Measurement of azimuthal asymmetries associated with deeply virtual Compton scattering on an unpolarized deuterium target

Abstract

Azimuthal asymmetries in exclusive electroproduction of a real photon from an unpolarized deuterium target are measured with respect to beam helicity and charge. They appear in the distribution of these photons in the azimuthal angle ϕ around the virtual-photon direction, relative to the lepton scattering plane. The extracted asymmetries are attributed to either the deeply virtual Compton scattering process or its interference with the Bethe–Heitler process. They are compared with earlier results on the proton target. In the measured kinematic region, the beam-charge asymmetry amplitudes and the leading amplitudes of the beam-helicity asymmetries on an unpolarized deuteron target are compatible with the results from unpolarized protons.

Full text

Page 1

arXiv:0911.0095v1 [hep-ex] 31 Oct 2009
Measurement of azimuthal asymmetries associated
with deeply virtual Compton scattering on an
unpolarized deuterium target
The HERMES Collaboration
A. Airapetianℓ,oN. AkopovzZ. AkopoveM. Amarianf,1
E.C. Aschenauerf,2W. AugustyniakyR. AvakianzA. Avetissianz
E. AvetisyaneB. BalloS. BelostotskirN. BianchijH.P. Blokq,x
H. B¨ ottcherfA. BorissoveJ. BowlesmV. BryzgalovsJ. Burnsm
M. CapiluppiiG.P. CapitanijE. CisbaniuG. CiulloiM. Contalbrigoi
P.F. DalpiaziW. Deconincke,o,3R. De LeobL. De Nardoe,v
E. De SanctisjM. DiefenthalerhP. Di NezzajJ. DreschlerqM. D¨ urenℓ
M. Ehrenfriedℓ,4G. ElbakianzF. Ellinghausd,5R. FabbrifA. Fantonij
L. FelawkavS. FrullaniuD. GabbertfG. GapienkosV. Gapienkos
F. GaribaldiuG. Gavrilove,r,vV. GharibyanzF. Giordanoe,iS. Gliskeo
C. Hadjidakisj,6M. Hartige,7D. HaschjT. HasegawawG. Hillm
A. HillenbrandfM. HoekmY. HollereI. HristovafY. Imazuw
A. IvanilovsA. IzotovrH.E. JacksonaA. JgounrH.S. JokS. Joostenn,k
R. KaisermG. KaryanzT. Kerim,ℓE. KinneydA. Kisselevr
N. KobayashiwV. KorotkovsV. KozlovpB. Kraussh,8P. Kravchenkor
V.G. KrivokhijinegL. LagambabR. LambnL. Lapik´ asqI. Lehmannm
P. LenisaiL.A. Linden-LevynA. L´ opez RuizkW. LorenzonoX.-G. Luf
X.-R. Luw,9B.-Q. MacD. MahonmN.C.R. MakinsnS.I. Manaenkovr
L. Manfr´ euY. MaocB. MarianskiyA. Martinez de la Ossad
H. MarukyanzC.A. MillervY. MiyachiwA. MovsisyanzV. Mucciforaj
D. M¨ uller10M. MurraymA. Mussgillere,hE. NappibY. Naryshkinr
A. NasshM. NegodaevfW.-D. NowakfL.L. Pappalardoi
R. Perez-BenitoℓN. Pickerth,8M. RaithelhP.E. ReimeraA.R. Reolonj
C. RiedlfK. RithhG. RosnermA. RostomyaneJ. RubinnD. Ryckboschk
Y. SalomatinsF. SanftltA. Sch¨ afertG. Schnellf,kK.P. Sch¨ ulere
B. SeitzmT.-A. ShibatawV. ShutovgM. StancariiM. Staterai
Preprint submitted to Nuclear Physics B31 October 2009

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E. SteffenshJ.J.M. SteijgerqH. StenzelℓJ. Stewartf,2F. Stinzingh
S. TaroianzA. TerkulovpA. TrzcinskiyM. TytgatkA. Vandenbrouckek,11
P.B. Van der NatqY. Van Haarlemk,12C. Van HulsekM. Varandae
D. VeretennikovrV. VikhrovrI. Vilardib,13C. Vogelh,14S. Wangc
S. Yaschenkof,hH. YecZ. Yee,15S. YenvW. YuℓD. Zeilerh
B. Zihlmanne,16P. Zupranskiy
aPhysics Division, Argonne National Laboratory, Argonne, Illinois 60439-4843, USA
bIstituto Nazionale di Fisica Nucleare, Sezione di Bari, 70124 Bari,Italy
cSchool of Physics, Peking University, Beijing 100871, China
dNuclear Physics Laboratory, University of Colorado, Boulder, Colorado 80309-0390, USA
eDESY, 22603 Hamburg, Germany
fDESY, 15738 Zeuthen, Germany
gJoint Institute for Nuclear Research, 141980 Dubna, Russia
hPhysikalisches Institut, Universit¨ at Erlangen-N¨ urnberg, 91058 Erlangen, Germany
iIstituto Nazionale di Fisica Nucleare, Sezione di Ferrara and Dipartimento di Fisica, Universit` a di
Ferrara, 44100 Ferrara, Italy
jIstituto Nazionale di Fisica Nucleare, Laboratori Nazionali di Frascati, 00044 Frascati, Italy
kDepartment of Subatomic and Radiation Physics, University of Gent, 9000 Gent, Belgium
ℓPhysikalisches Institut, Universit¨ at Gießen, 35392 Gießen, Germany
mDepartment of Physics and Astronomy, University of Glasgow, Glasgow G12 8QQ, United Kingdom
nDepartment of Physics, University of Illinois, Urbana, Illinois 61801-3080, USA
oRandall Laboratory of Physics, University of Michigan, Ann Arbor, Michigan 48109-1040, USA
pLebedev Physical Institute, 117924 Moscow, Russia
qNational Institute for Subatomic Physics (Nikhef), 1009 DB Amsterdam, The Netherlands
rPetersburg Nuclear Physics Institute, Gatchina, Leningrad region, 188300 Russia
sInstitute for High Energy Physics, Protvino, Moscow region, 142281 Russia
tInstitut f¨ ur Theoretische Physik, Universit¨ at Regensburg, 93040 Regensburg, Germany
uIstituto Nazionale di Fisica Nucleare, Sezione Roma 1, Gruppo Sanit` a and Physics Laboratory,
Istituto Superiore di Sanit` a, 00161 Roma, Italy
vTRIUMF, Vancouver, British Columbia V6T 2A3, Canada
wDepartment of Physics, Tokyo Institute of Technology, Tokyo 152, Japan
xDepartment of Physics and Astronomy, Vrije Universiteit, 1081 HV Amsterdam, The Netherlands
yAndrzej Soltan Institute for Nuclear Studies, 00-689 Warsaw, Poland
zYerevan Physics Institute, 375036 Yerevan, Armenia
Abstract
Azimuthal asymmetries in exclusive electroproduction of a real photon from an unpolarized
deuterium target are measured with respect to beam helicity and charge. They appear in the
distribution of these photons in the azimuthal angle φ around the virtual-photon direction, rel-
ative to the lepton scattering plane. The extracted asymmetries are attributed to either the
deeply virtual Compton scattering process or its interference with the Bethe-Heitler process.
They are compared with earlier results on the proton target. In the measured kinematic region,
the beam-charge asymmetry amplitudes and the leading amplitudes of the beam-helicity asym-
metries on an unpolarized deuteron target are compatible with the results from unpolarized
protons.
Key words: DIS, HERMES experiment, GPD, DVCS, deuteron, unpolarized deuterium target
PACS: 13.60.-r, 24.85.+p, 13.60.Fz, 14.20.Dh
2

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AA’
e
e’
* γ
γ
ξ
x+
ξ
x−
AA’
e
e’
* γ
γ
ξ
x+
ξ
x−
A A’
e e’
* γ
γ
A A’
e e’
* γ
γ
(a) (b)
Fig. 1. Leading order Feynman diagrams for (a) deeply virtual Compton scattering and (b) the
Bethe-Heitler process.
1. Introduction
Lepton-nucleon scattering experiments have long been an important tool in the de-
tailed study of nucleon structure [1]. Two complementary approaches have contributed
the most to our understanding of the nucleon. Elastic lepton-nucleon scattering has been
exploited to extract nucleon form factors, which reveal how the electromagnetic nucleon
structure differs from that of a point-like spin-1/2 particle. In another approach, Par-
ton Distribution Functions (PDFs) are extracted from Deeply Inelastic Scattering (DIS).
They represent distributions in the longitudinal momentum fraction carried by quarks
and gluons in a nucleon moving with “infinite” momentum. PDFs and form factors
present only one-dimensional pictures of nucleon structure. In recent years, a more com-
prehensive multi-dimensional description of the nucleon has emerged in the framework of
Generalized Parton Distributions (GPDs) [2,3,4]. Their dependence on three kinematic
1Now at: Old Dominion University, Norfolk, VA 23529, USA
2Now at: Brookhaven National Laboratory, Upton, New York 11772-5000, USA
3Now at: Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
4Now at: Siemens AG Molecular Imaging, 91052 Erlangen, Germany
5Now at: Institut f¨ ur Physik, Universit¨ at Mainz, 55128 Mainz, Germany
6Now at: IPN (UMR 8608) CNRS/IN2P3 - Universitet´ e Paris-Sud, 91406 Orsay, France
7Now at: Institut f¨ ur Kernphysik, Universit¨ at Frankfurt a.M., 60438 Frankfurt a.M., Germany
8Now at: Siemens AG, 91301 Forchheim, Germany
9Now at: Graduate University of Chinese Academy of Sciences, Beijing 100049, china
10Present address: Institut f¨ ur Theoretische Physik II, Ruhr-Universit¨ at Bochum, 44780 Bochum, Ger-
many
11Now at: Dept of Radiology, Stanford University, School of Medicine, Stanford, California 94305-5105,
USA
12Now at: Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA
13Now at: IRCCS Multimedica Holding S.p.A., 20099 Sesto San Giovanni (MI), Italy
14Now at: AREVA NP GmbH, 91058 Erlangen, Germany
15Now at: Fermi National Accelerator Laboratory, Batavia, Illinois 60510, USA
16Now at: Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606, USA
3

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Fig. 2. Definition of the azimuthal angle φ between the lepton scattering and photon production planes.
Note that the azimuthal angle defined in this work differs from that used in Ref. [15]: φ = π − φ[15].
quantities in addition to their evolution with the hard scale of the process carries informa-
tion on two-parton correlations and quark transverse spatial distributions [5,6,7,8,9,10].
GPDs embody PDFs as limiting cases, while elastic form factors appear as certain GPD
moments. Other moments are connected with the total parton angular momentum con-
tribution to the nucleon spin via the Ji relation [4].
GPDs can be constrained by measurements of hard exclusive leptoproduction of a pho-
ton or meson in ‘elastic’ processes that leave the target intact. In Deeply Virtual Compton
Scattering (DVCS), a quark absorbs a hard virtual photon, emits an energetic real photon
and joins the target remnant (see Fig. 1 (a)). DVCS is presently the only experimentally
feasible hard exclusive process for which the effects of next-to-leading order [11,12,13]
and next-to-leading twist [14,15,16] are under complete theoretical control [17].
The final state of the DVCS process cannot be experimentally distinguished from that
of the Bethe-Heitler (BH) process, i.e., radiative elastic scattering (see Fig. 1 (b)). Hence,
the two processes can interfere. Exclusive leptoproduction on a nucleon or nuclear target
A of a real photon with four-momentum q′is denoted by
e(k) + A(p) → e(k′) + A(p′) + γ(q′), (1)
where k (k′) and p (p′) are the four-momenta of the incoming (outgoing) lepton and
target, respectively. Averaged over the kinematic acceptance of the HERMES experiment,
the BH cross section is much larger than that of the DVCS process. However, the BH
cross section has a much weaker Q2dependence than the evolution of the DVCS cross
section [4], so that in the HERMES energy range they can become comparable near
Q2= 1GeV2, with −Q2≡ q2= (k − k′)2.
Even in kinematic conditions where the DVCS process makes only a small contribution
to the photon production cross section, its interference with the BH process provides
access to the DVCS amplitudes through measurements of cross section asymmetries with
respect to the charge and helicity of the incident lepton and the polarization of the target.
These asymmetries appear in the distribution of the real photons in the azimuthal angle
φ, defined as the angle between the lepton scattering plane, i.e., the plane defined by the
incoming and outgoing lepton direction and the photon production plane spanned by the
virtual and real photons (see Fig. 2). Significant azimuthal beam-helicity asymmetries in
hard electroproduction of photons on the proton were first reported in Refs. [18,19]. Later,
asymmetries with respect to longitudinal [20,21] and transverse [22] target polarization,
4

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as well as beam charge [23] and, with greater precision, beam helicity [24,25,26,27], were
also measured on the proton.
Measurements of azimuthal asymmetries for DVCS on nuclear targets [28] were ad-
vocated as a useful source of information about partonic behavior in nuclei and nuclear
binding forces [29]. If the target nucleus remains in its ground state the process is called
coherent, while it is called incoherent if the nucleus is broken up. The deuteron is a
spin-1 nucleus, with implications for DVCS observables for the coherent reactions, which
contribute mainly at very small values of the momentum transfer to the target. The
asymmetries from the incoherent process involve mainly hard exclusive electroproduc-
tion of a photon on the proton. The neutron contribution to the yield is typically small
due to the suppression of the BH amplitude on the neutron by the small elastic electric
form factor at low and moderate values of the momentum transfer to the target.
This paper reports the first observation of azimuthal asymmetries with respect to beam
helicity and charge for exclusive electroproduction of a real photon from an unpolarized
deuterium target (e±d → e±γ X). The dependence of these asymmetries on the kine-
matic conditions of the reaction is also presented and certain asymmetry amplitudes
are compared with the corresponding amplitudes obtained on an unpolarized hydrogen
target (e±p → e±γ X) at HERMES [27].
2. GPDs and DVCS
2.1. Generalized Parton Distributions
In the generalized Bjorken limit of large Q2at fixed values of the Bjorken scaling
variable xB = Q2/(2p · q) and small squared four-momentum transfer t = (p − p′)2to
the target, the DVCS process can be described by the leading (handbag) diagrams in
Fig. 1(a). Here, the process factorizes [3,12,30] into a hard photon-quark scattering part
calculable in quantum electrodynamics, and a soft part describing the nucleon structure,
which can be expressed in terms of GPDs [2,3,4].
Like PDFs, GPDs depend on x and on the factorization scale Q2. In addition, GPDs
depend on a skewness variable ξ and the Mandelstam variable t. The skewness ξ repre-
sents half the difference in the longitudinal momentum fractions of the quark before and
after the scattering, while x is their mean value (following the convention of Ref. [4]). In
leading order, ξ is directly accessible as it is related to the Bjorken scaling variable xB
by ξ ≃ xB/(2−xB). In contrast, x is not directly accessible in DVCS, and some observ-
ables appear as x-convolutions of GPDs. Hence x plays a role different from that of xB
in inclusive DIS. GPDs evolve logarithmically with Q2in analogy with PDFs [2,3,4,31].
This dependence on Q2is omitted for simplicity in the following.
DVCS on spin-1/2 targets, such as nucleons, is described by four leading-twist quark-
chirality conserving GPDs for each quark flavour q (and also for the gluon g), namely the
GPDs Hq, Eq,? Hqand?Eq[15]. The GPDs Hqand Eqare quark-helicity averagedwhereas
while Eqand?Eqare associated with a helicity flip of the nucleon. In contrast, the coherent
Hq
4— to describe all DVCS observables. In the forward limit of
vanishing momentum difference between the initial and final hadronic state (t → 0 and
? Hqand?Eqare quark-helicity dependent. The GPDs Hqand? Hqconserve nucleon-helicity
process on spin-1 nuclei, such as the deuteron, requires nine GPDs [32] — Hq
4, Hq
1, Hq
2, Hq
3,
5,? Hq
1,? Hq
2,? Hq
3and? Hq
5

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ξ → 0), the GPD Hq(x,0,0) reduces to fq
and? Hq(x,0,0) reduces to gq
limit:
1(x), the quark number density distribution,
1(x), the quark helicity distribution. Similarly, for spin-1
targets the GPDs H1,? H1and H5reduce to the following parton densities in the forward
1(x,0,0) =q1(x) + q−1(x) + q0(x)
Hq
3
≡ fq
≡ gq
≡ bq
1(x), (2)
? Hq
Hq
1(x,0,0) = q1
5(x,0,0) = q0(x) −q1(x) + q−1(x)
→(x) − q−1
→(x)
1(x), (3)
2
1(x), (4)
where qΛ
{x < 0} x > 0 and positive [negative] helicity in a rapidly moving deuteron target with
longitudinal spin projection Λ. The ‘unpolarized’ (polarization averaged) quark densi-
ties qΛare defined as qΛ(x) = qΛ
←(x). While the probabilistic interpretation of
polarization-averaged and polarization-difference structure functions f1(x) and g1(x) in
terms of quark densities is similar to that in the spin-1/2 case, the tensor structure func-
tion b1(x) does not exist for spin-1/2 targets. It has been measured in DIS on a polarized
spin-1 target [33]. Both H3 and H5 are associated with the 5% D-wave component of
the deuteron wave function in terms of nucleons [34]. H3is related to isoscalar currents
and probes the binding forces in the deuteron, and H5involves a tensor term [32,35], the
analog of which has no relationship to any local current due to Lorentz invariance.
→[←](x) represents the number density of a {anti} quark with momentum fraction
→(x) + qΛ
2.2. Deeply virtual Compton scattering amplitudes
For a target of atomic mass number A, the cross section for the hard exclusive lepto-
production of real photons is given by [35,36]
dσ
dxAdQ2d|t|dφ=
xAe6
32(2π)4Q4
|T |2
√1 + ε2, (5)
where xA ≡ Q2/(2MAν) is the nuclear Bjorken xB, where MAis the target mass and
ν ≡ p · q/MA, ε ≡ 2xAMA/?Q2, and |T | is the total reaction amplitude.
As the final states of the DVCS and BH processes are indistinguishable, the cross
section contains the square of the coherent sum of their amplitudes:
|T |2= |TBH+ TDVCS|2= |TBH|2+ |TDVCS|2+ TDVCST∗
BH+ T∗
??
DVCSTBH
??
I
.(6)
Here, I denotes the BH-DVCS interference term. The BH amplitude is calculable to
leading order in Quantum Electrodynamics (QED) using nuclear form factors measured
in elastic scattering.
The interference term I in Eq. 6 provides separate experimental access to the real and
imaginary parts of the DVCS amplitude through measurements of various cross-section
asymmetries as functions of the azimuthal angle φ [36]. Each of the three terms of Eq. 6
can be written as a Fourier series in φ [15], which in the case of an unpolarized target
reads
6

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|TBH|2=
KBH
P1(φ)P2(φ)×
2
?
n=0
cBH
n cos(nφ), (7)
|TDVCS|2= KDVCS×
?
cDVCS
0
+
2
?
n=1
cDVCS
n
cos(nφ) + λsDVCS
1
sinφ
?
, (8)
I = −
KIeℓ
P1(φ)P2(φ)×
?
cI
0+
3
?
n=1
cI
ncos(nφ) + λ
2
?
n=1
sI
nsin(nφ)
?
. (9)
Here, KBH, KDVCS, and KIare kinematic factors, eℓdenotes the lepton beam charge in
units of the elementary charge, and λ the helicity of the longitudinally polarized lepton
beam. The squared BH and interference terms have an additional cosφ dependence in the
denominator due to the lepton propagators P1(φ) and P2(φ) in the BH process [15,36].
The Fourier coefficients cI
nin Eq. 9 can be expressed as linear combinations of
Compton Form Factors F(ξ,t) (CFFs) [35], which in turn are convolutions of the corre-
sponding GPDs Fq(x,ξ,t) with the hard scattering coefficient functions C∓
?
q
−1
nand sI
q[11,12,13]:
F(ξ,t) =
?1
dx C∓
q(ξ,x)Fq(x,ξ,t), (10)
where the −{+} sign applies to Fq= Hq
target. The real and imaginary parts of the CFFs have different relationships to the flavor
sum over the respective quark GPDs. To leading order in αs,
?
q
1,...,Hq
5
?? Hq
1,...,? Hq
4
?
in the case of a spin-1
ℑm{F(ξ,t)} = −π
e2
q(Fq(ξ,ξ,t) ∓ Fq(−ξ,ξ,t)) .(11)
Hence measurements of cross-section asymmetries with respect to the beam helicity di-
rectly determine combinations of GPDs along the lines x = ±ξ. In contrast, the real
parts of the CFFs involve the full interval in x and constrain the x dependence of GPDs
through convolutions:
?
q
−1
to leading order in αs. Here, P denotes Cauchy’s principal value. Since the x dependence
of GPDs is thereby only weakly constrained, experimental asymmetries in beam charge
must be compared to the predictions of various GPD models.
At leading twist (twist-2), the coefficients cI
tion of GPDs. This is also true for the kinematically suppressed coefficient cI
The coefficients cI
1are sensitive to the ‘D-term’ [37,38], which contributes only in
the ‘ERBL’ region −ξ < x < ξ where quark GPDs have the characteristics of distribu-
tion amplitudes for the creation of a quark-antiquark pair. It does not contribute in the
complementary ‘DGLAP’ region |x| > ξ, where quark GPDs describe the emission and
reabsorption of an (anti-)quark in the infinite momentum frame, thereby having proper-
ties analogous to the familiar (anti-)quark distribution functions. The D-term provides
a convenient means of representing this profound difference in GPD properties between
the two regions, while, e.g., the absorption of this contribution into the double distri-
butions [2,3] would require the introduction of terms with unnatural divergence, having
ℜe{F(ξ,t)} =
e2
q
?
P
?1
dx Fq(x,ξ,t)
?
1
x − ξ±
1
x + ξ
??
,(12)
1and sI
1are related to the same combina-
0∝ −
√−t
QcI
1.
0and cI
7

Page 8

a severity beyond representation by delta functions or their derivatives. In addition to
cI
0, the only other Fourier coefficient related to only twist-2 quark GPDs is
cDVCS
0
. The coefficients cDVCS
1
, sDVCS
1
, cI
and cI
3arise from the gluonic transversity operator [39,40,41] at twist-2 level. The high-
est harmonics of the interference and squared DVCS terms may also receive a twist-4
contribution [42].
1, sI
1, and cI
2, and sI
2appear at the twist-3 level, while cDVCS
2
2.3. Azimuthal cross section asymmetries
The beam-helicity asymmetries for a longitudinally (L) polarized lepton beam and
an unpolarized (U) target, based on the difference and sum of yields for the two beam
charges, respectively, are defined as
AI
LU(φ) ≡[dσ+→(φ) − dσ+←(φ)] − [dσ−→(φ) − dσ−←(φ)]
(φ) ≡[dσ+→(φ) − dσ+←(φ)] + [dσ−→(φ) − dσ−←(φ)]
where → (←) denotes positive (negative) beam helicity and the superscript + (−) corre-
sponds to positron (electron) beam. These definitions serve to separate the sin(nφ) terms
in Eqs. 8 and 9. Similarly, the beam-charge asymmetry (BCA) for an unpolarized beam
scattering from this target is defined as
AC(φ) ≡dσ+(φ) − dσ−(φ)
dσ+(φ) + dσ−(φ)
=[dσ+→(φ) + dσ+←(φ)] − [dσ−→(φ) + dσ−←(φ)]
[dσ+→(φ) + dσ+←(φ)] + [dσ−→(φ) + dσ−←(φ)].
In terms of the Fourier coefficients of Eqs. 7–9 these equations read as
[dσ+→(φ) + dσ+←(φ)] + [dσ−→(φ) + dσ−←(φ)], (13)
ADVCS
LU
[dσ+→(φ) + dσ+←(φ)] + [dσ−→(φ) + dσ−←(φ)], (14)
(15)
AI
LU(φ) =
−
KI
P1(φ)P2(φ)
n=0cBH
n
?2
n=1sI
nsin(nφ)
?2
sinφ
?2
n=0cI
KBH
P1(φ)P2(φ)
?2
?2
?2
cos(nφ) + KDVCS
KDVCSsDVCS
1
n=0cBH
n
cos(nφ) + KDVCS
n=0cDVCS
n
cos(nφ),(16)
ADVCS
LU
(φ) =
KBH
P1(φ)P2(φ)
n=0cDVCS
n
cos(nφ), (17)
AC(φ) =
−
KI
P1(φ)P2(φ)
n=0cBH
n
?3
ncos(nφ)
?2
KBH
P1(φ)P2(φ)
cos(nφ) + KDVCS
n=0cDVCS
n
cos(nφ). (18)
At leading twist (twist-2, twist-3, and twist-2, respectively in the preceding three equa-
tions), and neglecting gluonic terms, they reduce to
AI
LU(φ) ≃
−
n=0cBH
KI
P1(φ)P2(φ)sI
1sinφ
KBH
P1(φ)P2(φ)
?2
?2
−
?2
n
cos(nφ) + KDVCScDVCS
0
,(19)
ADVCS
LU
(φ) ≃
KDVCSsDVCS
1
sinφ
KBH
P1(φ)P2(φ)
n=0cBH
n
cos(nφ) + KDVCScDVCS
0
,(20)
AC(φ) ≃
KI
P1(φ)P2(φ)(cI
n=0cBH
0+ cI
cos(nφ) + KDVCScDVCS
1cosφ)
KBH
P1(φ)P2(φ)
n
0
.(21)
8

Page 9

To the extent that the DVCS contributions to the common denominator can be ne-
glected at HERMES kinematics, the lepton propagators P1(φ) and P2(φ) cancel in
Eqs. 16, 18, 19, and 21. However, this approximation is not invoked in the following
because it would be subject to substantial model uncertainty.
2.4. From Compton form factors to asymmetries
Measured asymmetries are used to constrain GPD models by direct comparison of the
data with model predictions. However, it is instructive to consider certain approximations
relating CFFs and thereby GPDs to observed asymmetries. (These approximations are
not needed in the comparison of GPD model predictions with measured asymmetries.)
For an unpolarized nucleon target, the photon-helicity-conserving amplitude?
the Dirac and Pauli form factors F1and F2[15]:
xN
2 − xN(F1+ F2)? H −
where xN is the Bjorken variable for the nucleon and MN is the nucleon mass. At small
values of xN and −t,?
of the form factors F1 and F2 for the neutron. The leading Fourier coefficients of the
interference term can be approximated as sI
order in 1/Q and in HERMES kinematic conditions,
M1,1is
given at leading twist by a linear combination of the CFFs H,? H and E, together with
?
M1,1= F1H +
t
4M2
N
F2E ,(22)
M1,1≃ F1H for the proton. For the neutron, the term containing
the CFF E in Eq. 22 becomes substantial at large −t due to the relative magnitudes
1∝ ℑm?
M1,1and cI
1∝ ℜe?
M1,1. To leading
AI
AC(φ) ∝ℜeH
LU(φ) ∝ −ℑmH
F1
sinφ,(23)
F1
cosφ.(24)
For the coherent process on the deuteron, the relationship between the Fourier coeffi-
cients and the GPDs is complicated. However, the coefficients can be expanded in powers
of xD, the Bjorken variable for the deuteron target, and τ = t/(4M2
deuteron mass [35]. Then, to leading order in αsand 1/Q, AI
terms of the imaginary part of the deuteron CFFs H1, H3 and H5 and the deuteron
elastic form factors [43] G1and G3(see Fig. 3). The quantity |τ| is typically about 0.003
in the range of small −t where the coherent process is significant, extending up to values
of τ only as large as 0.01. However, as shown in Fig. 3, the magnitude of G3exceeds that
of G1by more than one order of magnitude. Hence certain terms leading in τ (but not
xD) are retained. Defining
D), where MDis the
LU(φ) can be expressed in
?D1,1
U≡3G1H1− 2τ[G1H3+ G3(H1−1
3G2
3H5)] + 4τ2G3H3
31− 4τG1G3+ 4τ2G2
, (25)
the kinematic expansion yields
AI
LU(φ) ≃ −
xD(2 − y)
?
−t
Q2(1 − y)
2 − 2y + y2
ℑm?D1,1
Usinφ,(26)
where y ≡ p · q/(p · k).
9

Page 10

Fig. 3. The deuteron elastic form factors according to Parameterization II of Ref. [43], and the relative
contributions to the denominator of, e.g., Eq. 25 of certain terms involving G3that are not leading in τ.
The relative contributions of those terms are also shown in Fig. 3; they are less than
10% at −t < 0.03GeV2. When these terms are neglected, Eq. 26 becomes
?
2 − 2y + y2
AI
LU(φ) ≃ −
xD(2 − y)
−t
Q2(1 − y)
ℑmH1
G1
sinφ.(27)
The deuteron AC(φ) is related to the real part of the same linear combination of CFFs
appearing in the deuteron AI
?
y
?
yG1
LU(φ):
AC(φ) ≃−
xD
−t
Q2(1 − y)
ℜe?D1,1
ℜeH1
Ucosφ (28)
≃−
xD
−t
Q2(1 − y)
cosφ.(29)
For the coherent process on the deuteron, the leading term in the expansion of coeffi-
cients sI
1lead respectively to Eqs. 27 and 29 which are analogous to Eqs. 23 and
24 for scattering on the nucleon.
1and cI
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Page 11

Table 1
The beam charge and polarization as well as the integrated luminosity in pb−1of the data sets used for
the extraction of the various asymmetries on the unpolarized deuterium target.
Beam BeamLuminosity
YearCharge Polarization [pb−1]
λ = −1 λ = +1
λ = −1 λ = +1
1996
e+
0.516 43.9
1997
e+
−0.51153.1
1998
e−
−0.30724.1
1999
e+
−0.5520.4180.9 5.1
2000
e+
−0.5840.55229.79.0
2005
e−
−0.3550.377 66.3 65.7
Sum174.1 123.7
3. The HERMES experiment
A detailed description of the HERMES experiment can be found in Ref. [44]. A longi-
tudinally polarized positron or electron beam of 27.6 GeV energy was scattered from an
unpolarized deuterium gas target internal to the HERA lepton storage ring at DESY.
The lepton beam was transversely polarized via the asymmetry in the emission of syn-
chrotron radiation (Sokolov-Ternov effect) [45] in the arcs of the HERA storage ring. The
transverse beam polarization was transformed locally into longitudinal polarization by
a pair of spin rotators located before and after the experiment [46]. The helicity of the
beam was typically reversed approximately every two months.
The beam polarization was continuously monitored by two Compton backscattering
polarimeters [47,48]. The average values of the beam polarization for various running
periods are given in Table 1; the average fractional systematic uncertainty was 2.4%.
The scattered leptons and produced particles were detected in the polar angle range
0.04 rad < θ < 0.22 rad. The lepton trigger required a coincidence of signals from
scintillator hodoscope planes and the local deposition of a minimum energy of 3.5 GeV
in the electromagnetic calorimeter. Lepton identification was accomplished using the
transition-radiation detector, the preshower scintillator counter, and the electromagnetic
calorimeter. The average lepton identification efficiency was at least 98% with hadron
contamination that was less than 1%. Photons were identified by the detection of energy
deposited in the calorimeter and preshower counter with no associated charged-particle
track.
4. Event selection and yield distributions
The data sets used in the extraction of the various asymmetries reported here are given
in Table 1. In this analysis, it was required that events contained exactly one charged-
particle track consistent with being the scattered beam lepton, and a single cluster in
the calorimeter with an energy deposit Eγ > 5.0GeV and with no associated charged
track. The following requirements were imposed on the event kinematics: 1GeV2<
11

Page 12

Data
Monte Carlo sum
semi-inclusive
BH with resonance exc.
MX
2 [GeV2]
1000 • Nγ / NDIS
0
0.05
0.1
0.15
0 102030
Fig. 4. The measured distribution (points) of electroproduced real-photon events versus the squared
missing mass M2
X. The solid curve represents a Monte Carlo simulation including coherent and incoher-
ent BH and DVCS processes, the BH processes with the excitation of resonant final states (represented
separately by the dashed-dotted curve), and the semi-inclusive background (dashed curve). The simula-
tions and data are both normalized to the number of DIS events. The region between the two vertical
lines indicates the selected exclusive events.
Q2< 10GeV2, W2
M2
MN was used in all kinematic constraints on event selection even at small values of
−t, where coherent reactions on the deuteron are dominant, because the experiment
did not distinguish between coherent and incoherent scattering and the latter dominates
over most of the kinematic range. Monte Carlo studies have shown that this choice has
little effect on the extracted asymmetries [49]. In order to reduce background from the
decay of neutral mesons, the angle between the laboratory 3-momenta of the real and
virtual photons was limited to θγ∗γ< 45mrad. The minimum angle requirement θγ∗γ>
5mrad was chosen according to Monte Carlo studies to be compatible with the effects of
instrumental resolution in determination of φ.
‘Exclusive’ single-photon events were selected by requiring the squared missing mass
M2
(q + PN− q′)2with PN= (MN,0,0,0). Due to the finite resolution of the spectrometer
and the calorimeter, M2
Xmay be negative. In Fig. 4, the squared missing mass distribu-
tion of the selected events is compared with the predictions of Monte Carlo simulations
of processes that contribute to both signal and background. One of the simulations uses
an exclusive-photon generator for the BH and DVCS processes, including coherent and
incoherent reactions as well as the excitation of resonant final states (a category known
N> 9GeV2, ν < 22GeV and 0.03 < xN < 0.35, where W2
N+ 2MNν − Q2, xN= Q2/(2MNν), and ν ≡ p · q/MN. The nucleonic (proton) mass
N=
Xto be close to the squared nucleon mass M2
N, where M2
Xis defined as M2
X=
12

Page 13

Data
Monte Carlo sum
incoherent BH + DVCS
coherent BH + DVCS
BH with resonance exc.
-t [GeV2]
1000 • Nγ / NDIS
10
-3
10
-2
10
-1
0 0.20.4 0.6
Fig. 5. Distribution in −t of events selected in the exclusive region of M2
perimental data while the continuous curve represents the simulation of real-photon production for all
exclusive final states including resonances. Background from π0decay is not included. The dotted and
dashed curves represent the BH plus DVCS contributions of the coherent and incoherent elastic process,
respectively. The dash-dotted curve shows the resonant BH contributions. The simulations and data are
both normalized to the number of DIS events.
X. The points represent ex-
as associated production). The DVCS simulation for incoherent reactions on the proton
is based on Ref. [50], while that for coherent reactions on the deuteron is based on the
model from Ref. [35]. Most of the background in the vicinity of the exclusive peak comes
from the decay of neutral pions. The dominant source of neutral pions is semi-inclusive
DIS, γ∗N → π0X → γγX, which is simulated using the Lepto event generator [51]
with a set of Jetset [52] fragmentation parameters tuned for HERMES kinematic con-
ditions [53]. In this simulation, the photon originates mainly from decay of π0s from DIS
fragmentation. Incoherent exclusive π0production, γ∗N → π0N, was simulated using
an exclusive Monte Carlo event generator based on the GPD models of Ref. [54] and
was found to be negligible [49,55]. HERMES data support this estimate [56]. The Monte
Carlo yield exceeds the data by approximately 2% in the exclusive region. This may be
due to the contribution of the DVCS process in the simulation of both coherent and
incoherent processes, which is highly model-dependent and can vary between 10% and
25% [55] for the incoherent processes. On the other hand, radiative effects not included
in the simulation would move events from the peak to the continuum [57].
Events were selected in the ‘exclusive region’, defined as −(1.5)2GeV2< M2
(1.7)2GeV2to minimize background from DIS fragmentation while maintaining rea-
sonable efficiency [58].
X<
13

Page 14

As the recoiling target nucleon or nucleus was undetected, the Mandelstam variable
t must be reconstructed from the measured kinematics of the scattered lepton and the
detected photon. The resolution in the photon energy from the calorimeter is inadequate
for a precise determination of t. Hence for events selected in the exclusive region in M2
the final state is assumed to be exclusive, leaving the target intact, thereby allowing t
to be reconstructed with improved resolution using only the photon direction and the
lepton kinematics [23]:
?ν2+ Q2cosθγ∗γ)
1 +
MN(ν −
The further restriction −t < 0.7GeV2is imposed in the selection of exclusive events in
order to reduce background from the decay of neutral mesons.
The t distribution of events for the deuterium target is shown in Fig. 5 and compared
with the Monte Carlo simulations discussed above. The simulated contributions of co-
herent and incoherent processes on the deuteron are also shown separately. Coherent
scattering on the deuteron occurs preferentially at small values of −t. The Monte Carlo
simulation shows that requiring −t < 0.06GeV2enhances the mean fractional contribu-
tion of the coherent process from 20% to 40% in the HERMES spectrometer acceptance.
Requiring −t < 0.01GeV2can further enhance the coherent contribution to 66%, but
only at the cost of a rapidly decreasing yield. In Sections 6.1 and 6.2, the first two −t
bins covering the range 0.00 − 0.06GeV2will provide a measure of coherent effects; in
Section 6.3, an attempt is made to isolate the coherent contribution.
X,
t =−Q2− 2ν (ν −
1
?ν2+ Q2cosθγ∗γ)
. (30)
5. Analysis of the data
5.1. Extraction of azimuthal asymmetry amplitudes
The distribution of the expectation value of the yield for scattering a polarized lepton
beam from an unpolarized deuterium target is given by
?N?(Pℓ,eℓ,φ) =L(Pℓ,eℓ)η(eℓ,φ)σUU(φ)
×?1 + PℓADVCS
Here, L denotes the integrated luminosity, Pℓthe longitudinal beam polarization, η the
detection efficiency, and σUU(φ) the cross section for an unpolarized target averaged over
both beam charges and both beam helicities, which can be expressed as
LU
(φ) + eℓAC(φ) + eℓPℓAI
LU(φ)?. (31)
σUU(φ) =
xD
32(2π)4Q4
1
√1 + ε2
×
?
KBH
P1(φ)P2(φ)
2
?
n=0
cBH
n
cos(nφ) + KDVCS
2
?
n=0
cDVCS
n
cos(nφ)
?
.(32)
The asymmetries AI
appearing in Eqs. 7–9, as illustrated by Eqs. 16–18. In analogy to the expansion of
the cross section in Eq. 7-9, these asymmetries are also expanded in terms of the same
harmonics in φ:
LU(φ), ADVCS
LU
(φ), and AC(φ) are related to the Fourier coefficients
14

Page 15

AI
LU(φ) ≃
2
?
n=1
Asin(nφ)
LU,I
sin(nφ) + Acos(0φ)
LU,I
, (33)
ADVCS
LU
(φ) ≃ Asinφ
3
?
n=0
LU,DVCSsinφ + Acos(0φ)
LU,DVCS, (34)
AC(φ) ≃
Acos(nφ)
C
cos(nφ), (35)
where the approximation is due to the truncation of the in general infinite Fourier series
caused by the azimuthal dependences in the denominators of Eqs. 16–18.
For each kinematic bin in −t, xB, or Q2, the sets of azimuthal asymmetry amplitudes
Asin(nφ)
LU,I
, Asin φ
C
, hereafter called ‘asymmetry amplitudes’, are simulta-
neously extracted from the observed exclusive sample using the method of maximum
likelihood (described in detail in Ref. [22]). Although these asymmetry amplitudes differ
somewhat from the coefficients given in Eqs. 7–9 and Eqs. 16–18, they are well defined
and can be computed in various GPD models for direct comparison with the data. Note
that in Eqs. 33 and 34, an additional constant term (n = 0) was introduced as a con-
sistency test. These terms must vanish as they are parity violating. Removing these
constant terms or also introducing additional harmonic terms in the fitting procedure do
not influence results for other asymmetry amplitudes [59].
LU,DVCSand Acos(nφ)
5.2. Background corrections and systematic uncertainties
In each kinematic bin, the results from the maximum likelihood fit are corrected for
photon background arising from semi-inclusive production of neutral mesons, mainly
pions. A corrected asymmetry amplitude is obtained as
Acorr=Araw− fsemi· Asemi
1 − fsemi
. (36)
Here, Arawstands for the extracted raw asymmetry amplitude, and fsemiand Asemithe
fractional contribution and corresponding asymmetry amplitude of the semi-inclusive
background. This fraction is obtained from a Monte Carlo simulation (see Section 4) and
ranges from 1% to 11%, depending on the kinematic conditions. As the semi-inclusive
process is only very weakly beam-charge dependent, its asymmetry with respect to the
beam charge or to the product of the beam charge and the beam polarization is assumed
to be zero. The asymmetry of the semi-inclusive π0background with respect to only
the longitudinal beam polarization is extracted from experimental data by requiring two
photons to be detected in the calorimeter with an invariant mass between 0.10 GeV
and 0.17 GeV and with no associated charged tracks. The restriction on the energy
deposition in the calorimeter of the less energetic cluster is relaxed to 1 GeV to improve
the statistical precision. The fractional energy z = Eπ/ν of the reconstructed neutral
pions is required to be larger than 0.8. After applying the correction of Eq. 36, the
resulting asymmetry amplitudes are expected to originate from elastic (coherent), and
incoherent photon production possibly including nucleon excitation.
The combined contribution to the systematic uncertainty from detector acceptance,
smearing, finite bin width, and alignment of the detector elements with respect to the
beam is determined from a Monte Carlo simulation using the GPD model described
15