Extended SAID Partial-Wave Analysis of Pion Photoproduction

A unified Chew-Mandelstam description of single-pion photoproduction data, together with pion- and eta-hadroproduction data, has been extended to include measurements carried out over the last decade. We consider photo-decay amplitudes evaluated at the pole with particular emphasis on ng couplings and the influence of weighting on our fits. Both energy-dependent and single-energy analysis (energy-binned data) are considered.


I. INTRODUCTION
Our knowledge of the baryon spectrum, as determined from analyses of experimental data, has advanced rapidly [24] over the past decade.The progress has been most significant for non-strange baryons, due largely to the wealth of new and more precise measurements made at electron accelerators worldwide.The majority of these new measurements have been performed at Jefferson Lab, USA (using the CLAS and Hall A detectors), with the MAMI accelerator in Mainz, Germany (the Crystal Ball/TAPS detector being particularly well suited for the measurement of neutral final states), and with the Crystal Barrel detector at ELSA in Bonn, Germany.While most of the early progress [1][2][3][4] in baryon spectroscopy was based on the analysis of meson-nucleon scattering data, particularly pion-nucleon scattering (πN → πN , πN → ππN ), photon-nucleon interactions offer the possibility of detecting unstable intermediate states with small branchings to the πN channel.Many groups have performed either single-channel or multi-channel analyses of these photon-induced reactions.In the more recent single-channel analyses, fits have typically used isobar models [5,6] with unitarity constraints at the lower energies, K-matrix-based formalisms, having built-in cuts associated with inelastic channels [7], and dispersionrelation constraints [6,8].Multi-channel fits have analyzed data (or, in some cases, amplitudes) from hadronic scattering experiments together with the photon-induced channels.These approaches have utilized unitarity more directly.Among others, analyses have been carried out by MAID [5], the Bonn-Gatchina [9], ANL-Osaka [10], Kent State [11], and JPAC [12] groups, SAID [7] (Scattering Analysis Interactive Database) and Jülich-Bonn [13].
Here we should also briefly mention the possibility of extracting reaction amplitudes directly from scattering data with minimal model input.Examples of this approach are described in the analyses of kaon photoproduction data by the Jefferson Lab [14] and Bonn-Gatchina [15] groups.The measurements required for an amplitude extraction with minimal model bias differ depending on whether the goal is to obtain helicity amplitudes (the usual complete experiment case [16]) or partial-wave amplitudes [17].A number of recent studies have shown the limits to model independence [18] and the convergence [19] of independent fits with the availability of more observables measured with high precision.The above studies have also recently been extended to pseudo-scalar-meson electroproduction [20].
An objective of this program is the determination of all relevant characteristics of these resonances, i.e., pole positions, widths, principal decay channels, and branching ratios.In order to compare directly with QCDinspired models and Lattice QCD predictions, there has also been a considerable effort to find "hidden" or "missing" resonances [21], predicted by quark models [22] and LQCD [23] but not yet confirmed.Actually, PDG [24] reports a third of predicted states by QMs and LQCD.
Knowledge of the N and ∆ resonance photodecay amplitudes has largely been restricted to the charged states.Apart from lower-energy inverse reaction π − p → γn measurements, the extraction of the two-body γn → π − p and γn → π 0 n observables requires the use of a modeldependent nuclear correction, which mainly comes from final state interaction (FSI) effects within the target deuteron [25][26][27].As a result, the observables for protontarget experiments are most thoroughly explored and, among neutron-target (deuteron) measurements, the π 0 n charge channel is least explored.This problem is less severe if isospin relations are used to express the four charge-channel amplitudes in terms of three isospin amplitudes [28].Then, in principle, the π 0 n production channel can be predicted in terms of the π 0 p, π + n and, π − p production channel amplitudes.This approach has been tested [29] with the improved availability of π 0 n data; we will consider this again in the fits to data that follow.
The GW SAID pion photoproduction analyses have been updated periodically since 1990 [30,31], with more frequent updates published through our GW website [32].Often, we present our results with CLAS and A2 Collaborations including determination of the resonance parameters (see, for instance, Refs.[33][34][35][36][37]) while our full analysis was reported 10 years ago [7,38].The present work updates our SAID partial-wave analysis (PWA) results and reports a new determination of photodecay amplitudes and pole positions in the complex energy plane.
High activity of worldwide electromagnetic facilities (JLab, MAMI, CBELSA, MAX-lab, SPring-8, and ELPH) increased the body of the SAID database by a significant amount (see Table I).60% of these are γp → π 0 p data.A review of the last two decades of using photon beams to measure the production of mesons, and in particular the information that can be obtained on the spectrum of light, non-strange baryons is given in Ref. [59].A wealth of γN → πN data, for single-and doublepolarization observables, have been anticipated over the past ten years.These data are pivotal in determining the underlying amplitudes in nearly complete experiments, and in discerning between various microscopic models of multichannel reaction theory.
The amplitudes from these analyses can be utilized, in particular, in evaluating contributions to the Gerasimov-Drell-Hearn (GDH) sum rule and related integrals, as was reported recently [60].
In the following section (Sec.II), we summarize changes to the SAID database since 2012.The changes reflected in our multipoles are displayed in Section III.A comparison of past and recent photo-decay amplitudes, for resonances giving a significant contribution to pion photoproduction, is made in Section IV.Finally, in Section V, we summarize our results and comment on possible changes due to further measurements and changes in our parametrization form.
Table I accumulates 21,190 γp → π 0 p, 1,502 γp → π + n, 10,923 γn → π − p, and 1,763 γn → π 0 n data published since 2012 [32].New measurements mostly cover the π 0 p sector.Then there are a lot of single (Σ, P, and T,) and double (E, G, F, and H) polarized data which came recently.It is an essential input for the amplitude reconstruction of the pion photoproduction and determination photocouplings.One can see that the "neutron" database grows rapidly which is important for the determination of the neutral photocouplings.
A full χ 2 /data contribution for each pion photoproduction reaction vs different PWAs reports in Table II

III. SAID MULTIPOLE AMPLITUDES
The SAID parametrization of the transition amplitude T αβ used in the hadronic fits to the πN scattering data is given as where α, β, and σ are channel indices for the πN , π∆, ρN , and ηN channels.Here K σβ are the Chew-Mandelstam K-matrices, which are parameterized as polynomials in the scattering energy.C α is the Chew-Mandelstam function, an element of a diagonal matrix C in channel space, which is expressed as a dispersion integral with an imaginary part equal to the two-body phase space [65].In Ref. [7], it was shown that this form could be extended to T αγ to include the electromagnetic channel as Here, the Chew-Mandelstam K-matrix elements associated with the hadronic channels are kept fixed from the previous SAID solution SP06 [2], and only the electromagnetic elements are varied.The resonance pole and cut structures are also fixed from hadronic scattering.This provides a minimal description of the photoproduction process, where only the N * and ∆ * states present in the SAID πN scattering amplitudes are included in this multipole analysis.
For each angular distribution, a normalization constant (X) and its uncertainty (ϵ X ) were assigned.The quantity ϵ X is generally associated with the normalization uncertainty (if known).The modified χ 2 function to be minimized is given by where the subscript i labels the data points within the distribution, θ exp i is an individual measurement, θ i is the corresponding calculated value, and ϵ i represents the total angle-dependent uncertainty.The total χ 2 is then found by summing over all measurements.This renormalization freedom is essential for obtaining the best SAID fit results.For other data analyzed in the fit, such as the total cross sections and excitation data, the statistical and systematic uncertainties were combined in quadrature and no re-normalization was allowed.
In the previous fits to differential cross sections, the unrestricted best fit gave re-normalization constants X significantly different from unity.As can be seen from Eq. (3), if an angular distribution contains many measurements with small statistical uncertainties, a change in the re-normalization may improve the fit with only a modest χ 2 penalty.Here, however, the weight of the second term in Eq. ( 3) has been adjusted by the fit for each dataset to keep the re-normalization constants approximately within X of unity.
With the new quality datasets (Table I), a new SAID multipole analysis has been completed.This new global energy-dependent solution has been labeled as SM22.
The overall fit quality of the present SM22 and previous SAID CM12 solutions are compared in Tables III and  IV.There are many cases where the CM12 fit produces a χ 2 per datum, for new measurements, which is significantly than greater than unity.The new best fit, SM22, includes these new measurements, reducing the χ 2 /data to more acceptable values.

Both energy-dependent (ED) and single-energy (SE)
TABLE II.Comparison of χ 2 per datum values for all charged and neutral channels covering fit energy range.The previous SAID fit, CM12, was published in Ref. [7] (and is valid up to Eγ = 2700 MeV).CM12 is compared to both the current database and data before 2012.All data are available in the SAID database (DB) [32].For the SM44 fit, π 0 n data were weighted by an arbitrary factor of 4. For the WM22 fit, all data with large χ 2 /data for the SM22 solution (data are listed in Table III) were weighted by an arbitrary factor of 4. The NM22 solution represents a fit without the inclusion of π 0 n data.The previous MAID2007 solution is valid up to Eγ = 1680 MeV (W = 2 GeV) [5].

Solution Observable
χ 2 /(π 0 p data) solutions were obtained from fits to the combined proton and neutron target database, extending from threshold to E γ = 2.7 GeV for the ED fit and to E γ = 2.2 GeV for SE fits.
Apart from the main ED result (SM22) several supplemental fits were done in order to gauge the importance of including π 0 n data (which can, in principle, be at least qualitatively predicted from the remaining more fully populated charge channels).Here fits were done with increased weight for the π 0 n data and conversely the removal of all such data.In addition, a fit was done more heavily weighting all data poorly fitted by SM22.[50].Notation for solutions is given in the caption of Table II.The SAID SM22 (WM22) fit is shown as a red solid (yellow dashed) curve.SAID CM12 [7] (MAID2007 [5]) predictions shown as blue dash-dotted (green dashed) curves.BG2019 [67] predictions are shown as magenta short dash-dotted curves.
For the amplitudes, the subscript l± gives the value of j = l ± 1/2, and the superscript gives the isospin index.Notation for solutions is given in the caption of Table II.New SAID SM22 fit is shown by red solid curves.Previous SAID CM12 [7] (MAID2007 [5], terminates at W = 2 GeV) predictions show by blue dash-dotted (green dashed) curves.BG2019 [67] predictions show by magenta short dash-dotted curves.

IV. RESONANCE COUPLINGS
Following the notation of Refs.[38,70], the (γ, π) Tmatrix element for helicity h is given by where α denotes the partial wave and k, q are the centerof-mass (c.m.) momenta of the photon and the pion.The factor C is 2/3 for isospin 3/2 and − √ 3 for isospin 1/2.The helicity multipoles A h α are given in terms of electric and magnetic multipoles with J = ℓ+1/2 for "+" multipoles and J = (ℓ+1)−1/2 for "−" multipoles, all having the same total spin J .
In Tables V to XIV, we list the pole positions together with the photo-decay amplitudes where the subscript p denotes quantities evaluated at the pole position and m N is the nucleon mass.In Ref. [38], the elastic residues, Res πN , and the pole positions, W p = M p −iΓ p /2, were taken from the GWU SAID PWA, SP06 [2] and each multipole was fitted separately, using the Laurent plus Pietarinen (L+P) method [38], to determine the corresponding residues.
Here, we have made a coupled multipole fit of all partial-wave amplitudes associated with particular resonances, including the pion-nucleon elastic scattering amplitudes.Thus, for example, the L+P fit of Ref. [38] for the E As in Ref. [38], the fitted partial waves are S 11 , P 11 , D 13 , F 15 , P 33 , D 33 , and F 37 with pion-nucleon partial waves taken from Ref. [71].

V. RESULTS AND CONCLUSIONS
The present results update the SAID fit (CM12) which first utilized a Chew-Mandelstam K-matrix approach (as opposed to the Heitler K-matrix formalism used in the original SAID analyses).The L+P method for pole parameter extraction has been extended to simultaneously incorporate all connected πN elastic and photoproduction amplitudes.
The amplitude tables give pole positions and helicity amplitudes at the pole where available.Values for the nγ amplitudes were not extracted in the 2014 SAID analysis; comparisons can now be made to multi-channel determinations.Complex amplitudes are given in terms of modulus and phase.In cases where a large phase is found, close to 180 degrees, a minus sign is commonly extracted to ease comparison with the real amplitudes found in older Breit-Wigner fits.The "modulus" then has a sign and a phase closer to zero.Here, however, the modulus remains positive.
In cases where the fitted multipoles have a clear canonical resonance variation, with a relatively small non-resonance contribution, comparison to the Bonn-Gatchina multi-channel analysis generally shows good agreement (to the 10% level).
Comparisons are more complicated for states associated with the low-angular momentum states E 1/2 0+ and M 1/2 1− .The N (1535)1/2 − and N (1650)1/2 − have some overlap and are close to the ηN threshold cusp.The N (1440) is complicated by the close proximity of its pole position to the π∆ threshold.We note that differences in N (1535)1/2 − pγ amplitudes disappear if one compares instead with the recent Jülich-Bonn analysis [75].For the nγ amplitudes, the agreement is qualitative and no Jülich-Bonn values are available.Qualitative agreement is also seen for the N (1650)1/2 − .Agreement for the ∆(1700)3/2 − is good for the moduli and at least qualitative for the phases.For the N (1720)3/2 + , within fairly large uncertainties, there is qualitative agreement of the helicity amplitude moduli, with less agreement at the level of phases.Hunt and Manley [11] note that the N (1675)5/2 − decays to pγ violate the Moorhouse selection rule [76].We see the moduli of pγ photo-decay amplitudes to be small but non-zero.

Resonance
Figures 1 and 2 plot representative comparisons of SAID fits to data.In addition, older MAID and more recent Bonn-Gatchina results are plotted for comparison.Numerical comparisons of the various SAID fits are given in Tables II to IV. Comparisons of the present SAID I = 3/2 and I = 1/2 multipoles amplitudes from threshold to W = 2.5 GeV (E γ = 2.7 GeV) shown in Figs. 3 -8.Also included, for comparison, are the BnGa and MAID multipoles.Comparisons of the present I = 3/2 and I = 1/2 ED and SE multipole amplitudes from threshold to W = 2.5 GeV (E γ = 2.7 GeV) shown on Figs. 9 -14.

FIG. 4 .FIG. 9 .
FIG.4.Comparison I = 3/2 multipole amplitudes (orbital momentum l = 2) from threshold to W = 2.5 GeV.Notation of the solutions is the same as in Fig.3.Additionally, the WM22 fit is shown by yellow dashed curves.

1 / 2 2− 2 −
multipole has been expanded to a simultaneous fit of the D 13 elastic amplitude, E (neutron target), yielding more self-consistent results.

FIG. 15 .FIG. 16 .FIG. 17 .
FIG.15.Samples of Laurent+Pietarinen (L+P) coupled fit of the S11 πN partial wave of the GWU-SAID fit WI08[71] and the SM22 ED GWU-SAID multipole solutions.Blue symbols are the GWU-SAID solutions, solid black curves are the L+P coupled-multipole fit, and thin red curves are the resonant contribution in the L+P coupled-multipole fit.

FIG. 18 .
FIG.18.Samples of Laurent+Pietarinen (L+P) coupled fit of the F37 πN partial wave of the GWU-SAID WI08[71] and SM22 SE4 GWU-SAID multipole solutions.Notation of the solutions is the same as in Fig.15.
. It presents a partial χ 2 /data contribution of data from Table III vs different PWAs.

TABLE I .
[32]ished data for γN → πN reactions since 2012 as given in the SAID database[32]: 1st column is the reaction, 2nd column is the observable, 3rd column is the number of energy bins, 4th column is the number of data points.

TABLE VI .
Photon-decay helicity amplitudes at the pole for pγ and nγ decays.Fit to pion-nucleon elastic amplitude P11 and multipole M Complex quantities given as modulus and phase.Results from present study (first row), PR2014

TABLE VII .
Photon-decay helicity amplitudes at the pole for pγ and nγ decays.Fit to pion-nucleon elastic amplitude P13 and multipoles E

TABLE VIII .
Photon-decay helicity amplitudes at the pole for pγ and nγ decays.Fit to pion-nucleon elastic amplitude D13 and multipoles E

TABLE X .
Photon-decay helicity amplitudes at the pole for pγ and nγ decays.Fit to pion-nucleon elastic amplitude F15 and multipoles E

TABLE XI .
Photon-decay helicity amplitudes at the pole for pγ decay.Fit to pion-nucleon elastic amplitude P33 and multipoles E

TABLE XIII .
Photon-decay helicity amplitudes at the pole for pγ decay.Fit to pion-nucleon elastic amplitude F35 and multipoles E

TABLE XIV .
Photon-decay helicity amplitudes at the pole for pγ decay.Fit to pion-nucleon elastic amplitude F37 and multipoles E