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Photoproduction of π+π− meson pairs on the proton

M. Battaglieri,1 R. De Vita,1 A. P. Szczepaniak,2 K. P. Adhikari,35 M.J. Amaryan,35 M. Anghinolfi,1

H. Baghdasaryan,45 I. Bedlinskiy,22 M. Bellis,7 L. Bibrzycki,29 A.S. Biselli,13, 36 C. Bookwalter,15

D. Branford,12 W.J. Briscoe,16 V.D. Burkert,42 S.L. Careccia,35 D.S. Carman,42 E. Clinton,28 P.L. Cole,18

P. Collins,4 V. Crede,15 D. Dale,18 A. D’Angelo,20, 38 A. Daniel,34 N. Dashyan,47 E. De Sanctis,19 A. Deur,42

S. Dhamija,14 C. Djalali,41 G.E. Dodge,35 D. Doughty,10, 42 V. Drozdov,1 H. Egiyan,30, 42 P. Eugenio,15

G. Fedotov,40 S. Fegan,17 A. Fradi,21 M.Y. Gabrielyan,14 L. Gan,32 M. Garcon,9 A. Gasparian,33 G.P. Gilfoyle,37

K.L. Giovanetti,24 F.X. Girod,9, ∗ O. Glamazdin,26 J. Goett,36 J.T. Goetz,5 W. Gohn,11 E. Golovatch,40, 1

R.W. Gothe,41 K.A. Griffioen,46 M. Guidal,21 L. Guo,42, † K. Hafidi,3 H. Hakobyan,44, 47 C. Hanretty,15

N. Hassall,17 K. Hicks,34 M. Holtrop,30 C.E. Hyde,35 Y. Ilieva,41, 16 D.G. Ireland,17 E.L. Isupov,40 J.R. Johnstone,17

K. Joo,11 D. Keller,34 M. Khandaker,31 P. Khetarpal,36 W. Kim,27 A. Klein,35 F.J. Klein,8 M. Kossov,22

A. Kubarovsky,35 V. Kubarovsky,42 S.V. Kuleshov,44, 22 V. Kuznetsov,27 J.M. Laget,42, 9 L. Lesniak,29

K. Livingston,17 H.Y. Lu,41 M. Mayer,35 M.E. McCracken,7 B. McKinnon,17 C.A. Meyer,7 K. Mikhailov,22

T Mineeva,11 M. Mirazita,19 V. Mochalov,23 V. Mokeev,40, 42 K. Moriya,7 E. Munevar,16 P. Nadel-Turonski,8

I. Nakagawa,39 C.S. Nepali,35 S. Niccolai,21 I. Niculescu,24 M.R. Niroula,35 M. Osipenko,1, 40 A.I. Ostrovidov,15

K. Park,41, 27, ∗ S. Park,15 M. Paris,16, 42 E. Pasyuk,4 S.Anefalos Pereira,19 S. Pisano,21 N. Pivnyuk,22

O. Pogorelko,22 S. Pozdniakov,22 J.W. Price,6 Y. Prok,45, ‡ D. Protopopescu,17 B.A. Raue,14, 42 G. Ricco,1

M. Ripani,1 B.G. Ritchie,4 G. Rosner,17 P. Rossi,19 F. Sabatie,9 M.S. Saini,15 C. Salgado,31 D. Schott,14

R.A. Schumacher,7 H. Seraydaryan,35 Y.G. Sharabian,42 D.I. Sober,8 D. Sokhan,12 A. Stavinsky,22 S. Stepanyan,42

S. S. Stepanyan,27 P. Stoler,36 I.I. Strakovsky,16 S. Strauch,41, 16 M. Taiuti,1 D.J. Tedeschi,41 A. Teymurazyan,25

S. Tkachenko,35 M. Ungaro,11, 36 M.F. Vineyard,43 A.V. Vlassov,22 D.P. Watts,17, § L.B. Weinstein,35

D.P. Weygand,42 M. Williams,7 E. Wolin,42 M.H. Wood,41 L. Zana,30 J. Zhang,35 B. Zhao,11, ¶ and Z.W. Zhao41

(The CLAS Collaboration)1Istituto Nazionale di Fisica Nucleare, Sezione di Genova, 16146 Genova, Italy

2Physics Department and Nuclear Theory CenterIndiana University, Bloomington, Indiana 47405

3Argonne National Laboratory, Argonne, Illinois 604394Arizona State University, Tempe, Arizona 85287-1504

5University of California at Los Angeles, Los Angeles, California 90095-15476California State University, Dominguez Hills, Carson, CA 90747

7Carnegie Mellon University, Pittsburgh, Pennsylvania 152138Catholic University of America, Washington, D.C. 20064

9CEA, Centre de Saclay, Irfu/Service de Physique Nucleaire, 91191 Gif-sur-Yvette, France10Christopher Newport University, Newport News, Virginia 23606

11University of Connecticut, Storrs, Connecticut 0626912Edinburgh University, Edinburgh EH9 3JZ, United Kingdom

13Fairfield University, Fairfield CT 0682414Florida International University, Miami, Florida 33199

15Florida State University, Tallahassee, Florida 3230616The George Washington University, Washington, DC 2005217University of Glasgow, Glasgow G12 8QQ, United Kingdom

18Idaho State University, Pocatello, Idaho 8320919INFN, Laboratori Nazionali di Frascati, 00044 Frascati, Italy

20INFN, Sezione di Roma Tor Vergata, 00133 Rome, Italy21Institut de Physique Nucleaire ORSAY, Orsay, France

22Institute of Theoretical and Experimental Physics, Moscow, 117259, Russia23Institute for High Energy Physics, Protvino, 142281, Russia

24James Madison University, Harrisonburg, Virginia 2280725University of Kentucky, Lexington, Kentucky 40506

26Kharkov Institute of Physics and Technology, Kharkov 61108, Ukraine27Kyungpook National University, Daegu 702-701, Republic of Korea

28University of Massachusetts, Amherst, Massachusetts 0100329Henryk Niewodniczanski Institute of Nuclear Physics PAN, 31-342 Krakow, Poland

30University of New Hampshire, Durham, New Hampshire 03824-356831Norfolk State University, Norfolk, Virginia 23504

32University of North Carolina, Wilmington, North Carolina 2840333North Carolina Agricultural and Technical State University, Greensboro, North Carolina 27455

34Ohio University, Athens, Ohio 45701

2

35Old Dominion University, Norfolk, Virginia 2352936Rensselaer Polytechnic Institute, Troy, New York 12180-3590

37University of Richmond, Richmond, Virginia 2317338Universita’ di Roma Tor Vergata, 00133 Rome Italy

39The Institute of Physical and Chemical Research, RIKEN, Wako, Saitama 351-0198, Japan40Skobeltsyn Nuclear Physics Institute, Skobeltsyn Nuclear Physics Institute, 119899 Moscow, Russia

41University of South Carolina, Columbia, South Carolina 2920842Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606

43Union College, Schenectady, New York 1230844Universidad Tecnica Federico Santa Marıa, Casilla 110-V Valparaıso, Chile

45University of Virginia, Charlottesville, Virginia 2290146College of William and Mary, Williamsburg, Virginia 23187-8795

47Yerevan Physics Institute, 375036 Yerevan, Armenia(Dated: July 6, 2009)

The exclusive reaction γp → pπ+π− was studied in the photon energy range 3.0 - 3.8 GeV andmomentum transfer range 0.4 < −t < 1.0 GeV2. Data were collected with the CLAS detector atthe Thomas Jefferson National Accelerator Facility. In this kinematic range the integrated lumi-nosity was about 20 pb−1. The reaction was isolated by detecting the π+ and proton in CLAS,and reconstructing the π− via the missing-mass technique. Moments of the di-pion decay angu-lar distributions were derived from the experimental data. Differential cross sections for the S,P , and D-waves in the Mπ+π− mass range 0.4 − 1.4 GeV were derived performing a partial waveexpansion of the extracted moments. Besides the dominant contribution of the ρ(770) meson inthe P -wave, evidence for the f0(980) and the f2(1270) mesons was found in the S and D-waves,respectively. The differential production cross sections dσ/dt for individual waves in the mass rangeof the above-mentioned mesons were extracted. This is the first time the f0(980) has been measuredin a photoproduction experiment.

I. INTRODUCTION

The two pion channel offers the possibility of investi-gating various aspects of the meson resonance spectrum.It couples to the scalar-isoscalar channel that containsthe σ, f0(980) and possibly a few more resonances withmasses below 2 GeV. It is the main decay mode of thelowest isoscalar-tensor f2(1270) resonance and it is theonly decay mode of the isovector-vector resonance, theρ(770). Among all these, the ρ-meson is by far the mostprominent and most extensively studied, both from thepoint of view of its production mechanisms and its inter-nal properties. Nowadays the other resonances too aresubjects of extensive theoretical and experimental inves-tigation. The σ meson is now established with pole massand width determined with good accuracy [1–3]. How-ever, its microscopic structure seems to be quite differentfrom that of the ρ and it is the subject of theoretical de-bate [4]. The f0(980) is even a more enigmatic state: itsexperimental determination is complicated by its proxim-

∗Current address:Thomas Jefferson National Accelerator Facility,

Newport News, Virginia 23606†Current address:Los Alamos National Laborotory, Los Alamos,

New Mexico 87545‡Current address:Christopher Newport University, Newport News,

Virginia 23606§Current address:Edinburgh University, Edinburgh EH9 3JZ,

United Kingdom¶Current address:College of William and Mary, Williamsburg, Vir-

ginia 23187-8795

ity to the KK threshold, and its QCD nature still awaitsan explanation [5]. Finally, the f2(1270) has been repre-sented so far as a Breit-Wigner resonance [2] and appearsto fit well into the quark model spectrum [6].

In this paper we focus on the scalar sector, using theρ meson as a benchmark for the analysis procedure. TheKK channel from the same data set is currently beinganalyzed and in the near future a coupled-channels anal-ysis will provide further constraints on the extraction ofthe meson properties.

For a long time most of our knowledge on the scalarmeson spectrum was obtained from hadron-induced re-actions, γγ collisions and studying the decays of vari-ous mesons, e.g. φ, J/Ψ, D and B. Very few studieswere attempted with electromagnetic probes, in particu-lar real photons, since their production cross sections arerelatively small compared to the dominant productionof vector mesons. On one hand, through vector mesondominance, the photon can be effectively described as avirtual vector meson. On the other hand, quark-hadronduality and the point-like-nature of the photon couplingmake it possible to describe photo-hadron interactions atthe QCD level. Recently, high-intensity and high-qualitytagged-photon beams, as the one available at JLab, haveopened a new window into this field.

In photoproduction processes, information about theS-wave strength can be extracted by performing a par-tial wave analysis. Angular distributions of photopro-duced mesons and related observables, such as the mo-ments of the angular distributions and the density matrixelements, are the most effective tools to look for inter-ference patterns. An interference between the S-wave

3

and the dominant P -wave was discovered in the momentanalysis of K+K− photoproduction on hydrogen, ana-lyzing the data collected in the experiments performedat DESY [7] and Daresbury [8]. In two-pion productionexperiments, such as reported in Refs. [9–11], momentsand density matrix elements were used to analyze theproperties of helicity amplitudes describing the photo-production process. Unfortunately, only the dominantspin-1 partial wave of the π+π− pair was taken into ac-count. No attempt to obtain information about the S-wave amplitude was made. More recently, the HERMESexperiment at DESY [12] investigated the interference ofthe P -wave in the π+π− system with the S and D-wavesin the π+π− electroproduction process, and showed thatsuch interference effects are measurable. The large pho-ton virtuality Q2 >3 GeV2 is, however, a crucial factorthat distinguishes this analysis from the photoproductionanalysis [9, 10].

Theoretical models for π+π− photoproduction havebeen investigated in a series of articles. A very success-ful approach is the one by Soding [13] and its numerousmodifications [14–17]. These models were able to de-scribe the shift of the maximum of the π+π− effectivemass distribution with respect to the nominal ρ massand the asymmetric shape observed in SLAC [9, 10] andDESY [11, 18] data. These properties are attributed tothe interference of the dominating diffractive ρ mesonproduction, with its subsequent decay into π+π−, withthe amplitudes corresponding to Drell-type diagrams inwhich the photon dissociates into π+ and π−, and oneof the pions is elastically scattered off the proton. Morerecently, Gomez Tejedor and Oset [19] applied an effec-tive Lagrangian to construct the photoproduction am-plitudes. Their approach is limited to photon energiesbelow 800 MeV and effective masses Mππ smaller than1 GeV. A two-stage approach for the π+π− S-wave pho-toproduction was proposed in the model of Ref. [20].First, a set of Born amplitudes, corresponding to pho-toproduction of π+π−, π0π0, K+K− and K0K0 pairsis calculated. Then the photoproduced meson pairs aresubject to final-state interactions resulting in the π+π−

system [21–24]. The coupled-channels calculations wereseparately performed for all isospin I components of thetransition matrix. Thus the S-wave amplitudes in thatmodel account for the existence of the isoscalar σ, f0(980)and f0(1500), and the isovector a0(980) and a0(1450) res-onances. The coupling of the KK isovector channel withthe πη amplitude is described in Ref. [25].

All theoretical approaches described above do not con-sider explicitly the s-channel production of baryon res-onances contributing to the pππ final state. Data fromRefs. [9, 11, 18], as well as from more recent experimen-tal studies [26], indicate that the contribution of baryonresonances, such as ∆++ and ∆0, dominate at lower inci-dent photon energies (below 2 GeV). Furthermore, dataobtained with the SAPHIR detector at ELSA for pho-ton energies between 0.5 GeV and 2.6 GeV show thatthe contribution of baryonic resonances to the π+p and

π−p mass distributions gradually decreases with photonenergy.

In this paper we review the results of the analysisof π+π− photoproduction in the photon energy range3.0 - 3.8 GeV and momentum transfer squared −t be-tween 0.4 GeV2 and 1 GeV2, where the di-pion effec-tive mass Mππ varies from 0.4 GeV to 1.4 GeV. Themain results were previously reported in Ref. [27]. Weare not aware of any previous evidence of scalar mesons,in particular of the f0(980), in photoproduction of pionpairs. This effective mass region is dominated by theproduction of the ρ(770) resonance in the P -wave. Fromother experiments, such as pion-nucleon collisions π−p →π+π−n [28, 29] or nucleon-antinucleon annihilation [30],there is some evidence that resonant states are formedin the S-wave. These resonances have been neglected inprevious experimental analyses of π+π− photoproduc-tion and, to our knowledge, the current analysis is thefirst one that explicitly takes into account the possibilitythat the S-wave is produced in the π+π− system.

In the following, some details are given on the exper-iment and data analysis (Sec. II), on the extraction ofthe angular moments of the di-pion system (Sec. III),and the fit of the moments using a dispersion relation(Sec. IV). Results of the partial wave analysis (differ-ential cross section for each partial wave and the spindensity matrix elements) and the physics interpretationare reported in Sec. V.

II. EXPERIMENTAL PROCEDURES AND

DATA ANALYSIS

A. The photon beam and the target

The measurement was performed using the CLASdetector [31] in Hall B at Jefferson Lab with abremsstrahlung photon beam produced by a continuous60-nA electron beam of energy E0 = 4.02 GeV imping-ing on a gold foil of thickness 8× 10−5 radiation lengths.A bremsstrahlung tagging system [32] with a photon en-ergy resolution of 0.1% E0 was used to tag photons inthe energy range from 1.6 GeV to a maximum energyof 3.8 GeV. In this analysis only the high-energy partof the photon spectrum, ranging from 3.0 to 3.8 GeV,was used. e+ e− pairs produced by the interaction ofthe photon beam on a thin gold foil were used to contin-uously monitor the photon flux during the experiment.Absolute normalization was obtained by comparing thee+ e− pair rate with the photon flux measured by a totalabsorption lead-glass counter in dedicated low-intensityruns. The energy calibration of the Hall-B tagger systemwas performed both by a direct measurement of the e+e−

pairs produced by the incoming photons [33] and by ap-plying an over-constrained kinematic fit to the reactionγp → pπ+π−, where all particles in the final state weredetected in CLAS [34]. The quality of the calibrationswas checked by looking at the mass of known particles,

4

as well as their dependence on other kinematic variables(photon energy, detected particle momenta and angles).

The target cell, a Mylar cylinder 4 cm in diameterand 40-cm long, was filled by liquid hydrogen at 20.4 K.The luminosity was obtained as the product of the tar-get density, target length and the incoming photon fluxcorrected for data-acquisition dead time. The overall sys-tematic uncertainty on the run luminosity was estimatedto be in the range of 10%, dominated by the uncertaintieson the photon flux.

MX2(γp→pπ+X) (GeV2)

Cou

nts

x 10

3 /(0.

07 G

eV2 )

0

50

100

150

200

250

300

-0.3 -0.2 -0.1 0 0.1 0.2

FIG. 1: Missing mass squared for the reaction γp → pπ+Xand the π− peak. The shaded area indicates the retainedevents.

B. The CLAS detector

Outgoing hadrons were detected in the CLAS spec-trometer. Momentum information for charged particleswas obtained via tracking through three regions of multi-wire drift chambers [35] within a toroidal magnetic field(∼ 0.5 T) generated by six superconducting coils. Thepolarity of the field was set to bend the positive particlesaway from the beam line into the acceptance of the de-tector. Time-of-flight scintillators (TOF) were used forcharged hadron identification [36]. The interaction timebetween the incoming photon and the target was mea-sured by the start counter (ST) [37]. This is made of 24strips of 2.2-mm thick plastic scintillator surrounding thehydrogen cell with a single-ended PMT-based read-out.A time resolution of ∼300 ps was achieved.

The CLAS momentum resolution, σp/p, ranges from0.5 to 1%, depending on the kinematics. The detectorgeometrical acceptance for each positive particle in therelevant kinematic region is about 40%. It is somewhat

less for low-energy negative hadrons, which can be lostat forward angles because their paths are bent towardthe beam line and out of the acceptance by the toroidalfield. Coincidences between the photon tagger and theCLAS detector triggered the recording of the events. Thetrigger in CLAS required a coincidence between the TOFand the ST in at least two sectors, in order to selectreactions with at least two charged particles in the finalstate. An integrated luminosity of 70 pb−1 (∼ 20 pb−1

in the range 3.0< Eγ <3.8 GeV) was accumulated in 50days of running in 2004.

1

10

10 2

10 3

Mπ+π-(GeV)

Mpπ

+(G

eV)

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

FIG. 2: Two dimensional plot of the invariant masses ob-tained combining pairs of particles of the exclusive reactionγp → pπ+π−.

C. Data analysis and reaction identification

The raw data were passed through the standard CLASreconstruction software to determine the four-momentaof detected particles. In this phase of the analysis, cor-rections were applied to account for the energy loss ofcharged particles in the target and surrounding materi-als, misalignments of the drift chamber’s positions, anduncertainties in the value of the toroidal magnetic field.

The reaction γp → pπ+π− was isolated detecting theproton and the π+ in the CLAS spectrometer, while theπ− was reconstructed from the four-momenta of the de-tected particles by using the missing-mass technique. Inthis way the exclusivity of the reaction is ensured, keep-ing the contamination from the multi-pion backgroundto a minimum. Figure 1 shows the π− missing masssquared. The background below the missing pion peakappears as a smooth contribution in the ππ invariantmass without creating narrow structures.

5

0

200

400

600

0 0.5 1 1.5 2

0

50

100

150

200

1 2

m_pipi

Mπ+π− (GeV)

Cou

nts

x 10

3

Mpπ+ (GeV)

Cou

nts

x 10

3

Mpπ− (GeV)

Cou

nts

x 10

3

0

50

100

150

200

1 2

FIG. 3: Invariant masses obtained combining pairs of particlesof the exclusive reaction γp → pπ+π−. Upper panel M

π+π− ;lower panel left Mpπ+ ; lower panel right Mpπ− . Spectra arenot corrected for the detector acceptance.

To avoid edge regions in the detector acceptance, onlyevents within a fiducial volume were retained in this anal-ysis. In the laboratory reference system, cuts were de-fined for the minimum hadron momentum (pproton >0.32 GeV and pπ+ > 0.125 GeV), and the minimumand maximum azimuthal angles (θproton,π+ > 10 andθπ+ < 120). The fiducial cuts were defined comparingin detail the experimental data distributions with the re-sults of the detector simulation. The minimum momen-tum cuts were tuned for different hadrons to take intoaccount the energy loss by ionization of the particles.

After all cuts, 41M events were identified as producedin the exclusive reaction γp → pπ+π−. The other eventtopologies, with at least two hadrons in the final state(pπ−, π+π−, pπ+π−), were not used since in the kine-matics of interest for this analysis (−t < 1 GeV2), thecollected data are about one order of magnitude less dueto the detector acceptance. Figures 2 and 3 show theinvariant mass spectra of the different combinations ofparticles in the final state. The ρ(770) dominates the ππspectrum and the ∆(1232)++ peak is clearly visible inthe pπ+ invariant mass. Figure 2 shows a small overlapbetween the ∆(1232)++ and the ππ spectrum. Bary-onic resonances in the pπ− invariant mass spectrum areless pronounced. It has to be noted that the projectionof the baryon resonance peaks in the ππ spectrum re-sults in a smooth contribution and cannot create narrowstructures. The effect of this background was extensivelystudied as discussed in Sec.VC.

III. MOMENTS OF THE DI-PION ANGULAR

DISTRIBUTION

In this section we consider the analysis of moments ofthe di-pion angular distribution defined as:

〈YLM 〉(Eγ , t, Mππ) =√

4π

∫

dΩπ

dσ

dtdMππdΩπ

YLM (Ωπ),

(1)where dσ is the differential cross section (in momentumtransfer t and di-pion invariant mass Mππ), YLM arespherical harmonic functions of degree L and order M ,and Ωπ = (θπ, φπ) are the polar and azimuthal angles ofthe π+ flight direction in the π+π− helicity rest frame.For the definition of the angles in the di-pion system wefollow the convention of Ref. [9]. It follows from Eq. 1that, for a given Eγ , t and di-pion mass Mππ, 〈Y00〉 corre-sponds to the di-pion production differential cross sectiondσ/dtdMππ.

There are many advantages in defining and analyz-ing moments rather than proceeding via a direct partialwave fit of the angular distributions. Moments can be ex-pressed as bi-linear in terms of the partial waves and, de-pending on the particular combination of L and M , showspecific sensitivity to a particular subset of them. In ad-dition, they can be directly and unambiguously derivedfrom the data, allowing for a quantitative comparisonto the same observables calculated in specific theoreticalmodels.

Extraction of moments requires that the measured an-gular distribution is corrected by the detector acceptance.We studied three methods for implementing acceptancecorrections. In the first two methods, the moments wereexpanded in a model-independent way in a set of basisfunctions and, after weighting with Monte Carlo events,they were compared to the data by maximizing a likeli-hood function. The first of these two parametrizes thetheory in terms of simplified amplitudes, while the sec-ond uses directly moments as defined above. The ap-proximations in these methods have to do with the choiceof the basis and depend on the number of basis functionsused. The systematic effect of such truncations was stud-ied and the main results are reported below. In the lastmethod, data and Monte Carlo were binned in all kine-matical variables. The data were then corrected by theacceptance defined as the ratio of reconstructed over gen-erated Monte Carlo events in that bin. Since it was foundto be not reliable in bins where the acceptance was smallor vanishing, this method was only used as a check of theothers and was not included in the final determination ofthe experimental moments.

A. Detector efficiency

The CLAS detection efficiency for the reaction γp →pπ+π− was obtained by means of detailed Monte Carlostudies, which included knowledge of the full detector

6

geometry and a realistic response to traversing particles.Events were generated according to three-particle phasespace with a bremsstrahlung photon energy spectrum.A total of 4 billion events were generated in the energyrange 3.0 GeV < Eγ < 3.8 GeV and covered the allowedkinematic range in −t and Mππ. About 700M eventswere reconstructed in the Mππ and −t ranges of inter-est (0.4 GeV < Mππ < 1.4 GeV , 0.1 GeV2 < −t < 1.0GeV2). This corresponds to more than fifteen times thestatistics collected in the experiment, thereby introduc-ing a negligible statistical uncertainty with respect to thestatistical uncertainty of the data.

B. Extraction of the moments via likelihood fit of

experimental data

Moments were derived from the data using detec-tor efficiency-corrected fitting functions. As mentionedabove, the expected theoretical yield was parametrized interms of appropriate physics functions: production am-plitudes in one case and moments of the cross section inthe other. The theoretical expectation, after correctionfor acceptance, was compared to the experimental yield.Parameters were extracted by maximizing a likelihoodfunction defined as:

L ∼ Πna=1

[

η(τa)I(τa)∫

dτη(τ)I(τ)

]

. (2)

Here a represents a data event, n = ∆N is the numberof data events in a given (Eγ ,−t, Mππ) bin (i.e. the fit isdone independently in each bin), τa represents the set ofkinematical variables of the ath event, η(τa) is the corre-sponding acceptance derived by Monte Carlo simulationsand I(τa) is the theoretical function representing the ex-pected event distribution. The measure dτ includes thephase space factor and the likelihood function is normal-ized to the expected number of events in the bin

n =

∫

dτη(τ)I(τ). (3)

The advantage of this approach lies in avoiding bin-ning the data and the large uncertainties related to thecorrections in regions of CLAS with vanishing efficien-cies. Comparison of the results of the two differentparametrizations allows one to estimate the systematicuncertainty related to the procedure. In the following,we describe the two approaches in more detail.

1. Parametrization with amplitudes

The expected theoretical yield in each bin is describedas:

I(τa) = 4π

∣

∣

∣

∣

∣

Lmax∑

L=0

L∑

M=−L

aLM (Eγ ,−t, Mππ)YLM (Ωπ)

∣

∣

∣

∣

∣

2

.(4)

This parametrization has the benefit that the intensityfunction I(τa) is by construction positive. However, itcan lead to ambiguous results since it has more parame-ters than can be determined from the data. In addition,for practical reasons, the parametrization involves a cut-off, Lmax, in the maximum number of amplitudes. For aspecific choice of Lmax, the number of fit parameters isgiven by 2(Lmax + 1)2. We also note that these ampli-tudes are not the same as the partial wave amplitudes inthe usual sense of a di-pion photoproduction amplitude,since the latter depends on the nucleon and photon spins.

After removing the irrelevant constants, the fit is per-formed minimizing the function:

− lnL = −∆N∑

a=1

ln η(τa)I(τa) (5)

+ ∆N ln∑

L′M ′;LM

a∗L′M ′ aLMΨL′M ′;L,M ,

where we have introduced the rescaled amplitudesaLM (Eγ ,−t, Mππ) defined by:

aLM =√

ηaLM , (6)

and the acceptance matrix Ψ(Eγ ,−t, Mππ) was com-puted using Monte Carlo events as:

ηΨL′M ′;LM =4π

∆NGen

∆NRec∑

a=1

Y ∗L′M ′ (Ωπ)YLM (Ωπ), (7)

where ∆NGen and ∆NRec are the number of generatedand reconstructed events, respectively.

Fits were done using MINUIT with the analytical ex-pression for the gradient, and using the SIMPLEX pro-cedure followed by MIGRAD [38]. After each fit, thecovariance matrix was checked and if it was not positivedefinite, the fit was restarted with random input param-eters. At the end, the uncertainties were computed fromthe full covariance matrix.

2. Parametrization with moments

The expected theoretical yield in each (Eγ ,−t, Mππ)bin is described as:

I(τa) =√

4π

Lmax∑

L=0

L∑

M=0

〈YLM 〉 ReYLM (Ωπ). (8)

The parametrization in terms of the moments directlygives the quantities we are interested in (moments

〈YLM 〉). However, the fit has to be restricted to makesure the intensity is positive. As in the amplitudeparametrization, a cutoff Lmax in the maximum num-ber of moments has to be used. The number of fit pa-rameters is given by (Lmax + 1)(Lmax + 2)/2. As Lmax

7

increases, moments with L close to Lmax show a signifi-cant variation, while moments with the lowest L remainunchanged.

The expected (acceptance-corrected) distribution isthen given by:

I(τa) =√

4π∑

L,M

[ηLM ReYLM (Ωπ)] 〈YLM 〉. (9)

The function to be minimized with respect to 〈YLM 〉 (L >0) is then given by:

−2 lnL = −2

∆N∑

a=1

ln I(τa), (10)

with the coefficients ηLM (Eγ ,−t, Mππ) computed usingMonte Carlo events

ηLM =

√4π

∆NGen

∆NRec∑

i

ReYLM (Ωi)

ǫL

, (11)

where ǫL = 1 for L = 0 and 1/2 for all other (LM).For Lmax ≤ 4, the results are similar to what was ob-tained with the previous method, showing the same sta-bility against Lmax truncation and a similar goodness ofthe fit. To check the sensitivity of the likelihood fit tothe parameter initialization, moments were extracted inthree different ways: 1) using random initialization for allparameters; 2) fixing the parameters up to L = 2 to theones obtained from a fit with Lmax = 2, and randomlyinitializing the others; 3) starting with parameters ob-tained in 2) and then releasing all parameters. The threedifferent methods gave consistent results and the differ-ence of moments obtained using the different procedureswas used to evaluate the systematic uncertainty relatedto the fit procedure.

3. Methods comparison and final results

Moments derived by the different procedures agreedqualitatively. The most stable results were obtained byusing the first parametrization, although we do find oc-casionally large bin-to-bin fluctuations. However, thereare no a priori reasons to prefer one of the two methodsand we consider the discrepancies between the fit resultsas a good estimate of the systematic uncertainty associ-ated with the moments extraction. The final results aregiven as the average of the first method (parametrizationwith amplitudes) and the second method (parametriza-tion with moments) with the three fit initializations:

Yfinal =1

4

∑

i=1,4 Methods

Yi, (12)

where Y stands for 〈YLM 〉(Eγ , t, Mππ).The total uncertainty on the final moments was eval-

uated adding in quadrature the statistical uncertainty,

Mππ (GeV)

<Y00

> (µ

b/G

eV3 )

0

10

20

30

40

0.4 0.6 0.8 1 1.2 1.4

Mππ (GeV)

<Y10

> (µ

b/G

eV3 )

-6

-4

-2

0

0.4 0.6 0.8 1 1.2 1.4

Mππ (GeV)

<Y11

> (µ

b/G

eV3 )

-4

-3

-2

-1

0

1

0.4 0.6 0.8 1 1.2 1.4

FIG. 4: Moments of the di-pion angular distribution in 3.2 <Eγ < 3.4 GeV and −t = 0.45±0.05 GeV2 (black), −t = 0.65±0.05 GeV2 (red) and −t = 0.95 ± 0.05 GeV2 (blue). Errorbars include both statistical and systematic uncertainties asexplained in the text.

δYMINUIT as given by MINUIT, and two systematic un-certainty contributions: δYsyst fit related to the momentextraction procedure, and δYsyst norm, the systematic un-certainty associated with the photon flux normalization(see Sec. II).

δYfinal =√

δY 2MINUIT + δY 2

syst fit + δY 2syst norm (13)

with:

δYsyst fit =

√

√

√

√

∑

i=1,4 Methods

(Yi − Yfinal)2

4 − 1(14)

δYsyst norm = 10% · Yfinal. (15)

For most of the data points, the systematic uncertain-ties dominate over the statistical uncertainty. Samples ofthe final experimental moments are shown in Figs. 4, 5, 6,

8

Mππ (GeV)

<Y20

> (µ

b/G

eV3 )

-10

-5

0

0.4 0.6 0.8 1 1.2 1.4

Mππ (GeV)

<Y21

> (µ

b/G

eV3 )

0

2

4

0.4 0.6 0.8 1 1.2 1.4

Mππ (GeV)

<Y22

> (µ

b/G

eV3 )

-2

0

2

4

0.4 0.6 0.8 1 1.2 1.4

FIG. 5: Moments of the di-pion angular distribution in 3.2 <Eγ < 3.4 GeV and −t = 0.45±0.05 GeV2 (black), −t = 0.65±0.05 GeV2 (red) and −t = 0.95 ± 0.05 GeV2 (blue). Errorbars include both statistical and systematic uncertainties asexplained in the text.

and 7. The whole set of moments resulting from thisanalysis is available at the Jefferson Lab [39] and theDurham [40] databases.

As a check of the whole procedure, the differential crosssection dσ/dt for the γp → pρ(770) meson has been ex-tracted by fitting the 〈Y00〉 moment in each −t bin with aBreit-Wigner plus a first-order polynomial background.The agreement within the quoted uncertainties with aprevious CLAS measurement [41], as well as the worlddata [11], gives us confidence in the analysis procedure.

IV. PARTIAL WAVE ANALYSIS

In the previous section we discussed how moments ofthe angular distribution of the π+π− system, 〈YLM 〉,were extracted from the data in each bin in photon en-ergy, momentum transfer and di-pion mass. In this sec-tion we describe how partial waves were parametrized

Mππ (GeV)

<Y30

> (µ

b/G

eV3 )

-1

0

1

2

3

0.4 0.6 0.8 1 1.2 1.4

Mππ (GeV)

<Y31

> (µ

b/G

eV3 )

-1.5

-1

-0.5

0

0.5

1

0.4 0.6 0.8 1 1.2 1.4

Mππ (GeV)

<Y32

> (µ

b/G

eV3 )

-1

-0.5

0

0.5

1

0.4 0.6 0.8 1 1.2 1.4

FIG. 6: Moments of the di-pion angular distribution in 3.2 <Eγ < 3.4 GeV and −t = 0.45±0.05 GeV2 (black), −t = 0.65±0.05 GeV2 (red) and −t = 0.95 ± 0.05 GeV2 (blue). Errorbars include both statistical and systematic uncertainties asexplained in the text.

and extracted by fitting the experimental moments.

Moments can be expressed as bi-linear in terms of theamplitudes alm = alm(λ, λ′, λγ , Eγ , t, Mππ) with angularmomentum l and z-projection m (in the chosen referencesystem m coincides with the helicity of the di-pion sys-tem) as:

〈YLM 〉 =∑

l′m′,lm,λ,λ′

C(l′m′, lm, LM)× alm a∗l′m′ , (16)

where C are Wigner’s 3jm coefficients, λγ is the helicity ofthe photon, and λ and λ′ are the initial and final nucleonhelicity, respectively. The explicit forms of the momentswith L ≤ 4 in terms of amplitudes with l = 0 (S-wave),l = 1 (P -wave), l = 2 (D-wave), and l = 3 (F -wave) aregiven in Appendix A.

9

Mππ (GeV)

<Y40

> (µ

b/G

eV3 )

-1

-0.5

0

0.5

1

1.5

0.4 0.6 0.8 1 1.2 1.4

Mππ (GeV)

<Y41

> (µ

b/G

eV3 )

-1

0

1

0.4 0.6 0.8 1 1.2 1.4

Mππ (GeV)

<Y42

> (µ

b/G

eV3 )

-1

-0.5

0

0.5

1

0.4 0.6 0.8 1 1.2 1.4

FIG. 7: Moments of the di-pion angular distribution in 3.2 <Eγ < 3.4 GeV and −t = 0.45±0.05 GeV2 (black), −t = 0.65±0.05 GeV2 (red) and −t = 0.95 ± 0.05 GeV2 (blue). Errorbars include both statistical and systematic uncertainties asexplained in the text.

A. Helicities, isospin and coupled-channels

dependence

The photon helicity was restricted to λγ = +1 sincethe other amplitudes are related by parity conservation.In addition, some approximations in the parametrizationof the partial waves were adopted to reduce the numberof free parameters in the fit and are discussed below.

• The number of waves was reduced restricting theanalysis to |m| ≤ 1 since m = 2 is only possiblefor l ≥ 2 (D and F waves), which are expected tobe small in the mass range considered [9, 10]. Inthe chosen reference system, m coincides with thehelicity of the di-pion system and, since we used asa reference the wave with λγ = +1, the three val-ues of m have a simple interpretation in terms ofhelicity transfer from the photon to the ππ-system:m = +1 corresponds to the non-helicity-flip am-

plitude (s-channel helicity conserving) that is ex-pected to be dominant [10], while m = 0,−1 corre-spond to one and two units of helicity flip, respec-tively. In the case of the S-wave (l = m = 0), onlyone amplitude is considered.

• The dependence on the nucleon helicity was sim-plified as follows. For a given l, m, Eγ, t set, thereare four independent partial wave amplitudes cor-responding to the four combinations of initial andfinal nucleon helicity, λ and λ′. It is expected thatthe dominant amplitudes require no nucleon helic-ity flip [10]. Without nucleon polarization informa-tion it is not possible to extract all four amplitudes.Thus our strategy is to consider in the analysis onlythe dominant ones or to exploit possible relationsamong them. For example, in the Regge ρ and ωexchange model, the following relations are satisfiedby the S-wave amplitudes: (λλ′) = (++) = (−−)and (+−) = −(−+), where ± corresponds to helic-ity ±1/2. More generally, by examining the exper-imental moments, we observe that the interferencebetween the dominant P -wave, seen in the 〈Y21〉moment in the ρ region, indicates that the Pm=+1

and the Pm=0 amplitudes are out of phase. For asingle nucleon-helicity amplitude, this would implya difference between the 〈Y11〉 and 〈Y10〉 moments,arising primarily from the interference between theS-wave and the Pm=+1 and Pm=0 waves, respec-tively, in the ρ region where the S amplitude doesnot vary substantially. The data suggests, however,that both 〈Y11〉 and 〈Y10〉 peak near the positionof the ρ. A possible explanation for the behaviorof the data is the following: the dominant Pm=+1

amplitude may originate from the helicity-non-flipdiffractive process and the Pm=0 amplitude from anucleon-helicity-flip vector exchange, which is alsoexpected to contribute to the S-wave production.This would also explain why the 〈Y11〉 and 〈Y10〉moments have comparable magnitudes. To accom-modate such behavior, at least two nucleon-helicityamplitudes are required. In addition, since stronginteractions conserve isospin, it is convenient towrite the ππ amplitudes in the isospin basis. Eachamplitude was then expressed as a linear combi-nation of ππ amplitudes of fixed isospin I (withI = 0, 1, 2).

• The coupling of the ππ system to other chan-nels was taken into account introducing a multi-dimension channel space: for a given isospin I inthe partial wave l, the amplitudes depend also onan index α that runs over different di-meson sys-tems. For example, α = 1 corresponds to ππ, α = 2to KK, α = 3 to ηη, etc. In the subsequent analy-sis we will restrict the channel space to include theππ and KK channels, which are the only channelsrelevant in the energy range considered.

According to these considerations, the moments

10

Mππ (GeV)

<Y00

> (µ

b/G

eV3 )

5

10

15

20

0.4 0.6 0.8 1 1.2 1.4

Mππ (GeV)

<Y10

> (µ

b/G

eV3 )

-4

-2

0

0.4 0.6 0.8 1 1.2 1.4

Mππ (GeV)

<Y11

> (µ

b/G

eV3 )

-2

-1

0

0.4 0.6 0.8 1 1.2 1.4

FIG. 8: Fit result (black line) of the final experimental mo-ments (in red) for 3.2 < Eγ < 3.4 GeV and 0.5 < −t <0.6 GeV2. The systematic uncertainty and fit uncertainty areadded in quadrature and are shown by the gray band.

were fitted to a set of amplitudes given by:

aI,αlm,i(Eγ , t, Mππ) (17)

for each l, m, |m| ≤ 1, with i = 1, 2 correspondingto the nucleon helicity non-flip and helicity-flip ofone unit, isospin I = 0, 1, 2 and channel α.

B. Amplitude parametrization

For each helicity state of the target λ, recoil nucleonλ′, and ππ system m, in a given Eγ and t bin, the cor-responding helicity amplitude alm(s = M2

ππ), was ex-pressed using a dispersion relation [42–48] as follows:

alm,I(s) =1

2[I + Slm,I(s)]alm,I(s) (18)

− 1

πD−1

lm,I(s)PV

∫

sth

ds′Nlm,I(s

′)ρ(s′)alm,I(s′)

s′ − s,

where PV represents the principal value of the integraland ρ corresponds to the phase space term. In this ex-pression, I and Slm,I are matrices in the multi-channelspace (ππ, KK), as mentioned above. Nlm,I and Dlm,I

can be written in terms of the scattering matrix of ππscattering, chosen to reproduce the known phase shifts,inelasticities [49, 50], and the isoscalar (l = S, D), isovec-tor (l = P, F ) and isotensor (l = S, D) amplitudes in therange 0.4 GeV <

√s < 1.4 GeV. Finally, the amplitude

alm,I represents our ignorance about the production pro-cess.

As a function of s = M2ππ, alm,I have cuts for s > 4m2

π

(right-hand cut) and for s < m2π (left-hand cut). The

left-hand cut reflects the nature of particle exchanges de-termining the ππ photoproduction amplitude, while theright-hand cut accounts for the final-state interactionsof the produced pions. In Eq. 18, these discontinuitiesare taken into account by the functions Nlm,I and Dlm,I ,while alm,I(s) does not have singularities for s > 4m2

π

and can be expanded in a Taylor series:

alm,I =[

A + Bs + Cs2 + · · ·]

[k] (19)

with A,B, . . . being matrices of numerical coefficients tobe determined by the simultaneous fit of the angular mo-ments defined in Eq. 16 and [k] = kl

αδα,β used to take intoaccount the threshold behavior in the l-th partial wave.All amplitudes but the scalar-isoscalar are saturated bythe ππ state. For the scalar-isoscalar amplitude, the KKchannel was also included. In addition, to reduce sensi-tivity to the large energy behavior of the (ππ,KK) am-plitudes, the real part of the integral was subtracted andreplaced by a polynomial in s, whose coefficients werealso fitted.

V. RESULTS

A. Fit of the moments

Using the parametrization of the partial waves de-scribed in the previous section, we fitted all moments〈YLM 〉 with L ≤ 4 and M ≤ 2 using amplitudes withl ≤ 3 (up to F -waves). In Figs. 8, 9, 10, and 11 wepresent a sample of the fit results for Eγ = 3.3±0.1 GeVand 0.5 < |t| < 0.6 GeV2.

To properly take into account the statistical and sys-tematic uncertainty contributions to the experimentalmoments described in Sec. III, the four sets of momentsresulting from the different fit procedures were individ-ually fitted and the results were averaged, obtaining thecentral value shown by the black line in the figures. Theerror band, shown as a gray area, was calculated fol-lowing the same procedure adopted for the experimentalmoments. The final uncertainty was computed as thesum in quadrature of the statistical uncertainty of the fitand the two systematic uncertainty contributions. Thefirst is related to the moment extraction procedure and

11

Mππ (GeV)

<Y20

> (µ

b/G

eV3 )

-4

-2

0

0.4 0.6 0.8 1 1.2 1.4

Mππ (GeV)

<Y21

> (µ

b/G

eV3 )

0

1

2

0.4 0.6 0.8 1 1.2 1.4

Mππ (GeV)

<Y22

> (µ

b/G

eV3 )

0

1

2

0.4 0.6 0.8 1 1.2 1.4

FIG. 9: Fit result (black line) of the final experimental mo-ments (in red) for 3.2 < Eγ < 3.4 GeV and 0.5 < −t <0.6 GeV2. The systematic uncertainty and fit uncertainty areadded in quadrature and are shown by the gray band.

is evaluated as the variance of the four fit results. Thelatter is associated with the photon flux normalizationand is estimated to be 10%. The central values and un-certainties for all the observables of interest discussed inthe following sections were derived from the fit resultswith the same procedure.

The moment 〈Y00〉, corresponding to the differentialproduction cross section dσ/dtdM , shows the dominantρ(770) meson peak. In the 〈Y10〉 and 〈Y11〉 moments, thecontribution of the S-wave is maximum and enters viainterference with the P -wave. In particular the struc-ture at Mππ ∼ 0.77 GeV in 〈Y11〉 is due to the interfer-ence of the S-wave with the dominant, helicity-non-flipwave, Pm=+1. In the 〈Y10〉 moment the same structureis due to the interference with the Pm=0 wave, whichcorresponds to one unit of helicity flip. A second dipnear Mππ = 1 GeV is clearly visible and corresponds tothe production of a resonance that we interpret as thef0(980).

Mππ (GeV)

<Y30

> (µ

b/G

eV3 )

-1

0

1

2

0.4 0.6 0.8 1 1.2 1.4

Mππ (GeV)

<Y31

> (µ

b/G

eV3 )

-0.5

-0.25

0

0.25

0.4 0.6 0.8 1 1.2 1.4

Mππ (GeV)

<Y32

> (µ

b/G

eV3 )

-0.5

0

0.5

0.4 0.6 0.8 1 1.2 1.4

FIG. 10: Fit result (black line) of the final experimental mo-ments (in red) for 3.2 < Eγ < 3.4 GeV and 0.5 < −t <0.6 GeV2. The systematic uncertainty and fit uncertainty areadded in quadrature and are shown by the gray band.

B. Partial wave amplitudes

The square of the magnitude of the S-, P -, D- andF -waves resulting from the fit, summed over the nucleonspin projections, is given by:

Ilm =∑

i=1,2

|alm,i(Eγ , t, Mππ)|2.

(20)

When summed over the di-pion helicity, this can be writ-ten as:

Il =∑

m

∑

i=1,2

|alm,i(Eγ , t, Mππ)|2,

(21)

where the sum is limited to m = −1, 0, 1 for l > 0 and tom = 0 for l = 0.

The resulting partial wave cross sections are shown inFigs. 12, 13, 14, and 15, for a selected photon energy

12

Mππ (GeV)

<Y40

> (µ

b/G

eV3 )

-0.5

0

0.5

1

0.4 0.6 0.8 1 1.2 1.4

Mππ (GeV)

<Y41

> (µ

b/G

eV3 )

-0.5

0

0.5

0.4 0.6 0.8 1 1.2 1.4

Mππ (GeV)

<Y42

> (µ

b/G

eV3 )

-0.5

-0.25

0

0.25

0.4 0.6 0.8 1 1.2 1.4

FIG. 11: Fit result (black line) of the final experimental mo-ments (in red) for 3.2 < Eγ < 3.4 GeV and 0.5 < −t <0.6 GeV2. The systematic uncertainty and fit uncertainty areadded in quadrature and are shown by the gray band.

Mππ (GeV)

dσ/d

tdM

ππ (

µb/G

eV3 )

0

0.5

1

1.5

2

0.4 0.6 0.8 1 1.2 1.4

FIG. 12: S-wave cross section derived by the fit in the3.2 < Eγ < 3.4 GeV and 0.5 < −t < 0.6 GeV2 bin. Thesystematic and the fit uncertainties are added in quadratureand are shown by the gray band.

5

10

15

0.4 0.6 0.8 1 1.2 1.4

0.5

1

1.5

0.4 1.4

1

2

3

0.4 1.4

Mππ (GeV)

dσ/d

tdM

ππ (

µb/G

eV3 )

Mππ (GeV)

dσ/d

tdM

ππ (

µb/G

eV3 )

Mππ (GeV)

dσ/d

tdM

ππ (

µb/G

eV3 )

Mππ (GeV)

dσ/d

tdM

ππ (

µb/G

eV3 )

5

10

15

0.4 1.4

FIG. 13: P -wave cross section derived by the fit in the3.2 < Eγ < 3.4 GeV and 0.5 < −t < 0.6 GeV2 bin. Bot-tom plots: the same amplitudes for the three possible valuesof λππ (from left to right -1, 0 and +1). The systematic andfit uncertainties are added in quadrature and are shown bythe gray band.

and −t bin. The whole set of partial wave amplitudesresulting from this analysis is available at the JeffersonLab [39] and the Durham [40] databases.

As expected, the dominant contribution from the ρ me-son is clearly visible in the P -wave, whose contributionis about one order of magnitude larger than the otherwaves. In particular the main contribution comes fromIlm=1,+1, corresponding to a non-helicity flip (s-channelhelicity conserving) transition. In the S-wave, a stronginterference pattern shows up around Mππ = 980 MeV,which reveals contributions from the f0(980) production.The contribution from the f2(1270) tensor meson is ap-parent in the D-wave, while no clear structures are seenin the F -wave.

C. Systematic studies

The error bands plotted in Figs. 12, 13, 14, and 15include the systematic uncertainties related to the mo-ment extraction and the photon flux normalization asdiscussed in Sec. III B 3. In addition, for the S-wave,where the f0(980) contribution is strongly affected by in-terference, detailed systematic studies using both MonteCarlo and data were performed.

In order to test the approximation introduced by thetruncation to Lmax=4 in the moment extraction, we firstverified the fit was able to reproduce the experimental

13

0.5

1

1.5

2

0.4 0.6 0.8 1 1.2 1.4

0

0.2

0.4

0.6

0.4 1.40

0.5

1

0.4 1.4

Mππ (GeV)

dσ/d

tdM

ππ (

µb/G

eV3 )

Mππ (GeV)

dσ/d

tdM

ππ (

µb/G

eV3 )

Mππ (GeV)

dσ/d

tdM

ππ (

µb/G

eV3 )

Mππ (GeV)

dσ/d

tdM

ππ (

µb/G

eV3 )

0

0.2

0.4

0.6

0.8

0.4 1.4

FIG. 14: As Fig. 13 for D-wave.

0.2

0.4

0.6

0.8

0.4 0.6 0.8 1 1.2 1.4

0

0.1

0.2

0.3

0.4

0.4 1.4

0.2

0.4

0.6

0.4 1.4

Mππ (GeV)

dσ/d

tdM

ππ (

µb/G

eV3 )

Mππ (GeV)

dσ/d

tdM

ππ (

µb/G

eV3 )

Mππ (GeV)

dσ/d

tdM

ππ (

µb/G

eV3 )

Mππ (GeV)

dσ/d

tdM

ππ (

µb/G

eV3 )

0.1

0.2

0.4 1.4

FIG. 15: As Fig. 13 for F -wave.

distributions in the kinematic range of interest. Figure 16shows the comparison between data and fit results forthe decay angles in the helicity system with Mππ in thef0(980) mass region (Mππ = 0.985±0.01 GeV). Figure 17shows the same comparison for the invariant mass Mpπ+

when three different regions of Mππ (Mππ = 0.475 ±0.01 GeV, Mππ = 0.775 ± 0.01 GeV, Mππ = 1.295 ±

0102030405060708090

100

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1cosθ

dN

/dco

sθ

φ(rad)

dN

/dφ

0102030405060708090

100

0 1 2 3 4 5 6

FIG. 16: Pion angles in the π+π− helicity rest frame for Mππ

in the f0(980) mass region (Mππ = 0.985 ± 0.01 GeV). Ex-perimental data are plotted in black and fit results in red.

0.01 GeV) were selected. The good agreement provesthe accuracy of the approximation.

As a second check, we applied the fit to pseudo-dataobtained with a realistic event generator, processed withthe CLAS GEANT-based simulation package and ana-lyzed with the same procedure used for the data. Sincethe event generator was tuned to previous two-pion pho-toproduction measurements, it does not include any ex-plicit limitation on the number of waves. The recon-structed moments showed that, with the chosen Lmax, allfits were capable of reproducing the generated momentsup to Mππ ∼ 1.1 GeV. Finally, we derived a quantitativeestimate of the truncation effect on the S-wave squaredamplitude as follows. The results of a Lmax = 8 fit of themoments was used as input for a new Monte Carlo eventgenerator. After being processed in the same way as dis-cussed above, pseudo-data were fitted with Lmax = 4 andthe S-wave amplitude was extracted. The difference be-tween the generated and the reconstructed partial wavecross section was found to be of the order of 25% that,added in quadrature to the other systematic uncertain-ties, was included in the gray band of Fig. 12.

We also demonstrated that no structures similar to thenarrow interference pattern we are interpreting as the ev-idence of the f0(980) were created by distortions inducedby the CLAS acceptance. This check was performed gen-erating events after removing the f0(980) contribution,and verifying that no spurious structures appeared in thespectra after the full GEANT simulation and reconstruc-tion.

14

0

10

20

30

40

50

1.2 1.4 1.6 1.8 2 2.2 2.4 2.6

0

50

100

150

200

1.2 1.4 1.6 1.8 2 2.2 2.4 2.6

Mpπ+ (GeV)

Cou

nts

/10

MeV

Mpπ+ (GeV)

Cou

nts

/10

MeV

Mpπ+ (GeV)

Cou

nts

/10

MeV

0

10

20

30

1.2 1.4 1.6 1.8 2 2.2 2.4 2.6

FIG. 17: Mpπ+ distribution in three different Mππ mass re-gions (bottom: Mππ = 0.475 ± 0.01 GeV, middle: Mππ =0.775 ± 0.01 GeV, top: Mππ = 1.295 ± 0.01 GeV). Experi-mental data are plotted in black and fit results in red.

In addition, the effects of baryon resonance contribu-tions to the di-pion mass spectrum were studied perform-ing the fit of the moments with the inclusion of an inco-herent background. In fact, the background in the di-pion mass spectrum introduced by the reflection of thebaryon resonances is expected to be smooth and struc-tureless, contributing to all waves. Therefore this wasparametrized as a second-order polynomial in Mππ thatwas summed to the parametrization of the moments interms of partial waves used in the standard fits. Fromthis study we concluded that the background contribu-tion is small, smooth and does not affect the quality ofthe fit. The comparison of the fit results with and with-out the inclusion of this additional background indicatesthat the P -wave and the S-wave in the f0(980) region areonly slightly affected, as shown in Fig. 18. On the con-trary, the low mass S-wave, corresponding to the σ(600)region, and the D-wave, corresponding to the f2(1270)region, show a significant variation and, therefore, a morecomplete analysis should be performed to extract reliableinformation in these mass ranges. A similar conclusionwas drawn by comparing the analysis results excluding

0

0.5

1

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4

0

5

10

15

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4

Mππ (GeV)

dσ/d

tdM

ππ (µ

b/G

eV3 )

Mππ (GeV)

dσ/d

tdM

ππ (µ

b/G

eV3 )

Mππ (GeV)

dσ/d

tdM

ππ (µ

b/G

eV3 )

0

0.5

1

1.5

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4

FIG. 18: S-, P - and D-wave cross sections in the 3.4 <Eγ < 3.6 GeV and 0.5 < −t < 0.6 GeV2 bin. The grayand red bands show the results of the standard fit and ofthe fit performed adding a second-order polynomial to thepartial wave expansion of the moments to account for thebaryon resonance contributions. The width of the bands rep-resents the fit uncertainties only. Fit results are shown for aspecific parametrization of the moments (second method, seeSec. III B 3).

the ∆(1232), the dominant baryon resonance contribu-tion for this final state, with the cut M(pπ+) > 1.4 GeV.A negligible effect was found on the rapid motion aroundthe narrow f0(980) meson, while a larger variation wasobserved at higher values of the M(ππ) mass.

To verify the stability of the fit of moments in theregion of the f0(980), the whole analysis was repeatedreducing the Mππ bin size from 10 to 5 MeV. The resultsobtained in the two cases were found to be consistent.

As a final check, the sensitivity to the specific choiceof the number of terms used in the Taylor expansion ofthe amplitudes aL (see Eq. 19) was tested performing thepartial wave analysis fits both with a second- and fourth-order polynomial. The effect was found to be negligiblecompared to the other systematic uncertainties.

D. The spin density matrix elements

From the production amplitudes derived by the fit, wecalculated the spin density matrix elements [51] for theP -wave and the interference between the S- and P -waves.

15

0

0.5

1

0.4 0.6 0.8 1 1.2 1.4

0

0.2

0.4

0.4 0.6 0.8 1 1.2 1.4

Mππ (GeV)

ρ0 0,0

Mππ (GeV)

ρ0 1,0

Mππ (GeV)

ρ0 1,-1

-0.2

0

0.2

0.4 0.6 0.8 1 1.2 1.4

FIG. 19: Spin density matrix elements for the P -wave in the3.0 < Eγ < 3.2 GeV and 0.4 < −t < 0.5 GeV2 bin. Theblack dots are data points from Ref. [9], taken in a similarkinematic bin (Eγ ∼ 2.8 GeV and 0.02 < −t < 0.4 GeV2).

Some selected results are shown in Figs. 19, 20 and21. Since these observables do not depend on the photonflux normalization, the error bands do not include the10% uncertainty mentioned above. The whole set of spindensity matrix elements resulting from this analysis isavailable at the Jefferson Lab [39] and the Durham [40]databases.

Comparisons of our measurements at 3.0 < Eγ <3.2 GeV and 0.4 < −t < 0.5 GeV2 with existing datafrom Refs. [9, 10] in a similar kinematic domain (Eγ ∼ 2.8GeV and 0.02 < −t < 0.4 GeV2) are shown in Fig. 19.As expected, the two matrix elements ρ10 and ρ11 agreevery well since they have a weak dependence on −t, whileρ00 shows a similar behavior, but with different values asit is more sensitive to the momentum transfer. If onecompares the larger −t bins we measured, the differencesincrease, showing that extrapolating our data to lower−t would probably give good agreement with previousmeasurements.

As shown in Fig. 21, around Mππ = 980 MeV an in-terference pattern clearly shows up in the S-P wave in-terference term, corresponding to the contribution fromthe f0(980) meson.

0.2

0.4

0.6

0.4 0.6 0.8 1 1.2 1.4

0

0.1

0.2

0.4 0.6 0.8 1 1.2 1.4

Mππ (GeV)

ρ0 0,0

Mππ (GeV)

ρ0 1,0

Mππ (GeV)

ρ0 1,-1

-0.1

0

0.4 0.6 0.8 1 1.2 1.4

FIG. 20: Spin density matrix elements for the P -wave in the3.2 < Eγ < 3.4 GeV and 0.5 < −t < 0.6 GeV2 bin.

-0.15

-0.1

-0.05

0

0.4 0.6 0.8 1 1.2 1.4Mππ (GeV)

ρ0 0,0

Mππ (GeV)

ρ0 1,0

-0.1

-0.05

0

0.05

0.4 0.6 0.8 1 1.2 1.4

FIG. 21: Spin density matrix elements for the interferencebetween S- and P -waves in the 3.2 < Eγ < 3.4 GeV and0.5 < −t < 0.6 GeV2 bin.

E. Differential cross sections

The differential cross sections [dσ/dt]l−wave for indi-vidual waves and mass resonance regions were obtainedintegrating the corresponding amplitudes. The cross sec-tions in the mass regions of the f0(980), ρ, and f2(1270)

16

mesons were obtained integrating the S-, P - and D-wavesin the mass ranges 0.98 ± 0.04 GeV, 0.4-1.2 GeV, and1.275 ± 0.185 GeV, respectively. These are shown inFigs. 22, 23 and 24 in the photon energy range 3.0-3.8 GeV. As mentioned previously, the P -wave is com-pletely dominated by the ρ meson production, and there-fore the integrated cross section can be directly comparedto the world’s data for the γp → pρ reaction [11, 41]. Itshould be noticed that the previous cross sections wereevaluated without performing a partial wave analysis butfitting the mass-dependent cross section with a relativis-tic Breit-Wigner plus a smooth polynomial function toseparate the resonance from the background. The goodagreement shown in Fig. 23 gives confidence in the partialwave analysis. As expected, the S-wave photoproductionis suppressed compared to the P -wave by more than anorder of magnitude, reflecting the different mechanismsthat lead to scalar and vector meson photoproduction:in Regge theory the latter is dominated by Pomeron ex-change, while the former is dominated by the exchange ofreggeons that become suppressed as the energy increases.

VI. SUMMARY

In summary, we have performed a partial wave anal-ysis of the reaction γp → pπ+π− in the photon en-ergy range 3.0-3.8 GeV and momentum transfer range−t = 0.4 − 1.0 GeV2. Moments of the di-pion angulardistribution, defined as bi-linear functions of partial waveamplitudes, were fitted to the experimental data with anunbinned likelihood procedure. Different parametriza-tion bases were used and detailed systematic checks wereperformed to insure the reliability of the analysis pro-cedure. We extracted moments 〈YLM 〉 with L ≤ 4 andM ≤ 2 using amplitudes with l ≤ 3 (up to F -waves). Us-ing a dispersion relation, unitarity constraint, and phaseshifts and inelasticities of ππ scattering, the productionamplitudes were expressed in a simplified form, wherethe unknown part was expanded in a Taylor series. Thecoefficients were fitted to the experimental moments toextract the S-, P -, D-, and F -waves in the Mππ range0.4-1.4 GeV.

The moment 〈Y00〉 is dominated by the ρ(770) mesoncontribution in the P -wave, while the moments 〈Y10〉and 〈Y11〉 show contributions of the S-wave throughinterference with the P -wave. The clear structure atMππ ∼ 1 GeV seen in such experimental moments andin the S-wave amplitude is evidence of a resonance con-tribution that we interpret as the f0(980). This is thefirst observation of the f0(980) scalar meson in photopro-duction. A contribution from the f2(1270) tensor mesonwas observed in the D-wave, while no resonant structureswere seen in the F -wave. The cross sections of individualpartial waves in the mass range of the ρ(770), f0(980),and f2(1270) were computed. Finally, the spin densitymatrix elements for the P -wave were evaluated, finding

-t (GeV2)

dσ/d

t (µ

b/G

eV2 )

10-3

10-2

10-1

1

0.4 0.5 0.6 0.7 0.8 0.9 1

FIG. 22: Differential cross section dσ/dt for the S-wave inthe Mππ range 0.98 ± 0.04 GeV and photon energy rangeEγ = 3.0 − 3.8 GeV.

CLAS This work Eγ= 3.4+-0.4 GeV CLAS Ref. 41 Eγ= 3.30+-0.10 GeV CLAS Ref. 41 Eγ= 3.55+-0.15 GeV CLAS Ref. 41 Eγ= 3.80+-0.10 GeV ABBHHM Ref. 11 Eγ= 3.0+-0.5 GeV ABBHHM Ref. 11 Eγ= 4.0+-0.5 GeV

-t (GeV2)

dσ/d

t (µ

b/G

eV2 )

1

10

0.4 0.5 0.6 0.7 0.8 0.9 1

FIG. 23: Differential cross section dσ/dt for the P -wave inthe Mππ range 0.4-1.2 GeV and photon energy range Eγ =3.0 − 3.8 GeV.

good agreement with previous measurements, and for thefirst time, the S − P interference term was extracted.

VII. ACKNOWLEDGMENTS

We would like to acknowledge the outstanding effortsof the staff of the Accelerator and the Physics Divi-sions at Jefferson Lab that made this experiment pos-sible. This work was supported in part by the ItalianIstituto Nazionale di Fisica Nucleare, the French CentreNational de la Recherche Scientifique and Commissariata l’Energie Atomique, the UK Science and TechnologyFacilities Research Council (STFC), the U.S. Departmentof Energy and National Science Foundation, and the Ko-rea Science and Engineering Foundation. The Southeast-ern Universities Research Association (SURA) operatesthe Thomas Jefferson National Accelerator Facility for

17

-t (GeV2)

dσ/d

t (µ

b/G

eV2 )

10-1

1

0.4 0.5 0.6 0.7 0.8 0.9 1

FIG. 24: Differential cross section dσ/dt for the D-wave inthe Mππ range 1.275 ± 0.185 GeV and photon energy rangeEγ = 3.0 − 3.8 GeV.

the United States Department of Energy under contractDE-AC05-84ER40150.

APPENDIX A

The explicit expressions for the moments, defined inEq. 1 in terms of partial waves, given Eq. 4, truncated tothe L = 3 (F ) wave are given by,

〈Y00〉 = |S|2 + |P−|2 + |P0|2 + |P+|2 + |D−|2 + |D0|2 + |D+|2 + |F−|2 + |F0|2 + |F+|2

〈Y10〉 = SP ∗0 + P0S

∗ +

√

3

5

(

P−D∗− + P ∗

−D− + P+D∗+ + D+P ∗

+

)

+

√

4

5(P0D

∗0 + D0P

∗0 )

+

√

24

35

(

D−F ∗− + F−D∗

− + D+F ∗+ + F+D∗

+

)

+

√

216

280(D0F

∗0 + F0D

∗0)

〈Y11〉 =(

−P−S∗ − SP ∗− + P+S∗ + SP ∗

+

)

+

√

1

20

(

P−D∗0 + D0P

∗− − P+D∗

0 − D0P∗+

)

+

√

3

20

(

−P0D∗− − D−P ∗

0 + P0D∗+ + D+P ∗

0

)

+

√

9

140

(

D−F ∗0 + F0D

∗− − D+F ∗

0 − F0D∗+

)

+

√

9

70

(

−D0F∗− − F−D∗

0 + D0F∗+ + F+D∗

0

)

〈Y20〉 = SD∗0 + D0S

∗ +

√

1

5

(

2|P0|2 − |P−|2 − |P+|2 + |F−|2 + |F+|2)

+

√

18

35

(

P−F ∗− + F−P ∗

− + P+F ∗+ + F+P ∗

+

)

+

√

27

35(P0F

∗0 + F0P

∗0 ) +

√

5

49

(

|D−|2 + |D+|2)

+

√

20

49|D0|2 +

√

16

45|F0|2

〈Y21〉 =1

2

(

SD∗+ + D+S∗ − SD∗

− − D−S∗)

+

√

3

20

(

P0P∗+ + P+P ∗

0 − P−P ∗0 − P0P

∗−

)

+

√

9

140

(

P−F ∗0 + F0P

∗− − P+F ∗

0 − F0P∗+

)

+

√

6

35

(

P0F∗+ + F+P ∗

0 − P0F∗− − F−P ∗

0

)

+

√

5

196

(

D0D∗+ + D+D∗

0 − D0D∗− − D−D∗

0

)

+

√

1

90

(

F0F∗+ + F+F ∗

0 − F0F∗− − F−F ∗

0

)

〈Y22〉 =

√

3

10

(

P−P ∗+ + P+P ∗

−

)

+

√

3

140

(

P−F ∗+ + F+P ∗

− + P+F ∗− + F−P ∗

+

)

+

√

4

30

(

−F+F ∗− − F−F ∗

+

)

+

√

3

196

(

−D−D∗+ − D+D∗

−

)

〈Y30〉 = SF ∗0 + F0S

∗ +

√

18

70

(

−P−D∗− − D−P ∗

− − P+D∗+ − D+P ∗

+

)

+

√

108

140(P0D

∗0 + D0P

∗0 )

+

√

2

45

(

D−F ∗− + F−D∗

− + D+F ∗+ + F+D∗

+

)

+

√

16

45(D0F

∗0 + F0D

∗0)

18

〈Y31〉 =1

2

(

SF ∗+ + F+S∗ − SF ∗

− − F−S∗)

+

√

18

140

(

P+D∗0 + D0P

∗+ − P−D∗

0 − D0P∗−

)

+

√

6

35

(

P0D∗+ + D+P ∗

0 − P0D∗− − D−P ∗

0

)

+

√

1

90

(

D+F ∗0 + F0D

∗+ − D−F ∗

0 − F0D∗−

)

+

√

1

20

(

D0F∗+ + F+D∗

0 − D0F∗− − F−D∗

0

)

〈Y32〉 =

√

3

14

(

−P+D∗− − D−P ∗

+ − P−D∗+ − D+P ∗

−

)

+

√

1

12

(

−D+F ∗− − F−D∗

+ − D−F ∗+ − F+D∗

−

)

〈Y40〉 =

√

2

7

(

−P+F ∗+ − F+P ∗

+ − P−F ∗− − F−P ∗

−

)

+

√

16

21(P0F

∗0 + F0P

∗0 ) +

√

16

49

(

−|D+|2 − |D−|2)

+

√

36

49|D0|2 +

√

36

121|F0|2 +

√

1

121

(

|F+|2 + |F−|2)

〈Y41〉 =

√

5

42

(

P+F ∗0 + F0P

∗+ − P−F ∗

0 − F0P∗−

)

+

√

5

28

(

P0F∗+ + F+P ∗

0 − P0F∗− − F−P ∗

0

)

+ .

√

30

196

(

D0D∗+ + D+D∗

0 − D−D∗0 − D0D

∗−

)

+

√

30

968

(

F0F∗+ + F+F ∗

0 − F0F∗− − F−F ∗

0

)

〈Y42〉 =

√

5

28

(

−P+F ∗− − F−P ∗

+ − P−F ∗+ − F+P ∗

−

)

+

√

10

49

(

−D−D∗+ − D+D∗

−

)

+

√

10

121

(

−F−F ∗+ − F+F ∗

−

)

It follows from Eq. 1 that the 〈Y00〉 moment is normal-ized by the differential cross section via,

〈Y00〉 =

∫

dΩπ

dσ

dtdMππdΩπ

. (A1)

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15, 397 (1970).

M (GeV)

Nor

m. M

om. (

nb/ 0

.1G

eV2 1

0MeV

)

<Y00>

2

4

6

8

10

12

14

16

18

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4

M (GeV)

Nor

m. M

om. (

nb/ 0

.1G

eV2 1

0MeV

)

<Y10>

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4

M (GeV)

Nor

m. M

om. (

nb/ 0

.1G

eV2 1

0MeV

)

<Y11>

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4

M (GeV)

Nor

m. M

om. (

nb/ 0

.1G

eV2 1

0MeV

)

<Y20>

-5

-4

-3

-2

-1

0

1

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4

M (GeV)

Nor

m. M

om. (

nb/ 0

.1G

eV2 1

0MeV

)

<Y21>

-0.5

0

0.5

1

1.5

2

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4

M (GeV)

Nor

m. M

om. (

nb/ 0

.1G

eV2 1

0MeV

)

<Y22>

-1

-0.5

0

0.5

1

1.5

2

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4

M (GeV)

Nor

m. M

om. (

nb/ 0

.1G

eV2 1

0MeV

)

<Y30>

-0.5

0

0.5

1

1.5

2

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4

M (GeV)

Nor

m. M

om. (

nb/ 0

.1G

eV2 1

0MeV

)

<Y31>

-0.4

-0.2

0

0.2

0.4

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4

M (GeV)

Nor

m. M

om. (

nb/ 0

.1G

eV2 1

0MeV

)

<Y32>

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4

M (GeV)

Nor

m. M

om. (

nb/ 0

.1G

eV2 1

0MeV

)

<Y33>

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4

M (GeV)

Nor

m. M

om. (

nb/ 0

.1G

eV2 1

0MeV

)

<Y40>

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4

M (GeV)

Nor

m. M

om. (

nb/ 0

.1G

eV2 1

0MeV

)

<Y41>

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4

M (GeV)

Nor

m. M

om. (

nb/ 0

.1G

eV2 1

0MeV

)

<Y42>

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4

M (GeV)

Nor

m. M

om. (

nb/ 0

.1G

eV2 1

0MeV

)

<Y43>

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4

M (GeV)

Nor

m. M

om. (

nb/ 0

.1G

eV2 1

0MeV

)

<Y44>

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4