Measurement of the virtual-photon asymmetry A2 and the spin-structure function g2 of the proton

Abstract

A measurement of the virtual-photon asymmetry A_2(x,Q^2) and of the
spin-structure function g_2(x,Q^2) of the proton are presented for the
kinematic range 0.004 < x < 0.9 and 0.18 GeV^2 < Q^2 < 20 GeV^2. The data were
collected by the HERMES experiment at the HERA storage ring at DESY while
studying inclusive deep-inelastic scattering of 27.6 GeV longitudinally
polarized leptons off a transversely polarized hydrogen gas target. The results
are consistent with previous experimental data from CERN and SLAC. For the
x-range covered, the measured integral of g_2(x) converges to the null result
of the Burkhardt-Cottingham sum rule. The x^2 moment of the twist-3
contribution to g_2(x) is found to be compatible with zero.

Full text

Page 1

arXiv:1112.5584v2 [hep-ex] 27 Feb 2012
EPJ manuscript No.
(will be inserted by the editor)
Measurement of the virtual-photon asymmetry A2 and the
spin-structure function g2of the proton
The HERMES Collaboration
A. Airapetian12,15, N. Akopov26, Z. Akopov5, E.C. Aschenauer6,a, W. Augustyniak25, R. Avakian26,
A. Avetissian26, E. Avetisyan5, S. Belostotski18, N. Bianchi10, H.P. Blok17,24, A. Borissov5,
J. Bowles13, V. Bryzgalov19, J. Burns13, M. Capiluppi9, G.P. Capitani10, E. Cisbani21, G. Ciullo9,
M. Contalbrigo9, P.F. Dalpiaz9, W. Deconinck5, R. De Leo2, L. De Nardo11,5, E. De Sanctis10,
M. Diefenthaler14,8, P. Di Nezza10, M. D¨ uren12, M. Ehrenfried12, G. Elbakian26, F. Ellinghaus4,
A. Fantoni10, L. Felawka22, S. Frullani21, D. Gabbert6, G. Gapienko19, V. Gapienko19, F. Garibaldi21,
G. Gavrilov5,18,22, V. Gharibyan26, F. Giordano5,9, S. Gliske15, M. Golembiovskaya6, C. Hadjidakis10,
M. Hartig5, D. Hasch10, A. Hillenbrand6, M. Hoek13, Y. Holler5, I. Hristova6, Y. Imazu23,
A. Ivanilov19, H.E. Jackson1, H.S. Jo11, S. Joosten14, R. Kaiser13,b, G. Karyan26, T. Keri13,12,
E. Kinney4, A. Kisselev18, V. Korotkov19, V. Kozlov16, P. Kravchenko8,18, V.G. Krivokhijine7,
L. Lagamba2, L. Lapik´ as17, I. Lehmann13, P. Lenisa9, A. L´ opez Ruiz11, W. Lorenzon15, B.-Q. Ma3,
D. Mahon13, N.C.R. Makins14, S.I. Manaenkov18, L. Manfr´ e21, Y. Mao3, B. Marianski25, A. Martinez
de la Ossa5,4, H. Marukyan26, C.A. Miller22, Y. Miyachi23,c, A. Movsisyan26, V. Muccifora10,
M. Murray13, A. Mussgiller5,8, E. Nappi2, Y. Naryshkin18, A. Nass8, M. Negodaev6, W.-D. Nowak6,
L.L. Pappalardo9, R. Perez-Benito12, A. Petrosyan26, P.E. Reimer1, A.R. Reolon10, C. Riedl6,
K. Rith8, G. Rosner13, A. Rostomyan5, J. Rubin1,14, D. Ryckbosch11, Y. Salomatin19, F. Sanftl23,20,
A. Sch¨ afer20, G. Schnell6,11,d, K.P. Sch¨ uler5, B. Seitz13, T.-A. Shibata23, V. Shutov7, M. Stancari9,
M. Statera9, E. Steffens8, J.J.M. Steijger17, J. Stewart6, F. Stinzing8, S. Taroian26, A. Terkulov16,
R. Truty14, A. Trzcinski25, M. Tytgat11, A. Vandenbroucke11, Y. Van Haarlem11, C. Van Hulse11,
D. Veretennikov18, V. Vikhrov18, I. Vilardi2, S. Wang3, S. Yaschenko6,8, Z. Ye5, S. Yen22,
V. Zagrebelnyy5,12, D. Zeiler8, B. Zihlmann5, P. Zupranski25
1Physics Division, Argonne National Laboratory, Argonne, Illinois 60439-4843, USA
2Istituto Nazionale di Fisica Nucleare, Sezione di Bari, 70124 Bari, Italy
3School of Physics, Peking University, Beijing 100871, China
4Nuclear Physics Laboratory, University of Colorado, Boulder, Colorado 80309-0390, USA
5DESY, 22603 Hamburg, Germany
6DESY, 15738 Zeuthen, Germany
7Joint Institute for Nuclear Research, 141980 Dubna, Russia
8Physikalisches Institut, Universit¨ at Erlangen-N¨ urnberg, 91058 Erlangen, Germany
9Istituto Nazionale di Fisica Nucleare, Sezione di Ferrara and Dipartimento di Fisica, Universit` a di Ferrara, 44100 Ferrara, Italy
10Istituto Nazionale di Fisica Nucleare, Laboratori Nazionali di Frascati, 00044 Frascati, Italy
11Department of Physics and Astronomy, Ghent University, 9000 Gent, Belgium
12Physikalisches Institut, Universit¨ at Gießen, 35392 Gießen, Germany
13SUPA, School of Physics and Astronomy, University of Glasgow, Glasgow G12 8QQ, United Kingdom
14Department of Physics, University of Illinois, Urbana, Illinois 61801-3080, USA
15Randall Laboratory of Physics, University of Michigan, Ann Arbor, Michigan 48109-1040, USA
16Lebedev Physical Institute, 117924 Moscow, Russia
17National Institute for Subatomic Physics (Nikhef), 1009 DB Amsterdam, The Netherlands
aNow at: Brookhaven National Laboratory, Upton, New
York 11772-5000, USA
bPresent address: International Atomic Energy Agency, A-
1400 Vienna, Austria
cNow at: Department of Physics, Yamagata University, Ya-
magata 990-8560, Japan
dNow at: Department of Theoretical Physics, University
of the Basque Country UPV/EHU, 48080 Bilbao, Spain and
IKERBASQUE, Basque Foundation for Science, 48011 Bil-
bao, Spain

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18Petersburg Nuclear Physics Institute, Gatchina, 188300 Leningrad Region, Russia
19Institute for High Energy Physics, Protvino, 142281 Moscow Region, Russia
20Institut f¨ ur Theoretische Physik, Universit¨ at Regensburg, 93040 Regensburg, Germany
21Istituto Nazionale di Fisica Nucleare, Sezione di Roma, Gruppo Collegato Sanit` a and Istituto Superiore di Sanit` a, 00161
Roma, Italy
22TRIUMF, Vancouver, British Columbia V6T 2A3, Canada
23Department of Physics, Tokyo Institute of Technology, Tokyo 152, Japan
24Department of Physics and Astronomy, VU University, 1081 HV Amsterdam, The Netherlands
25National Centre for Nuclear Research, 00-689 Warsaw, Poland
26Yerevan Physics Institute, 375036 Yerevan, Armenia
Received: February 28, 2012/ Revised version:
Abstract. A measurement of the virtual-photon asymmetry A2(x,Q2) and of the spin-structure function
g2(x,Q2) of the proton are presented for the kinematic range 0.004 < x < 0.9 and 0.18 GeV2< Q2< 20
GeV2. The data were collected by the HERMES experiment at the HERA storage ring at DESY while
studying inclusive deep-inelastic scattering of 27.6 GeV longitudinally polarized leptons off a transversely
polarized hydrogen gas target. The results are consistent with previous experimental data from CERN
and SLAC. For the x-range covered, the measured integral of g2(x) converges to the null result of the
Burkhardt–Cottingham sum rule. The x2moment of the twist-3 contribution to g2(x) is found to be
compatible with zero.

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2
The description of inclusive deep-inelastic scattering
of longitudinally polarized charged leptons off polarized
nucleons requires two nucleon spin-structure functions,
g1(x,Q2) and g2(x,Q2), in addition to the well-known
structure functions F1(x,Q2) and F2(x,Q2) [1]. Here,
−Q2is the squared four-momentum of the exchanged vir-
tual photon with laboratory energy ν, x = Q2/(2Mν)
is the Bjorken scaling variable, and M is the nucleon
mass. In the quark-parton model (QPM), the spin struc-
ture function g1(x,Q2) can be interpreted as a charge-
weighted sum of the quark helicity distributions ∆q(x,Q2)
describing a longitudinally polarized nucleon,
g1(x,Q2) =1
2
?
q
e2
q∆q(x,Q2). (1)
The spin structure function g2(x,Q2) does not have
such a probabilistic interpretation in the QPM. Its prop-
erties can be interpreted in the framework of the operator
product expansion (OPE) analysis [2,3,4], which shows
that g2(x,Q2) is related to the matrix elements of both
twist-2 and twist-3 operators. Neglecting quark mass ef-
fects, g2(x,Q2) can be written as a sum of two terms
g2(x,Q2) = gWW
2
(x,Q2) + ¯ g2(x,Q2). (2)
Here, gWW
and Wilczek [5]
2
(x,Q2) is the twist-2 part derived by Wandzura
gWW
2
(x,Q2) = −g1(x,Q2) +
?1
x
g1(y,Q2)dy
y
. (3)
The second term in Eq. (2), ¯ g2(x,Q2), is the twist-3 part
of g2(x,Q2). It arises from quark-gluon correlations in the
nucleon and is the most interesting part of the function.
The x2moment of ¯ g2(x,Q2),
d2(Q2) = 3
?1
0
x2¯ g2(x,Q2)dx, (4)
can be calculated on the lattice (see, e.g., [6,7], where d2
is defined with an additional factor of two with respect
to (4)). The moment d2has also been linked to the trans-
verse force acting on the quark that absorbed the virtual
photon in a transversely polarized nucleon, and thus to
the Sivers effect [8,9,10].
The Burkhardt–Cottingham sum rule [11] for g2 at
large Q2,
?1
0
does not follow from the OPE. Its validity relies on an
assumed Regge behaviour of g2at low x. In the absence of
higher twist contributions to the function g2, i.e., ¯ g2(x) ≡
0, the sum rule would automatically be fulfilled. Hence a
violation of the sum rule would indicate the presence of
higher-twist contributions.
The spin structure functions g1(x,Q2) and g2(x,Q2)
can be related to the virtual photon-absorption asymme-
tries A1(x,Q2) and A2(x,Q2) [1]
g2(x,Q2)dx = 0, (5)
A1=
σT
1/2− σT
σT
3/2
1/2+ σT
2σLT
σT
3/2
=g1− γ2g2
F1
, (6)
A2=
1/2+ σT
3/2
= γg1+ g2
F1
. (7)
Here, σT
sorption cross sections for total photon plus nucleon an-
gular momentum projection on the photon direction of
1/2 and 3/2, respectively. The cross-section σLTarises
from the interference between the transverse and longitu-
dinal photon-nucleon amplitudes, with γ = 2Mx/?Q2.
All of the σ’s are differential cross sections depending on
x and Q2, but this dependence was omitted for brevity.
The measurement of the structure function g2requires
a longitudinally polarized beam and a transversely po-
1/2and σT
3/2are the transverse virtual-photon ab-

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larized target. In this case, the inclusive differential cross
section can be represented as a sum of two terms, the
polarization-averaged part, σUU, and the polarization-
related part, σLT. Here, the subscript UU indicates that
both the beam and the target are unpolarized, while the
subscript LT indicates a longitudinally polarized beam
and a transversely polarized target. The polarization-
related part of the cross section at Born level, i.e., in
the one-photon approximation, is given by [3]
d3σLT
dxdydφ
= −hlcosφ4α2
Q2γ
?
1 − y −γ2y2
4
×
?y
2g1(x,Q2) + g2(x,Q2)
?
. (8)
Here, hl = +1 (−1) for a lepton beam with positive
(negative) helicity, α is the fine-structure constant, and
y = ν/E, where E is the incident lepton energy. The an-
gle φ is the azimuthal angle about the beam direction be-
tween the lepton scattering plane and the “upwards” tar-
get spin direction. The polarization-related cross section
σLT is significantly smaller than the polarization-avera-
ged part σUUand therefore its measurement requires high
statistical precision. Up to now, the function g2and the
asymmetry A2have been extracted [12,13,14] to less ac-
curacy than g1and A1.
A measurement of the inclusive cross sections (8) at
angles φ and φ+π allows one to construct the asymmetry
ALT,
ALT(x,Q2,φ) = hlσ(x,Q2,φ) − σ(x,Q2,φ + π)
σ(x,Q2,φ) + σ(x,Q2,φ + π)
σLT(x,Q2,φ)
σUU(x,Q2,φ)= −AT(x,Q2) cosφ, (9)
= hl
which defines the asymmetry amplitude AT(x,Q2). This
amplitude contains all information on the function g2and
the asymmetry A2. Their extraction requires the knowl-
edge of σUU(x,Q2), which can be expressed by the struc-
ture functions F1,2(x,Q2) or, equivalently, parameteriza-
tions of the function F2(x,Q2) and the ratio of longitudi-
nal to transverse virtual-photon absorption cross sections
R = R(x,Q2). The extraction of the structure function
g2(x,Q2) from the asymmetry amplitude AT is analo-
gous to the extraction of g1(x,Q2) from the longitudinal
asymmetry as described in [15]. The function g2can be
extracted from the measured asymmetry amplitude AT
and parameterizations of previous measurements of σUU
and g1, using (8) and (9). Also F1can be computed from
parameterizations of F2and R. This leads with (7) to the
following relations
g2=
F1
γ(1 + γξ)
?AT
d
− (γ − ξ)g1
F1
?
, (10)
A2=
1
1 + γξ
?AT
d
+ ξ(1 + γ2)g1
F1
?
, (11)
with
d =
?1 − y − γ2y2/4
(1 − y/2)
γ(1 − y/2)
(1 + γ2y/2),
D,(12)
ξ = (13)
D =
y(2 − y)(1 + γ2y/2)
y2(1 + γ2) + 2(1 − y − γ2y2/4)(1 + R). (14)
However, it is not obvious from these relations that the
extraction of g2is independent of correlated variations in
values of F1, F2 and R that conserve the directly mea-
sured values of σUU.
This paper reports a new measurement of the func-
tion g2and the asymmetry A2. The data were collected
during the years 2003 – 2005 with the HERMES spec-
trometer [16] using a longitudinally polarized positron or
electron beam of energy 27.6 GeV scattered off a trans-
versely polarized target [17] of pure hydrogen gas inter-
nal to the HERA lepton storage ring at DESY. The us-
age of a pure target avoids the complications of nuclear
corrections present in previous measurements. The open-
ended target cell was fed by an atomic-beam source [18]
based on Stern–Gerlach separation combined with radio-
frequency transitions between hydrogen hyperfine states.
The nuclear polarization of the atoms was flipped at 1–3
minute time intervals, while both the polarization mag-
nitude and the atomic fraction inside the target cell were
continuously measured [19]. The average magnitude of
the proton polarization was 0.78±0.04. The lepton beam
(positrons during 2003 – 2004 and electrons in 2005) was
self-polarized in the transverse direction due to the asym-
metry in the emission of synchrotron radiation [20] in the
arcs of the HERA storage ring. Longitudinal orientation
of the beam polarization was obtained by using a pair
of spin rotators [21] located before and after the inter-
action region of the HERMES spectrometer. The sign
of the beam polarization was reversed every few months.
The beam polarization was measured by two independent
HERA polarimeters [22,23,24]. The averagemagnitude of
the beam polarization was found to be 0.34 ± 0.01. The
scattered leptons were detected by the HERMES spec-
trometer within an angular acceptance of ±170 mrad
horizontally and ±(40 − 140) mrad vertically. The lep-
tons were identified using the information from an elec-
tromagnetic calorimeter, a transition-radiation detector,
a preshower scintillating counter and a dual-radiator ring-
imaging˘Cerenkov detector. The identification efficiency
for leptons with momentum larger than 2.5 GeV exceeds
98%, while the hadron contamination in the lepton sam-
ple is found to be less than 1%. The luminosity monitor
[25] measured e+e−(e−e−) pairs from Bhabha (Møller)
scattering of beam positrons (electrons) off the target gas
electrons, and γγ pairs from e+e−annihilation in two
NaBi(WO4)2 electromagnetic calorimeters, which were
mounted symmetrically on either side of the beam line.
Tracking corrections were applied for the deflections of
the scattered particles caused by the vertical 0.3 T target

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holding field, with little effect on the extracted asymme-
tries.
Most of the details of the analysis follow the inclusive
analysis described in [15]. The kinematic constraints im-
posed on the events were: 0.18 GeV2< Q2< 20 GeV2,
invariant mass of the virtual photon–nucleon system W >
1.8 GeV, 0.004 < x < 0.9, and 0.10 < y < 0.91. After
applying data quality criteria, 10.2 × 106events were
available for the asymmetry analysis. The kinematic re-
gion covered by the experiment in (x, Q2)-space was di-
vided into nine bins in x. Each of the seven x-bins in the
region x > 0.023 was subdivided into three logarithmi-
cally equidistant bins in Q2. The range in φ-space (2π)
was divided into 10 bins. Two of the φ-bins cover the
shielding steel-plate region of the spectrometer and thus
cannot be used for the analysis. The data were corrected
for the e+e−charge-symmetric background [15], which
amounted in total to about 1.8% of the events, reaching
the largest contribution of about 14% at small values of
x.
The measurement of the asymmetry ALT(x,Q2,φ)
given by (9) can be performed by either reversing the
transverse target polarization and comparing the num-
ber of events in the same part of the detector, or by com-
paring the number of events in the upper and lower part
of the detector for the same upward or downward target
polarization direction. The first method provides a bet-
ter cancellation of acceptance effects and was chosen to
obtain the asymmetry
ALT(x,Q2,φ,hl) =
hlNhl⇑(x,Q2,φ)Lhl⇓− Nhl⇓(x,Q2,φ)Lhl⇑
Nhl⇑(x,Q2,φ)Lhl⇓
p
+ Nhl⇓(x,Q2,φ)Lhl⇑
p
. (15)
Here, Nhl⇑(⇓)is the number of scattered leptons in one
bin of the 3-dimensional space (x,Q2,φ) for the case of
the incident lepton with helicity hl when the direction
of the proton spin points up (down). Lhl⇑(⇓)and Lhl⇑(⇓)
are the corresponding integrated luminosities and the in-
tegrated luminosities weighted with the absolute value of
the beam and target polarization product, respectively
p
Lhl⇑(⇓)=
?
?
dtLhl⇑(⇓)(t)τ(t),(16)
Lhl⇑(⇓)
p
= dtLhl⇑(⇓)(t) | PB(t)PT(t) | τ(t). (17)
Here, L(t) is the luminosity, τ(t) is the trigger live-time
factor, and PBand PT are the beam and target polariza-
tions, respectively. The asymmetries evaluated according
to (15) were found to be consistent for the two beam
helicity states. Therefore they were combined in the fur-
ther analysis. Finally, the asymmetry given by (15) was
unfolded for radiative and instrumental smearing effects
to obtain the asymmetry corresponding to single-photon
exchange in the scattering process. Radiative corrections
were calculated using a Monte-Carlo generator [26]. The
unfolding procedure is analogous to that used previously
in other HERMES analyses [15,27,28]. It inflates the size
of the statistical uncertainties especially in the lowest Q2-
bins at a given value of x. The magnitude of inflation
reaches almost a factor of two at low values of x. The
subdivision of x-bins in the range x > 0.023 into three
bins in Q2decreases the error inflation by about a fac-
tor of 1.5 because at larger Q2the amount of smearing
between x-bins is smaller and the prefactors of AT in
(10) and (11) are larger in magnitude. After the unfold-
ing procedure the central values of g2 and A2 changed
less than the initial statistical uncertainties. As a con-
sequence of the unfolding procedure, the resulting data
points are no longer correlated systematically through
radiative and instrumental smearing effects, but are only
statistically correlated [15]. The procedure generates a
statistical covariance matrix for the data points.
In every (x,Q2)-bin the amplitude AT(x,Q2) was ob-
tained by fitting the unfolded asymmetries with the func-
tion f(φ) = −AT(x,Q2)cosφ. Finally, the asymmetry
A2(x,Q2) and the function g2(x,Q2) were evaluated from
the amplitude AT and the previously measured function
g1, for which a world-data parameterization [29] was em-
ployed, using (10) and (11). The structure function
F1(x,Q2) = F2(x,Q2)(1 + γ2)/[2x(1 + R(x,Q2))](18)
was calculated using a parameterization of the structure
function F2(x,Q2) [30] and the ratio R(x,Q2) [31]. All
kinematic factors in (10) and (11), and the functions F1
and g1/F1were calculated at the average values of x and
Q2in each (x, Q2)-bin after unfolding.
The uncertainties in the measurements of the beam
and target polarizations produce in total a 10% scale un-
certainty on the value of AT. Other sources of system-
atic uncertainties such as acceptance effects, small beam
and spectrometer misalignments, uncertainties in the tar-
get polarization direction, correction for track deflection
in the vertical target holding field, the unfolding proce-
dure and a possible correlation between prefactors of AT
and AT itself in (10) and (11) were evaluated by Monte-
Carlo studies. Uncertainties stemming from parameteri-
zations of g1(x,Q2), F2(x,Q2), and R(x,Q2) were esti-
mated also. In the error propagation to g2, the uncer-
tainty in R(x,Q2) was not included in addition to that of
F2(x,Q2), since they are strongly correlated as explained
in [15]. The total systematic uncertainty was evaluated
as the quadratic sum of all the considered sources. Its
magnitude is less than the magnitude of the statistical
uncertainty.
Figure 1 shows the values of xg2as a function of Q2for
the bins with x > 0.1, which have sufficient coverage in
Q2, along with results from the E143 [13] and E155 [14]
experiments at SLAC. The entire set of measured data
and average values of x and Q2are presented in Table 1.
Within the accuracy of the data, they are in agreement
with the other experiments. Also shown is the Wandzura–
Wilczek term gWW
2
, which was evaluated according to (3).
A world data parameterization of g1(x,Q2) [29] was used
for the calculation.
In order to study the x dependence, A2(x,Q2) and
g2(x,Q2) in bins covering the same x range but with dif-
ferent Q2values were evolved to their mean value of Q2

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-0.1
-0.05
0
0.05
-0.1
-0.05
0
0.05
05 10
xg2
〈x〉 = 0.12
HERMES
E155
E143
xg2
WW
〈x〉 = 0.18
〈x〉 = 0.29
Q2 [GeV2]
〈x〉 = 0.50
05 1015
Fig. 1. The spin-structure function xg2(x,Q2) of the proton
as a function of Q2for selected values of x. Data from the
experiments E155 [14] and E143 [13] are presented also. The
average values of x for these two experiments are slightly dif-
ferent from the HERMES values of ?x? indicated in the panels.
The error bars represent the quadratic sum of the statistical
and systematic uncertainties. The solid curve is the result of
the Wandzura–Wilczek relation (3)
and then averaged. The evolution of A2(x,Q2) was car-
ried out assuming that the product
depend on Q2, which follows from (7), since g1/F1 is
known to vary only weakly over Q2. The structure func-
tion g2(x,Q2) was evolved assuming that its Q2depen-
dence is analogous to that for the Wandzura-Wilczek part
of g2.
The averaged results for xg2 and A2 and the statis-
tical and systematic uncertainties are listed in Table 2,
where the average values of x and Q2are also given. The
quoted statistical uncertainties correspond to the diago-
nal elements of the covariance matrix obtained from the
unfolding algorithm. The correlation matrix for xg2 in
nine x-bins is presented in Table 3.1
The results for the virtual-photon asymmetry A2and
the spin-structure function xg2as a function of x are pre-
sented in Fig. 2 together with data from the experiments
E155 [14], E143 [13], and SMC [12]. The HERMES data
are shown for two regions of Q2, ?Q2? > 1 GeV2(closed
symbols) and ?Q2? < 1 GeV2(open symbols). The exper-
?Q2A2 does not
1It is also available in 23 bins for the data in Table 1
at http://inspirehep.net/record/1082840 or from manage-
ment@hermes.desy.de.
A2
HERMES (〈Q2〉 <1 GeV2)
HERMES (〈Q2〉 >1 GeV2)
E155
E143
SMC
A2
WW
0
0.2
0.4
x
xg2
-0.05
0
0.05
0.1
0.15
10
-2
10
-1
1
Fig. 2.
of the proton as a function of x. Bottom panel: The spin-
structure function xg2 of the proton as a function of x. HER-
MES data are shown together with data from the E155 [14],
E143 [13], and SMC [12] experiments. The total error bars
for the HERMES, E155, and E143 experiments represent the
quadratic sum of the statistical and systematic uncertainties.
The statistical uncertainties are indicated by the inner error
bars. The error bars for the SMC experiment represent the sta-
tistical uncertainties only. The solid curve corresponds to the
Wandzura–Wilczek relation (3) evaluated at the average Q2
values of HERMES at each value of x. For the HERMES data,
the closed (open) symbols represent data with ?Q2? > 1 GeV2
(?Q2? < 1 GeV2)
Upper panel: The virtual-photon asymmetry A2

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6
iments have only slightly different values of average Q2
for a particular value of x. The results are within their un-
certainties in good agreement with each other. The solid
curves represent values of A2and xg2evaluated with the
Wandzura–Wilczek relation (3) using the g1(x,Q2) pa-
rameterization [29]. The values were calculated at the
average Q2of HERMES at each value of x. Within the
uncertainties the data satisfy the positivity bound [32]
for the asymmetry A2, |A2| ≤
all values of x in the kinematic range of the HERMES
experiment.
The Burkhardt–Cottingham integral (5) was evalu-
ated in the measured region of 0.023 ≤ x < 0.9 at Q2=
5 GeV2, resulting in?0.9
0.017. This result is to be compared with the combined
result from experiments E143 and E155 [14] in the region
0.02 ≤ x < 0.8:?0.8
Using the results measured by HERMES for the func-
tion g2, the twist-3 matrix element d2 given by (4) was
evaluated. For the unmeasured region 0.9 < x ≤ 1, the
ansatz g2(x) ∝ (1 − x)3was assumed. The uncertainty
in the extrapolated contribution was taken to be equal
to the contribution itself. The contribution from the re-
gion x < 0.023 was neglected because of the x2suppres-
sion factor. The result is d2 = 0.0148 ± 0.0096(stat) ±
0.0048(syst). This is to be compared with the combined
result from experiments E143 and E155[14]: d2= 0.0032±
0.0017.
In conclusion, HERMES measured the spin-structure
function g2and the virtual-photon asymmetry A2of the
proton in the kinematic range 0.004 < x < 0.9 and 0.18
GeV2< Q2< 20 GeV2. For the covered x-range the mea-
sured integral of g2(x) converges to the null result of the
Burkhardt–Cottingham sum rule. The x2moment of the
twist-3 contribution to g2(x) is found to be compatible
with zero, in agreement with expectations on its small-
ness from lattice calculations. The results on A2and g2
are overall in good agreement with measurements of SMC
at CERN, and E143 and E155 at SLAC, but they are not
statistically precise enough to detect a deviation of g2
from its Wandzura–Wilczek part, as seen by the SLAC
experiments.
?R(1 + A1)/2 ≃ 0.4, for
0.023g2(x,Q2)dx = 0.006±0.024±
0.02g2(x,Q2)dx = −0.042± 0.008.
We gratefully acknowledge the DESY management for its
support and the staff at DESY and the collaborating insti-
tutions for their significant effort. This work was supported
by the Ministry of Economy and the Ministry of Education
and Science of Armenia; the FWO-Flanders and IWT, Bel-
gium; the Natural Sciences and Engineering Research Coun-
cil of Canada; the National Natural Science Foundation of
China; the Alexander von Humboldt Stiftung, the German
Bundesministerium f¨ ur Bildung und Forschung (BMBF), and
the Deutsche Forschungsgemeinschaft (DFG); the Italian Is-
tituto Nazionale di Fisica Nucleare (INFN); the MEXT, JSPS,
and G-COE of Japan; the Dutch Foundation for Fundamenteel
Onderzoek der Materie (FOM); the Russian Academy of Sci-
ence and the Russian Federal Agency for Science and Inno-
vations; the U.K. Engineering and Physical Sciences Research
Council, the Science and Technology Facilities Council, and
the Scottish Universities Physics Alliance; the U.S. Depart-
ment of Energy (DOE) and the National Science Foundation
(NSF); the Basque Foundation for Science (IKERBASQUE)
and the UPV/EHU under program UFI 11/55; and the Eu-
ropean Community Research Infrastructure Integrating Ac-
tivity under the FP7 ”Study of strongly interacting matter
(HadronPhysics2, Grant Agreement number 227431)”.
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7
Table 1. The spin-structure function xg2(x,Q2) and the virtual-photon asymmetry A2(x,Q2) of the proton in bins of (x,Q2),
see text for details. Statistical and systematic uncertainties are presented separately
bin
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
?x?
0.009
0.018
0.033
0.039
0.044
0.067
0.069
0.076
0.116
0.118
0.124
0.182
0.183
0.187
0.282
0.298
0.311
0.458
0.482
0.484
0.630
0.658
0.678
?Q2?,GeV2
0.38
0.68
0.89
1.37
1.80
1.09
1.88
2.79
1.30
2.44
4.04
1.51
3.01
5.42
1.95
3.99
7.58
2.83
4.31
7.57
4.76
6.79
10.35
xg2
0.0799
0.0699
0.0450
-0.0047
0.3489
0.0044
0.0473
0.0202
-0.0094
0.0356
-0.0571
-0.0758
0.0121
-0.0334
0.0071
-0.0242
-0.0571
-0.0613
-0.0987
-0.0362
0.2413
-0.0129
0.0076
± stat.
0.0521
0.0513
0.0326
0.0652
0.1279
0.0421
0.0357
0.0674
0.0506
0.0301
0.0466
0.0642
0.0324
0.0440
0.0396
0.0195
0.0283
0.0582
0.0370
0.0183
0.1194
0.0320
0.0160
± syst.
0.0182
0.0111
0.0215
0.0080
0.0612
0.0097
0.0062
0.0323
0.0081
0.0099
0.0149
0.0230
0.0038
0.0041
0.0063
0.0055
0.0105
0.0129
0.0104
0.0045
0.0534
0.0081
0.0025
A2
± stat.
0.0163
0.0183
0.0165
0.0275
0.0507
0.0346
0.0210
0.0342
0.0603
0.0251
0.0311
0.1055
0.0375
0.0392
0.0925
0.0363
0.0437
0.2616
0.1704
0.0744
1.3295
0.4350
0.2551
± syst.
0.0057
0.0040
0.0109
0.0035
0.0243
0.0085
0.0041
0.0164
0.0111
0.0090
0.0102
0.0389
0.0074
0.0052
0.0167
0.0117
0.0166
0.0598
0.0500
0.0206
0.5969
0.1115
0.0419
0.0257
0.0269
0.0278
0.0033
0.1440
0.0190
0.0402
0.0225
0.0266
0.0584
-0.0137
-0.0466
0.0707
0.0143
0.1675
0.0718
0.0039
0.0064
-0.2064
0.0421
3.0231
0.1197
0.3672
Table 2. The spin-structure function xg2 and the virtual-photon asymmetry A2 of the proton after evolving to common Q2
and averaging over in each x-bin (see text for details). Statistical and systematic uncertainties are presented separately
bin
1
2
3
4
5
6
7
8
9
x range
0.004 - 0.014
0.014 - 0.023
0.023 - 0.050
0.050 - 0.090
0.090 - 0.150
0.150 - 0.220
0.220 - 0.400
0.400 - 0.600
0.600 - 0.900
?x?
0.009
0.018
0.036
0.069
0.118
0.183
0.291
0.473
0.654
?Q2?,GeV2
0.38
0.68
1.08
1.59
2.07
2.51
3.23
4.62
7.06
xg2
0.0794
0.0668
0.0456
0.0271
-0.0023
-0.0005
-0.0314
-0.0454
0.0107
±stat
0.0520
0.0509
0.0262
0.0236
0.0212
0.0086
0.0126
0.0154
0.0177
±syst
0.0153
0.0181
0.0157
0.0150
0.0085
0.0063
0.0043
0.0075
0.0073
A2
0.0256
0.0258
0.0261
0.0312
0.0289
0.0612
0.0629
0.0373
0.4275
±stat
0.0162
0.0182
0.0121
0.0154
0.0194
0.0109
0.0248
0.0665
0.2316
±syst
0.0049
0.0065
0.0074
0.0100
0.0088
0.0105
0.0104
0.0345
0.0970
Table 3. Correlation matrix for xg2 in 9 x-bins (as in Table 2)
123456789
1
2
3
4
5
6
7
8
9
1.0000
-0.1281
-0.0038
-0.0033
-0.0017
0.0005
0.0000
0.0000
0.0000
-0.1281
1.0000
-0.1584
-0.0083
-0.0007
0.0000
0.0000
0.0001
0.0000
-0.0038
-0.1584
1.0000
-0.1951
-0.0281
0.0077
-0.0016
0.0002
0.0000
-0.0033
-0.0083
-0.1951
1.0000
-0.2885
0.0312
-0.0107
0.0013
-0.0005
-0.0017
-0.0007
-0.0281
-0.2885
1.0000
-0.0102
-0.0654
0.0067
-0.0018
0.0005
0.0000
0.0077
0.0312
-0.0102
1.0000
-0.1829
0.0143
-0.0055
0.0000
0.0000
-0.0016
-0.0107
-0.0654
-0.1829
1.0000
-0.3539
0.0926
0.0000
0.0001
0.0002
0.0013
0.0067
0.0143
-0.3539
1.0000
-0.3947
0.0000
0.0000
0.0000
-0.0005
-0.0018
-0.0055
0.0926
-0.3947
1.0000