Experimental Verification of the GDH Sum Rule
A survey including the extension of the GDH integral to virtual photons
K. Helbing
on behalf of the GDH-Collaboration
Physikalisches Institut, Universität Erlangen-Nürnberg, 91058 Erlangen, Germany
Abstract. Sum rules involving the spin structure of the nucleon like those due to Bjorken, Ellis and
Jaffe and the one due to Gerasimov, Drell and Hearn offer the opportunity to study the structure of
strong interactions. At long distance scales in the confinement regime the Gerasimov-Drell-Hearn
(GDH) Sum Rule connects static properties of the nucleon like the anomalous magnetic moment κ
and the nucleon mass m, with the spin dependent absorption of real photons with total cross sections
σ3/2 and σ1/2 1 :
∞
2π 2 α
dν σ3/2 (ν ) − σ1/2 (ν ) =
· κ2
(1)
ν
m2
0
Hence the full spin-dependent excitation spectrum of the nucleon is related to its static properties. The sum rule has not been investigated experimentally until recently. For the first time this
fundamental sum rule is verified by the GDH-Collaboration with circularly polarized real photons
and longitudinally polarized nucleons at the two accelerators E LSA and M AMI. The investigation
of the response of the proton as well as of the neutron allows to perform an isospin decomposition.
Data from the resonance region up to the onset of the Regge regime are shown.
The “sum” on the left hand side of the GDH Sum Rule can be generalized to the case of virtual
photons. This allows to establish a Q 2 dependency and to study the transition to the perturbative
regime of QCD. This is the subject of several experiments e.g. at JL AB for the resonance region and
of the H ERMES experiment at D ESY for higher Q 2 .
Moreover, this paper covers the status of theory concerning the GDH Sum Rule, the different
experimental approaches and the results for the absorption of real and virtual photons will be
reviewed.
1. INTRODUCTION
The GDH Sum Rule has been derived in parallel by several authors in the second half
of the 1960ies. Today mostly Gerasimov [1], Drell and Hearn [2] are referenced. Both
works are based on a dispersion theoretic derivation. Hosoda and Yamamoto [3] in 1966
used the current algebra formalism to derive the same sum rule. Drell and Hearn rated
the sum rule to be of purely academic interest only since it would be impossible to
experimentally test it. In contrast, Hosoda and Yamamoto were convinced that it would
be straight forward to test it. Moreover, they thought that the sum rule only holds for
1
Here 3/2 and 1/2 identify relative spin orientation of the photon and the nucleon parallel or anti-parallel
respectively in the nucleon rest frame; α denotes the fine-structure constant and ν the energy of the photon.
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
33
a Dirac particle and hence a verification could prove that the nucleon complies to this
assumption. Iddings [4] in 1965 on the other hand was already all the way there to write
down the sum rule for Q2 = 0 but didn’t explicitly do so. Nonetheless, his work already
contains a version of what is called today a generalization of the GDH sum or integral.
For most of the further discussion we concentrate here on the dispersion theoretic
derivation used by Gerasimov, Drell and Hearn. Only fundamental constraints enter this
derivation: Lorentz invariance and gauge invariance allow to write the Compton-forward
amplitude in a simple form; unitarity provides the Optical Theorem; causality and the
so-called No-Subtraction-Hypothesis lead to the Kramers-Kronig dispersion relation;
the Low-Theorem [5, 6] is based again on Lorentz and gauge invariance.
Among these constraints the No-Subtraction-Hypothesis is the only assumption which
is open to reasonable questions. To reduce the dispersion relation for the spin-flip
Compton forward amplitude f 2 (ν ) from a contour integral in the complex plane to an
integration along the real axis one has to presume that f 2 (ν )/ν → 0 for ν → ∞. A
violation of this hypothesis would lead to a weird behavior of this spin-flip amplitude
and the corresponding differential forward cross sections, namely:
1 =∞
d σ3/2 − d σ1/2 ν →∞ dΩ
θ =0
lim
(2)
On the other hand for the total cross section Regge arguments related to the Froissart
bound ensure the following behavior:
tot
tot
− σ1/2
lim σ3/2
=0
(3)
ν →∞
A possible failure of the GDH Sum Rule would be related to a violation of the NoSubtraction-Hypothesis. There have been several attempts in the past to find reasons for
such a failure. Here some of them are reviewed:
Based on the current algebra derivation by Hosoda and Yamamoto the authors Chang,
Liang and Workman [8] have argued that an anomaly in the charge density commutator
gives rise to a modification of the GDH Sum Rule. Pantförder [9] was able to show that
the contribution from this anomaly cancels going to the infinite-momentum limit which
ultimately leads to the GDH Sum Rule.
It was questioned if the Low-Theorem holds to all orders of electromagnetic coupling.
While Low [5] showed the derivation only in the lowest order Gell-Mann and Goldberger stated in their original paper [6] that their derivation should be “exactly correct in
any known theory”. Later Roy and Singh [10] established the low theorem to the order
α 2.
Haim Goldberg [11] suspected that the photoproduction of gravitons violates the GDH
Sum Rule. Contributions at very high energies from photonuclear reactions other than
those of strong interactions may not be ignored a priori. On the other hand, the contribution of these effects at high energies to the sum rule will be largely damped due to the
weighting with the inverse of the photon energy.
Already in 1968 right after the discovery of the GDH Sum Rule Abarbanel and Goldberger [12] considered a J = 1 Regge fixed pole being a possible source for the failure of
the No-Subtraction-Hypothesis. Despite such a fixed pole is forbidden by the Froissart
34
theorem for purely hadronic processes and such a behavior has never been observed so
far it cannot be ruled out completely for electro-weak processes. Nevertheless, it should
be mentioned that it is not quite clear if such a fixed pole in the case of real Compton
scattering would not violate the Landau-Yang theorem which forbids two photons to
have a total angular momentum of J = 1.
Further examples for possible failures of the sum rule that have been considered can
be found in Ref. [7]. To summarize, today no stringent indication for a modification of
the GDH Sum Rule exists.
2. THE GDH-EXPERIMENT AT ELSA AND MAMI
2.1. Experimental concept
The primary aim of the GDH-Collaboration2 is obviously to verify the GerasimovDrell-Hearn Sum Rule. The central issue of the experimental conception of the GDHCollaboration is the reduction of systematic uncertainties in order to provide a setup
compatible with the fundamental character of the sum rule.
2.1.1. Region of integration
The GDH integrand on the left hand side of the GDH Sum Rule will be determined
from the resonance region up to the onset of the Regge regime. This is achieved by the
use of two electron accelerators with high duty cycle:
0.14 - 0.8 GeV
0.7 - 3.1 GeV
M AMI (Mainz)
E LSA (Bonn)
At E LSA a completely new experimental area was setup for the GDH measurements
while at M AMI the existing tagging facility in the A2-Hall was available. At M AMI two
primary electron energy settings are used to cover the energies from pion threshold up to
800 MeV. Five primary electron energy settings at E LSA allow to cover photon energies
up to 3.1 GeV. The circular polarization of the photons is given by the helicity transfer
of the longitudinal polarization of the electrons.
2.1.2. Beam polarization
At both accelerators the polarization of the electron beam is achieved by high intensity sources with strained super-lattice GaAs-crystals. The typical polarization of the
delivered electron beam is 65 − 75% [17, 18].
2
For a list of participants of the GDH-Collaboration be referred to the author list of Ref. [14].
35
The race-track of the electrons at M AMI is deterministic. Hence, almost no polarization is lost on the way from the source to the experiment. Møller polarimetry is provided
simultaneously to the photon tagging by a magnetic tagging spectrometer.
E LSA is a storage type accelerator with depolarizing resonances. The spin of the electrons has to be transported vertically in E LSA and rotated to the longitudinal direction in
the external beam line for the experiment. Because of these more delicate circumstances
of spin maintenance a dedicated 2-arm Møller spectrometer with large acceptance was
built. It enables fast spin diagnostics in all 3 vector components [19, 27].
2.1.3. Frozen spin target
A new solid state polarized frozen-spin target has been developed for the GDH
measurements [16]. The central part of this new target consists of a 3 He/4 He dilution
refrigerator that is installed horizontally along the beam axis. The refrigerator includes
an internal superconducting holding coil to maintain the nucleon polarization in the
frozen-spin mode longitudinally to the beam. The design of the dilution refrigerator
and the use of an internal holding coil enabled for the first time the measurement of
a spin-dependent total cross section in combination with a polarized solid state target.
Due to the low fringe field of the holding coil and the horizontal alignment allows the
detection of emitted particles with an angular acceptance of almost 4π (see below).
Butanol provided polarized protons. In addition, 6 LiD was used to obtain polarized
neutrons. Typical values of polarization of the protons in the butanol that have been
reached during data taking are 70-80%.
2.1.4. Detector concepts
Two detector concepts are used to meet the special requirements for the different
energy ranges: The DAPHNE detector at M AMI and the GDH-Detector at E LSA.
DAPHNE [20] is well suited for charged particle detection and for the identification
of low multiplicity states. It is essentially a charged particle tracking detector having a
cylindrical symmetry. In addition it has a useful detection efficiency for neutral pions. In
forward direction a silicon microstrip device called M IDAS [22] extends the acceptance
for charged particles.
The GDH-Detector [21] has been specifically designed for measurements of total
cross sections and is perfectly suited for situations where the contributing channels are
not well known and extrapolations due to unobserved final states are not advisable.
The concept of the GDH-Detector is to detect at least one reaction product from all
possible hadronic processes with almost complete acceptance concerning solid angle
and efficiency. This is achieved by an arrangement of scintillators and lead. The over all
acceptance for any hadronic process is better than 99 %.
Both detection systems have similar components in forward direction. The electromagnetic background is suppressed by about 5 orders of magnitude by means of a
threshold Č detector [21]. The Č detector is followed by the Far-Forward-Wall – a com-
36
ponent similar to the central parts of the GDH-Detector – to complete the solid angle
coverage [23].
2.2. Results
2.2.1. Systematic studies
Measurements of unpolarized total photoabsorption cross sections were performed [24, 25, 26] to ensure that both detection systems are operational even for
measurements of differences of cross sections. An unprecedented data quality has been
reached in unpolarized measurements on 1 H, 2 H and 3 He in the photon energy range
from pion threshold to 800 MeV as well as on Carbon and Beryllium in the energy
range from 250 MeV to 3100 MeV.
Systematic studies with respect to spin have been performed with an unpolarized
butanol target in frozen spin mode with all possible holding field configurations. In any
case the false asymmetry of σ3/2 − σ1/2 turned out to be less than 2 µ b [28].
2.2.2. Polarized cross sections in the resonance region
Fig. 1 shows the doubly polarized results for σ 3/2 − σ1/2 on the proton that have
been obtained until now. For comparison also the unpolarized cross section is plotted.
The data taken at M AMI have already been published [13] while the analysis of the data
collected at E LSA is preliminary [27, 28]. One observes that the data sets for the different
energy settings at the two accelerators match each other very well. The three major
resonances known from the unpolarized total cross section are present in the difference
as well - they are even more pronounced.
Here, it is not possible to summarize the wealth of information obtained for the single
resonances especially with respect to partial channels. Much of it is currently in the
process of being published. As examples Refs. [14, 15] might serve. A more detailed
view of our results for the second resonance region is shown in Fig. 2 on the left. The
differences of the total cross sections and the single pion contribution are compared
to the corresponding predictions of a unitary isobar model called M AID [29]. One
observes that the parameterization fails to describe our data. This might lead to a refined
1/2 , E 1/2 .
understanding of the involved resonance couplings to the multipoles M 2−
2−
In Fig. 2 on the left are shown the separate helicity states of the total photoabsorption
cross sections σ3/2 and σ1/2 . The separated helicity states are obtained by adding resp.
subtracting our polarized cross section difference from the unpolarized data. As already
visible in the cross section difference in Fig. 1 the third resonance is a clearly distinct
feature of the spectrum. Besides, there is another structure one might call the “forth”
resonance which comes more conspicuous in this representation than in the difference.
This structure might be due to the F35 and the F37 contributions.
37
Total real photoabsorption
Unpolarized Hydrogen (PDG)
GDH at MAMI, 0.8 GeV
GDH at ELSA 1.0 GeV, prel.
GDH at ELSA 1.4 GeV, prel.
GDH at ELSA 1.9 GeV, prel.
GDH at ELSA 2.4 GeV, prel.
GDH at ELSA 2.9 GeV, prel.
σ3/2 - σ1/2 , σunpol [µb]
500
400
300
200
100
0
-100
0
500
1000
1500
Eγ [MeV]
2000
2500
FIGURE 1. Difference of the polarized total photoabsorption of the proton in comparison to the
unpolarized cross section
2.2.3. High-energy behavior
Regge fits are able to describe many unpolarized total cross sections simultaneously [30]. All data follow a simple power law, namely σ T c1 · sαR (0)−1 + c2 · sαP (0)−1
with αR (0) being the ρ , ω trajectory intercept and αP (0) being that of the Pomeron. For
real photo absorption these fits are valid down to photon energies as low as 1.2 GeV. The
diagram in Fig. 3 on the left substantiates this statement. Shown is the behavior of the
normalized3 reduced χ 2 under inclusion of data points of lower photon energies. One
observes that for the unpolarized hydrogen data [31] the reduced χ 2 remains unchanged
as long as data below 1.2 GeV is not included. Hence, it appears that the Regge behavior
is established down to very low photon energies starting just above the third resonance.
In the polarized case Regge fits have recently been applied to deep inelastic scattering
data [32, 33, 34]. The extrapolation of these fits to Q 2 = 0 indicate that the integrand of
In order to have comparable values the χ 2 for the unpolarized data was scaled by a factor of 0.66 which
reflects that the error calculation in Ref. [31] was too optimistic. The polarized data from the GDHCollaboration does not show this problem. However, to have comparable variations of the reduced χ 2 –
for displaying purposes only – the polarized values were raised to the 10 th power. This in turn reflects the
inferior statistics of the polarized data.
3
38
Total photoabsorption and single pion production
MAID: single π
GDH: single π
GDH: total
σ3/2 GDH-Coll., prel.
σ1/2 GDH-Coll., prel.
260
240
cross section [µb]
σ3/2 - σ1/2 [µb]
150
Total photoabsorption
280
100
50
3rd resonace
220
200
4th resonace
180
160
0
140
-50
450 500 550 600 650 700 750 800 850 900
photon energy [MeV]
1000
1200 1400 1600 1800
photon energy [MeV]
2000
FIGURE 2. Left: Comparison of the differences of the total cross sections and the single pion contribution with the corresponding predictions from M AID ; Right: Separated helicity state total cross sections
σ3/2 and σ1/2 for the third and a “forth” resonance.
the GDH Sum Rule on the proton could be negative at higher energies. Our polarized
data up to 2.8 GeV photon energy disagree with these Regge fits (Fig. 3) but indicate a
sign change at the highest energies.
Bass and Brisudová [32] have argued that the polarized cross section difference for
the absorption of virtual photons can be described by the following Regge behavior:
c4
ln s
α f −1
αa −1
σ3/2 − σ1/2 = c1 s 1 · I + c2 s 1 + c3
+ 2 F(s, Q2)
s
ln s
where I denotes the isospin of the nucleon. The logarithmic terms are due to Regge
cuts and can be neglected at Q2 = 0 [35]. Also F(s, Q2 ) simplifies to a constant at the
real photon point and can be incorporated into c 1 and c2 . αa1 and α f are the Regge
1
intercepts of the respective trajectories. Hence in the case of real photons the expression
for the Regge behavior simplifies considerably to
σ3/2 − σ1/2 = c1 sαa1 −1 · I + c2 s
α f −1
1
(4)
The intercept of the f 1 trajectory is rather well defined by the deep inelastic scattering
data to be about -0.5. The situation is less clear with α a1 where the values from different
fits range from about -0.2 to +0.5. For the further calculations here we adopt a value
of +0.2.
The right hand diagram of Fig. 3 shows our fit via c 1 and c2 to the polarized data of
the GDH-Collaboration. It seems that the fit works well and it indicates a sign change
at about 2 GeV photon energy. The diagram on the left shows that indeed the reduced
χ 2 of the fit is about unity if only data above 1.3 GeV is included. In contrast to the
unpolarized case at 1.2 GeV the χ 2 already deviates from unity. We consider this as
another indication for a 4th resonance as discussed in section 2.2.2.
39
3
40
polarized proton
unpolarized proton
30
Bianchi, Thomas
GDH-Collaboration, prel.
a1,f1 fit to GDH proton data
20
2
σ3/2 - σ1/2 [µb]
normalized reduced χ
2
2.5
1.5
1
0
-10
0.5
0
1000
10
-20
-30
1100 1200 1300 1400 1500 1600
lower limit of Eγ included in fit [MeV]
1700
1400 1600 1800 2000 2200 2400 2600 2800 3000
photon energy [MeV]
FIGURE 3. Left: Normalized reduced χ 2 as a function of the lowest energy of the data included in the
fit; Right: Comparison of the difference of the total cross sections with a Regge fit to DIS data extrapolated
to Q2 = 0 and with our fit to the real photon data
2.2.4. Verification of the GDH Sum Rule for the proton
The GDH Sum Rule prediction for the proton amounts to 205 µ b. The experimental
value of the running GDH integral up to 2.8 GeV clearly overshoots this prediction
(Fig. 4). The preliminary value of the GDH sum up to 2.8 GeV is 225 ± 5 stat µ b. This
includes an unmeasured negative contribution at the pion threshold up to 200 MeV taken
from Ref. [29]. Since at threshold only a simple E0+ amplitude contributes this can be
regarded to be a reliable estimate.
The integrand σ3/2 − σ1/2 remains positive from about 230 MeV on up to about 2 GeV
as seen in Fig. 3 on the right. One has to assume a sign change of the integrand at
higher energies as indicated by the data and the Regge fit in order to obtain a better
agreement between the measurement and the GDH prediction. Indeed our Regge fit gives
a contribution ranging between about −20µ b and −35µ b above 3 GeV. The relatively
wide range is a result of the flexibility in choosing the a 1 trajectory intercept as discussed
in Sec. 2.2.3. This intercept can be determined much better with the neutron data already
taken by the GDH-Collaboration at E LSA (see Sec. 2.3).
To summarize, given the data and taking into account the extrapolation to higher
energies with our Regge fit we obtain agreement with the GDH Sum Rule prediction.
The level of precision obtained is of the order of 5% [13, 27, 28]. This represents the
first verification of the GDH Sum Rule ever.
2.3. Future measurements on the GDH Sum Rule with real photons
The measurements of the GDH-Collaboration on the proton have been extended up to
3.1 GeV photon energy at E LSA. The data are currently under analysis. It will be highly
interesting to see if the indications for a sign change of the GDH integrand consolidate
40
Running GDH integral [µb]
250
200
150
statistical errors only
100
GDH-Collaboration 2002, prel.
GDH sum rule value: 205 µb
50
0
-30 µb from MAID (E0+)
-50
0.2
0.5
1
photon energy [GeV]
2
3
FIGURE 4. Running GDH integral for the proton
and if our Regge fit still describes the data with higher precision. These data will also
allow to extrapolate to high energies with a decreased error.
There are several other experiments planed that can verify our findings. These experiments use the laser backscattering technique to obtain polarized photons instead of
bremsstrahlung produced by polarized electrons. The principal layout of the detection
systems are similar to that developed of the GDH-Experiment at E LSA (see Sec. 2.1.4).
The L EGS facility at BNL will use a polarized HD-target to cover photon energies
up to 470 MeV [36].
• The G RAAL facility also uses the HD-target technique and will cover photon
energies up to 550 MeV [37]. The currently envisaged start of the experiment is
in 2003.
• At SP RING -8 dynamically polarized PE-foils will be used to cover 1.8 –
2.8 GeV [38]. The anticipated beginning of data taking is at the end of 2003.
•
Beyond the energy coverage of the GDH-Collaboration there are experiments planed to
extend the measurements to higher energies:
At JL AB a proposal is being prepared to measure at photon energies starting from
2.5 to 6 GeV. A frozen spin target similar to that of the GDH-Collaboration is
envisaged to be developed.
• S LAC has an approved proposal to measure total cross section asymmetries in the
energy regime from 4 to 40 GeV. The anticipated starting date is in 2005.
•
41
200
σ3/2 - σ1/2 [µb]
150
GDH-Collaboration: proton, prel.
a1,f1 fit to GDH proton data
expectation for neutron
100
50
0
1400 1600 1800 2000 2200 2400 2600 2800 3000
photon energy [MeV]
FIGURE 5. Prediction for the neutron obtained from our fit to the proton data.
2.3.1. Neutron prospects
The GDH Sum Rule prediction for the neutron is 233 µ b which is almost 30 µ b
higher that the value for the proton. Moreover, the estimate using M AID [29] for the
GDH integral up to 500 MeV is about 25 µ b less than in the case of the proton. Relying
on this estimate – which already provided a good prediction for the proton in this energy
domain – one ends up with a total strength of about 50 µ b that the neutron should have
in excess over the proton at energies above the delta resonance. There are three possible
explanations:
A much larger difference in the Regge regime for the neutron than was seen for the
proton.
• Completely unanticipated behavior of the higher resonances.
• A failure of the isovector GDH Sum Rule.
•
Here, we will concentrate on the first option. Revisiting Eqn. 4 one observes that the
isospin I for the neutron has just the opposite sign and hence the contribution due to
the a1 trajectory simply flips sign. The effect, however, is large taking the coefficients
already obtained for our fit to the proton data of the GDH-Collaboration. Fig. 5 shows
that the prediction one obtains for the neutron is of the order of almost 100% asymmetry
for the neutron whereas that of the proton is close to zero. This would indeed compensate
the “missing” strength as argued above.
The GDH-Collaboration has already taken data using a polarized 6 LiD target. The
data will be analyzed within the next months. As is the case for the proton, there is no
other data from other experiments available for the neutron as well.
42
Q2 = 0
1
A1
0.5
0
-0.5
ELSA, MAMI
GDH-Collaboration
-1
1.2
1.4 1.6 1.8
W [GeV]
2
FIGURE 6. Evolution of A 1 with Q2
3. VIRTUAL PHOTON ABSORPTION
The spin structure of a virtual photon is more involved than that of real photon in that it
has a longitudinal polarization component:
√
dσ
1 − ε 2A1 σT cos ψ + 2ε (1 − ε )A2 σT sin ψ
= Γν · σT + εσL + Pe Pt
dE dΩ
(5)
where A1 = σ1/2 − σ3/2
σ1/2 + σ3/2 ≡ σT T /σT and A2 = σLT /σT . The observables σL , ε and σLT are due to the longitudinal polarization component only. Fig. 6
shows how the asymmetry A 1 evolves from the real photon point. The data are from
the GDH-Collaboration and from the Hall B experiment at JL AB [39]. One observes
that apart from the delta resonance no higher resonances are visible in the Hall B data.
The delta resonance looses strength when going to finite Q 2 and contributions above
W > 1.5 GeV change sign.
The GDH Sum Rule is valid only at the real photon point Q 2 = 0. There it is
a prediction with a high relative precision of 10 −6 because the anomalous magnetic
moment of the nucleon is very well known. There are several ways to generalize the
integral on left hand side of the GDH Sum Rule. Amongst these choices experimentalists
often favor a version which is a straight forward generalization of the GDH Sum Rule:
I GDH (Q2 ) = −
∞
dν 0
ν
· σ3/2 (ν , Q2 ) − σ1/2 (ν , Q2 )
(6)
On the other hand theorists prefer the following notation called first moment of g 1 :
Γ1 (Q2 ) =
1
0
dx g1 (x, Q2 ) = Q2 /2
∞
dν
0
ν
G1 ( ν , Q2 )
(7)
Going to Q2 → ∞ in the isovector case one obtains the Bjorken Sum Rule:
Γ1p − Γn1 = gA /6 known with a relative precision of 3 · 10 −3.
43
(8)
FIGURE 7. Left: Hall A measurements [43] for I(Q 2 ) vs. Q2 for the neutron with and without an
estimate of the DIS contribution; dotted (dot-dashed) line: ChPT calculations of Ref. [41] (Ref. [42]);
solid line: Ref. [29]; open boxes: data from H ERMES [44]; for 1 GeV 2 < Q2 a semi-log scale has been
adapted.
Right: H ERMES data from Ref. [45] for the proton, deuteron and the neutron in comparison
to a model by Soffer and Teryaev.
On the other end the GDH Sum Rule fixes the slope at Q2 = 0: Γ1 /Q2 = −κ 2 /8m2 .
Recently Ji and Osborne [40] have developed a unified formalism to describe the
generalized sum with respect to the virtual Compton forward scattering spin-amplitudes
S1 (ν , Q2 ), S2 (ν , Q2 ):
∞
dν
G1 ( ν , Q2 )
(9)
S̄1 (0, Q2 ) = 4
ν0 ν
were S̄1,2 are the amplitudes without the elastic intermediate state. The right hand side
of Eqn. 9 is measurable while the left hand side is calculable. At Q 2 > 1 GeV2 QCD
operator product expansions should yield the value of S̄1 while at Q2 < 0.1 GeV2 chiral
perturbation theory calculations are used. However, it turns out that these calculations
have by far not yet converged. A comparison of calculations in the heavy baryon
approach by Ji, Kao and Osborne [41] with a recent calculation by Bernard, Hemmert
and Meissner [42] shows that both approaches do not agree and moreover, that the chiral
expansion has not yet converged. The level of uncertainty for Γ 1 (Q2 ) at Q2 = 0.1 GeV2
is of the order of 50%.
Despite, there is no stringent sum rule established at finite Q 2 , one can study the
transition from hadronic degrees of freedom to partonic structure. Fig. 7 shows two
examples for experiments that have been performed in this respect. Plotted is in cases
I(Q2) as defined in Eqn. 6 as a function of Q2 . The left hand diagram mainly shows
the results obtained at Hall A at JL AB [43]. The right hand diagram shows data for the
H ERMES collaboration at D ESY [45]. The data predominantly show a 1/Q 2 behavior at
high Q2 while the interpretation at about Q2 < 0.3 is not clear.
44
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