Electroexcitation of nucleon resonances
We review recent progress in the investigation of the electroexcitation of nucleon resonances, both in experiment and in theory. We describe current experimental facilities, the experiments performed on and electroproduction off protons, and theoretical approaches used for the extraction of resonance contributions from the experimental data. The status of , , and electroproduction is also presented. The most accurate results have been obtained for the electroexcitation amplitudes of the four lowest excited states, which have been measured in a range of up to and GeV for the , and , , respectively. These results have been confronted with calculations based on lattice QCD, large- relations, perturbative QCD (pQCD), and QCD-inspired models. The amplitudes for the indicate large pion-cloud contributions at low and don’t show any sign of approaching the pQCD regime for GeV. Measured for the first time, the electroexcitation amplitudes of the Roper resonance, , provide strong evidence for this state as a predominantly radial excitation of a three-quark (3) ground state, with additional non-3-quark contributions needed to describe the low behavior of the amplitudes. The longitudinal transition amplitude for the was determined and has become a challenge for quark models. Explanations may require large meson-cloud contributions or alternative representations of this state. The clearly shows 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. The interpretation of the moments of resonance transition form factors in terms of transition transverse charge distributions in infinite momentum frame is presented.
The excitation of nucleon resonances in electromagnetic interactions has long been recognized as an important source of information for understanding strong interactions in the domain of quark confinement. Theoretical and experimental investigations of the electroexcitation of nucleon resonances have a long history. Early investigations in the 1960’s to 1980’s were based on experiments at the electron beam accelerators DESY at Hamburg in Germany, NINA at Daresbury in the UK, and at the University of Bonn in Germany. At the real photon point, systematic measurements were made at these facilities (including also the electron accelerators at Yerevan and Char’kov), which included a variety of polarization experiments along with measurements of differential cross sections. Phenomenological analyses of the data were able to extract information on the transition amplitudes for the well-established resonances with masses below GeV. The situation for virtual photons was different; only sparse data on differential cross sections for the reactions and were obtained for photon virtualities up to GeV. The data provided limited information, with large systematic differences among the various data sets, on the magnetic-dipole amplitude and on the transverse , , and amplitudes. One of the most interesting questions for the is its possible quadrupole deformation, which can be revealed through measurements of the electric-quadrupole and scalar amplitudes. The sensitivity of the data to these amplitudes was very limited, as was their sensitivity to the , , and longitudinal amplitudes; in fact, these quantities have not been determined. The theoretical scheme for the interpretation of the amplitudes extracted from experimental data in the 1960’s to the 1980’s was based on the constituent quark model (CQM) and the single quark transition model (SQTM). A review of these early data, the extracted amplitudes, and their theoretical interpretation at this stage of electroproduction experiments can be found in Refs. [1, 2].
The experimental situation changed dramatically with the advent of the new generation of electron beam facilities - the Continuous Electron Beam Accelerator Facility (CEBAF) at the Thomas Jefferson National Accelerator Facility (JLab), Mainz Microtron (MAMI) at Mainz, and the MIT/Bates out-of-plane scattering (OOPS) facility. Large amounts of significantly more precise and complete data were collected, in both pion and eta electroproduction off protons in the first, second, and third resonance regions in the range of GeV. For pion electroproduction, measurements of differential cross sections along with a variety of polarization experiments were performed. The list of new measurements of pion and eta electroproduction is given in Table 1. The majority of new data was obtained at JLab, in particular with the CEBAF Large Acceptance Spectrometer (CLAS) in Hall B. The MAMI and MIT/Bates experiments consist of measurements of and in the vicinity of the resonance at small GeV. Due to the new measurements, for the first time, electrocoupling amplitudes of the Roper resonance have been extracted from experimental data, as well the electric-quadrupole and scalar amplitudes and the and longitudinal amplitudes. Overall, accurate results have been obtained for the amplitudes of the and transitions up to GeV, and of the and transitions up to GeV. Experimental and theoretical advances on early stages of these investigations are reviewed in Refs. [3, 4].
Progress in the experimental investigation of the electroexcitation of nucleon resonances was accompanied by significant developments in understanding of QCD, including the domain of quark confinement. This made it possible in some cases to set relations between the properties of QCD found from first principles and the amplitudes extracted from experimental data. Below we list those relations that are directly connected to the results on the amplitudes discussed in this review.
Spontaneous chiral symmetry breaking in QCD leads to the existence of nearly massless Goldstone bosons (pions). As a consequence, there can be significant pion-loop contributions to the electromagnetic form factors at relatively small momentun transfer. These contributions are crucial for the description of the neutron electric form factor in CQM and bag models [32, 33, 34, 35, 36] and are essential for the transition amplitudes [37, 38, 39, 40]. The importance of the pion-cloud contributions to the transition form factors has been confirmed by lattice QCD calculations , where at small they modify the quenched results in agreement with expectations from chiral perturbation theory . The meson-cloud contribution is also identified as a source of the long-standing discrepancy between the data and CQM predictions for the magnetic-dipole amplitude within dynamical reaction models [43, 44, 45, 46]. From the results presented in this review it will be seen that complementing of the quark core contribution by that of the pion cloud can be necessary also for the correct description of the , , and amplitudes extracted from experimental data.
|JLab/Hall A||1.1 - 1.95|||
|for||1.17 - 1.35||[6, 7]|
|JLab/Hall B||0.16 - 0.36||1.1 - 1.38|||
|0.4 - 1.8||1.1 - 1.68|||
|3.0 - 6.0||1.1 - 1.39|||
|0.4, 0.65||1.1 - 1.66|||
|0.252, 0.385, 0.611||1.12 - 1.55|||
|0.3 - 0.6||1.1 - 1.55|||
|1.7 - 4.5||1.11 - 1.69|||
|0.4, 0.65||1.1 - 1.66|||
|0.375 - 1.385||1.5 - 1.86|||
|0.17 - 3.1||1.5 - 2.3|||
|JLab/Hall C||2.8, 4.2||1.115 - 1.385|||
|6.4, 7.7||1.11 - 1.39|||
|2.4 3.6||1.49 - 1.62|||
|5.7 7.0||1.5 - 1.8|||
|MAMI||1.22 - 1.3||[22, 23, 24]|
|MIT/Bates||1.23||[28, 29, 30]|
is a longitudinally polarized beam asymmetry for , and are longitudinal-target and beam-target asymmetries for , is a polarization of the final proton in the corresponding reactions, and is a longitudinal-transverse structure function.
The expansion introduced by ’t Hooft  and Witten  has been shown to be a powerful tool for exposing properties of QCD in the non-perturbative domain. It led to the understanding of baryon properties, such as ground-state and excited baryon masses, as well as their magnetic moments and electromagnetic transitions (see Refs. [49, 50, 51, 52] and references therein). In this review we will demonstrate good agreement between the amplitudes extracted from experimental data and recent predictions obtained in the large limit [53, 54, 55]. The predictions are made for a wide range of . In particular, for the magnetic-dipole amplitude, they extend up to GeV.
In recent years there has been significant progress in lattice QCD calculations by using a number of different fermion discretization schemes and pion masses reaching closer to the physical pion mass (the review can be found in Ref. ). Significant effort has been made to get consistent results for the benchmark transition. Recent predictions have been shown to be quite definite and in qualitative agreement with experimental data . There are also first exploratory calculations of the amplitudes , which need improvement using smaller pion mass values and employ an unquenched approximation.
Another approach, which can be considered as a tool that relates the first-principles properties of QCD to the amplitudes, is presented in Ref. . In this approach, the light-cone distribution amplitudes found through lattice calculations have been used to calculate the transition amplitudes by utilization of light-cone sum rules. At GeV, the predictions are in quite good agreement with the amplitudes extracted from experimental data.
The CQM remains a useful tool for understanding of the internal structure of hadrons and of their interactions. The majority of experimentally observed hadrons can be classified according to the group . The string model for confinement forces plus the associated spin-orbit interactions, as well as the interactions expected from the one-gluon exchange between quarks, approximately describe the mass spectrum of hadrons [59, 60] and their widths [59, 61]. However, there are well known shortcomings of this picture. These include the wrong mass ordering between the and , difficulties in the description of large width of the , and the large branching ratio of the to the channel. It was demonstrated in Refs. [62, 63] that extension of the quark model by inclusion of the lowest lying components can in principle overcome these problems. For example, agreement with the empirical value of the decay width for the can be reached with an component in this state . For the , it was found that the most likely lowest energy configuration is given by the component . This could solve the problem of mass ordering between the and , and explain the large couplings of the to , as well as the recently observed large couplings of this state to the and channels [64, 65].
To deal with the shortcomings of the CQM in the case of the , an alternative description of this resonance was proposed by treating it as a hybrid state [66, 67]. This possibility was motivated by the fact that in the bag model the lightest hybrid state has quantum numbers of the Roper resonance, and its mass can be GeV . Another alternative representation of the nucleon resonances, including the and , is the possibility that they are meson-baryon molecules generated in chiral coupled-channel dynamics [69, 70, 71, 72, 73].
In this review we present and discuss the predictions from alternative approaches for the and transitions, as well as the results of extended versions of the CQM. This will allow us to draw some conclusions as to the internal structure and nature of these resonances.
The information on the , , , and transition amplitudes, extracted from experimental data in a wide range of , is of great interest for understanding of the scale where the asymptotic domain of QCD may set in for these transitions. QCD in the asymptotic limit puts clear restrictions on the behavior of the transition amplitudes. They follow from hadron helicity conservation  and dimensional counting rules [75, 76, 77, 78, 79]. We compare the dependence of the amplitudes extracted from experimental data with the predictions of pQCD.
Empirical knowledge of the transition amplitudes in a wide range of also allows mapping out of the quark transverse charge distributions that induce these transitions [80, 81, 82]. These distributions will be presented and discussed in the review.
The results presented in this review are related mostly to the , , , and transition amplitudes extracted in and electroproduction. Recently published CLAS measurements [83, 84] present significant progress in the investigation of two-pion electroproduction, which is one of the biggest contributors to the process of electroproduction in the resonance energy region. This channel becomes increasingly important for high-lying resonances with masses above GeV. Evaluation of the electrocouplings from the CLAS two-pion electroproduction data is now in progress. There are already preliminary results that may be found in Refs. [85, 86, 87] and will be shown when presenting the results extracted from and electroproduction. Two-pion electroproduction as well electroproduction of and are intensively investigated with CLAS at JLab [88, 89, 90, 91]. These are channels with potential for the discovery of some of the so-called “missing” resonances, the states that are predicted by the CQM, however, are weakly coupled to and , and by this reason are not observed in and production. According to the quark model predictions [61, 93], some of these resonances may be more efficiently studied in the photo- and electroproduction of , and systems.
The paper is organized as follows. In section 2, we present the facilities and setups where the electroexcitation of nucleon resonances reported in this review have been investigated. In section 3, we present the definitions related to the kinematics and formalism of the reaction . Special attention is paid to the relations between different definitions of the helicity amplitudes: through the multipole amplitudes, through the matrix elements of the electromagnetic current, and through the form factors. It is known that the helicity amplitudes extracted from experimental data include the sign of the vertex. We present explicit relations that account for this sign. In section 4, we give a brief review of theoretical approaches that are employed in the analyses of photo- and electroproduction reactions in the resonance energy region. Approaches that have been used in the extraction of the electroexcitation amplitudes reported in this review are presented in more detail. In section 5, we describe the experiments performed on the new generation of electron accelerators, list the approaches used in the analyses of the experimental data, and present examples of the theoretical description of the data. The main results are discussed in section 6. Here we present the , , , and transition amplitudes as determined in the most recent analyses of the new data, and discuss the progress achieved due to the new experiments. We also perform some detailed comparison with theoretical models, including developments in understanding of QCD in the domain of quark confinement. We also discuss results related to the quark transverse charge distributions in the transitions and conclusions on the approach to the pQCD asymptotic regime. Finally, in section 7, we present and discuss results related to the third resonance region, before we conclude with some future prospects in section 8.
2 Experimental Facilities
2.1 Thomas Jefferson National Accelerator Facility
The Thomas Jefferson National Accelerator Facility in Newport News, Virginia, operates a CW electron accelerator with energies in the range up to 6 GeV . Three experimental Halls receive highly polarized electron beams with the same energies or with different but correlated energies. Beam currents in the range from 0.1 nA to 150 A can be delivered to the experiments simultaneously. In addition, the development of polarized nucleon targets that can be used in fairly intense electron beams, as well as use of recoil polarimeters in magnetic spectrometers, has provided access to a previously unavailable set of observables that are sensitive to the interference of resonant and non-resonant processes.
2.1.1 Experimental Hall A - HRS
Hall A houses a pair of identical focusing high resolution magnetic spectrometers (HRS) , each with a momentum resolution of ; one of them is instrumented with a gas Ĉerenkov counter and a shower counter for the identification of electrons. The hadron arm is instrumented with a proton recoil polarimeter. The detector package allows identification of charged pions, kaons, and protons. The pair of spectrometers can be operated at very high beam currents of up to 100 A. The HRS spectrometers have been used to measure the reaction in the resonance region [6, 7]. The excellent momentum resolution allows efficient use of the “missing mass” technique, where the undetected is inferred from the overdetermined kinematics. Due to the small angle and momentum acceptance, the angle and momentum settings have to be changed many times to cover the full kinematical range of interest. These data have been used to extract a large number of single and double polarization response functions for specific kinematics.
2.1.2 Experimental Hall B - CLAS
Hall B houses the CEBAF Large Acceptance Spectrometer (CLAS) detector and a photon energy tagging facility . CLAS can be operated with electron beams and with energy tagged photon beams. The detector system was designed with the detection of multiple particle final states in mind. The driving motivation for the construction of CLAS was the nucleon resonance () program, with emphasis on the study of the transition form factors and the search for missing resonances. Figure 1 shows the CLAS detector. At the core of the detector is a toroidal magnet consisting of six superconducting coils symmetrically arranged around the beam line. Each of the six sectors is instrumented as an independent spectrometer with 34 layers of tracking chambers allowing for the full reconstruction of the charged particle 3-momentum vectors. Charged hadron identification is accomplished by combining momentum and time-of-flight with the measured path length from the target to the plastic scintillation counters that surround the entire tracking region. The wide range of particle identification allows for study of the complete range of reactions relevant to the program. In the polar angle range of up to 70, photons and neutrons can be detected using the electromagnetic calorimeters. The forward angular range from about 10 to 50 is instrumented with gas Ĉerenkov counters for the identification of electrons.
In the program, CLAS is often used as a “missing mass” spectrometer, where all final state particles except one particle are detected. The undetected particle is inferred through the overdetermined kinematics, making use of the good momentum () and angle () resolution. Figure 2 shows an example of the kinematics covered in the reaction . It shows the invariant hadronic mass versus the missing mass . The undetected particles , , and are clearly visible as bands of constant . The correlation of certain final states with specific resonance excitations is also seen.
2.1.3 Experimental Hall C - HMS and SOS
Hall C houses the high momentum spectrometer (HMS) and the short orbit spectrometer (SOS). The HMS reaches a maximum momentum of 7 GeV, while the SOS is limited to about 1.8 GeV. The spectrometer pair has been used to measure the [18, 19] and transitions at high [20, 21]. For these kinematics the SOS was used as an electron spectrometer and the HMS to detect the proton. To achieve a large kinematic coverage, the spectrometers have to be moved in angle, and the spectrometer optics have to be adjusted to accommodate different particle momenta. This makes such a two-spectrometer setup most useful for studying meson production at high momentum transfer, or close to threshold. In either case, the Lorentz boost guarantees that particles are produced in a relatively narrow cone around the virtual photon, and can be detected in magnetic spectrometers with relatively small solid angles.
The MAMI-B microtron electron accelerator  at Mainz in Germany reaches a maximum beam energy of 850 MeV, and produces a highly polarized and stable electron beam with excellent beam properties. The recently upgraded MAMI-C machine reaches a maximum electron energy of 1.55 GeV. There are experimental areas for electron scattering experiments with three focusing magnetic spectrometers with high resolution [98, 99]. A two-spectrometer configuration has been used in cross section and polarization asymmetry measurements of electroproduction from protons in the region [22, 23, 24, 25, 26, 27].
The Bates 850 MeV linear electron accelerator has been used to study production in the resonance region using an out-of-plane spectrometer setup . A set of four independent focusing spectrometers was used to measure various response functions, including the beam helicity-dependent out-of-plane response function. Because of the small solid angles covered by this setup, a limited range of the polar angles in the center-of-mass frame of the subsystem could be covered. These spectrometers are no longer in use, but results recently published from earlier data taking are included in this review [28, 29, 30, 31].
3 Definitions and Conventions
The results on the electroexcitation of nucleon resonances reported in this review are based mostly on the experiments on pion and eta electroproduction off nucleons. We therefore only present the definitions that are important for extraction and presentation of the results for these reactions. Throughout we use natural units, , so that momenta and masses are expressed in units of GeV (rather than GeV or GeV). We also use the following conventions for the metric and -matrices: diag(1,-1,-1,-1), , , =2, . More explicitly, -matrices have the following form:
The differential cross section of the electroproduction of pions off nucleons in the one-photon approximation (Fig. 3) is related to the differential cross section of the production of pions by virtual photons in the standard way (see e.g. Refs. [101, 102]) through the virtual photon flux as:
is the fine structure constant, and are the initial and final electron energies in the laboratory frame, , is the polarization factor of the virtual photon, is the laboratory solid angle of the scattered electron, and is the pion solid angle in the c.m. system of the reaction , where is the angle between the pion and virtual photon in this system, and is the angle between the electron scattering and hadron production planes. The virtuality of the photon is given by . Since the photon is spacelike, i.e. , it is convenient to work with the positive quantity . The invariant mass squared of the final hadronic state (here, and ) is , where and are the target nucleon and virtual photon four momenta, and is the nucleon mass.
For unpolarized particles and for longitudinally polarized electron beam, the -dependence of the cross section can be specified in the following way:
Here we use notations of Ref. , , , , and are the so-called structure functions of the reaction that depend on , and describes the longitudinal polarization of the incident electron: if electrons are polarized parallel (anti-parallel) to the beam direction. For longitudinally polarized electron beam and polarized target and recoil nucleons, the relevant formulas can be found in Refs. [101, 102].
3.2 Expansion over multipole amplitudes
In order to extract resonance contributions from the data on the reaction , the observables should be defined through the multipole amplitudes. These are transverse amplitudes and and scalar(longitudinal) amplitudes (); they are related, respectively, to the photons of the magnetic, electric, and Coulombic type; is the angular monentum of pion in the c.m. system of the reaction . For this purpose, it is convenient to introduce transverse partial wave helicity amplitudes:
The amplitudes are related to the helicity amplitudes in the center-of-mass system (c.m.s.) of the reaction in the following way:
where are the elements of the matrices ,
Here and are the initial and final nucleon helicities, and is the photon helicity.
The structure functions given in Eq. (4) are related to the helicity amplitudes by:
where and and are, respectively, the photon equivalent energy and the virtual photon and pion 3-momenta in the c.m.s.
The total cross section can be written through partial wave helicity amplitudes in the compact way:
3.3 Definition of the helicity amplitudes
Experimental results on the helicity amplitudes (transverse amplitudes and and scalar (or longitudinal) amplitude ), extracted from the data on , correspond to the contribution of diagram (d) in Fig. 4 to this reaction. They are related to the resonant portions of the corresponding multipole amplitudes at the resonance positions in the following way:
, , and are, respectively, the total width, mass, spin and isospin of the resonance, for amplitudes, is the branching ratio of the resonance to the channel, and are the photon equivalent energy and the pion 3-momentum at the resonance position in the c.m.s. of , and are the isospin Clebsch-Gordon coefficients in the decay :
where we have taken into account that the pion isomultiplet is .
According to the definitions (23-25), the helicity amplitudes extracted from the data on the reaction contain the sign of the vertex; it defines the relative sign of the diagrams that correspond to the resonance (Fig. 4d) and Born terms (Figs. 4a,b,c) contributions to . The situation is analogous in other reactions. For example, the helicity amplitudes extracted from the data on the reaction contain the sign of the vertex.
In the calculations of the helicity amplitudes in theoretical approaches, the commonly used definition relates these amplitudes to the matrix elements of the electromagnetic current:
where , and it is assumed that the -axis is directed along the photon 3-momentum () in the rest frame, and are the projections of the nucleon and resonance spins on this axis, and
where is the sign that reflects the presence of the vertex in Fig. 4d. The relation between and the sign of the ratio of the and coupling constants was found in Ref.  using the results of covariant calculations of the Fig. 4 diagrams in Ref. . With the definitions
3.4 The helicity amplitudes in terms of the form factors
Many theoretical approaches, e.g. light-front relativistic quark models, QCD sum rules, and lattice QCD, use the definition of the helicity amplitudes in terms of the form factors. In this section, we present the definition introduced in Ref. , where the form factors were defined in a unified way for all resonances with The definition of Ref.  for the resonances coincides with the widely used definition by Jones and Scadron .
For the resonances, the definition of Ref.  for the matrix element is:
For the ,… resonances the definitions are more combersome: