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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
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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
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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)
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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
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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-
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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