HERMES Measurements of the Generalized
GDH Integral and of Quark-Hadron Duality
W.-D. Nowak
on behalf of the HERMES Collaboration
DESY Zeuthen, D-15738 Zeuthen, Germany
Abstract. The physics impact of measuring the generalized Drell-Hearn-Gerasimov integral in
inclusive deep inelastic scattering (DIS) is discussed. Studies of the transition region from the realphoton point into the DIS region have been performed simultaneously in and beyond the nucleonresonance region, thereby testing the duality concept. HERMES measurements on the proton and
deuteron show consistently that the hard regime holds down to photon virtualities of about 1.5 GeV 2 .
PHYSICS MOTIVATION
The Gerasimov-Drell-Hearn sum rule
∞
ν0
→
→
[σ ⇐ (ν ) − σ ⇒ (ν )]
dν
4π 2 I α 2
=−
κ ,
ν
Mt2
(1)
establishes a connection between the helicity-dependent dynamics and ground state
properties of a nucleus [1] (or nucleon [1, 2]). The dynamics is expressed through the
difference of the photoabsorption cross-sections for anti-parallel and parallel beam (→)
→
→
and target (⇒) spin orientation, σ ⇐ (ν ) and σ ⇒ (ν ), respectively, while the ground state
µM
is described in terms of the anomalous contribution κ = eI t −Z to the magnetic moment
µ of the target with atomic number Z. Here I is the spin of the nucleus, ν the photon
energy in the target rest frame, ν0 the photoabsorption threshold, Mt the nucleus mass,
α the electromagnetic fine–structure constant and e the elementary charge.
The GDH integral on the l.h.s. of Eq.(1) can be generalized to the case of non–zero
photon virtuality Q2 . In terms of the helicity–dependent virtual–photon absorption cross→
→
sections σ ⇐ (ν , Q2 ) and σ ⇒ (ν , Q2 ) [3, 4] it reads:
IGDH (Q2 ) =
∞
ν0
→
→
[σ ⇐ (ν , Q2 ) − σ ⇒ (ν , Q2 )]
dν
.
ν
(2)
This integral is well defined over the entire Q2 -range and so allows the study of the
transition of the helicity-dependent dynamics from the photoabsorption point (Q 2 = 0)
into the region of Deep Inelastic Scattering (DIS). Above Q2 ≈ 1 GeV2 perturbative
Quantum Chromodynamics (pQCD) can be applied taking twist-2 and twist-3 contributions into account. Below this value there exists no field-theoretical description yet.
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
591
For Q2 -values below 0.3 GeV2 the QCD-inspired chiral quark-soliton model [5] delivers
promising descriptions. In between, one has to resort to phenomenological models [6].
The generalized GDH integral can be described in pQCD by the helicity-dependent
nucleon structure functions g1 (x, Q2 ) and g2 (x, Q2 ). For a spin- 12 target it reads:
IGDH (Q2 ) =
8π 2 α
M
x
2
2
2
0 g (x, Q ) − γ g (x, Q ) dx
1
2
K
0
x
,
(3)
γ 2 = Q2 /ν 2 , x = Q2 /2M ν , and x0 = Q2 /2M ν0 . The virtual–photon flux factor
where K = ν 1 + γ 2 was chosen in the Gilman notation [7]. In the Q 2 -region under consider
ation, the Burkhardt-Cottingham sum rule 01 g2 (x, Q2 )dx = 0 is expected to hold so that
IGDH (Q2 ) can be entirely described through Γ1 (Q2 ), the first moment of g1 :
IGDH (Q2 ) =
16π 2 α
Γ1 (Q2 ).
Q2
(4)
At large Q2 , the first moment Γ1 (Q2 ) = 01 g1 (x, Q2 )dx exhibits only the logarithmic
Q2 -dependence from QCD evolution.
Two regions in the invariant mass W 2 of the virtual-photon nucleon system are to be
distinguished: the DIS region above, and the nucleon-resonance region below W 2 ≈ 4
GeV2 . While IGDH (Q2 ) is defined over the entire W 2 -range, nucleon structure functions
were originally introduced only in the DIS region; their interpretation in the nucleonresonance region requires additional assumptions, as e.g. quark-hadron duality (see
below). The GDH integral may thus be considered to describe the ‘transition’ of Γ 1 (Q2 )
down to low virtualities including the nucleon-resonance region. To this end it allows to
determine at which virtuality the hard 1/Q 2 regime breaks down.
The difference between the generalized GDH integrals for proton and neutron is of
special interest. At large virtualities it is directly related to the Bjorken sum rule:
Q2 large
p
n
(Q2 ) − IGDH
(Q2 ) −→
IGDH
16π 2 α p 2
[Γ1 (Q ) − Γn1 (Q2 )].
Q2
(5)
This fundamental sum rule is derived from current algebra and isospin symmetry,
Γ1p (Q2 ) − Γn1 (Q2 ) = 16 · ga · Cns (αs (Q2 )), where ga = |gA /gV | = 1.2670 ± 0.0035 is the
neutron beta–decay coupling constant. Its Q 2 -dependence is given by DGLAP evolution. Measurements of the generalized GDH integral on both proton and neutron thus
connect two important quantities characterizing the ‘static’ nucleon, g A and κ .
The concept of quark-hadron duality, in the interpretation of Bloom and Gilman [8],
connects experimental results obtained in two different regimes. It conjectures a structure function (or a ratio) measured in the DIS region to represent an average of the same
function (or ratio) measured in the resonance region. In the pQCD re-formulation of
this concept [9] the leading terms describe the non-interacting quarks, i.e. scaling, and
the non-leading ones the quark-gluon interactions, characterized by a 1/Q 2 -behaviour.
Duality is expected to break down at low virtuality as does the whole pQCD picture.
Note that duality for polarized structure functions is non-trivial because of the different
helicity structure of polarized DIS and nucleon resonances.
592
DISCUSSION OF EXPERIMENTAL RESULTS
(µb)
I
p
30
60
GDH
35
GDH
Id (µb)
The HERMES results discussed in this contribution were derived from measurements
with a longitudinally polarized positron beam of 27.57 GeV incident on a gas target
internal to the HERA storage ring at DESY. The gas target was filled with longitudinally polarized atomic Hydrogen (Deuterium) in 1997 (1998-2000). The cross-section
asymmetry A1 for the absorption of virtual photons was calculated from the measured
cross-section asymmetry A|| . The latter was obtained from differences of luminosityweighted numbers of events for parallel and anti-parallel orientation of virtual-photon
and nucleus(on) spin, respectively, normalized by their sum. For details on the experiment as well as a complete description of the analysis procedure necessary to calculate
GDH integrals see Ref. [10] and references therein. The large kinematic coverage of
the HERMES experiment made it possible to measure for the first time the generalized
GDH integrals for deuteron, proton and neutron in the nucleon-resonance region and in
the DIS region, simultaneously.
50
25
40
20
15
30
10
20
5
10
0
0
-5
-10
0
2
4
6
-10
8
2
2
Q (GeV )
0
2
4
6
8
2
2
Q (GeV )
FIGURE 1. Q2 -dependence of the generalized GDH integrals [10] for the deuteron nucleus (left) and
the proton (right), shown for the nucleon-resonance region (triangles), DIS region (squares), and full
region (circles) including the extrapolation of the unmeasured part above W 2 = 45 GeV2 . Solid and
dashed curves are predictions for the full integral [6] and the integral in the nucleon-resonance region [11],
respectively. The error bars represent the statistical uncertainties. The systematic uncertainties of the full
integral are given as a band; the hatched area inside represents the contribution of the nucleon-resonance
region alone.
From the HERMES data shown in Fig. 1 it can be seen that the resonance contribution
decreases rapidly with increasing Q2 while the DIS contribution is sizeable over the
entire range measured, even down to the lowest measured value of Q2 = 1.5 GeV2 .
This holds for both proton and deuteron, as well as for the neutron (not shown here).
From Fig. 2 (left panel), which shows HERMES results on the full integral for all
three targets, the agreement with previous measurements of the first moment of g 1
can be seen. Good agreement exists also with the model of Ref. [6] which is based
only on a leading-twist Q2 -evolution of the first moments of g 1 and g2 without any
593
(µb)
30
GDH
- In
40
60
GDH
50
Ip
IGDH (µb)
60
20
10
40
0
-10
-20
20
-30
-40
-50
-60
0
2
4
6
0
8
2
2
Q (GeV )
0
2
4
6
8
2
2
Q (GeV )
FIGURE 2. Q2 -dependence of the full generalized GDH integral [10]. Left: deuteron (squares), proton
(circles), and neutron (triangles) obtained from deuteron and proton. Right: proton-neutron difference.
The curves are predictions from Refs. [6] (left) and [12] (right). The error bars represent statistical
uncertainties. The bands describe systematic uncertainties; in the left panel open for neutron, lined for
deuteron, and cross-hatched for proton. The open symbols at Q 2 = 5 GeV2 show measurements from
Refs. [13, 14]. The stars represent the three highest Q 2 bins of the neutron measurement from Ref. [15]
including an extrapolation of the unmeasured DIS region.
resonance contribution. The proton-neutron difference (right panel of Fig. 2) is equally
well described by the prediction of Ref. [12] and a 1/Q2 fit which at Q2 = 5 GeV2 shows
agreement of the HERMES data (14.3 ± 0.9stat. ± 1.3syst. µ b) with the Bjorken sum
rule prediction (16.33 ± 0.45 µ b) and with earlier measurements at SLAC, E-143 [13]
and E155 [14]. The main result from HERMES with regard to the four measured full
integrals (proton, deuteron, neutron and proton-neutron difference) is that a clear 1/Q 2 behaviour is observed, i.e. no effects from higher twists or resonance form factors are
seen down to the lowest measured value of Q2 = 1.5 GeV2 . This demonstrates that the
hard regime is intact down to a surprisingly low virtuality. Note that at the real-photon
point (Q2 = 0) the values for the four full integrals are given by the GDH sum rule to be
-204, -0.65, -233 and + 29 µ b. Hence at a rather low value of Q2 a turn-over must occur
for proton and deuteron which is not seen in the HERMES kinematics.
Duality in polarized lepton-nucleon scattering was studied for the first time at HERMES [16]. In the left panel of Fig. 3 the spin asymmetry A 1 , measured in the resonance
region, is shown to well describe the world DIS data on average. The average ratio
of the measured Ares
1 to the DIS fit shown is 1.11 ± 0.16 stat. ± 0.18syst. . The effect of
target-mass corrections was found to be smaller than 5%. In the right panel is shown
the Q2 -dependence of the ratio of the first moments of g1 , measured in the nucleonresonance and in the DIS region, respectively. There is clear evidence that quark-hadron
duality holds down to Q 2 -values of 1.6 GeV2 .
Acknowledgements. I am indebted to A. Fantoni, H. Jackson, and B. Seitz for
valuable comments and discussions.
594
A1
Γ1 res / Γ1 DIS
res
A1 HERMES
1.4
DIS
A1 HERMES
DIS
A1 E143
DIS
A1 SMC
DIS
A1 E155
1.2
1
10
2
W = 1.0 - 4.0 GeV
2
HERMES preliminary
SLAC
1
0.8
0.6
SU(6)
10
-1
0.4
syst. unc. ≤ 20% (±8% exp, ±14% A2,
0.2
0
DIS
res
±7% A1 , ±10% F2 )
10
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
-2
1
x
1
10
Q [GeV ]
2
2
FIGURE 3. Left panel: Spin asymmetry A 1 as a function of x measured at HERMES in the resonance
region (full circles) [16]. The error bars represent the statistical uncertainties; the systematic uncertainty
for the data in the resonance region is about 16 %. Data obtained in the DIS region (open symbols [17, 13,
18, 14]) are described by a power law fit for x > 0.3 (curve). Right panel: Preliminary HERMES data (full
circles) on the Q2 -dependence of the ratio of the first moments of g 1 , calculated in the nucleon-resonance
region (W 2 = 1 ÷ 4 GeV2 ) and in the DIS region (using the NMC parameterization [19]), respectively. For
comparison SLAC data (open circles) at lower Q 2 [13] are shown, as well.
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