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

00.20.40.60.811.21.4

)

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

-0.100.10.20.30.4

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

00.20.40.60.811.21.4

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.

MissingmassMX[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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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

00.1 0.20.30.40.50.60.70.8

)

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.

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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. value max. 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

00.20.40.6 0.811.21.41.61.82

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

0 0.20.40.60.811.21.41.61.82

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

00.2 0.40.60.811.21.41.61.82

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