Page 1
arXiv:0807.3834v1 [hep-ex] 24 Jul 2008
EPJ manuscript No.
(will be inserted by the editor)
Exclusive ρ0electroproduction on the proton at CLAS
S.A. Morrow1,2, M. Guidal1a, M. Gar¸ con2, J.M. Laget2,3, E.S. Smith3, G. Adams4, K.P. Adhikari5, M. Aghasyan6,
M.J. Amaryan5, M. Anghinolfi7, G. Asryan8, G. Audit2, H. Avakian3, H. Bagdasaryan8,5, N. Baillie9, J.P. Ball10,
N.A. Baltzell11, S. Barrow12, M. Battaglieri7, I. Bedlinskiy13, M. Bektasoglu14,5, M. Bellis15, N. Benmouna16,
B.L. Berman16, A.S. Biselli17, L. Blaszczyk18, B.E. Bonner19, C. Bookwalter18, S. Bouchigny1, S. Boiarinov13,3,
R. Bradford15, D. Branford20, W.J. Briscoe16, W.K. Brooks3,21, S. B¨ ultmann5, V.D. Burkert3, C. Butuceanu9,
J.R. Calarco22, S.L. Careccia5, D.S. Carman3, B. Carnahan23, L. Casey23, A. Cazes11, S. Chen18, L. Cheng23,
P.L. Cole3,24, P. Collins10, P. Coltharp12, D. Cords3, P. Corvisiero7, D. Crabb25, H. Crannell23, V. Crede18,
J.P. Cummings4, D. Dale24, N. Dashyan8, R. De Masi1,2, R. De Vita7, E. De Sanctis6, P.V. Degtyarenko3,
H. Denizli26, L. Dennis18, A. Deur3, S. Dhamija27, K.V. Dharmawardane5, K.S. Dhuga16, R. Dickson15,
J.-P. Didelez1, C. Djalali11, G.E. Dodge5, D. Doughty28, M. Dugger10, S. Dytman26, O.P. Dzyubak11, H. Egiyan22,9,3,
K.S. Egiyan8, L. El Fassi29, L. Elouadrhiri3, P. Eugenio18, R. Fatemi25, G. Fedotov30, R. Fersch9, R.J. Feuerbach15,
T.A. Forest24, A. Fradi1, G. Gavalian22,5, N. Gevorgyan8, G.P. Gilfoyle31, K.L. Giovanetti32, F.X. Girod3,2,
J.T. Goetz33, W. Gohn34, C.I.O. Gordon35, R.W. Gothe11, L. Graham11, K.A. Griffioen9, M. Guillo11, N. Guler5,
L. Guo3, V. Gyurjyan3, C. Hadjidakis1, K. Hafidi29, H. Hakobyan8, C. Hanretty18, J. Hardie28,3, N. Hassall35,
D. Heddle28,3, F.W. Hersman22, K. Hicks14, I. Hleiqawi14, M. Holtrop22, E. Hourany1, C.E. Hyde-Wright5,
Y. Ilieva16, D.G. Ireland35, B.S. Ishkhanov30, E.L. Isupov30, M.M. Ito3, D. Jenkins36, H.S. Jo1, J.R. Johnstone35,
K. Joo34,3, H.G. Juengst5, N. Kalantarians5, D. Keller14, J.D. Kellie35, M. Khandaker37, P. Khetarpal4, W. Kim38,
A. Klein5, F.J. Klein23, A.V. Klimenko5, M. Kossov13, L.H. Kramer27,3, V. Kubarovsky3, J. Kuhn4,15, S.E. Kuhn5,
S.V. Kuleshov13,21, V. Kuznetsov38, J. Lachniet15,5, J. Langheinrich11, D. Lawrence39, Ji Li4, K. Livingston35,
H.Y. Lu11, M. MacCormick1, C. Marchand2, N. Markov34, P. Mattione19, S. McAleer18, M. McCracken15,
B. McKinnon35, J.W.C. McNabb15, B.A. Mecking3, S. Mehrabyan26, J.J. Melone35, M.D. Mestayer3, C.A. Meyer15,
T. Mibe14, K. Mikhailov13, R. Minehart25, M. Mirazita6, R. Miskimen39, V. Mokeev30,3, L. Morand2,
B. Moreno1, K. Moriya15, M. Moteabbed27, J. Mueller26, E. Munevar16, G.S. Mutchler19, P. Nadel-Turonski16,
R. Nasseripour16,27,11, S. Niccolai1, G. Niculescu32,14, I. Niculescu32,16,3, B.B. Niczyporuk3, M.R. Niroula5,
R.A. Niyazov4,5, M. Nozar3, G.V. O’Rielly16, M. Osipenko7,30, A.I. Ostrovidov12, K. Park38,11, S. Park18,
E. Pasyuk10, C. Paterson35, S. Anefalos Pereira6, S.A. Philips16, J. Pierce25, N. Pivnyuk13, D. Pocanic25,
O. Pogorelko13, E. Polli6, I. Popa16, S. Pozdniakov13, B.M. Preedom11, J.W. Price40, S. Procureur2, Y. Prok25,28,3,
D. Protopopescu22,35, L.M. Qin5, B.A. Raue27,3, G. Riccardi18, G. Ricco7, M. Ripani7, B.G. Ritchie10, G. Rosner35,
P. Rossi6, P.D. Rubin31, F. Sabati´ e2, M.S. Saini18, J. Salamanca24, C. Salgado37, J.P. Santoro23, V. Sapunenko3,
D. Schott27, R.A. Schumacher15, V.S. Serov13, Y.G. Sharabian3, D. Sharov30, N.V. Shvedunov30, A.V. Skabelin41,
L.C. Smith25, D.I. Sober23, D. Sokhan20, A. Stavinsky13, S.S. Stepanyan38, S. Stepanyan3, B.E. Stokes18, P. Stoler4,
I.I. Strakovsky16, S. Strauch11,16, M. Taiuti7, D.J. Tedeschi11, A. Tkabladze14,16, S. Tkachenko5, L. Todor31,15,
C. Tur11, M. Ungaro34,4, M.F. Vineyard42,31, A.V. Vlassov13, D.P. Watts20,35, L.B. Weinstein5, D.P. Weygand3,
M. Williams15, E. Wolin3, M.H. Wood11, A. Yegneswaran3, M. Yurov38, L. Zana22, J. Zhang5, B. Zhao34, and
Z.W. Zhao11
(CLAS Collaboration)
1Institut de Physique Nucleaire ORSAY, Orsay, France
2CEA-Saclay, Service de Physique Nucl´ eaire, 91191 Gif-sur-Yvette, France
3Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606
4Rensselaer Polytechnic Institute, Troy, New York 12180-3590
5Old Dominion University, Norfolk, Virginia 23529
6INFN, Laboratori Nazionali di Frascati, 00044 Frascati, Italy
7INFN, Sezione di Genova, 16146 Genova, Italy
8Yerevan Physics Institute, 375036 Yerevan, Armenia
9College of William and Mary, Williamsburg, Virginia 23187-8795
10Arizona State University, Tempe, Arizona 85287-1504
11University of South Carolina, Columbia, South Carolina 29208
12Florida State University, Tallahassee, Florida 32306
aCorresponding author: guidal@ipno.in2p3.fr
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2
13Institute of Theoretical and Experimental Physics, Moscow, 117259, Russia
14Ohio University, Athens, Ohio 45701
15Carnegie Mellon University, Pittsburgh, Pennsylvania 15213
16The George Washington University, Washington, DC 20052
17Fairfield University, Fairfield CT 06824
18Florida State University, Tallahassee, Florida 32306
19Rice University, Houston, Texas 77005-1892
20Edinburgh University, Edinburgh EH9 3JZ, United Kingdom
21Universidad T´ ecnica Federico Santa Mar´ ıa, Casilla 110-V, Valpara´ ıso, Chile
22University of New Hampshire, Durham, New Hampshire 03824-3568
23Catholic University of America, Washington, D.C. 20064
24Idaho State University, Pocatello, Idaho 83209
25University of Virginia, Charlottesville, Virginia 22901
26University of Pittsburgh, Pittsburgh, Pennsylvania 15260
27Florida International University, Miami, Florida 33199
28Christopher Newport University, Newport News, Virginia 23606
29Argonne National Laboratory, Illinois 60439
30Moscow State University, General Nuclear Physics Institute, 119899 Moscow, Russia
31University of Richmond, Richmond, Virginia 23173
32James Madison University, Harrisonburg, Virginia 22807
33University of California at Los Angeles, Los Angeles, California 90095-1547
34University of Connecticut, Storrs, Connecticut 06269
35University of Glasgow, Glasgow G12 8QQ, United Kingdom
36Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061-0435
37Norfolk State University, Norfolk, Virginia 23504
38Kyungpook National University, Daegu 702-701, Republic of Korea
39University of Massachusetts, Amherst, Massachusetts 01003
40California State University, Dominguez Hills, Carson, CA 90747
41Massachusetts Institute of Technology, Cambridge, Massachusetts 02139-4307
42Union College, Schenectady, NY 12308
Received: date / Revised version: date
Abstract. The ep → e′pρ0reaction has been measured, using the 5.754 GeV electron beam of Jefferson Lab
and the CLAS detector. This represents the largest ever set of data for this reaction in the valence region.
Integrated and differential cross sections are presented. The W, Q2and t dependences of the cross section
are compared to theoretical calculations based on t-channel meson-exchange Regge theory on the one hand
and on quark handbag diagrams related to Generalized Parton Distributions (GPDs) on the other hand.
The Regge approach can describe at the ≈ 30% level most of the features of the present data while the
two GPD calculations that are presented in this article which succesfully reproduce the high energy data
strongly underestimate the present data. The question is then raised whether this discrepancy originates
from an incomplete or inexact way of modelling the GPDs or the associated hard scattering amplitude
or whether the GPD formalism is simply inapplicable in this region due to higher-twists contributions,
incalculable at present.
PACS. 13.60.Le Production of mesons by photons and leptons – 12.40.Nn Regge theory – 12.38.Bx
Perturbative calculations
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1 Introduction
The exclusive electroproduction of photons and mesons
on the nucleon is an important tool to better understand
nucleon structure and, more generally, the transition be-
tween the low energy hadronic and high energy partonic
domains of the Quantum Chromodynamics (QCD) theory.
Among all such exclusive processes, the ep → e′pρ0
reaction bears some particular advantages. It is a pro-
cess for which numerical calculations and predictions are
available both in terms of hadronic degrees of freedom,
via Reggeized meson exchanges, and in terms of partonic
degrees of freedom, via Generalized Parton Distributions
(GPDs). We refer the reader to refs. [1,2,3] and refs. [4,
5,6,7,8,9,10] for the original articles and general reviews
of Regge theory and GPDs respectively. Defining Q2as
the absolute value of the squared mass of the virtual pho-
ton that is exchanged between the electron and the target
nucleon, partonic descriptions are expected to be valid at
large Q2, while hadronic descriptions dominate in photo-
and low-Q2electroproduction. Fig. 1 illustrates these two
approaches at the electron beam energies available at Jef-
ferson Laboratory (JLab). Concerning the Reggeized me-
son exchange approach, the total and differential cross
sections associated with the exchanges of the dominant
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3
Regge σ and f2trajectories have been calculated by Laget
et al. [11,12]. Concerning the GPDs approach, the so-
called “handbag” diagram, with recent modelings of the
unpolarized GPDs, has been calculated by two groups:
Goloskokov-Kroll[13] and Vanderhaeghen et al. [14,15,16,
17]. Let us note here that in the GPD approach the leading
twist handbag calculation is valid only for the longitudi-
nal part of the cross section and that, experimentally, it
is important to separate the longitudinal and transverse
parts of the cross sections when measuring this process.
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Fig. 1. The mechanisms for ρ0electroproduction at JLab ener-
gies for low Q2(left diagram) through the exchange of mesons
and for high Q2(right diagram) through the quark exchange
“handbag” mechanism (valid for longitudinal photons) where
H and E are the unpolarized GPDs.
This article presents results for the exclusive electro-
production of the ρ0vector meson on the proton mea-
sured with the 5.754 GeV electron beam of the CEBAF
accelerator and the CEBAF Large Acceptance Spectrom-
eter (CLAS) at JLab. The aim of this analysis is to com-
pare the integrated and differential cross sections of the
ep → e′pρ0reaction that have been extracted over the
intermediate Q2region accessible at CLAS, with the two
Regge and GPD theoretical approaches, and thus deter-
mine their domain of validity and constrain their various
inputs.
There are a few existing electroproduction data in a
similar kinematical regime: early data with the 7.2 GeV
beam at DESY [18] and with the 11.5 GeV beam at Cor-
nell [19], and more recently with the 27 GeV beam at
HERMES [20] and the 4.2 GeV beam of JLab [21]. The
present work explores new phase-space regions and, in re-
gions of overlap, has much finer binning and precision.
In section 2 we present the experimental procedure we
have adopted to extract our integrated and differential
cross sections. In section 3, after briefly describing the
Regge and GPDs models, we compare these calculations
to our data. Finally, we draw our conclusions in section 4.
2 Experimental procedure
The CLAS detector [22] is built around six superconduc-
tion coils that generate a toroidal magnetic field primarily
in the azimuthal direction. Each sector is equipped with
three regions of multi-wire drift chambers (DC) and time-
of-flight scintillator counters (SC) that cover the angular
range from 8◦to 143◦. In the forward region (8◦< θ <
45◦), each sector is furthermore equipped with a gas-filled
threshold Cerenkov counter (CC) and a lead-scintillator
sandwich type electromagnetic calorimeter (EC). Azimuthal
coverage for CLAS is limited by the magnet’s six coils and
is approximatively 90% at large polar angles and 50% at
forward angles.
The data were taken with an electron beam having an
energy of 5.754 GeV impinging on an unpolarized 5-cm-
long liquid-hydrogen target. The integrated luminosity of
this data set was 28.5 fb−1. The data were taken from Oc-
tober 2001 to January 2002. The kinematic domain of the
selected sample corresponds approximately to Q2from 1.5
to 5.5 GeV2. We analyzed data with W, the γ∗−p center-
of-mass energy, greater than 1.8 GeV, which corresponds
to a range of xB approximatively from 0.15 to 0.7. Here
xB is the standard Bjorken variable equal to
with mpthe mass of the proton.
The ρ0decays into two pions (π+π−), with a branching
ratio of 100% [23]. To select the channel ep → e′pρ0, we
based our analysis on the identification of the scattered
electron, the recoil proton and the positive decay pion
(because of the polarity of the magnetic field, negative
pions are bent toward the beam pipe and in general escape
the acceptance of CLAS); we then used the missing mass
ep → e′pπ+X for the identification of the ep → e′pπ+π−
final state.
Once this final state is identified and its yield normal-
ized, the reduced γ∗p → pρ0cross section is extracted
by fitting in a model-dependent way the (π+π−) invari-
ant mass using a parametrized ρ0shape, which will be
described later. The longitudinal and transverse cross sec-
tions are then further extracted by analyzing the decay
pion angular distribution in the ρ0center-of-mass frame.
We detail all these steps in the following sections.
Q2
W2−m2
p+Q2
2.1 Particle identification
The electron is identified as a negative track, determined
from the DC, having produced a signal in the CC and
the EC. Pions, potentially misidentified as electrons, were
rejected by cutting on the CC amplitude (> 2 photoelec-
trons), imposing a minimum energy deposition in the EC
(60 MeV) and correlating the measurements of the mo-
mentum from the DC and of the energy from the EC. In
order to minimize radiative corrections and residual pion
contamination, a further cut E′≥ 0.8 GeV was also ap-
plied, where E′is the scattered electron energy. Finally,
vertex and geometric fiducial cuts, which select only re-
gions of well understood acceptance, were included.
The efficiencies of the CC and EC cuts, respectively
ηCCand ηEC, were determined from data samples, select-
ing unambiguous electrons with very tight CC or EC cuts.
The CC-cut efficiencies range from 86 to 99% and the EC-
cut efficiencies from 90 to 95%, depending on the electron
kinematics. The efficiencies of the geometric fiducial cuts
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4
were derived from CLAS GEANT-based Monte-Carlo sim-
ulations.
Pions and protons are identified by the correlation be-
tween the momentum measured by the DC and the ve-
locity measured by the SC. This identification procedure
is unambiguous for particles with momenta up to 2 GeV.
Particles with momenta higher than 2 GeV were there-
fore discarded. The efficiencies of the cuts imposed for
this momentum-velocity correlation and of the geometric
fiducial cuts were determined from CLAS GEANT-based
Monte-Carlo simulations.
Once the electron, the proton and the positive pion
are identified, the ep → e′pπ+π−final state is identi-
fied through the missing mass technique. Fig. 2 shows
the square of the missing mass for the system e′pπ+(i.e.
M2
system ep (i.e. MX[e′pX]). One distinguishes the ρ0and
ω loci quite clearly. A cut on the M2
is required in order to separate the ρ0and the associated
π+π−continuum from the ω and the three–pion contin-
uum background. The optimum value of this cut:
X[e′pπ+X]) as a function of the missing mass for the
X[e′pπ+X] variable
− 0.05 ≤ M2
X[e′pπ+X] ≤ 0.08 GeV2
(1)
was determined from a study whereby we estimated the
number of ρ0events, from fits to the M2
tion, as a function of the cut values. The cuts were chosen
in the region where the number of ρ0events began to
vary only very weakly with these cut values.The simula-
tion used to calculate acceptances reproduces the features
of fig. 3. The position of this cut relative to M2
is shown in fig. 3.
X[e′pX] distribu-
X[e′pπ+X]
[e’pX] (GeV)
X
MM
00.2 0.40.60.81 1.2 1.4
)
2
X] (GeV
+
π
[e’p
2
MM
X
-0.1
0
0.1
0.2
0.3
0.4
0.5
0
200
400
600
800
1000
1200
1400
Fig. 2. Squared missing mass M2
W ≥ 1.8 GeV and E′≥ 0.8 GeV.
X[e′pπ+X] vs MX[e′pX] for
The missing mass distribution for the system ep, ob-
tained after this cut, is shown in fig. 4. The ρ0peak is
very broad : Γth
ρ0 ≈ 150 MeV from ref. [23]. and sits on
top of a background of a non-resonant two pion contin-
uum, which originates from other processes leading to the
e′pπ+π−final state, such as ep → e′∆++π−→ e′pπ+π−.
In fig. 4, one can additionnally distinguish two bumps
at masses around 950 MeV and 1250 MeV correspond-
ing respectively to the scalar f0(980) and tensor f2(1270)
)
2
0.5
(GeV
+X
π
ep
2
MM
-0.10 0.10.2 0.3 0.4
0
10000
20000
30000
40000
50000
60000
Fig. 3. Missing mass M2
E′≥ 0.8 GeV. The red lines show the cut used (see eq. 1) to
select the e′pπ+π−final state.
X[e′pπ+X] for W ≥ 1.8 GeV and
mesons. These will be even more evident when we look at
the differential spectra later on.
(GeV)
epX
MM
00.2 0.40.60.811.21.4
0
2000
4000
6000
8000
10000
12000
14000
16000
18000
20000
22000
24000
Fig.
M2
The ρ0(770), as well as the f0(980) and f2(1270) resonances
which can be distinguished, sits on top of a background of
non-resonant two-pion continuum.
4.
Missing massMX[e′pX] for-0.05
≤
X[e′pπ+] ≤ 0.08 GeV2, W ≥ 1.8 GeV and E′≥ 0.8 GeV.
2.2 Acceptance calculation
The acceptance of the CLAS detector for the e′pπ+X
process has been determined with the standard GEANT-
based code developed for CLAS. Our event generator [24]
contains the three main channels leading to the e′pπ+π−
final state: ep → e′pρ0֒→ π+π−, ep → e′π−∆++֒→ pπ+,
and the non-resonant (phase space) ep → e′pπ+π−. This
event generator is based on tables of total and differen-
tial cross sections of double pion photoproduction data
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5
that have been extrapolated to electroproduction. This
has been done by multiplying these tables by a virtual
photon flux factor and a dipole form factor in order to
obtain a relatively realistic Q2dependence of the cross
section. We also have tuned the relative weight of all the
aforementioned channels in order to reproduce the main
kinematical distributions of our experimental data.
Eight independent kinematical variables are necessary
to describe a reaction with four particles in the final state.
However, in unpolarized electroproduction, the cross sec-
tion does not depend on the azimuthal angle of the scat-
tered electron. The following seven variables are then cho-
sen: Q2, xB, t, Mπ+π−, Φ, cos(θHS) and φHS. Here Q2
and xBare respectively the absolute value of the squared
electron four-momentum transfer and the Bjorken vari-
able, which describe the kinematics of the virtual pho-
ton γ∗. At some stages, W, the γ∗− p center-of-mass en-
ergy will equivalently be used. Then t is the squared four-
momentum transferred to the ρ0, Φ is the azimuthal angle
between the electron scattering plane and the hadronic
production plane, and Mπ+π− is the invariant mass of the
π+π−system. Finally, cos(θHS) and φHSdescribe the de-
cay of the ρ0into two pions and are respectively the polar
and azimuthal angles of the π+in the so-called Helicity
Frame (HS) where the ρ0is at rest and the z-axis is given
by the ρ0direction in the γ∗− p center-of-mass system.
All these variables are illustrated in fig. 5.
θ
θ∗
p
electron scattering plane
y
z
γ∗(Q )
(t)
(E)
Laboratory frame
c.m. frame
Helicity frame
( at rest)
ρ
(E’)
e
e’
φ
x
hadronic production plane
Virtual photoproduction
θ
2
ρ
ρ
HS
HS
Fig. 5. Reference frames and relevant variables for the descrip-
tion of the ep → e′pρ0֒→ π+π−reaction.
The procedure we have followed has been to calculate
an acceptance for each of the 7-dimensional bins. In the
limit of small bin-size and unlimited statistics, this proce-
dure is independent of the model used to generate events.
Our event generator has nonetheless been tuned to repro-
duce the experimental data. The binning in the 7 indepen-
dent variables is defined in table 1 and its 2-dimensional
(Q2, xB) projection is shown in fig. 6.
More than 200 million Monte-Carlo (MC) events were
generated using CLAS GEANT to calculate the accep-
Variable
Q2
Unit
GeV2
Range
1.60 – 3.10
3.10 – 5.60
0.16 – 0.7
0.10 – 1.90
1.90 – 4.30
0.00 – 360.00
-1.00 – 1.00
0.00 – 360.00
0.22 – 1.87
# of bins
5
5
9
6
3
9
8
8
15
Width
0.30
0.50
0.06
0.30
0.80
40.00
0.25
45.00
0.22
xB
−t
–
GeV2
Φ
cos(θHS
φHS
π+
MMep
deg.
–
deg.
GeV
π+)
Table 1. Binning in the 7 independent variables for the ac-
ceptance table.
B
x
00.10.20.30.40.50.60.70.8
)
2
(GeV
2
Q
1
2
3
4
5
6
0
1000
2000
3000
4000
5000
6000
7000
8000
Fig. 6. Binning in (Q2,xB) for our experimental data (with
W > 1.8 GeV and E′≥ 0.8 GeV).
tance in each of the individual 7-dimensional bins. Each
“real data” event was then weighted by the ratio of the
number of MC generated to accepted events for each 7-
dimensional bin. The acceptances are at most a few per-
cent. Bins that have a very small acceptance (< 0.16%)
have a very high weight, which produces an unphysically
high and narrow peak in the weighted event distributions,
and have to be cut away. The efficiency of this cut is evalu-
ated by MC computation of the ratio of weighted accepted
events to generated events mapped onto 1-dimensional dis-
tributions. This correction factor ηwis therefore model de-
pendent since it is 1-dimensional and thus integrated over
the remaining variables. It is on average 15%.
Radiative corrections were part of our event generator
and were calculated according to ref. [25]. The MC accep-
tance calculation presented above therefore took into ac-
count the effects of the radiation of hard photons and the
corresponding losses due to the application of the cut of
eq. 1. The contribution of soft photons and virtual correc-
tions were determined by turning on and off the radiative
effects in our event generator, defining an Fradfactor for
each (Q2, xB) bin for the integrated cross sections, or for
each (Q2, xB, X) bin for the differential cross sections, X
being one of t, Φ, cos(θHS) or φHS.
Page 7
6
2.3 γ∗p → pπ+π−total cross section
The total reduced cross section for the ep → e′pπ+π−
reaction can then be obtained from:
σγ∗p→pπ+π−(Q2,xB,E) =
1
ΓV(Q2,xB,E)
d2σep→e′pπ+π−
dQ2dxB
(2)
with:
d2σep→e′pπ+π−
dQ2dxB
=
nw(Q2,xB)
Lint∆Q2∆xB
×
Frad
ηCCηECηw,(3)
where
– nw(Q2,xB) is the weighted number of ep → e′pπ+π−
events in a given bin (Q2, xB),
– Lint is the effective integrated luminosity (that takes
into account the correction for the data acquisition
dead time),
– ∆Q2and ∆xB are the corresponding bin widths (see
table 1); for bins not completely filled, because of W
or E′cuts on the electron for instance (see fig. 6), the
phase space ∆Q2∆xBincludes a surface correction and
the Q2and xBcentral values are modified accordingly.
– Fradis the correction factor due to the radiative effects
(see section 2.2),
– ηCCis the CC-cut efficiency (see section 2.1),
– ηECis the EC-cut efficiency (see section 2.1),
– ηwis the efficiency of the cut on the weight in the ac-
ceptance calculation (see section 2.2).
We adopted the Hand convention [26] for the definition
of the virtual photon flux ΓV:
ΓV(Q2,xB,E) =
α
8π
Q2
pE2
m2
1 − xB
x3
B
1
1 − ǫ
(4)
with
ǫ =
1
1 + 2Q2+(E−E′)2
4EE′−Q2
(5)
and α ≈
stant.
Fig. 7 shows the total reduced cross section σγ∗p→pπ+π−
as a function of Q2for constant W bins compared with
the world’s data [18,19,27].
Relatively good agreement between the various exper-
iments can be seen. It is important to realize that what
is plotted is the unseparated cross section, i.e. a linear
combination of the transverse (σT) and longitudinal (σL)
cross sections : σ = σT+ ǫσL. This means that, due to ǫ
(eq. 5), there is a dependence on the beam energy in this
observable. Since the CORNELL data have been taken
with an 11.5 GeV electron beam energy [19], the DESY
data with a 7.2 GeV electron beam energy [18] and the
previous CLAS data with a 4.2 GeV beam [27], the data
sets, although at approximatively equivalent Q2and W
values, are not directly comparable and are not expected
1
137the standard electromagnetic coupling con-
to fully match each other. We will come back to this issue
in section 2.5 when we are comparing the ρ0cross sections.
The next step is to extract the γ∗p → pρ0cross section
from the γ∗p → pπ+π−cross section, which requires a
dedicated fitting procedure.
2.4 Fitting procedure for the γ∗p → pρ0cross section
Fig. 8 shows the acceptance-weighted Mπ+π− spectra for
all our (Q2,xB) bins. The ρ0peak (along with the f0(980)
and f2(1270) peaks clearly visible in some (Q2,xB) bins)
sits on top of a π+π−continuum background (see also
fig. 4 where all data have been integrated).
This background can be decomposed, in a first approxi-
mation, into the non-resonant ep → e′pπ+π−phase space
and the exclusive electroproduction of a pion and a nu-
cleon resonance, the latter decaying into a pion and a nu-
cleon, such as ep → e′π−∆++֒→ pπ+. Evidence for this
can be seen in figs. 9 and 10, which show for all our (Q2,
xB) bins the acceptance-corrected pπ−and pπ+invariant-
mass spectra where structures are clearly seen. Most of
these nucleon resonances (N∗) are rather well known, the
most prominent being the ∆0,++(1232), the D13(1520)
and the F15(1680). However, their production amplitudes
with an associated pion (i.e. ep → e′πN∗֒→ pπ) are
mostly unknown.
At low energies (W <1.8 GeV) where very few N∗
can be produced, a phenomenological model has been de-
veloped [28,27] based on an effective Lagrangian where a
few N∗’s are superposed along with the production of the
ρ0. Such a model could be a strong constraint and guide
to extract the ρ0cross section from all the other mecha-
nisms. However, at the present higher energies, numerous
new higher mass N∗’s appear as shown by the spectra of
figs. 9 and 10. For theoretical calculations, interference ef-
fects between all these channels are virtually impossible to
control and drastically complicate the analysis. Therefore
this approach cannot be pursued in our case.
At present, it is unrealistic to describe simultaneously
the π+π−, pπ−and pπ+invariant mass spectra over our
entire phase space because there are too many structures
varying independently with (Q2, xB) in each invariant
mass distribution.
Therefore, since a complete description of the di-pion
mass spectra is not available, we have adopted an em-
pirical description of the data using non-interfering con-
tributions that, together with a model for the ρ0shape,
reproduce the π+π−invariant mass spectrum.
The ρ0peak is broad and the strength of the non-
resonant π+π−background under it is quite significant,
and, even more importantly, its nature is unknown. There-
fore we must carry out a non-trivial and model-dependent
fitting procedure in order to extract the γ∗p → pρ0cross
section.
The procedure we have followed consisted of fitting
only the Mπ+π− distributions for each (Q2, xB) bin in
the case of the integrated cross sections and for each (Q2,
xB, X) bin, where X can be t, Φ, cos(θHS) or φHS, in
Page 8
7
b)
µ
) (
-π
+
π
p
→
p
*
γ (
σ 1
10
1.80 < W (GeV) < 2.00
)
2
(GeV
2
Q
123456
b)
µ
) (
-π
+
π
p
→
p
*
γ (
σ 1
10
2.40 < W (GeV) < 2.60
2.00 < W (GeV) < 2.20
)
2
(GeV
2
Q
123456
2.60 < W (GeV) < 2.80
2.20 < W (GeV) < 2.40
CORNELL
DESY
CLAS (4.2 GeV)
CLAS (5.754 GeV)
Fig. 7. Reduced cross sections γ∗p → pπ+π−as a function of Q2for constant W bins, in units of µbarn from the current
analysis. Also shown are earlier data from CLAS with a 4.2 GeV beam energy [27] as well as the data from DESY [18] and
Cornell [19] with, respectively, a 7.2 GeV and 11.5 GeV beam energy.
the case of the differential cross sections. It was possi-
ble to fit the Mπ+π− spectra with five contributions: three
Breit-Wigner shapes to describe the three evident mesonic
π+π−resonant structures of the ρ0(770), f0(980) and f2(1270),
where the masses in MeV indicated in parentheses are
the values given by the Particle Data Group (PDG) [23],
and two smoothed histograms that are the Mπ+π− pro-
jections of the reactions ep → e′π−∆++֒→ pπ+and of
the non-resonant continuum ep → e′pπ+π−. These two
latter spectra are calculated by our aforementioned event
generator [24]. We now detail these five contributions and
explain why they are necessary (and sufficient).
We first discuss the contribution of the ρ0(770) and
the way to model it. It is well known that simple sym-
metric Breit-Wigner line shapes which are, to first order,
used to describe resonances, are too naive to reproduce
the ρ shape, because of, among other aspects, interference
effects with the non-resonant π+π−continuum. Several
methods can be found in the literature for treating the ρ0
shape (see for instance ref. [29] for such a discussion). The
procedure we adopted was the following:
– Introduction of an energy-dependent width in order
to take into account that the ρ0is an unstable spin-
1 particle that decays into two spin-0 particles; it is
also called a p-wave Breit-Wigner [30]. This modified
Breit-Wigner reads:
BWρ(Mπ+π−) =
MρΓ(Mπ+π−)
π+π−)2+ M2
(M2
ρ− M2
ρΓ2
ρ(Mπ+π−)
(6)
with the energy-dependent width:
Γρ(Mπ+π−) = Γρ
?q
qρ
?2l+1
Mρ
Mπ+π−,(7)
where l = 1 for a p-wave Breit-Wigner, q is the momen-
tum of the decay pion in the ρ0center-of-mass frame
and qρis equal to q for Mπ+π− = Mρ:
q =
?
M2
π+π−− 4M2
2
π
, qρ=
?
M2
ρ− 4M2
2
π
. (8)
– Ross and Stodolsky [31] and S¨ oding [32] have shown
that the interferences between the broad ρ0peak and
the important non-resonant π+π−contribution under-
neath leads to a skewing of the Breit-Wigner. Accord-
ing to Ross-Stodolsky, one way to take account of this
effect is to introduce a correction term that consists of
multiplying the Breit-Wigner formula by an empirical
factor that shifts the centroid of the Mπ+π− distribu-
tion:
BWsk.
ρ (Mπ+π−) = BWρ(Mπ+π−)
?
Mρ
Mπ+π−
?nskew
. (9)
Page 9
8
0 0.4 0.8 1.2 1.6 2 0.4 0.8 1.2 1.6 2 0.4 0.8 1.2 1.6 2
0.16 0.22
Fig. 8. Acceptance-corrected MX[e′pX] (in GeV) missing mass spectra for all our (Q2, xB) bins. The three red lines are located
at MX = 0.770, 0.980, 1.275 GeV corresponding to the three well-known resonant states in the (π+π−) system.
0.28 0.34 0.40 0.46 0.52 0.58 0.64 0.70
1.60
1.90
2.20
2.50
2.80
3.10
3.60
4.10
4.60
5.10
5.60
B
x
)
2
(GeV
2
Q
where nskew is the “skewing” parameter. Although
Ross-Stodolsky have predicted the value of nskew to
be 4, it is often a parameter that is fitted to the data
since so little is known concerning the interference be-
tween the ρ0signal and the π+π−continuum.
As is evident in fig. 8, in addition to the ρ0(770), there
are two well-known resonant structures in the π+π−sys-
tem: the f0(980) and f2(1270). Due to the large widths
of these mesonic resonances (40 to 100 MeV [23] for the
f0(980) and ≈ 180 MeV [23] for the f2(1270)), it is clearly
necessary to include them in our Mπ+π− fit because their
contribution can extend into the ρ0region, which is itself
also broad. We also used the formulas of eqs. 6-9 for these
two other mesonic resonant states f0(980) and f2(1270)
with appropriate parameters Mf0, Γf0, Mf2, Γf2and took
into account their l = 0 and l = 2 nature.
In principle, the only free parameter to vary in eq. 9
should be nskew. However, we have also allowed the cen-
tral masses and widths of the three mesons to vary in a
very limited range of at most 20 MeV from their nominal
values (see table 2). The motivation for this is that, be-
sides the largely unknown interference effects between the
meson and the π+π−continuum, several other effects can
shift or distort the meson shapes: radiative corrections,
binning, acceptance corrections, imprecise values for the
central masses and widths of some of these mesons, etc.
Finally, besides the ρ0, f0(980) and f2(1270) mesons,
the two other contributions entering our fit are: the Mπ+π−
projections of the reactions ep → e′π−∆++֒→ pπ+and
the non-resonant continuum (phase space) ep → e′pπ+π−.
The shapes of these distributions are given by our event
generator. In particular, the ∆++has the shape of a stan-
dard Breit-Wigner in the pπ+distribution with a centroid
at 1.232 GeV and a width of 111 MeV [23]. As is obvious
from fig. 9, the ∆0contribution can be neglected.
In principle, of course, other processes contribute to
the Mπ+π− continuum, for instance all of the ep → eπ−N∗֒→
pπ+reactions, as already mentioned. As a test, we mod-
eled the pπ−and pπ+invariant mass distributions of figs. 9
and 10 by adding, at the cross section level, several Breit-
Wigners matching the structures seen in these figures and
identifying them with known N∗masses (and widths) that
can be found in the PDG. Like for the ∆++, we introduced
their contribution into our fit of the Mπ+π− distribution
using our event generator. The conclusion we reached was
twofold. Firstly, this procedure introduced a large number
of additional free parameters: for each of the extra N∗’s,
Page 10
9
0.7 1.11.5 1.9 2.3 2.7 1.1 1.5 1.9 2.3 2.7 1.1 1.5 1.9 2.3 2.7
0.16 0.22
Fig. 9. Acceptance-corrected Mpπ− (in GeV) invariant mass distributions for all our (Q2, xB) bins. The three red lines are
located at MX = 1.232, 1.520, 1.680 GeV corresponding to three well-known resonance regions in the (pπ−) system.
0.28 0.34 0.40 0.46 0.52 0.58 0.64 0.70
1.60
1.90
2.20
2.50
2.80
3.10
3.60
4.10
4.60
5.10
5.60
B
x
)
2
(GeV
2
Q
parameterPDG valuemin. valuemax. value
ρ0mass Mρ (MeV)
ρ0width Γρ (MeV)
f0 mass Mf0(MeV)
f0 width Γf0(MeV)
f2 mass Mf2(MeV)
f2 width Γf2(MeV)
≈ 770
≈ 150
≈ 980
40-100
≈ 1275
≈ 185
750
140
970
40
1260
170
790
170
990
120
1280
200
Table 2. Range of variations permitted for the parameters to be fitted in formula 9.
two parameters for the central mass and width to vary in
the approximate ranges given by the PDG and one more
for the weight/normalization. Secondly, we found that the
Mπ+π− projected shape of these high mass N∗’s was very
similar to the phase-space Mπ+π− distribution. In other
words, in a first approximation, the phase-space contri-
bution can reflect and absorb the high mass N∗’s. How-
ever, the ∆++contribution to the Mπ+π− distribution was
found to be sufficiently different from the phase space dis-
tribution to be kept as an individual contribution.
To summarize, each (Q2, xB) bin of fig. 8 was fit with
the following formula:
dN
dMπ+π−= BWρ(Mπ+π−)
+BWf0(Mπ+π−) + BWf2(Mπ+π−)
+M∆++π−(Mπ+π−) + Mpπ+π−(Mπ+π−)
(10)
It involves 14 parameters that are:
– 1) weight (normalization), 2) central mass, 3) width
and 4) nskewof ρ0;
– 5) weight (normalization), 6) central mass, 7) width
and 8) nskewof f0;
Page 11
10
1 1.4 1.8 2.2 2.6 3 1.4 1.8 2.2 2.6 3 1.4 1.8 2.2 2.6 3
0.16 0.22
Fig. 10. Acceptance-corrected Mpπ+ (in GeV) invariant mass distributions for all our (Q2, xB) bins. The three red lines are
located at MX = 1.232, 1.600, 1.900 GeV corresponding to the three well-known resonance regions in the (pπ+) system.
0.28 0.34 0.40 0.46 0.52 0.58 0.64 0.70
1.60
1.90
2.20
2.50
2.80
3.10
3.60
4.10
4.60
5.10
5.60
B
x
)
2
(GeV
2
Q
– 9) weight (normalization), 10) central mass, 11) width
and 12) nskewof f2;
– 13) weight (normalization) of Mπ+π− projection of the
ep → e′π−∆++֒→ pπ+process; and
– 14) weight (normalization) of ep → e′pπ+π−phase
space.
Fourteen parameters might appear a lot to fit only a
1-dimensional distribution. However, on the one hand, six
of these (the central mass and width of the ρ0, f0(980) and
f2(1270) mesons) are quite constrained and are allowed to
vary in a very limited range. On the other hand this simply
reflects the complexity and our lack of knowledge of the
ep → e′pπ+π−reaction, to which many unknown, inde-
pendent though interfering processes contribute; namely:
meson production ep → e′pM0֒→ π+π−, N∗production
ep → e′π−N∗++֒→ pπ+, ep → e′π+N∗0֒→ pπ−, non-
resonant ep → e′pπ+π−, etc.
Fig. 11 shows the result of our fits to the Mπ+π− distri-
butions, normalized in terms of the reduced cross sections
of eq. 3 for all of our (Q2, xB) bins. In a few cases, the
fits do not fully describe the data. For instance, for 0.46
< xB < 0.52, 3.10 < xB < 3.60, the data tend to show
a “structure” around Mπ+π−=0.9 GeV, i.e. between the
known ρ0and f0resonances, which cannot be reproduced
by our fit formula of eq. 10. We attribute this discrepancy
to interference effects not taken into account by our sim-
ple fit procedure. As discussed in more detail in the next
subsections, a systematic uncertainty of 25% is assigned
to this whole fit procedure which is meant to account,
among other aspects, for the inadequacies in the model.
On all the figures that are going to be presented from now
on, unless explicitely stated otherwise, all the error bars
associated to our data points will represent the quadratic
sum of the statistical and systematic errors.
2.5 Integrated ρ0cross section
We use the ρ0strength (green line) extracted from the
distributions shown in fig. 11 to calculate the cross section.
Fig. 12 shows the resulting reduced cross section σγ∗p→pρ0
compared with the world’s data presented as a function
of W for constant Q2bins. Fig. 13 shows the reduced
cross section σγ∗p→pρ0 compared with the world’s data
presented as a function of Q2for constant W bins.
With respect to the γ∗p → pπ+π−cross section that
we extracted in the previous section, there is an additional
Page 12
11
0
4
1
2
3
4
5
b)
µ
= 0.91 +/- 0.05 (
ρ
σ
< 0.22
2
B
0.16<x
< 1.90 1.60<Q
0
0.5
1
1.5
2
2.5
3
3.5
b)
µ
= 0.55 +/- 0.08 (
ρ
σ
< 0.28
< 2.50
B
2
0.22<x
2.20<Q
0.5
1
1.5
2
2.5
3
3.5
4
4.5
b)
µ
= 0.54 +/- 0.03 (
ρ
σ
< 0.34
< 2.50
B
2
0.28<x
2.20<Q
1
2
3
4
5
6
7
8
b)
µ
= 0.77 +/- 0.06 (
ρ
σ
< 0.40
< 2.50
B
2
0.34<x
2.20<Q
b/GeV)
µ
(
/dM
σ
d
0
6
0.5
1
1.5
2
2.5
b)
µ
= 0.30 +/- 0.02 (
ρ
σ
< 0.40
< 3.60
B
2
0.34<x
3.10<Q
0
1
2
3
4
5
b)
µ
= 0.60 +/- 0.05 (
ρ
σ
< 0.46
< 3.10
B
2
0.40<x
2.80<Q
2
4
6
8
10
b)
µ
= 1.12 +/- 0.12 (
ρ
σ
< 0.52
< 3.10
B
2
0.46<x
2.80<Q
0
0.5
1
1.5
2
2.5
b)
µ
= 0.35 +/- 0.02 (
ρ
σ
< 0.52
2
B
0.46<x
< 4.60 4.10<Q
[epX] (GeV)
X
M
00.20.40.60.811.21.4 1.61.82
0
0.5
1
1.5
2
2.5
3
3.5
4
b)
µ
= 0.49 +/- 0.04 (
ρ
σ
< 0.64
2
B
0.58<x
< 4.60 4.10<Q
0
1
2
3
4
5
6
7
8
b)
µ
= 1.01 +/- 0.07 (
ρ
σ
< 0.28
2
B
0.22<x
< 1.90 1.60<Q
2
4
6
8
10
12
14
16
b)
µ
= 1.96 +/- 0.47 (
ρ
σ
< 0.34
< 1.90
B
2
0.28<x
1.60<Q
0
5
0.5
1
1.5
2
2.5
3
b)
µ
= 0.41 +/- 0.02 (
ρ
σ
< 0.34
< 2.80
B
2
0.28<x
2.50<Q
1
2
3
4
b)
µ
= 0.60 +/- 0.04 (
ρ
σ
< 0.40
< 2.80
B
2
0.34<x
2.50<Q
b/GeV)
µ
(
/dM
σ
d
2
4
6
8
10
12
14
16
b)
µ
= 1.51 +/- 0.17 (
ρ
σ
< 0.46
< 2.50
B
2
0.40<x
2.20<Q
0
0.5
1
1.5
2
2.5
3
3.5
b)
µ
= 0.37 +/- 0.02 (
ρ
σ
< 0.46
< 3.60
B
2
0.40<x
3.10<Q
1
2
3
4
5
b)
µ
= 0.52 +/- 0.04 (
ρ
σ
< 0.52
< 3.60
B
2
0.46<x
3.10<Q
1
2
3
4
5
b)
µ
= 0.66 +/- 0.06 (
ρ
σ
< 0.58
2
B
0.52<x
< 4.10 3.60<Q
[epX] (GeV)
X
M
00.20.40.60.811.21.41.61.82
0
0.5
1
1.5
2
2.5
3
b)
µ
= 0.34 +/- 0.03 (
ρ
σ
< 0.64
2
B
0.58<x
< 5.10 4.60<Q
0
7
1
2
3
4
b)
µ
= 0.64 +/- 0.04 (
ρ
σ
< 0.28
2
B
0.22<x
< 2.20 1.90<Q
0
1
2
3
4
5
6
b)
µ
= 0.83 +/- 0.06 (
ρ
σ
< 0.34
< 2.20
B
2
0.28<x
1.90<Q
2
4
6
8
10
12
14
b)
µ
= 1.44 +/- 0.15 (
ρ
σ
< 0.40
< 2.20
B
2
0.34<x
1.90<Q
0
9
0.5
1
1.5
2
2.5
3
3.5
b)
µ
= 0.40 +/- 0.02 (
ρ
σ
< 0.40
< 3.10
B
2
0.34<x
2.80<Q
b/GeV)
µ
(
/dM
σ
d
1
2
3
4
5
6
7
8
b)
µ
= 0.86 +/- 0.07 (
ρ
σ
< 0.46
< 2.80
B
2
0.40<x
2.50<Q
0
0.5
1
1.5
2
2.5
b)
µ
= 0.31 +/- 0.02 (
ρ
σ
< 0.46
< 4.10
B
2
0.40<x
3.60<Q
0
0.5
1
1.5
2
2.5
3
3.5
b)
µ
= 0.32 +/- 0.02 (
ρ
σ
< 0.52
< 4.10
B
2
0.46<x
3.60<Q
0
2
0.5
1
1.5
2
2.5
3
3.5
b)
µ
= 0.39 +/- 0.03 (
ρ
σ
< 0.58
2
B
0.52<x
< 4.60 4.10<Q
[epX] (GeV)
X
M
00.20.40.60.811.21.41.6 1.82
0
0.5
1
1.5
b)
µ
= 0.29 +/- 0.03 (
ρ
σ
< 0.70
B
2
0.64<x
< 5.60 5.10<Q
Fig. 11. Acceptance-corrected MX[e′pX] missing mass distributions , showing our fits. In red: total fit result; in green: ρ0
contribution; in blue: f0 contribution; in purple: f2 contribution and in dotted green: π+π−continuum, which is the sum of the
ep → e′π−∆++֒→ pπ+projections on Mπ+π− and of the phase space ep → e′pπ+π−contributions. The error bars on the data
points are purely statistical. The uncertainties on the cross sections given by the fit is also purely statistical.
Page 13
12
b)
µ
) (
0
ρ
p
→
p
*
γ (
σ
-1
10
1
) < 1.90
2
(GeV
2
1.60 < Q
b)
µ
) (
0
ρ
p
→
p
*
γ (
σ
-1
10
1
) < 3.10
2
(GeV
2
2.80 < Q
W (GeV)10
b)
µ
) (
0
ρ
p
→
p
*
γ (
σ
-1
10
1
) < 5.10
2
(GeV
2
4.60 < Q
) < 2.20
2
(GeV
2
1.90 < Q
) < 3.60
2
(GeV
2
3.10 < Q
W (GeV)10
) < 5.60
2
(GeV
2
5.10 < Q
) < 2.50
2
(GeV
2
2.20 < Q
) < 4.10
2
(GeV
2
3.60 < Q
CLAS (5.754 GeV)
CLAS (4.2 GeV)
CORNELL
HERMES
E665
) < 2.80
2
(GeV
2
2.50 < Q
) < 4.60
2
(GeV
2
4.10 < Q
Fig. 12. Reduced cross sections γ∗p → pρ0as a function of W for constant Q2bins, in units of µbarn. The error bars of
the CLAS data result from the quadratic sum of the statistical and systematic uncertainties. The horizontal error bars of the
Cornell data indicate their W range. The 4.2 GeV CLAS, CORNELL, HERMES and E665 data are respectively from refs. [21],
[19], [20] and [33].
source of systematic uncertainty for the σγ∗p→pρ0 cross
section that arises from the subtraction procedure de-
scribed in the previous section. This contribution is quite
difficult to evaluate. It is not so much the quality of the
fit in fig. 11 that matters; we have varied the minimum
and maximum limits imposed on the parameters in ta-
ble 2 and found that the results of the fits are very stable.
The uncertainty arises more from the reliability and confi-
dence we can assign to the modeling that we have adopted
for the ρ0, f0(980) and f2(1270) mesons with the skewed
Breit-Wigners and for the non-resonant continuum π+π−
distribution. We have tried several shapes for this latter
continuum. As mentioned in the previous subsection, we
introduced N∗states other than the ∆++. Ultimately, we
ended up finding the fits to be stable at the ≈ 20 % level
on average. Overall, we cannot take account of any inter-
ference effects between the ρ0peak and the non-resonant
π+π−continuum. This uncertainty is of a theoretical na-
ture, and in the absence of sufficient guidance at present,
we have decided to assign a relatively conservative 25%
systematic uncertainty to our extracted ρ0yields. We will
find some relative justification for this estimation in the
next section when we study the differential distributions,
in particular those of t and cosθHS.
Recently, a partial wave analysis of data of exclusive
π+π−photoproduction on the proton from CLAS, has
been carried out [34]. This study showed that the ρ0cross
sections resulting from this sophisticated method were
consistent with those resulting from simple fits of the two-
pion invariant mass as we have just described, to a level
much lower than 25%. Although no such partial wave anal-
ysis has been done to the present electroproduction data,
this photoproduction comparison gives relative confidence
that the 25% systematic uncertainty that we presently as-
sign, is rather conservative.
Coming back to fig. 12, we find that our data are in
general agreement with the other world’s data in regions
of overlap. In the upper left plot of fig. 12 (1.60< Q2<1.90
GeV2), our CLAS (5.754 GeV) data seem to overestimate
the CLAS (4.2 GeV) results, but this can certainly be
attributed to a kinematic effect due to the different beam
energies of the two data sets. Indeed, we are comparing
the total reduced cross sections : σ = σT+ǫσL. However,
at W=2.1 GeV and Q2=1.7 GeV2, ǫ=0.53 for a 4.2 GeV
beam energy but ǫ=0.77 for a 5.754 GeV beam energy.
This can readily explain the lower CLAS (4.2 GeV) data
with respect to the CLAS (5.754 GeV) data.
On this general account, we could have expected that
the Cornell data stand to some extent above the CLAS
(5.754 GeV) data since they have been obtained with an
11.5 GeV beam energy. This is not the case which might
indicate a slight incompatibility between the Cornell and
CLAS data. This point, as well as the compatibility of
the CLAS (4.2 GeV) and CLAS (5.754 GeV) data, will be
Page 14
13
b)
µ
) (
0
ρ
p
→
p
*
γ (
σ
1
10
1.80 < W (GeV) < 2.00
)
2
(GeV
2
Q
123456
b)
µ
) (
0
ρ
p
→
p
*
γ (
σ
1
10
2.40 < W (GeV) < 2.60
2.00 < W (GeV) < 2.20
)
2
(GeV
2
Q
123456
2.60 < W (GeV) < 2.80
2.20 < W (GeV) < 2.40
CLAS (5.754 GeV)
CLAS (4.2 GeV)
CORNELL
DESY
Fig. 13. Reduced cross sections γ∗p → pρ0as a function of Q2for constant W bins, in units of µbarn. The 4.2 GeV CLAS,
CORNELL and DESY data are respectively from refs. [21], [19] and [18].
confirmed in section 3.3 where we compare the separated
longitudinal and transverse cross sections for which this
beam energy kinematical effect is removed.
2.6 Differential ρ0cross sections
After having obtained the total ρ0cross section, we now
extract the differential cross sections in t, Φ, cos(θHS) and
φHS.
Since the data are now binned in an additional vari-
able, each bin has fewer statistics, not only for the real
data but also for the MC data that are necessary to cal-
culate the acceptance correction. Bins for which ηwis less
than 0.6 were rejected, where, we recall, ηwis the correc-
tion factor in the acceptance calculation that was intro-
duced in section 2.2. This explains why some holes occur
at several instances, in particular in the Φ and cos(θHS)
distributions.
We start by extracting the dσ/dt cross section. Defin-
ing t′as t − t0, where t0 is the maximum t value kine-
matically allowed for a given (Q2, xB) bin, we divided the
data into 6 bins for 0 < −t′< 1.5 GeV2and 3 bins for
1.5 < −t′< 3.9 GeV2. For each of the (t, Q2, xB) bins, we
extracted the ρ0signal from the (π+,π−) invariant mass
spectra using the fitting procedure previously described.
Fig. 14 shows dσ/dt for all our (Q2,xB) bins as a func-
tion of t. The general feature of these distributions is that
they are of a diffractive type, i.e. proportional to ebt. The
values of the slope b are between 0 and 3 GeV−2. They
are plotted as a function of W in fig. 15 along with the
world’s data. For the sake of clarity, only the world’s data
for Q2> 1.5 GeV2are displayed. For Q2< 1.5 GeV2,
the data show the same trend but with more dispersion.
The data exhibit a rise with W until they reach a plateau
around W= 6 GeV at a b value of ≈ 7 GeV−2. The high-
energy experiments (H1 and ZEUS) have shown that this
saturating value tends to decrease with Q2, which is il-
lustrated by the H1 points in fig. 15 that correspond to
different Q2values.
By integrating the dσ/dt cross section, we are able to
recover at the ≈ 20% level the integrated cross sections
that were presented in section 2.5. The agreement is not
perfect since for the integrated cross section one fits a sin-
gle full statistics Mπ+π− spectrum, whereas for the differ-
ential cross section, one fits several lower statistics Mπ+π−
spectra, that are then summed. This relatively good agree-
ment serves, among other arguments, to justify the 25%
systematic uncertainties that we have applied in the non-
resonant π+π−background subtraction procedure.
We note that the integrated cross sections that we have
presented so far (and which will be presented in the re-
mainder of this article) have been summed over only the
domain where we had data and acceptance. We have not
extrapolated our cross sections beyond the t domain ac-
cessed in this experiment, which we deem unsafe and very
model-dependent. Fig. 14 indicates that this might under-
estimate some integrated cross sections for a (very limited)
Page 15
14
-t
0
0.16
123
/dt
L
σ
d0.1
1
b = 2.83+/- 0.44
b = 3.08+/- 0.40
b = 2.47+/- 0.40
-t
123
b = 2.63+/- 0.44
b = 2.08+/- 0.44
b = 2.28+/- 0.45
b = 2.21+/- 0.43
-t
123
b = 1.48+/- 0.40
b = 1.69+/- 0.38
b = 1.63+/- 0.35
b = 1.71+/- 0.38
b = 1.70+/- 0.36
b = 1.51+/- 0.38
b = 1.06+/- 0.37
b = 1.34+/- 0.38
b = 1.46+/- 0.32
b = 1.25+/- 0.36
b = 0.78+/- 0.31
b = 1.07+/- 0.30
b = 0.97+/- 0.29
b = 1.14+/- 0.32
b = 0.32+/- 0.28
b = 0.56+/- 0.29
b = 0.46+/- 0.32
b = 0.86+/- 0.35
b = -0.25+/- 0.34
b = 0.58+/- 0.48
0.22 0.28 0.34 0.40 0.46 0.52 0.58 0.64 0.70
1.90
2.20
2.50
2.80
3.10
3.60
4.10
4.60
5.10
5.60
B
x
)
2
(GeV
2
Q
Fig. 14. Cross section dσ/dt (in µb/GeV2) for all bins in (Q2,xB) as a function of −t (in GeV2). The red line shows the fit to
the function ebtover the limited range 0 < −t′< 1.5 GeV2.
number of (Q2,xB) bins at large xB, where the t depen-
dence appears rather flat.
We proceeded in the same way to extract dσ/dΦ. All of
our (Q2,xB) bins are shown in fig. 16. Several of the bins
near Φ=180oare empty or have large error bars because
of very low acceptance in CLAS in this region.
These distributions were fitted with the expected Φ
dependence for single meson electroproduction:
dσ
dΦ=
1
2π( σT+ ǫσL
+ǫcos2Φ σTT+
?
2ǫ(1 + ǫ)cosΦ σTL)
(11)
from which we could extract the interference terms σTT
and σTL. The curves in fig. 16 show the corresponding
fits, and σT+ ǫσL, σTT and σTLare displayed in fig. 17.
If helicity is conserved in the s channel (SCHC), the in-
terference terms σTT and σTLwould vanish. Most of our
extracted values are consistent with 0 within (large) er-
ror bars, although one clearly cannot make strong claims
about SCHC at this point.
Turning to the pion decay angles of the ρ0, θHS and
φHS, they are expected to follow the general and model
independent distribution [40]:
W(Φ,cosθHS,ϕHS) =
3
4π
−√2Rer04
−ǫcos2Φ(r1
−√2Rer1
−ǫsin2Φ(√2Imr2
+Imr2
?1
2(1 − r04
00) +1
2(3r04
00− 1)cos2θHS
?
10sin2θHScosϕHS− r04
11sin2θHS+ r1
10sin2θHScosϕHS− r1
10sin2θHSsinϕHS
1−1sin2θHSsin2ϕHS)
2ǫ(1 + ǫ)cosΦ(r5
−√2Rer5
+
?
+Imr6
1−1sin2θHScos2ϕHS
00cos2θHS
1−1sin2θHScos2ϕHS)
+
?
11sin2θHS+ r5
00cos2θHS
1−1sin2θHScos2ϕHS)
10sin2θHScosϕHS− r5
2ǫ(1 + ǫ)sinΦ(√2Imr6
1−1sin2θHSsin2ϕHS) ],
10sin2θHSsinϕHS
(12)
where:
r04
ij=ρ0
ij+ ǫRρ4
1 + ǫR
ρα
ij
1 + ǫR
ij
rα
ij=
α = 1,2
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