Results on DIS Exclusives and Generalized
Parton Distributions
M. Vanderhaeghen
Institut für Kernphysik, Johannes Gutenberg-Universität, D-55099 Mainz, Germany
Abstract. We discuss how generalized parton distributions (GPDs) enter in a variety of hard exclusive processes such as deeply virtual Compton scattering (DVCS) and hard meson electroproduction
reactions on the nucleon. We discuss the links between GPDs and elastic nucleon form factors as
well as the information contained in the second moment of (generalized) parton distributions. We
subsequently show some key observables which are sensitive to the various hadron structure aspects
of the GPDs, and discuss their experimental status.
INTRODUCTION
Generalized parton distributions (GPDs), are universal non-perturbative objects entering
the description of hard exclusive electroproduction processes (see Refs. [1, 2, 3] for reviews and references). In leading twist there are four GPDs for the nucleon, i.e. H, E,
H̃ and Ẽ, which are defined for each quark flavor (u, d, s). These GPDs depend upon
the different longitudinal momentum fractions x ξ (x ξ ) of the initial (final) quark
and upon the overall momentum transfer t ∆ 2 to the nucleon (see Fig. 1). As the mo-
R
k ; 2 P
q0
q
-
; 2
N
0
0
q R q0
R+
P
(a)
k ; 2 R k + 2
N
P
2
-
; 2
N
R k + 2
R+
P
(b)
N
2
FIGURE 1. “Handbag” diagrams for the DVCS process, containing the GPDs.
mentum fractions of initial and final quarks are different, one accesses quark momentum
correlations in the nucleon. Furthermore, if one of the quark momentum fractions is negative, it represents an antiquark and consequently one may investigate qq̄ configurations
in the nucleon. Therefore, these functions contain a wealth of new nucleon structure information, generalizing the information obtained in inclusive deep inelastic scattering.
In particular, the GPDs allow to access the fraction of the nucleon spin carried by the
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
132
quark total angular momentum (J u J d , etc.), q̄q components of the nucleon wave function (in particular the D-term [4]), the strength of the skewedness effects in the GPDs
(encoded in their ξ -dependence), the quark structure of N N £ ∆ transitions, flavor
SU 3 breaking effects, and others. Furthermore, it has been shown that by a Fourier
transform of the t-dependence of GPDs, it is conceivable to access the distributions of
parton in the transverse plane [5, 6].
NUCLEON ELECTROMAGNETIC FORM FACTORS
We start by discussing the t-dependence of the GPDs which is directly related to nucleon
elastic form factors (FFs) through sum rules. In particular, the nucleon Dirac and Pauli
FFs F1 t and F2 t can be calculated from the GPDs H and E through the following
sum rules for each quark flavor (q u d)
1
1
dx H q x ξ t F2q t dx E q x ξ t (1)
F1q t 1
1
We can choose ξ 0 in the previous equations, and model H x 0 t and E x 0 t subsequently. For the GPD H x 0 t , a plausible ansatz at low t is a Regge form as
discussed in [3]. This leads to the following integrals to calculate the Dirac FFs for uand d-quark flavors :
1
1
1
1
F1u t dx uv x
F1d t dx dv x
(2)
α
t
α
0
0
x 1
x 1t
¼
¼
where uv x and dv x are the u- and d-quark valence distributions, and where α1 is the
slope of the leading Regge trajectory. The proton and neutron Dirac FFs then follow
from
¼
F1p t eu F1ut ed F1d t F1nt eu F1d t ed F1u t with eu 2 3 ed 1 3 the u (d) quark charges respectively.
(3)
In the ansatz of Eq. (2), the Regge slope α1 is the only free parameter, which can be
determined from the proton Dirac radius r12 , yielding α1 11 GeV 2 . Such value is
close to the expectation from Regge slopes for meson trajectories, therefore supporting
the ansatz of Eq. (2).
To calculate the electric and magnetic nucleon FFs, one also needs to calculate the
Pauli FF F2 . For F2 , we use an ansatz based on a valence quark distribution for the
valence part of E x 0 t entering in (1) as :
¼
¼
F2u t 1
0
1
1
dx κu uv x
2
xα2 t
¼
F2d t 1
0
dx κd dv x
1
xα2 t
¼
(4)
where κu and κd are given by κu 2 κ p κn , and κd κ p 2 κn .
In Fig. 2, the predictions of the above Regge ansatz are shown for the nucleon FFs
133
(taking α1 α2 , supported by the universality of meson Regge slopes). For both proton
and neutron magnetic FFs, the Regge parametrization catches the basic features of
the data for t 05 GeV2 . It also reproduces the decreasing trend with t for the
ratio of electric to magnetic proton FFs, and yields a remarkable good description for
the neutron electric FF up to t 1 GeV2 . At larger values of t, the Regge form
expectedly falls short of the data as one expects a transition to the perturbative behavior.
For t 1 GeV2 , an overlap representation linking the nucleon Dirac FF to GPDs has
been given [10, 11]. A topic for further study is to incorporate both small-t and large-t
regimes in a unified parametrization, needed to perform the Fourier transform for the
t-dependence of GPDs in order to map out the distribution of partons in the transverse
plane.
¼
1.2
1
p
GE / G D
0.6
1
0.8
1
0.2
n
GE
p
0
1
p
p
0.6
0.4
0.5
µ GE / G M
1.2
n
0.8
n
GM / µ G D
p
p
GM / µ GD
¼
0.5
0
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1
0
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1
2
2
-t (GeV )
-t (GeV )
FIGURE 2. Left side : proton magnetic (upper panel) and electric (middle panel) form factors compared
to the dipole form G D t 11 t 0712, as well as the ratio of both form factors (lower panel). Right
side : neutron magnetic (upper panel) and electric (lower panel) form factors. The curves correspond to
the Regge ansatz of Eqs.(2) and (4) , with α 1 11 GeV 2 , α2 11 GeV 2 .
¼
¼
ANGULAR MOMENTUM SUM RULE AND PROTON
MOMENTUM FRACTIONS CARRIED BY VALENCE QUARKS
The second moments of the quark helicity independent GPDs are given by the nucleon
form factors of the symmetric energy momentum tensor. At zero momentum transfer
(t 0), this leads to a sum rule [12] for the fraction of the nucleon angular momentum
134
(J q ) carried by a quark of the flavor q :
Jq 1
2
1
1
dx x
H qx ξ 0 E qx ξ 0 12 ∆Σ Lq
(5)
As the quark spin part ∆Σ is measured through polarized DIS experiments as ∆Σ 30%,
the knowledge of the second moment of the GPDs H and E provides an access to the
quark orbital angular momentum (Lq ).
A relation involving second moments of quark distributions has been proposed [3] for
the ratio of proton to neutron anomalous magnetic moments :
κp
κn
12
4 M2dval M2uval
M2dval M2uval
(6)
in terms of the momentum fractions carried by valence u- and d-quarks :
M2qval
1
0
dx x qval x (7)
In Fig. 3, the rhs of Eq. (6), is seen to be scale independent and the relation (6) is
numerically verified, for all parton distributions, to an accuracy at the one percent level!
It may be interesting to investigate this further within different nucleon structure models
(see e.g. [15]).
-0.92
-0.925
MRST01 NLO
MRST01 NNLO
-0.93
CTEQ5M NLO
-0.935
CTEQ6M NLO
GRV98 NLO
κ /κ
p
-0.94
n
MRST98 NLO
-0.945
-0.95
1
10
µ ( GeV )
2
2
FIGURE 3. Scale dependence of the rhs of (6) for various parton distributions. Dotted curves : MRST98
NLO , MRST01 NLO , MRST01 NNLO [7]. Dashed curves : CTEQ5M NLO , CTEQ6M NLO [13].
Dashed-dotted curve : GRV98 NLO(MS) [14]. The constant solid curve shows the experimental value
for κ p κ n .
135
DVCS BEAM-HELICITY ASYMMETRY
We next turn to the DVCS observables and their dependence on the GPDs. At intermediate lepton beam energies, one can extract the imaginary part of the DVCS amplitude
through the ep epγ reaction with a polarized lepton beam, by measuring the out-ofplane angular dependence (in the angle φ ) of the produced photon [16]. It was found in
Refs. [17, 18] that the resulting electron single spin asymmetry (SSA)
σeh12 σeh 12
SSA σeh12 σeh 12
(8)
ALU
with σeh the cross section for an electron of helicity h, can be sizeable for HERMES
(Ee = 27 GeV) and JLab (Ee = 4 - 11 GeV) beam energies. The SSA for the ep epγ
reaction has recently been measured in pioneering experiments at HERMES [19] and
JLab/CLAS [20], as shown in Fig. 4. They display already at the accessible values of
Q2 1 25 GeV2 predominantly a sin φ dependence, indicating a dominance of the
twist-2 DVCS amplitude. Furthermore, the observed magnitude is in good agreement
with the theoretical calculations [21, 22] in terms of GPDs. Once the leading order
mechanism is confirmed by experiment, the measured helicity difference is directly proportional to the GPDs along the line x ξ .
0.4
0.6
0.3
0.4
0.2
0.2
0.1
A
0
0
-0.2
-0.1
-0.4
-0.2
-0.3
-0.6
-3
-2
-1
0
1
2
3
φ (rad)
-0.4
0
50
100
150
200
φ (deg)
250
300
350
FIGURE 4. The DVCS beam helicity asymmetry as measured at HERMES [19] (left) and JLab/CLAS
[20] (right). Full curves : twist-2 + twist-3 predictions of Ref. [21].
Dedicated experiments to measure the SSA with improved accuracy in a large kinematic range are already planned and underway both at JLab and HERMES.
136
DVCS BEAM-CHARGE ASYMMETRY
Besides the beam-helicity asymmetry for the ep epγ reaction, which accesses the
imaginary part of the DVCS amplitude, one gets access to the real part of the DVCS amplitude by measuring both e p e pγ and e p e pγ processes. In those reactions,
besides the mechanism where the photon originates from a quark (handbag diagrams of
Fig. 1), the photon can also be emitted by the lepton lines, in the so-called Bethe-Heitler
(BH) process. However, in the difference σe σe , the BH drops out, and one measures
the real part of the BH-DVCS interference [23]
σe σe
ℜ
T BH T DVCS
£
(9)
which is sensitive to the GPDs away from the line x ξ .
It has been shown in [21] that this beam-charge asymmetry (BCA) gets a sizeable
contribution from the D-term. The latter encodes qq̄ scalar-isoscalar correlations in the
nucleon as shown in Fig. 5, and has been estimated in the chiral quark soliton model
[24].
γ*
γ
π
π
FIGURE 5. Model contribution to the D-term entering the GPDs H and E.
The DVCS BCA has been accessed experimentally at HERMES [25], and the preliminary data are shown in Fig. 6, together with the theoretical predictions. The measured
asymmetry shows a cos φ dependence with magnitude 010 015, and favors the
calculations which include the D-term. This opens up the perspective to study systematically (mesonic) qq̄ components in the nucleon. Further measurements, with improved
statistics, of the DVCS BCA are planned at HERMES.
DOUBLE DEEPLY VIRTUAL COMPTON SCATTERING
In the DVCS observables, as discussed above, the GPDs enter in general in convolution
integrals over the average quark momentum fraction x, and only ξ can be accessed experimentally. A particular exception is when one measures an observable proportional
137
e /e + p → e /e + p + γ
-
+
-
2
+
2
2
(σe+ - σe-) / (σe+ + σe-)
Ee = 27 GeV, Q = 2.5 GeV , xB = 0.11, t = -0.25 GeV
0.5
0.4
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
-0.5
-150
-100
-50
0
50
100
150
Φ (deg)
FIGURE 6. The DVCS beam-charge asymmetry with preliminary HERMES data [25]. Theoretical
predictions from [21]. Dashed-dotted (dashed) curves : twist-2 DVCS with (without) D-term. Full curve :
twist-3 effects in addition to the D-term.
to the imaginary part of the amplitude, such as discussed for the beam helicity asymmetry in DVCS. Then, one actually measures directly the GPDs at some specific point,
x ξ , which is certainly an important gain of information but clearly not sufficient
to map out the GPDs independently in both quark momentum fractions. In absence of
any model-independent “deconvolution" procedure at this moment, existing analyses of
DVCS experiments have to rely on some global model fitting procedure.
The double DVCS (DDVCS) process, i.e. the scattering of a spacelike virtual photon from the nucleon with the production of a virtual photon in the final state, i.e. the
l p l pe e reaction, provides a way around this problem of principle. Compared to
the DVCS process with a real photon in the final state, the virtuality of the final photon in
DDVCS yields an additional lever arm, which allows to vary both quark momenta x and
ξ independently [26]. In particular, the imaginary part of the DDVCS amplitude maps
out the GPD where its first argument lies in the range x ξ . Although one does not
access the whole range in x, clearly, the gain of information on the GPDs is tremendous
as no deconvolution is involved to access this region of the GPDs. Furthermore, x ξ
is just the range where the GPDs contain wholly new information on mesonic (qq̄) components of the nucleon, which is absent in the forward limit (where ξ 0). However
to construct sum rules, one also needs information in the region x ξ . To access the
range x ξ one would need two spacelike virtual photons, necessitating to select the
two-photon exchange process in elastic electron nucleon scattering.
138
In Fig. 7,the dependence of the estimated cross section and SSA for the ep epe e process is shown [26] as function of the virtuality q ¼2 of the outgoing lepton pair, in kinematics accessible at JLab. As the twist-2 SSA basically displays a sin Φ structure, we
show its value at Φ 90o . As is seen from Fig. 7, the ep epe e cross section scaled
with the factor N 1 q¼2, with N αem 4π 43 reduces to the ep epγ cross section
when approaching the real photon point. Similarly, the SSA for the ep epe e process reduces to the corresponding SSA for the ep epγ process. When going to larger
virtualities q¼2 , the DDVCS shows a growing deviation from the 1q ¼2 behavior and the
magnitude of the SSA decreases. The strong sensitivity of the SSA on q ¼2 , as seen from
Fig. 7, should therefore allow to map out the GPDs in the range x ξ . Although the
cross sections are small, their measurement seems feasible with a dedicated experiment
at JLab and at a future high-energy, high-luminosity lepton facility.
e +p→e +p+ee
-
-
- +
Ee = 11 GeV, Q = 4 GeV , xB = 0.25, t = -0.2 GeV , Φ = 90
2
2
o
0.07
0.06
0.05
2
4
N * q’ * dσ/(dQ dxBdtdΦdq’ ) (nb/GeV )
2
0.04
0.03
-1
2
2
0.02
0.01
0
-2
10
0.4
10
-1
1
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
-2
10
SSA
10
-1
1
2
2
q’ ( GeV )
FIGURE 7. Cross section (upper panel) and SSA (lower panel) of the ep epe e process as function
of the e e virtuality q¼2 . Dashed curves : BH; dashed-dotted curves : BH + DDVCS, full curves : BH +
DDVCS + ρL0 , the latter being a background process. The dotted curves are the corresponding results for
ep epγ . The ep epe e cross section is scaled with N 1 q¼2 , in order to yield the ep epγ cross
section in the limit q¼2 0. Calculations from Ref. [26].
139
HARD MESON ELECTROPRODUCTION
The GPDs reflect the structure of the nucleon independently of the reaction which probes
the nucleon. In this sense, they are universal quantities and can also be accessed, in
different flavor combinations, through the hard exclusive electroproduction of mesons
- π 0¦ η ρ 0¦ ω φ - for which a QCD factorization proof was given [27]. This
factorization theorem applies when the virtual photon is longitudinally polarized, which
corresponds to a small size configuration compared to a transversely polarized photon.
For the longitudinal vector meson (VL ) electroproduction processes γL£ N VL N
at large Q2 , the GPDs enter in different isospin combinations for VL = ρL0 , ρL , ωL ,
allowing for a flavor decompostion of GDPs [17, 28].
An γL£ N VL N observable of particular interest is the transverse spin asymmetry
(TSA) for a nucleon polarized perpendicular to the reaction plane [3]. The TSA is
proportional to the imaginary part of the interference of the amplitudes which contain
the GPDs H and E respectively. Therefore, the TSA provides a unique observable to
extract the GPD E. Besides, one may expect that the theoretical uncertainties for the
meson electroproduction cross sections largely disappear for the TSA, as it involves a
ratio of cross sections, suggesting that the leading order expression is already accurate at
accessible values of Q2 (of a few GeV2 ). Due to its linear dependence on the GPD E, the
TSA for longitudinally polarized vector mesons opens up the perspective to extract the
total angular momentum contributions J u and J d of the u and d-quarks to the proton
spin. Due to the different u- and d-quark content of the vector mesons, the asymmetries
for the ρL0 , ωL and ρL channels are sensitive to different combinations of J u and J d , with
ρL0 production sensitive to 2J u J d , ωL to 2J u J d , and ρL to J u J d .
γL + p → ρL + p
TRANSVERSE SPIN ASYMMETRY
*
0
0.05
d
2
J =0
2
Q = 5 GeV
0
2
-t = 0.5 GeV
-0.05
-0.1
u
J = 0.1
-0.15
u
J = 0.2
-0.2
u
J = 0.3
-0.25
u
J = 0.4
-0.3
-0.35
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
xB
FIGURE 8. xB dependence of the transverse spin asymmetry for the γ L£p ρL0 p reaction. The estimates
are given using the model of Ref. [3] for the GPDs E u and E d . The sensitivity is shown to different values
of J u (for a value J d 0).
140
In Fig. 8, the TSA for ρL0 production is shown. One observes that it displays a
pronounced sensitivity to J u . It will therefore be very interesting to provide a first
measurement of this asymmetry in the near future, for a transversely polarized target,
such as it currently available at HERMES.
OUTLOOK
We have seen some very promising first glimpses of GPDs entering hard exclusive
reactions at existing facilities. A dedicated program aiming at the extraction of the full
physics potential contained in the GPDs will require a dedicated facility combining high
luminosity and a good resolution.
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