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1996

Deep Inelastic Scattering of Polarized by Polarized ³He and the Study of the Spin Structure

P. L. Anthony

R. G. Arnold

H. R. Band

H. Borel

P. E. Bosted

See next page for additional authors

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Original Publication Citation Anthony, P.L., Arnold, R.G., Band, H.R. ... Kuhn, S.E., Young, CC., Zapalac, G., E142 Collaboration (1996). Deep inelastic scattering of polarized electrons by polarized ³He and the study of the neutron spin structure Physical Review D, 54(11), 6620-6650. https://doi.org/10.1103/PhysRevD.54.6620

This Article is brought to you for free and open access by the Physics at ODU Digital Commons. It has been accepted for inclusion in Physics Faculty Publications by an authorized administrator of ODU Digital Commons. For more information, please contact [email protected]. Authors P. L. Anthony, R. G. Arnold, H. R. Band, H. Borel, P. E. Bosted, V. Breton, G. D. Cates, T. E. Chupp, F. S. Dietrich, J. Dunne, R. Erbacher, J. Felbauum, H. Fonvielle, R. Gearhart, R. Holmes, E. W. Hughes, J. R. Johnson, D. Kawall, C. Keppel, S. E. Kuhn, R. M. Lombard-Nelsen, J. Marroncle, T. Maruyama, W. Meyer, Z.- E. Meziani, H. Middleton, J. Morgenstern, N. R. Newbury, G. G. Petratos, R Pitthan, R. Prepost, Y. Robin, S. E. Rock, S. H. Rokni, G. Shapiro, T. Smith, P. A. Souder, M. Spengos, F. Staley, L. M. Stuart, Z. M. Szlata, Y. Terrien, A. K. Thompson, J. L. White, M. Woods, J. Xu, C. C. Young, G. Zaplac, and E142 Collaboration

This article is available at ODU Digital Commons: https://digitalcommons.odu.edu/physics_fac_pubs/472 PHYSICAL REVIEW D VOLUME 54, NUMBER 11 1 DECEMBER 1996

Deep inelastic scattering of polarized electrons by polarized 3He and the study of the neutron spin structure

P. L. Anthony, 7,12 R. G. Arnold, 1 H. R. Band, 16 H. Borel, 5 P. E. Bosted, 1 V. Breton, 2 G. D. Cates, 11 T. E. Chupp, 8 F. S. Dietrich, 7 J. Dunne, 1 R. Erbacher, 13 J. Fellbaum, 1 H. Fonvieille, 2 R. Gearhart, 12 R. Holmes, 14 E. W. Hughes, 4,12 J. R. Johnson, 16 D. Kawall, 13 C. Keppel, 1 S. E. Kuhn, 13,10 R. M. Lombard-Nelsen, 5 J. Marroncle, 5 T. Maruyama, 16,12 W. Meyer, 12 Z.-E. Meziani, 13,15 H. Middleton, 11 J. Morgenstern, 5 N. R. Newbury, 11 G. G. Petratos, 6,12 R. Pitthan, 12 R. Prepost, 16 Y. Roblin, 2 S. E. Rock, 1 S. H. Rokni, 12 G. Shapiro, 3 T. Smith, 8 P. A. Souder, 14 M. Spengos, 1 F. Staley, 5 L. M. Stuart, 12 Z. M. Szalata, 1 Y. Terrien, 5 A. K. Thompson, 9 J. L. White, 1,12 M. Woods, 12 J. Xu, 14 C. C. Young, 12 and G. Zapalac 16 ͑E142 Collaboration͒ 1 American University, Washington, D.C. 20016 2 Laboratoire de Physique Corpusculaire, Institut National de Physique Nucle´aire et de Physique des Particules, Centre National de la Recherche Scientifique, Universite´ Blaise Pascal, F-63170 Aubie´re Cedex, France 3 University of California and Lawrence Berkeley Laboratory, Berkeley, California 94720 4 California Institute of Technology, Pasadena, California 91125 5 Centre d’Etudes de Saclay, Departement d’Astrophysique, de Physique des Particules de Physique Nucle´aire et d’Instrumentation Associe´e/Service de Physique Nucle´aire, F-91191 Gif-sur-Yvette, France 6 Kent State University, Kent, Ohio 44242 7 Lawrence Livermore National Laboratory, Livermore, California 94550 8 University of Michigan, Ann Arbor, Michigan 48109 9 National Institute of Standards and Technology, Gaithersburg, Maryland 20899 10 Old Dominion University, Norfolk, Virginia 23529 11 Princeton University, Princeton, New Jersey 08544 12 Stanford Linear Accelerator Center, Stanford, California 94309 13 Stanford University, Stanford, California 94305 14 Syracuse University, Syracuse, New York 13210 15 Temple University, Philadelphia, Pennsylvania 19122 16 University of Wisconsin, Madison, Wisconsin 53706 ͑Received 23 May 1996͒

n n The neutron longitudinal and transverse asymmetries A1 and A2 have been extracted from deep inelastic scattering of polarized electrons by a polarized 3He target at incident energies of 19.42, 22.66, and 25.51 GeV. n 2 The measurement allows for the determination of the neutron spin structure functions g1(x,Q ) and n 2 2 2 g2(x,Q ) over the range 0.03ϽxϽ0.6 at an average Q of 2 ͑GeV/c) . The data are used for the evaluation n 2 of the Ellis-Jaffe and Bjorken sum rules. The neutron spin structure function g1(x,Q ) is small and negative 0.6 n within the range of our measurement, yielding an integral ͐0.03g1(x)dxϭϪ0.028Ϯ0.006 ͑stat͒Ϯ0.006 ͑syst͒. n 1 n Assuming Regge behavior at low x, we extract ⌫1ϭ͐0g1(x)dxϭϪ0.031Ϯ0.006 ͑stat͒Ϯ0.009 ͑syst͒. Com- p n bined with previous integral results from SLAC experiment E143, we find ⌫1Ϫ⌫1ϭ0.160Ϯ0.015 in p n 2 2 agreement with the Bjorken sum rule prediction ⌫1Ϫ⌫1ϭ0.176Ϯ0.008 at a Q value of 3 ͑GeV/c) evaluated using ␣sϭ0.32Ϯ0.05. ͓S0556-2821͑96͒03923-9͔

PACS number͑s͒: 13.60.Hb, 13.88.ϩe, 29.25.Bx

I. INTRODUCTION Pioneering experiments ͓6–8͔ with polarized electrons and , performed at SLAC in the late 1970’s and early During the past twenty five years, experiments measuring 1980’s in a limited x range, revealed large spin dependence spin-averaged deep inelastic scattering of electrons, , in deep inelastic e-p scattering. These large effects were pre- and have provided a wealth of knowledge about dicted by Bjorken ͓9͔ and by simple SU͑6͒ models. the nature of QCD and the structure of the in terms More recently, results from the CERN European Col- of and . Among the highlights are the determi- laboration ͑EMC͒ experiment ͓10͔ over a wider x range have nation of scaling violations ͓1–3͔ from structure functions as sparked considerable interest in the field because the data predicted by QCD, leading to a value of the strong coupling suggest surprisingly that quarks contribute relatively little to constant ␣s ͓4͔, and the test of the Gross-Lewellyn-Smith the spin of the proton and that the strange sea quark polar- sum rule ͓5͔, in which QCD radiative corrections are verified ization is significant. within experimental errors. More recently, polarized deep in- A central motivation for these experiments is a pair of elastic scattering experiments, which probe the spin orienta- sum rules. The first, due to Bjorken ͓9͔, is a QCD prediction tion of the nucleon’s constituents, are providing a new win- that invokes isospin symmetry to relate the spin-dependent dow on QCD and the structure of the nucleon. structure functions to the neutron beta-decay axial coupling

0556-2821/96/54͑11͒/6620͑31͒/$10.00 -54 6620 © 1996 The American Physical Society 54 DEEP INELASTIC SCATTERING OF POLARIZED . . . 6621 constant gA . The experimental test of this sum rule requires QCD arguments ͓13,14͔. Nucleon models have been con- data on both the proton and the neutron. Advances in tech- structed incorporating these general features and yield rea- nology for producing highly polarized beams and targets sonable fits to the data ͓15–17͔ for the proton. make possible increasingly precise measurements. A second The Bjorken sum rule, however, applies to the integrals of sum rule, due to Ellis and Jaffe ͓11͔, which has more theo- g1: retical uncertainty, applies to the proton and neutron sepa- rately. Assuming SU͑3͒ symmetry, data from either the neu- 1 p͑n͒ 2 p͑n͒ 2 tron or proton can be used to determine the contributions of ⌫1 ͑Q ͒ϭ ͵ g1 ͑x,Q ͒dx. ͑4͒ each quark flavor to the spin of the nucleon. It is the apparent 0 disagreement of the EMC proton data with the prediction of (p)n the Ellis-Jaffe sum rule that led to the striking conclusions The goal of the experiments is to measure g1 over as wide (p)n mentioned above. This paper reports on a precision determi- a kinematic range as possible to extract a value for ⌫1 . n The paper is organized as follows. Section II defines the nation of the neutron spin structure function g1 using a po- larized 3He target. theoretical framework used in polarized deep inelastic scat- In spin-dependent deep inelastic scattering ͓12͔ one mea- tering and tests of the Bjorken and Ellis-Jaffe sum rules. sures the quantity Section III discusses the experimental method and data col- lection. Section IV describes the analysis leading to the raw Ϫ asymmetries used to extract the physics asymmetries A and 2 ␴1/2 ␴3/2 1 A1͑x,Q ͒ϭ ͑1͒ A and spin structure functions g and g of 3He. Section V ␴1/2ϩ␴3/2 2 1 2 reports on dilution factor studies and radiative corrections, 3 where ␴(3/2)(/1/2) is the absorption cross section for virtual while section VI reports on the He results and the nuclear photons with total J ϭ 3 ( 1 ) for the final state. corrections used to extract the virtual photon-neutron asym- z 2 2 n n n In the case of a target of Dirac particles, A is unity. In metries A1 and A2 and the spin structure functions g1 and 1 n QCD, the nucleon may be described in terms of a set of g2 . Section VII describes the physics implications of the 2 results and conclusions are presented in Section VIII. quark momentum distributions qi(x,Q ), where x is the frac- tion of nucleon momentum carried by the struck quark and 2 Q is the squared four-momentum transfer to the nucleon. II. THEORETICAL FRAMEWORK The index i includes u, d, and s quarks and antiquarks. Thus A. Bjorken sum rule for the nucleon, A1 measures how the individual quarks, weighted by the square of their charges, are aligned relative The Bjorken sum rule prediction, which relates high en- to the nucleon as a whole. ergy electromagnetic scattering to the low energy beta decay 2 2 The qi(x,Q ) have been evaluated from the measured of the neutron, was derived in the high Q limit by Bjorken 2 values of the spin averaged structure function F1(x,Q ) for ͓9͔ using current algebra, and later shown to be a rigorous various leptons and nucleon targets. However, the determi- QCD prediction ͓3͔ with calculable radiative corrections for 2 2 nation of A1(x,Q ) requires that quark momentum distribu- finite Q ͓18–20͔. This sum rule may be derived in QCD by tions be decomposed in terms of spin. Thus using the operator product expansion ͑OPE͓͒21͔. The OPE 2 2 qi↑(x,Q )„qi↓(x,Q )… give the probability that a quark of type relates integrals of quark momentum distributions 2 i has a fraction x of the nucleon’s momentum with its spin ⌬qi(x,Q ) to matrix elements of single operators such as parallel ͑antiparallel͒ to that of the nucleon. Then qi 1 G s␮ϵ 2 ͗n,s͉¯qi␥␮␥5qi͉n,s͘, ͑5͒ 2 2 2 2 2 An ⌬qi͑x,Q ͒ϵqi↑͑x,Q ͒Ϫqi↓͑x,Q ͒ϩ¯qi↑͑x,Q ͒Ϫ¯qi↓͑x,Q ͒ ͑2͒ qi where ͉n,s͘ represents a neutron (n) with spin s. The GAn and are constants independent of Q2, although they do depend on the choice of renormalization scale ␮. There are different ͚e2⌬q ͑x,Q2͒ g x,Q2 G ’s corresponding to each combination of quarks and bary- 2 i i 1͑ ͒ A A1͑x,Q ͒ϭ 2 2 2 Ӎ 2 , ons in the lowest octet. They are related by isospin and ͚ei ͓qi͑x,Q ͒ϩ¯qi͑x,Q ͔͒ F1͑x,Q ͒ ͑3͒ SU͑3͒ symmetry so that in the limit that SU͑3͒ is exact only three independent quantities remain, where ei is the charge of the ith quark. The latter equation defines the spin-dependent structure Gu ϭ⌬u, 2 Ap function g1(x,Q ). A more precise definition of g1 and the relevant kinematics is given in Sec. IV below. d ϭ In the nonrelativistic quark model, the spins of the quarks GAp ⌬d, relative to the spin of the nucleon are given by the SU͑6͒ p 5 s wave functions, resulting in the predictions A1ϭ 9 and GApϭ⌬s. ͑6͒ n A1ϭ0. This simple picture holds approximately at xϷ0.3. At p low x, A1 decreases due to the dominance of sea quarks These are the matrix elements of the proton. With the latter which one might naively expect to have small polarization. notation, one must be very careful to distinguish ⌬q from ␣ 2 In this region, Regge theory suggests that A1ϳx with ⌬q(x,Q ). 1Ͻ␣Ͻ1.5. At large x, A 1 according to perturbative Useful linear combinations of the matrix elements are: 1→ 6622 P. L. ANTHONY et al. 54

u d where the upper ͑lower͒ numbers are for three ͑four͒ quark a3ϭgAϭGApϪGAp ; flavors, and higher twist terms have been neglected. The u d s number of active quark flavors is determined by the number a8ϭGApϩGApϪ2GAp ; of quarks with mqϽQ, taking mcϭ1.5 GeV and mbϭ4.5 u d s GeV. For our case we use three flavors, since the effects of a ϭ⌬⌺ϭG ϩG ϩG . ͑7͒ 0 Ap Ap Ap charm are expected to turn on slowly. Bj 2 Similar combinations for the quark momentum distributions Measuring ⌫ at different Q provides a sensitive test of are QCD and its radiative corrections. It is one of two QCD sum rules ͑the Gross-Llewellyn-Smith being the other͒ where the 2 2 2 right hand side of the sum rule is accurately known. With ⌬q3͑x,Q ͒ϭ⌬u͑x,Q ͒Ϫ⌬d͑x,Q ͒; sufficiently precise data, one can extract ␣S based on the 2 2 2 2 Bjorken sum rule and compare with the determination of ⌬q8͑x,Q ͒ϭ⌬u͑x,Q ͒ϩ⌬d͑x,Q ͒Ϫ2⌬s͑x,Q ͒; ␣S from other processes. 2 2 2 2 ⌬q0͑x,Q ͒ϭ⌬u͑x,Q ͒ϩ⌬d͑x,Q ͒ϩ⌬s͑x,Q ͒. ͑8͒ B. The nucleon sum rule 2 Here ⌬q3(8)(x,Q ) is the nonsinglet combination and 2 The sum rule for a single nucleon is ͓24͔: ⌬q0(x,Q ) is the singlet combination. An immediate advan- 2 tage of this notation is that a3 and a8 are constants indepen- 1 ␣s 3.58 ␣s 2 2 ⌫ p͑n͒͑Q2͒ϭ ͑Ϯg ϩ 1 a ͒ 1Ϫ Ϫ dent of the renormalization scale ␮ . ⌬⌺(␮ ) on the other 1 12 ͭ A 3 8 ͫ ␲ ͩ 3.25ͪͩ ␲ ͪ hand depends on the scale and special care must be used 3 when interpreting this quantity. Using the above notations, 20.2 ␣s ␣s Ϫ ϩ 4 ⌬⌺͑␮2ϭQ2͒ 1Ϫ the main result of OPE gives ͩ13.8ͪͩ ␲ ͪ •••ͬ 3 ͫ ␲

ϱ 2 n 2 1 ␣ ͑Q ͒ 1.10 ␣s 2 s Ϫ Ϫ , ͑12͒ ⌬q3͑8͒͑x,Q ͒dxϭa3͑8͒ 1ϩ͚ Cn 0.07 ••• ͵0 ͫ nϭ1 ͩ ␲ ͪ ͬ ͩ ͪͩ ␲ ͪ ͬͮ ϱ where the upper ͑lower͒ coefficients are for three ͑four͒ fla- Km ϩ 2m. ͑9͒ vors. To leading order, this expression requires a new m͚ϭ1 Q nucleon matrix element, (a8ϩ4⌬⌺), which can only be es- timated from nucleon models. As pointed out by Gourdin The perturbative QCD series in ␣ (Q2) describes high s ͓25͔, a ϭ3FϪD as determined from baryon beta decay if energy or short distance effects and has been recently evalu- 8 flavor SU͑3͒ symmetry is assumed. Then ⌬⌺ may be deter- ated exactly up to third order in QCD. The power series in mined from the sum rule. ⌬⌺ is an important input for 1/Q2,inEq.͑9͒contains the ‘‘higher twist’’ terms. These nucleon models. Much of the excitement in the field arises terms describe long-distance, nonperturbative behavior that from the unexpected EMC ͓10͔ result that ⌬⌺Ϸ0. In the involves, among other effects, the details of the wave func- nonrelativistic quark model, ⌬⌺ϭ1. However the motion of tions of the quarks in the nucleon. The calculation of some of the quarks should give a suppression similar to the suppres- these terms has been the subject of recent literature ͓22͔.Itis sion of g from 5/3 to the experimental value of 1.2, yielding expected that the contributions of these terms are small for A ⌬⌺Ϸ0.7. In addition, gluons and orbital angular momentum the Q2 range discussed in this paper. may make substantial contributions to the spin of the proton. The spin dependent structure function g (x,Q2) measures 1 The present world average of ⌬⌺ϳ0.3 was not anticipated the difference in number of partons with helicity parallel by most authors prior to the measurements. versus antiparallel to the helicity of the nucleon weighted by In addition, ⌬⌺ is also needed for predicting elastic scat- the square of the parton charge. Explicitly, we have tering cross sections. Two examples of physical processes involving ⌬⌺ are scattering and the scattering of p͑n͒ 2 1 2 2 possible supersymmetric particles. g1 ͑x,Q ͒ϭ 2 ͚ ei ⌬qi͑x,Q ͒ Equation 12 involves singlet operators, which in leading 1 4 1 2 1 4 2 order of QCD includes the Adler-Bell-Jackiw ABJ ϭ ͓ ͑ ͒⌬u͑x,Q ͒ϩ ͑ ͒⌬d͑x,Q ͒ ͑ ͒ 2 9 9 9 9 anomaly ͓26,27͔ because the relevant anomalous dimension 1 2 2 ϩ 9 ⌬s͑x,Q ͔͒. ͑10͒ is nonzero. One result is the scale dependence of ⌬⌺(␮ ). Any physical process dependent on ⌬⌺ must also involve The Bjorken sum rule follows from Eqs. ͑9͒ and ͑10͒ other ␮2-dependent factors such that the result is indepen- ͓23͔: dent of ␮2. For example, part of the neutrino-proton elastic scattering cross section arises from the current ͓28͔ Bj 2 p 2 n 2 ⌫ ͑Q ͒ϵ⌫1͑Q ͒Ϫ⌫1͑Q ͒ J0ϭz0͑␮2͒⌬⌺͑␮2͒s . ͑13͒ 2 2 2 ␯ ␯ 1 ␣s͑Q ͒ 3.58 ␣s͑Q ͒ ϭ 6 gA 1Ϫ Ϫ 0 1 2 ͫ ␲ ͩ3.25ͪͩ ␲ ͪ Here z ϭ 2͓1ϩ⌬(␮ )͔, the weak charge of the proton. This quantity is just 1/2 in the SU͑2͒ϫU͑1͒ electroweak theory 2 3 0 20.2 ␣s͑Q ͒ and is scale dependent. The scale dependencies of z and Ϫ ϩ ͑11͒ ͩ13.8ͪͩ ␲ ͪ •••ͬ ⌬⌺ cancel so that J0 is scale independent and thus a mea- ␯ 54 DEEP INELASTIC SCATTERING OF POLARIZED . . . 6623

2 2 surable physical quantity. Moreover, choosing ␮ ϭM Z from a8 so that the error ␦a8 becomes important. The deter- 2 2 s helps minimize ⌬(␮ ). Strikingly, it is ⌬⌺(␮ ) with mination of GAϭ⌬s from the nucleon sum rule also suffers ␮2Ͼ2 ͑GeV/c) 2 that is needed to compute neutrino scatter- from this problem. 2 ing at Q Ϸ0. Given the importance of a8, it is worth going into some For our results, we have chosen to use the scale ␮2ϭQ2 detail about its origin. By assuming flavor SU͑3͒ for the to avoid passing over quark thresholds. If experiments per- nucleon octet, we have formed at different Q2 are compared, the scale of one or both ¯ Ϫ ¯ of the experiments should be changed according to the for- ͗n,s͉␺␥␮␥5Vϩ␺͉⌬⌺ ,s͘Ϸ͗n,s͉␺␥␮␥5V3␺͉n,s͘, mula ͑18͒

2 Ϫ 2 or in terms of quark operators 2 2 0.667 ␣s͑␮1͒ ␣s͑␮2͒ ⌬⌺͑␮1͒ϭ⌬⌺͑␮2͒ 1ϩ ͫ ͩ0.960ͪͩ ␲ ͪ g ͑⌺Ϫ n͒2s ϵ͗n,s͉¯u␥ ␥ s͉⌺Ϫ,s͘ A → ␮ ␮ 5 2 2 2 2 1.21 ␣s͑␮1͒Ϫ␣s͑␮2͒ Ϸ͗n,s͉u␥ ␥ uϪ¯s␥ ␥ s͉n,s͘ ϩ . ͑14͒ ␮ 5 ␮ 5 ͩ1.97ͪͩ ␲2 ͪͬ u s ϵGAnϪGAnϷ͗p,s͉d␥␮␥5dϪ¯s␥␮␥5s͉p,s͘ 2 2 Ideally, one would choose a universal scale: ␮ ϭM P or d s 2 2 ϭ͑GApϪGAp͒2s␮ . ͑19͒ ␮ ϭM Z , however, the first option suffers from the fact that ␣s(M P) is not well known and the second option requires Thus the axial matrix elements for ⌺ decay can be related to running the scale across several mass thresholds which yields proton matrix elements. Similar results hold for the other even more complex expressions. Another option commonly hyperon decays. To average over many hyperon decay mea- used is to define a quantity ⌬⌺inv and use the formula ͓24͔ surements, the following relation is used:

1 2 1 u s p 1 ␣s 3.58 ␣s Fϭ ͑ G Ϫ G ͒ and ⌫ ͑n͒͑Q2͒ϭ ͑Ϯg ϩ a ͒ 1Ϫ Ϫ 2 ͗ Ap͘ ͗ Ap͘ 1 12 ͭ A 3 8 ͫ ␲ ͩ 3.25ͪͩ ␲ ͪ 1 u d s 3 Dϭ 2 ͑͗GAp͘Ϫ2͗GAp͘ϩ͗GAp͒͘. ͑20͒ 20.2 ␣s 0.33 ␣s Ϫ ϩ 4 ⌬⌺ 1Ϫ ͩ13.8ͪͩ ␲ ͪ •••ͬ 3 invͫ ͩ 0.04ͪ ␲ Jaffe and Manohar ͓30͔ assign a generous error 3FϪDϭ0.60Ϯ0.12 while Ratcliffe ͓31͔ quotes a range of 2 0.55 ␣s values form 0.53 to 0.83 based on various assumptions about Ϫ Ϫ••• . ͑15͒ ͩ Ϫ0.54ͪͩ ␲ ͪ ͬͮ SU͑3͒ f breaking and which decays to use. For the purpose of this paper, we will use a8ϭ3FϪDϭ0.58Ϯ0.12, which is We will also quote ⌬⌺inv for our data. the updated central value of Close and Roberts ͓32͔ but has The individual quark contributions can be extracted using the generous error of Jaffe and Manohar ͓30͔. The net result 2 either Eq. ͑12͒ or Eq. ͑15͒ along with a8ϭ3FϪD: is that the prediction of the Ellis-Jaffe sum rule for Q ϭ2 ͑GeV/c)2 is ⌫nϭϪ0.011Ϯ0.016. The 0.12 uncertainty on ⌬uϭ 2⌬⌺ϩa ϩ3g /6; 1 ͓ 8 A͔ 3FϪD translates to a 0.06 uncertainty on ⌬s, which is not negligible compared to typical world averages of ⌬dϭ͓2⌬⌺ϩa Ϫ3g ͔/6; 8 A ⌬sϳϪ0.1. An alternative definition, equivalent in the limit of exact ⌬sϭ͓⌬⌺Ϫa ͔/3. ͑16͒ 8 SU͑3͒, that is often used in the literature is FϩDϵgA(n p)ϭ1.2573Ϯ0.0028. Then hyperon data are C. The Ellis-Jaffe sum rule used to obtain→F/D. The world average, based on the analy- Prior to the establishment of QCD, Ellis and Jaffe ͓11͔ sis of Close and Roberts ͓32͔,isF/Dϭ0.575Ϯ0.016. This n made a numerical prediction for the nucleon sum rule by result yields ⌫1ϭϪ0.011Ϯ0.005, with a substantially s arguing that ⌬sϭGAϭ0. This gives the additional relation smaller uncertainty, while the uncertainty on ⌬s changes from 0.06 to 0.04. a8ϭ⌬⌺ ͑17͒ The above two alternatives illustrate how the prediction of the Ellis-Jaffe sum rule is sensitive to the various assump- which, when combined with the value for a8 extracted from tions chosen. hyperon decay, provides values for all of the needed matrix elements. As pointed out by Jaffe 29 , this relation is rather ͓ ͔ III. THE E142 EXPERIMENT curious in the context of QCD because a8 is scale indepen- dent and ⌬⌺(␮2) is scale dependent. Hence the prediction of The experiment discussed in this paper was performed to the Ellis-Jaffe sum rule depends upon the scale chosen. Ex- measure for the first time the virtual photon-nucleon spin 2 2 n n periment E142 has used ␮ ϭ2 ͑GeV/c) . A more common asymmetries, A1 and A2 , in deep inelastic scattering of po- 3 choice is to set a8ϭ⌬⌺inv . The difference is about 0.005 in larized electrons by polarized He. From these asymmetries, n n n ⌫1 and should be accounted for when comparing different the neutron spin structure functions g1 and g2 are extracted. experiments. The experiment relied on the production and delivery of a A second issue with the Ellis-Jaffe sum rule is that Eq. high energy polarized beam at the Stanford Linear p(n) ͑17͒ implies a large contribution to ⌫1 in Eq. ͑12͒ coming Accelerator Center ͑SLAC͒. The polarized incident electrons 6624 P. L. ANTHONY et al. 54

TABLE I. Table of parameters for the E142 experiment.

Description

Average beam polarization using a AlGaAs source 36% Average current 1.5 ␮A High density polarized 3He target Pressure ϭ 8.6 atm, Volumeϭ 90 cm 3, Polarization ϭ 33% Thin windows to minimize the dilution factor 110 ␮m/window High counting rate due to low spectrometer angle 4.5° and 7° Large statistics with a polarized beam on polarized target 300ϫ106 events Target windows cooling in vacuum No target explosions due to glass radiation damage

were delivered to End Station A where they scattered off a 24.5° geϪ2 E 3 ⌬␪ϭ␲ ͑21͒ polarized He target and were detected in magnetic spec- 180° 2 m trometers. The experiment, named SLAC E142, collected data over a period of six weeks in November and December where E is the beam energy, m is the electron mass, g is the of 1992. An overview of the primary technical achievements e electron gyromagnetic ratio, and ⌬␪ is the angle between the of the experiment are presented in Table I. Details of the electron spin and momentum at the target. When ␪ is an polarized electron beam polarimetry, polarized target and integer multiple of ␲, the electron spin is longitudinal at the magnetic spectrometers are discussed in subsequent sections. n target. The energies 19.42 and 22.66 GeV satisfy this condi- Previous results on the spin structure function g1 from tion exactly, while the energy 25.51 corresponds to 93% of this experiment have been published ͓33͔. This paper reports the maximum available polarization. This latter energy was on a more thorough analysis of the results leading to a the maximum energy that the beam line magnets could sup- n change in the previously published results for g1 . Among port at the time. the new information presented in the present paper are re- The beam spot size was typically 2 to 4 mm at the target. n sults on the neutron transverse spin structure function g2 , Studies of the spot position and radius near the target as and the results on the Q2 dependence of the neutron longi- measured by a wire array found no dependence on beam tudinal spin structure function gn . helicity at the level of better than Ϯ 0.01 mm. From models 1 of the variation of the target window thicknesses, it was de- termined that this implied that false asymmetries due to a A. The polarized electron beam possible helicity-dependent motion of the electron beam po- sition would be significantly less than 10 Ϫ4. The SLAC polarized electron source ͓34͔, using an The electron beam polarization was determined using AlGaAs photocathode at a temperature of 0 °C, produced the single-arm Mo”ller polarimetry. The high peak beam currents polarized electron beam for this experiment ͓35,36͔. Polar- precluded the detection of double arm coincidences. The ized electrons were produced by illuminating the photocath- cross section for spin dependent elastic electron-electron ode with circularly polarized light at a wavelength near the scattering is given by ͓37͔: band-gap edge of the photocathode material. AlGaAs with 13% Al, rather than GaAs, was chosen as the photocathode since the larger band gap of the AlGaAs cathode was a better i j d␴/d⍀ϭ͑d␴0 /d⍀͒ 1ϩ͚ PBAijPT ͑22͒ match to the available flashlamp pumped dye laser operating ͩ i, j ͪ at a wavelength of 715 nm. The electron helicity was i changed randomly pulse by pulse by controlling the circular where PB are the components of the beam polarization and polarization of the excitation light. Using this cathode, the j PT are the components of the target polarization. The z axis polarized source produced an electron beam polarization of is along the beam direction and the y axis is chosen normal about 36%. to the scattering plane. The cross section is given by the Electrons from the source were accelerated to energies unpolarized cross section d␴0 /d⍀, and the asymmetry terms ranging from 19 to 26 GeV and directed onto the polarized Aij .IfPT is independently known, the above expression 3He target. The SLAC accelerator operated with pulses of may be used to determine the beam polarization PB . approximately 1 ␮sec duration at a rate of 120 Hz. The beam To lowest order, the fully relativistic unpolarized labora- 11 current was quite high, operating at typically 2ϫ10 elec- tory cross section is given by: trons per pulse. The spectrometers collected typically 2 events per pulse from the polarized 3He target, yielding ap- ␣͑1ϩcos␪ ͒͑3ϩcos2␪ ͒ 2 ϫ 6 c.m. c.m. proximately 300 10 events for the experiment. ͑d␴0 /d⍀͒Lϭ 2 . ͑23͒ The experiment collected data at three discrete energies of ͫ 2msin ␪c.m. ͬ 19.42, 22.66, and 25.51 GeV with an energy acceptance of typically 0.5%. Since the primary beam undergoes a 24.5° For the measurement of longitudinal polarization with a bend before reaching the experimental target, the electron longitudinally polarized target foil, the only relevant asym- spin precesses more than the momentum by an amount metry term is Azz given by 54 DEEP INELASTIC SCATTERING OF POLARIZED . . . 6625

0.10 20 Detector

0.08 10 -o.os e J\a? -3- 0 >< 20 z (m) ------a.. 0.04 Beampipe -10 0.02

(a) 0 -20 0.90 0.95 1.00 1.05 1.10 (8~:b) 20 e 0 -3- FIG. 1. The calculated probability distribution P(␪lab /␪0) for >, laboratory scattering angles due to atomic electron motion. The -20 horizontal axis is the laboratory scattering angle (␪lab) in units of the central scattering angle (␪ ). The dashed curve is for M shell 0 -40 polarized target electrons. The solid curve is an average for all target electrons. -60 2 2 ͑7ϩcos ␪c.m.͒sin ␪c.m. AzzϭϪ 2 2 . ͑24͒ ͑3ϩcos ␪c.m.͒ FIG. 2. Top ͑a͒ and side ͑b͒ views of the E-142 Mo”ller polar- imeter. The mask selects Mo”ller scattered electrons near the hori- zontal plane which are then dispersed vertically by the magnet. The Here ␪c.m. is the center-of-mass scattering angle, m is the electron mass, and ␣ is the fine structure constant. The asym- detector uses gas proportional tubes embedded in lead to sample the metry maximum is at ␪ ϭ90° where the unpolarized labo- Mo”ller signal over a specific momentum range and to measure the c.m. scattering angle. ratory cross section is 0.179 b/sr and AzzϭϪ7/9. Most if not all Mo”ller polarimeters utilize thin ferromag- netic foils as the polarized electron target. The distinction tered electrons, a segmented detector array to detect the scat- between the free target electrons of the previous formulas tered electrons, and a data acquisition system. and the bound atomic electrons of the physical target was The magnetized target foils were made of Vacoflux ͓41͔, ignored until recently when Levchuk ͓38͔ pointed out that an alloy of 49% Co, 49% Fe, and 2% Va by weight. The the analyzing power of Mo”ller polarimeters may have sig- foils were 3 cm wide by 35 cm long and were mounted at a nificant corrections due to the electron orbital motion of the 20.7° angle with respect to the beam. Three foils of approxi- target foil electrons. Atomic electrons have momentum dis- mate thickness 20, 30, and 50 ␮m were installed. Nearly all tributions which are different for different atomic shells. the Mo”ller data were taken with the two thinner foils. The Electrons in the outer shells have small momenta but those foils were magnetized by Helmholtz coils providing a 100 G from the inner shells have momenta up to 100 keV. Although field along the beam direction. The polarity of the coils was small compared to a beam energy of 22.66 GeV, these mo- typically reversed between Mo”ller data runs to alternate the menta are not small compared to the electron rest mass and sign of the foil polarization and to minimize systematic er- can alter the center of mass energy and thus the scattering rors. angle in the laboratory frame by up to 10%. The relative The polarization PT of the target electrons was deter- angular smearing correction for polarized and unpolarized mined from the relation: electrons is shown in Fig. 1. The effect causes different line shapes for scatters from different shells. Since the polarized target electrons are only in the 3d (M) shell, the fraction of M gЈϪ1 ge P ϭ , ͑25͒ signal from the polarized target electrons and thus the ex- T n ␮ ͩ gЈ ͪͩg Ϫ1ͪ e B e pected Mo”ller asymmetry varies over the Mo”ller scattering elastic peak. Inclusion of this effect has been shown to modify the analyzing power of Mo”ller polarimeters by up to where M is the bulk magnetization in the foil, ne is the 15% ͓38–40͔ depending on the exact geometry of the polar- electron density, geϭ2.002 319 is the free electron gyromag- Ϫ21 3 imeter. Inclusion of this effect modifies the analyzing power netic ratio, and ␮Bϭ9.273ϫ10 Gcm is the Bohr mag- of the E142 Mo”ller polarimeter by 5%. neton. The factor involving the magnetomechanical ratio The E-142 Mo”ller polarimeter shown in Fig. 2 consisted (gЈ) includes the correction for the orbital contribution to the of a scattering target chamber containing several magnetized magnetization. Interpolating between the measured gЈ values foils, a collimator to define the scattering angle and angular of Fe and Co, the gЈ of Vacoflux was calculated to be acceptance, a magnet to measure the momentum of the scat- 1.889Ϯ0.005. Substituting into the above equation yields: 6626 P. L. ANTHONY et al. 54

M PTϭ ϫ͑0.940 11Ϯ0.002 80͒. ͑26͒ ne␮B 1500 R+L (a) The magnetization M was determined by direct flux mea- 2l surements using a precise integrating voltmeter ͑IVM͒ con- C ::J 0 1000 nected to a pickup coil placed around the foils. From Fara- (.) day’s law, as the external H field is swept between ϪH and (]) > ϩH, the integral of the induced voltage over time can be ~ ai 500 related to the B and H fields and hence the magnetization a: M through 0 10 20 30 ͐inVdtϪ͐outVdt 4␲MϭBϪHϭ , ͑27͒ 40 2ϫNTϫAfoil R-L (b) where in and out refer to flux measurements with the foil in 2l 30 X C and out, and Afoil is the cross-section area of the foil. Rec- ::J 0 ognizing that the foil density can be determined from the (.) X (]) 20 measured mass of the foil, length, and area Afoil , the foil .e: X polarization PT can be determined from Eq. ͑26͒ and Eq. 1ii ai X ͑27͒. a: 10 X A 21 radiation length thick tungsten mask located 7.11 X X X meter from the Moller target restricted the scattering angles -"")()( )()()()( ” 0 ~- of the particles entering the polarimeter detector to 10 20 30 5.0р␪р10.5 mrad. The azimuthal acceptance of the rectan- Channel gular mask opening depended on the scattering angle and varied from Ϯ0.14 to Ϯ0.068 rad with respect to the hori- FIG. 3. RϩL ͑a͒ and RϪL ͑b͒ scattering distributions for an zontal plane. The scattered particles next passed through a average of 12 Mo”ller runs at Eϭ22.66 GeV/c with a 30 ␮m thick 0.25 mm thick Mylar vacuum window and entered a large target. This solid line in ͑a͒ is the fitted RϩL line shape and the aperture spectrometer magnet. The 1.83 meter long magnet dotted line is the fitted background. was typically run at a ͐Bdlϭ14.5 kG m for the 22.66 GeV data. The spectrometer setting selects Moller scattered elec- ” detector length of 4 cm corresponded to a momentum accep- trons at 10 GeV/c corresponding to a center of mass scatter- tance of 2.9%. The entire detector package was mounted on ing angle of 97°. The field integral was adjusted during the a vertical mover allowing different momenta to be selected. experiment to position the Mo”ller peak in the detector, to compensate for different beam energies and to maximize sig- The signal in each tube was integrated over the 1 ␮sec nal and reduce background. long beam pulse by a charge integrating preamplifier. The The main beam passed through a 33 mm round hole in the data, together with the sign of the beam polarization, were mask and continued down the E-142 beam line. The beam recorded by a peak sensing analog to digital converter exiting the central hole in the mask contained large numbers ͑ADC͒ system. The beam polarization was randomly re- of low energy bremsstrahlung electrons produced by the tar- versed between pulses to reduce systematic errors. The num- get foil. Large magnetic fields would bend these particles out ber of Mo”ller electrons detected per pulse varied with current of the beamline generating unacceptable backgrounds in the and target thickness, but was typically 70 per pulse. A typi- 6 detector. To reduce the field along the beamline a 7.6 cm by cal Mo”ller run lasted 150 seconds and contained 10 Mo”ller 30.5 cm soft iron septum witha5cmby5cmhole for the electron scatters. beam was inserted in the magnet gap. The septum reduced the B field seen by the beam by about a factor of 100 reduc- ing ͐Bdl to approximately 150 G m. B. Beam polarization analysis After exiting the magnet the Moller scattered electrons ” For each Moller run an average pulse height for each traveled through a He bag to the detector located 22.4 meters ” detector channel was calculated for both right (R) and left from the target. The detector consisted of 37 gas proportional tubes embedded in lead. Each 4 mm diameter brass tube ͑L͒ handed incident beam. The pulse to pulse variance of the contained a 40 micron wire strung through the center. The ADC values was used to estimate the error in the average tubes were placed in two parallel rows 7.9 mm apart. The pulse height. These averages and errors were recorded with first row was behind 36.8 mm of lead. The second row was relevant beam currents, detector and target positions, and 6.9 mm behind the front row and offset by 3.9 mm giving an magnet settings. Typical measured distributions for RϩL effective segmentation of 3.9 mm in the horizontal ͑scatter- and RϪL are shown in Fig. 3. The RϩL distribution ͓Fig. ing͒ plane. The lead absorbed soft photon backgrounds and 3͑a͔͒ shows an elastic scattering peak with a radiative tail on amplified the Mo”ller signal. Since the momenta and scatter- top of an unpolarized background. The signal to background ing angle of the Mo”ller scatters are correlated, the scatters ratio varied with shielding conditions and beam parameters fall in a tilted stripe at the detector. The detector was oriented from Ϸ2 at the beginning of the experiment to Ϸ7 after so that the tubes were parallel to the Mo”ller stripe. The active shielding improvements made during the experiment. The 54 DEEP INELASTIC SCATTERING OF POLARIZED . . . 6627

TABLE II. Systematic uncertainties contributing to the beam 60 polarization measurement.

40 Value ͑%͒

5 20 Foil magnetization Ϯ1.7 Kinematic acceptance Ϯ0.1 ias 0 Model dependence Ϯ1.0 ~ Gain and nonlinearity correction Ϯ2.2 E -20 22.66GeV gJ Fit range Ϯ1.0 IXl -40 ,, \11fttt,_ Total Ϯ3.1

-60'--'------'----'------'--- 300 400 500 600 700 Moller Run from the 30 ␮m foil data was 0.354Ϯ0.001 where the errors are statistical only. The foil averages differ by 1.5%, within FIG. 4. The measured longitudinal beam polarization P vs the 1.7% systematic error on the foil polarization. The beam B polarization did not exhibit any time dependence over the Mo”ller polarimeter run number for runs passing the cuts described in the text. The sign is relative to the polarization direction at the duration of the experiment. source. In addition to the systematic error in the foil polarization there is a contribution to the overall systematic error from the uncertainty in the modeling of the scattering kinematics, RϪL distribution ͑Fig. 3b͒ is to a good approximation pure line shapes, asymmetries, detector linearity, and preamp- Mo”ller electron scattering and shows only a radiative tail ADC linearity. The various contributions to the systematic with no background. error are summarized in Table II. Adding the systematic The beam polarization was determined by fitting the ob- uncertainties in quadrature yields an overall systematic served elastic scattering and asymmetry distributions to line error of 3.1% relative. The resulting longitudinal beam shapes based on Mo”ller scattering plus a background com- polarization averaged over the target foils is then ponent. Although several techniques were used as cross PBϭ(0.357Ϯ0.001Ϯ0.011)cos͓␲E (GeV)/3.237͔. checks, the full data sets were analyzed with a technique which derived the Moller component of the line shapes from ” C. The polarized 3He target the measured RϪL distributions and used this shape together with a quadratic background component to fit the observed The experiment used a polarized 3He target to extract the scattering distributions. In this technique, all of the observed neutron spin structure function. The polarized 3He target re- lies on the technique of spin-exchange optical pumping 42– RϪL line shape is attributed to Mo”ller scattered electrons ͓ and the background is assumed to be unpolarized. 44͔. Spin-exchange optical pumping refers to a two step pro- The analysis technique used the observed RϪL line shape cess in which, ͑1͒ rubidium ͑Rb͒ atoms are polarized by and the angular smearing functions shown in Fig. 1 to gen- optical pumping, and ͑2͒ the electronic polarization of the Rb atoms is transferred to the nuclei of the 3He atoms by spin- erate a predicted RϩL line shape for Mo”ller scatters. For zero target momenta the RϪL line shape and the RϩL line exchange collisions. The optical pumping is accomplished 2 shapes are identical except for backgrounds. The trial RϩL by driving transitions from the Rb 5 S1/2 ground state to the 2 line shape was generated from the observed RϪL line shape 5 P1/2 first excited state using circularly polarized light from by first correcting for the angular smearing due to the polar- lasers. The wavelength of this transition, often referred to as ized target electrons and then convoluting the result with the the Rb D1 line, is 795 nm. Within a timescale of millisec- smearing correction for all ͑polarized and unpolarized͒ target onds, one of the two substates of the ground state is selec- electrons. Additional corrections were made for the variation tively depopulated, resulting in very high atomic polarization of the cross section and change of azimuthal acceptance with ͓45͔. The spin-exchange takes place when the polarized Rb scattering angle and for the variation in the value of the atoms undergo binary collisions with the 3He atoms. The 3He electrons, being paired in the 1S ground state, do not Mo”ller scattering asymmetry over the angular acceptance of 0 the detector. The observed RϩL distribution was then fit by participate in the collision from a spin point of view. The 1 3 the predicted line shape plus a quadratic background. The spin- 2 He nucleus, however, interacts with the Rb valence solid line is Fig. 3͑a͒ shows the resultant fitted line shape for electron through hyperfine interactions, which can result in a the typical runs. mutual spin flip. As long as the Rb vapor is continually being The measured longitudinal beam polarization PB is shown polarized, this results in a gradual transfer of angular mo- mentum to the 3He nuclei. in Fig. 4 for Mo”ller polarimeter data runs covering the last 5 weeks of the experiment. Only runs with the Moller peak The time evolution of the 3He polarization, assuming the ” 3 well centered and with statistical errors less than 5% are He polarization PHeϭ0attϭ0, is given by displayed. The lower polarization of the 25.5 GeV data is ␥ evident showing the effect of the nonoptimal beam energy. SE Ϫ͑␥ ϩ⌫ ͒t P ͑t͒ϭ͗P ͘ ͑1Ϫe SE R ͒ ͑28͒ Correcting for the beam energy, an average beam polariza- He Rb ͩ ␥ ϩ⌫ ͪ SE R tion was calculated for each of the target foils averaging over 3 the different beam energies. The average beam polarization where ␥SE is the spin-exchange rate per He atom between 3 3 determined from the 20 ␮m foil data was 0.360Ϯ0.002 and the Rb and He, ⌫R is the relaxation rate of the He nuclear 6628 P. L. ANTHONY et al. 54

50 / ~ TI q Ti: Sapphire II, 40 ...... • 4 l ...... : ~ Main Coils C: : Q~~!llil~!llil~~;;;:, : 0 30 ~ N -~ 20 0 a. e­ I"~ I 10 !--Ce~"-! --➔ Beam : : : Pickup Coils : : : : : 100 150 200 : : Time (hours) j RF Drive Coils j : Q~~~~~~~;;;:, : : : •...... -• FIG. 5. Typical polarization spin-up curve for the longitudinal Vacuum Chamber 3He polarization in one of the target cells used in the experiment. FIG. 6. Schematic of the E142 3He target system. polarization through all channels other than spin exchange with Rb, and ͗PRb͘ is the average polarization of a Rb atom. that accounts for all relaxation mechanisms that are associ- ͓46͔ The spin-exchange rate ␥SE is defined by ated with a specific cell. For the three cells actually used in Ϫ1 our experiment, ⌫cell varied between 53 and 65 hours. ␥ ϵ ␴ ␷ ͓Rb͔ ͑29͒ SE ͗ SE ͘ A In addition to ⌫cell there are interactions not inherent to the target cell which further increase the nuclear relaxation where ϭ1.2ϫ10Ϫ19 cm 3/sec is the velocity-averaged ͗␴SE␷͘ rate. Inhomogeneities in the magnetic field that provides an spin-exchange cross section for Rb – 3He collisions 46,47 ͓ ͔ alignment axis for the 3He nuclear polarization induce relax- and Rb is the average Rb number density seen by a 3He ͓ ͔ A ation according to atom. We operated at Rb number densities such that the spin- Ϫ1 exchange time constant ␥SE was typically 10 to 30 hrs and ٌ͉B ͉2ϩٌ͉B ͉2 3 x y the time constant for build-up of He nuclear polarization, ⌫ٌBϭD 2 , ͑32͒ Ϫ1 ͩ B ͪ 0 (␥SEϩ⌫R) ranged from about 9 to 20 hours. A typical spin-up polarization curve is shown in Fig. 5. where D is the diffusion constant of the 3He in the target, In order to achieve the highest 3He polarization, we at- B0 is the magnitude of the alignment field, assumed to lie tempted to maximize ␥SE and minimize ⌫R . From Eq. ͑29͒, along the z axis, and Bx and By are the components of the maximizing ␥SE implies increasing the alkali-metal number magnetic field transverse to Bz ͓50͔. This effect was very density, which in turn requires more laser power ͓46,48͔. For Ϫ1 .a fixed volume of polarized Rb, the number of photons small and we calculated ⌫ٌB ϳ 400 hours in our target needed per second must compensate for the number of Rb During the experiment, nuclear relaxation was also in- spins destroyed per second. In total, we used five lasers, duced by the presence of ionizing radiation from the electron which collectively provided about 16–22 W and achieved a beam, a phenomenon which is well understood both theoreti- cally ͓51͔ and experimentally ͓52͔. When a 3He atom is value of ␥SEϷ1/12 hours. There are several processes which contribute to the 3He ionized, the hyperfine interaction couples the nuclear spin to the unpaired electron spin which can in general be depolar- relaxation rate ⌫R . An important example is relaxation that 3 occurs during 3He– 3He collisions due to the dipole interac- izing. Furthermore, electrons from other He atoms can be transferred to the original ion, creating the potential for de- tion between the two 3He nuclei ͓49͔. Dipole induced relax- polarizing other atoms. The depolarization rate ⌫beam there- ation provides a lower bound to ⌫R , and has been calculated 3 to be fore depends on the ionization rate of the He and the aver- age number of 3He nuclear depolarizations per 3He ion Ϫ1 1 created. The relaxation time ⌫beam for our experiment in- ⌫ ϭ ͓ 3He͔, ͑30͒ 3 dipolar 744 hours ferred from the ϳ 10% relative drop in He polarization at our maximum beam current of 3.3 ␮A is 100–200 hours, at 23 °C where ͓ 3He͔ is the number density of 3He in ama- consistent with the predicted time constant of 170 hours. gats ͑an amagat is a unit of density corresponding to 1 atm at When all the relaxation mechanisms are included, the to- 0°C͓͒49͔. The relaxation rate varies with temperature, im- tal 3He nuclear relaxation rate is given by plying a maximum relaxation time constant of ϳ100 hours 3 for the He densities and temperatures found in our target. ⌫Rϭ⌫cellϩ⌫beamϩ⌫ٌB . ͑33͒ Another important contribution to ⌫ is relaxation that oc- R Ϫ1 curs during wall collisions, a relaxation rate we will desig- From the previous discussion we see that ⌫R was in the Ϫ1 nate ⌫wall . Both ⌫dipolar and ⌫wall are intrinsic to a given range of about 40 hours. With ␥SE Ϸ25 hours, Eq. ͑28͒ pre- target cell, making it useful to define the quantity dicts that the maximum polarization, given by ␥SE /(␥SEϩ⌫R), is about 0.62. ⌫cellϭ⌫dipolarϩ⌫wall ͑31͒ A schematic of the target system is shown in Fig. 6. The -54 DEEP INELASTIC SCATTERING OF POLARIZED . . . 6629 central feature of the polarized 3He target is the glass cell The glass target cell, the oven, the rf coils, the pickup containing ϳ8.4 atm of 3He ͑as measured at 20 °C͒, several coils, and other assorted target components were located in- milligrams of Rb metal, and ϳ65 Torr of N 2. The N 2 aids in side a vacuum chamber in order to reduce the background the optical pumping by causing radiationless quenching of event rates from nontarget materials. Small cooling jets of the Rb atoms when they are in the excited state ͓45͔. The 4He were directed at the thin entrance and exit windows of target cells were based on a double chamber design, ͓53͔ the target chamber as a precaution against the thin glass comprising an upper ‘‘pumping chamber’’ in which the op- breaking due to excessive heating from the electron beam. tical pumping and spin exchange took place, and a lower The production of target cells with long intrinsic relax- ‘‘target chamber’’ through which the electron beam passed. Ϫ1 ation times ⌫cell proved to be a challenging task. The cells The Rb was contained almost entirely in the upper pumping were made out of aluminosilicate glass ͑Corning 1720͒ since chamber, which was the only chamber that was heated ͑to such glass is known to have favorable spin-relaxation prop- achieve the desired Rb number density͒ when the target was erties ͓56,57͔. The use of aluminosilicate glass, however, in operation. The upper and lower chambers had roughly the Ϫ1 was not sufficient for obtaining long ⌫ ’s. We found it same volume ͑70 cm 3 and 90 cm 3, respectively͒ for a total cell necessary, for instance, to ‘‘resize’’ tubing before incorpo- volume of about 160 cm 3, and were connected by a 60 mm rating the tubing into the final cell construction. In this pro- long, 9 mm inside diameter ‘‘transfer tube.’’ The target cess, tubing of some initial diameter is brought to a molten chamber had a length of about 30 cm, a diameter that was roughly 2 cm, and thinned rounded convex end windows. state and blown to a new diameter while being turned on a The average window thickness for the cells used in the ex- glass lathe. It is our belief that this process results in a more periment was 112 ␮m per window. pristine surface, presumably with fewer contaminants and The pumping chamber was enclosed by an oven, with defects. 3 heating supplied by flowing hot air, the temperature of which For filling with He gas, the cells were attached to a high Ϫ7 Ϫ8 determined the Rb number density. The oven, and all other vacuum system (ϳ 10 to 10 Torr͒ and given long items which were near the target cell, were made of nonmag- bake-outs under vacuum for 3 to 6 days at 475 °C. The Rb netic materials so as not to interfere with the NMR polarim- was distilled into the cell with a hand-held torch from a side etry. The oven was made of a high temperature plastic called arm of the vacuum system. During the distillation, the cells Nylatron GS, and was operated at temperatures of about 160 remained open to the vacuum pumps so that any material to 165 °C, which corresponds to a Rb number density of outgassed due to the heat of the torch was pumped away. 1.7Ϫ2.2ϫ1014 atoms/cm 3 ͓54͔. It was found that higher Next, a small amount of nitrogen ͑99.9995% pure͒ was fro- temperatures resulted in leaks forming in the oven. The zen into the cell. Finally, the initially 99.995% chemically colder target chamber, at ϳ 65 °C, had a Rb density that was pure 3He was introduced into the cell through a trap at liquid 4 about three orders of magnitude lower. The quantity ͓Rb͔ A He temperature. This cryogenic trap further purifies the that was referred to earlier is the volume weighted average 3He by condensing out any contaminants. The cells were of the Rb number density over both chambers. For the tem- cooled with liquid 4He during filling in order to achieve a peratures at which we operated, the pressure in the cell was high density of 3He while maintaining a pressure of less than 11 atmospheres, and the density in the target chamber was one atmosphere. The cryogenic filling technique ensures that about 8.9 Amagats. when the tube through which the cell is filled is heated, the The optical pumping was accomplished using five glass will collapse on itself, thereby sealing the cell. titanium-sapphire lasers pumped by five argon ion lasers. Out of ten cells produced with the techniques described The beams were passed through ␭/4 plates to achieve circu- Ϫ1 above, ⌫cell was carefully characterized in five. In these cases lar polarization, and were arranged to get a reasonable filling Ϫ1 of the pumping chamber’s cross section. The laser system ⌫cell was always in excess of 30 hours, and for the cells used Ϫ1 was housed in a protective ‘‘laser hut’’ in a high radiation in the experiment, ⌫cell was in the range of 50 to 65 hours at area near the target. Access for laser tuning during the ex- room temperature. These numbers, compared to the 95 hour Ϫ1 periment was limited, but was generally necessary only once upper limit on ⌫dipolar at 20 °C, imply that most of the relax- every few days. ation was caused by the unavoidable 3He- 3He dipole inter- Ϫ1 A set of 1.4 m diameter Helmholtz coils, coaxial with the action, although some improvement in ⌫cell is still possible. electron beam, produced a 20 to 40 G alignment field for the Polarimetry was accomplished by comparing the AFP sig- 3He nuclear polarization. The field strength was chosen to be nals of the 3He with the AFP signals from water samples. large enough to ͑1͒ suppress the effects of ambient magnetic The AFP scans involved applying rf at 92 kHz using the field inhomogeneities and ͑2͒ to facilitate a nuclear magnetic drive coils while simultaneously sweeping the main mag- resonance ͑NMR͒ measurement of the Boltzmann equilib- netic field through the resonance condition. When passing rium polarization of protons in water for polarimetry pur- through the resonance, the nuclear spins reverse their direc- poses. The 3He nuclear polarization was measured using the tion, creating a measureable NMR signal in the pickup coils. NMR technique of adiabatic fast passage ͑AFP͓͒55͔. The Two resonance curves were obtained during each AFP mea- AFP system used, in addition to the main field coils, a set of surement — one as the field was swept up, and the other as 46 cm diameter Helmholtz rf drive coils and an orthogonal the field was swept down. Examples of resonance curves for set of smaller pick-up coils, both of which are pictured in both 3He and water are shown in Fig. 7. In the case of the Fig. 6. A second set of Helmholtz coils transverse to the water signal, the average of 25 scans is shown. Target polar- electron beam axis was used to rotate the target polarization ization losses during a measurement were typically less than and for operation with a polarization transverse to the beam. 0.1% relative. 6630 P. L. ANTHONY et al. 54

200 ,_ - 50 3He Signal (a) 2.5 Target Polarization(%) > 150 - - > 40 .§. -:::2.0 ,_ ♦ G) :-, ~ + .., .. ~.... {• ♦• "iii "iii .c .. ♦ •• .,_,,..._,. C: C: E 30 ~ + • + Cl 100 - - ::, ... .oo ~1.5 z t- a: a: C 20 :i: :i: ::, z 50 z 1.0 cc f:._,.~ 10 _J Target Helicity 0 ' 0.5 5 6 7 8 9 10 -6.0 -5.5 -5.0 0 Nov. 7 Dec. 22 DAC Voltage (V) -10 1200 1400 1600 1800 FIG. 7. ͑a͒ 3He NMR-AFP signal obtained using one sweep of Run Number the main holding field. ͑b͒ Water NMR-AFP average signal ob- tained using twenty five sweeps of the main holding field. Note the FIG. 8. 3He target polarization vs time. difference in scale between the two signals. The curve corresponds to a Lorentzian fit to the data. line of the NMR signal before and after passing through resonance, an effect that is clearly visible in Fig. 7. The When studying water, some care needs to be taken to interpretation of the water signal, that is, the calculation of ␰, was another important contribution. Electronic gains, the interpret the AFP signals properly. The average proton po- comparison of the exact shapes of the different 3He and larization that occurs during the two AFP peaks can be writ- water cells, and knowledge of the exact density of the 3He ten were also important contributions. Finally, the lock-in ampli- fier that was used in the NMR set-up had a time constant that h␯ gave a small distortion to the AFP resonance shape, for P ϭ␰tanh , ͑34͒ p ͩ 2k Tͪ which a small correction had to be applied. The various er- B rors are summarized in Table III. 3 where h is Planck’s constant, ␯ϭ92 kHz is the frequency of During the experimental run, the He polarization was the applied rf, k is Boltzmann’s constant, and T is the tem- measured roughly every four hours. The results of these mea- B surements are shown in Fig. 8. The average 3He polarization perature of the sample. Basically, P p is the thermal equilib- rium Boltzmann polarization that is expected at the field cor- over the entire experiment was about 33%. During the first responding to the point at which resonance occurs. There is a three weeks of the experiment, there were a few precipitous drops in the polarization due to a variety of problems, as caveat, however, in that the proton spins relax toward the indicated in Fig. 8. Later, however, the target polarization effective magnetic field experienced in the rotating frame of was very stable, running for three weeks with only slow the rf, which on resonance is given by the magnitude of the drifts. Toward the end of the experiment, the slight drop in applied rf ͑in our case about 76 mG, much less than the polarization that is evident in Fig. 8 is due to an increase in applied static field͒. This effect is accounted for by the pa- the beam current from an average of 2.1 ␮Ato3.4␮A. rameter ␰, which we have calculated to be 0.966Ϯ0.014. If the longitudinal relaxation time T1 for the protons were in- D. The electron spectrometers finite, ␰ would be equal to one. As it is, however, the mea- 3 sured proton signal corresponds to a slightly lower polariza- Electrons scattered from the He target were detected in tion than one would naively expect. two single-arm spectrometers. The spectrometers were cen- Through a careful comparison with water signals, we de- termined a calibration of 1.61Ϯ 0.11% polarization per 10 SLAC E142 Spectrometers 3 mV of signal. As Fig. 7 shows, the He signals were ex- TOP VIEW Dipole Magnets Hodoscope tremely clean. The uncertainty in the polarimetery was thus dominated by the uncertainty in the calibration constant. The largest contribution was from the determination of the mag- ¾ nitude of the water signals, which were about 1.9 ␮V. The Target error here was dominated by a systematic shift in the base 8eeam 70

Hodoscope Pb-glass TABLE III. Target polarimetry systematic uncertainties. Shower SIDE VIEW (7°) Counter Proton signal magnitude Ϯ5.6% Proton polarization (␰) Ϯ1.5% Electronic gain Ϯ1.5% Floor I Cell geometry Ϯ2.6% 30 Lock-in time constant correction Ϯ1.0% meter 3He density Ϯ2.5% Total Ϯ7.1% FIG. 9. Layout of the magnets and detectors used in E142 ex- periment. 54 DEEP INELASTIC SCATTERING OF POLARIZED . . . 6631

10 O.B SLAC 8 GeV/c (b) Spectrometer 0.6 8 0.4 SLAC 20 GeV/c i-i Spectrometer i ! ~ 6 i 0.2 -e I > j Cl) '! g 0 ~~--~---=---~ 5 10 15 20 5 10 15 20 4 "'0 MOMENTUM (GeV/c) MOMENTUM (GeV/c)

FIG. 11. Momentum acceptance of ͑a͒ the 4.5° and ͑b͒ the 2 7 ° spectrometer. Acceptances for the SLAC 20 GeV ͑a͒, the 8 GeV ͑b͒, and the E130 spectrometers are presented for comparison. 0 0.01 0.05 0.1 0.5 spectrometer (xр0.4 in the 4.5° spectrometer, and xр0.6 in X the 7° spectrometer͒. The spectrometer design ͓58͔ used two dipoles bending in FIG. 10. Q2 vs x range covered by the experiment for opposite directions, providing a large solid angle acceptance Eϭ22.66 GeV. This range is defined by the beam incident energy, which remains constant over a very large momentum inter- scattering angle, and spectrometers momentum and angular accep- val. The solid angle of the ‘‘reverse-bend’’ dipole doublet tances. The density of points corresponds to the relative electron configuration, when integrated over the 7 to 20 GeV/c mo- scattering rates. mentum interval, is twice that of previous ‘‘conventional’’ designs with the two dipoles bending in the same direction. tered at 4.5° and 7.0° with respect to the beam line in order The maximum solid angle of the two spectrometers is shown to maximize the kinematic coverage for an electron beam as a function of momentum in Fig. 11. The reverse bend also energy of 22.7 GeV and an event selection criteria of fulfills the ‘‘two-bounce’’ requirement by optimizing the de- Q2Ͼ1.1 (GeV/c)2. The momentum acceptance ranged flecting angles and the separation of the two dipoles. In the from 7 to 20 GeV/c for both arms. A schematic of the two 7.0° spectrometer the distance between the two dipoles was systems is shown in Fig. 9. Both arms used magnetic ele- 2 m and the two vertical deflection angles were 7° ͑down͒ ments from the existing SLAC 8 and 20 GeV/c spectrom- for the front dipole and 12° ͑up͒ for the rear dipole for 12 eters. GeV particles. This combination makes the spectrometer The design of the spectrometer was driven by several re- a‘‘two bounce’’ system for photons and at the same time quirements. The cross sections to be measured were known provides sufficient dispersion for determining the scattered to be small, typically of the order of 10Ϫ32 cm 2/͑sr GeV͒. particle momenta. In the 4.5° arm the deflection angles of the The raw counting ratio asymmetry of the two different spin dipoles are the same as for the 7.0° arm but their separation orientations was also predicted to be small, of the order of is 4 m resulting also in a ‘‘two-bounce’’ system. 10Ϫ3 to 10Ϫ4. In order to minimize beam running time, the Another advantage of the reverse bend configuration is spin structure function measurements required spectrometers that the detector package is located at approximately the pri- with the largest possible solid angle over a momentum ac- mary beam height. This convenient elevation makes the con- ceptance range extending from 7 to 20 GeV/c. Such a mo- crete structure required for shielding the detectors from room mentum acceptance gives a rather wide coverage over x with background considerably less massive compared to the con- Q2Ͼ1.1 (GeV/c)2 ͑see Fig. 10͒. ventional design with both dipoles bending up. The reduced In addition, these small scattering angle spectrometers mechanical complexity translates to significant economic were designed to suppress an expected large photon back- benefits as both the setup time and apparatus costs are mini- ground coming from the target due to bremsstrahlung, radia- mized. tive Mo”ller scattering and the decay of photoproduced ␲° In the 7° arm, the bend plane position of the scattered mesons. Background rate calculations indicated the need for particles at the detectors depends weakly on their momenta at least a ‘‘two-bounce system’’ ͑the configuration of the as shown in Fig. 12. The particle momenta are correlated spectrometer should allow a photon to reach the detectors with the divergence of their trajectory at the exit of the spec- only after scattering at least twice on the magnet poles or trometer. This results in a loss in momentum resolution, not vacuum walls͒ in order to keep this background at a tolerable critical to the experiment, but spreads out the pion back- level. ground, which is highly peaked at 7 GeV/c, onto a large The energy resolution of the spectrometers was defined detector area, allowing measurements at a fairly large pion solely by the required x resolution. The cross section asym- rate. metries were not expected to exhibit any sizable dependence The purpose of the quadrupole in the 4.5° spectrometer on momentum transfer. The energy resolution ranged from was to increase the angular magnification in the nonbend Ϯ5% at EЈϭ7 GeV to Ϯ4% at EЈϭ 18 GeV for each plane and spread the scattered particles onto a larger detector spectrometer. The resulting resolution in ⌬x/x ranged from area in this direction as can be seen in Fig. 13. In the bend Ϯ8% at low x up to Ϯ15% at the highest x covered by each plane the quadrupole focusing improves the momentum reso- 6632 P. L. ANTHONY et al. 54

___ 7 GeV/c ___ 13 GeV/c ...... 7.5 GeV/c ___ 10 GeV/c __ 19 GeV/c _. _ 8.5 GeV/c 13 GeV/c 19 GeV/c -60 10 GeV/c -60 (a) -40 (a) -40 -20 -20 E ~ 0 20 >, +16 mr 20 +16 mr 40 40

60 L_ __ 5 20 0 30 30 ~--,K0.~'1~SSj-- 1 ~. {b) 20 Pb - (b) ~ -- 20 10 +10 mr ··.• ------10 ~E 01 ~ E ~--.~~~-~~-======7~ 0 >< -10 -10 mr >< -10 ~ -5 mr -20 -20

-30 L-~~~~~-;2005 0 -30 z (m) 0 10 30 z (m)

FIG. 12. Bend plane ͑a͒ and nonbend plane ͑b͒ raytrace for the 7.0° spectrometer for rays of different momenta originating from FIG. 13. Bend plane ͑a͒ and nonbend plane ͑b͒ ray trace for the the center of the polarized target. All rays are drawn with respect to 4.5° spectrometer for rays of different momenta originating from the central trajectory of the system ͓͑a͒ ␾0ϭ0 mr, ͑b͒ ␪0 ϭ 0mr the center of the polarized target. All rays are drawn with respect to and p0 ϭ 10 GeV/c͔. Also shown are the iron magnet poles and the central trajectory of the system ͓͑a͒ ␾0 ϭ 0 mr, ͑b͒ ␪0 ϭ 0mr lead collimators. and p0ϭ10 GeV/c͔. Also shown are the iron magnet poles and lead collimators. lution of the system as both the position and divergence of the scattered particles at the exit of the spectrometer are cor- scintillator hodoscopes and the known optical properties of related with momentum. The introduction of the quadrupole the magnetic spectrometer. The second one relied on energy reduces the highly peaked solid angle in the range of 7 to 10 deposition in the lead-glass calorimeter. GeV/c and relaxes the instantaneous counting rates in the The two Cˇ erenkov counters of each spectrometer ͓59͔ em- detectors allowing accumulation of data in parallel and at ployed N 2 radiator gas with an effective length of 2 and 4 about the same rate with the 7° spectrometer. meters, respectively. Two spherical mirrors in each of the Each spectrometer was instrumented with a pair of gas two-meter tanks and three mirrors in each of the four-meter threshold Cˇ erenkov detectors, a segmented lead-glass calo- tanks collected Cˇ erenkov radiation over the active area of rimeter of 24 radiation lengths in a fly’s eye arrangement, six light emission. Each set of mirrors focused the Cˇ erenkov planes of segmented scintillation counters grouped into two light on one 5Љ R1584 Hamamatsu photomultiplier per tank. hodoscopes ͑front and rear͒ and two planes of lucite trigger The glass mirrors were manufactured at CERN by slumping counters. The electrons were distinguished from the large a 3 mm thick, 836 mm diameter disk of float glass into a pion background using the pair of Cˇ erenkov counters in co- stainless steel mold. The glass was cut to the appropriate incidence. The scattered electron energies were measured by dimensions, cleaned and then coated with 80 nm of Al fol- two methods. The first used the track information from the lowed by a protective coating of 30 nm of MgF 2 which is 54 DEEP INELASTIC SCATTERING OF POLARIZED . . . 6633

plane, it was ϳϮ0.5 mr for the 4.5° arm and ϳϮ0.3 mr for o.oa-~-~------~ the 7° arm. • 7° Spectrometer • The momentum resolution depends on the absolute value V 4.5° Spectrometer of momentum and varied from Ϯ0.5% to Ϯ2.5% for the • • • V. 0.06 4.5° spectrometer and from Ϯ0.6% to Ϯ3.5% for the 7.0° '7 ~ ~ ~ V V spectrometer as can be seen in Fig. 14. The figure also dis- odo + LG (7°) plays the energy resolution of the shower counter. The initial ͑at the target͒ production coordinates x0 , ␪0 , ~ 0.04 y 0 and ␾0 , and the momentum of the particles transported through the spectrometers were reconstructed by means of reverse-order TRANSPORT ͓64͔ matrix elements using the 0.02 final ͑at the rear hodoscope location͒ coordinates x f , ␪ f , y f , and ␾ f of the detected particles. The very large momen- tum bites of the spectrometers required using a fourth-order reverse TRANSPORT expansion in y f and ␾ f for recon- 5 10 15 20 25 structing the particle momenta. E (GeV) The shower counter calorimeter for each spectrometer was assembled from a selected subset of 200 ͑20 rows of 10 FIG. 14. Data for the resolution of the ratio of energy in the blocks͒ lead glass bars from a previous experiment ͓65͔. shower counter divided by momentum from tracking ␴(EЈ/P)vs Each bar consisted of Schott type F2 ͑refractive index of EЈfor each spectrometer. Expected contributions from tracking 1.58͒ lead glass with dimension 6.2 ϫ 6.2 ϫ75 cm 3 provid- ͑‘‘hodoscope’’͒, from energy deposition in the lead glass ͑LG͒ and ing for 24 radiation lengths along the direction of the de- from both combined ͑hodoscope ϩ lead glass calorimeter͒ are also tected electrons. The blocks were arranged in a fly’s eye shown by the curves. configuration, stacked upon each other with a segmentation that allowed for an accumulation of data at a maximum ˇ transparent down to 115 nm ͓60͔. The measurement of the ␲/e ratio of about 20. With the two Cerenkov counters in the reflectivities for all ten mirrors used in the detectors yielded trigger, the contamination of the shower signals by pions was an average of 80% at 160 nm and 89% at 200 nm. To en- small ͑on the order of a few percent͒. The shower counter resolution for electrons was measured hance the electron detection efficiency, each of the photo- in a test beam at CERN to be ͓66͔ multiplier UV glass surfaces was coated with 2400nm of p-terphenyl wavelength shifter followed by a protective ␴/EЈϷϮ͑2.5ϩ6.5/ͱEЈ͒%. ͑35͒ coating of 25 nm of MgF 2 ͓61͔. The fluorescence maximum of p-terphenyl of about 370 nm ͓62͔ matched well the region The counters were calibrated with a sample of scattered elec- of high quantum efficiency of the photomultipliers. More- trons of 5 GeV energy in a special elastic electron-proton over, the short 1–2 ns decay time of this emission enabled us scattering run using a gaseous hydrogen target. Extrapolation to retain accurate timing information from the Cˇ erenkovs. of the calibration algorithm to higher energies was per- The2mCˇerenkov counters operated at a threshold for pions formed using the scintillator hodoscopes and the known op- of 9 GeV/c andthe4mCˇerenkov counters at a threshold of tical properties of the spectrometers. The detailed study of 13 GeV/c. The measured number of photoelectrons per inci- the performance of the detector is described elsewhere ͓66͔. The spectrometer setup and detector packages proved to dent electron was ϳ 7.5, resulting in a detection efficiency of ˇ over 99.5%. be robust. All Cerenkov counters ran with an acceptable av- erage photoelectron yield, typically greater than six photo- The two scintillator hodoscopes ͓63͔ provided data for an electrons. The simple hodoscope tracking system was able to evaluation of possible systematic errors in the lead-glass and reconstruct tracks with an efficiency of greater than 80%. Cˇ erenkov counter data. They were used to identify back- Subsequent sections describe in some detail the analysis and grounds and to measure the pion asymmetry in order to sub- spectrometer performance. tract contaminations in the electron sample. The fine hodo- scope segmentation (ϳ185 scintillator elements per spectrometer͒ was chosen to tolerate the large expected pho- E. The electron trigger ton and neutron backgrounds and to reconstruct with suffi- The main electron trigger ͓67,68͔ for each spectrometer cient resolution the production coordinates of the scattered consisted of a triple coincidence between the two Cˇ erenkovs particles. Both horizontal and vertical planes consist of scin- and the sum of the shower counter signals. This trigger was tillator elements of 3 cm width with a ‘‘2/3’’ overlap result- 96% efficient for electron events and had a contamination of ing in a bin width of 1 cm. 13% of nonelectrons events. Secondary triggers, prescaled to The separation of the two hodoscopes was ϳ6.5 m in the reduce the rates, consisted of various combinations of the 4.5° arm and ϳ4.5 m in the 7.0° arm. The angular tracking detector elements designed to measure efficiencies and pion resolution of the hodoscopes was Ϯ0.7 mrad for the 4.5° backgrounds. spectrometer and Ϯ0.9 mrad for the 7.0° spectrometer; the Up to four triggers were allowed per spectrometer per position tracking resolution was Ϯ0.3 cm for both spectrom- beam spill. There was a 30 ns deadtime after each trigger. eters. The angular resolutions in the non-bend plane were Each trigger gated a separate set of 205 ADC’s ͑Lecroy 2280 ϳϮ0.5 mr for both spectrometers, whereas for the bend system͒ for the shower counter. Each shower counter signal 6634 P. L. ANTHONY et al. 54

250 (a) 4.5°

>Q) 200

0 0 10 15 20 10 15 20 Scattered Electron Energy (GeV) Scattered Electron Energy (GeV)

FIG. 15. Comparison between measured ͑solid circles͒ and Monte Carlo simulated ͑highlighted band͒ deep inelastic cross sections for the 4.5° spectrometer ͑a͒ and for the 7° spectrometer ͑b͒. The data error bars are dominated by uncertainties in the spectrometer solid angle. The expected cross section is based on a model which relies on previous SLAC and CERN measurements. The width of the band is due to uncertainty in the target density. was sent to four ADC’s corresponding to the four possible trigger were studied by comparing the energy and momen- triggers, making a total of over 800 ADC channels per spec- tum determination of the event from the measurements in the trometer. The hodoscope signals along with the selected el- shower counter and hodoscope, respectively. Overall, the ements of the trigger went to multihit TDC’s ͑LRS 2277 contamination from high energy pion events was less than system͒ which had a 20 ns deadtime. To reduce the load on 1%, since the cross section for this process is low; whereas, the data aquisition, the signals to the hodoscope TDC’s had a backgrounds from events with an electron and pion in coin- 100 ns gate provided by the trigger system. Thus each elec- cidence was also relatively small, since the electron energy tron candidate had a 100 ns window of activity in the detec- cluster would typically deposit more energy than the pions, tor. The deadtime correction to the asymmetry was deter- and only the highest energy cluster per trigger was kept for mined from a Monte Carlo model of the trigger and the analysis. Backgrounds from neutral particles were mini- instantaneous beam intensity ͓59,67,68͔. The average correc- mal, since the spectrometer was designed to accept only tion was about 10% in the 4.5° spectrometer and less than charged particles. 4% in the 7° spectrometer. Electron events were selected if they triggered both Cˇ er- enkov counters with an ADC signal greater than one and a IV. ANALYSIS half photoelectrons, and deposited a minimum energy ͑typi- Data analysis was directed at extracting the electron scat- cally greater than 5.5 GeV͒ as a cluster of3x3blocks in the tering asymmetries and structure functions with a high rate segmented shower counter. An algorithm based on artificial spectrometer. The highest rates occurred in the 4.5° spec- intelligence techniques ͑cellular automaton͒ was developed trometer arm, ranging from 1.5 to 2.5 electron triggers per to cluster hit blocks belonging to the same shower ͓69͔. pulse on average. The rate was typically less than 1 electron Events with a shower contained entirely in one block were trigger per pulse in the 7° arm. Since the SLAC Linac deliv- rejected. This single-block event cut served to reject a large ers pulses at a rate of 120 Hz, the experiment recorded a fraction of pion events as determined both from a GEANT large sample of deep inelastic scattering electron events. In simulation ͓70͔ and from studying results from the energy total, approximately 350ϫ106 electron events were collected versus momentum comparison. A trigger from the Cˇ erenkov of which 300ϫ106 were used to determine the asymmetries counters opened a 100 nsec gate ͑during the 1 ␮sec spill͒ and and spin structure functions. the pulse heights from the lead glass blocks and from the The analysis focused on identifying electrons and deter- phototubes of the Cˇ erenkovs were recorded using zero- mining their momentum in a high rate environment. Charged suppressing LeCroy 2282 ADCs. Only clusters with the pions were the main source of background. The electron trig- maximum energy in the 100 nsec gate were accepted in the ger consisted of a triple coincidence in the signals coming analysis. The ability to reject pions is studied by comparing from the two Cˇ erenkov counters and summed shower the energy and momentum determinations of an event. Typi- counter per spectrometer arm. This method served to reject cally, pion events registered a higher momentum than en- the majority of charged pions, which typically enter the spec- ergy, since the pion shower energy is usually not entirely trometer out of time compared to the Cˇ erenkov signal. visible in the electromagnetic calorimeter. The remaining pion background originated from either For extracting the asymmetries, once an electron event very high momentum pions ͑typically greater than 13 GeV͒ was selected, the shower counter was used both to identify or from pions that enter the spectrometer in time with an its energy and to determine the scattering angle ␪ from the electron. Backgrounds from pion contamination within the centroid position of the shower in the calorimeter. The posi- 54 DEEP INELASTIC SCATTERING OF POLARIZED . . . 6635

4000 (a) 4.5° (c) 4.5° 3000 0.4 23 3000 C: 2000 8 2000 ..= 0.2 1000 <( 1000 ~ ai 0 0 E E 0 + + ~ (b) 7° (d) 7° <( 4000 C: 2000 0 ii: -0.2 .l/3 3000 C: ::, 0 (.) 2000 1000 -0.4 1000

0 0 0.01 0.05 0.1 0.5 0 2 0 1 2 E/P E/P X

␲ FIG. 16. Plots ͑a͒ and ͑b͒ represent the ratio of energy ͑deter- FIG. 17. Inclusive pion asymmetry Aʈ vs x. The results are 2 mined by the calorimeter͒ to the momentum ͑determined by the consistent with zero (␹ /d.f.ϭ0.6͒. hodoscopes͒ of events detected in the 4.5° and 7.0° spectrometers. The electrons are identified by the peak centered around check was useful for systematic studies. Extraction of the EЈ/Pϭ1, whereas the pions, which deposit less energy in the calo- total cross section requires detailed knowledge of the spec- rimeters, are in the region EЈ/PϽ0.8. Plots ͑c͒ and ͑d͒ show events trometer acceptance, central momentum deadtime and target with the highest energy cluster for a given trigger and requiring an thickness. Figure 15 present a comparison of the results on electron hardware trigger. These events define greater than 99% the total cross section for the 4.5° and 7° spectrometer, re- pure electrons sample and are those used in the physics analysis. spectively. Systematic errors on the measurement are large ͑typically 15%͒. The data are especially sensitive to the tion determination of the shower centroid was accurate to knowledge of the radiative corrections which can change the approximately Ϯ 10 mm, significantly better than needed for shape significantly. Within systematic uncertainties, there is sufficient x resolution. The hodoscope tracking system was reasonable agreement between the data and the Monte Carlo developed to calibrate the shower counter, perform system- determination, which uses cross section measurements from atic studies of the backgrounds and to monitor the shower previous SLAC experiments ͓71͔. counter and Cˇ erenkov efficiencies throughout the experi- After event selection, contamination of the electron event ment. Typically, hodoscope tracking efficiencies varied from samples was relatively small. Figure 16 presents distribu- 80 to 95% depending on the trigger rate. Noise sources from tions of events comparing the energy measured in the shower random hits due to photon background with the associated counters, EЈ, compared to the momentum, p, measured by electronic deadtime were the main cause of hodoscope inef- tracking using the hodoscopes. Low-energy tails ficiency. For this reason, the shower counter was the primary (EЈ/pϽ0.8) come largely from pion contamination, whereas detector used for the analysis. high-energy tails (EЈ/pϾ1.2) come from overlapping events Calibration of the shower counter energy was performed with typically either two electron interactions or an electron using two methods. First, knowledge of the magnetic field and pion interactions. A neural network analysis was devel- and spatial positions of the spectrometer magnets allowed for oped to study pion rejection using only the calorimeter infor- the determination of the particle’s momenta via tracking. mation ͓70͔, but was not used in the final asymmetry analy- Tracking with pristine events was used to calibrate the sis. The pion contamination of the electron event sample at shower counter. The primary uncertainty in this method low energy (EЈϷ7 GeV͒ was found to be approximately 3% comes from the finite width of the hodoscope fingers and the and decreasing at higher energies to less than 1%. knowledge of the position of these fingers relative to the Corrections to the electron asymmetries from pion con- spectrometer magnets. The uncertainty in the momentum tamination are performed assuming zero asymmetry coming measurement is energy dependent and estimated to be below from the pions with an uncertainty of Ϯ0.15. A special study 2.5% over the range of energies detected ͑Fig. 14͒. The sec- using out of time pion events within the trigger gate revealed ond method employed a special test run performed ahead of no evidence for a significant pion asymmetry as shown in the experiment and used a 5 GeV electron beam scattering Fig. 17. The final effect due to pion contamination is small. off a hydrogen target ͑H 2) to observe the elastic peak. The An additional source of contamination to the deep inelas- location of the peak in both the hodoscope and the shower tic scattering electron event sample arises from de- counter checked the absolute energy scale to Ϯ3% as deter- cays producing secondary electrons. For example, if a neu- mined from magnetic measurements. tral pion is produced in the final state of an interaction, it will In order to investigate possible inefficiencies in the detec- decay into two photons which themselves can scatter and tor package, some special low rate runs were performed to produce an electron which enters the spectrometer and simu- measure the total cross section. Although the spectrometer lates a true deep inelastic scattering event. Contamination was not designed for cross section measurements, such a due to this process is measured by reversing the polarity of 6636 P. L. ANTHONY et al. 54

TABLE IV. Polarized target parameters used in calculating the dilution factor. A run corresponds typi- cally to one or two hours of data taking.

Target

123 Runs used 1000-1117,1320-1771 1118-1181 1182-1319 Length ͑mm͒ 295Ϯ2 297Ϯ2 303Ϯ2 Front window (␮m͒ 110Ϯ7.5 % 110Ϯ7.5% 110Ϯ7.5 % Rear window (␮m͒ 124Ϯ7.5 % 107Ϯ7.5 % 110Ϯ7.5 % Glass density ͑g/cm 2) 2.52Ϯ1% 2.52Ϯ1% 2.52Ϯ1% 3He density ͑amagats͒ 8.63Ϯ2.5% 8.90Ϯ2.0% 8.74Ϯ2.0%

N 2 density ͑amagats͒ 0.070Ϯ1.7% 0.069Ϯ 1.8% 0.082Ϯ 1.8% 3He density ͑cm Ϫ3) 2.32ϫ1020Ϯ2.5% 2.39ϫ1020Ϯ2.0% 2.35ϫ1020Ϯ 2.0% Ϫ3 18 18 18 N 2 density ͑cm ) 1.88ϫ10 Ϯ1.7% 1.85ϫ10 Ϯ 1.8% 2.20ϫ10 Ϯ1.8% the spectrometer magnets and collecting dedicated data on 1 Aʈ the positron rates. The measurement serves as a valid sub- A1ϭ ͓A /DϪ͑␩/d͒AЌ͔ϭ Ϫ␩A2 , ͑1ϩ␩␨͒ ʈ D traction as long as the hadronic decay process is charge sym- metric. We found that approximately 5% of the low x events 1 were contaminated from such a process. The effect decreases A2ϭ ͓AЌ /dϩ͑␨/D͒Aʈ͔. ͑39͒ rapidly as x increases. The behavior is similar to the pion ͑1ϩ␩␨͒ contamination, with a larger contamination at low EЈ and 3He 2 dying off quickly at higher EЈ. The asymmetry in the posi- Furthermore, the spin structure functions g1 (x,Q ) and 3He 2 tron rates is measured to be zero with large uncertainties g2 (x,Q ) are extracted using the asymmetries given above, (ϷϮ 30%͒. The largest systematic uncertainty in the lowest x bin comes from this effect ͑see Tables XI and XII͒. F2 g ϭ ͓A ϩ␥A ͔, For the asymmetry analysis, electron events which passed 1 2x͑1ϩR͒ 1 2 the event selection cuts were divided into bins of scattered energy EЈ, scattering angle ␪, and relative target and beam F 1 ( ) 2 helicities, N↑↓ ↑↑ (EЈ,␪). From these counts ͑normalized by g ϭ A ϪA 2 2x͑1ϩR͒ ͩ 2 ␥ 1 ͪ incident charge Ne), raw asymmetries are formed: 1ϩ␥2 y EϩEЈcos␪ ϭF2 AЌϪsin␪Aʈ . ͑N/Ne͒↑↓Ϫ͑N/Ne͒↑↑ 2xDЈ͑1ϩR͒ 2sin␪ ͩ EЈ ͪ raw Aʈ ϭ ͑36͒ ͑N/Ne͒↑↓ϩ͑N/Ne͒↑↑ ͑40͒

2 and for data collected with transverse target polarization, Here R(x,Q )ϭ␴L /␴T is the ratio of the longitudinal to 2 transverse cross sections and F2(x,Q ) is the unpolarized deep inelastic structure function. Both are functions of x and ͑N/N ͒ Ϫ͑N/N ͒ 2 raw e ↑→ e ↓→ Q . The other kinematic variables are related to the incoming AЌ ϭ . ͑37͒ ͑N/Ne͒↑→ϩ͑N/Ne͒↓→ and outgoing scattered electron energy (Eand EЈ, respec- tively͒ via On these measured raw asymmetries, corrections for the ␯ϭEϪEЈ, beam polarization Pb , target polarization Pt , dilution factor f He , electronic dead time ⌬dt , radiative corrections ⌬RC , and kinematics are performed to extract the 3He parallel and yϭ␯/E, ͑41͒ transverse asymmetries EϪEЈ⑀ Dϭ , raw E 1ϩ⑀R͒ ͑A ϩ⌬ ͒ ͑ ʈ dt ʈ Aʈϭ ϩ⌬RC , PbPt f He ͑1Ϫ⑀͒͑2Ϫy͒ DЈϭ , ͑42͒ y͓1ϩ⑀R͔ raw ͑AЌ ϩ⌬dt͒ Ќ AЌϭ ϩ⌬RC . ͑38͒ and PbPt f He 2⑀ 3 dϭD . ͑43͒ From these asymmetries, the virtual photon- He asymme- ͱ1ϩ⑀ 3He 2 3He 2 tries A1 (x,Q ) and A2 (x,Q ) are found as a function of x and Q2: The factors ␩, ␨, and ␥ are found via 54 DEEP INELASTIC SCATTERING OF POLARIZED . . . 6637

12

2.5 10 lu ..-. 4.5° Spectrometer

0.5 0.5 (a) 4.5" spectrometer (b) T spectrometer 0 0.4 ... 0.4 u('(I 0 IL u('(I C ft IL 0.3 f 0 0.3 t ~ ~ ~ ~ C ~ ~ i5 .2 Cl) 0.2 i5 0.2 Method Method Cl) J=- :::c el el (') 0.1 0.1 OIi OIi

0.0 0 0.01 0.05 0.1 0.5 0.1 0.2 0.3 0.4 0.5 0.6 X X

FIG. 20. Material ͑method I͒ and reference cell ͑method II͒ results for the dilution factor of the reference cell as measured using the 4.5° arm ͑a͒ and the 7° arm ͑b͒ for a 3He gas pressure of 147 psi.

The target cell glass ͑Corning 1720͒ has the composition where PND(x) is an estimate of the nuclear dependence ef- 3 57% SiO 2, 20.5% Al 2, 12% MgO, 5.5% CaO, 4% B 2O 3 fect in He from Ref. ͓72͔ and differs from unity by less than and 1% Na 2, yielding within 1% the same number of protons 2% in our kinematic range. We have assigned an additional and . Assuming that the ratio Rϭ␴ /␴ is the same 3He L T 1% uncertainty to F2 due to the nuclear dependence effect. for the proton and the neutron ͓71͔, we can express the dilu- The overall systematic uncertainty from this method is domi- tion factor as nated by the knowledge of the thickness of the glass win- dows. The windows of the target cells were measured using 3 1ϩRnp PND X n a precision tooling gauge with an accuracy of 7%. Uncertain- 2 ͑ ͒ g g g f He͑x,Q ͒ϭ 1ϩ np ND ties in the measurement due to variations of the glass thick- ͫ 2͑2ϩR ͒ ͩP XHe nHe He nesses are included in estimating this uncertainty. Other con- PND X n Ϫ1 tributions to the uncertainty from the nuclear dependence N2 N2 N2 ϩ ND ͑50͒ effect and F2 are negligible. P XHe nHeͪͬ He A limitation of the method described above is that events

2 originating from beam halo interactions with the 30 cm long where X(x,Q ) is a radiative correction factor which relates side walls of the 1 cm radius target cell are not taken into the Born cross section to the experimental cross section for np n p account. The electron beam was centered on the target cell. different species. The quantity R ϭF2/F2 is the ratio of Primary electrons from the beam passing 1 cm from the cen- unpolarized structure functions for neutrons and protons. We ter could interact with the target glass walls producing addi- determined the neutron structure function using tional scattered electrons. n D p F2ϭ2F2 ϪF2 where the proton and deuteron structure func- During the experiment, several dedicated runs were per- tions per nucleon are taken from the NMC fits ͓73͔ to the formed in which the beam was steered away from the target BCDMS ͓74͔, SLAC ͓75͔, and NMC data. Because no un- center. Figure 18 presents the average event rate per pulse in certainties are included in the NMC fit, we used the relative the spectrometer as a function of the central beam position. point-to-point uncertainties to the SLAC data ͓71͔ which is An increase in the event rate is evident as the beam is moved at similar kinematics as the present experiment. We also in- more than 3 mm from its nominal position. The variation in cluded a normalization uncertainty from the SLAC data of event rate is well-described by a quadratic function and is p D 2.1% for F2 and 1.7% for F2 . Furthermore, we included a 2% uncertainty for the proton and 0.6% for the deuteron TABLE V. Radiation lengths tout seen by electrons exiting the arising from the maximum deviation of the SLAC and NMC target. fits in the range 0.08ϽxϽ0.6. For xϽ0.08 where there is very little SLAC data, a 5% error is placed on the NMC Spectrometer Fraction of events Radiation lengths (tout) structure functions defining the maximum deviation between 4.5° 36.50% 0.001 the NMC data and the NMC fit. For Rϭ␴L /␴T we used the 54.55% 0.085 central values and errors given by a SLAC global analysis 6.85% 0.169 71 . ͓ ͔ 2.10% 0.291 For this section and throughout this paper the spin- 7.0° 23.39% 0.001 independent 3He structure function in the deep inelastic re- 48.36% 0.055 gion is evaluated as follows: 18.75% 0.269

3He 2 ND D 2 p 2 9.50% 0.399 F2 ͑x,Q ͒ϭP ͑x͓͒2F2 ͑x,Q ͒ϩF2͑x,Q ͔͒ ͑51͒ 54 DEEP INELASTIC SCATTERING OF POLARIZED . . . 6639

3He 2 TABLE VI. The radiative corrections to A1 (x) at the average Q of each x bin are tabulated here, with the corrections to the 4.5° spectrometer given first, then those of the 7.0°. The ⌬’s are the additive correction to be made to the asymmetry for internal ͑int͒, external ͑ext͒, and f i are the fractions of events in a particular bin coming from quasielastic and inelastic tails including internal and external radiative effects. The elastic contribution is small.

int ext x ⌬RC ⌬RC f quel f inel f ext 0.035 -0.003 -0.000 0.08 0.15 0.10 0.050 -0.003 -0.001 0.04 0.13 0.08 0.080 -0.004 -0.002 0.02 0.10 0.08 0.125 -0.004 -0.003 0.01 0.07 0.08 0.175 -0.004 -0.003 0.00 0.04 0.05 0.250 -0.004 -0.003 0.00 0.01 0.04 0.350 -0.003 -0.002 0.00 0.02 0.02

0.080 -0.004 -0.001 0.04 0.11 0.09 0.125 -0.004 -0.002 0.01 0.07 0.09 0.175 -0.004 -0.003 0.01 0.04 0.06 0.250 -0.004 -0.003 0.00 0.00 0.05 0.350 -0.003 -0.002 0.00 0.03 0.02 0.466 -0.002 -0.000 0.00 0.06 0.01 attributed to the beam passing through an increasingly thick P 2 ␣ TC part of the target end caps. During the data taking the target f 3He ͑x,Q ͒ϭ . ͑53͒ ͑␣P ϩ␤͒ was positioned at the event rate minimum. From such stud- TC ies, we concluded that as long as the beam position was Since the 3He pressure was directly measured and the stable at the center of the target, the beam halo effect on the reference cell glass window thicknesses known to better than dilution factor should be small, except for the possibility of 7%, this measurement could be compared directly to the first flat tails. If the beam halo has long, flat tails, then if the beam method described above Fig. 20 . The direct measurement is moved off center the event rate may not change, but a ͑ ͒ of the dilution factor from these special runs would naturally constant background of unpolarized events may be present take into account any possible beam halo effects. From these from interactions with the glass side walls. studies, we conclude that there was no observation of any A second independent method for determining the dilu- large beam halo effects on the dilution factor. The final sys- tion factor was performed. Periodically throughout the ex- tematic uncertainty on the dilution factor is taken to be 8%, periment, data were collected in which a reference cell was where the dominant uncertainty comes from the knowledge placed in the electron beam. The cell consisted of the same of the window thicknesses 7% , needed for the first method glass type and had the same dimensions as the polarized ͑ ͒ described above. target cells. The cell was filled with 3He gas at various con- trolled pressures. At zero pressure the events are due to the glass end caps, B. Radiative corrections but as the cell is filled to different pressures the event rate 2 Due to the real and virtual radiation of electrons during increases. The event rate r(x,Q ) normalized to incident the scattering process, the longitudinal and transverse mea- charge is expressed as a function of pressure as follows: sured asymmetries (Aʈ ,AЌ) need further corrections known as the electromagnetic radiative corrections. The latter are LPr 2 r͑x,Q2͒ϭC n ␴ ͑x,Q2͒ϩ3N ␴ ͑x,Q2͒ , ͑52͒ performed to extract the structure functions g1,2(x,Q ) and ͫ g g RT He ͬ photon-nucleon asymmetries A (x,Q2) as defined in the 1,2 Born approximation where the scattering process is de- where C is a proportionality constant, R the 3He gas con- scribed by the exchange of a single virtual photon. These stant, T the temperature of the gas, is Avogadro’s number, corrections are cast into two categories: internal and external. N L the length, and Pr the pressure of the reference cell, re- The internal effects are those occurring at the nucleus re- spectively. When the pressure in the reference cell Pr equals sponsible for the deep inelastic scattering under investigation that of the target cell PTC , the number density and therefore need to be performed even for an infinitely thin nHeϭ3 LPr /RT matches that of Eq. ͑48͒. The rates were target. The external effects are those which modify the en- correctedN for rate-dependent electron detection efficiencies ergy of the incident and scattered electron via bremsstrah- and changes in the external unpolarized radiative corrections lung and ionization losses from interactions with other atoms as a function of helium pressure. before and after the deep inelastic process has occurred. The Figure 19 presents an example of a sequence of the refer- external corrections depend on target thickness. While the ence cell runs where the slope ␣ can be interpreted as formalism for the spin-independent deep inelastic scattering 2 2 nHe␴He(x,Q )/PTC and intercept ␤ as Cng␴g(x,Q ). The was developed by Mo and Tsai ͓76,77͔, that of the spin- dilution factor is extracted as dependent formalism was developed by Kuchto, Shumeiko, 6640 P. L. ANTHONY et al. 54

The internal corrections were calculated using the pro- 0.02 I I I I gram POLRAD version 14 which uses the new iterative A 3He 1 • Nor.c . method ͓79͔. In this method, the best fit to the experimental □ Internal r.c. 3He ◊ lntemal+extemal r.c. asymmetry A1 (x) is used to build the polarized structure 0 3He functions g1 (x). The cross sections for specific states of polarizations are then constructed and used to evaluate all the contributions of Eq. ͑54͒. Here all quantities refer to 3He. -0.02 ~ From this result a new A1 is produced and used as an input to the next iteration step by constructing a new model 3 for g He -0.04 ,_ i\ \ \ 1 F ͑x,Q2͒ \ ͑k͒ 2 measured ͑kϪ1͒ -0.06 g1 ϭ 2 ͓A1 ϩ⌬A1͑g1 ͔͒, 0 0.1 0.2 0.3 0.4 0.5 2x͓1ϩR͑x,Q ͔͒ ͑55͒ X

3He where k is the iteration index. FIG. 21. Change in the asymmetry A1 ͑averaged over E and ␪) as the radiative corrections ͑r.c.͒ are added. Only the statistical The process is then repeated until convergence is reached, error on the final results are shown for comparison. The x values of which occurs within three to four iterations. POLRAD was first each data set are the same but they have been shifted for clarity. checked against a program we developed based on the work by Kuchto and Shumeiko ͓78͔ and also against the Tsai ͓77͔ formalism for the unpolarized case. Similar results to POL- and Akushevich ͓78,79͔ and implemented in their code POL- RAD were found with both checks as expected. RAD. The internally radiated helicity-dependent deep inelas- The nuclear coherent elastic tail is evaluated using differ- tic scattering cross section can be decomposed into its com- 3 ponents following Ref. ͓79͔: ent best fits to the elastic form factors of He and found to be small. This leaves only three physical regions of significant contribution to the total internal radiative correction to be ␣ ␴r ͑x,y͒ϭ␴B ͑x,y͒ 1ϩ ͑␦IRϩ␦ ϩ␦l ϩ␦h ͒ considered: the quasielastic region which starts a few MeV Ϯ Ϯ ͩ ␲ R vert vac vac ͪ after the elastic peak ͑since no nuclear excited states are 3 F qel el bound in He͒; the resonance region which partially overlaps ϩ␴Ϯ,in͑x,y͒ϩ␴Ϯ ͑x,y͒ϩ␴Ϯ͑x,y͒ ͑54͒ the quasielastic tail; and finally the deep inelastic region which we have assumed to start at W2 ϭ 4 ͑GeV/c)2.InEq. ͑54͒ resonance and inelastic contributions are both included where r is the internally radiated helicity-dependent differ- F ␴Ϯ in ␴in . The internal and external radiative corrections require 2 ential cross section (d ␴/dxdy)Ϯ and (Ϯ) refers to the he- the knowledge of the spin independent structure functions B He 2 He 2 licity of the electron relative to that of the target, and ␴Ϯ is F1 (x,Q ) and F2 (Q ,␯) and spin dependent structure the helicity-dependent Born cross section of interest. The functions gHe(x,Q2) and gHe(x,Q2) over the canonical tri- IR l h 1 2 quantities ␦vert , ␦R , ␦vac and ␦vac are the electron vertex angle region ͓76,77͔. The lowest x bin in this measurement contribution, the soft photon emission, the electron vacuum (xϭ0.035) determines the largest kinematic range of Q2 and polarization, and hadronic vacuum polarization contribu- x over which the structure functions have to be known. It F 2 2 tions, respectively. The quantity ␴Ϯ,in is the infrared extends in the range 0.31рQ р18.4 ͑GeV/c) and qel divergence-free part of the inelastic radiative tail, and ␴Ϯ 0.03рxр1. The variables of integration which define the el and ␴Ϯ are the quasielastic and elastic radiative tail contri- canonical triangle are given by M x and t, in the range 2 butions, respectively. M nϩm␲рM xрW and tminрtрtmax ; tϵQ . Here M x is the

3He 3He TABLE VII. Results for A1 and g1 . x range x Q2 3He 3He ͗ ͗͘͘ A1Ϯ␦(stat)Ϯ␦(syst) g1 Ϯ␦(stat)Ϯ␦(syst) ͑GeV/c)2

0.03-0.04 0.035 1.1 Ϫ0.0264Ϯ0.0168Ϯ0.0054 Ϫ0.248Ϯ0.159Ϯ0.055 0.04-0.06 0.050 1.2 Ϫ0.0238Ϯ0.0102Ϯ0.0039 Ϫ0.168Ϯ0.072Ϯ0.027 0.06-0.10 0.082 1.8 Ϫ0.0317Ϯ0.0081Ϯ0.0048 Ϫ0.146Ϯ0.038Ϯ0.020 0.10-0.15 0.124 2.5 Ϫ0.0447Ϯ0.0083Ϯ0.0068 Ϫ0.141Ϯ0.027Ϯ0.018 0.15-0.20 0.175 3.1 Ϫ0.0463Ϯ0.0098Ϯ0.0094 Ϫ0.105Ϯ0.023Ϯ0.014 0.20-0.30 0.246 3.7 Ϫ0.0333Ϯ0.0099Ϯ0.0121 Ϫ0.051Ϯ0.016Ϯ0.007 0.30-0.40 0.343 4.4 Ϫ0.0003Ϯ0.0172Ϯ0.0220 0.000Ϯ0.017Ϯ0.004 0.40-0.60 0.466 5.5 Ϫ0.0145Ϯ0.0301Ϯ0.0199 Ϫ0.007Ϯ0.014Ϯ0.002 54 DEEP INELASTIC SCATTERING OF POLARIZED . . . 6641

3He 3He TABLE VIII. Results for A2 and g2 . Note that the systematic uncertainties are small compared to the statistical uncertainties. x range x Q2 3He 3He ͗ ͗͘͘ A2Ϯ␦(stat)Ϯ␦(syst) g2 Ϯ␦(stat)Ϯ␦(syst) ͑GeV/c)2

0.03-0.04 0.036 1.1 Ϫ0.0619Ϯ0.0738Ϯ0.0169 Ϫ9.314Ϯ10.62Ϯ2.593 0.04-0.06 0.050 1.2 0.0222Ϯ0.0516Ϯ0.0165 2.106Ϯ4.280Ϯ1.399 0.06-0.10 0.082 1.8 Ϫ0.0158Ϯ0.0425Ϯ0.0189 Ϫ0.648Ϯ1.705Ϯ0.756 0.10-0.15 0.124 2.5 Ϫ0.0193Ϯ0.0453Ϯ0.0216 Ϫ0.256Ϯ0.934Ϯ0.428 0.15-0.20 0.175 3.1 Ϫ0.0885Ϯ0.0559Ϯ0.0263 Ϫ0.764Ϯ0.641Ϯ0.283 0.20-0.30 0.246 3.7 Ϫ0.0349Ϯ0.0585Ϯ0.0286 Ϫ0.088Ϯ0.336Ϯ0.153 0.30-0.40 0.342 4.4 0.0756Ϯ0.1002Ϯ0.0347 0.140Ϯ0.246Ϯ0.080 0.40-0.60 0.466 5.5 Ϫ0.1875Ϯ0.1683Ϯ0.0344 Ϫ0.188Ϯ0.170Ϯ0.036

m invariant mass of all possible contributing scattering and W cross sections. The measured cross section ␴Ϯ is expressed the invariant mass of the scattering of interest. as the convolution of the internally radiated Born cross sec- The 3He spin-independent structure functions used in the tion with the radiation effect due to the finite thickness of the quasielastic region were those of de Forest and Walecka target: ͓80͔. These structure functions allow for a convenient param- eterization in the evaluation of the unpolarized radiative tail. Es Emax ␴m ͑E ,E ͒ϭ dEЈ p dEЈI͑E ,EЈ,t ͒ In the resonance region we chose the spin independent struc- Ϯ s p min s p s s in ͵E ͵E ture functions obtained by fitting the data in the resonance s p r region given in Ref. ͓81͔, while for the deep inelastic region ϫ␴Ϯ͑EsЈ,EЈp͒I͑EЈp,E p ,tout͒ ͑56͒ we used the same models for the proton and deuteron struc- ture functions as described in Sec. V.A. to build the spin where Es ,E p are the incident and detected electron energies, independent structure functions of 3He see Eq. 51 . ͓ ͑ ͔͒ and I(Ein ,Eout ,t) is the probability that an electron of energy The spin dependent structure functions used in the reso- Ein will have an energy Eout due to bremsstrahlung emission nance region were obtained from the AO program ͓82͔ after having passed through t radiation lengths ͑rl͒ of mate- which is based on an analysis of electromagnetic transition rial. For the polarized case, this probability function is spin- amplitudes in that region. In the deep inelastic region, as independent for a thin target because it is dominated by for- 3He described previously, a fit to the extracted A1 from this ward ͑charge͒ scattering off target atoms. All kinematics experiment was used to build the first spin dependent struc- parameters and the function I are well-described and dis- ture functions input to the iterative method. cussed in ͓77͔. The entrance glass window plus half of the 3 The external corrections were performed by extending the He thickness accounts for tinϭ0.00125 rl. For tout , the elec- procedure developed by Mo and Tsai ͓76,77͔ for the unpo- trons exit the target through four discrete sections in which larized scattering cross sections to that of the helicity depen- the amount of material the electrons traverses after scattering dent scattering cross sections. It used an iterative unfolding is different ͑see Table V͒. These four contributions to tout procedure on the internal and external corrections together from each region are summed. Note that tout is much larger until convergence is reached. The procedure requires the than tin , especially in the region where the scattered elec- knowledge of the internally radiated Born helicity-dependent trons passed through the NMR pick-up coils.

n n 2 n 2 2 n TABLE IX. Results on A1 and g1 at the measured average Q , along with g1 evaluated at Q ϭ2(GeV/c) assuming that A1 does not depend on Q2.

2 n n n x range ͗x͗͘Q͘A1Ϯ␦(stat)Ϯ␦(syst) g1Ϯ␦(stat)Ϯ␦(syst) g1Ϯ␦(stat)Ϯ␦(syst) ͑GeV/c)2 ͓Q2ϭ2͑GeV/c)2͔

0.03-0.04 0.035 1.1 Ϫ0.092Ϯ0.061Ϯ0.022 Ϫ0.269Ϯ0.182Ϯ0.065 Ϫ0.311Ϯ0.207Ϯ0.074 0.04-0.06 0.050 1.2 Ϫ0.082Ϯ0.038Ϯ0.017 Ϫ0.177Ϯ0.083Ϯ0.033 Ϫ0.195Ϯ0.090Ϯ0.036 0.06-0.10 0.081 1.8 Ϫ0.109Ϯ0.031Ϯ0.021 Ϫ0.151Ϯ0.044Ϯ0.025 Ϫ0.154Ϯ0.044Ϯ0.026 0.10-0.15 0.124 2.5 Ϫ0.162Ϯ0.033Ϯ0.030 Ϫ0.146Ϯ0.031Ϯ0.022 Ϫ0.142Ϯ0.031Ϯ0.022 0.15-0.20 0.174 3.1 Ϫ0.170Ϯ0.041Ϯ0.042 Ϫ0.105Ϯ0.026Ϯ0.017 Ϫ0.099Ϯ0.026Ϯ0.016 0.20-0.30 0.245 3.7 Ϫ0.113Ϯ0.044Ϯ0.055 Ϫ0.045Ϯ0.018Ϯ0.009 Ϫ0.042Ϯ0.018Ϯ0.009 0.30-0.40 0.341 4.4 ϩ0.050Ϯ0.083Ϯ0.107 0.011Ϯ0.019Ϯ0.005 ϩ0.010Ϯ0.020Ϯ0.006 0.40-0.60 0.466 5.5 ϩ0.006Ϯ0.159Ϯ0.108 0.000Ϯ0.016Ϯ0.003 ϩ0.000Ϯ0.020Ϯ0.003 6642 P. L. ANTHONY et al. 54

n n TABLE X. Results on A2 and g2 . Note that the systematic uncertainties are small compared to the statistical uncertainties.

2 n n x range ͗x͗͘Q͘A2Ϯ␦(stat)Ϯ␦(syst) g2Ϯ␦(stat)Ϯ␦(syst) ͑GeV/c)2

0.03-0.04 0.036 1.1 Ϫ0.225Ϯ0.270Ϯ0.068 Ϫ10.68Ϯ12.22Ϯ3.17 0.04-0.06 0.050 1.2 0.084Ϯ0.191Ϯ0.062 2.44Ϯ4.92Ϯ1.63 0.06-0.10 0.081 1.8 Ϫ0.058Ϯ0.163Ϯ0.073 Ϫ0.74Ϯ1.96Ϯ0.87 0.10-0.15 0.124 2.5 Ϫ0.074Ϯ0.181Ϯ0.087 Ϫ0.29Ϯ1.07Ϯ0.49 0.15-0.20 0.174 3.1 Ϫ0.365Ϯ0.234Ϯ0.119 Ϫ0.88Ϯ0.74Ϯ0.34 0.20-0.30 0.245 3.8 Ϫ0.147Ϯ0.261Ϯ0.128 Ϫ0.11Ϯ0.39Ϯ0.18 0.30-0.40 0.341 4.4 0.365Ϯ0.480Ϯ0.171 0.16Ϯ0.28Ϯ0.09 0.40-0.60 0.466 5.5 Ϫ0.975Ϯ0.885Ϯ0.218 Ϫ0.22Ϯ0.20Ϯ0.05

The procedure is applied for each helicity case separately ever, they do not include vertex and vacuum polarization ͑similar to the unpolarized case͒ and then the asymmetry terms which are considered as non-physical background con- 3He tributions. A1 is formed from the result of each helicity case: The systematic errors in the radiative corrections are esti- mated by changing the input model for the asymmetries and 1 ␴B Ϫ␴B 3He ↓↑ ↑↑ unpolarized cross sections in the unmeasured kinematics de- A1Bornϭ B B D ͩ␴ ϩ␴ ͪ fined by the ‘‘canonical’’ triangle in POLRAD. The unmea- ↓↑ ↑↑ m m sured region, for example, includes the resonance region at 1 ␴ Ϫ␴ 2 , , ext int low Q . In order to estimate the sensitivity of the corrections ϵ ↓ ↑ ↑ ↑ ϩ⌬ ϩ⌬ . ͑57͒ D ͩ␴m ϩ␴m ͪ RC RC to these models, we tested our results assuming a flat asym- , , ↓ ↑ ↑ ↑ 3He metry input for our raw A1 data and compare the results to The statistical uncertainty of the radiative corrections at a quadratic fit input to the same data with weak constraints at each measured kinematics point follows from the statistical the low and high x regions. From the variation of the results, uncertainty of the measured rate at that point and the as- we estimate a 25% relative uncertainty in the internal and sumption of exact knowledge of the radiative background. external radiative corrections for the 7° spectrometer data Using Eq. ͑54͒ we can generalize this expression to the full and 25% for the internal corrections to the 4.5° spectrometer radiative corrections where internal and external radiative ef- data. The external corrections to the 4.5° were particularly fects are convoluted. In order to evaluate the total statistical sensitive to the cross section shape at high x. This correction error for each corrected kinematics point during the correc- is assigned a 35% relative uncertainty. In Table VI we present the radiative corrections to the tion procedure, a table of fractions f i of absolute ‘‘back- ground’’ contributions to the total cross sections due to the data as well as the fractions necessary to evaluate the radiative elastic, quasielastic and deep inelastic tails was changes in statistical uncertainty on the results, while Fig. 21 stored. These fractions include photon tails from the internal shows the effect of the internal and external electromagnetic 3He corrections as well as those of external contributions. How- radiative corrections on the measured asymmetry A1

n TABLE XI. Table of systematic uncertainties on A1 for each x point.

Parameter xϭ0.035 xϭ0.05 xϭ0.08 xϭ0.125 xϭ0.175 xϭ0.25 xϭ0.35 xϭ0.47

Pb 0.0027 0.0024 0.0032 0.0048 0.0051 0.0036 0.0008 0.0023 Pt 0.0063 0.0054 0.0072 0.0110 0.0117 0.0083 0.0018 0.0052 f 0.0071 0.0061 0.0081 0.0124 0.0131 0.0093 0.0020 0.0059

⌬dt 0.0010 0.0011 0.0012 0.0012 0.0013 0.0015 0.0019 0.0013 ⌬RC 0.0026 0.0039 0.0055 0.0068 0.0079 0.0082 0.0068 0.0022 R 0.0038 0.0034 0.0038 0.0051 0.0055 0.0035 0.0006 0.0021

F2 0.0061 0.0054 0.0070 0.0063 0.0072 0.0051 0.0073 0.0090 ␳n 0.0021 0.0019 0.0025 0.0037 0.0039 0.0026 0.0012 0.0001 p A1 0.0021 0.0018 0.0017 0.0019 0.0024 0.0035 0.0058 0.0116 ␳ p 0.0010 0.0013 0.0019 0.0027 0.0036 0.0051 0.0076 0.0131 AЌ 0.0082 0.0093 0.0123 0.0191 0.0335 0.0518 0.1064 0.1053 Ϫ A␲ 0.0079 0.0037 0.0024 0.0000 0.0000 0.0000 0.0000 0.0000 ϩ Ae 0.0123 0.0044 0.0030 0.0033 0.0000 0.0000 0.0000 0.0000 3He corr. 0.0046 0.0041 0.0054 0.0081 0.0085 0.0057 0.0025 0.0003 Total 0.0216 0.0165 0.0209 0.0295 0.0413 0.0551 0.1074 0.1075 54 DEEP INELASTIC SCATTERING OF POLARIZED . . . 6643

n TABLE XII. Table of systematic uncertainties on g1 for each x point.

Parameter xϭ0.035 xϭ0.05 xϭ0.08 xϭ0.125 xϭ0.175 xϭ0.25 xϭ0.35 xϭ0.47

Pb 0.0090 0.0055 0.0044 0.0042 0.0030 0.0014 0.0001 0.0002 Pt 0.0207 0.0125 0.0099 0.0097 0.0069 0.0031 0.0003 0.0005 f 0.0233 0.0141 0.0112 0.0109 0.0077 0.0035 0.0003 0.0006

⌬dt 0.0034 0.0027 0.0017 0.0011 0.0008 0.0006 0.0004 0.0002 ⌬RC 0.0089 0.0093 0.0078 0.0061 0.0047 0.0031 0.0014 0.0002 R 0.0200 0.0130 0.0102 0.0077 0.0054 0.0024 0.0014 0.0009

F2 0.0131 0.0085 0.0047 0.0039 0.0028 0.0013 0.0000 0.0002 ␳n 0.0071 0.0045 0.0035 0.0033 0.0023 0.0010 0.0002 0.0000 p A1 0.0061 0.0039 0.0024 0.0018 0.0015 0.0014 0.0012 0.0011 ␳ p 0.0028 0.0028 0.0026 0.0025 0.0022 0.0020 0.0016 0.0012 AЌ 0.0310 0.0165 0.0096 0.0075 0.0068 0.0050 0.0047 0.0023 Ϫ A␲ 0.0268 0.0087 0.0035 0.0000 0.0000 0.0000 0.0000 0.0000 ϩ Ae 0.0413 0.0104 0.0045 0.0029 0.0000 0.0000 0.0000 0.0000 3He corr. 0.0155 0.0098 0.0076 0.0072 0.0050 0.0021 0.0005 0.0000 Total 0.0737 0.0363 0.0255 0.0218 0.0161 0.0088 0.0055 0.0031 needed to obtain the Born asymmetry. One sees that the cor- yield a symmetric wave function, implying that the two pro- rections are quite small ͑typically shifting the data by 1/3 the tons are paired antisymmetrically in a spin singlet state. In size of the statistical error͒. this picture, the two proton spins line up anti-parallel to one another, resulting in a cancellation of spin dependent effects 3 VI. EXPERIMENTAL RESULTS coming from the protons. Naturally, the He nucleus is not exactly a system of nucleons in a spatially pure S state, and 3 A. Measured He results corrections due to the other states must be implemented in 3He 3He 3He 3He order to extract the result for a pure neutron. Fairly extensive Results on A1 , A2 , g1 , and g2 versus x are given 3 3He work on the He wave functions has been performed and in Tables VII and VIII. The asymmetry A2 is determined using Eq. ͑39͒. Within our limited statistical precison, the 3He values of A2 are consistent with zero. Experiment E143 d p n ͓83͔ measured A2 and A2 and found that the difference A2 is consistent with zero with small uncertainties. For the rest of 3He 3He the analysis, we extract A1 and g1 using Eq. ͑39͒ and Eq. 3He ͑40͒ with A2 set to be identically zero with a systematic uncertainty equivalent to our measured statistical uncertain- ties shown in Table VIII or the positivity constraint n A2ϽͱR, whichever is smaller. The uncertainty provided by the measurement of AЌ was smaller than the ͱR limit except for the results in the two x bins, xϭ0.35 and xϭ0.47. Within 2 3He our precision there is no obvious Q dependence of A1 at 2 3He fixed x. The Q averaged A1 are given in Table VII. From 3He the asymmetry results of A1 , the spin structure function 3He 3He g1 is obtained assuming A2 ϭ0, namely

3He 2 3 F2 ͑x,Q ͒ 3 g He͑x,Q2͒ϭ A He͑x,Q2͒. ͑58͒ 1 2x͓1ϩR͑x,Q2͔͒ 1

B. Extracting the neutron result from 3He A polarized 3He target can be used to extract information on polarized neutrons. The main reason is that in the naive approximation the 3He nucleus is considered to be a system of three nucleons in a spatially symmetric S state. The Pauli n 2 FIG. 22. The asymmetry A1 is plotted vs Q for five different n principle constrains the overall wave function to be antisym- values of x. The results are consistent with A1 being independent of metric, and therefore the spin-isospin wave function must Q2. The data comes from the two spectrometer arms and three then be antisymmetric. Exchanging the two protons must beam energies. 6644 P. L. ANTHONY et al. 54

n 2 TABLE XIII. Results on A1 vs Q . Systematic uncertainties are small and have been neglected. See Table IX for other values of x.

2 n x range ͗x͗͘Q͘A1Ϯ␦(stat) ͑GeV/c)2

0.06-0.10 0.082 1.3 Ϫ0.13Ϯ0.08 1.6 Ϫ0.11Ϯ0.05 1.9 Ϫ0.18Ϯ0.07 2.0 0.01Ϯ0.10 2.5 Ϫ0.08Ϯ0.10 2.9 Ϫ0.07Ϯ0.13 0.10-0.15 0.124 1.5 Ϫ0.24Ϯ0.10 2.0 Ϫ0.04Ϯ0.07 2.4 Ϫ0.12Ϯ0.10 2.5 Ϫ0.16Ϯ0.09 3.1 Ϫ0.24Ϯ0.07 3.6 Ϫ0.20Ϯ0.09 0.15-0.20 0.175 1.7 0.01Ϯ0.15 2.2 Ϫ0.05Ϯ0.09 2.7 Ϫ0.14Ϯ0.15 2.9 Ϫ0.22Ϯ0.10 3.7 Ϫ0.23Ϯ0.07 4.4 Ϫ0.24Ϯ0.10 0.20-0.30 0.246 1.8 0.05Ϯ0.17 2.4 Ϫ0.05Ϯ0.10 3.0 Ϫ0.34Ϯ0.17 3.4 Ϫ0.13Ϯ0.12 4.3 Ϫ0.16Ϯ0.07 5.2 Ϫ0.02Ϯ0.11 0.30-0.40 0.343 2.0 0.30Ϯ0.34 2.6 0.00Ϯ0.18 3.3 Ϫ0.25Ϯ0.38 3.8 Ϫ0.12Ϯ0.24 5.0 Ϫ0.05Ϯ0.13 6.1 0.47Ϯ0.20

these wave functions are used to estimate magnetic moments 3He p F2 1 3 F2 and to extract the degree of polarization of the neutron in An ϭ A HeϪ2 ␳ A p , ͑60͒ 1,2 n 1,2 3He p 1,2 3 F2 ␳n ͩ F ͪ He ͓84͔. Furthermore, the determination of the neutron spin 2 structure function from a measurement on 3He relies on an n p 3He understanding of the reaction mechanism for the virtual pho- where g1,2 , g1,2 , and g1,2 are the spin structure functions of ton absorption combined with the use of a realistic 3He wave an effective free neutron, a free proton, and 3He, respec- 3 n p 3He function. Detailed investigations of the He inelastic spin tively. Similarly A1,2 , A1,2 , and A1,2 are the photon-target response functions versus that of a free neutron have been asymmetries for an effective free neutron, a free proton, and 3 carried out by three groups ͓85–87͔. They examined the ef- He, respectively. The studies yield ␳nϭ(87Ϯ2)% and fect of the Fermi motion of nucleons and their binding in ␳ ϭ(Ϫ2.7Ϯ0.4)% for the polarizations of the neutron and 3 p He along with the study of the electromagnetic vertex using proton in 3He due to the S, SЈ and D states of the wave 3 the most realistic He wave function. Consistent findings function ͓84,85͔. The calculations using the ‘‘exact’’ 3He have been reached among these groups, and we summarize wave function including the full treatment of Fermi motion here those relevant to our experiment. and binding effects show negligible differences with the In the deep inelastic region a neutron spin structure re- above approximation in the deep inelastic region. A precise 3 sponse and asymmetry can be extracted from that of He proton measurement is important to minimize the error on using a procedure in which S, SЈ, and D states of the 3He the correction. We point out that our measurements have a wave function are included, but no Fermi motion or binding lower limit in missing mass W2 of 4 GeV 2, already beyond effects are introduced: the quasielastic and resonance region which were found to be more sensitive to nuclear effects ͓87͔. 1 3 gn ϭ ͑g HeϪ2␳ g p ͒, ͑59͒ In this analysis we have used Eqs. 59 and 60 to extract the 1,2 ␳ 1,2 p 1,2 n n neutron asymmetry A1 and spin-dependent structure function 54 DEEP INELASTIC SCATTERING OF POLARIZED . . . 6645

0.2 2

0.1 1 +.JR 0

A~ -0.1 I t t I l -1 -0.2 t

-2 -o.~0-2 0.05 0.5

X X

n n FIG. 25. A2 vs x averaged over both spectrometers is shown. FIG. 23. The neutron asymmetry A1 vs x. The error bars are The error bars are statistical only ͑systematic errors are small in statistical, with the enclosed region at the bottom representing the comparison͒. size of the systematic errors. detailed contribution of every correction parameter to the n p n n g1 , where the proton asymmetry A1 and spin-dependent overall systematic uncertainty on A1 and g1 at each x point. p n structure function g1 used are those measured in experiment In order to extract the values of g1 at one unique value of 2 n 2 E143 ͓88͔. The uncertainties in the measured proton results Q , we assumed that the asymmetry A1 is Q independent n 2 are taken into account in extracting the results on g1 . The and used A2ϭ0, consistent with the study of the Q depen- relative impact on the overall error bars is small for all x dence of our data. From the two spectrometers and the three n bins. beam energies used in this experiment, we extracted A1 at No further corrections due to possible final-state effects six different values of Q2. Over this modest range of Q2 and n have been incorporated. Nevertheless, placing limits on pos- within the statistical errors, we find that A1 is consistent with sible contaminations from final state nuclear effects has been being independent of Q2 as seen in Fig. 22 and enumerated one of the significant motivations for measuring the neutron in Table XIII. This trend is confirmed by the recent precision spin structure function with different nuclear targets ͑i.e., E143 results ͓89͔ on the proton and neutron in the equivalent polarized and 3He͒. Q2 range. Figures 23 and 24 show the compiled results on Results on An , gn , An, and gn are presented in Tables IX n n 1 1 2 2 A1(x) and xg1(x). Figure 25 presents the results for and X. In Table IX the values of gn are also given at constant n 1 A2(x). 2 2 Q0 ϭ 2.0 ͑GeV/c) . Table XI and Table XII present the VII. NEUTRON FIRST MOMENT AND PHYSICS IMPLICATIONS 0.02 Integrating gn(x,Q2) over the measured range of x at a xg1 n(x) 1 0 fixed value of Q2 ϭ 2 ͑GeV/c)2, one obtains 0.01 0 • E142 Q 2= 2 (GeV/c) 2 0.02 0.00 2 xg1 n(x) • E142 a2 = 2 (GeV/c)2 0.01 □ E143 a2 = 3 (GeV/c) -0.01 t t t 0 -0.02 I t t -0.01 -0.03 ! !H j -0.02 I 1H ~~~\"@ l -0.04 -0.03 0.01 0.1 X -0.04 0.01 0.1 n 2 X FIG. 24. The spin structure function xg1 evaluated at fixed Q0 ϭ 2 ͑GeV/c)2. The error bars are statistical while the band at the n bottom represents the size of the systematic uncertainties (1␴). FIG. 26. Comparison of xg1 between E142 and E143. 6646 P. L. ANTHONY et al. 54

n n TABLE XIV. Systematic uncertainties on ⌫1 . The total uncertainty on ⌫1 coming from systematics is 0.0060.

p 3 Pb Pt f ⌬RC RF2 A1 ␳p AЌ He corr. 0.0012 0.0021 0.0024 0.0016 0.0021 0.0010 0.0009 0.0010 0.0032 0.0015

8 0.6 Here we assumed a Regge behavior up to x0ϭ0.1 and gn͑x,Q2͒dxϭ gn͑x ,Q2͒⌬x n ͵ 1 0 ͚ 1 i 0 i used a weighted fit of g1 to the lowest three x bins to reduce 0.03 iϭ1 the statistical uncertainties. The low x extrapolation yields 2 2 ϭϪ0.0284Ϯ0.0061͑stat͒Ϯ0.0059͑syst͒ the result at Q ϭ2 ͑GeV/c) ,

͑61͒ 0.03 n ͵ g1͑x͒dxϭϪ0.0053Ϯ0.0053 ͑64͒ n 2 0 where ⌬xi are the bin widths and the g1(xi ,Q0) are evalu- ated using Eq. ͑59͒. The first error is statistical and the sec- in which we assign a 100% uncertainty to the extrapolation. ond is systematic. The total systematic error is evaluated by n If one uses an alternative low x behavior of g1 ͓94͔, namely adding all contributions in quadrature, assuming they come n g1Ϸaln(1/x), and perform the extrapolation using only the from uncorrelated sources. Table XIV lists the dominant 0.03 n contributions to the total systematic error of the integral in lowest x bin data point, one obtains ͐0 g1(x)dx Eq. ͑61͒. ϭϪ0.012. The assigned error encompasses this result, sta- Since the data are only measured over part of the interval tistical errors on the low x points and the result obtained 0ϽxϽ1, we must extrapolate in order to evaluate the inte- using the other boundary of the Regge intercept ␣ϭϪ0.5. n n 2 2 1 n 2 Our fit of g1 is also consistent with the low x results from the gral ⌫1͓Q ϭ2 ͑GeV/c) ]ϭ͐0g1(x,Q0)dx over the full x range. There are two regions to consider, large x Spin Muon Collaboration ͑SMC͓͒95,96,93͔ over the mea- sured range within the statistical uncertainties. (0.6Ͻxр1) and small x (0рxϽ0.03). For large x, within a 2 2 three constituent quark description of the nucleon, the as- The total neutron integral at Q ϭ2 ͑GeV/c) becomes the sumption of single flavor dominance at high x and Q2 leads sum of the three integrals Eqs. ͑61͒, ͑62͒, and ͑64͒, to the prediction that A 1asx 1. This phenomenologi- 1 1 cal result has also been→ derived→ from arguments based on n n ⌫1ϭ g1͑x͒dxϭϪ0.031Ϯ0.006͑stat͒Ϯ0.009͑syst͒. perturbative QCD and a nonperturbative wave function de- ͵0 scribing the nucleon ͓13,14͔. ͑65͒ In order to evaluate this integral we assumed that for n n xϾ0.6, A1ϭ0.5Ϯ0.5 and used the F2 results from SLAC A comparison of the E142 data with those of E143 ͓97͔ ͓71͔ rather than NMC ͓73͔ since it is based on data closer in and SMC ͓98͔ shows no significant disagreement, though kinematic range to the present experiment. The error as- there is some interesting behavior. Figure 26 presents a n signed to A1 was chosen to cover all possible behaviors of comparison of xg 1 versus x for E142 and E143. The data n n from both experiments are in reasonable agreement over the A1 in this region including that of g1 suggested by the quark n 3 2 measured range. The integrals over x of gn are counting rules ͓90͔ g1(x)ϵ(1Ϫx) . We find for Q ϭ2 1 2 n 2 GeV , ⌫1ϭϪ0.031Ϯ0.006͑stat͒Ϯ0.009͑syst͒ for E142 at Q ϭ 2 2 n ͑GeV/c) and ⌫1ϭϪ0.037Ϯ0.008͑stat͒ Ϯ0.011͑syst͒ for 2 2 1 E143 Q ϭ 3 ͑GeV/c) . Figure 27 compares the spin struc- gn x dxϭϩ0.003Ϯ0.003. 62 n 2 1͑ ͒ ͑ ͒ ture function xg1 extracted from SMC at Q ϭ 10 ͵0.6 ͑GeV/c)2 and from E142. When the SMC and E142 neutron results are combined, the shape of the structure function is For small x one must rely heavily upon theory, especially n interesting, with small negative results for g1 over the range noting that if A Þ0asx 0, ͐g dx ϱ. We assume the 1 → 1 → in x covered by E142 followed by relatively large negative Regge theory prediction for the behavior of the nucleon g1, values at low x measured by SMC just below the kinematic namely g (x 0)ϰxϪ␣1 where the Regge intercept ␣ can 1 → 1 range accessible to E142. The behavior is a strong motivator vary in the range Ϫ0.5Ͻ␣1Ͻ0 ͓91,92͔, although there are n for future measurements at low x. The integrals of g1 over no strong theoretical grounds for these limits. In this region the mid x range common to E142 and SMC differ, however, dominated by the sea and contributions to the nucleon by approximately two standard deviations. We extract structure, it is thought that no difference should exist be- ͐gn(x)dxϭϪ0.027Ϯ0.004͑stat͒Ϯ0.006͑syst͒ from the tween the proton and neutron behavior. Therefore, we used 1 E142 data over the x range from 0.04 to 0.3, where the the same value of ␣ as the previous proton spin structure 1 statistical error bars are relatively small. We compare this function experiments ͓88,96͔, ␣ ϭ0. The low x contribution 1 result to the SMC result over the same range, to the spin structure function integral becomes n ͐g1(x)dxϭϩ0.007Ϯ0.015͑stat͒. Systematic errors are ne- glected from the SMC data, since they are expected to be x 0 n n small compared to the statistical uncertainty. We do not as- g1͑x͒dxϭx0g1͑x0͒. ͑63͒ ͵0 sign any special significance to the difference but point out 54 DEEP INELASTIC SCATTERING OF POLARIZED . . . 6647

0.06 0.3 -~-----,---,---,-----,----,--,--r----,-----,

• E142 a2=2 (GeV/c)2 3rd order PQCD No corrections 0.04 □ SMC 0 2=2 (GeV/c)2 ~~r~~t~~s______\ ___ _ 0.2 0.02

:E l C: ~ 0 0.1 ~ □ E143 (p, d) t::,SMC(p,d) ~ t ! t • E142 (He), E143 (p) .._EMC {p), EMC (d) -0.02 f t 0 L__j_ _ __L__.J..._-----1._ _ _J__.i...___.__~_~~ 2 4 6 8 10 -0.04 a2 (GeV/c)2

-0.06 p n FIG. 28. Comparison of data for ⌫1Ϫ⌫1 using different experi- 0.001 0.01 0.1 1 ments compared to the Bjorken sum rule with 3rd order QCD cor- X rections and no higher twist corrections. The E142 results are at Q2ϭ3(GeV/c)2 but have been shifted for clarity. n FIG. 27. Comparison of xg1 between E142 and SMC. CHTϭϪ0.09Ϯ0.06 ͓22͔ in one case or that it should not be ignored and needs to be monitored in CHTϭϪ0.015Ϯ0.02 ͓100͔ in another. The sensitivity of the 2 future measurements where the Q of the measured data is result can be described by the change in sign in CHT found investigated. when the estimate is made using a bag model ͓101͔. There- In order to test the Bjorken sum rule, the proton and the fore, an additional theoretical uncertainty equal in magnitude p n neutron first moments ⌫1 and ⌫1 are evaluated at the same to the present size of the higher twist correction should be 2 Q . The available experimental proton data span a different included in the theoretical estimate of ␣s . range of Q2, making it necessary to evolve the proton or the We use the Bjorken sum rule with perturbative QCD cor- 2 neutron data to a common value of Q . rections up to third order in ␣s without higher twist term 2 2 Since our neutron results and the E143 proton results ͓88͔ corrections to extract a value of ␣s at Q ϭ3 ͑GeV/c) for are at similar Q2, we combine the two to test the Bjorken polarized deep inelastic scattering, p sum rule. The E143 proton results reads ⌫1ϭ0.127Ϯ0.011 2 2 2 2 ϩ0.070 at Q ϭ 3 ͑GeV/c) , while evolving the E142 neutron result ␣s͓Q ϭ3 ͑GeV/c͒ ͔ϭ0.408Ϫ0.085 . ͑67͒ 2 2 n to Q ϭ 3 ͑GeV/c) ,wefind⌫1ϭϪ0.033Ϯ0.011. These Bj B j p n If we consider the higher twist corrections to ⌫ and choose results lead to the Bjorken integral ⌫expϭ⌫1Ϫ⌫1 an average value and error of CHT from the QCD sum rules, ϭ0.160Ϯ0.015 where correlations between the two experi- that is C ϭϪ0.10Ϯ0.05, we find ments, primarily from the beam polarization determination, HT have been taken into account. ␣ ͓Q2ϭ3 ͑GeV/c͒2͔ϭ0.312ϩ0.098 . ͑68͒ There is agreement with the Bjorken sum rule prediction s Ϫ0.130 B j ⌫th ϭ0.176Ϯ0.008 using Eq. ͑11͒ from Sec. II A. Assuming Both results ͑with or without a higher twist correction͒ for 2 2 three flavors and choosing ␣s͓Q ϭ3 (GeV/c) ͔ ␣s are in agreement with the world average ͓99͔. ϭ0.32Ϯ0.05 ͓99͔. Figure 28 shows tests of the Bjorken sum For the Ellis-Jaffe sum rule test, we compare at the rule from different experiments. The present determination is 2 2 2 average Q of the experiment, namely Q0ϭ2 ͑GeV/c) . the most accurate test of the Bjorken sum rule to date. From Eq. ͑12͒, and using ␣sϭ0.35Ϯ0.05 and We can rewrite the Bjorken sum rule including the higher 3FϪDϭ0.58Ϯ0.12, we obtain the theoretical value of twist contributions and extract a value for ␣ . n s ⌫1ϭϪ0.016Ϯ0.016 where the error on the result is domi- nated by the error on the quantity 3FϪD. We see that the 2 2 2 2 3 ␣s͑Q ͒ ␣s͑Q ͒ ␣s͑Q ͒ Ellis-Jaffe sum rule is one standard deviation away from the ⌫Bjϭ1 g 1Ϫ Ϫ3.58 Ϫ20.2 6 Aͫ ␲ ͩ ␲ ͪ ͩ ␲ ͪ ͬ experimental result, and the experimental result is consistent with the results of E143 and SMC. 1 CHT ϩ ͑66͒ Higher order perturbative QCD corrections to this sum 6 Q2 rule have had a significant impact on the interpretation of the experimental results. For the Q2 at which the SLAC experi- where CHT is the higher twist contribution to the Bjorken ment E142 is performed, these corrections are quite large. At sum rule. Recent estimates of CHT show that it is very model this time, these corrections have been given up to third order 2 dependent. For example, using QCD sum rules methods sev- in the expansion of ␣s(Q ). For example, at the average 2 2 eral authors have evaluated CHT and found it to be Q of the E142 experiment ͓2 ͑GeV/c) ͔, the corrections 6648 P. L. ANTHONY et al. 54

TABLE XV. Total quark flavor contributions to the nucleon’s spin using a conservative uncertainty on F/D as described in the text.

⌬u ⌬d ⌬s ⌬⌺ Q2ϭ 2 ͑GeV/c)2

0.87Ϯ0.04 Ϫ0.39Ϯ0.04 Ϫ0.05Ϯ0.06 0.43Ϯ0.12

⌬u ⌬d ⌬s ⌬⌺inv Invariant quantities 0.86Ϯ0.04 Ϫ0.40Ϯ0.04 Ϫ0.06Ϯ0.06 0.39Ϯ0.11 change the Ellis-Jaffe sum rule prediction for the neutron results are in agreement with an extraction of the neutron from Ϫ0.020 ͑without corrections͒ to Ϫ0.011 ͑with correc- spin structure function from the deuteron as performed by 2 tions͒, assuming a value of ␣sϭ0.35. SLAC experiment E143. We see no dependence on Q n Using ⌫1 experimental results and the same values for within the limited precision of the data sample. In addition, n gA and 3FϪD as earlier, one can extract a value for the total we present results on A2(x) which are compatible with zero quark spin contribution to the nucleon using the E142 re- and significantly better than the unitarity limit given by ͱR sults, ⌬⌺ ͓2 ͑GeV/c)2͔ϭ0.43 Ϯ 0.12 and similarly, we can n in the range from x of 0.03 to x of 0.3. The A2(x) results, extract the fraction of polarized strange sea contribution ⌬s however, are less precise than what one could extract from 2 ϭϪ Ϯ ϭ Ϯ ͑2 GeV/c) 0.05 0.06. We also find ⌬⌺inv 0.39 the E143 proton and deuteron data. 0.11 with the corresponding fraction of polarized strange sea From our measurement we proceed to extract the first contribution ⌬sinv ϭϪ0.06Ϯ0.06. Table XV gives the total n n moment of g1 , namely ͐g1(x)dx. Since our asymmetries quark flavors contributions to the nucleon’s spin using Eq. n ͑12͒ in one case and Eq. ͑15͒ in the other assuming a con- over the measured region are small, ͐g1(x)dx is small. We servative uncertainty on F/D. The uncertainty on the deter- proceed to use the results for the neutron integral to extract mination of ⌬s is still large even when we take a more the quark flavor distributions, ⌬u, ⌬d, ⌬s, and ⌬⌺ with n optimistic uncertainty on the value of F/D. For F/D ϭ some caveats. In our extraction of the g1 integral, we assume 0.576Ϯ0.059 as quoted by Close and Roberts ͓32͔, and using that one can do a Regge theory extrapolation of the contri- Eq. ͑15͒ we find ⌬sϭϪ0.06Ϯ0.04 while the uncertainty on bution to the integral between x of 0 and the lowest values of the other quark flavors contributions remains the same ͑see x measured in the experiment. The implications of this fit are Table XVI͒. that the integral contribution at low x is itself small. On the Including the higher order perturbative QCD corrections, other hand, recent data from the SMC collaboration appears this result is in agreement with the extraction of ⌬⌺ from the to indicate that there may be a large negative contribution to E143 (⌬⌺invϭ0.30Ϯ0.06) and SMC (⌬⌺invϭ0.20Ϯ0.11) the neutron integral in the x range below where we measure. experiments ͓97,98͔. Care must be taken when comparing If this is true, then the assumption of Regge behavior up to these numbers since different authors make different as- x of approximately 0.1 underestimates the neutron contribu- sumption in extracting ⌬⌺ from their data. In addition, the tion at low x. A major motivator for future measurements of good agreement does depend on the validity of the perturba- spin structure functions at either higher energies or with tive QCD corrections in the low Q2 region and the estimated higher precision comes from studying the spin structure small size of the higher twist corrections. functions at lower x. With the Regge theory assumption, we extract a value of ⌬⌺, the total quark contribution to the VIII. CONCLUSIONS nucleon’s spin of approximately 40%. We note that this re- sult has a sensitive dependence on higher order perturbative We report final results on the first determination of the and nonperturbative QCD corrections. In addition, the result neutron spin structure function using a polarized 3He depends on the scale at which ⌬⌺ is evaluated and the num- nucleus. Over the kinematic range accessible to the experi- ber of quark flavors used in the evaluation, typically three or ment, we find small negative asymmetries similar to the pre- four. We can tune for different values of ⌬⌺ ranging from dictions from the quark parton model ͓16,17͔. Within the ⌬⌺ of 0.36 to ⌬⌺ of 0.43 with different theoretical assump- statistics of the experiment, we are not able to distinguish tions. any clear shape as a function of x to the neutron spin asym- We combine the proton results from experiment E143 metries, although significant deviations are expected, par- with the neutron results from this experiment to test the ticularly at low and high x. For example, as x approaches Bjorken sum rule. Ignoring the unlikely possibility for large n unity, the neutron asymmetry A1 is predicted to approach nonsinglet contributions to the proton and neutron integrals n unity. As x approaches zero, A1 should approach zero. The at low x, this comparison still stands as the most precise test

TABLE XVI. Total quark flavor contributions to the nucleon’s spin using an uncertainty as quoted by Close and Roberts ͑see text͒.

⌬u ⌬d ⌬s ⌬⌺inv Invariant quantities 0.86Ϯ0.04 Ϫ0.40Ϯ0.04 Ϫ0.06Ϯ0.04 0.39Ϯ0.11 54 DEEP INELASTIC SCATTERING OF POLARIZED . . . 6649 of the Bjorken sum rule to date. We find that the sum rule is with higher precision and investigations with an increasing satisfied at the 10% level. attention to detail. Future measurements of the proton and neutron spin structure functions will increase the kinematic coverage, par- ACKNOWLEDGMENTS ticularly at low x. SLAC experiments E154 ͓102͔ and E155 We are indebted to G. S. Bicknell, G. Bradford, R. Boyce, ͓103͔ will extract the proton and neutron spin structure func- B. Brau, J. Davis, S. Dyer, R. Eisele, C. Fertig, J. Hrica, C. tions using a higher energy 50 GeV polarized electron beam. Hudspeth, M. Jimenez, G. Jones, J. Mark, J. McDonald, W. These two experiments aim to measure the spin structure Nichols, M. Racine, B. Smith, M. Sousa, R. Vogelsang, and 2 functions at a higher average Q and will extract data at J. White for their exceptional efforts in the prepartion of this lower values of x with higher statistical precision. Additional experiment, to the SLAC Accelerator Operations Group for measurements from the CERN SMC program will continue delivering the polarized beam, and to the technical staff of to increase the statistical precision, needed to draw decisive CE-Saclay and LPC-Clermont. We also would like to thank conclusions at low x. A collider experiment like HERA with N.M. Shumeiko for providing us with the latest version of a polarized electron and a polarized proton beam would, in POLRAD to perform the electromagnetic internal radiative principle, be ideal for reaching very low x (Ϸ 10 Ϫ4)to corrections. This work was supported by U.S. Department of extract the proton spin structure function. In addition, preci- Energy Contracts No. DE-AC03-76SF00098 ͑LBL͒, No. sion measurements of the proton and neutron spin structure W-4705-Eng-48 ͑LLNL͒, No. DE-FG02-90ER40557 functions at high x (xϾ0.5) are useful for testing Quark ͑Princeton͒, No. DE-AC03-76SF00515 ͑SLAC͒, No. DE- Parton Model predictions. The rising behavior of the neutron FG03-88ER40439 ͑Stanford͒, No. DE-FG02-84ER40146 n asymmetry A1 at high x still needs to be confirmed. Experi- ͑Syracuse͒, No. DE-FG02-94ER40844 ͑Temple͒, No. DE- ments at HERMES ͓104͔ and Thomas Jefferson National AC02-76ER00881 ͑Wisconsin͒, National Science Founda- Acceleration Facility ͑TJNAF͓͒105͔ are likely to be the best tion Grants No. 9014406, No. 9114958 ͑American͒, No. grounds for these tests along with future precision measure- 8914353, No. 9200621 ͑Michigan͒; and the Bundesministe- ments of g2. rium fuer Forschung und Technologie ͑W.M.͒, the Centre We conclude by noting that this first measurement of the National de la Recherche Scientifique ͑LPC-Clermont- neutron spin structure function does not complete the study, Ferrand͒, and the Commissariat a` l’Energie Atomique but instead has helped pave the way for future measurements ͑Saclay͒.

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