.099 We find an increase in the statistical error by a factor 1/␳ 0.150–0.600 0.226 11.8 0.149Ϯ0.161 which reaches 1.5 at small x ͑Table V͒, but the central values 0.0035–0.006 0.005 0.7 Ϫ0.066Ϯ0.167 of the asymmetries remain unaffected by the change in the 0.006–0.015 0.01 1.3 0.086Ϯ0.097 radiative correction procedure ͓88͔. The remaining differ- 5348 D. ADAMS et al. 56

p 2 FIG. 15. Results for the asymmetry A2(x) extrapolated to Q0 2 2 p ϭ5 GeV assuming ͱQ A2 scales ͓10͔. The solid and dashed curves show the limit ͉A2͉ϽͱR and the prediction corresponding to ¯ p d 2 g2ϭ0, respectively. Also shown are data from the E143 experiment FIG. 16. The structure functions g1 and g1 at the measured Q 2 2 n ͓41͔ extrapolated to the same Q0 assuming that ͱQ A2 scales. The and the corresponding g1 . The upper and lower shaded areas rep- p d errors are statistical only. resent the systematic error for g1 and g1 , respectively.

p 2. Comparison with earlier experiments The sources of systematic errors in A1 are time- p dependence instabilities of the acceptance ratios r and rЈ, In Fig. 14, we compare our results for A1 with data from uncertainties in the beam and target polarizations, in the ef- earlier experiments ͓1,2,6,96͔. Good agreement is observed fective dilution factor f Ј, the radiative corrections, and in in the kinematic region of overlap. A consistency test be- Rϭ␴L /␴T , and the neglect of A2 . The individual errors tween the SLAC E80/E130, EMC, SLAC E143, and SMC ͑Table VI͒ are combined in quadrature to obtain the total data yields a ␹2ϭ11.4 for 16 degrees of freedom. Since the 2 systematic error ͑Table V͒. average Q of SMC and E143 differ by a factor of 7, the p 2 good agreement confirms the earlier conclusion that no Q2 Table VII and Fig. 13 show A1 as a function of Q and x, including the data with Q2р1 GeV2. In Fig. 13, a small dependence is observed within the present accuracy of the correction is applied to the data to display them at the same data. average x in each bin. A study of the Q2 dependence which includes the SMC data ͓9,12͔ was first made by the E143 E. Transverse cross section asymmetry 2 2 Collaboration for 0.03рxр0.6 and Q Ͼ0.3 GeV , and 1. Results for Ap showed no significant Q2 dependence for Q2Ͼ1 GeV2 ͓96͔. 2 2 p We study here the Q dependence for 0.003рxр0.03. A The asymmetry A2 is obtained from our measurements of 2 p p parametrization A1ϭaϩb ln Q is fitted to the data and b is AЌ ͓10͔ and of Aʈ ͓1,2,9͔, using Eq. ͑2.17͒. It is seen from 2 found to be consistent with zero for all x in this range. When Eq. ͑2.9͒ that A2 has an explicit 1/ͱQ dependence, and 2 2 p fitting a parametrization aЈϩc/Q to account for possible hence it is convenient to evaluate ͱQ A2 assuming that it is higher-twist effects, we again find no significant Q2 depen- independent of Q2 in Eq. ͑4.9͒. Our results do not depend on dence. this assumption ͓97͔.

p TABLE IX. Results for the spin-dependent structure function g1 . The ﬁrst error is statistical and the second is systematic. The third error in the last column is the uncertainty associated with the QCD evolution.

2 2 p 2 p 2 2 x range ͗x͗͘Q(GeV )͘ g1(x,Q ) g1(x,Q0ϭ10 GeV ) 0.003–0.006 0.005 1.3 1.97Ϯ0.97Ϯ0.15 2.37Ϯ0.97Ϯ0.15Ϯ0.66 0.006–0.010 0.008 2.1 0.73Ϯ0.61Ϯ0.06 1.03Ϯ0.61Ϯ0.06Ϯ0.17 0.010–0.020 0.014 3.6 0.63Ϯ0.33Ϯ0.05 0.79Ϯ0.33Ϯ0.05Ϯ0.04 0.020–0.030 0.025 5.7 0.45Ϯ0.29Ϯ0.03 0.51Ϯ0.29Ϯ0.03Ϯ0.02 0.030–0.040 0.035 7.8 0.20Ϯ0.26Ϯ0.02 0.22Ϯ0.26Ϯ0.02Ϯ0.01 0.040–0.060 0.049 10.4 0.38Ϯ0.17Ϯ0.02 0.37Ϯ0.17Ϯ0.02Ϯ0.00 0.060–0.100 0.077 15.0 0.42Ϯ0.10Ϯ0.02 0.40Ϯ0.10Ϯ0.02Ϯ0.01 0.100–0.150 0.122 21.4 0.41Ϯ0.08Ϯ0.03 0.39Ϯ0.08Ϯ0.02Ϯ0.01 0.150–0.200 0.173 27.8 0.26Ϯ0.08Ϯ0.02 0.25Ϯ0.08Ϯ0.02Ϯ0.01 0.200–0.300 0.242 35.5 0.15Ϯ0.05Ϯ0.01 0.15Ϯ0.05Ϯ0.01Ϯ0.01 0.300–0.400 0.342 45.5 0.15Ϯ0.04Ϯ0.01 0.17Ϯ0.04Ϯ0.01Ϯ0.00 0.400–0.700 0.482 57.1 0.06Ϯ0.02Ϯ0.00 0.08Ϯ0.02Ϯ0.00Ϯ0.00 56 SPIN STRUCTURE OF THE PROTON FROM POLARIZED... 5349

2 2 TABLE X. Parameters of the polarized parton distributions at Qrefϭ1 GeV , obtained from the QCD ﬁt discussed in the text.

a ␣␤ ␩

⌬qNS 25.4 Ϯ39.1 Ϫ0.67Ϯ0.25 2.12Ϯ0.28 proton: 1.087Ϯ0.006 ͑ﬁxed͒ deuteron: 0.145Ϯ0.002 ͑ﬁxed͒ ⌬⌺ Ϫ1.30Ϯ 0.16 0.71Ϯ0.33 1.56Ϯ1.00 0.40Ϯ0.04

⌬ga⌬⌺ Ϫ0.70Ϯ0.27 4 ͑ﬁxed͒ 0.98Ϯ0.61

p 2 The results for the asymmetry A2 are shown in Table VIII ͑2.15͒ and ͑2.16͒. This analysis is restricted to Q 2 and Fig. 15. They are significantly smaller than the positivity Ͼ1 GeV . For F2 , we use the parametrization of Ref. ͓98͔ p limit ͉A2͉рͱR and are consistent with A2ϭ0 and with the and for R the parametrization of Ref. ͓99͔. The parametriza- WW assumption that g2ϭg2 , i.e., ¯g2ϭ0. Also shown in Fig. 15 tion of R is based on data for xϾ0.01 only and therefore are the E143 data ͓41͔. They confirm our results, with better must be extrapolated to cover smaller values of x. However, 2 statistical accuracy, for xϾ0.03. the structure function g1 at the average Q of the measure- The main systematic uncertainties are due to the param- ment is nearly independent of R due to a partial cancellation p etrizations of Aʈ /D and R. The effects due to time variations between the R dependence of D,ofF2, and of the explicit 2 p of the acceptance are negligible as expected, since the results term ͓1ϩR(x,Q )͔. The results for g1 are shown in Table depend on the ratio of acceptances for muons scattered into IX and, together with our deuteron data ͓13͔, in Fig. 16. the top and the bottom halves of the spectrometer, which p 2 should be affected in the same way by typical variations of B. Evolution of g1 to a fixed Q0 chamber efficiencies. The errors from the dilution factor and p 1 p To evaluate the first moment ⌫1ϭ͐0g1dx, the measured the beam and target polarizations are also very small. The 2 2 p g1(x,Q ) must be evolved to a common Q0 for all x.In total systematic error on A2 is at least one order of magni- 2 previous analyses, g1(x,Q0) was obtained assuming A1 tude smaller than the statistical error at all values of x. 2 Ӎg1 /F1 to be independent of Q . This assumption is con- p sistent with the data. However, perturbative QCD predicts V. RESULTS FOR g1 AND ITS FIRST MOMENT 2 the Q dependences of g1 and F1 to differ by a considerable p 2 A. Evaluation of g1„x,Q … amount at small x. The evolution of g1 /F1 is poorly con- The spin-dependent structure function g p is evaluated strained by the data in this region, where the data cover a 1 very narrow Q2 range. Recent experimental and theoretical from the virtual photon-proton asymmetry A p using Eqs. 1 progress allows us to perform a QCD analysis of polarized structure functions in next-to-leading order ͑NLO͒, and therefore a realistic evolution of g1 can be obtained. Three groups have published such analyses ͓31,100,101͔. They all use the splitting and coefficient functions calculated to NLO in the MS scheme ͓23–25͔, but the choices made for the 2 reference scales Qref at which the polarized parton distributions are parametrized and the forms of the parametrization are different. Also the selections of data sets used for the fits differ. In Ref. ͓31͔ the splitting and coefficient functions are transformed from the MS scheme to different factorization schemes before the fits are performed. We shall refer to the results obtained in the Adler-Bardeen scheme. We used the method2 of Ref. ͓31͔ to fit the present data and those of Refs. ͓2, 11–13, 6, 96, 7͔. The quark-singlet, quark nonsinglet, and gluon polarized distributions are parametrized as

2 ␣ f ␤ f ⌬ f ͑x,Qref͒ϭN f ␩ f x ͑1Ϫx͒ ͑1ϩa f x͒, ͑5.1͒

where the normalization factors N f are chosen such that ͐⌬ fdxϭ␩f . We have assumed that agϭa⌬⌺ . The normal- izations of the nonsinglet quark densities are ﬁxed using neu-

p d n tron and hyperon ␤ decay constants and assuming SU͑3͒ FIG. 17. The structure functions g1 , g1 , and g1 at the measured ﬂavor symmetry. We use g /g ϭFϩDϭϪ1.2601 Q2 for the SMC ͓13͔, E143, ͓6,7͔, and E142 ͓107͔ data. The solid A V curves correspond to our NLO ﬁts at the Q2 of the data points, the 2 2 2 dashed curve at Q0ϭ10 GeV , and the dot-dashed curve at Q0 ϭ1 GeV2. 2The code was kindly provided by the authors. 5350 D. ADAMS et al. 56

p 2 2 FIG. 18. The structure function g1 evolved to Q0ϭ10 GeV 2 using the scaling assumption that g1 /F1 is independent of Q , and using NLO evolution according to our analysis and those of BFR ͓31͔, GRSV ͓100͔, and GS ͓101͔.

Ϯ0.0025 ͓102͔ and F/Dϭ0.575Ϯ0.016 ͓103͔. The parameters of the polarized parton distributions obtained from this fit are given in Table X, and the fit is shown in Fig. 17. We have fixed the exponent ␤ of the gluon distribution to ␤ p d n 2 ϭ4 as expected from QCD counting rules ͓104,105͔, while FIG. 19. Measurements of g1 , g1 , and g1 evolved to Q0 2 n p the fitted values of ␤ for the quark-singlet and -nonsinglet ϭ5 GeV . The SMC and E143 g1 data are obtained from g1 and d components are found to be close to the expectation ␤ϭ3. g1 . Only statistical errors are shown. The ␹2 for the fit is 284 for 295 degrees of freedom. Results n of E142 on g1 were not included in the fit, but used as a Ref. ͓31, 100, 101͔ and by assuming scaling for g1 /F1 . For n(fit) cross-check. In Fig. 17 their data and g1 calculated from the smallest x bin, the latter results in a considerably larger p d the fit to g1 and g1 are presented and found to be in very value of g1 . good agreement. 2 2 2 The measured g1(x,Q ) are then evolved from Q to Q0 C. First moment of gp by adding the correction 1 p 2 From the evolved structure function g1(x,Q0), its first 2 2 fit 2 fit 2 p 2 2 ␦g1͑x,Q ,Q0͒ϭg1 ͑x,Q0͒Ϫg1 ͑x,Q ͒, ͑5.2͒ moment ⌫1 is evaluated at Q0ϭ10 GeV , which is close to 2 fit the average Q of our data. The integral over the measured x where g1 is calculated by evolving the fitted parton distribu- range is p 2 tions. The resulting g1(x,Q0) is shown in Table IX and Fig. p 2 0.7 18. Also shown is the g1(x,Q0) obtained by using the fits of p 2 g1͑x,Q0͒dxϭ0.130Ϯ0.013Ϯ0.008Ϯ0.005, ͵0.003 p TABLE XI. Contributions to the error of ⌫1 . ͑5.3͒

Soure of the error ⌬⌫ 1 where the ﬁrst error is statistical, the second systematic, and Beam polarization 0.0048 the third is the uncertainty due to the Q2 evolution. The Extrapolation at small x 0.0042 individual contributions to the systematic error are summa- Target polarization 0.0036 rized in Table XI. The error from the evolution is mainly due

Uncertainty on F2 0.0030 to the uncertainties in the factorization and renormalization Dilution factor 0.0025 scales, in the parametrizations chosen for the parton distribu- Acceptance variation ⌬r 0.0014 tions, the error in ␣s(M Z), and mass threshold effects. In Momentum measurement 0.0014 addition, we varied the values of F and D used as inputs to Kinematic resolution 0.0010 the ﬁt, of the A2 , Afalse , f , P␮ , Pt , and F2 , and of the Radiative corrections 0.0008 radiative corrections used to calculate g1 . The uncertainty in Extrapolation at large x 0.0007 the ﬁtted parameters of the parton distributions is also included, but is found to be relatively small. These errors on Neglect of A2 0.0004 2 Uncertainty on R 0.0000 ␦g1(x,Q0) are treated as correlated from bin to bin, but uncorrelated amongst each other. Total systematic error 0.0087 The resulting g1 using the different phenomenological analyses of the Q2 evolution ͓31,100,101͔ are shown in Fig. Evolution error 0.0045 18. Despite their different procedures, the differences in their Statistical error 0.0125 results are small and are covered by the error that we quote for the evolution uncertainty. 56 SPIN STRUCTURE OF THE PROTON FROM POLARIZED... 5351

p 2 2 TABLE XII. ⌫1 and the contributions from different x regions at Q0ϭ5 GeV . The results of our analysis of the SMC and E143 data, as well as the combined analysis of the SLAC-E80/130 ͓1͔, EMC ͓2͔, SMC, and SLAC-E143 ͓6͔ data are given with the statistical and systematic errors added in quadrature. Results of extrapolations are marked with an asterisk.

x range 0–0.003 0.003–0.03 0.03–0.7 0.7–0.8 0.8–1 0–1

SMC 0.004(2)* 0.022͑7͒ 0.104͑13͒ 0.0018(4)* 0.0006(2)* 0.132͑17͒ E143 0.0012(1)* 0.010(1)* 0.115͑7͒ 0.0020͑6͒ 0.0006(2)* 0.129͑8͒ All 0.004(2)* 0.021͑6͒ 0.114͑6͒ 0.0020͑6͒ 0.0006(2)* 0.141͑11͒

p 2 Ϫ1 0.003 p 2 To estimate the integral for 0.7ϽxϽ1.0, assume that A1 model, g1(x)Ϸ(x ln x) , gives ͐0.0 g1(x,Q0)dxϭ0.036 p ϭ0.7Ϯ0.3 in this region. This is consistent with the large x Ϯ0.016. This model results in a larger ⌫1 , but cannot simul- data and with the expectation from perturbative QCD that n taneously accommodate the negative values of g1 found g /F 1asx 1͓104͔. We obtain 1 1→ → from our combined deuteron ͓13͔ and proton data ͑Fig. 16͒. In principle the small-x contribution to the integral can be 1.0 p 2 2 obtained from the fit to g , i.e., gfit . However, as known g1͑x,Q0ϭ10 GeV ͒dxϭ0.0015Ϯ0.0007. ͑5.4͒ 1 1 ͵0.7 from unpolarized parton distribution functions, the behavior of the fitted distribution below the measured region is unre- The results from our fit shown in Fig. 17 are used to liable since it depends strongly on the choice of the function, 1.0 p 0 evaluate ͐0.003g1(x,Q2)dx and found to be consistent with renormalization, and factorization scales. the sum of Eqs. ͑5.3͒ and ͑5.4͒. p 2 The result for the first moment of g1(x,Q0)is The contribution to the first moment from the unmeasured region 0ϽxϽ0.003 is evaluated assuming a constant g p at p 2 2 1 ⌫1͑Q0ϭ10 GeV ͒ϭ0.136Ϯ0.013Ϯ0.009Ϯ0.005. Q2ϭ10 GeV2, in agreement with a Regge-type behavior ͑5.6͒ ͓27͔. Using the average of the two smallest x data points in Table IX, we obtain Using the results of the NLO evolutions of p 2 Refs. ͓31,100,101͔,wefind⌫1(Q0) between 0.133 and 0.003 p 2 p 2 0.136 ͑Fig. 18͒. If we evaluate g (x,Q ) assuming that g ͑x,Q ϭ10 GeV2͒dxϭ0.0042Ϯ0.0016. 1 0 ͵ 1 0 2 p 2 0 g1 /F1 is independent of Q , we obtain ⌫1(Q0)ϭ0.139 ͑5.5͒ Ϯ0.014Ϯ0.010. We conclude that within the experimental accuracy of our data the different NLO QCD analyses yield However, to evaluate the systematic error on ⌫ p , we have 1 consistent results for the evolution of g1 and that g1 /F1 assumed an error of 100% in this integral ͑Table XI͒.It deviates significantly from scaling at small x. should be noted that we have assumed constant Regge-type behavior at Q2ϭ10 GeV2. If we apply the same procedure at D. Combined analysis of ⌫p Q2ϭ1 GeV2 and then evolve the resulting extrapolation to 1 2 2 p Q ϭ10 GeV using the NLO fits, we obtain a value which is We present a combined analysis of ⌫1 which includes the within 1.5␴ of the assumed error. Other models describing proton spin asymmetries for Q2Ͼ1 GeV2 from our data and the small-x behavior of g1 ͑Sec. II D͒ are also considered to those of Refs. ͓1, 2, 6͔ shown in Fig. 14. The EMC and SMC check the sensitivity of our result to the small-x extrapola- data were taken at an average Q2 of 10 GeV2, while for the 2 2 tion. A g1(x)Ϸln x dependence is compatible with the error SLAC data the average Q is 3 GeV . The combined result given in Eq. ͑5.5͒, while the x behavior in the diffractive is evaluated at an intermediate Q2 of 5 GeV2 to avoid large-

TABLE XIII. The Ellis-Jaffe sum rule calculated with NLO QCD corrections compared to our result for p 2 2 ⌫1 at Q0ϭ10 and 5 GeV and to the combined analysis of ﬁt the E80/E130 ͓1͔, EMC ͓2͔, SMC, and E143 2 2 ͓6͔ data at Q0ϭ5 GeV . The Bjorken sum rule calculated with NNLO QCD corrections and compared to our p n p d results on ⌫1Ϫ⌫1 from the SMC, the combined analysis of ⌫1 and ⌫1 ͑SMC ͓13͔ and E143 ͓7͔͒, and the p d n combined analysis of ⌫1 , ⌫1 , and ⌫1 ͑E142 ͓107͔͒.

p n d p n Experiment/theory ⌫1 ⌫1 ⌫1 ⌫1Ϫ⌫1

2 2 Q0ϭ10 GeV SMC 0.136Ϯ0.016 Ϫ0.046Ϯ0.021 0.041Ϯ0.007 0.183Ϯ0.034 Ellis-Jaffe/Bjorken 0.170Ϯ0.004 Ϫ0.016Ϯ0.004 0.071Ϯ0.004 0.187Ϯ0.002 2 2 Q0ϭ5 GeV SMC 0.132Ϯ0.017 Ϫ0.048Ϯ0.022 0.039Ϯ0.008 0.181Ϯ0.035 Combined (p,d) 0.141Ϯ0.011 Ϫ0.065Ϯ0.017 0.039Ϯ0.006 0.199Ϯ0.025 Combined (p,d,n) 0.142Ϯ0.011 Ϫ0.061Ϯ0.016 0.038Ϯ0.006 0.202Ϯ0.022 Ellis-Jaffe/Bjorken 0.167Ϯ0.005 Ϫ0.015Ϯ0.004 0.070Ϯ0.004 0.181Ϯ0.003 5352 D. ADAMS et al. 56

2 p If A1 is assumed to be independent of Q , we obtain ⌫1 ϭ0.140Ϯ0.012. It should be noted that the error quoted by the E143 Col- laboration ͓6͔ from their data alone and the error obtained from our combined analysis are comparable. The statistical uncertainties of the SMC data for 0.003ϽxϽ0.03 introduce p a larger error to ⌫1 than the uncertainty assumed by the E143 Collaboration for their extrapolation from xϭ0.03 to xϭ0. We also calculated the extrapolations from the evolved E143 and SMC data separately. The results are compared in Table XII. p The results for ⌫ p from SMC and from the combined FIG. 20. Comparison of the experimental results for ⌫1 to the 1 prediction of the Ellis-Jaffe sum rule. analysis are compared with the Ellis-Jaffe sum rule in Table XIII. The Ellis-Jaffe prediction is calculated from Eq. ͑2.44͒.

Q2 evolutions. Corrections to g1 /F1 calculated at NLO are The higher-order QCD corrections are applied assuming p 2 three active quark flavors, and using ␣ (5 GeV2)ϭ0.287 found to be up to 20–25 %. The evolution of g1 to Q0 s 2 2 ϭ5 GeV ͑Fig. 19͒ is performed using the procedure of Sec. Ϯ0.020 and ␣s(10 GeV )ϭ0.249Ϯ0.015 corresponding to 2 2 2 VB. ␣s(M Z)ϭ0.118Ϯ0.003 ͓102͔.AsQ0ϭ10 GeV is close to The data are combined on a bin-by-bin basis. The inte- the charm threshold, a small uncertainty has been included to grals ⌬⌫i ϭ͐ gp(x,Q2)dx are computed for the x bins of account for the difference between the perturbative QCD 1 ⌬xi 1 0 each experiment individually, starting from the published corrections for three and four flavors. This uncertainty is also i included in the error estimate for the Bjorken sum rule pre- asymmetries. The ⌬⌫ which fall into the same SMC x bin 1 diction presented in the next section. are first summed for each experiment and then the integral We reevaluated the first moments for all experiments at for this bin is obtained as the weighted average of these 2 their average Q using the g1 evolution described in Sec. sums. The weights are calculated by adding the statistical 2 errors and systematic errors uncorrelated between the experi- V B. In Fig. 20 the results are shown as a function of Q . All experimental results are smaller than the Ellis-Jaffe sum rule ments in quadrature. The error and the central value of the p integral in the measured region is computed using a Monte prediction. From the combined analysis of ⌫1 the Ellis-Jaffe Carlo method, which takes into account the bin-to-bin corre- sum rule is violated by more than two standard deviations. lation of the systematic errors within each experiment as well The implications of this result on the spin content of the as correlations between the experiments. These correlated proton will be discussed in Sec. VIII. contributions are due to the polarizations of the beam and the VI. RESULTS FOR gp AND ITS FIRST MOMENT target, the dilution factor, the neglect of A2 , the time depen- 2 dence of the acceptance ratio, the radiative corrections, and p 2 A. Evaluation of g2„x,Q … the parametrizations of F 98 ,ofR 99 , and of the parton 2 ͓ ͔ ͓ ͔ p distribution functions used to evolve g . Correlations be- The spin-dependent structure function g2 is evaluated 1 p tween the experiments arise mainly from the latter three from the A2 data ͑Table VIII͒ using sources. The error distributions in the Monte Carlo sampling F1 ␥͑␥Ϫ␩͒ Aʈ 2Mx are assumed to be Gaussian. g ϭ ͱQ2A 1Ϫ Ϫ , The x range of the combined data is 0.003ϽxϽ0.8. The 2 2Mx ͫ 2ͩ 1ϩ␥2 ͪ D ͩ1ϩ␥2ͪͬ extrapolations at large and small x are performed using the ͑6.1͒ procedures described in Sec. V C. The contributions to the from Eqs. ͑2.7͒ and ͑2.9͒ and a parametrization of Aʈ /D integral from the measured and extrapolated regions of x are 2 p p from Refs. ͓2, 9, 6͔. We assume that ͱQ A2 and Aʈ /D are shown in Table XII. 2 p independent of Q , which is consistent with the data. The The combined result for the first moment of g1 is p p new analyses of g1 or F2 do not affect the g2 results that we p 2 2 ⌫1͑Q0ϭ5 GeV ͒ϭ0.141Ϯ0.011 ͑all proton data͒. published in Ref. ͓10͔ due to the limited accuracy of the data. p ͑5.7͒ The g2 values are given in Table XIV. The expected values

p WW TABLE XIV. Results for the spin-dependent structure function g2 . The predicted twist-2 term for g2 ͓Eq. ͑2.48͔͒ and the upper limit obtained from ͉A2͉ϽͱR are also given. Only statistical errors are shown.

2 2 WW upper x range ͗x͗͘Q͘(GeV ) ͗y͘ g2 g2 g2 0.006–0.015 0.010 1.36 0.72 0.8Ϯ75.8 0.72Ϯ0.22 429Ϯ61 0.015–0.050 0.026 2.66 0.57 7.1Ϯ13.9 0.45Ϯ0.07 101Ϯ12 0.050–0.100 0.069 5.27 0.42 1.1Ϯ4.8 0.19Ϯ0.02 17.4Ϯ4.6 0.100–0.150 0.121 7.65 0.34 Ϫ1.0Ϯ2.9 0.04Ϯ0.02 6.1Ϯ2.8 0.150–0.300 0.199 10.86 0.30 0.2Ϯ1.7 Ϫ0.08Ϯ0.01 1.9Ϯ1.2 0.20–0.600 0.378 17.07 0.25 0.6Ϯ0.6 Ϫ0.10Ϯ0.01 0.2Ϯ0.5 56 SPIN STRUCTURE OF THE PROTON FROM POLARIZED... 5353

ability of the deuteron to be in a D state, we have taken ␻Dϭ0.05Ϯ0.01, which covers most of the published values ͓106͔. Using the method described in Sec. V D to account for the correlations between errors, we obtain

p n 2 2 ⌫1Ϫ⌫1ϭ0.183Ϯ0.034 ͑Q0ϭ10 GeV ͒, ͑7.2͒ where statistical and systematic errors are combined in quadrature. The theoretical prediction at the same Q2, in- 3 cluding perturbative QCD corrections up to O(␣s ) and as- p n suming three quark flavors ͑Sec. II E 1͒,is⌫1Ϫ⌫1ϭ0.187 Ϯ0.002. We have also performed a combined analysis of all proton 2 2 and deuteron data at Q0ϭ5 GeV ͑Fig. 19͒. The combined d ⌫1 is obtained using the same method as described in Sec. V D for ⌫ p . We find FIG. 21. Comparison of the combined experimental results for 1 p n d ⌫1 , ⌫1 , and ⌫1 with the predictions for the Bjorken and the Ellis- p n Jaffe sum rules. The Ellis-Jaffe prediction is shown by the black ⌫1Ϫ⌫ ϭ0.199Ϯ0.025 ellipse inside the Bjorken sum rule band. ͑Q2ϭ5 GeV2, all proton and deuteron data͒. WW 0 of g2 and the upper bound of g2 , based on the positivity ͑7.3͒ limit of A2 are also included. The statistical accuracy on g2 2 2 p n is poor since the error is proportional to 1/x and ͱQ , and The corresponding theoretical expectation is ⌫1Ϫ⌫1 the data are characterized by small x and high Q2. All values ϭ0.181Ϯ0.003, which agrees with the experimental result as are consistent with zero. shown in Fig. 21. n The structure function g1 of the neutron has also been p measured by scattering polarized electrons on a polarized B. First moment of g2 3He target ͓5͔. The reanalyzed neutron data on gn from E142 The Burkhardt-Cottingham sum rule predicts that the first 1 p ͓107͔ are included in the combined analysis. This requires moment of g2 vanishes ͑Sec. II G͒. This integral is evaluated p n d 2 2 the combination of ⌫1 , ⌫1 , and ⌫1 via a fit constrained by over the measured x range at the mean Q of the data (Q0 the integral of Eq. ͑7.1͒ and the use of a Monte Carlo method 2 2 ϭ5 GeV ) by assuming a constant value of ͱQ A2(x) p n to compute the 3ϫ3 correlation matrix between ⌫1 , ⌫1 , within each x bin. We obtain d p d n and ⌫1 . The ⌫1 and ⌫1 are obtained as before; ⌫1 is ob- 0.6 tained from the E142 data in their measured region, but the Ϫ1.0Ͻ g p x,Q2͒dxϽ2.1, ͑6.2͒ n ͵ 2͑ 0 small-x extrapolation is determined from the g1 values ob- 0.006 tained from the SMC proton and deuteron data. The result is at 90% confidence level. Our measurement of g2 is not ac- ⌫ pϪ⌫nϭ0.202Ϯ0.022 curate enough to perform a meaningful extrapolation to x 1 1 ϭ0 using the expected g2 Regge behavior, g2(x 0) 2 2 Ϫ1ϩ␣ → ͑Q0ϭ5 GeV , all proton, deuteron, and neutron data͒. ϳx ͓56͔ and to test the sum rule. The first moment ͑7.4͒ 2 2 2 WW 2 ⌫2(Q0) can be divided into ⌫2(Q0)ϭ⌫2(Q0) ϩ⌫2(Q0), WW WW where ⌫2 is obtained from g2 ͓Eq. ͑2.48͔͒ and ⌫2 from As discussed in Ref. ͓108͔, the central value and the error of p p ⌫n is very sensitive to the SMC proton and deuteron data. the g2 component. Using a parametrization of all g /F data 1 1 1 p d n ͓2,9,6͔, we find that the twist-2 part is, as expected, compat- The relation between ⌫1 , ⌫1 , and ⌫1 and the Bjorken 2 WW sum rule is illustrated in Fig. 21, and the results are given in ible with zero (⌫2(Q0) Ӎ0.001Ϯ0.008). A violation of Table XIII.3 Proton, deuteron, and neutron results confirm the sum rule caused by the g2 term cannot be excluded by the present data. the Bjorken sum but disagree with the Ellis-Jaffe sum rule.

p n VIII. SPIN STRUCTURE OF THE PROTON VII. EVALUATION OF ⌫1؊⌫1 AND TEST OF THE BJORKEN SUM RULE n p A. x dependence of g1 and g1 2 2 p d We ﬁrst test the Bjorken sum rule at Q0ϭ10 GeV as- In Fig. 16 we show our results for g1 and g1 , together n p d suming with g1 obtained from g1 and g1 using Eq. ͑7.1͒.Weﬁnd n p Ϫ d that the ratio g1/g1 is close to 1 at small x, in contrast to g1 g pϪgnϭ2 g pϪ . ͑7.1͒ 1 1 1 3 ͩ 1Ϫ 2 ␻D ͪ 3 n The error on ⌫1 given in this paper is different from what appears For this test we employ our present proton data and our in Ref. ͓13͔ where the correlations between errors were not taken previously published deuteron data ͓11–13͔. For the prob- into account properly. 5354 D. ADAMS et al. 56

TABLE XV. Results for a0 and individual quark contributions from proton data. The results based on 2 2 2 2 2 SMC data only are given at the average Q of the data, Q0ϭ10 GeV , and at Q0ϭ5 GeV for a direct comparison with the combined analysis of all proton data.

Data used a0 au ad as

p 2 SMC ⌫1 (10 GeV ) 0.28Ϯ0.16 0.82Ϯ0.05 Ϫ0.44Ϯ0.05 Ϫ0.10Ϯ0.05 p 2 SMC ⌫1 (5 GeV ) 0.28Ϯ0.17 0.82Ϯ0.06 Ϫ0.44Ϯ0.06 Ϫ0.10Ϯ0.06 p 2 All ⌫1 (5 GeV ) 0.37Ϯ0.11 0.85Ϯ0.04 Ϫ0.41Ϯ0.04 Ϫ0.07Ϯ0.04

n p the ratio F2/F2 which is close to ϩ1 for xϽ0.01 The results are given in Table XV. They indicate that as is p n ͓87,109,110͔. In the QPM the difference between g1 and g1 negative, in agreement with the measurement of elastic ␯-p can be written as scattering ͓114,115͔. In the QPM, aiϭ⌬qi . However, as discussed in Sec. II F, p n 1 ¯ g1Ϫg1ϭ 6 ͓⌬uv͑x͒Ϫ⌬dv͑x͒ϩ2⌬¯u͑x͒Ϫ2⌬d͑x͔͒. due to the U͑1͒ anomaly of the singlet axial vector current ͑8.1͒ the axial charges receive a gluon contribution. In the AB scheme ͓31͔ used in our QCD fit for three flavors, we have Under the assumption of flavor symmetry in the polarized ¯ n p 2 sea (⌬¯uϭ⌬d) ͓111,112͔, the small-x behavior of g1/g1 in- ␣s͑Q ͒ a ϭ⌬q Ϫ ⌬g͑Q2͒ ͑iϭu,d,s͒. ͑8.7͒ dicates a dominant contribution from the valence quarks. i i 2␲ This is consistent with our results from semi-inclusive spin asymmetries ͓14͔, which show that ⌬u (x)Ϫ⌬d (x) is 2 ͓ v v ͔ In this scheme ⌬qi is independent of Q . For this reason positive and that ⌬uv(x) and ⌬dv(x) have opposite signs. some authors consider this to be the correct scheme when Fits of polarized parton distributions in the NLO analysis assuming ⌬sϭ0 ͓47–49͔. lead to the same conclusion ͓100,101͔. The relation between the matrix element a3 and the neutron ␤-decay constant gA /gV relies only on the assumption B. Axial quark charges of isospin invariance. However, in order to relate a8 to the When only three flavors contribute to the nucleon spin, semileptonic hyperon decay constants F and D, we assume p SU͑3͒ flavor symmetry and hence conclusions on a0 depend the first moment of g1 can be expressed in terms of the proton matrix elements of the axial vector currents ͑Sec. on its validity. SU͑3͒-symmetry-breaking effects do not van- IIE1͒ ish at first order for axial vector matrix elements ͓116͔,as they do for vector matrix elements ͓117͔. It has been sug- NS 2 S 2 gested that in order to reproduce the experimental values of C1 ͑Q ͒ 1 C1͑Q ͒ ⌫p͑Q2͒ϭ a ϩ a ϩ a ͑Q2͒. F and D, the QPM requires large relativistic corrections 1 12 3 3 8 9 0 ͫ ͬ which depend on the quark masses; since the s-quark mass is ͑8.2͒ much larger than that of u and d quarks, these corrections 2 p 2 We obtain a0(Q ) from ⌫1(Q ) and the experimental should break SU͑3͒ symmetry. Similarly, the relations between the baryon magnetic moments predicted by SU 3 are nonsinglet matrix elements a3 and a8 , which are calculated ͑ ͒ badly broken 118 . from gA /gV and F/D, as presented in Sec. V B. The singlet ͓ ͔ S NS The uncertainty on a propagates into a and a accord- ͑nonsinglet͒ coefficient function S1(C1 ) is the same as pre- 8 0 s S ing to sented in Sec. II E 1, and C1 is computed with the coefficients cS in the last column of Table I. If instead the coeffi- a CNSץ i cient from the third column were used, we would get aϱ . 0 1 0 ϭϪ S ӍϪ0.23, ͑8.8͒ a8 4C1ץ ϱ 2 2 Numerically, a0 is smaller than a0(Q ) by 10% at Q ϭ10 GeV2. NS S as C1 ϩ4C1ץ From the combined analysis of all proton data, we find ϭϪ ӍϪ0.44. ͑8.9͒ a S 12C1 8 ץ 2 2 2 a0͑Q0͒ϭ0.37Ϯ0.11 ͑Q0ϭ5 GeV , all proton data͒. ͑8.3͒ The smaller magnitude of as and its larger derivative with In Table XV we compare the results with those based on respect to a8 make it much more sensitive to uncertainties in SMC data only. Calculations in lattice QCD ͓113͔ agree with a8 than a0 ͓119͔. For instance, the experimental test of SU͑3͒ from the compatibility of different hyperon ␤ decays the measured values of both a0 and gA /gV . Using a3ϭau 2 allows for a 15% modification of a8 ; this would change as Ϫad , a8ϭauϩadϪ2as , and a0(Q )ϭauϩadϩas , the individual contributions from quark flavors are evaluated from by as much as 50%, while a0 changes by less than 10%. A result for a8 has been obtained from a leading-order 1 2 auϭ 6 ͓2a0͑Q ͒ϩa8ϩ3a3͔, ͑8.4͒ 1/Nc expansion ͓120͔ which is much smaller than the value based on the SU͑3͒ analysis. The use of this smaller value of 1 2 adϭ 6 ͓2a0͑Q ͒ϩa8Ϫ3a3͔, ͑8.5͒ a8 causes a0 to become larger, while as becomes positive. In principle, another source of uncertainty arises from the 1 2 asϭ 3 ͓a0͑Q ͒Ϫa8͔. ͑8.6͒ possible contributions of heavier quarks. The heavy quark 56 SPIN STRUCTURE OF THE PROTON FROM POLARIZED... 5355

Skyrme model also assumes ⌬gϭLgϭ0. In a recent version of this model, where gA /gV is calculated to within 4% of the experimental value, ⌬⌺ is found to be between 0.18 and 0.32 ͓127͔. In QCD, a0 differs from ⌬⌺ in a scheme-dependent way. In the AB scheme the determination of ⌬⌺ and the various ⌬qi from the measured a0 and ai requires an input value for ⌬g. The allowed values for ⌬⌺ and for the ⌬qi are shown in Fig. 22 as a function of ⌬g ͓Eq. ͑8.7͔͒. We see that a value 2 2 of ⌬sϭ0 and ⌬⌺ϳ0.57 corresponds to ⌬g(Q )Ϸ2atQ0 ϭ5 GeV2. However, the gluon contribution ⌬g could be smaller than indicated in Eq. ͑8.7͒ due to finite quark masses and a possibly non-negligible contribution from charm, according to the authors of Ref. ͓128͔. In the absence of direct measurements of ⌬g, our results can only be compared with the estimate of ⌬g(Q2) obtained from NLO GLAP fits to the 2 g1 data as in Sec. V B. Different estimates of ⌬g(Q ) have FIG. 22. Quark spin contributions to the proton spin as a func- been obtained. The factorization scheme used in the fit of 2 2 2 2 tion of the gluon contribution at Q ϭ5 GeV in the Adler-Bardeen Ref. ͓31͔ and Sec. V B provides ⌬⌺ ⌬g(Q ), while a0(Q ) scheme. All the available proton data samples are taken into ac- and ⌬g(Q2) are obtained in the scheme used for the fits of count. Refs. ͓100͔ and ͓101͔. While the singlet distribution depends on the factorization scheme, the gluon distribution is the 2 2 2 axial current has a nonzero matrix element because it can same in both ͓51͔. For Q0ϭ5 GeV we find ⌬g(Q0)ϭ1.7 2 mix with light quark operators ͓121͔. This mixing is closely Ϯ1.1, and Refs. ͓129͔ and ͓100͔ find ⌬g(Q0)ϭ2.6 and 1.4, related to the U͑1͒ anomaly and is directly calculable in respectively. The results of Ref. ͓129͔ are based on the 2 2 QCD ͓122,123͔, where the heavy quark contributions can be method of Ref. ͓31͔. Similarly, at Q0ϭ10 GeV it is found 2 expressed in terms of light quark contributions. Following that ⌬g(Q0) is equal to 2.0Ϯ1.3, 3.1, and 1.7, respectively. the analysis of Ref. ͓121͔ and using the result for a0 of Eq. 2 2 ͑8.3͒, the expected values for ab and ac for Q Ӷmb are D. Combined analysis of a0 from all proton, neutron, Ϫ0.003Ϯ0.001 and Ϫ0.006Ϯ0.002, respectively. In view of and deuteron data the current accuracy for a and of the Q2 range covered by 0 The analysis used to test the Bjorken sum rule can be the data, the contribution from heavier quarks can be ne- extended to evaluate a0 , giving ͑proton, deuteron, and neu- glected. 2 2 p tron data, Q0ϭ5 GeV ͒, Another possible explanation of the low value of ⌫1 can be given by the formalism developed by ͓124͔ based on a a0ϭ0.29Ϯ0.06, auϭ0.82Ϯ0.02, U͑1͒ Golberger-Treiman relation for the singlet axial current. In this approach the data can indicate a violation of the a ϭϪ0.43Ϯ0.02, a ϭϪ0.10Ϯ0.02. p d s Okubo-Zweig-Izuka ͑OZI͒ rule. Their predicted value of ⌫1 is very close to the measured one. An analysis of a0 based on a different selection and treatment of experimental data has been presented in Ref. ͓130͔, C. Spin content of the proton with similar results. The nucleon spin can be written IX. CONCLUSION S ϭ 1 ⌬⌺ϩL ϩ⌬gϩL ϭ 1 , ͑8.10͒ z 2 q g 2 A. Summary in which ⌬⌺ϭ⌬uϩ⌬dϩ⌬s and ⌬g are the contributions We have presented a complete analysis of our measure- of the quark and gluon spins to the nucleon spin, and Lq and ment of the spin-dependent structure function g1 of the pro- Lg are the components of the orbital angular momentum of ton from deep-inelastic scattering of high-energy polarized the quarks and the gluons along the quantization axis ͓125͔. muons on a polarized target. The data cover the kinematic The Q2 dependence of the angular momentum terms ana- range 0.003ϽxϽ0.7 for Q2Ͼ1 GeV2, with an average Q2 lyzed in LO was studied in Ref. ͓126͔. It is observed that the ϭ10 GeV2. In addition to these data, we have also shown for 2 1 asymptotic limits (Q ϱ) of the terms ( 2 ⌬⌺ϩLq) and the first time virtual photon-proton asymmetries in the kine- → 2 2 (⌬gϩLg) are about the same and equal to ϳ1/4. matic range 0.0008ϽxϽ0.003 and Q Ͼ0.2 GeV . In the ki- In the naive QPM, ⌬gϭLgϭ0 and ⌬⌺ϭa0 . In this nematic range xϽ0.03, our data are the only available mea- framework earlier experiments concluded that only a small surements of the spin-dependent asymmetries. p p p 2 fraction of the nucleon spin is carried by the quark spins and The virtual photon asymmetry A1Ӎg1/F1 shows no Q that the strange quark spin contribution is negative. This con- dependence over the x range of our data within the experi- clusion is in disagreement with the Ellis-Jaffe assumption of mental uncertainty. This observation holds when we com- ⌬sϭasϭ0, which corresponds to ⌬⌺ϭa8Ӎ0.57, with Lq bine our results with those from electron scattering experi- 2 carrying about half of the total angular momentum. The ments performed at smaller Q . However, g1 and F1 are 5356 D. ADAMS et al. 56 predicted to evolve differently and the difference should be larized muon-nucleon scattering by COMPASS ͓132͔ at observable at small x in future precise measurements. CERN, and a similar semi-inclusive polarized electron- p From our data on g1 together with our deuteron data, we nucleon experiment at SLAC ͓133͔. Furthermore, a polarized n p electron-proton collider experiment at HERA to study the find that the ratio g1/g1 is close to Ϫ1 at small x n p inclusive and semi-inclusive scattering is also under consid- (ϳ0.005), in contrast to F2/F2 , which approaches ϩ1. This suggests that either the valence quarks give a significant con- eration ͓134͔. The non-Regge behavior of the unpolarized tribution to the net quark polarization in this region or that structure function F2 has been observed at HERA in agree- the spin distribution functions of the u and d sea quarks are ment with perturbative QCD predictions ͓135,136͔. The cor- different, i.e., ⌬¯u(x)Þ⌬¯d(x). The data suggest a rise in responding behavior predicted for the polarized spin structure function g is particularly interesting due to the fact that g p(x)asxdecreases from 0.03 to 0.0008. A small-x extrapo- 1 1 the higher-order corrections in the polarized case are ex- lation of g beyond the measured region is necessary to com- 1 pected to be stronger ͓129,137͔. Also, unlike the unpolarized pute its first moment ⌫ and test sum rule predictions. Pre- 1 case where only the gluon distribution is important at small cise data at small x are crucial for constraining this x, in the polarized case the singlet quark, the nonsinglet extrapolation. quark, and the gluon distributions all play a role. The new data have initiated much theoretical activity in In conclusion, the study of the spin structure of the recent years, resulting in an extensive discussion of the NLO nucleon appears certain to remain active well into the next QCD analyses of the x and Q2 dependence of g and of the 1 century. interpretation of a0 in terms of the spin content of the nucleon. As a result, we have used new methods for the 2 ACKNOWLEDGMENTS evaluation of the structure function g1 at fixed Q . From this evolved structure function we determined the first moment of We wish to thank our host laboratory CERN for providing g1 and confirmed the violation of the Ellis-Jaffe sum rule for major and efficient support for our experiment and an excit- the proton by more than 2␴. We obtain for the singlet axial ing and pleasant environment in which to do it. In particular, 2 2 2 charge of the proton a0(Q0)ϭ0.28Ϯ0.16 at Q0ϭ10 GeV . we thank J. V. Allaby, P. Darriulat, F. Dydak, L. Foa, G. From the fit to all currently available data, we obtain Goggi, H. J. Hilke, and H. Wenninger for substantial support 2 ⌬g(Q0)ϭ2.0Ϯ1.3, which in the Adler-Bardeen renormal- and constant advice. We also wish to thank L. Gatignon and ization scheme implies the value ⌬⌺Ӎ0.5. The new data and the SPS Division for providing us with an excellent beam, theoretical developments now afford a first glimpse of the the LHC-ECR group for efficient cryogenics support, and J. polarized gluon distribution and its first moment. M. Demolis for all his technical support. We also thank all The Bjorken sum rule is fundamental and must hold in those people in our home institutions who have contributed 3 perturbative QCD. When corrections up to O(␣s ) are in- to the construction and maintenance of our equipment, espe- p n 2 2 cially A. Dae¨l, J. C. Languillat, and C. Cure´ from DAPNIA/ cluded, it predicts ⌫1Ϫ⌫1ϭ0.187Ϯ0.002 at Q0ϭ10 GeV . Saclay for providing us with the high-performance target su- Using the first moments of the structure functions g1 evalu- p n perconducting magnet, Y. Lefe´vre and J. Homma from ated from our proton and deuteron data, we find ⌫1Ϫ⌫1 2 2 NIKHEF for their contributions to the construction of the ϭ0.183Ϯ0.034 at Q0ϭ10 GeV , in excellent agreement with the theoretical prediction. Combining our data with all dilution refrigerator, and E. Kok for his contributions to the available data results in a somewhat more precise confirma- electronics and the data taking. It is a pleasure to thank G. tion of the Bjorken sum rule. Altarelli, R. D. Ball, J. Ellis, S. Forte, and G. Ridolfi, for valuable discussions. This work was supported by the Bundesministerium fu¨r B. Outlook Bildung, Wissenschaft, Forschung und Technologie, par- New data on the spin-dependent structure functions g1 tially supported by TUBITAK and the Center for Turkish- and g2 of the nucleon are expected in the next two years Balkan Physics Research and Application ͑Bogaziçi Univer- from the SMC, the E154, and E155 Collaborations at SLAC sity͒, supported by the U.S. Department of Energy, the U.S. and from the HERMES Collaboration at HERA. However, National Science Foundation, Monbusho Grant-in-Aid for further knowledge is needed of the small-x behavior of g1 Scientific Research ͑International Scientific Research Pro- and of the polarized gluon distribution ⌬g(x) due to the gram and Specially Promoted Research͒, the National Sci- limited coverage in x and Q2 of these experiments. ence Foundation ͑NWO͒ of the Netherlands, the Commis- Future experiments are planned at various experimental sariat a` l’Energie Atomique, Comision Interministerial de facilities, including semi-inclusive polarized proton-proton Ciencia y Tecnologia, the Israel Science Foundation, and scattering by RHIC SPIN ͓131͔ at BNL, semi-inclusive po- Grant No. KBN SPUB/P3/209/94.

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