Nucleon Spin Structure Functions g1 and g2 from
Polarized Inclusive Scattering
Todd D. Averett
College of William and Mary, Department of Physics, Williamsburg, VA 23187
Abstract. This paper will present a survey of recent experimental results for the g 1 and g2 spin
structure functions. Over the past decade, these structure functions (and the virtual photon asymmetries A1 and A2 ) have been well-measured in the large-Q 2 scaling region using inclusive polarized
deep-inelastic scattering. New precision results from Jefferson Lab are now becoming available
which cover kinematic regions that were previously poorly measured or completely unmeasured.
Topics covered will include: recent experiments at Jefferson Lab which have made precise measurements of g 1 in the resonance region for both proton and neutron to investigate the Q 2 evolution
of the GDH sum rule, a new measurement of A 1 for the neutron in the large-x region where valence
quark dynamics dominate, new precision results for g 2 from Jefferson Lab, a review of the g 1 results and NLO analyses in the scaling region for the SLAC, HERMES, and SMC data. Finally, a
survey of results expected from several new experiments and longer-term experimental programs
are discussed.
INTRODUCTION AND FORMALISM
Polarized inclusive lepton scattering has been extensively used as a tool for probing the
internal structure of the nucleon. Early experiments at SLAC and CERN discovered that
the spin of the proton could not be accounted for by the contribution from quark spin
alone. Since this early work, the idea of using polarized inclusive lepton scattering as
a probe of nucleon structure has matured into a mainstay of current nucleon structure
research. Experimental results over the last decade from SLAC, CERN, and DESY have
confirmed that the quark contribution to the nucleon spin is only 23% and have also
confirmed the prediction of the fundamental Bjorken Sum Rule. New experiments at
Jefferson lab continue to explore the spin structure of the nucleon by making precise
measurements of the spin structure function g 1 and g2 as well as the asymmetries A1 and
A2 . The experiments provide new information about nucleon structure through precise
measurements in both the deep inelastic and the resonance regions.
The basic inclusive process involves the exchange of a virtual photon between a
polarized lepton beam and a polarized proton, deuteron, or 3 He target. Only the scattered
electron is detected and uniquely determines the kinematics of the exchanged photon.
The virtual photon energy, ν , and momentum, q, are defined by the difference in initial
and final electron energy and momentum as ν E E and q k k . It is this
photon which probes the electromagnetic structure of the nucleon. The Lorentz invariant
Q2 q q ν 2 and the Bjorken scaling variable x Q2 2M ν are typically used to
describe the scattering process.
If the spin of the lepton and target are known before scattering, then the following
CP675, Spin 2002: 15th Int'l. Spin Physics Symposium and Workshop on Polarized Electron
Sources and Polarimeters, edited by Y. I. Makdisi, A. U. Luccio, and W. W. MacKay
© 2003 American Institute of Physics 0-7354-0136-5/03/$20.00
88
cross section differences can be used to extract the two nucleon spin structure functions
g1 x Q2 and g2 x Q2 as follows:
dσ
4α 2 E Q2
E E cos θ g1 x Q2 g2 x Q2 2
dΩdE
MQ ν E
ν
(1)
4α 2 E sin θ dσ
ν g1 x Q2 2Eg2 x Q2 (2)
2
dΩdE
MQ ν E
where the arrows refer to the lepton and nucleon spin respectively. Because the virtual
photon is the real probe of the nucleon, the spin structure functions g 1 and g2 can be
related to the asymmetries for virtual photon absorption as follows:
A1 x Q2 A2 x Q2 σT12 σT32
γ g2 x Q g1 x QF
32
x Q2
(3)
g2 x Q γ g1 x QF 32
x Q2 (4)
2
σT12 σT
2σT L
2
2
1
2
σT12 σT
2
1
where γ 2 Q2 ν and the σT refer to the absorption of a transverse virtual photon with
the total helicity of the photon-nucleon system of 12 or 32. The quantity σ T L is an
interference cross section for absorption of a transverse photon in the initial state and
longitudinal in the final state.
DEEP INELASTIC SCATTERING AND THE NUCLEON SPIN
PUZZLE
At large Q2 , the photon-nucleon interaction is less sensitive to QCD effects and may
be treated perturbatively. In the scaling limit, Q 2 ∞ and ν ∞ while Q2 ν remains
finite, the g1 structure function has simple interpretation in the quark parton model,
g1 x Q
2
N
1 f 2
ei ∆qi x Q2 ∆q̄i x Q2 ∑
2 i
(5)
where the sum is over quark flavors, ∆qi qi qi , and the qi represent the probability
of finding a quark of a given flavor with its spin parallel or anti-parallel to the spin of the
nucleon. By integrating g1 over all x we obtain (for the proton)
1
4
1
1
p
2
(6)
Γ1 Q g1p x Q2 dx ∆u ∆d ∆s
18
18
18
0
where the ∆u, ∆d, and ∆s are the net contributions of the quark flavors to the overall spin
of the nucleon. Using angular momentum conservation, a sum rule exists for the total
spin of the nucleon:
1
1
∆Σ ∆G Lz (7)
2
2
89
where ∆Σ, ∆G, and Lz are the total quark, gluon, and angular momentum contributions
to the spin of the nucleon. Taking the difference between proton and neutron yields the
famous Bjorken sum rule [1] (in the scaling limit),
Γ1pn Q2 0
1
g1p x Q2 gn1 x Q2 dx 1 gA
6 gV
(8)
where gA gV is the ratio of axial to vector coupling constants. This sum rule can also
be extended to finite Q2 using perturbative QCD and is considered a rigorous test of the
quark-parton model and QCD.
Experimental and Theoretical Results
Over the past two decades there has been an extensive experimental program at SLAC,
DESY, and CERN to accurately measure g1 for the proton, deuteron, and 3 He over
the widest kinematic range possible. Published results are summarized in the paper by
Blümlein and Böttcher [2] and are reproduced in Table 1. Data from the most recent
SLAC measurement [3] on proton and deuteron are shown in Figure 1.
TABLE 1. Summary of experimental programs and their kinematic coverage measuring the spin structure functions g 1 and g2
in the deep inelastic region.
Experiment
x–range
Q 2 –range (GeV 2 )
Ref.
0.027 – 0.749
0.028 – 0.660
0.015 – 0.750
0.005 – 0.480
0.015 – 0.466
1.17 – 9.52
1.13 – 7.46
1.22 – 34.72
1.30 – 58.0
3.50 – 29.5
[5]
[6]
[3]
[7]
[8]
0.027 – 0.749
0.015 – 0.750
0.005 – 0.479
1.17 – 9.52
1.22 – 34.79
1.30 – 54.8
[5]
[9]
[7]
0.035 – 0.466
0.033 – 0.464
0.017 – 0.564
1.10 – 5.50
1.22 – 5.25
1.20 – 15.0
[10]
[11]
[12]/[13]
Proton
E143(p)
HERMES(p)
E155(p)
SMC(p)
EMC(p)
Deuteron
E143(d)
E155(d)
SMC(d)
Neutron
E142(n)
HERMES(n)
E154(n)
One of the best tools for interpreting this data is through a next-to-leading-order
(NLO) analysis. In this type of analysis, functional forms are chosen to describe the
spin dependent parton distributions as a function of x. The world data is then fit and
Q2 dependence is accounted for using the DGLAP evolution method [4]. Recent results
from the NLO analysis by Blümlein and Böttcher [2] were presented.
90
FIGURE 1. SLAC results for g 1p , gn1 , and g1p n (using proton and deuteron data) at Q 2 5 GeV2 are
shown as solid circles. Data from E143 (open circles), SMC (squares), and HERMES (stars) are also
shown. The solid and dashed curves are NLO QCD fits and a simple parameterization, respectively [3].
A1 in the valence region
The measurements presented above provide a wealth of data and information about
the spin content of the nucleon. However, in the large x region, which is dominated
by valence quark dynamics, the data quality remains poor, particularly for the neutron.
This is because 1) the large-x region is typically also at large Q 2 where the cross
section is small, and 2) previous experiments have focused on the low-x region to test
the convergence of the Bjorken sum rule as x 0. The large-x region provides an
important opportunity to test basic models of nucleon structure which are largely free
from the effects of sea quark pairs. In this region, it is more interesting to study the
asymmetry A1 which is essentially dominated by the absorption of a virtual photon by
a single quark which flips its spin. The simplest model for the nucleon in this region
is the SU(6) symmetric wave function which predicts A n1 0 for all x. More realistic
models of the nucleon will allow SU(6) symmetry breaking by the addition of some sort
of interaction between the valence quarks. Popular models include constituent quark
models (CQM) [14] where quarks interact by a hyperfine, or spin-spin, interaction.
Models based on pQCD analyses, LSS(BBS,HHC) [15] and LSS 2001 [16] use polarized
parton distributions and DGLAP evolution to predict the large-x behavior. Both the CQM
and pQCD models predict An1 1 as x 1. Other models include a statistical models
91
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
FIGURE 2. An1 results compared with theoretical predictions and existing world data. The nuclear
target used by each experiment is shown in brackets. Curves: predictions of g n1 F1n from pQCD HHC
based LSS(BBS) parameterization (1) and BBS parameterization (2); g n1 F1n from chiral soliton model
at Q2 3 (GeV/c)2 (3); predictions of A n1 from constituent quark model (shaded band) (4); g n1 F1n
from LSS 2001 parameterization at Q 2 5 (GeV/c)2 (5); predictions of A n1 from statistical model at
Q2 4 (GeV/c)2 (6), meson-cloud bag model at Q 2 3 (GeV/c)2 (7), and from basic SU(6) symmetry
(8).
based on a NLO fit of world data [17], chiral soliton model [18], and meson cloudy
bag model [19]. All models predict A n1 will be positive for x 04, and most predict a
dramatic rise as x 1. The existing world data give no clear indication of an increase as x
increases, and instead seem to show a slightly negative trend. Because the structure of the
nucleon should become less complicated in the valence region, a precision measurement
of the x dependence of An1 at large x provides fertile ground for testing basic models of
the nucleon.
Experiment E99-117 ran in Hall A at Jefferson Lab in the summer of 2001. Polarized
electrons were scattered from a polarized 3 He target and were detected in one of two
independent high-resolution spectrometers at a scattering angle of 155 Æ . From this
data, An1 was determined at x 033, 048, and 061 with corresponding Q 2 27, 36,
and 49 GeV2 . The preliminary data, shown in Figure 2 are most consistent with the
statistical [17] and LSS 2001 pQCD [16] models.
92
The g2 structure function
All of the results presented up to this point have been focused on g 1 or A1 which have
relative simple parton model interpretations in the deep-inelastic region. The g 2 spin
structure function however, does not. This structure function is sensitive to higher-twist
effects, such as quark-gluon correlations, and is best interpreted in the framework of
the Operator Product Expansion (OPE) [20, 21]. Here, the unknown hadronic current
is expanded in a series of operators with coefficients which are not easily calculated
due to their non-perturbative nature. The terms are grouped according to their twist,
with terms of increasing twist successively suppressed by powers of 1Q. The leading
term is twist-2 and is the dominant contribution to both g 1 and g2 . It corresponds to
scattering from a single, non-interacting, quark. The next term is twist-3, and is only
present in g2 at leading order in αS . The twist-3 term includes contributions from
scattering from a quark which is simultaneously exchanging a gluon with the rest of the
nucleon. It is an inherently non-perturbative effect and can be isolated by measuring g 2
and subtracting the twist-2 contribution using the following relationship from Wandzura
and Wilczek [22]:
2
2
gww
2 x Q g1 x Q 1
x
g1 y Q2
dy
y
(9)
This expression provides a method of calculating the twist-2 part of g 2 only using
measured data from g1 due to the fact that g1 and g2 both contain the same twist-2
matrix element. By making a precise measurement of g 2 and comparing it to the gww
2
calculation, one can extract the higher twist contributions to the nucleon structure.
Jefferson Lab experiment E97-103 also ran during the summer of 2002 in Hall A. A
transversely polarized 3 He target was used to make a series of precise measurements
of g2 at x 02 in the range 058 Q2 136 GeV2 . This measurement made an
improvement in statistical errors by a factor of 15 over previous measurements and
will allow us, for the first time, to make definitive statements about the size of these
higher twist effects. Preliminary data (errors only) are shown in Figure 3 along with
n
calculations of the twist-2 gww
2 contribution using models of g 1 [3, 23, 2]. Recent g2 data
for proton and deuteron from SLAC experiment E155x [24] is also now available and
was presented at the conference. It is further discussed in these proceedings by S. Rock.
RESONANCE REGION
The experiments presented above were all measurements in the deep-inelastic region
where QCD effects can be largely treated perturbatively. As one begins to lower Q2 , the
nucleon begins to respond more as a whole object. Specifically, at low Q 2 , the nucleon
resonances begin to become prominent and the scattering can no longer be interpreted
using simple quark models and pQCD. At Q 2 0, the photon which is exchanged
between the electron and nucleon is real and the following fundamental sum rule, first
derived by Gerasimov, and also by Drell and Hearn [25], allows one to relate the photonnucleon absorption cross sections to bulk properties of the nucleon such as its mass M
93
FIGURE 3. Expected errors for g n2 as a function of Q 2 from Jefferson Lab experiment E97-103. Twist-2
n
gww
2 predictions using various models for g 1 are also shown.
and anomalous magnetic moment, κ .
I Q2 0 ∞
ν0
σ12 ν σ32 ν dν
ν
π α 2
2M
κ
2
2
(10)
The left side of the GDH sum rule can be generalized to Q2 0 by replacing the real
photon cross sections by the virtual photon absorption cross sections, σ T12 ν Q2 and
σT32 ν Q2 . To generalize the right hand side, it was pointed out by Ji and Osborne [26]
that the both the Bjorken and the GDH sum rules can be related to the the virtual
Compton amplitude S1 Q2 . This quantity can be calculated at very low Q2 using chiral
perturbation theory and at very large Q2 using perturbative QCD. This important relation
provides the formalism needed to connect the very low Q 2 behavior where the nucleon
is described by hadronic degrees of freedom, to the large Q2 region where the quarks are
the relevant degrees of freedom. Clearly, a measurement of the Q2 dependence of the
GDH integral for Q2 1 GeV2 provides a powerful tool to investigate the behavior of
the nucleon in the transition region between hadron-like and quark-like behavior.
Experiment E94-010 at Jefferson Lab recently published the first results [27] on
the Q2 evolution of the GDH sum rule for the neutron in the region 01 Q 2 09
GeV2 . This measurement was made by inclusively scattering longitudinally polarized
electrons from a longitudinal or transversely polarized 3 He target, from the quasi-elastic,
through resonance, and into the deep-inelastic regions. Figure 4 shows the results of this
experiment, which for the first time show the expected decrease towards the real GDH
sum rule prediction as Q2 0. This decrease is largely due to the increasing asymmetry
from the ∆1232 resonance as Q2 decreases.
94
This data provides clear motivation to continue this measurement to even lower Q 2
to study the approach to Q2 0. Experiment E97-110 will soon measure the neutron
GDH sum rule from Q2 03 GeV2 down to Q2 002 GeV2 . Expected results from
this experiment are shown in Figure 5.
50
0
2
I(Q ) (µb)
−100
−200
Resonance
Resonance+DIS
GDH Sum Rule
Hermes (DIS)[30]
−300
0
0.2
0.4
0.6
0.8
2
2
Q (GeV )
1
10
FIGURE 4. Measurements of the neutron GDH integral from JLab experiment E94-010 are shown
with open circles. Black squares include a contribution from the deep inelastic region. Also shown are
the model predictions of Drechsel [28] (solid line), Ji [26] (dashed), and Bernard [29] (dash-dotted). The
HERMES data [30] is shown by open diamonds on a log scale.
Preliminary results from Jefferson Lab Hall B were presented at the conference where
the low Q2 behavior of the GDH sum rule for the proton was measured using a polarized
ammonia target. These results are presented by G. Dodge in these proceedings.
Spin Duality
A new experiment is now underway at Jefferson Lab, again using the polarized 3 He
to look for quark-hadron duality in the neutron spin structure function g 1 . Duality has
been observed in the unpolarized case [31] where the quark-like behavior of the nucleon
(F2 x structure function ) in the deep inelastic region is an accurate average of the
resonance structure seen at low Q2 . Experiment E01-012 is currently measuring g 1 x in
the resonance region up to Q2 54 GeV2 . This resonance data will be compared to g1
in the deep-inelastic region where smooth, non-resonant behavior is observed. If duality
holds, the curves for g1 in the resonance region will follow the trend of the smooth deepinelastic curve. A plot showing an example of expected results is shown in Figure 6.
95
GDH Integral (µb)
0
Bernard et al. (Delta included)
Bernard et al.
Drechsel et al.
Ji et al.
GDH Sum Rule
Resonance+DIS (JLab E94010)
JLab E97110
−100
−200
−300
0.001
0.010
0.100
2
1.000
2
Q (GeV )
g1n
FIGURE 5. Expected errors for the GDH integral from Jefferson Lab experiment E97-110 are shown
(diamonds) as a function of Q 2 (log scale) at low Q2 . Published data from the previous JLab measurement
(E94-010) at larger Q 2 are shown as blue squares [27]. Model predictions from Bernard [29], Ji [26] and
Drechsel [28] are also shown.
0
-0.025
-0.05
-0.075
-0.1
-0.125
-0.15
-0.175
-0.2
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
x
FIGURE 6. Exected errors only for from JLab experiment E01-012 at
13 GeV2 are shown as
black triangles. Also shown is the resonance and deep inelastic data from SLAC experiment E143 [5] and
E155 [3].
g n1
Q2
96
REFERENCES
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
J.D. Bjorken, Phys. Rev. 148 (1966) 1467; Phys. Rev. D1 (1970) 1376.
J. Blümlein and H. Böttcher, hep-ph/0203155, and Nuc. Phys. B636 (2002) 225.
P.L. Anthony et al., E155, Phys. Lett. B493 (2000) 19.
V.N. Gribov and L.N. Lipatov, Sov. J. Nucl. Phys. 15 (1972) 438, 675; Yu L. Dokshitzer, Sov. Phys.
JTEP 46 (1977) 641; G. Altarelli and G. Parisi, Nucl. Phys. B126 (1977) 298.
K. Abe et al., E143, Phys. Rev. D58 (1998) 120003.
A. Airapetian et al., HERMES collaboration, Phys. Lett. B442 (1998) 484
B. Adeva et al., SMC collaboration, Phys. Rev. D58 (1998) 112001.
J. Ashman et al., EMC collaboration, Phys. Lett. B206 (1988) 364; Nucl. Phys. B328 (1989) 1.
P.L. Anthony et al., E155, Phys. Lett. B463 (1999) 339.
P.L. Anthony et al., E142, Phys. Rev. D54 (1996) 6620.
K. Ackerstaff et al., HERMES collaboration, Phys. Lett. B404 (1997) 383.
K. Abe et al., E154, Phys. Rev. Lett. 79 (1997) 26.
K. Abe et al. (E154), Phys. Lett. B405 (1997) 180.
N. Isgur, Phys. Rev. D59 (1999) 034013.
E. Leader, A.V. Sidorov, and D.B. Stamenov, hep-ph/9708335.
E. Leader, A.V. Sidorov, and D.B. Stamenov, hep-ph/0111267.
C, Bourrely, J. Soffer, and F. Buccella, Eur. Phys. J C23 (2002) 479; hep-ph/0109160v1.
H. Weigel, L. Gamberg, and H. Reinhardt, Phys. Lett. B399 (1997) 287; Phys. Rev. D55 (1997) 6910.
F.M. Steffens, H. Holtmann, and A.W. Thomas, Phys. Lett. B358 (1995) 139.
E. Shuryak and A. Vainshtein, Nuc. Phys. B201 (1982) 141.
R. Jaffee and X. Ji, Phys. Rev. D43 (1991) 724.
S. Wandzura and F. Wilczek, Phys. Lett. B72 (1977) 195.
M. Gluck, E. Reya, and A. Vogt, Eur. Phys. J. C5 (1998) 461.
P.L. Anthony, et al., Phys. Lett. B553 (2003) 18.
S.B. Gerasimov, Sov. J. of Nucl. Phys. 2 (1966) 430; S. D. Drell and A.C. Hearn, Phys. Rev. Lett. 16
(1966) 908.
X. Ji, C. Kao, and J. Osborne, Phys. Lett. B472 (2000) 1.
M. Amarian, et al., Phys. Rev. Lett. 89 (2002) 242301.
D. Drechsel, SS. Kamalov, and L. Tiator, Phys. Rev. D3 (2001) 114010. Note, model I c is plotted.
V. Bernard, T.R. Hemmert, and Ulf.-G. Meissner, hep-ph/0203167.
K. Ackerstaff et al., HERMES collaboration, Phys. Lett. B444 (1998) 531.
I. Niculescu et al., Phys. Rev. Lett. 85 (2000) 1186; I. Niculescu et al., Phys. Rev. Lett. 85 (2000)
1182.
97