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Precision Measurement of the Neutron Spin Asymmetries and Spin-dependent Structure Functionsin the Valence Quark Region

X. Zheng,13 K. Aniol,3 D. S. Armstrong,22 T. D. Averett,8,22 W. Bertozzi,13 S. Binet,21 E. Burtin,17 E. Busato,16

C. Butuceanu,22 J. Calarco,14 A. Camsonne,1 G. D. Cates,21 Z. Chai,13 J.-P. Chen,8 Seonho Choi,20 E. Chudakov,8

F. Cusanno,7 R. De Leo,7 A. Deur,21 S. Dieterich,16 D. Dutta,13 J. M. Finn,22 S. Frullani,7 H. Gao,13 J. Gao,2

F. Garibaldi,7 S. Gilad,13 R. Gilman,8,16 J. Gomez,8 J.-O. Hansen,8 D. W. Higinbotham,13 W. Hinton,15 T. Horn,11

C.W. de Jager,8 X. Jiang,16 L. Kaufman,12 J. Kelly,11 W. Korsch,10 K. Kramer,22 J. LeRose,8 D. Lhuillier,17

N. Liyanage,8 D.J. Margaziotis,3 F. Marie,17 P. Markowitz,4 K. McCormick,9 Z.-E. Meziani,20 R. Michaels,8

B. Moffit,22 S. Nanda,8 D. Neyret,17 S. K. Phillips,22 A. Powell,22 T. Pussieux,17 B. Reitz,8 J. Roche,22 R. Roché,5

M. Roedelbronn,6 G. Ron,19 M. Rvachev,13 A. Saha,8 N. Savvinov,11 J. Singh,21 S. Širca,13 K. Slifer,20

P. Solvignon,20 P. Souder,18 D.J. Steiner,22 S. Strauch,16 V. Sulkosky,22 A. Tobias,21 G. Urciuoli,7 A. Vacheret,12

B. Wojtsekhowski,8 H. Xiang,13 Y. Xiao,13 F. Xiong,13 B. Zhang,13 L. Zhu,13 X. Zhu,22 P.A. Żołnierczuk,10

The Jefferson Lab Hall A Collaboration1Université Blaise Pascal Clermont-Ferrand et CNRS/IN2P3LPC 63, 177 Aubière Cedex, France

2California Institute of Technology, Pasadena, California91125, USA3California State University, Los Angeles, Los Angeles, California 90032, USA

4Florida International University, Miami, Florida 33199, USA5Florida State University, Tallahassee, Florida 32306, USA

6University of Illinois, Urbana, Illinois 61801, USA7Istituto Nazionale di Fisica Nucleare, Sezione Sanità, 00161 Roma, Italy

8Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606, USA9Kent State University, Kent, Ohio 44242, USA

10University of Kentucky, Lexington, Kentucky 40506, USA11University of Maryland, College Park, Maryland 20742, USA

12University of Massachusetts Amherst, Amherst, Massachusetts 01003, USA13Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

14University of New Hampshire, Durham, New Hampshire 03824, USA15Old Dominion University, Norfolk, Virginia 23529, USA

16Rutgers, The State University of New Jersey, Piscataway, New Jersy 08855, USA17CEA Saclay, DAPNIA/SPhN, F-91191 Gif sur Yvette, France

18Syracuse University, Syracuse, New York 13244, USA19University of Tel Aviv, Tel Aviv 69978, Israel

20Temple University, Philadelphia, Pennsylvania 19122, USA21University of Virginia, Charlottesville, Virginia 22904,USA

22College of William and Mary, Williamsburg, Virginia 23187,USA

We report on measurements of the neutron spin asymmetriesAn1,2 and polarized structure functionsgn1,2 at

three kinematics in the deep inelastic region, withx = 0.33, 0.47 and0.60 andQ2 = 2.7, 3.5 and4.8 (GeV/c)2,respectively. These measurements were performed using a5.7 GeV longitudinally-polarized electron beam anda polarized3He target. The results forAn1 andg

n1 atx = 0.33 are consistent with previous world data and, at

the two higherx points, have improved the precision of the world data by about an order of magnitude. ThenewAn1 data show a zero crossing aroundx = 0.47 and the value atx = 0.60 is significantly positive. Theseresults agree with a next-to-leading order QCD analysis of previous world data. The trend of data at highxagrees with constituent quark model predictions but disagrees with that from leading-order perturbative QCD(pQCD) assuming hadron helicity conservation. Results forAn2 andg

n2 have a precision comparable to the best

world data in this kinematic region. Combined with previousworld data, the momentdn2 was evaluated andthe new result has improved the precision of this quantity byabout a factor of two. When combined with theworld proton data, polarized quark distribution functionswere extracted from the newgn1 /F

n1 values based on

the quark parton model. While results for∆u/u agree well with predictions from various models, results for∆d/d disagree with the leading-order pQCD prediction when hadron helicity conservation is imposed.

PACS numbers: 13.60.Hb,24.85.+p,25.30.-c

I. INTRODUCTION

Interest in the spin structure of the nucleon became promi-nent in the 1980’s when experiments at CERN [1] and

SLAC [2] on the integral of the proton polarized structurefunctiongp1 showed that the total spin carried by quarks wasvery small,≈ (12±17)% [1]. This was in contrast to the sim-ple relativistic valence quark model prediction [3] in whichthe spin of the valence quarks carries approximately75% of

http://arxiv.org/abs/nucl-ex/0405006v5

2

the proton spin and the remaining25% comes from their or-bital angular momentum. Because the quark model is verysuccessful in describing static properties of hadrons, thefactthat the quark spins account for only a small part of the nu-cleon spin was a big surprise and generated very productiveexperimental and theoretical activities to the present. Currentunderstanding [4] of the nucleon spin is that the total spin isdistributed among valence quarks,qq̄ sea quarks, their orbitalangular momenta, and gluons. This is called the nucleon spinsum rule:

SNz = Sqz + L

qz + J

gz =

1

2, (1)

whereSNz is the nucleon spin,Sqz andL

qz represent respec-

tively the quark spin and orbital angular momentum (OAM),andJgz is the total angular momentum of the gluons. Onlyabout(20− 30)% of the nucleon spin is carried by the spin ofthe quarks. To further study the nucleon spin, one thus needsto know more precisely how it decomposes into the three com-ponents and to measure their dependence onx. Herex is theBjorken scaling variable, which in the quark-parton model [5]can be interpreted as the fraction of the nucleon momentumcarried by the quark. For a fixed target experiment one hasx = Q2/(2Mν), with M the nucleon mass,Q2 the four mo-mentum transfer squared andν the energy transfer from theincident electron to the target. However, due to experimen-tal limitations, precision data have been collected so far onlyin the low and moderatex regions. In these regions, one issensitive to contributions from a large amount ofqq̄ sea andgluons and the nucleon is hard to model. Moreover, at largedistances corresponding to the size of a nucleon, the theoryof the strong interaction – Quantum Chromodynamics (QCD)– is highly non-perturbative, which makes the investigationof the roles of quark orbital angular momentum (OAM) andgluons in the nucleon spin structure difficult.

Our focus here is the first precise neutron spin structure datain the largex regionx >∼ 0.4. For these kinematics, the va-lence quarks dominate and the ratios of structure functionscan be estimated based on our knowledge of the interactionsbetween quarks. More specifically, the virtual photon asym-metryA1, defined as

A1(x,Q2) ≡ σ1/2 − σ3/2

σ1/2 + σ3/2

(the definitions ofσ1/2,3/2 are given in Appendix A), whichat largeQ2 is approximately the ratio of the polarized andthe unpolarized structure functionsg1/F1, is expected toapproach unity asx → 1 in perturbative QCD (pQCD). Thisis a dramatic prediction, not only because this is the onlykinematic region where one can give an absolute predictionfor the structure functions based on pQCD, but also becauseall previous data on the neutron asymmetryAn1 in the regionx >∼ 0.4 have large uncertainties and are consistent withAn1 6 0. Furthermore, because bothqq̄ sea and gluoncontributions are small in this region, it is a relatively cleanregion to test the valence quark model and to study the roleof valence quarks and their OAM contribution to the nucleon

spin.

Deep inelastic scattering (DIS) has served as one of the ma-jor experimental tools to study the quark and gluon structureof the nucleon. The formalism of unpolarized and polarizedDIS is summarized in Appendix A. Within the quark par-ton model (QPM), the nucleon is viewed as a collection ofnon-interacting, point-like constituents, one of which carriesa fractionx of the nucleon’s longitudinal momentum and ab-sorbs the virtual photon [5]. The nucleon cross section is thenthe incoherent sum of the cross sections for elastic scatteringfrom individual charged point-like partons. Therefore theun-polarized and the polarized structure functionsF1 andg1 canbe related to the spin-averaged and spin-dependent quark dis-tributions as [6]

F1(x,Q2) =

1

2

∑

i

e2i qi(x,Q2) (2)

and

g1(x,Q2) =

1

2

∑

i

e2i∆qi(x,Q2) , (3)

whereqi(x,Q2) = q↑i (x,Q

2) + q↓i (x,Q2) is the unpolarized

parton distribution function (PDF) of theith quark, definedas the probability that theith quark inside a nucleon carriesa fractionx of the nucleon’s momentum, when probed with aresolution determined byQ2. The polarized PDF is definedas∆qi(x,Q2) = q

↑i (x,Q

2) − q↓i (x,Q2), whereq↑i (x,Q

2)

(q↓i (x,Q2)) is the probability to find the spin of theith quark

aligned parallel (anti-parallel) to the nucleon spin.The polarized structure functiong2(x,Q2) does not have a

simple interpretation within the QPM [6]. However, it can beseparated into leading twist and higher twist terms using theoperator expansion method [7]:

g2(x,Q2) = gWW2 (x,Q

2) + ḡ2(x,Q2) . (4)

Here gWW2 (x,Q2) is the leading twist (twist-2) contribu-

tion and can be calculated using the twist-2 component ofg1(x,Q

2) and the Wandzura-Wilczek relation [8] as

gWW2 (x,Q2) = −g1(x,Q2) +

∫ 1

x

g1(y,Q2)

ydy . (5)

The higher-twist contribution tog2 is given byḡ2. When ne-glecting quark mass effects, the higher-twist term representsinteractions beyond the QPM,e.g., quark-gluon and quark-quark correlations [9]. The moment ofḡ2 can be related to thematrix elementd2 [10]:

d2 =

∫ 1

0

dx x2[

3g2(x,Q2) + 2g1(x,Q

2)]

= 3

∫ 1

0

dx x2ḡ2(x,Q2) . (6)

Henced2 measures the deviations ofg2 fromgWW2 . The valueof d2 can be obtained from measurements ofg1 andg2 and

3

can be compared with predictions from Lattice QCD [11],bag models [12], QCD sum rules [13] and chiral soliton mod-els [14].

In this paper we first describe available predictions forAn1at largex. The experimental apparatus and the data analysisprocedure will be described in Section III, IV and V. In Sec-tion VI we present results for the asymmetries and polarizedstructure functions for both3He and the neutron, a new exper-imental fit forgn1 /F

n1 and a result for the matrix elementd

n2 .

Combined with the world proton and deuteron data, polarizedquark distribution functions were extracted from ourgn1 /F

n1

results. We conclude the paper by summarizing the results forAn1 and∆d/d and speculating on the importance of the roleof quark OAM on the nucleon spin in the kinematic regionexplored. Some of the results presented here were publishedpreviously [15]; the present publication gives full details onthe experiment and all of the neutron spin structure resultsforcompleteness.

II. PREDICTIONS FOR An1 AT LARGE x

From Section II A to II F we present predictions ofAn1 atlargex. Data onAn1 from previous experiments did not havethe precision to distinguish among different predictions,aswill be shown in Section II G.

A. SU(6) Symmetric Non-Relativistic Constituent QuarkModel

In the simplest non-relativistic constituent quarkmodel (CQM) [16], the nucleon is made of three con-stituent quarks and the nucleon spin is fully carried by thequark spin. Assuming SU(6) symmetry, the wavefunction of aneutron polarized in the+z direction then has the form [17]:

|n ↑〉 = 1√2

∣

∣d↑(du)000〉

+1√18

∣

∣d↑(du)110〉

(7)

−13

∣

∣d↓(du)111〉

− 13

∣

∣u↑(dd)110〉

+

√2

3

∣

∣u↓(dd)111〉

,

where the three subscripts are the total isospin, total spinSand the spin projectionSz along the+z direction for the ‘di-quark’ state. For the case of a proton one needs to exchangetheu andd quarks in Eq. (7). In the limit where SU(6) sym-metry is exact, both diquark spin states withS = 1 andS = 0contribute equally to the observables of interest, leadingto thepredictions

Ap1 = 5/9 and An1 = 0 ; (8)

∆u/u → 2/3 and ∆d/d → −1/3. (9)

We defineu(x) ≡ up(x), d(x) ≡ dp(x) ands(x) ≡ sp(x)as parton distribution functions (PDF) for the proton. For aneutron one hasun(x) = dp(x) = d(x), dn(x) = up(x) =u(x) based on isospin symmetry. The strange quark distribu-tion for the neutron is assumed to be the same as that of the

proton,sn(x) = sp(x) = s(x). In the following, all PDF’sare for the proton, unless specified by a superscript ‘n’.

In the case of DIS, exact SU(6) symmetry implies the sameshape for the valence quark distributions,i.e. u(x) = 2d(x).Using Eq. (2) and (A4), and assuming thatR(x,Q2) is thesame for the neutron and the proton, one can write the ratio ofneutron and protonF2 structure functions as

Rnp ≡ Fn2

F p2=

u(x) + 4d(x)

4u(x) + d(x). (10)

Applyingu(x) = 2d(x) gives

Rnp = 2/3 . (11)

However, data on theRnp ratio from SLAC [18], CERN [19,20, 21] and Fermilab [22] disagree with this SU(6) predic-tion. The data show thatRnp(x) is a straight line starting withRnp|x→0 ≈ 1 and dropping to below1/2 asx → 1. In ad-dition, Ap1(x) is small at lowx [23, 24, 25]. The fact thatRnp|x→0 ≈ 1 may be explained by the presence of a domi-nant amount of sea quarks in the lowx region and the fact thatAp1|x→0 ≈ 0 could be because these sea quarks are not highlypolarized. At largex, however, there are few sea quarks andthe deviation from SU(6) prediction indicates a problem withthe wavefunction described by Eq. (7). In fact, SU(6) sym-metry is known to be broken [26] and the details of possi-ble SU(6)-breaking mechanisms is an important open issue inhadronic physics.

B. SU(6) Breaking and Hyperfine Perturbed Relativistic CQM

A possible explanation for the SU(6) symmetry breakingis the one-gluon exchange interaction which dominates thequark-quark interaction at short-distances. This interactionwas used to explain the behavior ofRnp nearx → 1 andthe ≈ 300-MeV mass shift between the nucleon and the∆(1232) [26]. Later this was described by an interaction termproportional to~Si · ~Sj δ3(~rij), with ~Si the spin of theithquark, hence is also called the hyperfine interaction, or chro-momagnetic interaction among the quarks [27]. The effect ofthis perturbation on the wavefunction is to lower the energyof theS = 0 diquark state, causing the first term of Eq. (7),|d ↑ (ud)000〉n, to become more stable and to dominate thehigh energy tail of the quark momentum distribution that isprobed asx → 1. Since the struck quark in this term has itsspin parallel to that of the nucleon, the dominance of this termasx → 1 implies (∆d/d)n → 1 and(∆u/u)n → −1/3 forthe neutron, while for the proton one has

∆u/u → 1 and ∆d/d → −1/3 as x → 1 . (12)One also obtains

Rnp → 1/4 as x → 1 , (13)which could explain the deviation ofRnp(x) data from theSU(6) prediction. Based on the same mechanism, one canmake the following predictions:

Ap1 → 1 and An1 → 1 as x → 1 . (14)

4

The hyperfine interaction is often used to break SU(6) sym-metry in the relativistic CQM (RCQM). In this model, theconstituent quarks have non-zero OAM which carries≈ 25%of the nucleon spin [3]. The use of RCQM to predict the largex behavior of the nucleon structure functions can be justifiedby the valence quark dominance,i.e., in the largex regionalmost all quantum numbers, momentum and the spin of thenucleon are carried by the three valence quarks, which cantherefore be identified as constituent quarks. PredictionsofAn1 andA

p1 in the largex region using the hyperfine-perturbed

RCQM have been achieved [28].

C. Perturbative QCD and Hadron Helicity Conservation

In the early 1970’s, in one of the first applications of per-turbative QCD (pQCD), it was noted that asx → 1, the scat-tering is from a high-energy quark and thus the process canbe treated perturbatively [29]. Furthermore, when the quarkOAM is assumed to be zero, the conservation of angular mo-mentum requires that a quark carrying nearly all the momen-tum of the nucleon (i.e.x → 1) must have the same helicity asthe nucleon. This mechanism is called hadron helicity conser-vation (HHC), and is referred to as the leading-order pQCDin this paper. In this picture, quark-gluon interactions causeonly theS = 1, Sz = 1 diquark spin projection componentrather than the fullS = 1 diquark system to be suppressed asx → 1, which gives

∆u/u → 1 and ∆d/d → 1 as x → 1 ; (15)

Rnp → 37, Ap1 → 1 and An1 → 1 as x → 1 . (16)

This is one of the few places where pQCD can make an abso-lute prediction for thex-dependence of the structure functionsor their ratios. However, how low inx andQ2 this pictureworks is uncertain. HHC has been used as a constraint in amodel to fit data on the first moment of the protongp1 , giv-ing the BBS parameterization [30]. TheQ2 evolution was notincluded in this calculation. Later in the LSS(BBS) parame-terization [31], both proton and neutronA1 data were fitted di-rectly and theQ2 evolution was carefully treated. Predictionsfor An1 using both BBS and LSS(BBS) parameterizations havebeen made, as shown in Fig. 1 and 2 in Section II G.

HHC is based on the assumption that the quark OAMis zero. Recent experimental data on the tensor polariza-tion in elastic e−2H scattering [32], neutral pion photo-production [33] and the proton electro-magnetic form fac-tors [34, 35] disagree with the HHC predictions [36]. It hasbeen suggested that effects beyond leading-order pQCD, suchas quark OAM [37, 38, 39, 40], might play an important rolein processes involving quark spin flips.

D. Predictions from Next-to-Leading Order QCD Fits

In a next-to-leading order (NLO) QCD analysis of theworld data [41], parameterizations of the polarized and un-polarized PDFs were performed without the HHC constraint.

Predictions ofgp1/Fp1 andg

n1 /F

n1 were made using these pa-

rameterizations, as shown in Fig. 1 and 2 in Section II G.In a statistical approach, the nucleon is viewed as a gas of

massless partons (quarks, antiquarks and gluons) in equilib-rium at a given temperature in a finite volume, and the partondistributions are parameterized using either Fermi-DiracorBose-Einstein distributions. Based on this statistical pictureof the nucleon, a global NLO QCD analysis of unpolarizedand polarized DIS data was performed [42]. In this calcula-tion∆u/u ≈ 0.75, ∆d/d ≈ −0.5 andAp,n1 < 1 atx → 1.

E. Predictions from Chiral Soliton and Instanton Models

While pQCD works well in high-energy hadronic physics,theories suitable for hadronic phenomena in the non-perturbative regime are much more difficult to construct. Pos-sible approaches in this regime are quark models, chiral effec-tive theories and the lattice QCD method. Predictions forAn,p1have been made using chiral soliton models [43, 44] and theresults of Ref. [44] giveAn1 < 0. The prediction thatA

p1 < 0

has also been made in the instanton model [45].

F. Other Predictions

Based on quark-hadron duality [46], one can obtain thestructure functions and their ratios in the largex region bysumming over matrix elements for nucleon resonance tran-sitions. To incorporate SU(6) breaking, different mechanismsconsistent with duality were assumed and data on the structurefunction ratioRnp were used to fit the SU(6) mixing param-eters. In this picture,An,p1 → 1 asx → 1 is a direct result.Duality predictions forAn,p1 using different SU(6) breakingmechanisms were performed in Ref. [47]. There also existpredictions from bag models [48], as shown in Fig. 1 and 2 inthe next section.

G. Previous Measurements ofAn1

A summary of previousAn1 measurements is given in

TABLE I: Previous measurements ofAn1 .

Experiment beam target x Q2

(GeV/c)2

E142 [51] 19.42, 22.66, 3He 0.03-0.6 225.51 GeV; e−

E154 [52] 48.3 GeV; e− 3He 0.014-0.7 1-17HERMES [50] 27.5 GeV; e+ 3He 0.023-0.6 1-15

E143 [25] 9.7, 16.2, NH3, ND3 0.024-0.75 0.5-1029.1 GeV; e−

E155 [53] 48.35 GeV; e− NH3, LiD3 0.014-0.9 1-40SMC [49] 190 GeV;µ− C4H10O 0.003-0.7 1-60

C4D10O

5

[25][53][51][52]

[49][50]

FIG. 1: Previous data onAn1 [25, 49, 50, 51, 52, 53] and vari-ous theoretical predictions:An1 from SU(6) symmetry (solid line atzero) [17], hyperfine-perturbed RCQM (shaded band) [28], BBS pa-rameterization atQ2 = 4 (GeV/c)2 (higher solid) [30], LSS(BBS)parameterization atQ2 = 4 (GeV/c)2 (dashed) [31], statistical modelat Q2 = 4 (GeV/c)2 (long-dashed) [42], quark-hadron duality us-ing two different SU(6) breaking mechanisms (dash-dot-dotted anddash-dot-dot-dotted)[47], and non-meson cloudy bag model(dash-dotted) [48]; gn1 /F

n1 from LSS2001 parameterization atQ

2 =5 (GeV/c)2 (lower solid) [41] and from chiral soliton models [43] atQ2 = 3 (GeV/c)2 (long dash-dotted) and [44] atQ2 = 4.8 (GeV/c)2

(dotted).

Table I. The data onAn1 andAp1 are plotted in Fig. 1 and 2

along with theoretical calculations described in previoussec-tions. Since theQ2-dependence ofA1 is small andg1/F1 ≈A1 in DIS, data forgn1 /F

n1 andg

p1/F

p1 are also shown and all

data are plotted without evolving inQ2. As becomes obvi-ous in Fig. 1, the precision of previousAn1 data atx > 0.4from SMC [49], HERMES [50] and SLAC [25, 51, 52] is notsufficient to distinguish among different predictions.

III. THE EXPERIMENT

We report on an experiment [55] carried out at in the Hall Aof Thomas Jefferson National Accelerator Facility (JeffersonLab, or JLab). The goal of this experiment was to provideprecise data onAn1 in the largex region. We have measuredthe inclusive deep inelastic scattering of longitudinallypolar-ized electrons off a polarized3He target, with the latter beingused as an effective polarized neutron target. The scatteredelectrons were detected by the two standard High ResolutionSpectrometers (HRS). The two HRS were configured at thesame scattering angles and momentum settings to double thestatistics. Data were collected at threex points as shown inTable II. Both longitudinal and transverse electron asymme-

[54][24][23][25][53]

FIG. 2: World data onAp1 [23, 24, 25, 53, 54] and predictions forgp1/F

p1 at Q

2 = 5 (GeV/c)2 from the E155 experimental fit (longdash-dot-dotted) [53] and a new fit as described in Section VIB (longdash-dot-dot-dotted). The solid curve corresponds to the predictionfor gn1 /F

n1 from LSS(2001) parameterization atQ

2 = 5 (GeV/c)2.Other curves are the same as in Fig. 1 except that there is no predic-tion for the proton from BBS and LSS(BBS) parameterizations.

tries were measured, from whichA1, A2, g1/F1 andg2/F1were extracted using Eq. (A22–A25).

TABLE II: Kinematics of the experiment. The beam energy wasE = 5.734 GeV. E′ andθ are the nominal momentum and angleof the scattered electrons.〈x〉, 〈Q2〉 and〈W 2〉 are values averagedover the spectrometer acceptance.

〈x〉 0.327 0.466 0.601E′ 1.32 1.72 1.455θ 35◦ 35◦ 45◦

〈Q2〉 (GeV/c)2 2.709 3.516 4.833〈W 2〉 (GeV)2 6.462 4.908 4.090

A. Polarized 3He as an Effective Polarized Neutron

As shown in Fig. 1, previous data onAn1 did not have suf-ficient precision in the largex region. This is mainly due totwo experimental limitations. Firstly, high polarizationandluminosity required for precision measurements in the largexregion were not available previously. Secondly, there exists nofree dense neutron target suitable for a scattering experiment,mainly because of the neutron’s short lifetime (≈ 886 sec).Therefore polarized nuclear targets such as2~H or 3 ~He arecommonly used as effective polarized neutron targets. Con-

6

sequently, nuclear corrections need to be applied to extractneutron results from nuclear data. For a polarized deuteron,

np p pp

n

S state D state

n

p p

(89.93%) (8.75%) (1.26%)S’ state

FIG. 3: An illustration of3He wavefunction. TheS, S′ andD statecontributions are from calculations using the AV18 two-nucleon in-teraction and the Tucson-Melbourne three-nucleon force, as given inRef. [56].

approximately half of the deuteron spin comes from the pro-ton and the other half comes from the neutron. Therefore theneutron results extracted from the deuteron data have a signifi-cant uncertainty coming from the error in the proton data. Theadvantage of using3 ~He is that the two protons’ spins cancelin the dominantS state of the3He wavefunction, thus the spinof the3He comes mainly (> 87%) from the neutron [56, 57],as illustrated in Fig. 3. As a result, there is less model de-pendence in the procedure of extracting the spin-dependentobservables of the neutron from3He data. At largex, the ad-vantage of using a polarized3He target is more prominent inthe case ofAn1 . In this region almost all calculations showthatAn1 is much smaller thanA

p1, therefore theA

n1 results ex-

tracted from nuclear data are more sensitive to the uncertaintyin the proton data and the nuclear model being used.

In the largex region, the cross sections are small becausethe parton densities drop dramatically asx increases. In ad-dition, the Mott cross section, given by Eq. A3, is small atlargeQ2. To achieve a good statistical precision, high lumi-nosity is required. Among all laboratories which are equippedwith a polarized3He target and are able to perform a mea-surement of the neutron spin structure, the polarized electronbeam at JLab, combined with the polarized3He target in HallA, provides the highest polarized luminosity in the world [58].Hence it is the best place to study the largex behavior of theneutron spin structure.

B. The Accelerator and the Polarized Electron Source

JLab operates a continuous-wave electron accelerator thatrecirculates the beam up to five times through two super-conducting linear accelerators. Polarized electrons are ex-tracted from a strained GaAs photocathode [59] illuminatedby circularly polarized light, providing a polarized beam of(70 − 80)% polarization and≈ 200µA maximum current toexperimental halls A, B and C. The maximum beam energyavailable at JLab so far is 5.7 GeV, which was also the beamenergy used during this experiment.

C. Hall A Overview

The basic layout of Hall A during this experiment is shownin Fig. 4. The major instrumentation [60] includes beamlineequipment, the target and two HRSs. The beamline starts af-

Laser Hut

TargetPol. 3He

Q3D

Q1

PolarimeterCompton

MollerPolarimeter

Raster

BCM BPMARC eP

Left HRS

Right HRS

Q2

Pb glasscounters

Pb glasscounters

Scintillators

To Beam Dump

Cerenkov

VDCs

FIG. 4: (Color online) Top-view of the experimental hall A (not toscale).

ter the arc section of the accelerator where the beam is bentinto the hall, and ends at the beam dump. The arc sectioncan be used for beam energy measurement, as will be de-scribed in Section III D. After the arc section, the beamlineis equipped with a Compton polarimeter, two Beam CurrentMonitors (BCM) and an Unser monitor for absolute beam cur-rent measurement, a fast raster, the eP device for beam en-ergy measurement, a Møller polarimeter and two Beam Posi-tion Monitors (BPM). These beamline elements, together withspectrometers and the target, will be described in detail inthefollowing sections.

D. Beam Energy Measurement

The energy of the beam was measured absolutely by twoindependent methods - ARC and eP [60, 61]. Both methodscan provide a precision ofδEbeam/Ebeam ≈ 2 × 10−4. Forthe ARC method [60, 62], the deflection of the beam in the arcsection of the beamline is used to determine the beam energy.In the eP measurement [60, 63] the beam energy is determinedby the measurement of the scattered electron angleθe and therecoil proton angleθp in 1H(e, e′p) elastic scattering.

E. Beam Polarization Measurement

Two methods were used during this experiment to measurethe electron beam polarization. The Møller polarimeter [60]measures Møller scattering of the polarized electron beam off

7

polarized atomic electrons in a magnetized foil. The crosssection of this process depends on the beam and target po-larizations. The polarized electron target used by the Møllerpolarimeter was a ferromagnetic foil, with its polarization de-termined from foil magnetization measurements. The Møllermeasurement is invasive and typically takes an hour, provid-ing a statistical accuracy of about0.2%. The systematic errorcomes mainly from the error in the foil target polarization.An additional systematic error is due to the fact that the beamcurrent used during a Møller measurement (≈ 0.5µA) is lowerthan that used during the experiment. The total relative sys-tematic error was≈ 3.0% during this experiment.

During a Compton polarimeter [60, 64] measurement, theelectron beam is scattered off a circularly polarized photonbeam and the counting rate asymmetry of the Compton scat-tered electrons or photons between opposite beam helicitiesis measured. The Compton polarimeter measures the beampolarization concurrently with the experiment running in thehall.

The Compton polarimeter consists of a magnetic chicanewhich deflects the electron beam away from the scattered pho-tons, a photon source, an electromagnetic calorimeter and anelectron detector. The photon source was a 200 mW laser am-plified by a resonant Fabry-Perot cavity. During this experi-ment the maximum gain of the cavity reachedGmax = 7500,leading to a laser power of1500 W inside the cavity. Thecircular polarization of the laser beam was> 99% for bothright and left photon helicity states. The asymmetry measuredin Compton scattering at JLab with a1.165 eV photon beamand the5.7 GeV electron beam used by this experiment had amean value of≈ 2.2% and a maximum of9.7%. For a 12µAbeam current, one hour was needed to reach a relative statis-tical accuracy of(∆Pb)stat/Pb ≈ 1%. The total systematicerror was(∆Pb)sys/Pb ≈ 1.6% during this experiment.

The average beam polarization during this experiment wasextracted from a combined analysis of 7 Møller and 53 Comp-ton measurements. A value of(79.7± 2.4)% was used in thefinal DIS analysis.

F. Beam Helicity

The helicity state of electrons is regulated every33ms at theelectron source. The time sequence of the electrons’ helicitystate is carried by helicity signals, which are sent to exper-imental halls and the data acquisition (DAQ) system. Sincethe status of the helicity signal (H+ or H- pulses) has eitherthe same or the opposite sign as the real electron helicity, theabsolute helicity state of the beam needs to be determined byother methods, as will be described later.

There are two modes – toggle and pseudorandom – whichcan be used for the pulse sequence of the helicity signal. Inthe toggle mode, the helicity alternates every33 ms. In thepseudorandom mode, the helicity alternates randomly at thebeginning of each pulse pair, of which the two pulses musthave opposite helicities in order to equalize the numbers oftheH+ and H- pulses. The purpose of the pseudorandom mode isto minimize any possible time-dependent systematic errors.

Fig. 5 shows the helicity signals and the helicity states of the

+ −+ −+ −+ − + −+ −DAQ

33 ms

−− + − − − −+ + +DAQ

one pulse pair

the helicity alternates randomlybetween pulse pairs

FIG. 5: Helicity signal and the helicity status of DAQ in toggle (top)and pseudorandom (bottom) modes.

DAQ system for the two regulation modes.There is a half-wave plate at the polarized source which

can be inserted to reverse the helicity of the laser illuminatingthe photocathode hence reverse the helicity of electron beam.During the experiment this half-wave plate was inserted forhalf of the statistics to minimize possible systematic effectsrelated to the beam helicity.

The scheme described above was used to monitor the rela-tive changes of the helicity state. The absolute sign of the elec-trons’ helicity states during each of the H+ and H- pulses wereconfirmed by measuring a well known asymmetry and com-paring the measured asymmetry with its prediction, as will bepresented in Section V B and V C.

G. Beam Charge Measurement and Charge AsymmetryFeedback

The beam current was measured by the BCM system lo-cated upstream of the target on the beamline. The BCM sig-nals were fed to scaler inputs and were inserted in the datastream.

Possible beam charge asymmetry measured at Hall A canbe caused by the timing asymmetry of the DAQ system, orby the timing and the beam intensity asymmetries at the po-larized electron source. The beam intensity asymmetry orig-inates from the intensity difference between different helicitystates of the circularly polarized laser used to strike the pho-tocathode. Although the charge asymmetry can be correctedfor to first order, there may exist unknown non-linear effectswhich can cause a systematic error in the measured asymme-try. Thus the beam charge asymmetry should be minimized.This was done by using a separate DAQ system initially devel-oped for the parity-violation experiments [65], called thepar-ity DAQ. The parity DAQ used the measured charge asymme-

8

try in Hall A to control the orientation of a rotatable half-waveplate located before the photocathode at the source, such thatintensities for each helicity state of the polarized laser used tostrike the photocathode were adjusted accordingly. The parityDAQ was synchronized with the two HRS DAQ systems sothat the charge asymmetry in the two different helicity statescould be monitored for each run. The charge asymmetry wastypically controlled to be below2 × 10−4 during this experi-ment.

H. Raster and Beam Position Monitor

To protect the target cell from being damaged by the effectof beam-induced heating, the beam was rastered at the target.The raster consists a pair of horizontal and vertical air-coredipoles located upstream of the target on the beamline, whichcan produce either a rectangular or an elliptical pattern. Weused a raster pattern distributed uniformly over a circularareawith a radius of 2 mm.

The position and the direction of the beam at the targetwere measured by two BPMs located upstream of the tar-get [60]. The beam position can be measured with a precisionof 200µm with respect to the Hall A coordinate system. Thebeam position and angle at the target were recorded for eachevent.

I. High Resolution Spectrometers

The Hall A High Resolution Spectrometer (HRS) systemswere designed for detailed investigations of the structureofnuclei and nucleons. They provide high resolution in mo-mentum and in angle reconstruction of the reaction prod-uct as well as being able to be operated at high luminos-ity. For each spectrometer, the vertically bending design in-cludes two quadrupoles followed by a dipole magnet and athird quadrupole. All quadrupoles and the dipole are super-conducting. Both HRSs can provide a momentum resolutionbetter than2 × 10−4 and a horizontal angular resolution bet-ter than 2 mrad with a design maximum central momentumof 4 GeV/c [60]. By convention, the two spectrometers areidentified as the left and the right spectrometers based on theirposition when viewed looking downstream.

The basic layout of the left HRS is shown in Fig. 6. Thedetector package is located in a large steel and concretedetector hut following the last magnet. For this experimentthe detector package included (1) two scintillator planes S1and S2 to provide a trigger to activate the DAQ electronics;(2) a set of two Vertical Drift Chambers (VDC) [66] forparticle tracking; (3) a gas̆Cerenkov detector to provideparticle identification (PID) information; and (4) a set of leadglass counters for additional PID. The layout of the right HRSis almost identical except a slight difference in the geometryof the gas̆Cerenkov detector and the lead glass counters.

S1

S2

CherenkovGas

CountersLead Glass

2nd VDC

Q2Dipole

Q3

Q1

1st VDC

45o

RayCentral

Target PivotCenter of

FIG. 6: (Color online) Schematic layout of the left HRS and detectorpackage (not to scale).

J. Particle Identification

For this experiment the largest background came fromphoto-produced pions. We refer to PID in this paper as theidentification of electrons from pions. PID for each HRS wasaccomplished by a CO2 threshold gas̆Cerenkov detector anda double-layered lead glass shower detector.

The twoC̆erenkov detectors, one on each HRS, were oper-ated with CO2 at atmospheric pressure. The refraction indexof the CO2 gas was 1.00041, giving a threshold momentum of≈ 17 MeV/c for electrons and≈ 4.8 GeV/c for pions. Theincident particles on each HRS were also identified by theirenergy deposits in the lead glass shower detector.

SinceC̆erenkov detectors and lead glass shower detectorsare based on different mechanisms and their PID efficienciesare not correlated [67], we extracted the PID efficiency of thelead glass counters by using electron events selected by theC̆erenkov detector, andvice versa. Fig. 7 shows a spectrum ofthe summed ADC signal of the left HRS gasC̆erenkov detec-tor, without a cut on the lead glass signal and after applyingsuch lead glass electron and pion cuts. The spectrum fromthe right HRS is similar. Fig. 8 shows the distribution of theenergy deposit in the two layers of the right HRS lead glasscounters, without ăCerenkov cut, and after̆Cerenkov electronand pion cuts.

Detailed PID analysis was done both before and duringthe experiment. The PID performance of each detector ischaracterized by the electron detection efficiencyηe and thepion rejection factorηπ,rej, defined as the number of pionsneeded to cause one pion contamination event. In the HRScentral momentum range of0.8 < p0 < 2.0 (GeV/c), the PIDefficiencies for the left HRS were found to be⋄ GasC̆erenkov: ηπ,rej > 770 atηe = 99.9%;⋄ Lead glass counters:ηπ,rej ≈ 38 atηe = 98%;⋄ Combined:ηπ,rej > 3× 104 atηe = 98%.

and for the right HRS were⋄ GasC̆erenkov: ηπ,rej = 900 atηe = 99%;⋄ Lead glass counters:ηπ,rej ≈ 182 atηe = 98%;⋄ Combined:ηπ,rej > 1.6× 105 atηe = 97%.

9

w/ cutπ−

single photo−electron

PID cut applied

peak

w/ e− cut

no cutY

ield

(ar

b. u

nits

)

ADC sum (channels)

FIG. 7: (Color online) Summed ADC signal of the left HRS gasC̆erenkov detector: without cuts, after lead glass counters electroncut and after pion cut. The vertical line shows a cut

∑

ADCi > 400applied to select electrons.

ESHOWER(channels)

EP

reS

HO

WE

R(ch

anne

ls)

FIG. 8: (Color online) Energy deposited in the first layer (preshower)vs that in the second layer (shower) of lead glass counters in theright HRS. The two blobs correspond to the spectrum with a tightgasC̆erenkov ADC electron cut and with a pion cut applied. Thelines show the boundary of the two-dimensional cut used to selectelectrons in the data analysis.

K. Data Acquisition System

We used the CEBAF Online Data Acquisition (CODA) sys-tem [68] for this experiment. In the raw data file, data fromthe detectors, the beamline equipment, and from the slow con-trol software were recorded. The total volume of data ac-cumulated during the two-month running period was about0.6 TBytes. Data from the detectors were processed usingan analysis package called Experiment Scanning Program forhall A Collaboration Experiments (ESPACE) [69]. ESPACEwas used to filter raw data, to make histograms for recon-structed variables, to export variables into ntuples for further

analysis, and to calibrate experiment-specific detector con-stants. It also provided the possibility to apply conditions onthe incoming data. The information from scaler events wasused to extract beam charge and DAQ deadtime corrections.

IV. THE POLARIZED TARGET

Polarized3He targets are widely used at SLAC, DESY,MAINZ, MIT-Bates and JLab to study the electromagneticstructure and the spin structure of the neutron. There existtwo major methods to polarize3He nuclei. The first one usesthe metastable-exchange optical pumping technique [70]. Thesecond method is based on optical pumping [71] and spin ex-change [72]. It has been used at JLab since 1998 [73], andwas used here.

The3 ~He target at JLab Hall A uses the same design as theSLAC 3 ~He target [74]. The first step to polarize3He nucleiis to polarize an alkali metal vapor (rubidium was used atJLab as well as at SLAC) by optical pumping [71] with cir-cularly polarized laser light. Depending on the photon helic-ity, the electrons in the Rb atoms will accumulate at eithertheF = 3,mF = 3 or theF = 3,mF = −3 level (hereF is the atom’s total spin andmF is its projection along themagnetic field axis). The polarization is then transfered tothe3He nuclei through the spin exchange mechanism [72] duringcollisions between Rb atoms and the3He nuclei. Under oper-ating conditions the3He density is about1020 nuclei/cm3 andthe Rb density is about1014 atoms/cm3.

To minimize depolarization effects caused by the unpolar-ized light emitted from decay of the excited electrons, N2buffer gas was added to provide a channel for the excitedelectrons to decay to the ground state without emitting pho-tons [71]. In the presence of N2, electrons decay throughcollisions between the Rb atoms and N2 molecules, which isusually referred to as non-radiative quenching. The numberdensity of N2 was about1% of that of3He.

A. Target Cells

The target cells used for this experiment were 25-cm longpressurized glass cells with∼ 130-µm thick end windows.

TABLE III: Target cell characteristics. Symbols are:Vp pumpingchamber volume in cm3; Vt target chamber volume in cm3; Vtrtransfer tube volume in cm3; V0 total volume in cm3; Ltr transfertube length in cm;n0: 3He density in amg at room temperature (1amg= 2.69 × 10−19/cm3 which corresponds to the gas density atthe standard pressure andT = 0◦C); lifetime is in hours.

Name Vp Vt Vtr V0 Ltr n0 lifetimeCell #1 116.7 51.1 3.8 171.6 6.574 9.10 49Cell #2 116.1 53.5 3.9 173.5 6.46 8.28 44

uncertainty 1.5 1.0 0.25 1.8 0.020 2% 1

10

Target Chamber

11.8Transfer Tube

64.5

Pumping chamber

68.8

66.5

250

e’ to left HRS

e’ to right HRS

19.3

Laser

incident e beam

FIG. 9: (Color online) JLab target cell, geometries are given in mmfor cell #2 used in this experiment.

The cell consisted of two chambers, a spherical upper cham-ber which holds the Rb vapor and in which the optical pump-ing occurs, and a long cylindrical chamber where the electronbeam passes through and interacts with the polarized3He nu-clei. Two cells were used for this experiment. Figure 9 is apicture of the first cell with dimensions shown in mm. Ta-ble III gives the cell volumes and densities.

B. Target Setup

Figure 10 is a schematic diagram of the target setup. Therewere two pairs of Helmholtz coils to provide a 25 G mainholding field, with one pair oriented perpendicular and theother parallel to the beamline (only the perpendicular pairisshown). The holding field could be aligned in any horizon-tal direction with respect to the incident electron beam. Thecoils were excited by two power supplies in the constant volt-age mode. The coil currents were continuously measured andrecorded by the slow control system. The cell was held atthe center of the Helmholtz coils with its pumping chambermounted inside an oven heated to170◦C in order to vapor-ize the Rb. The lasers used to polarize the Rb were three30 W diode lasers tuned to a wavelength of 795 nm. The tar-get polarization was measured by two independent methods –the NMR (Nuclear Magnetic Resonance) [60, 73, 75] and theEPR (Electro Paramagnetic Resonance) [58, 60, 73, 76] po-larimetry. The NMR system consisted of one pair of pick-upcoils (one on each side of the cell target chamber), one pairof RF coils and the associated electronics. The RF coils wereplaced at the top and the bottom of the scattering chamber, ori-ented in the horizontal plane, as shown in Fig. 10. The EPRsystem shared the RF coils with the NMR system. It consistedof one additional RF coil to induce light signal emission fromthe pumping chamber, a photodiode and the related optics tocollect the light, and associated electronics for signal process-ing.

Mai

n H

oldi

ng H

elm

holtz

Coi

l

Mai

n H

oldi

ng H

elm

holtz

Coi

l

Pick−Up Coils

EPR RF Coil

oven

beame−

RF Drive Coil

RF Drive Coil

Lasers (795nm)30W DiodeThree (four)

EPR optics

opticsLaser

FIG. 10: (Color online) Target setup overview (schematic).

C. Laser System

The laser system used during this experiment consisted ofseven diode lasers – three for longitudinal pumping, three fortransverse pumping and one spare. To protect the diode lasersfrom radiation damage from the electron beam, as well asto minimize the safety issues related to the laser hazard, thediode lasers and the associated optics system were located ina concrete laser hut located on the right side of the beamlineat90◦, as shown in Fig. 4. The laser optics had seven individuallines, each associated with one diode laser. All seven opticallines were identical and were placed one on top of the otheron an optics table inside the laser hut. Each optical line con-sisted of one focusing lens to correct the angular divergenceof the laser beam, one beam-splitter to linearly polarize thelasers, two mirrors to direct them, three quarter waveplatesto convert linear polarization to circular polarization, and twohalf waveplates to reverse the laser helicity. Figure 11 showsa schematic diagram of one optics line.

Under the operating conditions for either longitudinal ortransverse pumping, the original beam of each diode laser wasdivided into two by the beam-splitter. Therefore there wereatotal of six polarized laser beams entering the target. The di-ameter of each beam was about 5 cm which approximatelymatched the size of the pumping chamber. The target wasabout 5 m away from the optical table. For the pumping ofthe transversely polarized target, all these laser beams wentdirectly towards the pumping chamber of the cell through awindow on the side of the target scattering chamber enclo-sure. For longitudinal pumping, they were guided towards thetop of the scattering chamber, then were reflected twice andfinally reached the cell pumping chamber.

11

S

P

P

S

T > 95%, R > 99.8%P S

90 lineo

Beam splitter:

λ/4 waveplate

laser fiber

performance:

p

ps

p p

laser

controlhelicity

3" mirror

waveplateλ/4

waveplateλ/2

focusing lensFL=8.83cm

splitterbeam

Holding posts

(back)

λ/2

2" mirror

polarizedcircularly

Optics Table

Con

cret

e W

all

Lase

r H

ut

polarizing

λ/4polarizing

To target

(left) (right)

(perpendicular to the beamline)

FIG. 11: (Color online) Laser polarizing optics setup (schematic) forthe Hall A polarized3He target.

D. NMR Polarimetry

The polarization of the3He was determined by measuringthe 3He Nuclear Magnetic Resonance (NMR) signal. Theprinciple of NMR polarimetry is the spin reversal of3He nu-clei using the Adiabatic Fast Passage (AFP) [77] technique.At resonance this spin reversal will induce an electromagneticfield and a signal in the pick-up coil pair. The signal mag-nitude is proportional to the polarization of the3He and canbe calibrated by performing the same measurement on a wa-ter sample, which measures the known thermal polarizationof protons in water. The systematic error of the NMR mea-surement was about3%, dominated by the error in the watercalibration [75].

E. EPR Polarimetry

In the presence of a magnetic field, the Zeeman splittingof Rb, characterized by the Electron-Paramagnetic ResonancefrequencyνEPR, is proportional to the field magnitude. When3He nuclei are polarized (P ≈ 40%), their spins generate asmall magnetic fieldB3He of the order of≈ 0.1 Gauss, super-imposed on the main holding fieldBH = 25 Gauss. Duringan EPR measurement [76] the spin of the3He is flipped byAFP, hence the direction ofB3He is reversed and the changein the total field magnitude causes a shift inνEPR. This fre-

quency shiftδνEPR is proportional to the3He polarizationin the pumping chamber. The3He polarization in the targetchamber is calculated using a model which describes the po-larization diffusion from the pumping chamber to target cham-ber. The value of the EPR resonance frequencyνEPR can alsobe used to calculate the magnetic field magnitude. The sys-tematic error of the EPR measurement was about3%, whichcame mainly from uncertainties in the cell density and tem-perature, and from the diffusion model.

F. Target Performance

The target polarizations measured during this experimentare shown in Fig. 12. Results from the two polarimetries arein good agreement and the average target polarization in beam

Days Since Beginning of Experiment0 10 20 30 40 50 60

Per

cent

Pol

ariz

atio

n

0

10

20

30

40

50

EPR Measurements

NMR Measurements

Mai

nten

ance

Target Polarizations for E99−117

Cell #2Cell #1

FIG. 12: Target polarization, starting June 1 of 2001, as measured byEPR and NMR polarimetries.

was(40.0 ± 2.4)%. In a few cases the polarization measure-ment itself caused an abrupt loss in the polarization. This phe-nomenon may be the so-called “masing effect” [74] due tonon-linear couplings between the3He spin rotation and con-ducting components inside the scattering chamber,e.g., theNMR pick-up coils, and the “Rb-ring” formed by the rubid-ium condensed inside the cell at the joint of the two cham-bers. This masing effect was later suppressed by adding coilsto produce an additional field gradient.

V. DATA ANALYSIS

In this section we present the analysis procedure leadingto the final results in Section VI. We start with the analysisof elastic scattering, the∆(1232) transverse asymmetry, andthe check for false asymmetry. Next, the DIS analysis andradiative corrections are presented. Finally we describe nu-clear corrections which were used to extract neutron structurefunctions from the3He data.

12

A. Analysis Procedure

The procedure to extract the electron asymmetries from ourdata is outlined in Fig. 13. From the raw data one first ob-

A rawA 2

A 1

g /F1 1

12g /F

Nuclear

correction

A 1n

g /F1 1n n

12g /Fnn

An2

A

AData

+−

+−+−N

ARC

ARC

Radiative corrections

PbeamP

targ2Nf

(1232)∆asymmetry

Signconvention

Elasticanalysis

Q

LT

detector cutsPID cuts

cutsacceptanceHRS

Relativeyield

FIG. 13: Procedure for asymmetry analysis.

tains the helicity-dependent electron yieldN± using accep-tance and PID cuts. The efficiencies associated with thesecuts are not helicity-dependent, hence are not corrected for inthe asymmetry analysis. The yield is then corrected for thehelicity-dependent integrated beam chargeQ± and the live-time of the DAQ systemLT±. The asymmetry of the cor-rected yield is the raw asymmetryAraw. Next, to go fromAraw to the physics asymmetriesA‖ andA⊥, four factorsneed to be taken into account: the beam polarizationPb, thetarget polarizationPt, the nitrogen dilution factorfN2 due tothe unpolarized nitrogen nuclei mixed with the polarized3Hegas, and a sign based on the knowledge of the absolute stateof the electron helicity and the target spin direction:

A‖,⊥ = ±Araw

fN2PbPt(17)

The results of the beam and the target polarization measure-ments have been presented in previous sections. The nitrogendilution factor is obtained from data taken with a referencecell filled with nitrogen. The sign of the asymmetry isdescribed by “the sign convention”. The sign convention forparallel asymmetries was obtained from the elastic scatteringasymmetry and that for perpendicular asymmetries was fromthe ∆(1232) asymmetry analysis, as will be described inSections V B and V C. The physics asymmetriesA‖ andA⊥,after corrections for radiative effects, were used to calculateA1 andA2 and the structure function ratiosg1/F1 andg2/F1using Eq. (A22—A25). Then the last step is to apply nuclearcorrections in order to extract the neutron asymmetries andthe structure function ratios from the3He results, as will bedescribed in Section V F.

Although the main goal of this experiment was to provideprecise data on the asymmetries, cross sections were also ex-tracted from the data. The procedure for the cross sectionanalysis is outlined in Fig. 14. One first determines the ab-solute yield of~e− ~3He inclusive scattering from the raw data.Unlike the asymmetry analysis, corrections need to be madefor the detector and PID efficiencies and the spectrometer ac-ceptance. A Monte-Carlo simulation is used to calculate the

spectrometer acceptance based on a transport model for the

Data

N2 data

N2

LT

QAbsoluteyield σData

MCA( )

σ MC

detector eff.PID eff.

subtract

HRSacceptance

HRS modelstruct. func. Monte−Carlo Simulation

(w/ radiation effects)

FIG. 14: Procedure for cross section analysis.

HRS [60] with radiative effects taken into account. One thensubtracts the yield ofe−N scattering caused by the N2 nucleiin the target. The clean~e − ~3He yield is then corrected forthe helicity-averaged beam charge and the DAQ livetime togive cross section results. Using world fits for the unpolarizedstructure functions (form factors) of3He, one can calculatethe expected DIS (elastic) cross section from the Monte-Carlosimulation and compare to the data.

B. Elastic Analysis

Data for~e −3 ~He elastic scattering were taken on a lon-gitudinally polarized target with a beam energy of 1.2 GeV.The scattered electrons were detected at an angle of20◦. Theformalism for the cross sections and asymmetries are summa-rized in Appendix B. Results for the elastic asymmetry wereused to check the product of beam and target polarizations,as well as to determine the sign convention for differentbeam-helicity states and target spin directions.

The raw asymmetry was extracted from the data by

Araw =

N+

Q+LT+ − N−

Q−LT−

N+

Q+LT+ +N−

Q−LT−

(18)

with N±, Q± andLT± the helicity-dependent yield, beamcharge and livetime correction, respectively. The elasticasymmetry is

Ael‖ = ±Araw

fN2fQEPbPt(19)

with fN2 = 0.975± 0.003 the N2 dilution factor determinedfrom data taken with a reference cell filled with nitrogen, andPb andPt the beam and target polarization, respectively. Acut in the invariant mass|W −M3He| < 6 (MeV) was used toselect elastic events. Within this cut there are a small amountof quasi-elastic events andfQE > 0.99 is the quasi-elasticdilution factor used to correct for this effect.

The sign on the right hand side of Eq. (19) depends on theconfiguration of the beam half-wave plate, the spin precessionof electrons in the accelerator, and the target spin direction. Itwas determined by comparing the sign of the measured rawasymmetries with the calculated elastic asymmetry. We found

13

l l

l l

FIG. 15: (Color online) Elastic parallel asymmetry resultsfor the twoHRS. The kinematics areE = 1.2 GeV andθ = 20◦. A cut in theinvariant mass|W −M3He| < 6 (MeV) was used to select elasticevents. Data from runs with beam half-wave plate inserted are shownas triangles. The error bars shown are total errors including a4.5%systematic uncertainty, which is dominated by the error of the beamand target polarizations. The combined asymmetry and its total errorfrom ≈ 20 runs are shown by the horizontal solid and dashed lines,respectively, as well as the solid circle as labeled [58].

that for this experiment the electron helicity was aligned to thebeam direction during H+ pulses when the beam half-waveplate wasnot inserted. Since the electron spin precession inthe accelerator can be well calculated using quantum electro-dynamics and the results showed that the beam helicity dur-ing H+ pulses was the same for the two beam energies usedfor elastic and DIS measurements, the above convention alsoapplies to the DIS data analysis.

A Monte-Carlo simulation was performed which took intoaccount the spectrometer acceptance, the effect of the quasi-elastic scattering background and radiative effects. Resultsfor the elastic asymmetry and the cross section are shown inFig. 15 and 16, respectively, along with the expected valuesfrom the simulation. The data show good agreement with thesimulation within the uncertainties.

C. ∆(1232) Transverse Asymmetry

Data on the∆(1232) resonance were taken on a trans-versely polarized target using a beam energy of1.2 GeV. Thescattered electrons were detected at an angle of20◦ and thecentral momentum of the spectrometers was set to0.8 GeV/c.The transverse asymmetry defined by Eq. (A15) was extractedfrom the raw asymmetry using Eq. (17). A cut in the invariantmass|W − 1232| < 20 (MeV) was used to select∆(1232)events. The sign on the right hand side of Eq. (17) depends onthe beam half-wave plate status, the spin precession of elec-trons in the accelerator, the target spin direction, and in which

cross section

cross section

combined

combined

FIG. 16: (Color online) Elastic cross section results for the two HRS.The kinematics wereE = 1.2 GeV andθ = 20◦. A systematic er-ror of 6.7% was assigned to each data point, which was dominatedby the uncertainty in the target density and the HRS transport func-tions [58].

T

T

extrapolated from

extrapolated from

FIG. 17: (Color online) Measured raw∆(1232) transverse asymme-try, with beam half-wave plate inserted and target spin pointing to theleft side of the beamline. The kinematics areE = 1.2 GeV,θ = 20◦

andE′ = 0.8 GeV/c. The dashed lines show the expected valueobtained from previous3He data extrapolated inQ2.

(left or right) HRS the asymmetry is measured. Since datafrom a previous experiment [73] in a similar kinematic regionshowed thatA∆‖ < 0 andA

∆⊥ > 0 [78], A

∆⊥ can be used

to determine the sign convention of the measured transverseasymmetries. The raw∆(1232) transverse asymmetry mea-sured during this experiment was positive on the left HRS,as shown in Fig. 17, with the beam half-wave plate insertedand the target spin pointing to the left side of the beamline.Also shown is the expected value obtained from previous3Hedata extrapolated inQ2. Similar to the longitudinal configu-

14

ration, this convention applied to both the∆(1232) and DISmeasurements.

D. False Asymmetry and Background

False asymmetries were checked by measuring the asym-metries from a polarized beam scattering off an unpolarized12C target. The results show that the false asymmetry wasless than2× 10−3, which was negligible compared to the sta-tistical uncertainties of the measured3He asymmetries. To es-timate the background from pair productionγ → e−+e+, thepositron yield was measured atx = 0.33, which is expectedto have the highest pair production background. The positroncross section was found to be≈ 3% of the total cross sectionat x = 0.33, and the positron contribution atx = 0.48 andx = 0.61 should be even smaller. The effect of pair produc-tion asymmetry is negligible compared to the statistical uncer-tainties of the measured3He asymmetries and is not correctedfor in this analysis.

E. DIS Analysis

The longitudinal and transverse asymmetries defined byEq. (A13) and (A15) for DIS were extracted from the rawasymmetries as

A‖,⊥ = ±Araw

fN2PbPt(20)

where the sign on the right hand side was determined by theprocedure described in Sections V B and V C. The N2 dilutionfactor, extracted from runs where a reference cell was filledwith pure N2, was found to befN20.938± 0.007 for all threeDIS kinematics.

Radiative corrections were performed for the3He asym-metriesA

3He‖ andA

3He⊥ . We denote byA

obs the observed

asymmetry,ABorn the non-radiated (Born) asymmetry,∆Air

the correction due to internal radiation effects and∆Aer theone due to external radiation effects. One hasABorn =Aobs +∆Air +∆Aer for a specific target spin orientation.

Internal corrections were calculated using an improved ver-sion of POLRAD 2.0 [79]. External corrections were calcu-lated with a Monte-Carlo simulation based on the procedurefirst described by Mo and Tsai [80]. Since the theory of radia-tive corrections is well established [80], the accuracy of theradiative correction depends mainly on the structure functionsused in the procedure. To estimate the uncertainty of bothcorrections, five different fits [81, 82, 83, 84, 85] were usedfor the unpolarized structure functionF2 and two fits [86, 87]were used for the ratioR. For the polarized structure functiong1, in addition to those used in POLRAD 2.0 [88, 89], we fit toworld gp1/F

p1 andg

n1 /F

n1 data including the new results from

this experiment. Both fits will be presented in Section VI B.For g2 we used bothgWW2 and an assumption thatg2 = 0.The variation in the radiative corrections using the fits listedabove was taken as the full uncertainty of the corrections. For

TABLE IV: Total radiation lengthX0 and thicknessd of the ma-terial traversed by incident (before interaction) and scattered (afterinteraction) electrons. The cell is made of glass GE180 which hasX0 = 7.04 cm and densityρ = 2.77 g/cm3. The radiation lengthand thickness after interaction are given by left/right depending onby which HRS the electrons were detected.

x 0.33, 0.48 0.61 0.61θ 35◦ 45◦ 45◦

Cell #2 #2 #1Cell window (µm) 144 144 132

X0 (before) 0.00773 0.00773 0.00758d (g/cm2, before) 0.23479 0.23479 0.23317Cell wall (mm) 1.44/1.33 1.44/1.33 1.34/1.43

X0 (after) 0.0444/0.04160.0376/0.03540.0356/0.0374d (g/cm2, after) 0.9044/0.85060.7727/0.72930.7336/0.7687

TABLE V: Internal radiative corrections toA3He

‖ andA3He

⊥ .

x ∆Air,3He

‖ (×10−3) ∆Air,

3He

⊥ (×10−3)

0.33 -5.77± 0.47 2.66± 0.030.48 -3.28± 0.13 1.47± 0.050.61 -2.66± 0.15 1.28± 0.07

TABLE VI: External radiative corrections toA3He

‖ andA3He

⊥ . Errorsare from uncertainties in the structure functions and in thecell wallthickness.

x ∆Aer,3He

‖ (×10−3) ∆Aer,

3He

⊥ (×10−3)

0.33 -0.67± 0.10 -0.05± 0.110.48 -1.16± 0.15 0.80± 0.460.61 -0.39± 0.03 0.29± 0.04

external corrections the uncertainty also includes the contribu-tion from the uncertainty in the target cell wall thickness.Thetotal radiation length and thickness of the material traversedby the scattered electrons are given in Table IV for each kine-matic setting. Results for the internal and external radiativecorrections are given in Table V and VI, respectively.

By measuring DIS unpolarized cross sections and using theasymmetry results, one can calculate the polarized cross sec-tions and extractg1 andg2 from Eq. (A5) and (A6). We useda Monte-Carlo simulation to calculate the expected DIS un-polarized cross sections within the spectrometer acceptance.This simulation included internal and external radiative cor-rections. The structure functions used in the simulation werefrom the latest DIS world fits [83, 87] with the nuclear effectscorrected [90]. The radiative corrections from the elasticandquasi-elastic processes were calculated in the peaking approx-imation [91] using the world proton and neutron form factordata [92, 93, 94]. The DIS cross section results agree with

15

the simulation at a level of10%. Since this is not a dedicatedcross section experiment, we obtained the values forg1 andg2by multiplying ourg1/F1 andg2/F1 results by the world fitsfor unpolarized structure functionsF1 [83, 87], instead of theF1 from this analysis.

F. From 3He to Neutron

Properties of protons and neutrons embedded in nuclei areexpected to be different from those in free space because ofa variety of nuclear effects, including that from spin depo-larization, binding and Fermi motion, the off-shell natureofthe nucleons, the presence of non-nucleonic degrees of free-dom, and nuclear shadowing and antishadowing. A coherentand complete picture of all these effects for the3He structurefunctiong

3He1 in the range of10

−4 ≤ x 6 0.8 was presentedin [97]. It gives

g3He1 = Png

n1 + 2Ppg

p1 − 0.014

[

gp1(x) − 4gn1 (x)]

+a(x)gn1 (x) + b(x)gp1(x) (21)

where Pn(Pp) is the effective polarization of the neutron(proton) inside3He [57]. Functionsa(x) and b(x) areQ2-dependent and represent the nuclear shadowing and antishad-owing effects.

From Eq.(A12), the asymmetryA1 is approximately the ra-tio of the spin structure functiong1 andF1. Noting that shad-owing and antishadowing are not present in the largex region,using Eq. (21) one obtains

An1 =F

3He2

[

A3He1 − 2

Fp2

F3He2

PpAp1(1 − 0.0142Pp )

]

PnFn2 (1 +0.056Pn

). (22)

The two terms0.056Pn and0.0142Pp

represent the corrections toAn1associated with the∆(1232) component in the3He wave-function. Both terms causeAn1 to increase in thex rangeof this experiment, and to turn positive at lower values ofxcompared to the situation when the effect of the∆(1232) isignored. ForFn2 andF

3He2 , we used the world proton and

deuteronF2 data and took into account the EMC effects [90].We used the world proton asymmetry data forAp1. The ef-fective nucleon polarizationsPn,p can be calculated using3He wavefunctions constructed from N-N interactions, andtheir uncertainties were estimated using various nuclear mod-els [56, 57, 98, 99], giving

Pn = 0.86+0.036−0.02 and Pp = −0.028+0.009−0.004 . (23)

Eq. (22) was also used for extractingAn2 , gn1 /F

n1 andg

n2 /F

n1

from our 3He data. The uncertainty inAn1 due to the uncer-tainties inF p,d2 , in the correction for EMC effects, inA

p1 data

and inPn,p is given in Table X. Compared to the convolutionapproach [98] used by previous3He experiments [50, 51, 52],in which only the first two terms on the right hand side ofEq. (21) are present, the values ofAn1 extracted from Eq. (22)are larger by(1− 2)% in the region0.2 < x < 0.7.

G. Resonance Contributions

Since there are a few nucleon resonances with masses above2 GeV and our measurement at the highestx point has an in-variant mass close to2 GeV, the effect of possible contribu-tions from baryon resonances were evaluated. This was doneby comparing the resonance contribution togn1 with that toFn1 . For our kinematics atx = 0.6, data on the unpolarizedstructure functionF2 andR [95] show that the resonance con-tribution toF1 is less than5%. The resonance asymmetry wasestimated using the MAID model [96] and was found to beapproximately0.10 at W = 1.7 (GeV). Since the resonancestructure is more evident at smallerW , we took this value asan upper limit of the contribution atW = 2 (GeV). The res-onance contribution to ourAn1 andg

n1 /F

n1 results atx = 0.6

were then estimated to be at most0.008, which is negligiblecompared to their statistical errors.

VI. RESULTS

A. 3He Results

Results of the electron asymmetries for~e−3 ~He scattering,A

3He‖ andA

3He⊥ , the virtual photon asymmetriesA

3He1 and

A3He2 , structure function ratiosg

3He1 /F

3He1 and g

3He2 /F

3He1

and polarized structure functionsg3He1 andg

3He2 are given in

Table VII. Results forg3He1,2 were obtained by multiplying

the g3He1,2 /F

3He1 results by the unpolarized structure function

F3He1 , which were calculated using the latest world fits of

DIS data [83, 87] and with nuclear effects corrected [90].Results forA

3He1 andg

3He1 are shown in Fig. 18 along with

SLAC [51, 100] and HERMES [101] data.

16

[101]

[100]

[51]

[100][51]

FIG. 18: Results for the3He asymmetryA3He

1 and the structure functionsg3He

1 as a function ofx, along with previous data from SLAC [51,100] and HERMES [101]. Error bars of the results from this work include both statistical and systematic uncertainties.

TABLE VII: Results for 3He asymmetriesA3He

1 andA3He

2 , structure function ratiosg3He

1 /F3He

1 andg3He

2 /F3He

1 , and polarized structure

functionsg3He

1 andg3He

2 . Errors are given as± statistical± systematic.

〈x〉 0.33 0.47 0.60〈Q2〉 (GeV/c)2 2.71 3.52 4.83

A3He

‖ −0.020± 0.005 ± 0.001 −0.012 ± 0.005 ± 0.000 0.007 ± 0.007 ± 0.001

A3He⊥ 0.000 ± 0.010 ± 0.000 0.016 ± 0.008 ± 0.001 −0.010 ± 0.016 ± 0.001

A3He

1 −0.024± 0.006 ± 0.001 −0.019 ± 0.006 ± 0.001 0.010 ± 0.009 ± 0.001

A3He

2 −0.004± 0.014 ± 0.001 0.020 ± 0.012 ± 0.001 −0.013 ± 0.023 ± 0.001

g3He

1 /F3He

1 −0.022± 0.005 ± 0.001 −0.008 ± 0.008 ± 0.001 0.003 ± 0.009 ± 0.001

g3He

2 /F3He

1 0.010 ± 0.036 ± 0.002 0.050 ± 0.022 ± 0.003 −0.028 ± 0.038 ± 0.002

g3He

1 −0.024± 0.006 ± 0.001 −0.004 ± 0.004 ± 0.000 0.001 ± 0.002 ± 0.000

g3He

2 0.011 ± 0.039 ± 0.001 0.026 ± 0.012 ± 0.002 −0.006 ± 0.009 ± 0.001

B. Neutron Results

Results for the neutron asymmetriesAn1 andAn2 , structure function ratiosg

n1 /F

n1 andg

n2 /F

n1 and polarized structure functions

gn1 andgn2 are given in Table VIII.

TABLE VIII: Results for the asymmetries and spin structure functions for the neutron. Errors are given as±statistical±systematic.

〈x〉 0.33 0.47 0.60

〈Q2〉 (GeV/c)2 2.71 3.52 4.83

An1 −0.048± 0.024+0.015−0.016 −0.006 ± 0.027

+0.019−0.019 0.175 ± 0.048

+0.026−0.028

An2 −0.004± 0.063+0.005−0.005 0.117 ± 0.055

+0.012−0.021 −0.034 ± 0.124

+0.014−0.014

gn1 /Fn1 −0.043± 0.022

+0.009−0.009 0.040 ± 0.035

+0.011−0.011 0.124 ± 0.045

+0.016−0.017

gn2 /Fn1 0.034 ± 0.153

+0.010−0.010 0.207 ± 0.103

+0.022−0.021 −0.190 ± 0.204

+0.027−0.027

gn1 −0.012± 0.006+0.003−0.003 0.005 ± 0.004

+0.001−0.001 0.006 ± 0.002

+0.001−0.001

gn2 0.009 ± 0.043+0.003−0.003 0.026 ± 0.013

+0.003−0.003 −0.009 ± 0.009

+0.001−0.001

17

TheAn1 , gn1 /F

n1 andg

n1 results are shown in Fig. 19, 20 and

21, respectively. In the region ofx > 0.4, our results haveimproved the world data precision by about an order of mag-nitude, and will provide valuable inputs to parton distributionfunction (PDF) parameterizations. Our data atx = 0.33 are ingood agreement with previous world data. For theAn1 results,this is the first time that the data show a clear trend thatAn1turns to positive values at largex. As x increases, the agree-ment between the data and the predictions from the constituentquark models (CQM) becomes better. This is within the ex-pectation since the CQM is more likely to work in the valencequark region. It also indicates thatAn1 will go to higher valuesat x > 0.6. However, the trend of theAn1 results does notagree with the BBS and LSS(BBS) parameterizations, whichare from leading-order pQCD analyses based on hadron he-licity conservation (HHC). This indicates that there mightbeproblem in the assumption that quarks have zero orbital angu-lar momentum which is used by HHC.

x

A1n

E142 [51]

E154 [52]

HERMES [50]

This work

−0.5

0

0.5

1

0 0.2 0.4 0.6 0.8 1

FIG. 19: OurAn1 results along with theoretical predictions andprevious world data obtained from polarized3He targets [50, 51,52]. Curves: predictions ofAn1 from SU(6) symmetry (x axisat zero) [17], constituent quark model (shaded band) [28], statisti-cal model atQ2 = 4 (GeV/c)2 (long-dashed) [42], quark-hadronduality using two different SU(6) breaking mechanisms (dash-dot-dotted and dash-dot-dot-dotted), and non-meson cloudy bagmodel(dash-dotted) [48]; predictions ofgn1 /F

n1 from pQCD HHC based

BBS parameterization atQ2 = 4 (GeV/c)2 (higher solid) [30] andLSS(BBS) parameterization atQ2 = 4 (GeV/c)2 (dashed) [31],LSS 2001 NLO polarized parton densities atQ2 = 5 (GeV/c)2

(lower solid) [41] and chiral soliton models [43] atQ2 = 3 (GeV/c)2

(long dash-dotted) and [44] atQ2 = 4.8 (GeV/c)2 (dotted).

The sources for the experimental systematic uncertaintiesare listed in Table IX. Systematic uncertainties for theAn1 re-sults include that from experimental systematic errors, uncer-tainties in internal radiative corrections∆An,ir1 and externalradiative corrections∆An,er1 as derived from the values in Ta-

[25][53]

FIG. 20: Results forgn1 /Fn1 along with previous world data from

SLAC [25, 53]. The curves are the prediction forgn1 /Fn1 from the

LSS 2001 NLO polarized parton densities atQ2 = 5 (GeV/c)2 [41],the E155 experimental fit atQ2 = 5 (GeV/c)2 (long dash-dot-dotted) [53] and the new fit as described in the text (long dash-dot-dot-dotted).

This work[51][52]

[50][53]

FIG. 21: Results forgn1 along with previous world data fromSLAC [51, 52, 53] and HERMES [50].

TABLE IX: Experimental systematic errors for theAn1 result.

source errorBeam energyEb ∆Eb/Eb < 5× 10−4

HRS central momentump0 ∆Ee/Ee < 5× 10−4 [103]HRS central angleθ0 ∆θ0 < 0.1◦ [104]Beam polarizationPb ∆Pb/Pb < 3%Target polarizationPt ∆Pt/Pt < 4%Target spin directionαt ∆αt < 1◦

18

bles V and VI, and that from nuclear corrections as describedin Section V F. Table X gives these systematic uncertaintiesfor the An1 results along with their statistical uncertainties.The total uncertainties are dominated by the statistical uncer-tainties.

TABLE X: Total uncertainties forAn1 .

〈x〉 0.33 0.47 0.60Statistics 0.024 0.027 0.048Experimental syst. 0.004 0.003 0.004∆An,ir1 0.012 0.013 0.015∆An,er1 0.002 0.002 0.003F p2 , F

d2 0.006 0.008

+0.005−0.010

EMC effect 0.001 0.000 0.009Ap1 0.001 0.005 0.011Pn, Pp +0.005−0.012

+0.009−0.020

+0.018−0.037

We used five functional forms,xαPn(x)(1 + β/Q2), to fitour gn1 /F

n1 results combined with data from previous experi-

ments [25, 53]. HerePn is thenth-order polynomial,n = 1, 2for a finiteα orn = 1, 2, 3 if α is fixed to be0. The total num-ber of parameters is limited to6 5. For theQ2-dependence ofg1/F1, we used a term1+ β/Q2 as in the E155 experimentalfit [53]. No constraints were imposed on the fit concerningthe behavior ofg1/F1 asx → 1. The function which givesthe smallestχ2 value isgn1 /F

n1 = (a+ bx+ cx

2)(1+β/Q2).The new fit is shown in Fig. 20. Results for the fit parametersare given in Table XI and the covariance error matrix is

ǫ =

1.000 −0.737 0.148 0.960−0.737 1.000 −0.752 −0.5810.148 −0.752 1.000 −0.0390.960 −0.581 −0.039 1.000

.

TABLE XI: Result of the fitgn1 /Fn1 = (a+ bx+ cx

2)(1 + β/Q2).

a = −0.049 ± 0.052b = −0.162 ± 0.217c = 0.698 ± 0.345β = 0.751 ± 2.174

Similar fits were performed to the proton worlddata [25, 53, 54] and functiongp1/F

p1 = x

α(a+bx)(1+β/Q2)was found to give the smallestχ2 value. The new fit is shownin Fig. 2 of Section II G. Results for the fit parameters aregiven in Table XII and the covariance error matrix is

ǫ =

1.000 0.908 −0.851 0.7230.908 1.000 −0.967 0.401−0.851 −0.967 1.000 −0.3690.723 0.401 −0.369 1.000

.

TABLE XII: Result of the fitgp1/Fp1 = x

α(a+ bx)(1 + β/Q2).

α = 0.813 ± 0.049a = 1.231 ± 0.122b = −0.413 ± 0.216β = 0.030 ± 0.124

Figures 22 and 23 show the results forAn2 andxgn2 , respec-

tively. The precision of our data is comparable to the datafrom E155x experiment at SLAC [102], which is so far theonly experiment dedicated to measuringg2 with published re-sults.

To evaluate the matrix elementdn2 , we combined ourgn2 re-

sults with the E155x data [102]. The averageQ2 of the E155xdata set is about5 (GeV/c)2. Following a similar procedure asused in Ref. [102], we assumed thatḡ2(x,Q2) is independentof Q2 andḡ2 ∝ (1 − x)m with m = 2 or 3 for x >∼ 0.78 be-yond the measured region of both experiments. We obtainedfrom Eq. (6)

dn2 = 0.0062± 0.0028 . (24)

Compared to the value published previously [102], the uncer-tainty ondn2 has been improved by about a factor of two. Thelarge decrease in uncertainty despite the small number of ourdata points arises from thex2 weighting of the integral whichemphasizes the largex kinematics. The uncertainties on theintegrand has been improved in the regionx > 0.4 due to ourgn2 results at the two higherx points being more precise thanthat of E155x. While a negative value was predicted by latticeQCD [11] and most other models [12, 13, 14], the new resultfor dn2 suggests that the higher twist contribution is positive.

[102]

FIG. 22: Results forAn2 along with the best previous worlddata [102]. The curve gives the twist-2 contribution atQ2 =4 (GeV/c)2 calculated using the E155 experimental fit [53] andgWW2of Eq. (5).

19

[102]

FIG. 23: Results forxgn2 along with the best previous worlddata [102]. The curve gives the twist-2 contribution atQ2 =4 (GeV/c)2 calculated using the E155 experimental fit [53] andgWW2of Eq. (5).

C. Flavor Decomposition using the Quark-Parton Model

Assuming the strange quark distributionss(x), s̄(x),∆s(x)and∆s̄(x) to be small in the regionx > 0.3, and ignoringany Q2-dependence of the ratio of structure functions, onecan extract polarized quark distribution functions based on thequark-parton model as

∆u +∆ū

u+ ū=

4gp1(4 + Rdu)

15F p1− g

n1 (1 + 4R

du)

15Fn1(25)

and

∆d+∆d̄

d+ d̄=

4gn1 (1 + 4Rdu)

15Fn1 Rdu

− gp1(4 + R

du)

15F p1Rdu

, (26)

with Rdu ≡ (d+ d̄)/(u+ ū). Results for(∆u+∆ū)/(u+ū) and(∆d+∆d̄)/(d+ d̄) are given in Table XIII. As in-puts we used our own results forgn1 /F

n1 , the world data on

gp1/Fp1 [58], and the ratioR

du extracted from proton anddeuteron unpolarized structure function data [105]. In a sim-ilar manner as for Eq.(25) and (26) and ignoring nuclear ef-fects, one can also add the world data ong

2H1 /F

2H1 to the fit-

ted data set and extract these polarized quark distributions.The results are, however, consistent with those given in Ta-ble XIII and have very similar error bars because the data onthe deuteron in general have poorer precision than the data onthe proton and the neutron data from this experiment. Theresults presented here have changed compared to the valuespublished previously in Ref. [15] due to an error discoveredin our fitting ofRdu from Ref. [105]. The analysis procedureis consistent with what was used in Ref. [15].

Figure 24 shows our results along with semi-inclusivedata on(∆q + ∆q̄)/(q + q̄) obtained from recent resultsfor ∆q and ∆q̄ [106] by the HERMES collaboration, andthe CTEQ6M unpolarized PDF [107]. To estimate the ef-fect of the s and s̄ contributions, we used two unpolar-ized PDF sets, CTEQ6M [107] and MRST2001 [108], and

TABLE XIII: Results for the polarized quark distributions.The threeuncertainties are those due to thegn1 /F

n1 statistical error,g

n1 /F

n1

systematic uncertainty and the uncertainties of thegp1/Fp1 data, the

Rdu fit and the correction fors andc quark contributions.

〈x〉 (∆u+∆ū)/(u+ ū) (∆d+∆d̄)/(d+ d̄)

0.33 0.545 ± 0.004 ± 0.002+0.024−0.025 −0.352 ± 0.035 ± 0.014+0.017−0.031

0.47 0.649 ± 0.006 ± 0.002+0.058−0.058 −0.393 ± 0.063 ± 0.020+0.041−0.049

0.60 0.728 ± 0.006 ± 0.002+0.114−0.114 −0.440 ± 0.092 ± 0.035+0.107−0.142

[106,107]

FIG. 24: Results for(∆u+∆ū)/(u+ ū) and(∆d+∆d̄)/(d+ d̄)in the quark-parton model, compared with semi-inclusive data fromHERMES [106] and CTEQ unpolarized PDF [107] as described inthe text, the RCQM predictions (dash-dotted) [28], predictions fromLSS 2001 NLO polarized parton densities atQ2 = 5 (GeV/c)2

(solid) [41], the statistical model atQ2 = 4 (GeV/c)2 (long-dashed) [42], the pQCD-based predictions with the HHC constraint(dashed) [31], the duality model using two different SU(6) break-ing mechanisms (dash-dot-dotted and dash-dot-dot-dotted) [47], andpredictions from chiral soliton model atQ2 = 4.8 (GeV/c)2 (dot-ted) [44]. The error bars of our data include the uncertainties givenin Table XIII. The shaded band near the horizontal axis showsthedifference between∆qV /qV and(∆q +∆q̄)/(q + q̄) that needs tobe added to the data when comparing with the RCQM calculation.

three polarized PDF sets, AAC2003 [109], BB2002 [110] andGRSV2000 [111]. Forc andc̄ contributions we used the twounpolarized PDF sets [107, 108] and the positivity conditions

20

that |∆c/c| 6 1 and |∆c̄/c̄| 6 1. To compare with theRCQM predictions, which are given for valence quarks, thedifference between∆qV /qV and(∆q +∆q̄)/(q+ q̄) was es-timated using the two unpolarized PDF sets [107, 108] andthe three polarized PDF sets [109, 110, 111] and is shownas the shaded band near the horizontal axis of Fig. 24. HereqV (∆qV ) is the unpolarized (polarized) valence quark distri-bution foru or d quark. Results shown in Fig. 24 agree wellwith the predictions from the RCQM [28] and the LSS 2001NLO polarized parton densities [41]. The results agree rea-sonably well with the statistical model calculation [42]. Butresults for thed quark do not agree with the predictions fromthe leading-order pQCD LSS(BBS) parameterization [31] as-suming hadron helicity conservation.

VII. CONCLUSIONS

We have presented precise data on the neutron spinasymmetryAn1 and the structure function ratiog

n1 /F

n1 in the

deep inelastic region at largex obtained from a polarized3He target. These results will provide valuable inputs to theQCD parameterizations of parton densities. The new datashow a clear trend thatAn1 becomes positive at largex. Ourresults forAn1 agree with the LSS 2001 NLO QCD fit to theprevious data and the trend of thex-dependence ofAn1 agreeswith the hyperfine-perturbed RCQM predictions. Data onthe transverse asymmetry and structure functionAn2 andg

n2

were also obtained with a precision comparable to the bestprevious world data in this kinematic region. Combined withprevious world data, the matrix elementdn2 was evaluated andthe new value differs from zero by more than two standarddeviations. This result suggests that the higher twist con-tribution is positive. Combined with the world proton data,the polarized quark distributions(∆u + ∆ū)/(u + ū) and(∆d+∆d̄)/(d+ d̄) were extracted based on the quark partonmodel. While results for(∆u+∆ū)/(u+ ū) agree well withpredictions from various models and fits to the previous data,results for(∆d + ∆d̄)/(d + d̄) agree with the predictionsfrom RCQM and from the LSS 2001 fit, but do not agreewith leading order pQCD predictions that use hadron helicityconservation. Since hadron helicity conservation is basedonthe assumption that quarks have negligible orbital angularmomentum, the new results suggest that the quark orbitalangular momentum, or other effects beyond leading-orderpQCD, may play an important role in this kinematic region.

APPENDIX A: FORMALISM FOR ELECTRON DEEPINELASTIC SCATTERING

The fundamental quark and gluon structure of strongly in-teracting matter is studied primarily through experimentsthatemphasize hard scattering from the quarks and gluons at suf-ficiently high energies. One important way of probing the dis-tribution of quarks and antiquarks inside the nucleon is elec-tron scattering, where an electron scatters from a single quark

or antiquark inside the target nucleon and transfers a largefraction of its energy and momentum via exchanged photons.In the single photon exchange approximation, the electron in-teracts with the target nucleon via only one photon, as shownin Fig. 25 [6], and probes the quark structure of the nucleonwith a spatial resolution determined by the four momentumtransfer squared of the photonQ2 ≡ −q2. Moreover, if a po-larized electron beam and a polarized target are used, the spinstructure of the nucleon becomes accessible. In the following

( , q)νq = E

E’

FIG. 25: (Color online) Electron scattering in the one-photon ex-change approximation.

we denote the incident electron energy byE, the energy of thescattered electron byE′ thus the energy transfer of the photonis ν = E − E′, and the three-momentum transfer from theelectron to the target nucleus by~q.

1. Structure Functions

In the case of unpolarized electrons scattering off an unpo-larized target, the differential cross-section for detecting theoutgoing electron in a solid angledΩ and an energy range(E′, E′ + dE′) in the laboratory frame can be written as

d2σ

dΩdE′=

( dσ

dΩ

)

Mott·

[ 1

νF2(x,Q

2) +2

MF1(x,Q

2) tan2θ

2

]

,(A1)

whereθ is the scattering angle of the electron in the laboratoryframe. The four momentum transferQ2 is given by

Q2 = 4EE′ sin2θ

2, (A2)

and the Mott cross section,

( dσ

dΩ

)

Mott=

α2 cos2 θ24E2 sin4 θ2

=α2 cos2 θ2

Q4E′

E(A3)

with α the fine structure constant, is the cross section for scat-tering relativistic electrons from a spin-0 point-like infinitelyheavy target.F1(x,Q2) andF2(x,Q2) are the unpolarizedstructure functions of the target, which are related to eachother as

F1(x,Q2) =

F2(x,Q2)(1 + γ2)

2x(

1 +R(x,Q2)) (A4)

with γ2 = (2Mx)2/Q2. HereR is defined asR ≡ σL/σTwith σL andσT the longitudinal and transverse virtual photon

21

cross sections, which can also be expressed in terms ofF1 andF2.

Note that for a nuclear target, there exists an alternativepernucleondefinition (e.g. as used in Ref. [83]) which is1/Atimes the definition used in this paper, hereA is the numberof nucleons inside the target nucleus.

A review of doubly polarized DIS was given in Ref. [112].When the incident electrons are longitudinally polarize