Electroexcitation of nucleon resonances from CLAS data on single pion electroproduction

Electroexcitation of nucleon resonances from CLAS data on single pion electroproduction

I.G. Aznauryan Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606 Yerevan Physics Institute, 375036 Yerevan, Armenia    V.D. Burkert Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606    A.S. Biselli Fairfield University, Fairfield CT 06824    H. Egiyan Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606 University of New Hampshire, Durham, New Hampshire 03824-3568    K. Joo University of Connecticut, Storrs, Connecticut 06269 University of Virginia, Charlottesville, Virginia 22901    W. Kim Kyungpook National University, Daegu 702-701, Republic of Korea    K. Park Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606 Kyungpook National University, Daegu 702-701, Republic of Korea    L.C. Smith University of Virginia, Charlottesville, Virginia 22901    M. Ungaro Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606 University of Connecticut, Storrs, Connecticut 06269 Rensselaer Polytechnic Institute, Troy, New York 12180-3590    K. P.  Adhikari Old Dominion University, Norfolk, Virginia 23529    M. Anghinolfi INFN, Sezione di Genova, 16146 Genova, Italy    H. Avakian Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606    J. Ball CEA, Centre de Saclay, Irfu/Service de Physique Nucléaire, 91191 Gif-sur-Yvette, France    M. Battaglieri INFN, Sezione di Genova, 16146 Genova, Italy    V. Batourine Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606    I. Bedlinskiy Institute of Theoretical and Experimental Physics, Moscow, 117259, Russia    M. Bellis Carnegie Mellon University, Pittsburgh, Pennsylvania 15213    C.  Bookwalter Florida State University, Tallahassee, Florida 32306    D. Branford Edinburgh University, Edinburgh EH9 3JZ, United Kingdom    W.J. Briscoe The George Washington University, Washington, DC 20052    W.K. Brooks Universidad Técnica Federico Santa María, Casilla 110-V Valparaíso, Chile Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606    S.L. Careccia Old Dominion University, Norfolk, Virginia 23529    D.S. Carman Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606    P.L. Cole Idaho State University, Pocatello, Idaho 83209    P. Collins Arizona State University, Tempe, Arizona 85287-1504    V. Crede Florida State University, Tallahassee, Florida 32306    A. D’Angelo INFN, Sezione di Roma Tor Vergata, 00133 Rome, Italy Universita’ di Roma Tor Vergata, 00133 Rome Italy    A. Daniel Ohio University, Athens, Ohio 45701    R. De Vita INFN, Sezione di Genova, 16146 Genova, Italy    E. De Sanctis INFN, Laboratori Nazionali di Frascati, 00044 Frascati, Italy    A. Deur Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606    B Dey Carnegie Mellon University, Pittsburgh, Pennsylvania 15213    S. Dhamija Florida International University, Miami, Florida 33199    R. Dickson Carnegie Mellon University, Pittsburgh, Pennsylvania 15213    C. Djalali University of South Carolina, Columbia, South Carolina 29208    D. Doughty Christopher Newport University, Newport News, Virginia 23606 Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606    R. Dupre Argonne National Laboratory, Argonne, Illinois 60441    A. El Alaoui [ Institut de Physique Nucléaire ORSAY, Orsay, France    L. Elouadrhiri Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606    P. Eugenio Florida State University, Tallahassee, Florida 32306    G. Fedotov Skobeltsyn Nuclear Physics Institute, 119899 Moscow, Russia    S. Fegan University of Glasgow, Glasgow G12 8QQ, United Kingdom    T.A. Forest Idaho State University, Pocatello, Idaho 83209 Old Dominion University, Norfolk, Virginia 23529    M.Y. Gabrielyan Florida International University, Miami, Florida 33199    G.P. Gilfoyle University of Richmond, Richmond, Virginia 23173    K.L. Giovanetti James Madison University, Harrisonburg, Virginia 22807    F.X. Girod Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606 CEA, Centre de Saclay, Irfu/Service de Physique Nucléaire, 91191 Gif-sur-Yvette, France    J.T. Goetz University of California at Los Angeles, Los Angeles, California 90095-1547    W. Gohn University of Connecticut, Storrs, Connecticut 06269    E. Golovatch Skobeltsyn Nuclear Physics Institute, 119899 Moscow, Russia    R.W. Gothe University of South Carolina, Columbia, South Carolina 29208    M. Guidal Institut de Physique Nucléaire ORSAY, Orsay, France    L. Guo [ Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606    K. Hafidi Argonne National Laboratory, Argonne, Illinois 60441    H. Hakobyan Universidad Técnica Federico Santa María, Casilla 110-V Valparaíso, Chile Yerevan Physics Institute, 375036 Yerevan, Armenia    C. Hanretty Florida State University, Tallahassee, Florida 32306    N. Hassall University of Glasgow, Glasgow G12 8QQ, United Kingdom    D. Heddle Christopher Newport University, Newport News, Virginia 23606 Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606    K. Hicks Ohio University, Athens, Ohio 45701    M. Holtrop University of New Hampshire, Durham, New Hampshire 03824-3568    C.E. Hyde Old Dominion University, Norfolk, Virginia 23529    Y. Ilieva University of South Carolina, Columbia, South Carolina 29208 The George Washington University, Washington, DC 20052    D.G. Ireland University of Glasgow, Glasgow G12 8QQ, United Kingdom    B.S. Ishkhanov Skobeltsyn Nuclear Physics Institute, 119899 Moscow, Russia    E.L. Isupov Skobeltsyn Nuclear Physics Institute, 119899 Moscow, Russia    S.S. Jawalkar College of William and Mary, Williamsburg, Virginia 23187-8795    J.R. Johnstone University of Glasgow, Glasgow G12 8QQ, United Kingdom Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606    D.  Keller Ohio University, Athens, Ohio 45701    M. Khandaker Norfolk State University, Norfolk, Virginia 23504    P. Khetarpal Rensselaer Polytechnic Institute, Troy, New York 12180-3590    A. Klein [ Old Dominion University, Norfolk, Virginia 23529    F.J. Klein Catholic University of America, Washington, D.C. 20064    L.H. Kramer Florida International University, Miami, Florida 33199 Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606    V. Kubarovsky Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606    S.E. Kuhn Old Dominion University, Norfolk, Virginia 23529    S.V. Kuleshov Universidad Técnica Federico Santa María, Casilla 110-V Valparaíso, Chile Institute of Theoretical and Experimental Physics, Moscow, 117259, Russia    V. Kuznetsov Kyungpook National University, Daegu 702-701, Republic of Korea    K. Livingston University of Glasgow, Glasgow G12 8QQ, United Kingdom    H.Y. Lu University of South Carolina, Columbia, South Carolina 29208    M. Mayer Old Dominion University, Norfolk, Virginia 23529    J. McAndrew Edinburgh University, Edinburgh EH9 3JZ, United Kingdom    M.E. McCracken Carnegie Mellon University, Pittsburgh, Pennsylvania 15213    B. McKinnon University of Glasgow, Glasgow G12 8QQ, United Kingdom    C.A. Meyer Carnegie Mellon University, Pittsburgh, Pennsylvania 15213    T Mineeva University of Connecticut, Storrs, Connecticut 06269    M. Mirazita INFN, Laboratori Nazionali di Frascati, 00044 Frascati, Italy    V. Mokeev Skobeltsyn Nuclear Physics Institute, 119899 Moscow, Russia Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606    B. Moreno Institut de Physique Nucléaire ORSAY, Orsay, France    K. Moriya Carnegie Mellon University, Pittsburgh, Pennsylvania 15213    B. Morrison Arizona State University, Tempe, Arizona 85287-1504    H. Moutarde CEA, Centre de Saclay, Irfu/Service de Physique Nucléaire, 91191 Gif-sur-Yvette, France    E. Munevar The George Washington University, Washington, DC 20052    P. Nadel-Turonski Catholic University of America, Washington, D.C. 20064    R. Nasseripour [ University of South Carolina, Columbia, South Carolina 29208 Florida International University, Miami, Florida 33199    C.S. Nepali Old Dominion University, Norfolk, Virginia 23529    S. Niccolai Institut de Physique Nucléaire ORSAY, Orsay, France The George Washington University, Washington, DC 20052    G. Niculescu James Madison University, Harrisonburg, Virginia 22807    I. Niculescu James Madison University, Harrisonburg, Virginia 22807    M.R.  Niroula Old Dominion University, Norfolk, Virginia 23529    M. Osipenko INFN, Sezione di Genova, 16146 Genova, Italy    A.I. Ostrovidov Florida State University, Tallahassee, Florida 32306 University of South Carolina, Columbia, South Carolina 29208    S. Park Florida State University, Tallahassee, Florida 32306    E. Pasyuk Arizona State University, Tempe, Arizona 85287-1504    S. Anefalos Pereira INFN, Laboratori Nazionali di Frascati, 00044 Frascati, Italy    S. Pisano Institut de Physique Nucléaire ORSAY, Orsay, France    O. Pogorelko Institute of Theoretical and Experimental Physics, Moscow, 117259, Russia    S. Pozdniakov Institute of Theoretical and Experimental Physics, Moscow, 117259, Russia    J.W. Price California State University, Dominguez Hills, Carson, CA 90747    S. Procureur CEA, Centre de Saclay, Irfu/Service de Physique Nucléaire, 91191 Gif-sur-Yvette, France    Y. Prok [ University of Virginia, Charlottesville, Virginia 22901    D. Protopopescu University of Glasgow, Glasgow G12 8QQ, United Kingdom University of New Hampshire, Durham, New Hampshire 03824-3568    B.A. Raue Florida International University, Miami, Florida 33199 Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606    G. Ricco INFN, Sezione di Genova, 16146 Genova, Italy    M. Ripani INFN, Sezione di Genova, 16146 Genova, Italy    B.G. Ritchie Arizona State University, Tempe, Arizona 85287-1504    G. Rosner University of Glasgow, Glasgow G12 8QQ, United Kingdom    P. Rossi INFN, Laboratori Nazionali di Frascati, 00044 Frascati, Italy    F. Sabatié CEA, Centre de Saclay, Irfu/Service de Physique Nucléaire, 91191 Gif-sur-Yvette, France    M.S. Saini Florida State University, Tallahassee, Florida 32306    J. Salamanca Idaho State University, Pocatello, Idaho 83209    R.A. Schumacher Carnegie Mellon University, Pittsburgh, Pennsylvania 15213    H. Seraydaryan Old Dominion University, Norfolk, Virginia 23529    N.V. Shvedunov Skobeltsyn Nuclear Physics Institute, 119899 Moscow, Russia    D.I. Sober Catholic University of America, Washington, D.C. 20064    D. Sokhan Edinburgh University, Edinburgh EH9 3JZ, United Kingdom    S.S. Stepanyan Kyungpook National University, Daegu 702-701, Republic of Korea    P. Stoler Rensselaer Polytechnic Institute, Troy, New York 12180-3590    I.I. Strakovsky The George Washington University, Washington, DC 20052    S. Strauch University of South Carolina, Columbia, South Carolina 29208 The George Washington University, Washington, DC 20052    R. Suleiman Massachusetts Institute of Technology, Cambridge, Massachusetts 02139-4307    M. Taiuti INFN, Sezione di Genova, 16146 Genova, Italy    D.J. Tedeschi University of South Carolina, Columbia, South Carolina 29208    S. Tkachenko Old Dominion University, Norfolk, Virginia 23529    M.F. Vineyard Union College, Schenectady, NY 12308    D.P. Watts [ University of Glasgow, Glasgow G12 8QQ, United Kingdom    L.B. Weinstein Old Dominion University, Norfolk, Virginia 23529    D.P. Weygand Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606    M. Williams Carnegie Mellon University, Pittsburgh, Pennsylvania 15213    M.H. Wood Canisius College, Buffalo, NY    L. Zana University of New Hampshire, Durham, New Hampshire 03824-3568    J. Zhang Old Dominion University, Norfolk, Virginia 23529    B. Zhao [ University of Connecticut, Storrs, Connecticut 06269
Abstract

We present results on the electroexcitation of the low mass resonances , , , and in a wide range of . The results were obtained in the comprehensive analysis of JLab-CLAS data on differential cross sections, longitudinally polarized beam asymmetries, and longitudinal target and beam-target asymmetries for electroproduction off the proton. The data were analysed using two conceptually different approaches, fixed- dispersion relations and a unitary isobar model, allowing us to draw conclusions on the model sensitivity of the obtained electrocoupling amplitudes. The amplitudes for the show the importance of a meson-cloud contribution to quantitatively explain the magnetic dipole strength, as well as the electric and scalar quadrupole transitions. They do not show any tendency of approaching the pQCD regime for GeV. For the Roper resonance, , the data provide strong evidence for this state as a predominantly radial excitation of a 3-quark ground state. Measured in pion electroproduction, the transverse helicity amplitude for the allowed us to obtain the branching ratios of this state to the and channels via comparison to the results extracted from electroproduction. The extensive CLAS data also enabled the extraction of the and longitudinal helicity amplitudes with good precision. For the , these results became a challenge for quark models, and may be indicative of large meson-cloud contributions or of representations of this state different from a 3q excitation. The transverse amplitudes for the clearly show the rapid changeover from helicity-3/2 dominance at the real photon point to helicity-1/2 dominance at GeV, confirming a long-standing prediction of the constituent quark model.

pacs:
11.55.Fv, 13.40.Gp, 13.60.Le, 14.20.Gk

Current address:]LPSC-Grenoble, France Current address:]Los Alamos National Laborotory, New Mexico, NM Current address:]Los Alamos National Laborotory, New Mexico, NM Current address:]The George Washington University, Washington, DC 20052 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, Virginia 23187-8795

The CLAS Collaboration

I Introduction

The excitation of nucleon resonances in electromagnetic interactions has long been recognized as an important source of information to understand the strong interaction in the domain of quark confinement. The CLAS detector at Jefferson Lab is the first large acceptance instrument designed for the comprehensive investigation of exclusive electroproduction of mesons with the goal to study the electroexcitation of nucleon resonances in detail. In recent years, a variety of measurements of single pion electroproduction on protons, including polarization measurements, have been performed at CLAS in a wide range of photon virtuality from 0.16 to 6 GeV Joo1 (); Joo2 (); Joo3 (); Egiyan (); Ungaro (); Smith (); Park (); Biselli (). In this work we present the results on the electroexcitation of the resonances , , , and , obtained from the comprehensive analysis of these data.

Theoretical and experimental investigations of the electroexcitation of nucleon resonances have a long history, and along with the hadron masses and nucleon electromagnetic characteristics, the information on the transitions played an important role in the justification of the quark model. However, the picture of the nucleon and its excited states, which at first seemed quite simple and was identified as a model of non-relativistic constituent quarks, turned out to be more complex. One of the reasons for this was the realization that quarks are relativistic objects. A consistent way to perform the relativistic treatment of the transitions is to consider them in the light-front (LF) dynamics Drell (); Terentev (); Brodsky (). The relevant approaches were developed and used to describe the nucleon and its excited states Aznquark (); Aznquark1 (); Weber (); Capstick (); Simula (); Simula1 (); Bruno (); AznRoper (). However, much more effort is required to obtain a better understanding of what are the and LF wave functions and what is their connection to the inter-quark forces and to the QCD confining mechanism. Another reason is connected with the realization that the traditional picture of baryons built from three constituent quarks is an oversimplified approximation. In the case of the and , the mass ordering of these states, the large total width of , and the substantial coupling of to the channel PDG () and to strange particles Liu (); Xie (), are indicative of posible additional components in the wave functions of these states Riska (); An () and (or) of alternative descriptions. Within dynamical reaction models Yang (); Kamalov (); Sato (); Lee (), the meson-cloud contribution is identified as a source of the long-standing discrepancy between the data and constituent quark model predictions for the magnetic-dipole amplitude. The importance of pion (cloud) contributions to the transition form factors has also been confirmed by the lattice calculations Alexandrou (). Alternative descriptions include the representation of as a gluonic baryon excitation Li1 (); Li2 () and the possibility that nucleon resonances are meson-baryon molecules generated in chiral coupled-channel dynamics Weise (); Krehl (); Nieves (); Oset1 (); Lutz (). Relations between baryon electromagnetic form factors and generalized parton distributions (GPDs) have also been formulated that connect these two different notions to describe the baryon structure GPD1 (); GPD2 ().

The improvement in accuracy and reliability of the information on the electroexcitation of the nucleon’s excited states over a large range in photon virtuality is very important for the progress in our understanding of this complex picture of the strong interaction in the domain of quark confinement.

Our goal is to determine in detail the -behavior of the electroexcitation of resonances. For this reason, we analyse the data at each point separately without imposing any constraints on the dependence of the electroexcitation amplitudes. This is in contrast with the analyses by MAID, for instance MAID2007 MAID (), where the electroexcitation amplitudes are in part constrained by using parameterizations for their dependence.

The analysis was performed using two approaches, fixed- dispersion relations (DR) and the unitary isobar model (UIM). The real parts of the amplitudes, which contain a significant part of the non-resonant contributions, are built in these approaches in conceptually different ways. This allows us to draw conclusions on the model sensitivity of the resulting electroexcitation amplitudes.

The paper is organized as follows. In Sec. II, we present the data and discuss the stages of the analysis. The approaches we use to analyse the data, DR and UIM, were successfully employed in analyses of pion-photoproduction and low--electroproduction data, see Refs. Azn0 (); Azn04 (); Azn065 (). In Sec. III we therefore discuss only the points that need different treatment when we move from low to high . Uncertainties of the background contributions related to the pion and nucleon elastic form factors, and to transition form factors are discussed in Sec. IV. In Sec. V, we present how resonance contributions are taken into account and explain how the uncertainties associated with higher resonances and with the uncertainties of masses and widths of the , , and are accounted for. All these uncertainties are included in the total model uncertainty of the final results. So, in addition to the uncertainties in the data, we have accounted for, as much as possible, the model uncertainties of the extracted , , , and amplitudes. The results are presented in Sec. VI, compared with model predictions in Sec. VII, and summarized in Sec. VIII.

Ii Data analysis considerations

The data are presented in Tables 1-4. They cover the first, second, and part of the third resonance regions. The stages of our analysis are dictated by how we evaluate the influence of higher resonances on the extracted amplitudes for the and for the resonances from the second resonance region.

In the first stage, we analyse the data reported in Table 1 (GeV) where the richest set of polarization measurements is available. The results based on the analysis of the cross sections and longitudinally polarized beam asymmetries () at and GeV were already presented in Refs. Azn04 (); Azn065 (). However, recently, new data have become available from the JLab-CLAS measurements of longitudinal target () and beam-target () asymmetries for at GeV Biselli (). For this reason, we performed a new analysis on the same data set, including these new measurements. We also extended our analysis to the available data for the close values of and GeV. As the asymmetries have relatively weak dependences, the data on asymmetries at nearby were also included in the corresponding sets at and GeV. Following our previous analyses Azn04 (); Azn065 (), we have complemented the data set at GeV with the DESY cross sections data Alder (), since the corresponding CLAS data extend over a restricted range in .

In Ref. Azn065 (), the analysis of data at GeV was performed in combination with JLab-CLAS data for double-pion electroproduction off the proton Fedotov (). This allowed us to get information on the electroexcitation amplitudes for the resonances from the third resonance region. This information, combined with the amplitudes known from photoproduction data PDG (), sets the ranges of the higher resonance contributions when we extract the amplitudes of the , , , and transitions from the data reported in Table 1.

In the next step, we analyse the data from Table 2 which present a large body of differential cross sections and longitudinally polarized electron beam asymmetries at large GeV Park (). As the isospin nucleon resonances couple more strongly to the channel, these data provide large sensitivity to the electrocouplings of the , , and states. Until recently, the information on the electroexcitation of these resonances at GeV was based almost exclusively on the (unpublished) DESY data Haidan () on ( and GeV) which have very limited angular coverage. Furthermore, the final state is coupled more weakly to the isospin states, and is dominated by the nearby isospin resonance. For the , which has a large branching ratio to the channel, there is also information on the transverse helicity amplitude found from the data on electroproduction off the proton Armstrong (); Thompson (); Denizli ().

In the range of covered by the data Park () (Table 2), there is no information on the helicity amplitudes for the resonances from the third resonance region. The data Park () cover only part of this region and do not allow us to extract reliably the corresponding amplitudes (except those for ). For the , , and amplitudes extracted from the data Park (), the evaluation of the uncertainties caused by the lack of information on the resonances from the third resonance region is described in Sec. V.

Finally, we extract the amplitudes from the data reported in Tables 3 and 4. These are low data for and electroproduction differential cross sections Smith () and data for electroproduction differential cross sections at GeV Joo1 () and GeV Ungaro (). In the analysis of these data, the influence of higher resonances on the results for the was evaluated by employing the spread of the , , and amplitudes obtained in the previous stages of our analysis of the data from Tables 1 and 2.

Although the data for GeV (Table 4) cover a wide range in , the absence of electroproduction data for these , except GeV, does not allow us to extract the amplitudes for the , , resonances with model uncertainties comparable to those for the amplitudes found from the data of Tables 1 and 2. For GeV, there are DESY electroproduction data Alder (), which cover the second and third resonance regions, allowing us to extract amplitudes for all resonances from the first and second resonance regions at GeV. To evaluate the uncertainties caused by the higher mass resonances, we have used for GeV the same procedure as for the data from Table 2.

Number
of data
Obser- points Ref.
vable (GeV) (GeV) () DR UIM
0.3 1.1-1.55 2364 2.06 1.93 Egiyan ()
0.252 1.125-1.55 594 1.36 1.48 Biselli ()
0.252 1.125-1.55 598 1.19 1.23 Biselli ()
0.4 1.1-1.68 3530 1.23 1.24 Joo1 ()
0.4 1.1-1.55 2308 1.92 1.64 Egiyan ()
0.4 1.1-1.66 956 1.24 1.18 Joo2 ()
0.4 1.1-1.66 918 1.28 1.19 Joo3 ()
0.385 1.125-1.55 696 1.40 1.61 Biselli ()
0.385 1.125-1.55 692 1.22 1.25 Biselli ()
0.525 1.1-1.66 3377 1.33 1.35 Joo1 ()
0.5 1.1-1.51 2158 1.51 1.48 Egiyan ()
0.65 1.1-1.68 6149 1.09 1.14 Joo1 ()
0.6 1.1-1.41 1484 1.21 1.24 Egiyan ()
1.4-1.76 477 1.72 1.74 Alder ()
0.65 1.1-1.66 805 1.09 1.13 Joo2 ()
0.65 1.1-1.66 812 1.09 1.04 Joo3 ()
0.611 1.125-1.55 930 1.38 1.40 Biselli ()
0.611 1.125-1.55 923 1.26 1.28 Biselli ()
Table 1: The data sets included in the first stage of the analysis, as discussed in the text. The columns corresponding to DR and UIM show the results for per data point obtained, respectively, using fixed- dispersions relations and the unitary isobar model described in Sec. III.
Number of
Obser- data points
vable (GeV) (GeV) () DR UIM
1.72 1.11-1.69 3530 2.3 2.5
2.05 1.11-1.69 5123 2.3 2.2
2.44 1.11-1.69 5452 2.0 2.0
2.91 1.11-1.69 5484 1.9 2.1
3.48 1.11-1.69 5482 1.3 1.4
4.16 1.11-1.69 5778 1.1 1.1
1.72 1.12-1.68 699 2.9 3.0
2.05 1.12-1.68 721 3.0 2.9
2.44 1.12-1.68 725 3.0 3.0
2.91 1.12-1.68 767 2.7 2.7
3.48 1.12-1.68 623 2.4 2.3
Table 2: The data from Ref. Park ().
Number of
Obser- data points
vable (GeV) (GeV) () DR UIM
0.16 1.1-1.38 3301 1.96 1.98
0.16 1.1-1.38 2909 1.69 1.67
0.20 1.1-1.38 3292 2.29 2.24
0.20 1.1-1.38 2939 1.76 1.78
0.24 1.1-1.38 3086 1.86 1.82
0.24 1.1-1.38 2951 1.49 1.46
0.28 1.1-1.38 2876 1.56 1.59
0.28 1.1-1.38 2941 1.47 1.44
0.32 1.1-1.38 2836 1.51 1.48
0.32 1.1-1.38 2922 1.39 1.37
0.36 1.1-1.38 2576 1.46 1.42
0.36 1.1-1.38 2611 1.35 1.38
Table 3: The low data from Ref. Smith () analysed in the third stage of the analysis.
Number of
Obser- data points
vable (GeV) (GeV) () DR UIM
0.75 1.1-1.68 3555 1.16 1.18
0.9 1.1-1.68 3378 1.22 1.25
1.36-1.76 725 1.62 1.66
1.15 1.1-1.68 1796 1.09 1.15
1.45 1.1-1.62 1878 1.15 1.18
3 1.11-1.39 1800 1.41 1.37
3.5 1.11-1.39 1800 1.22 1.24
4.2 1.11-1.39 1800 1.16 1.19
5 1.11-1.39 1800 0.82 0.88
6 1.11-1.39 1800 0.66 0.67
Table 4: The data included in the third stage of the analysis: the data for at and GeV are from Refs. Joo1 () and Park (), respectively; the data for are from Ref. Alder ().

Iii Analysis approaches

The approaches we use to analyse the data, DR and UIM, are described in detail in Refs. Azn0 (); Azn04 () and were successfully employed in Refs. Azn0 (); Azn04 (); Azn065 () for the analyses of pion-photoproduction and low--electroproduction data. In this Section we discuss certain aspects in these approaches that need a different treatment as we move to higher .

iii.1 Dispersion relations

We use fixed- dispersion relations for invariant amplitudes defined in accordance with the following definition of the electromagnetic current for the process Devenish ():

(1)
(2)

where are the four-momenta of the virtual photon, pion, and initial and final nucleons, respectively; are the invariant amplitudes that are functions of the invariant variables ; , are the Dirac spinors of the initial and final state nucleon, and is the pion field.

The conservation of leads to the relations:

(3)
(4)

where . Therefore, only six of the eight invariant amplitudes are independent. In Ref. Azn0 (), the following independent amplitudes were chosen: . Taking into account the isotopic structure, we have 18 independent invariant amplitudes. For the amplitudes , unsubtracted dispersion relations at fixed can be written. The only exception is the amplitude , for which a subtraction is neccessary:

(5)

where , , is the pion form factor, is the nucleon Pauli form factor, and and are the nucleon and pion masses, respectively.

At , using the relation , which follows from Eq. (3), and DR for the amplitude , one obtains:

(6)

where .

This expression for was successfully used for the analysis of pion photoproduction and low 0.4, 0.65 GeV electroproduction data Azn0 (); Azn04 (). However, it turned out that it is not suitable at higher . Using a simple parametrization:

(7)

a suitable subtraction was found from the fit to the data for GeV Park (). The linear parametrization in is also consistent with the subtraction found from Eq. (6) at low . Fig. 1 demonstrates smooth transition of the results for the coefficients found at low GeV using Eq. (6) to those at large GeV found from the fit to the data Park ().

Figure 1: dependence of the coefficients (solid curve) and (dashed curve) from Eq. (7). The results for GeV were found using Eq. (6), whereas the results for GeV are from the fit to the data Park ().

Fig. 2 shows the relative contribution of compared with the pion contribution in Eq. (5) at and GeV. It can be seen that the contribution of is comparable with the pion contribution only at large , where the latter is small. At small , is very small compared to the pion contribution.

Figure 2: The pion contribution in GeV units (solid curves) to the DR for the amplitude , Eq. (5), compared to at (a) and GeV (b). The dashed curves represent taken in the form of Eq. (6), the dash-dotted curve corresponds to the results for obtained by fitting the data Park (). At GeV, the physical region is located on the right side of the dotted vertical line.

iii.2 Unitary isobar model

The UIM of Ref. Azn0 () was developed on the basis of the model of Ref. Drechsel (). One of the modifications made in Ref. Azn0 () consisted in the incorporation of Regge poles with increasing energies. This allowed us to describe pion photoproduction multipole amplitudes GWU0 (); GWU3 () with a unified Breit-Wigner parametrization of resonance contributions in the form close to that introduced by Walker Walker (). The Regge-pole amplitudes were constructed using a gauge invariant Regge-trajectory-exchange model developed in Refs. Laget1 (); Laget2 (). This model gives a good description of the pion photoproduction data above the resonance region and can be extended to finite Laget3 ().

The incorporation of Regge poles into the background of UIM, built from the nucleon exchanges in the - and -channels and -channel , and exchanges, was made in Ref. Azn0 () in the following way:

(8)

Here the Regge-pole amplitudes were taken from Refs. Laget1 (); Laget2 () and consisted of reggeized , , , , and -channel exchange contributions. This background was unitarized in the -matrix approximation. The value of GeV was found in Ref. Azn0 () from the description of the pion photoproduction multipole amplitudes GWU0 (); GWU3 (). With this value of , we obtained a good description of electroproduction data at and GeV in the first, second and third resonance regions Azn04 (); Azn065 (). The modification of Eq. (8) was important to obtain a better description of the data in the second and third resonance regions, but played an insignificant role at GeV.

When the relation in Eq. (8) was applied for GeV, the best description of the data was obtained with GeV. Consequently, in the analysis of the data Park (), the background of UIM was built just from the nucleon exchanges in the - and -channels and -channel , and exchanges.

Iv contributions

In both approaches, DR and UIM, the non-resonant background contains Born terms corresponding to the - and -channel nucleon exchanges and -channel pion contribution, and therefore depends on the proton, neutron, and pion form factors. The background of the UIM also contains the and -channel exchanges and, therefore, the contribution of the form factors . All these form factors, except the neutron electric and ones, are known in the region of that is the subject of this study. For the proton form factors we used the parametrizations found for the existing data in Ref. Melnitchouk (). The neutron magnetic form factor and the pion form factor were taken from Refs. Lung (); Lachniet () and Bebek1 (); Bebek2 (); Horn (); Tadevos (), respectively. The neutron electric form factor, , is measured up to GeV Madey (), and Ref. Madey () presents a parametrization for all existing data on , which we used for the extrapolation of to GeV. In our final results at high , we allow for up to a deviation from this parametrization that is accounted for in the systematic uncertainty. There are no measurements of the form factors ; however, investigations made using both QCD sum rules Eletski () and a quark model AznOgan () predict a dependence of close to the dipole form . We used this dipole form in our analysis and introduced in our final results a systematic uncertainty that accounts for a deviation from GeV. All uncertainties, including those arising from the measured proton, neutron and pion form factors, were added in quadrature and will be, as one part of our total model uncertainties, referenced as model uncertainties (I) of our results.

V Resonance contributions

We have taken into account all well-established resonances from the first, second, and third resonance regions. These are 4- and 3-star resonances: , , , , , , , , , , , , and . For the masses, widths, and branching ratios of these resonances we used the mean values of the data from the Review of Particle Physics (RPP) PDG (). They are presented in Table 5. Resonances of the fourth resonance region have no influence in the energy region under investigation and were not included.

Resonance contributions to the multipole amplitudes were parametrized in the usual Breit-Wigner form with energy-dependent widths Walker (). An exception was made for the resonance, which was treated differently. According to the phase-shift analyses of scattering, the amplitude corresponding to the resonance is elastic up to GeV (see, for example, the latest GWU analyses GWU1 (); GWU2 ()). In combination with DR and Watson’s theorem, this provides strict constraints on the multipole amplitudes , , that correspond to the resonance Azn0 (). In particular, it was shown Azn0 () that with increasing , the -dependence of remains unchanged and close to that from the GWU analysis GWU3 () at , if the same normalizations of the amplitudes at the resonance position are used. This constraint on the large amplitude plays an important role in the reliable extraction of the amplitudes for the transition. It also impacts the analysis of the second resonance region, because resonances from this region overlap with the .

(MeV) (MeV) (MeV) (MeV)
Table 5: List of masses, widths, and branching ratios of the resonances included in our analysis. The quoted ranges are taken from RPP PDG (). The quantities labeled by tildes (, , ) correspond to the values used in the analysis and in the extraction of the helicity amplitudes.

The fitting parameters in our analyses were the helicity amplitudes, , , . They are related to the resonant portions of the multipole amplitudes at the resonance positions. For the resonances with , these relations are the following:

(9)
(10)
(11)

For the resonances with :

(12)
(13)
(14)

where and are the spin and parity of the resonance, , and

(15)

Here are the isospin Clebsch-Gordon coefficients in the decay ; , , and are the total width, mass, and isospin of the resonance, respectively, is its branching ratio to the channel, and are the photon equivalent energy and the pion momentum at the resonance position in c.m. system. For the transverse amplitudes and , these relations were introduced by Walker Walker (); for the longitudinal amplitudes, they agree with those from Refs. Arndt (); Capstick (); Kamalov1 ().

The masses, widths, and branching ratios of the resonances are known in the ranges presented in Table 5. The uncertainties of masses and widths of the , , and are quite significant and can affect the resonant portions of the multipole amplitudes for these resonances at the resonance positions. These uncertainties were taken into account by refitting the data multiple times with the width (mass) of each of the resonances changed within one standard deviation111The standard deviations were defined as and , with the maximum and minimum values as shown in Table V. while keeping those for other resonances fixed. The resulting uncertainties of the , , amplitudes were added in quadrature and considered as model uncertainties (II).

In Sec. II, we discussed that in the analysis of the data reported in Table 2, there is another uncertainty in the amplitudes for the , , and , which is caused by the limited information available on magnitudes of resonant amplitudes in the third resonance region. To evaluate the influence of these states on the extracted , , amplitudes, we used two ways of estimating their strength.

(i) Directly including these states in the fit, taking the corresponding amplitudes , , as free parameters.

(ii) Applying some constraints on their amplitudes. Using symmetry relations within the multiplet given by the single quark transition model SQTM (), we have related the transverse amplitudes for the members of this multiplet (, , , , and ) to the amplitudes of and that are well determined in the analysis. The longitudinal amplitudes of these resonances and the amplitudes of the resonances and , which have small photocouplings PDG () and are not seen in low and 2 electroproduction Azn065 (), were assumed to be zero.

The results obtained for , , and using the two procedures are very close to each other. The amplitudes for these resonances presented below are the average values of the results obtained in these fits. The uncertainties arising from this averaging procedure were added in quadrature to the model uncertainties (II).

Vi Results

Results for the extracted , , , amplitudes are presented in Tables 6-12. Here we show separately the amplitudes obtained in the DR and UIM approaches. The amplitudes are presented with the fit errors and model uncertainties caused by the , and contributions to the background, and those caused by the masses and widths of the , , and , and by the resonances of the third resonance region. These uncertainties, discussed in Sections IV and V, and referred to as model uncertainties (I) and (II), were added in quadrature and represent model uncertainties of the DR and UIM results.

The DR and UIM approaches give comparable descriptions of the data (see values in Tables 1-4), and, therefore, the differences in are related only to the model assumptions. We, therefore, ascribe the difference in the results obtained in the two approaches to model uncertainty, and present as our final results in Tables 6-10 and 12 the mean values of the amplitudes extracted using DR and UIM. The uncertainty that originates from the averaging is considered as an additional model uncertainty - uncertainty (III). Along with the average values of the uncertainties (I) and (II) obtained in the DR and UIM approaches, it is included in quadrature in the total model uncertainties of the average amplitudes.

In the fit we have included the experimental point-to-point systematics by adding them in quadrature with the statistical error. We also took into account the overall normalization error of the CLAS cross sections data which is about 5%. It was checked that the overall normalization error results in modifications of all extracted amplitudes, except , that are significantly smaller than the fit errors of these amplitudes. For , this error results in the overall normalization error which is larger than the fit error. It is about 2.5% for low , and increases up to 3.2-3.3% at GeV. For , the fit error given in Table 6 includes the overall normalization error added in quadrature to the fit error.

Examples of the comparison with the experimental data are presented in Figs. 3-12. The obtained values of in the fit to the data are presented in Tables 1-4. The relatively large values of for at GeV and for at GeV and GeV are caused by small statistical errors, which for each data set Smith (), Egiyan () and Park (), increase with increasing . The values of for at GeV are somewhat large. However, as demonstrated in Figs. 5,6, the description on the whole is satisfactory.

(), W=1.232 GeV
(GeV)
DRUIM Final results
0.3
0.4
0.525
0.65
0.75
0.9
1.15