Deeply virtual and exclusive electroproduction of ωmesons
ABSTRACT The exclusive ω electroproduction off the proton was studied in a large kinematical domain above the nucleon resonance region and for the highest possible photon virtuality (Q2) with the 5.75 GeV beam at CEBAF and the CLAS spectrometer. Crosssections were measured up to large values of the fourmomentum transfer ( t < 2.7="">2) to the proton. The contributions of the interference terms σ{TT} and σ{TL} to the crosssections, as well as an analysis of the ω spin density matrix, indicate that helicity is not conserved in this process. The tchannel π0 exchange, or more generally the exchange of the associated Regge trajectory, seems to dominate the reaction γ*p↦ωp, even for Q2 as large as 5 GeV2. Contributions of handbag diagrams, related to Generalized Parton Distributions in the nucleon, are therefore difficult to extract for this process. Remarkably, the hight behaviour of the crosssections is nearly Q2independent, which may be interpreted as a coupling of the photon to a pointlike object in this kinematical limit.
 Citations (21)
 Cited In (0)
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ABSTRACT: We outline in detail the properties of generalized parton distributions (GPDs), which contain new information on the structure of hadrons and which enter the description of hard exclusive reactions. We highlight the physics content of the GPDs and discuss the quark GPDs in the large Nc limit and within the context of the chiral quarksoliton model. Guided by this physics, we then present a general parametrization for these GPDs. Subsequently we discuss how these GPDs enter in a wide variety of hard electroproduction processes and how they can be accessed from them. We consider in detail deeply virtual Compton scattering and the hard electroproduction of mesons. We identify a list of key observables which are sensitive to the various hadron structure aspects contained in the GPDs and which can be addressed by present and future experiments.Progress in Particle and Nuclear Physics 06/2001; · 2.26 Impact Factor  SourceAvailable from: arxiv.org[Show abstract] [Hide abstract]
ABSTRACT: At moderate and large momentum transfer, high energy vector meson photoproduction and Compton scattering can be consistently explained by mechanisms based on interactions and exchanges of constituent quarks and gluons. This opens an original window on the partonic structure of hadronic matter by emphasizing the role of quark correlations, quark exchange as implemented through saturating Regge trajectories and nonperturbative effects in the gluon propagator, which provides us with a bridge with lattice calculations. Compton scattering is directly related to vector meson photoproduction and enlarges the data set. Comment: 4 pages, 4 figuresPhysical Review D 11/2001; · 4.69 Impact Factor  SourceAvailable from: arxiv.org
Article: Generalized parton distributions
[Show abstract] [Hide abstract]
ABSTRACT: We give an overview of the theory for generalized parton distributions. Topics covered are their general properties and physical interpretation, the possibility to explore the threedimensional structure of hadrons at parton level, their potential to unravel the spin structure of the nucleon, their role in smallx physics, and efforts to model their dynamics. We review our understanding of the reactions where generalized parton distributions occur, to leading power accuracy and beyond, and present strategies for phenomenological analysis. We emphasize the close connection between generalized parton distributions and generalized distribution amplitudes, whose properties and physics we also present. We finally discuss the use of these quantities for describing soft contributions to exclusive processes at large energy and momentum transfer.Physics Reports. 12/2003; 388(s 2–4):41–277.
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arXiv:hepex/0504057v1 28 Apr 2005
EPJ manuscript No.
(will be inserted by the editor)
Deeply virtual and exclusive electroproduction of ω mesons
The CLAS collaboration
L. Morand1, D. Dor´ e1, M. Gar¸ con1a, M. Guidal2, J.M. Laget1,3, S. Morrow1,2, F. Sabati´ e1, E. Smith3,
G. Adams31, P. Ambrozewicz11, M. Anghinolfi17, G. Asryan39, G. Audit1, H. Avakian3, H. Bagdasaryan29,39,
J. Ball1, J.P. Ball4, N.A. Baltzell34, S. Barrow12, V. Batourine22, M. Battaglieri17, M. Bektasoglu29, M. Bellis31,
N. Benmouna14, B.L. Berman14, A.S. Biselli6,31, S. Boiarinov3,20, B.E. Bonner32, S. Bouchigny2, R. Bradford6,
D. Branford10, W.J. Briscoe14, W.K. Brooks3, S. B¨ ultmann29, V.D. Burkert3, C. Butuceanu38, J.R. Calarco26,
S.L. Careccia29, D.S. Carman28, A. Cazes34, S. Chen12, P.L. Cole3,18, D. Cords3, P. Corvisiero17, D. Crabb37,
J.P. Cummings31, E. De Sanctis16, R. DeVita17, P.V. Degtyarenko3, H. Denizli30, L. Dennis12, A. Deur3,
K.V. Dharmawardane29, K.S. Dhuga14, C. Djalali34, G.E. Dodge29, J. Donnelly15, D. Doughty3,8, M. Dugger4,
S. Dytman30, O.P. Dzyubak34, H. Egiyan3,38, K.S. Egiyan39, L. Elouadrhiri3, P. Eugenio12, R. Fatemi37,
G. Feldman14, R.G. Fersch38, R.J. Feuerbach3, H. Funsten38, G. Gavalian26, G.P. Gilfoyle33, K.L. Giovanetti21,
F.X. Girod1, J.T. Goetz5, C.I.O. Gordon15, R.W. Gothe34, K.A. Griffioen38, M. Guillo34, N. Guler29, L. Guo3,
V. Gyurjyan3, C. Hadjidakis2, R.S. Hakobyan7, J. Hardie3,8, D. Heddle3, F.W. Hersman26, K. Hicks28, I. Hleiqawi28,
M. Holtrop26, C.E. HydeWright29, Y. Ilieva14, D.G. Ireland15, M.M. Ito3, D. Jenkins36, H.S. Jo2, K. Joo3,9,
H.G. Juengst14, J.D. Kellie15, M. Khandaker27, W. Kim22, A. Klein29, F.J. Klein7, A.V. Klimenko29, M. Kossov20,
V. Kubarovski31, L.H. Kramer3,11, S.E. Kuhn29, J. Kuhn6,31, J. Lachniet6, J. Langheinrich34, D. Lawrence24,
T. Lee26, Ji Li31, K. Livingston15, C. Marchand1, L.C. Maximon14, S. McAleer12, B. McKinnon15, J.W.C. McNabb6,
B.A. Mecking3, S. Mehrabyan30, J.J. Melone15, M.D. Mestayer3, C.A. Meyer6, K. Mikhailov20, R. Minehart37,
M. Mirazita16, R. Miskimen24, V. Mokeev25, J. Mueller30, G.S. Mutchler32, J. Napolitano31, R. Nasseripour11,
S. Niccolai2,14, G. Niculescu21,28, I. Niculescu3,14,21, B.B. Niczyporuk3, R.A. Niyazov3,29M. Nozar3, G.V. O’Rielly14,
M. Osipenko17, A.I. Ostrovidov12K. Park22, E. Pasyuk4, S.A. Philips14, N. Pivnyuk20, D. Pocanic37, O. Pogorelko20,
E. Polli16, I. Popa14, S. Pozdniakov20, B.M. Preedom34, J.W. Price5, Y. Prok37, D. Protopopescu15,26, B.A. Raue3,11,
G. Riccardi12, G. Ricco17, M. Ripani17, B.G. Ritchie4, F. Ronchetti16, G. Rosner15, P. Rossi16, P.D. Rubin33,
C. Salgado27, J.P. Santoro3,36, V. Sapunenko3, R.A. Schumacher6, V.S. Serov20, Y.G. Sharabian3, J. Shaw24,
A.V. Skabelin23, L.C. Smith37, D.I. Sober7, A. Stavinsky20, S. Stepanyan3,29, S.S. Stepanyan22, B.E. Stokes12,
P. Stoler31, I.I. Strakovsky14, S. Strauch14, M. Taiuti17, D.J. Tedeschi34, U. Thoma3,13,19, A. Tkabladze28,
L. Todor6,33, C. Tur34, M. Ungaro9,31, M.F. Vineyard33,35, A.V. Vlassov20, L.B. Weinstein29, D.P. Weygand3,
M. Williams6, E. Wolin3, M.H. Wood34, A. Yegneswaran3, and L. Zana26
1CEASaclay, Service de Physique Nucl´ eaire, F91191 GifsurYvette, France
2Institut de Physique Nucl´ eaire, F91406 Orsay, France
3Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606, USA
4Arizona State University, Tempe, Arizona 852871504, USA
5University of California at Los Angeles, Los Angeles, California 900951547, USA
6Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA
7Catholic University of America, Washington, D.C. 20064, USA
8Christopher Newport University, Newport News, Virginia 23606, USA
9University of Connecticut, Storrs, Connecticut 06269, USA
10Edinburgh University, Edinburgh EH9 3JZ, United Kingdom
11Florida International University, Miami, Florida 33199, USA
12Florida State University, Tallahassee, Florida 32306, USA
13Physikalisches Institut der Universit¨ at Gießen, 35392 Gießen, Germany
14The George Washington University, Washington, DC 20052, USA
15University of Glasgow, Glasgow G12 8QQ, United Kingdom
16INFN, Laboratori Nazionali di Frascati, Frascati, Italy
17INFN, Sezione di Genova, 16146 Genova, Italy
18Idaho State University, Pocatello, Idaho 83209, USA
19Institute f¨ ur Strahlen und Kernphysik, Universit¨ at Bonn, Germany
20Institute of Theoretical and Experimental Physics, Moscow, 117259, Russia
21James Madison University, Harrisonburg, Virginia 22807, USA
aCorresponding author: mgarcon@cea.fr
Page 2
2
22Kyungpook National University, Daegu 702701, The Republic of Korea
23Massachusetts Institute of Technology, Cambridge, Massachusetts 021394307, USA
24University of Massachusetts, Amherst, Massachusetts 01003, USA
25Moscow State University, General Nuclear Physics Institute, 119899 Moscow, Russia
26University of New Hampshire, Durham, New Hampshire 038243568, USA
27Norfolk State University, Norfolk, Virginia 23504, USA
28Ohio University, Athens, Ohio 45701, USA
29Old Dominion University, Norfolk, Virginia 23529, USA
30University of Pittsburgh, Pittsburgh, Pennsylvania 15260, USA
31Rensselaer Polytechnic Institute, Troy, New York 121803590, USA
32Rice University, Houston, Texas 770051892, USA
33University of Richmond, Richmond, Virginia 23173, USA
34University of South Carolina, Columbia, South Carolina 29208, USA
35Union College, Schenectady, NY 12308, USA
36Virginia Polytechnic Institute and State University, Blacksburg, Virginia 240610435, USA
37University of Virginia, Charlottesville, Virginia 22901, USA
38College of William and Mary, Williamsburg, Virginia 231878795, USA
39Yerevan Physics Institute, 375036 Yerevan, Armenia
Received: date / Revised version: date
Abstract. The exclusive ω electroproduction off the proton was studied in a large kinematical domain
above the nucleon resonance region and for the highest possible photon virtuality (Q2) with the 5.75 GeV
beam at CEBAF and the CLAS spectrometer. Cross sections were measured up to large values of the
fourmomentum transfer (−t < 2.7 GeV2) to the proton. The contributions of the interference terms σTT
and σTL to the cross sections, as well as an analysis of the ω spin density matrix, indicate that helicity is
not conserved in this process. The tchannel π0exchange, or more generally the exchange of the associated
Regge trajectory, seems to dominate the reaction γ∗p → ωp, even for Q2as large as 5 GeV2. Contributions
of handbag diagrams, related to Generalized Parton Distributions in the nucleon, are therefore difficult to
extract for this process. Remarkably, the hight behaviour of the cross sections is nearly Q2independent,
which may be interpreted as a coupling of the photon to a pointlike object in this kinematical limit.
PACS. 13.60.Le Production of mesons by photons and leptons – 12.40.Nn Regge theory – 12.38.Bx
Perturbative calculations
Page 3
2
1 Introduction
The exclusive electroproduction of vector mesons is a pow
erful tool, on one hand to understand the hadronic prop
erties of the virtual photon (γ∗) which is exchanged be
tween the electron and the target nucleon [1], and on the
other hand to probe the quarkgluon content of the pro
ton (p) [2,3,4]. At moderate energies in the γ∗p system,
but large virtuality of the photon, quarkexchange mech
anisms become significant in the vector meson production
reactions γ∗p → pρ/ω, thus shedding light on the quark
structure of the nucleon.
The interaction of a real photon with nucleons is dom
inated by its hadronic component. The exchange in the
tchannel of a few Regge trajectories permits a descrip
tion of the energy dependence as well as the forward an
gular distribution of many, if not all, realphotoninduced
reactions (see e.g. Ref. [5]). For instance, this approach re
produces the photoproduction of vector mesons from the
CEBAF energy range to the HERA range (a few to 200
GeV) [6]. The exchange of the Pomeron (or its realiza
tion into two gluons) dominates at high energies, while
the exchange of meson Regge trajectories (π, σ, f2) takes
over at low energies. At γp energies of a few GeV, ω pho
toproduction off a proton is dominated by π0exchange
in the tchannel (fig. 1). The use of a saturating Regge
trajectory [4] is very successful in describing recent pho
toproduction data [7] at large angles (large momentum
transfer t). This is a simple and economical way to pa
rameterize hard scattering mechanisms. Extending these
measurements to the virtual photon sector opens the way
to tune the hadronic component of the exchanged pho
ton, to explore to what extent π0exchange survives, and
to observe hard scattering mechanisms with the help of a
second hard scale, the virtuality Q2of the photon.
The study of such reactions in the Bjorken regime1
holds promise, through perturbative QCD, to access the
socalled Generalized Parton Distributions (GPD) of the
nucleon [8,9]. These structure functions are a generaliza
tion of the parton distributions measured in the deep in
elastic scattering experiments and their first moment links
them to the elastic form factors of the nucleon. Their sec
ond moment gives access to the sum of the quark spin and
the quark orbital angular momentum in the nucleon [8].
The process under study may be represented by the so
called handbag diagram (fig. 1). Its amplitude factorizes [10]
1Q2and ν large and xB finite, where −Q2and ν are the
squared mass and the laboratoryframe energy of the virtual
photon, while xB = Q2/2Mpν is the usual Bjorken variable.
Page 4
3
p
∗
ρ, ω ...
γ
p’
e’
x x’
t
e
p
L
factorization
L
L
GPD’s
ρ , ω ...
∗
γ
Fig. 1. Schematic representations of the tchannel exchange
(left) and of the handbag diagram (right) for exclusive vector
meson electroproduction.
into a “hard” process where the virtual photon is ab
sorbed by a quark and a “soft” one containing the new
information on the nucleon, the GPD (which are func
tions of x and x′, the momentum fraction carried by the
quark in the initial and final states, and of t, the squared
fourmomentum transfer between the initial and final pro
tons). The factorization applies only to the transition,
at small values of −t, between longitudinal photons (L)
and helicity0 mesons, which is dominant in the Bjorken
regime. Because of the necessary gluon exchange to pro
duce the meson in the hard process (see fig. 1), the domi
nance of the handbag contribution is expected to be reached
at a higher Q2for meson production than for photon pro
duction (DVCS). Nevertheless, recent results on deeply
virtual ρ production show a qualitative agreement with
calculations based on the handbag diagram [11,12]. Vector
meson production is an important complement to DVCS,
since it singles out the quark helicity independent GPD H
and E which enter Ji’s sum rule [8] and allows, in princi
ple, for a flavor decomposition of these distributions (see
e.g. Ref. [13]).
Apart from early, low statistics, muon production ex
periments at SLAC [14,15], the leptoproduction of ω mesons
was measured at DESY [16], for 0.3 < Q2< 1.4 GeV2,
W < 2.8 GeV (xB< 0.3), and then at Cornell [17], in a
wider kinematical range (0.7 < Q2< 3 GeV2, W < 3.7
GeV) but with larger integration bins. These two experi
ments yielded cross sections differing by a factor of about 2
wherever they overlap (around Q2≃ 1 GeV2). The DESY
experiment also provided the only analysis so far, in elec
troproduction, of the ω spin density matrix elements, av
eraged over the whole kinematical range. This analysis in
dicated that, in contrast with ρ electroproduction, there is
little increase in the ratio R of longitudinal to transverse
cross sections (σL/σT) when going from photoproduction
to low Q2electroproduction. More recently, ω electropro
duction was measured at ZEUS [18], at high Q2and very
low xB, in a kinematical regime more sensitive to purely
diffractive phenomena and to gluons in the nucleon. Fi
nally, there is also unpublished data from HERMES [19].
The main goal of the present experiment was to reach
the highest achievable Q2values in exclusive meson elec
troproduction in the valence quark region. In the specific
case of the ω production, it is to test which of the two de
scriptions — with hadronic or quark degrees of freedom,
B
x
00.10.20.30.40.50.60.70.8
)
2
(GeV
2
Q
0
1
2
3
4
5
6
7
DESY
Cornell
HERMES
ZEUS
JLab (this work)
Fig. 2. (Color online) Kinematical range covered by this and
previous [16,17,18,19] ω electroproduction experiments. The
lines are indicative of the total coverage in Q2and xB of pre
vious experiments.
more specifically tchannel Regge trajectory exchange or
handbag diagram — applies in the considered kinematical
domain (see fig. 2). For this purpose, the reduced cross
sections σγ∗p→ωp were measured in fine bins in Q2and
xB, as well as their distribution in t and φ (defined be
low). In addition, parameters related to the ω spin density
matrix were extracted from the analysis of the angular dis
tribution of the ω decay products. If the vector meson is
produced with the same helicity as the virtual photon,
schannel helicity conservation (SCHC) is said to hold.
From our results, the relevance of SCHC and of natural
parity exchange in the tchannel was explored in a model
independent way. These properties have been established
empirically in the case of photo and electroproduction of
the ρ meson (see e.g. Ref. [20]), but may not be a general
feature of all vector meson production channels.
This paper is based on the thesis work of Ref. [21],
where additional details on the data analysis may be found.
2 Experimental procedure
We measured the process ep → epω, followed by the decay
ω → π+π−π0. The scattered electron and the recoil pro
ton were detected, together with at least one charged pion
from the ω decay. At a given beam energy E, this process
is described by ten independent kinematical variables. In
the absence of polarization in the ep initial state, the ob
servables are independent of the electron azimuthal angle
in the laboratory. Q2and xB are chosen to describe the
γ∗p initial state. The scattered electron energy E′and,
for ease of comparison with other data, the γ∗p center
ofmass energy W will be used as well. t is the squared
fourmomentum transfer from the γ∗to the ω, and φ the
angle between the electron (ee′γ∗) and hadronic (γ∗ωp)
planes. Since t is negative and has a kinematical upper
bound t0(Q2,xB) corresponding to ω production in the di
rection of the γ∗, the variable t′= t0−t will also be used.
The ω decay is described in the socalled helicity frame,
Page 5
4
φ
p
x
hadronic production plane
Virtual photoproduction
c.m. frame
θ
2
N
N
θ
θ∗
ω
electron scattering plane
y
z
ω
γ∗(Q )
(t)
(E)
Laboratory frame
Helicity frame
ω
( at rest)
(E’)
e
e’
φ
Fig. 3. (Color online) Reference frames and relevant variables
used for the description of the reaction ep → epω, followed by
ω → π+π−π0.
where the ω is at rest and the zaxis is given by the ω
direction in the γ∗p centerofmass system. In this helicity
frame, the normal to the decay plane is characterized by
the angles θN and ϕN (fig. 3). Finally the distribution of
the three pions within the decay plane is described by two
angles and a relative momentum. This latter distribution
is known from the spin and parity of the ω meson [22]
and is independent of the γ∗p → ωp reaction mechanism.
The purpose of the present study is to characterize as
completely as possible the distributions of cross sections
according to the six variables Q2, xB, t, φ, cosθNand ϕN.
2.1 The experiment
The experiment was performed at the Thomas Jefferson
National Accelerator Facility (JLab). The CEBAF 5.754
GeV electron beam was directed at a 5cm long liquid
hydrogen target. The average beam intensity was 7 nA,
resulting in an effective integrated luminosity of 28.5 fb−1
for the data taking period (October 2001 to January2002).
The target was positioned at the center of the CLAS spec
trometer. This spectrometer uses a toroidal magnetic field
generated by six superconducting coils for the determina
tion of particle momenta. The field integral varied approx
imately from 2.2 to 0.5 Tm, in average over charges and
momenta of different particles, for scattered angles be
tween 14◦and 90◦. All the spectrometer components are
arranged in six identical sectors. Charged particle trajec
tories were detected in three successive packages of drift
chambers (DC), the first one before the region of magnetic
field (R1), the second one inside this region (R2), and the
third one after (R3). ThresholdˇCerenkov counters (CC)
were used to discriminate pions from electrons. Scintilla
tors (SC) allowed for a precise determination of the par
ticle timeofflight. Finally, a segmented electromagnetic
calorimeter (EC) provided a measure of the electron en
ergy. This geometry and the event topology are illustrated
Fig. 4. (Color online) Schematic view of the CLAS spectrome
ter components (see text for description) and of typical particle
tracks, viewed in projection. The torus coils are not shown.
in fig. 4. A detailed description of the CLAS spectrometer
and of its performance is given in Ref. [23].
The data acquisition was triggered by a coincidence
CC·EC corresponding to a minimal scattered electron en
ergy of about 0.575 GeV. The trigger rate was 1.5 kHz,
with a data acquisition dead time of 6%. A total of 1.25×
109events was recorded.
2.2 Particle identification
After calibration of all spectrometer subsystems, tracks
were reconstructed from the DC information. The identi
fication of particles associated with each track proceeded
differently for electrons and hadrons.
Electrons were identified from the correlation between
momentum (from DC) and energy (from EC). In addi
tion pions were rejected from the electron sample by a
cut in the CC amplitude and imposing a condition on
the energy sharing between EC components compatible
with the depth profile of an electromagnetic shower. Ge
ometrical fiducial cuts ensured that the track was inside
a high efficiency region for both CC and EC. The effi
ciencies of the electron identification cuts (ηCCand ηEC)
depended on the electron momentum and angle (or on
Q2and xB). ηECwas calculated from data samples using
very selective CC cuts in order to unambiguously select
electrons. ηCCwas extracted from an extrapolation of the
CC amplitude Poisson distribution into the low amplitude
region. These efficiencies varied respectively between 0.92
and 0.99 (CC), and 0.86 to 0.96 (EC). At low electron en
ergies, a small contamination of pions remained, which did
not however satisfy the ω selection criteria to be described
below.
The relation between momentum (from DC) and ve
locity (from path length in DC and timeofflight in SC)
allowed for a clean identification of protons (p) and pi
ons (π+and π−). However, for momenta larger than 2
GeV/c, ambiguities arose between p and π+identification,
which led us to the discarding of events corresponding to
Page 6
5
[epX] (GeV)
X
M
00.20.40.60.811.21.41.6
)
2
X] (GeV
+
π
[ep
2
M
X
0
0.2
0.4
0.6
0.8
1
0
500
1000
1500
2000
2500
3000
3500
Fig. 5. (Color online) Identification of the ω channel in the
case of the detection of one charged pion: M2
MX[epX]. Events corresponding to ρ (horizontal locus) and
other twopion production channels are clearly separated from
those corresponding to ω production (vertical locus).
X[epπ+X] vs
t < −2.7 GeV2. Fiducial cuts were applied to hadrons
as well. The efficiency for the hadron selection cuts was
accounted for in the acceptance calculation described in
sect. 2.4.
2.3 Event selection and background subtraction
Two configurations of events were studied, with one or two
detected charged pions: ep → epπ+X and ep → epπ+π−X.
The former benefits from a larger acceptance and is ade
quate to determine cross sections, while the latter is nec
essary to measure in addition the distribution of the ω
decay plane orientation and deduce from it the ω spin
density matrix. The final selection of events included cuts
in W and E′: W > 1.8 GeV to eliminate the threshold
region sensitive to resonance production [24] and E′> 0.8
GeV to minimize radiative corrections and residual pion
contamination in the electron tracks. The first configu
ration was selected requiring a missing mass MX larger
than 0.316 GeV to eliminate two pion production chan
nels (M2
cut was chosen slightly above the two pion mass in or
der to minimize background. The corresponding losses in
ep → epω events were very small and accounted for in the
acceptance calculation to be discussed below. Events cor
responding to the ω production appear as a clear peak in
the ep → epX missing mass spectrum (fig. 6). The width
of this peak (σ ≃ 16 MeV) is mostly due to the experi
mental resolution.
After proper weighting of each event with the accep
tance calculated as indicated in sect. 2.4, a background
subtraction was performed for each of 34 bins (Q2,xB)
and, for differential cross sections, for each bin in t or φ.
The background was determined by a fit to the acceptance
weighted distributions with a secondorderpolynomial and
a peak shape as modeled by simulations (a skewed gaus
sian shape taking into account the experimental resolution
and radiative tail). At the smallest values of W, the fitted
background shape was modified to account for kinemati
X> 0.1 GeV2on the vertical axis of fig. 5). This
0.6 0.650.70.750.80.85
M
0.9
pX] (GeV)
0.951
0
5000
10000
15000
20000
25000
30000
35000

[e
0.60.650.70.750.8 0.85
M
0.9
pX] (GeV)
0.951
0
5000
10000
15000
20000
25000

[e
Fig. 6. (Color online) Missing mass MX[epX] distributions for
the ep → epπ+X event configuration, for two (Q2,xB) bins,
after selection cuts and event weighting discussed in the text.
Left : 2.2 GeV2≤ Q2≤ 2.5 GeV2and 0.34 ≤ xB ≤ 0.40.
Right : 3.1 GeV2≤ Q2≤ 3.6 GeV2and 0.52 ≤ xB ≤ 0.58 (at
the edge of kinematical acceptance). The two lines indicate the
subtracted background (green) and the fitted distribution (ω
peak + background in red).
pX] (GeV)

[e
X
M
0 0.20.40.6 0.81 1.21.41.6
)
2
X] (GeV
π
+
π
p

[e
2
M
X
0.05
0
0.05
0.1
0.15
0.2
0
100
200
300
400
500
600
700
800
Fig. 7. (Color online) Identification of the ω channel in the
case of the detection of two charged pions: M2
MX[epX] for M2
0.02) corresponds to the ω.
X[epπ+π−X] vs
X[epπ±X] ≥ 0.1 GeV2. The spot at ( 0.78,
cal acceptance cuts. The acceptanceweighted numbers of
ep → epω events were computed using the sum of weighted
counts in the MX[epX] distributions for .72 < MX< .85
GeV, diminished by the fitted background integral in the
same interval.
Likewise, events from the second configuration (ep →
epπ+π−X) were selected with cuts in missing masses:
MX[epπ+X] and MX[epπ−X] > 0.316 GeV, 0.01 GeV2≤
M2
MX[epX] spectrum after these cuts is illustrated in fig. 8.
The background subtraction in the spectrum of weighted
events proceeded in the same way for each of 64 (Q2, xB,
cosθN) bins or for each of 64 (Q2, xB, ϕN) bins in order
to analyze the ω decay distribution (see sect. 4).
X[epπ+π−X] ≤ 0.045 GeV2(fig. 7). The resulting
2.4 Acceptance calculation
The tracks reconstruction and the event selection were
simulated using a GEANTbased Monte Carlo (MC) sim
ulation of the CLAS spectrometer. We used an event gen
erator tuned to reproduce photoproduction and low Q2
Page 7
6
[epX] (GeV)
X
M
00.20.40.60.811.21.41.61.82
0
5000
10000
15000
20000
25000
30000
Fig. 8. Unweighted MX[epX] spectrum for all ep → epπ+π−X
events after selection cuts discussed in the text.
data in the resonance region and extrapolated into our
kinematical domain [25]. The acceptance was defined in
each elementary bin in all relevant variables as the ratio
of accepted to generated MC events. At the limit of small
sixdimensional bins, it is independent of the model used
to generate the MC events. The MC simulation included
a tuning of the DC and SC time resolutions to reproduce
the observed widths of the hadron particle identification
spectra and of the missing mass spectra, so that the effi
ciency of the corresponding cuts described above could be
correctly determined.
For the extraction of cross sections from the ep →
epπ+X configuration, acceptance calculations were per
formed in 1837 fourdimensional bins (Q2, xB, t and φ)
with two different assumptions about the event distribu
tion in cosθN and ϕN. The two different MC calcula
tions were used for an estimate of the corresponding sys
tematic uncertainties (see sect. 3.1). For the analysis of
the decay plane distribution W(cosθN,ϕN,φ) from the
ep → epπ+π−X configuration, the acceptance calculation
was performed in 3575 sixdimensional bins (Q2, xB, t, φ,
cosθNand ϕN). The binning is defined in table 1 and the
numbers above correspond to kinematically allowed bins
that have significant statistics.
The calculated acceptances are, on average, of the or
der of 2% and 0.2% respectively for the two event con
figurations of interest. They vary smoothly for all vari
ables except φ, where oscillations, due to the dead zones
in the CLAS sectors, reproduce the physical distributions
of events (fig. 9). Each event was then weighted with the
inverse of the corresponding acceptance. Events belong
ing to bins with either very large or poorly determined
weights were discarded (for the ep → epπ+X configura
tion, acceptance smaller than 0.25% or associated MC sta
tistical uncertainty larger than 35%). The corresponding
losses (a few percent) were quantified through the MC ef
ficiency ηMC by applying these cuts to MC events and
computing the ratio of weighted accepted MC events to
generated events. No attempt was made to calculate the
acceptance for the nonresonant threepion background,
so that the background shape in fig. 6 differs from the
physical distribution dσ/dMX when MX differs from the
ω mass.
050100150200250300
(deg)
350
0
0.01
0.02
Φ
050100 150200 250300
(deg)
350
0
0.001
Φ
Fig. 9. φ dependence of calculated acceptance, integrated over
other kinematical variables, for the one and two detected pion
configurations: ep → epπ+X (left), ep → epπ+π−X (right).
2.5 Radiative corrections
Radiative corrections were calculated following Ref. [26].
They were dealt with in two separate steps. The MC ac
ceptance calculation presented above took into account ra
diation losses due to the emission of hard photons, through
the application of the cut MX[epX] < 0.85 GeV. Correc
tions due to soft photons, and especially the virtual pro
cesses arising from vacuum polarization and vertex cor
rection, were determined separately for each bin in (Q2,
xB, φ). The same event generator employed for the com
putation of the acceptance was used, with radiative effects
turned on and off, thus defining a corrective factor Frad.
The tdependence of Fradis smaller than all uncertainties
discussed in sect. 3.1 and was neglected.
3 Cross sections for γ∗p → ωp
The total reduced cross sections were extracted from the
data through :
σγ∗p→ωp(Q2,xB,E) =
1
ΓV(Q2,xB,E)×
Frad
ηCCηECηMC
nw(Q2,xB)
BLint∆Q2· ∆xB
×
.(1)
The Hand convention [27] was used for the definition
of the virtual transverse photon flux ΓV, which includes
here a Jacobian in order to express the cross sections in
the chosen kinematical variables :
ΓV(Q2,xB,E) =
α
8π
Q2
pE2
M2
1 − xB
x3
B
1
1 − ε,
(2)
with the virtual photon polarization parameter being de
fined as :
ε =
1 + 2Q2+(E−E′)2
4EE′−Q2
1
.(3)
In eq. (1), nw(Q2,xB) is the acceptanceweighted num
ber of ep → epω events after background subtraction.
The branching ratio of the ω decay into three pions is
Page 8
7
Table 1. Definition of binning for cross section (1) and ω
polarization (2) data. N refers to the number of bins in the
specified range for each variable.
VariableRange(1)N(1)Range(2)N(2)
Q2(GeV2)1.6  3.1
3.1  5.1
0.16  0.64
0.1  1.9
1.9  2.7
0  2π


5
4
8
6
1
9


1.7  4.1
4.1  5.2
0.18  0.62
0.1  2.1
2.1  2.7
0  2π
−1  1
0  2π
4
1
4
4
1
xB
−t (GeV2)
φ (rd)
cosθN
ϕN (rd)
6/9/12
8
8
B = 0.891 [28]. The integrated effective lumimosity Lint
includes the data acquisition dead time correction. ∆Q2
and ∆xB are the corresponding bin widths; for bins not
completely filled (because of W or E′cuts on the electron,
or of detection acceptance), the phase space ∆Q2· ∆xB
includes a surface correction and the Q2and xB central
values are modified accordingly. The radiative correction
factor and the various efficiencies not included in the MC
calculation were discussed in previous sections.
Differential cross sections in t or φ were extracted in a
similar manner. Cross section data and corresponding MC
data for the acceptance calculation were binned according
to table 1.
3.1 Systematic uncertainties
Systematic uncertainties in the cross section measurements
arise from the determination of the CLAS acceptance, of
electron detection efficiencies, of the luminosity, and from
the background subtraction. They are listed in table 2 and
discussed hereafter.
Errors in the acceptance calculation may be due to
inhomogeneity in the detectors response, such as faulty
channels in DC or SC, to possible deviations between
experimental and simulated resolutions in spectra where
cuts were applied, to the input of the event generator
(both in cross section and in decay distribution Wgen),
to radiative corrections and finally to the event weight
ing procedure. The most significant of these uncertainties
(8%) was quantified by performing a separate complete
MC simulation varying inputs for the parameters describ
ing the decay distribution Wgen.
Systematic uncertainties on the electron detection effi
ciencies were estimated with experimental data, by vary
ing the electron selection cuts or the extrapolated CC am
plitude distribution (see sect. 2.2).
Systematic background subtraction uncertainties were
estimated by varying the assumed background functional
shapes. In particular, the background curvature under the
ω peak was varied between extreme values compatible
with an equally good fit to the distributions in fig. 6. For
bins corresponding to low values of W, the acceptance cut
Table 2. Pointtopoint and normalization systematic uncer
tainties, for integrated and differential cross sections.
Source of uncertaintyσdσ/dtdσ/dφ
CLAS acceptance
 inhomogeneities
 resolutions
 σgen(Q2,xB,t)
 Wgen(cosθN,ϕN,φ)
 radiative corrections
 binning
 ηMC
Electron detection
 ηCC
 ηEC
Background subtraction
6%
2%
5%
8%
4%
5%
4%
6%
2%
5%
8%

5%
27%
6%



2%

220%
1.5%
2%
711%






Pointtopoint1618%1314%721%
Normalization 3%912% 1416%
to the right of the MX[epX] peak induced an additional
uncertainty.
Finally, overall normalization uncertainties were due to
the knowledge of target thickness (2%) and density (1%)
and of beam integrated charge (2%).
Errors contributing pointtopoint and to the overall
normalization are separately added in quadrature in ta
ble 2. For t or φ distributions, the same uncertainties ap
ply, but may contribute to the overallnormalization uncer
tainty instead of pointtopoint. For example, the shape of
the MX[epX] distributions depends mostly on Q2and xB,
not on t and φ; the background subtraction uncertainties
are then considered as a normalization uncertainty for the
dσ/dt and dσ/dφ distributions. The uncertainties on ηMC
are largest for t and φ bins with the smallest acceptance
(lowest and highest t values, as well as φ ≃ 180◦).
3.2 Integrated reduced cross sections
Results for σγ∗p→ωp(Q2,xB) are given in table 3 and fig. 10.
For the purpose of comparison with previous data, fig. 11
shows cross sections as a function of Q2for fixed, approx
imately constant, values of W. When comparing different
data sets, note that σ = σT+ εσLdepends on the beam
energy through ε. However, as will be shown, it is likely
that the difference of longitudinal contributions between
two different beam energies (ε2− ε1)σL is much smaller
than the total cross section σ. In addition, the range of
integration in t is different for all experiments, larger in
this work, but most of the total cross section comes from
small −t values. A direct comparison of the cross sections
is then meaningful.
There is no direct overlap between the present data
and the DESY data [16], but they seem to be compatible
with a common trend. The Cornell data [17] are roughly a
factor 2 lower than ours. Where they overlap, the Cornell
data are also a factor 2 lower than the DESY data. We
Page 9
8
Table 3. Cross sections σ = σT + εσL and interference terms σTT and σTL for the reaction γ∗p → ωp, integrated over
−2.7 GeV2< t < t0. Slope b of dσ/dt for −1.5 GeV2< t < t0. Quoted uncertainties are obtained from the addition in
quadrature of statistical uncertainties and of pointtopoint systematic uncertainties as discussed in sect. 3.1.
xB
Q2
Wεt0
σ ± ∆σ
(nb)
σTT ± ∆σTT
(nb)
σTL± ∆σTL
(nb)
b ± db
(GeV−2)(GeV2)(GeV)(GeV2)
0.203
0.250
0.252
0.265
0.308
0.310
0.310
0.313
0.327
0.370
0.370
0.370
0.370
0.378
0.429
0.430
0.430
0.430
0.430
0.436
0.481
0.490
0.490
0.490
0.490
0.494
0.538
0.549
0.550
0.550
0.557
0.601
0.610
0.610
1.725
1.752
2.042
2.320
1.785
2.050
2.350
2.639
2.914
2.050
2.350
2.650
2.950
3.295
2.055
2.350
2.650
2.950
3.350
3.807
2.371
2.651
2.950
3.350
3.850
4.307
2.968
3.357
3.850
4.350
4.765
3.882
4.352
4.850
2.77
2.48
2.63
2.70
2.21
2.33
2.47
2.58
2.62
2.09
2.21
2.32
2.43
2.51
1.90
2.00
2.10
2.19
2.31
2.41
1.85
1.91
1.99
2.09
2.21
2.30
1.85
1.91
2.01
2.11
2.16
1.86
1.91
2.00
0.37
0.59
0.43
0.32
0.72
0.63
0.50
0.37
0.28
0.74
0.65
0.55
0.43
0.31
0.81
0.74
0.67
0.58
0.45
0.30
0.79
0.74
0.68
0.58
0.43
0.29
0.73
0.66
0.55
0.41
0.31
0.61
0.52
0.40
0.09
0.15
0.14
0.14
0.25
0.23
0.21
0.20
0.22
0.37
0.34
0.31
0.30
0.30
0.59
0.53
0.49
0.46
0.43
0.42
0.79
0.75
0.69
0.64
0.60
0.58
0.99
0.96
0.88
0.83
0.83
1.26
1.24
1.16
536 ± 96
661 ± 118
421 ± 75
344 ± 62
1139 ± 205
551 ± 98
395 ± 71
287 ± 52
226 ± 43
1002 ± 180
581 ± 104
380 ± 68
273 ± 49
230 ± 42
2203 ± 348
1013 ± 182
626 ± 113
427 ± 78
265 ± 48
191 ± 35
1660 ± 265
1113 ± 177
644 ± 116
397 ± 72
272 ± 50
187 ± 37
894 ± 148
514 ± 84
327 ± 59
258 ± 48
222 ± 44
292 ± 57
221 ± 43
150 ± 26
60 ± 87
156 ± 61
104 ± 47
58 ± 74
310 ± 122
121 ± 42
103 ± 41
111 ± 48
138 ± 84
91 ± 75
150 ± 48
85 ± 40
95 ± 42
17 ± 55
54 ± 158
181 ± 82
90 ± 63
56 ± 48
105 ± 37
128 ± 60
34 ± 173
291 ± 109
174 ± 69
84 ± 49
72 ± 46
169 ± 79
111 ± 148
83 ± 67
95 ± 54
29 ± 62
42 ± 121
91 ± 89
75 ± 54
110 ± 48
2 ± 30
35 ± 24
18 ± 18
14 ± 26
175 ± 60
66 ± 20
49 ± 17
9 ± 18
46 ± 27
25 ± 39
36 ± 23
33 ± 17
44 ± 17
27 ± 19
143 ± 85
102 ± 43
41 ± 32
24 ± 23
12 ± 15
54 ± 21
241 ± 97
15 ± 62
154 ± 35
49 ± 24
5 ± 20
14 ± 27
8 ± 69
12 ± 31
52 ± 26
13 ± 27
55 ± 51
74 ± 50
53 ± 23
39 ± 20
2.44 ± 0.18
1.93 ± 0.16
2.28 ± 0.16
1.88 ± 0.17
1.23 ± 0.17
1.90 ± 0.16
2.03 ± 0.16
1.90 ± 0.17
1.87 ± 0.23
0.97 ± 0.17
1.35 ± 0.17
1.62 ± 0.17
1.93 ± 0.18
1.46 ± 0.18
1.03 ± 0.25
0.78 ± 0.22
0.81 ± 0.22
0.92 ± 0.23
1.34 ± 0.23
1.47 ± 0.24
0.92 ± 0.50
0.68 ± 0.37
0.23 ± 0.34
1.39 ± 0.25
1.41 ± 0.26
0.78 ± 0.32
−
−
0.65 ± 0.49
0.90 ± 0.43
1.36 ± 0.67
−
−
−
can only make the following conjectures as to the origin
of this discrepancy: the Cornell results do not appear to
have been corrected for internal virtual radiative effects
(about 15%); their overall systematic uncertainty in ab
solute cross sections is 25%; their acceptance calculation
has a model dependence which was not quantified and in
particular the decay distribution, given in eq. (6) below,
was assumed flat; finally, the estimate of average values of
kinematical variables ?Q2? and ?W? may be an additional
source of uncertainty since the corresponding bins are at
least 5 times larger than in the present work.
3.3 t dependence of cross sections
Four of the 34 distributions of differential cross sections
are illustrated in fig. 12. The general features of these dis
tributions are of a diffractive type (dσ/dt ∝ ebt) at small
values of −t. The values of the slope b, as determined from
a fit of the distributions in the interval 1.5 GeV2< t < t0,
are between 0.5 and 2.5 GeV−2and are compiled in ta
ble 3. They are also plotted as a function of the formation
length (distance of fluctuation of the virtual photon in a
real meson) [1]:
c∆τ =
1
?ν2+ Q2+ M2
ω− ν
(4)
in fig. 13. They are compatible with those obtained for
the reaction γ∗p → ρ0p [12]. There exists only one pre
vious determination [17] of this quantity for γ∗p → ωp,
integrated over a wide kinematical range corresponding
to 0.6 < c∆τ < 2.5 fm, with a value b = 6.1±0.8 GeV−2.
This larger value of b is consistent with the observed dis
crepancy between the Cornell experiment and the present
work. For larger values of −t, the slope of the cross sec
tions becomes much smaller and dσ/dt becomes nearly
Page 10
9
0.22 < xB < 0.28
0.1
1.0
0.16 < xB < 0.22
0.28 < xB < 0.34 0.34 < xB < 0.40
2345
Q
2 (GeV
2)
0.1
1.0
0.40 < xB < 0.46
2345
Q
2 (GeV
2)
0.46 < xB < 0.52
2345
Q
2 (GeV
2)
0.52 < xB < 0.58
2345
Q
2 (GeV
2)
0.58 < xB < 0.64
Fig. 10. (Color online) Reduced cross sections γ∗p → ωp as
a function of Q2for differents bins in xB, in units of µb. Full
circles: this work; open circles: Ref. [17]. The red cross and
curves correspond to the JML model [29] discussed in sect. 5.
12345
Q
2 (GeV
2)
0.1
1.0
σ (µbarn)
Cornell 1.7 < W < 2.3
JLab 1.95 < W < 2.05
Cornell 2.3 < W < 3.7
JLab 2.7 < W < 2.8
0.1
1.0
σ (µbarn)
DESY 1.7 < W < 2
JLab 1.82 < W < 1.88
DESY 2 < W < 2.2
JLab 2.05 < W < 2.15
DESY 2.2 < W < 2.8
JLab 2.35 < W < 2.5
Fig. 11. (Color online) Total cross sections for the reaction
γ∗p → ωp, as a function of Q2and at fixed W : this work
in full symbols, DESY [16] (top) and Cornell [17] (bottom) in
open symbols. Each symbol (or color) corresponds to a given
central value of W (GeV). Note the range of integration in W
for each data set.
independent of Q2, except for the lowest values of W (see
fig. 14). This is certainly a new finding from this experi
ment, which may indicate a pointlike coupling of the vir
tual photon to the target constituents in this kinematical
regime. This behaviour will be discussed quantitatively in
Sect. 5.
Fig. 12. dσ/dt for the reaction γ∗p → ωp, at W ≃ 2.45 GeV for
different bins in Q2: our data and the JML model (discussed
in sect. 5) with Fπωγ given by eq. (14) (full lines) and without
the tdependence in this equation (dashed lines).
0.30.40.5
c∆τ (fm)
0.60.70.8
1
2
3
b (GeV
−2)
Fig. 13. Slope b of dσ/dt, for the reaction γ∗p → ωp, as a
function of the formation length c∆τ, cf. eq. (4).
Page 11
10
012
2 (GeV
345
Q
2)
0.1
1.0
dσ/dt (µb/GeV
2)
0.1
1.0
dσ/dt (µb/GeV
2)
W = 2.45 GeV
W = 2 GeV
Fig. 14. dσ/dt at fixed values of t and W, as a function of Q2,
for the reaction γ∗p → ωp : t = −0.55 (full circles), t = −1.45
(empty circles) and t = −2.30 (squares) GeV2.
0.22 < xB < 0.28
−0.2
−0.1
0.0
0.1
0.2
0.3
0.16 < xB < 0.220.28 < xB < 0.340.34 < xB < 0.40
2345
Q
2 (GeV
2)
−0.2
−0.1
0.0
0.1
0.2
0.3
0.40 < xB < 0.46
2345
Q
2 (GeV
2)
0.46 < xB < 0.52
2345
Q
2 (GeV
2)
0.52 < xB < 0.58
2345
Q
2 (GeV
2)
0.58 < xB < 0.64
Fig. 15. (Color online) σTT (open circles) and σTL (full cir
cles), in units of µb, for the reaction γ∗p → ωp as a function of
Q2for different bins in xB, integrated over −2.7 GeV2< t < t0.
The dashed blue and full red curves are the corresponding cal
culations in the JML model [29] discussed in sect. 5.
3.4 φ dependence of cross sections
The 34 φ distributions have the expected φ dependence :
dσ
dφ=
1
2π
?
σ + εcos2φ σTT+
?
2ε(1 + ε)cosφ σTL
?
.
(5)
The interference terms σTT and σTLwere extracted from
a fit of each distribution with eq. (5). The results appear
in fig. 15 and in table 3. If helicity were conserved in the
schannel (SCHC), these interference terms σTT and σTL
would vanish. It does not appear to be the case in fig. 15.
The φ distributions do not support the SCHC hypothesis.
4 Analysis of ω decay distribution
In the absence of polarization in the initial state, the dis
tribution of the pions from ω decay is characterized by
eq. (6) [30]. The quantities ρα
ijare defined from a decompo
sition of the ω spin density matrix on a basis of 9 hermitian
matrices. The superscript α refers to this decomposition
and it is related to the virtual photon polarization (α = 0–
2 for transverse photons, α = 4 for longitudinal photons,
and α = 5–6 for interference between L and T terms). For
example, ρ0
00is related to the probability of the transition
between a transverse photon and a longitudinal meson.
All elements ρα
ijcan be expressed as bilinear combina
tions of helicity amplitudes which describe the γ∗p → ωp
transition [5,30]. An analysis of the W distribution can
then be used to test whether helicity is conserved in the
schannel (SCHC), that is between the virtual photon and
ω. If SCHC applies, ρ0
leads to a direct relation between the measured r04
ratio R = σL/σT. In that case, the longitudinal and trans
verse cross sections may be extracted from data without
a Rosenbluth separation.
The matrix elements r04
using onedimensional projections of the W distribution.
Note that r04
1−1should be zero if SCHC applies. Integrating
eq. (6) over φ and then respectively over ϕNor cosθN, one
gets:
00= 0 and ρ4
00= 1. Then eq. (7)
00and the
00and r04
1−1were first extracted
W(cosθN) =3
4
?(1 − r04
1
2π
00) + (3r04
00− 1)cos2θN
?, (8)
W(ϕN) =
?1 − 2r04
1−1cos2ϕN
?.(9)
The background subtraction in the cosθNor ϕNdistribu
tions was performed in 8 bins of the corresponding vari
ables, and for 8 bins in (Q2, xB). The number of acceptance
weighted events was extracted from the correspondingMX[epX]
distribution, as in sect. 2.3. See figs. 16 and 17 for results,
together with fits to eqs. (8) and (9) respectively.
Alternatively, the 15 matrix elements rα
pressed in terms of moments of the decay distribution
W(cosθN,ϕN,φ) [30]. This method of expressing moments
includes the background contribution under the ω peak
(about 25%). It yields compatible results with the (back
ground subtracted) 1D projection method for r04
It was used to study the t dependence of rα
uate the systematic uncertainties in their determination.
Results for forward γ∗p → ωp reaction (t′< 0.5 GeV2)
are given in fig. 18. Systematic uncertainties originate
from the determination of the MC acceptance. The main
source of uncertainties was found to be the finite bin size in
φ. Calculations with different bin sizes (see table 1) and
checks of higher, unphysical, moments in the event dis
tribution led to systematic uncertainties of 0.02 to 0.08,
depending on the rα
ijmatrix element. In addition, cuts in
the event weights were varied, resulting in a systematic
uncertainty of about 0.03 for all matrix elements.
Finally, the rα
ijmatrix elements were also extracted
using an unbinned maximum likelihood method. Results
were compatible with the first two methods. In view of the
ijmay be ex
00and r04
ijand to eval
1−1.
Page 12
11
W(cosθN,ϕN,φ) =
3
4π
?1
2(1 − r04
−εcos2φ(r1
−εsin2φ(√2Imr2
+?
+?
00) +1
2(3r04
11sin2θN + r1
00− 1)cos2θN −
00cos2θN −
10sin2θNsinϕN + Imr2
√2Rer04
√2Rer1
10sin2θNcosϕN − r04
10sin2θNcosϕN − r1
1−1sin2θNsin2ϕN)
√2Rer5
1−1sin2θNcos2ϕN
1−1sin2θNcos2ϕN)
2ε(1 + ε)cosφ(r5
2ε(1 + ε)sinφ(√2Imr6
11sin2θN + r5
00cos2θN −
10sin2θNcosϕN − r5
1−1sin2θNsin2ϕN)
1−1sin2θNcos2ϕN)
10sin2θNsinϕN + Imr6
?
(6)
where the parameters rα
ij, hereafter referred to as matrix elements, are related to the ω spin density matrix:
ρα
ij
1 + εRfor α = 1,2 ;
r04
ij =ρ0
ij+ εRρ4
1 + εR
ij
;rα
ij=
rα
ij=
√R
ρα
ij
1 + εRfor α = 5,6. (7)
−1
0.18
01
0
10
5
10
5
0101
−101
cosθN
0.290.40
xB
0.510.62
1.7
2.3
2.9
3.5
4.1
4.7
5.3
Q
2 (GeV
2)
0.432 +− 0.063
0.389 +− 0.052
6.10
4
4.10
4
2.10
4
0.298 +− 0.028
0.362 +− 0.045
0.330 +− 0.029
0.332 +− 0.028
0.375 +− 0.064
0.286 +− 0.073
Fig. 16. (Color online) Distributions of acceptanceweighted
and backgroundsubtracted counts as a function of cosθN, for
8 bins in (Q2, xB). The location and size of each graph cor
respond to the (Q2, xB) range over which the data is inte
grated. On all graphs, one division on the vertical axis rep
resents 2 × 104(arbitrary units). All data are integrated in
t (−t < 2.7 GeV2). The blue curves correspond to fits with
eq. (8), with the resulting r04
indicated on each distribution. The systematic uncertainty on
this matrix element is estimated at 0.042.
00and its statistical uncertainty
φ dependence of the acceptance (see fig. 9), this method
was used for checking the validity of the rα
when restricting the φ range taken into consideration in
the fit.
These studies lead to the conclusion that SCHC does
not hold for the reaction γ∗p → ωp, not only when con
sidering the whole t range (fig. 17), but also, though in a
lesser extent, in the forward direction (fig. 18). For SCHC,
all matrix elements become zero, except five: r04
Im r2
10and these are not all independent;
they satisfy [5]: r1
The quantity
ijdetermination
00, r1
1−1,
1−1, Re r5
10, Im r6
1−1= −Imr2
1−1and Rer5
10= −Imr6
10.
χ2=
1
12
?10
?
1
?r
∆r
?2
+
(r1
1−1)2+ (∆Imr2
1−1+ Imr2
1−1)2
(∆r1
1−1)2
0
π
2π
0.29
0
10
5
10
5
π
2π
0.40
xB
π
2π
0.51
0
π
2π
φN
0.18 0.62
1.7
2.3
2.9
3.5
4.1
4.7
5.3
Q
2 (GeV
2)
0.055 +− 0.059
−0.060 +− 0.045
6.10
4
4.10
4
2.10
4
−0.074 +− 0.027
−0.064 +− 0.045
−0.102 +− 0.025
−0.174 +− 0.026
−0.146 +− 0.053
−0.176 +− 0.056
Fig. 17. (Color online) Distributions of acceptanceweighted
and backgroundsubtracted counts as a function of ϕN. The
blue curves correspond to fits with eq. (9), with the resulting
r04
bution. The systematic uncertainty on this matrix element is
estimated at 0.042. See also legend of fig. 16.
1−1and its statistical uncertainty indicated on each distri
+
(Rer5
10+ Imr6
10)2+ (∆Imr6
10)2
(∆Rer5
10)2
?
(10)
where the sum is carried over the ten matrix elements
which would be zero if SCHC applies, may be used as a
measure of SCHC violation. Including in the denomina
tors ∆r the systematic uncertainties added in quadrature
to the statistical uncertainties, the 7 χ2values (excluding
the distributions at the lowest xBbin where SCHC viola
tion is the most manifest in fig. 18) range from 2.3 to 7.7
when including all data, and drop only to 1.7 to 5.1, in
spite of doubled statistical uncertainties, when consider
ing only the forward production (t′< 0.5 GeV2). Further
more, when examining the relation between these matrix
elements and helicityflip amplitudes, it does not appear
possible to ascribe the SCHC violation to a small subset
of these amplitudes. It is therefore not justified to cal
culate R from eq. (7) and separate the longitudinal and
transverse cross sections from this data.
Page 13
12
−0.5
0.0
0.5
1.0
0.5
1.0
0.5
1.0
−0.5
0.0
0.0
0.5
1.0
0.18 0.29 0.40
xB
0.51 0.62
1.7
2.3
2.9
3.5
4.1
4.7
5.3
Q
2 (GeV
2)
Fig. 18. (Color online) rα
ments for 8 bins in (Q2, xB) and for t′< 0.5 GeV2. The loca
tion and size of each graph correspond to the (Q2, xB) range
over which the data is integrated, but the scale is the same
on all graphs. The abscissa on each graph corresponds to the
following list of matrix elements: r04
Re r1
Im r6
which are zero if SCHC applies. The 16th entry (blue empty
circle, in some cases off scale) is the combination of rα
by eq. (11). Error bars include systematic uncertainties added
in quadrature.
ijextracted with the method of mo
00, Re r04
00, r5
10, r04
10, r5
1−1, r1
1−1, Im r6
00, r1
11,
10,
10, r1
1−1. The red filled symbols indicate those matrix elements
1−1, Im r2
10, Im r2
1−1, r5
11, Re r5
ijgiven
When one retains only those amplitudes which corre
spond to a natural parity exchange in the tchannel, then
the following relation should hold [19] :
1 − r04
00+ 2r04
1−1− 2r1
11− 2r1
1−1= 0(11)
This particular combination is plotted as the 16th point
on each of the graphs of fig. 18. The fact that it is not zero
points to the importance of the unnatural parity (presum
ably pion) exchange.
It is also possible to estimate qualitatively the role of
pion exchange through the U/N asymmetry of the trans
verse cross section, where U and N refer to unnatural and
natural parity exchange contributions [5]:
P ≡σN
T− σU
σN
T
T+ σU
T
= (1 + εR)(2r1
1−1− r1
00) .(12)
Our results yield r1
kinematical range, and thus:
1−1< 0 and r1
00≥ 0 over the whole
P < −(2r1
1−1 + r1
00) .(13)
Hence P is large and negative, which means that most
of the transverse cross section is due to unnatural parity
exchange.
5 Comparison with a Regge model
Regge phenomenology was applied with success to the
photoproduction of vector mesons in our energy range
Table 4. Meson and Pomeron (or twogluon) exchanges con
sidered in the JML model for vector meson production.
Produced
vector meson
Exchanged
Regge trajectories
ρ
ω
φ
σ, f2, P/2g
π0, f2, P/2g
P/2g
and at higher energies [4,31]. Laget and coworkersshowed
that the introduction of saturating Regge trajectories pro
vides an excellent simultaneous description of the high
−t behaviour of the γp → pρ,ω,φ cross sections, given
an appropriate choice of the relevant coupling constants.
The tchannel exchanges considered in this JML model are
indicated in table 4. Saturating trajectories have a close
phenomenological connection to the quarkantiquark in
teraction which governs the mesonic structure [32]. They
provide an effective way to implement gluon exchange be
tween the quarks forming the exchanged meson.
This model was extended to the case of electroproduc
tion [29]. The Q2dependence of the f2 and P exchange
is built in the model. In the case of ω production, the
only additional free parameters come from the electro
magnetic form factor which accounts for the finite size
of the vertex between the virtual photon, the exchanged
π0trajectory and the ω meson. This form factor could
be chosen as the usual parameterization of the pion elec
tromagnetic form factor: Fωπγ = Fπ = (1 + Q2/Λ2
with Λ2
fails to account for the observed t dependence (see dashed
lines in fig. 12). From the observation that the differential
cross section becomes nearly Q2independent at high −t,
an adhoc modification of the form factor
π)−1,
π= 0.462 GeV2. As described so far, the model
Fωπγ(Q2) → Fωπγ(Q2,t) =
1
1 +Q2
Λ2
π
?
1+απ(t)
1+απ(0)
?2
(14)
was proposed [29]. The saturating π0Regge trajectory
obeys the relation limt→−∞απ(t) = −1, so that the form
factor becomes flat at high −t. Thus, eq. (14) associates
the pointlike coupling of the virtual photon with the satu
ration of the π0Regge trajectory which accounts for hard
scattering in this kinematical limit [29]. Note that this
modification of the form factor does not violate gauge
invariance, which holds separately for each contribution
from Table 4 and, in the case of π0exchange, from the
spin and momentum structure of the ωπγ vertex.
The t dependence of the differential cross sections is
then well described (solid lines in fig. 12). The Q2de
pendence of the cross sections is illustrated in fig. 10. At
high xB, which corresponds to the lowest values of W,
schannel resonance contributions are not taken into ac
count in the model and may explain the observed disagree
ment. Finally the interference terms σTT and σTL agree
in sign and trend, but not in magnitude, with our results
(fig. 15).
Page 14
13
So, within this model, π0exchange, or rather the ex
change of the associated saturating Regge trajectory, con
tinues to dominate the cross section at high Q2and the
cross section is mostly transverse. This is consistent with
our observations of the dominance of unnatural parity ex
change in the tchannel in the previous section.
6 Relevance of the handbag diagram
Let us recall that the handbag diagram of fig. 1 is expected
to be the leading one in the Bjorken regime. In this picture,
the transition γ∗
L→ ωLwould dominate the process. This
is clearly antinomic to the findings in sect. 5, where our
results are interpreted as dominated by the π0exchange,
which is mostly due to transverse photons. In addition, π0
exchange is of a pseudoscalar nature, while the H and E
GPD which enter the handbag diagram amplitude are of
a vector nature.
Independent of the model interpretation presented in
sect. 5, our results point to the nonconservation of helicity
in the schannel (figs. 15, 17 and 18), meaning that the
handbag diagram does not dominate the process, even for
small values of −t and Q2as large as 4.5 GeV2.
As a consequence, σLcould not be extracted from our
data for a direct comparison with models based on the
GPD formalism. It is however instructive to consider here
the predictions of a GPDbased model [3,33], denoted
hereafter VGG. This is a twist2, leading order calcula
tion, where the GPD are parameterized in terms of dou
ble distributions (DD) and include the socalled Dterm
(see Ref. [33] for definitions): H,E ∼ DD(x,ξ)eb(ξ,Q2)t/2,
where b is taken from the data (see sect. 3.3 and table 3).
An effective way of incorporating some of the higher twist
effects is to introduce a “frozen” strong coupling constant
αS= 0.56. This model is described in some more details
in Ref. [12] and is applied here to the specific case of ω
production. The model calculations (VGG and JML) of
εσLare plotted in fig. 19. The sharp drop of the curves at
high Q2is due to the decrease of ε, at our given beam en
ergy, as Q2reaches its kinematical limit. When compared
to our results, εσLis calculated to be only 1/6 to 1/4 of
the measured cross sections, thus explaining the difficulty
in extracting this contribution.
The ω channel thus appears to be a challenging reac
tion channel to study the applicability of the GPD formal
ism. This is attributed to the tchannel π0exchange, which
remains significant even at high values of Q2. In contradis
tinction, the π0exchange is negligible in the case of the ρ
production channel, where SCHC was found to hold, and
σLcould be extracted and compared successfully to GPD
models [11,12].
7 Summary
An extensive set of data on exclusive ω electroproduction
has been presented, for Q2from 1.6 to 5.1 GeV2and W
from 1.8 to 2.8 GeV (xB from 0.16 to 0.64). Total and
12
Q
34
2 (GeV
2)
0.1
1.0
σ (µbarn)
0.1
1.0
σ (µbarn)
W = 2.1 GeV
W = 2.8 GeV
Fig. 19. (Color online) Total cross sections for the reaction
γ∗p → ωp, for ?W? = 2.1 (top) and 2.8 (bottom) GeV : this
work (full red circles), DESY data (empty diamonds), Cornell
data (empty circles), and JML model (dotted curves). The lon
gitudinal contribution, ε(E,Q2)σL, is calculated according to
the JML (solid lines) and VGG (dashed blue lines) models.
differential cross sections for the reaction γ∗p → ωp were
extracted, as well as matrix elements linked to the ω spin
density matrix.
The t differential cross sections are surprisingly large
for high values of −t (up to 2.7 GeV2). This feature can
be accounted for in a Reggebased model (JML), provided
a t dependence is assumed for the ωπγ vertex form factor,
with a prescription inspired from saturating Regge trajec
tories. It appears that the virtual photon is more likely to
couple to a pointlike object as −t increases.
The analysis of the φ differential cross sections and of
the ω decay matrix elements indicate that the schannel
helicity is not conserved in this process. As a first conse
quence, the longitudinal and transverse contributions to
the cross sections could not be separated. Furthermore,
the values of some decay matrix elements point to the im
portance of unnatural parity exchange in the tchannel,
such as π0exchange. This behaviour had been previously
established in the case of ω photoproduction, but not for
the large photon virtuality obtained in this experiment.
The results on these observables also support the JML
model, where the exchange of the saturating Regge tra
jectory associated with the π0is mostly transverse and
dominates the process.
Page 15
14
Finally, the experiment demonstrated that exclusive
vector meson electroproduction can be measured with high
statistics in a wide kinematical range. The limitations at
high Q2were not due to the available luminosity of the
CEBAF accelerator or to the characteristics of the CLAS
spectrometer, but to the present beam energy. With the
planned upgrade of the beam energy up to 12 GeV [34],
such reactions will be measured to still higher values of Q2.
In the specific case of the ω meson, as was shown in this
paper, this will be a necessary condition for the extrac
tion of a longitudinal contribution of the handbag type,
related at low values of −t to generalized parton distribu
tions. More generally, this experiment opens a window on
the high Q2and high −t behaviour of exclusive reactions,
which needs further exploration.
We would like to acknowledge the outstanding efforts of the
staff of the Accelerator and the Physics Divisions at JLab
that made this experiment possible. This work was supported
in part by the Italian Istituto Nazionale di Fisica Nucleare,
the French Centre National de la Recherche Scientifique, the
French Commissariat ` a l’Energie Atomique, the U.S. Depart
ment of Energy and National Science Foundation, the Emmy
Noether grant from the Deutsche Forschungs Gemeinschaft
and the Korean Science and Engineering Foundation. The South
eastern Universities Research Association (SURA) operates the
Thomas Jefferson National Accelerator Facility for the United
States Department of Energy under contract DEAC0584ER
40150.
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