Differential Cross Section Measurements for $\gamma n\to\pi^-p$ Above the First Nucleon Resonance Region

TL;DR

This paper reports extensive measurements of the differential cross sections for gamma-neutron to pion-proton reactions above the first nucleon resonance, providing new data that refine resonance analysis and determine photon decay amplitudes for the first time.

Contribution

It presents a large set of new differential cross section data for gamma-neutron to pion-proton reactions, significantly increasing the available statistics and enabling improved resonance analysis and extraction of decay amplitudes.

Findings

8428 new data points for cross sections

First determination of photon decay amplitudes at resonance poles

Enhanced constraints on N* resonance spectrum

Abstract

The quasi-free $\gamma d\to\pi^{-}p(p)$ differential cross section has been measured with CLAS at photon beam energies $E_\gamma$ from 0.445 GeV to 2.510 GeV (corresponding to $W$ from 1.311 GeV to 2.366 GeV) for pion center-of-mass angles $\cos\theta_\pi^{c.m.}$ from -0.72 to 0.92. A correction for final state interactions has been applied to this data to extract the $\gamma n\to\pi^-p$ differential cross sections. These cross sections are quoted in 8428 $(E_\gamma,\cos\theta_\pi^{c.m.})$ bins, a factor of nearly three increase in the world statistics for this channel in this kinematic range. These new data help to constrain coupled-channel analysis fits used to disentangle the spectrum of $N^*$ resonances and extract their properties. Selected photon decay amplitudes $N^* \to \gamma n$ at the resonance poles are determined for the first time and are reported here.

Tables

Table 1. Table 1: Comparison of χ 2 superscript 𝜒 2 \chi^{2} per data point ( d . p . formulae-sequence 𝑑 𝑝 d.p. ) below E γ subscript 𝐸 𝛾 E_{\gamma} = 2.7 GeV ( W 𝑊 W = 2.5 GeV) for the γ ​ n → π − ​ p → 𝛾 𝑛 superscript 𝜋 𝑝 \gamma n\to\pi^{-}p channel using predictions for the recent SAID PR15  pr15 and the current MA27 solution. The first row of solutions compares the fit quality to the available data not including the CLAS g13 data. The second row compares the solutions to the available data including the g13 data. The last row compares the solutions only to the g13 data.

Data	Solution	${\chi }^2/({\pi }^{-}pd.p.)$
Existing data	PR15	6541/3162 = 2.07
without g13	MA27	7112/3162 = 2.25
Existing data	PR15	24052/11590 = 2.08
with g13	MA27	16442/11590 = 1.42
Only	PR15	17511/8452 = 2.07
g13	MA27	9330/8452 = 1.10

Table 2. Table 2: Comparison of χ 2 superscript 𝜒 2 \chi^{2} per data point ( d . p . formulae-sequence 𝑑 𝑝 d.p. ) below E γ subscript 𝐸 𝛾 E_{\gamma} = 2.7 GeV ( W 𝑊 W = 2.5 GeV) for all γ ​ N → π ​ N → 𝛾 𝑁 𝜋 𝑁 \gamma N\to\pi N channels using predictions for the recent SAID PR15  pr15 and the current MA27 solution. The fit quality for the π 0 ​ p superscript 𝜋 0 𝑝 \pi^{0}p , π + ​ n superscript 𝜋 𝑛 \pi^{+}n , π − ​ p superscript 𝜋 𝑝 \pi^{-}p , and π 0 ​ n superscript 𝜋 0 𝑛 \pi^{0}n channels is compared to the available data including the g13 data.

Data	Solution	${\chi }^2/({\pi }^{0}pd.p.)$	${\chi }^2/({\pi }^{+}nd.p.)$	${\chi }^2/({\pi }^{-}pd.p.)$	${\chi }^2/({\pi }^{0}nd.p.)$
Existing data	PR15	54985/25540 = 2.15	23558/9859 = 2.39	24052/11590 = 2.08	1152/364 = 3.16
with g13	MA27	55530/25540 = 2.17	20736/9859 = 2.10	16442/11590 = 1.42	1540/364 = 4.23

Table 3. Table 3: Moduli (in ( GeV ) − 1 / 2 superscript GeV 1 2 \rm(GeV)^{-1/2} ) and phases (in degrees) of the neutron helicity amplitudes A 1 / 2 ​ ( n ) subscript 𝐴 1 2 𝑛 A_{1/2}(n) and A 3 / 2 ​ ( n ) subscript 𝐴 3 2 𝑛 A_{3/2}(n) from the SAID MA27 solutions (third column). The Breit-Wigner neutron photo-decay amplitudes are compared from SAID GB12  gb12 (fourth column) from BG2013  BnGa13 (fifth column), and from MAID2007 MAID07 (sixth column). The relativized quark model predictions from Ref.  Capstick (seventh column) are included along with the PDG values (eighth column)  PDG16 .

Resonance	Coupling	MA27	GB12	BG2013	MAID2007	Capstick	PDG 2016
		modulus, phase
$N\left(1440\right)1/2^{+}$	$A_{1/2}\left(n\right)$	0.065$\pm$0.005, 5∘$\pm$3∘	0.048$\pm$ 0.004	0.043$\pm$0.012	0.054	-0.006	0.040$\pm$0.010
$N\left(1535\right)1/2^{-}$	$A_{1/2}\left(n\right)$	-0.055$\pm$0.005, 5∘$\pm$2∘	-0.058$\pm$ 0.006	-0.093$\pm$0.011	-0.051	-0.063	-0.075$\pm$0.020
$N\left(1650\right)1/2^{-}$	$A_{1/2}\left(n\right)$	0.014$\pm$0.002, -30∘$\pm$10∘	-0.040$\pm$ 0.010	0.025$\pm$0.020	0.009	-0.035	-0.050$\pm$0.020
$N\left(1720\right)3/2^{+}$	$A_{1/2}\left(n\right)$	-0.016$\pm$0.006, 10∘$\pm$5∘		-0.080$\pm$0.050	-0.003	0.004	-0.080$\pm$0.050
$N\left(1720\right)3/2^{+}$	$A_{3/2}\left(n\right)$	0.017$\pm$0.005, 90∘$\pm$10∘		-0.140$\pm$0.065	-0.031	0.011	-0.140$\pm$0.065

Equations

$${\epsilon }_{\mathrm{Ratio}}=\frac{{\epsilon }_{\mathrm{Simulation}}-{\epsilon }_{\mathrm{Experiment}}}{{\epsilon }_{\mathrm{Experiment}}}.$$

$$N_{\gamma }=ϵN_e,$$

$$ϵ=\frac{N_{\mathrm{TAC}}}{N(1-\alpha )},$$

$$\begin{gathered} \frac{d\sigma }{d\Omega }(E_{\gamma },\cos {\theta }_{\pi }^{c.m.})= \\ \frac{1}{2\pi (\Delta \cos {\theta }_{\pi }^{c.m.})}\frac{A_r}{\rho L{N}_A}\frac{Y(E_{\gamma },\cos {\theta }_{\pi }^{c.m.})Y_{\mathrm{CF}}}{\Phi (E_{\gamma })A(E_{\gamma },\cos {\theta }_{\pi }^{c.m.})}, \end{gathered}$$

$$\frac{d\sigma }{d\Omega }(\gamma n)=R(E_{\gamma },{\theta }_{\pi }^{c.m.}{)}^{-1}\frac{d\sigma }{d\Omega }(\gamma d),$$

$$m_{eff}=\sqrt{(p_d-p_s{)}^2}\approx m_n-{ϵ}_d-{\vec{p}}_s^2/m_N$$

$$\sqrt{s_{\pi N}}=\sqrt{s_{\gamma N}}=\sqrt{[(E_{\gamma }+m_d-E_s{)}^2-({\vec{p}}_{\gamma }-{\vec{p}}_s{)}^2]},$$

$$\frac{d\sigma (W,\cos {\theta }_{\pi }^{c.m.})}{d\Omega }=\sum _{J=0}^{10}{A}_J(W)P_J(\cos {\theta }_{\pi }^{c.m.}),$$

$$T_{\alpha \beta }=\sum _{\sigma }[1-\overline{K}C{]}_{\alpha \sigma }^{-1}{\overline{K}}_{\sigma \beta },$$

$$T_{\alpha \gamma }=\sum _{\sigma }[1-\overline{K}C{]}_{\alpha \sigma }^{-1}{\overline{K}}_{\sigma \gamma }.$$

$${\chi }^2=\sum _i{\left(\frac{X{\theta }_i-{\theta }_i^{exp}}{{ϵ}_i}\right)}^2+{\left(\frac{X-1}{{ϵ}_X}\right)}^2,$$

$$M(W)=\sum _{i=1}^k\frac{a_{-1}^{(i)}}{W-W_i}+B^L(W).$$

$$B^L(W)$$

$$X(W)$$

$$Y(W)$$

$$Z(W)$$

Full text

††preprint: Phys. Rev. C

I Introduction

The determination of the resonance properties for all accessible baryon states is a central objective in nuclear physics. The extracted resonance parameters provide a crucial body of information for understanding the nucleon excitation spectrum and for testing models of the nucleon inspired by Quantum Chromodynamics (QCD) and, more recently, lattice QCD calculations. The spectrum of $N^{*}$ and $\Delta^{*}$ baryon resonances has been extensively studied through meson-nucleon scattering and meson photoproduction experiments. Properties of the known resonances continue to become better determined as experiments involving polarized beams, targets, and recoil measurements are expanded and refined joint . Extracted quantities include resonance masses, widths, branching fractions, pole positions, and associated residues, as well as photo-decay amplitudes PDG16 . New states have also been found, mainly through multi-channel analyses that are sensitive to states having a relatively weak coupling to the $\pi N$ decay channel Julich ; BnGa14 ; EBAC .

Knowledge of the $N^{*}$ and $\Delta^{*}$ resonance photo-decay amplitudes has largely been restricted to the charged states. Apart from lower-energy inverse reaction $\pi^{-}p\to\gamma n$ measurements, the extraction of the two-body $\gamma n\to\pi^{-}p$ and $\gamma n\to\pi^{0}n$ observables requires the use of a model-dependent nuclear correction, which mainly comes from final state interaction (FSI) effects within the target deuteron. Most $\gamma n$ data are unpolarized and cover fairly narrow energy ranges. Of these, only about 400 $\pi^{0}n$ measurement data points exist, spanning the full nucleon resonance region SAID .

The importance of improving the $\gamma n$ database relative to the $\gamma p$ database is directly related to the fact that the electromagnetic interaction does not conserve isospin symmetry. The amplitude for the reactions $\gamma N\to\pi X$ factors into distinct $I=1/2$ and $I=3/2$ isospin components, $A_{\gamma,\pi^{\pm}}=\sqrt{2}(A_{p/n}^{I=1/2}\mp A^{I=3/2})$. This expression indicates that the excitation of the $I=3/2$ $\Delta^{*}$ states can be entirely determined from proton target data. However, measurements from datasets with both neutron and proton targets are required to determine the isospin $I=1/2$ amplitudes and to separate the $\gamma pN^{*}$ and $\gamma nN^{*}$ photocouplings.

This work focuses on negative pion photoproduction off the neutron using a deuteron target. A large body of new precision $\gamma n\to\pi^{-}p$ differential cross sections for $E_{\gamma}$ = 0.445 GeV to 2.510 GeV in laboratory photon energy, corresponding to an invariant energy range from $W$ = 1.311 GeV to 2.366 GeV, are reported. Pion center-of-mass (c.m.) production angles, ranging from $\theta_{\pi}^{c.m.}$ = 26*∘* to 135*∘*, have been measured during the CLAS Collaboration g13 run period g13_Report . These new cross section data have nearly tripled the world $\gamma n\to\pi^{-}p$ database below $E_{\gamma}$ = 2.700 GeV SAID .

The $\gamma n\to\pi^{-}p$ differential cross section was previously measured by the CLAS g10 gb12 experiment. Those measurements contained 855 data points in 50- and 100-MeV-wide bins of beam energy $E_{\gamma}$ from 1.050 GeV to 3.500 GeV, corresponding to a $W$ range from 1.690 GeV to 2.731 GeV. However, the 8428 data points from g13 are a precision measurement of this cross section, with a factor of $\sim$10 increase in data points. These data are reported in 10- and 20-MeV-wide bins of beam energy $E_{\gamma}$, with overall normalization uncertainties of $\sim$3.4%, compared to the $\sim$6% to $\sim$10% overall normalization uncertainties achieved by g10. Also, unlike the g10 measurements, the g13 data cover the $W$ range of the low-mass $N^{*}$ resonances, and can be used to investigate their helicity amplitudes and resonance parameters.

The present dataset, together with completed polarized measurements for both $\pi^{-}p$ and $\pi^{0}n$ from Jefferson Lab FR-HD and MAMI A2 , are expected to lead to the determination of well-constrained $\gamma n$ decay amplitudes in the near future. However, these new CLAS $\gamma n\to\pi^{-}p$ data allow for the first determination of selected photon decay amplitudes $N^{*}\to\gamma n$ at their pole on the complex plane.

The organization for this paper is as follows. In Section II, details of the g13 experiment and the CLAS detector are provided. Section III outlines the event selection and Section IV provides the tracking and triggering efficiency corrections. Section V describes the extraction of the event yields and the acceptance corrections, and Section VI describes how the beam-target luminosity was determined. Section VII presents and discusses the measured differential cross sections for the reaction $\gamma n\to\pi^{-}p$, while Section VIII reviews the approach for determining the final state interaction corrections. Sections IX and X describe the Legendre fits and multipole fit results, respectively. Finally, Section XI provides a summary of this work and the conclusions.

II Experiment

The CLAS g13 experiment g13_Report ran from October 2006 to June 2007 in Hall B at the Continuous Electron Beam Accelerator Facility (CEBAF) at Jefferson Lab in Newport News, Virginia. Circularly and linearly polarized tagged bremsstrahlung photon beams were incident on a liquid-deuterium ($LD_{2}$) target located near the center of the CLAS detector NIM_CLAS . The circularly polarized photon beam portion of this experiment, called g13a, was used for this analysis.

For g13a, the CEBAF electron beam was supplied at two different energies, 1.990 GeV and 2.655 GeV. These electrons were delivered at currents between 33 nA and 45 nA in beam bunches separated by about 2 ns. The electron beam was incident on a $10^{-4}$ radiation-length-thick gold foil radiator to produce the bremsstrahlung photon beam.

The dipole magnet of the Hall B photon tagger deflected the electron beam and post-bremsstrahlung electrons in order to tag photons produced with energies between $\sim$20% and $\sim$95% of the incident electron beam energy NIM_Tagger . The tagging system provided a photon beam energy resolution of $\sim$0.1% of the electron beam energy with a 150 ps timing resolution. A 6.4 mm diameter nickel collimator downstream of the radiator provided $\sim$90% beam transmission to the 40-cm-long $LD_{2}$ target, which was centered 20 cm upstream from the center of the CLAS detector. This resulted in a tagged photon flux on the order of $10^{7}$ Hz on the target.

The CLAS detector, shown in Fig. 1, was designed around six superconducting coils arranged in a hexagonal configuration that produced an approximately toroidal magnetic field surrounding the beamline. The magnetic field bent charged particles through the three regions of multi-layer drift chambers for momentum measurements. The drift chambers were positioned between the superconducting coils within six sectors in azimuthal $\phi$, each spanning roughly $60^{\circ}$. Charged particles produced at a momentum of 1 GeV/c were measured with a momentum resolution of $\sigma(p)/p$ $\leq$ 0.5%, and with average angular resolutions in the fiducial volume of $\sigma(\theta)$, $\sigma(\phi)$ $\sim$2 mrad NIM_DC . For the g13 experiment, the torus magnet operated at $\sim$40% of its maximum current with reversed field polarity (such that negatively charged particles were bent away from the beamline), producing an integrated magnetic field of 0.972 T m along the track path length at forward angles and 0.233 T m at $90^{\circ}$.

The start counter (ST) surrounding the target had a timing resolution of 260 ps, and was used to determine which of the 2 ns electron beam bunches was associated with the recorded physics event NIM_StartCounter . The time-of-flight (TOF) scintillator paddles had a timing resolution between 150 ps - 250 ps, depending on the length of the paddle, and were used for particle identification NIM_TOF . At forward angles, Cherenkov counters (not used for this experiment) could be used to identify electrons NIM_CC , and the electromagnetic calorimeters could be used to detect electrons and neutral particles NIM_EC .

A coincidence between the start counter and TOF scintillators in at least two of the six CLAS sectors was required for triggering of the data acquisition. With slightly more than two months of running, 20 billion physics events were recorded in the g13a dataset.

III Event Selection

The $\gamma d\to\pi^{-}p(p)$ differential cross section was measured separately for the 1.990 GeV and 2.655 GeV beam data, and these cross section results were combined as discussed in Section VII. The yields of the $\gamma d\to\pi^{-}p(p)$ reaction were determined by reconstructing the $\pi^{-}$ and scattered (higher momentum) proton, with the lower momentum proton missing. The proton in the deuteron typically has a momentum from Fermi-motion of less than 200 MeV/c ParisPotential (and peaks at $\sim$50 MeV/c), and was often stopped before it could escape the $LD_{2}$ target.

The reconstructed beam energy and track momenta were slightly distorted by effects not taken into account during the event reconstruction. These effects included uncertainties in the incident electron beam energy, unaccounted for energy losses as the tracks traversed the detector, and drift chamber misalignments and inaccuracies in the magnetic field map that affected the reconstructed track momenta. Each of these effects was studied and resulted in beam energy and track momentum corrections on the order of a few percent CLAS_g13_Studies .

III.1 Particle Identification

Initially, all reconstructed positively and negatively charged tracks were treated as candidates for the proton and $\pi^{-}$, respectively. Then, for each combination of proton and $\pi^{-}$ candidates, their start counter hits were used to select the beam bunch corresponding to the event. The arrival time of the beam bunches was known to a resolution of 50 ps, and was used as a reference time for the particle identification. Fig. 2 shows the $\Delta\beta$ vs. momentum distributions of the proton and $\pi^{-}$ candidates, where $\Delta\beta$ is the difference between $\beta=p/E$ using the candidate mass, and $\beta$ determined from the track path length (from the event vertex to the TOF system) and the track hit time from the TOF paddle. $\Delta\beta$ is centered at zero for the protons and $\pi^{-}$’s, and the neighboring bands are from other particle types, such as $\pi^{+}$’s, or from choosing the wrong beam bunch.

For proton identification, a momentum-dependent $\pm 5\sigma$ cut was applied on $\Delta\beta$. No particle identification cut was used to identify $\pi^{-}$’s since the background from electron, muon, and kaon events was negligible, as seen in Fig. 2(b). Poorly performing or miscalibrated TOF counters were excluded from the analysis.

III.2 Vertex Cuts and Missing Momentum

The vertex-$z$ distribution of the reconstructed tracks, defined as their distance-of-closest-approach to the nominal beamline (defined as the $z$-axis), is shown in Fig. 3. A cut was applied requiring that the reconstructed vertex-$z$ of both the proton and the $\pi^{-}$ be less than 5 cm. The target extended from -40 cm to 0 cm in vertex-$z$, so this cut was used to remove backgrounds from beam photons striking the aluminum endcap of the target assembly.

To illustrate the missing momentum distribution of $\gamma d\to\pi^{-}p(p)$ events, a $\pm 3\sigma$ cut was applied around the missing mass peak of the proton. Fig. 4(a) shows the missing momentum distribution after this cut. The missing momentum is primarily peaked at low momenta due to Fermi motion, and the high-momentum tail is primarily from rescattering events. Fig. 4(b) shows that the slow-proton momentum is uniformly distributed in $\cos\theta$, where $\theta$ is the angle between the missing momentum and the beam in the laboratory frame. A cut was applied at 200 MeV/c to reject the majority of the rescattering events. Since there are still rescattering effects present after this cut, the $\gamma d\to\pi^{-}p(p)$ cross section is quoted as “quasi-free.”

IV CLAS Efficiency Studies

IV.1 Tracking Efficiency

To determine the charged particle tracking efficiency, the CLAS drift chamber wire hit efficiencies were studied by determining how often a given sense wire recorded a hit when a reconstructed track passed nearby. To make sure that the study was unbiased, the efficiencies were only evaluated when there were significantly more hits on the track than the minimum needed for reconstruction.

These studies allowed issues associated with missing wires due to bad high-voltage connections and amplifier low-voltage shorts to be taken into account. Furthermore, these studies were able to determine tracking inefficiencies due to readout electronics problems, cable disconnects, and cable swaps. Groups of wires that were correlated with a common problem were grouped together in the simulation so that they were either kept or rejected as a whole. The efficiency calculated and applied for the regions with cable swaps or disconnects does not properly model the experimental data, so these regions were eliminated from the analysis. In this manner a very good match of the tracking efficiency in the simulation code to the CLAS hardware was possible. A comparison of the tracking efficiencies for each CLAS drift chamber sector are available in Ref. CLAS_g13_Studies .

The track reconstruction efficiencies for protons and $\pi^{-}$’s were studied by analyzing the $\gamma d\to\pi^{-}p(p)$ and $\gamma d\to pp(\pi^{-})$ topologies, respectively, and determining how often the missing particle was reconstructed when it was in the fiducial region of the detector. These studies were performed with both experimental and phase-space Monte Carlo (MC) simulated data as a function of track momentum and direction. For the g13 experiment, these reconstruction efficiencies were 95% or higher in the nominal fiducial regions of the detector. Fig. 5 shows the ratio of these reconstruction efficiencies $\varepsilon$ for the proton, which was computed as:

[TABLE]

Thus, regions with an efficiency ratio significantly greater (less) than zero are regions where the reconstruction efficiency was much lower (higher) in the experiment than in the MC simulation. The discrepancies seen in the figure at the edges of the acceptance are due to a mismatch between the simulated and experimental geometry. These regions were cut from the analysis so that only regions that were accurately modeled in the simulation were included in the cross section measurement. The efficiency ratio distributions for the $\pi^{-}$’s, which were bent differently in the CLAS magnetic field, are similar but required separate cuts. The absolute minimum accepted proton and $\pi^{-}$ momenta were 360 MeV/c and 100 MeV/c, respectively.

IV.2 Triggering Efficiency

As discussed in Section II, the g13a trigger was designed to record events with a ST and TOF coincidence in at least two sectors of CLAS. To determine the triggering efficiency, the $\gamma d\to\pi^{-}pp$ topology was studied, with the requirement that the three final state particles be in different sectors. For every pair of particles that registered as contributing to the trigger, the triggering rate of the third particle was studied.

Figure 6 shows the proton and $\pi^{-}$ triggering efficiencies in a representative sector, as a function of the track angle and momentum. Because these efficiencies were studied as a function of all kinematics, they include both TOF and ST efficiency effects. The proton triggering was efficient in general, but was low in a few of the TOF paddles, due to one or both of the photomultiplier tubes (PMTs) at the end of the scintillators having low gain.

However, the $\pi^{-}$ efficiencies were significantly worse than those of the proton. This was because $\pi^{-}$’s deposited much less energy than protons of the same momentum in the scintillators, due to their higher velocity. A number of inefficient channels were present due to low gain TOF PMTs (even though they were set at their maximum voltage). These PMTs were still efficient for hit readout, as the 100 mV triggering threshold was much higher than the 20 mV detection threshold. The efficiencies for the other sectors are available in Ref. CLAS_g13_Studies . These triggering efficiencies were applied to the MC simulation to model these event losses.

V Yields and Acceptance

The $\gamma d\to\pi^{-}p(p)$ data were separated into 10- and 20-MeV-wide $E_{\gamma}$ bins and 0.02- to 0.04-wide bins in $\cos\theta_{\pi}^{c.m.}$, where $\theta_{\pi}^{c.m.}$ is the angle between the $\pi^{-}$ and the beam in the $\pi^{-}p$ c.m. frame. These data spanned the range from 0.440 GeV to 2.520 GeV in beam energy and $-0.72$ to $0.92$ in $\cos\theta_{\pi}^{c.m.}$ for a total of 8428 bins. In each bin, the missing-proton peaks were fit to double-Gaussian functions over a linear background, an example of which is shown in Fig. 7. A double-Gaussian function is defined as the sum of two Gaussians with identical means, but different heights and widths. The larger, primary Gaussian was used to model the Gaussian-scattering portion of the signal distribution, and the smaller, secondary Gaussian was used to fit the tails of the signal distribution.

The backgrounds are primarily due to misidentified protons and $\pi^{-}$’s, or selection of the wrong beam photon. The $\gamma d\to\pi^{-}p(p)$ yield was defined as the number of events above the background within the $\pm 4.5\sigma$ fit range about the missing-proton peak. There were over $400$ million $\gamma d\to\pi^{-}p(p)$ events in the g13a experimental data sample used for this analysis.

A total of 1.8 billion MC $\gamma d\to\pi^{-}pp$ events were simulated for each electron beam energy to calculate the acceptance corrections. These data were evaluated separately in order to individually compute the cross sections for the different run ranges. The George Washington University (GWU) SAID GB12 cross section predictions gb12 , based on the world data of the $\gamma n\to\pi^{-}p$ reaction, were used to generate the event distributions. After a preliminary quasi-free $\gamma d\to\pi^{-}p(p)$ cross section measurement was obtained from the g13 data, this measurement was used to generate the final simulated data. A comparison of the reconstructed $\gamma d\to\pi^{-}p(p)$ yield between the experimental data and the MC is shown in Fig. 8. This shows that the inefficient regions of the detector are well modeled by the simulation. Thus, any variations of the detector acceptance across the widths of the narrow yield-extraction bins did not cause an incorrect modeling of the CLAS acceptance.

The same analysis procedure and cuts used to select the $\gamma d\to\pi^{-}p(p)$ final state in the experimental data were used for the simulated data. However, a $\sim$5% yield correction factor $Y_{\mathrm{CF}}$ was applied to the experimental yields to correct for event losses from choosing the incorrect beam bunch, which was not modeled in the simulation. This correction factor was determined by studying the $\gamma d\to\pi^{-}p(p)$ yield from all other beam bunches recorded in the event. The uncertainties on this correction factor were $\sim$0.003% from statistics and $\sim$0.88% from systematics, determined by studying the variation in the correction factor with beam energy.

Since the CLAS acceptance rapidly falls off near the edges of the detector, cross section measurements in these regions had systematic uncertainties that were difficult to quantify. In addition, small mismatches between the generated MC distribution and the experimental data could cause large uncertainties in regions of low acceptance. To remove these regions, bins with an acceptance less than 20% of the maximum acceptance within each $E_{\gamma}$ bin were rejected from the analysis. The CLAS acceptance of the $\gamma d\to\pi^{-}p(p)$ reaction for the CLAS g13 experiment after this cut is shown in Fig. 9 for selected beam energy bins. Overall, the acceptance varied between 5% and 50%, and the large dips were primarily due to triggering inefficiencies and drift chamber problem areas.

The systematic uncertainty due to event selection was typically less than 2%, although it increased to 10% near the edges of the detector. The uncertainty due to the yield extraction was less than 5%. These uncertainties were determined by varying the widths of the cuts used, and the range and starting parameters of the missing mass fits.

The systematic uncertainties due to the acceptance corrections were typically less than 5%, but increased to 10% in problematic regions with low triggering or drift chamber acceptance. These uncertainties were determined by studying how the acceptance-corrected yields changed when individual CLAS sectors and target vertex-$z$ bins were removed from the analysis. A small number of bins had large ($\geq 5\%$) systematic uncertainties for half of the CLAS sectors, and were removed from the results. Overall, the angular-dependent systematic uncertainties varied between 1% and 15%.

VI Luminosity Determination

The number of tagged photons incident on the target while the data acquisition (DAQ) system was ready to record events, $N_{\gamma}$, was calculated separately for each tagger counter as gflux :

[TABLE]

where $\epsilon$ is the tagging ratio of the given tagger counter and $N_{e}$ is the number of detected electron hits in that counter while the DAQ was ready. $N_{e}$ was calculated from the rate of “out-of-time” electron hits and the livetime of the DAQ. “Out-of-time” hits are from electrons that did not coincide in time with the trigger, and were used so that the rate calculation was not biased by the trigger.

The tagging ratios $\epsilon$ were determined by taking several normalization runs throughout the g13 experiment. During these runs, a total absorption counter (TAC) was inserted into the beamline to determine the number of photons incident on the target. The TAC was positioned about 25 m downstream of CLAS and consisted of a single lead-glass block. A PMT attached to the block was used to count the number of photons incident on the TAC, which was 100% efficient NIM_CLAS . A low beam current of 0.1 nA was necessary to prevent radiation damage to the TAC during these normalization runs.

For each normalization run the tagging ratios were calculated for each tagger counter as gflux :

[TABLE]

where $N$ is the total number of electron hits in a given tagger counter, $N_{\mathrm{TAC}}$ is the total number of these hits that represent coincident matched photon hits in the TAC for that tagger counter, and $\alpha$ is the photon attenuation factor. This factor takes into account the fraction of the photons incident on the target that did not reach the TAC. This photon attenuation factor was $\sim$4% and was energy-independent PhotonLoss_ToTAC . These losses were primarily due to electron-positron pair production and Compton scattering as the photons interacted with the target and the beamline components. The tagging ratios were typically 60% - 72% for the 1.990 GeV data and between 73% and 82% for the 2.655 GeV data. Because the beam was collimated through a 6.4 mm opening, the tagging ratios were lower for the 1.990 GeV data due to the larger beam dispersion of the lower-energy beam.

In total, approximately 46.8 trillion tagged photons were incident on the CLAS target in this analysis. The statistical uncertainty on the flux measurement ranged between 0.0024% and 0.14%, and are reported as energy-dependent normalization uncertainties. The systematic uncertainty of the photon flux was determined by examining the stability of the flux-normalized yields of $\gamma d\to\pi^{-}p(p)$ throughout the experimental run. These systematic uncertainties were 0.4% and 0.7% for the 1.990 GeV and 2.655 GeV data, respectively, and are reported as energy-independent normalization uncertainties.

In addition, the systematic uncertainties on the target length and density determinations were each 0.4%, and were dominated by thermal contraction and temperature variation, respectively. These uncertainties are reported as energy-independent normalization uncertainties.

VII Differential Cross Sections

The data from the 1.990 GeV and 2.655 GeV electron beam energies were merged together to produce the final set of measured $\gamma d\to\pi^{-}p(p)$ differential cross sections. This merging was performed by calculating an uncertainty-weighted average of the two cross section measurements in bins where both were available. In bins where data was only available from one beam energy, only that result was used.

The differential cross section of the $\gamma d\to\pi^{-}p(p)$ reaction was calculated for each bin of photon beam energy $E_{\gamma}$ and $\cos\theta_{\pi}^{c.m.}$ as:

[TABLE]

where $\Delta\cos\theta_{\pi}^{c.m.}$ is the bin width in $\cos\theta_{\pi}^{c.m.}$, $A_{r}$ is the effective atomic weight of the neutrons in the deuterium target, $\rho$ is the target density, $L$ is the target length, $N_{A}$ is Avogadro’s number, $\Phi$ is the photon flux in the given photon energy bin, $Y$ is the experimental yield in the given bin, $A$ is the simulated acceptance in the given bin, and $Y_{\mathrm{CF}}$ is the yield correction factor discussed in Section V. The factor of $2\pi$ is due to the integration over the azimuthal angle $\phi$ in the binning used for the cross section calculation. The statistical uncertainty of the cross section was calculated for each bin by combining the statistical uncertainties of the experimental yield and simulated acceptance in quadrature, and ranged between 0.3% and 5%. These uncertainties were dominated by the yield uncertainties. All data from this measurement are included in the CLAS physics database database .

To study the stability of the overall normalization of the $\gamma d\to\pi^{-}p(p)$ cross section measurements, it was calculated separately for several different run ranges throughout both beam energy settings of the experiment. Overall, the total spread between the measurements was 2.4%, and is reported as an energy-independent normalization uncertainty. This uncertainty takes into account any systematic differences between the 1.990 GeV and 2.655 GeV data that were merged together. The total normalization uncertainties were about 3.4%, and were primarily due to this run range-dependent variation in the cross section measurements and the FSI corrections, which are discussed in Section VIII. The total uncertainty on the $\gamma d\to\pi^{-}p(p)$ cross sections is typically between 4.2% and 15%.

To extract the $\gamma n\to\pi^{-}p$ differential cross sections, model-dependent final state interaction corrections were applied to the $\gamma d\to\pi^{-}p(p)$ data, as discussed in Section VIII. These data were split up into 157 photon energy bins from 0.440 GeV to 2.520 GeV, 10-MeV-wide below 1.5 GeV and 20-MeV-wide above. The $\gamma n\to\pi^{-}p$ differential cross section measurements are shown for 40 of these $E_{\gamma}$ bins in Figs. 10 and 11, compared against previous measurements and available partial wave analysis solutions. They are also shown in Fig. 12 vs. $W$ in four bins of $\cos\theta_{\pi}^{c.m.}$. These figures include $\gamma n\to\pi^{-}p$ measurements from CLAS g10 gb12 , SLAC Scheffler , DESY Benz , MAMI-B Briscoe , and Frascati Beneventano , and $\pi^{-}p\to\gamma n$ measurements from BNL Shafi , LBL Weiss , and LAMPF Kim . Only the angle-dependent uncertainties are shown for all measurements. All non-CLAS g13 data shown in Figs. 10 and 11 are within $\pm$10 MeV of the selected g13 energy bin, and all non-CLAS g13 data shown in Fig. 12 are within $\cos\theta_{\pi}^{c.m.}$ of $\pm$0.05 of the g13 angle bin.

The differential cross section peaks at low energy due to $\Delta(1232)$ and $N^{*}$ resonance production, and at forward angles due to $t$-channel pion exchange. Below $E_{\gamma}$ = 1 GeV, the new CLAS g13 data dominate the previous world measurements, with $\cos\theta_{\pi}^{c.m.}$ bins 0.02-wide and total uncertainties typically less than 10% in this range. The CLAS g13 data are systematically lower than the DESY Benz , BNL Shafi , and SLAC Scheffler measurements in several energy bins, and each of these measurements quote normalization uncertainties of about 5%. There is also a discrepancy in the trend of the data at forward angles between the CLAS g13 and SLAC measurements below $E_{\gamma}$ = 0.800 GeV, with the g13 data rising more sharply at forward angles.

Above $E_{\gamma}=1$ GeV, the g13 data are reported in bins that are 0.03-wide in $\cos\theta_{\pi}^{c.m.}$ up to 1.5 GeV, and 0.04-wide in $\cos\theta_{\pi}^{c.m.}$ above 1.5 GeV. Here, the CLAS g10 data gb12 were the previous highest-statistics measurement, reported in 50- and 100-MeV-wide beam energy bins. The g13 data are in excellent agreement with these measurements, as the g10 data have normalization uncertainties of $\sim$6% to $\sim$10% that are not shown in the figures.

The SAID PR15 pr15 , Bonn-Gatchina BG2014-02 BnGa14 , and MAID2007 MAID07 curves shown in these figures did not include the new CLAS g13 data in their fits, and the MAID2007 fit does not include the CLAS g10 measurements either. The data in these previous fits, and in the new SAID MA27 fit that includes the g13 data, are discussed in Section X.

VIII Final State Interactions

The $\gamma n\to\pi^{-}p$ cross sections were extracted on a free neutron from the deuteron data in the quasi-free kinematic region of the $\gamma d\to\pi^{-}pp$ reaction, which has a fast knocked-out proton $p_{1}$ and a slow proton spectator $p_{2}$, assumed not to be involved in the pion production process. In this quasi-free region, the reaction mechanism corresponds to the “dominant” Impulse Approximation (IA) diagram in Fig. 13(a) with the slow proton $p_{2}$ emerging from the deuteron vertex. Here, the differential cross section on the deuteron can be related to that on the neutron target in a well understood way (see, e.g., Eq. (22) of Ref. FSI and references therein). Fig. 13(a) illustrates this “dominant” IA diagram, as well as the “suppressed” IA diagram with the protons interchanged. This approximation, with the additional assumption that the neutron is at rest in the deuteron, allows for the identification of the quasi-free cross section $\frac{d\sigma}{d\Omega}$ on the deuteron with that on the neutron, where $d\Omega$ is the solid angle of the outgoing pion in the $\gamma n$ rest frame. The $\gamma n$ cross section can be calculated as

[TABLE]

where $\frac{d\sigma}{d\Omega}(\gamma d)$ is the quasi-free CLAS g13 measurement on the deuteron and $R(E_{\gamma},\theta_{\pi}^{c.m.})$ is the FSI correction factor that takes into account the FSI effects discussed below, as well as the identity of the two protons in the $\gamma d$ reaction. This factor is defined as the ratio between the full contribution of the three diagrams in Fig. 13 and that of the “dominant” IA diagram in Fig. 13(a). There are two critical factors to consider when using this approach:

the neutron is bound in the deuteron and not at rest, and 2. 2)

there are $NN$- and $\pi N$-FSI effects.

Factor 1) means that the effective mass of the neutron

[TABLE]

is not equal to the mass of the free neutron $m_{n}$. Here, $p_{d}$, $p_{s}$, $\vec{p}_{s}$, $\epsilon_{d}$, and $m_{N}$ are the deuteron 4-momentum, 4- and 3-momenta of the spectator proton, the deuteron binding energy, and the nucleon mass, respectively. Also, the invariant mass $\sqrt{s_{\pi N}}$ of the final $\pi N$-system,

[TABLE]

depends on the proton-spectator momentum $\vec{p}_{s}$ ($s_{\gamma N}$ is the invariant mass squared of the initial $\gamma N$ state). Here, $E_{\gamma}$ ($E_{s}$), $m_{d}$, and $\vec{p}_{\gamma}$ are the total energy of the initial photon (proton-spectator), the deuteron mass, and the photon 3-momentum, respectively, and $E_{\gamma}=|\vec{p}_{\gamma}|$.

Since $\sqrt{s_{\pi N}}$ depends on $\vec{p}_{s}$, the $\gamma N\to\pi N$ cross section extracted from the deuteron data, with an undetected nucleon-spectator, is averaged over an energy range that depends on the kinematic cuts employed for $\vec{p}_{s}$. Thus, the effective photon laboratory energy $E_{\gamma n}$ (defined through the relation $s_{\gamma N}=m_{n}^{2}+2m_{n}E_{\gamma n}$ for the $\gamma n\to\pi^{-}p$ reaction) and the pion c.m. angle $\theta_{\pi}^{c.m.}$ are smeared due to the deuteron wave function (DWF). This smearing has been estimated from a simplified calculation, where the $\gamma d\to\pi^{-}pp$ amplitude is proportional to the DWF and depends only on the laboratory momentum of one of the final protons, say $p_{2}$. Here, $E_{\gamma n}$ is determined through the above-mentioned relation with the effective mass of the pion-proton pair with the other proton $p_{1}$. The distortion of the extracted $\gamma n\to\pi^{-}p$ cross sections due to the smearing effect is negligible, as was shown in Ref. Briscoe .

Factor 2) corresponds to the inclusion of the FSI corrections. Their leading terms correspond to the Feynman diagrams shown in Figs. 13(b,c). The GWU SAID database contains phenomenological amplitudes for the reactions $\pi N\to\pi N$ piN , $NN\to NN$ NN , and $\gamma N\to\pi N$ du07 , which were used as inputs to calculate the dominant diagrams of the GWU-ITEP FSI approach. The full Bonn potential Bonn was used for the deuteron description.

Calculations of the $\gamma d\to\pi^{-}pp$ differential cross sections with the FSI taken into account (including all diagrams in Fig. 13) were done for the present g13 data as they were done previously for the CLAS g10 data ($E_{\gamma}=1.050$ GeV to 2.700 GeV and $\theta_{\pi}^{c.m.}$ = 32∘ to 157∘) gb12 and MAMI-B data ($E_{\gamma}=0.301$ GeV to 0.455 GeV and $\theta_{\pi}^{c.m.}=58^{\circ}$ to 141∘) Briscoe .

The GWU-ITEP FSI calculations FSI are available over a broad energy range (threshold to $E_{\gamma}=2.700$ GeV) and for the full c.m. angular range ($\theta_{\pi}^{c.m.}$ = 0*∘* to 180*∘*). Fig. 14 shows the FSI correction factor $R=R(E_{\gamma},\theta_{\pi}^{c.m.})$ for the $\gamma n\to\pi^{-}p$ differential cross section as a function of $\theta_{\pi}^{c.m.}$ for different energies over the range of the CLAS g13 experiment. Overall, the FSI correction factor $R<1$, while the value of $R$ varied from 70% to 90% depending on the kinematics. The behavior of $R$ is very smooth vs. pion production angle. Note that $R(E_{\gamma},\theta_{\pi}^{c.m.})$ is the FSI correction factor for the CLAS quasi-free $\gamma d\to\pi^{-}pp$ cross section averaged over the laboratory photon energy $E_{\gamma}$ bin width.

The contribution of FSI calculations FSI to the overall systematic normalization uncertainty is estimated to be about 2.2% (the sensitivity to the DWF is 1% and to the number of steps in the integration of the five-fold integrals is 2%). No sensitivity was found to the kinematic cuts used for the detected protons in CLAS.

IX Legendre Analysis

Legendre expansions provide a model-independent approach suitable for presentation of modern detailed (high-precision, high-statistics, and narrow energy and angular binning) data for pion photoproduction reactions leg . This approach is applicable both to cross sections and to polarization observables; it is much more compact and visual than traditional methods (see, for instance, Figs. 10 to 12), at least at energies within the nucleon resonance region. The Legendre coefficients reveal specific correlations and interferences between resonant states of definite parities.

The small statistical uncertainties of the g13 data obtained here allow a correspondingly robust determination of the Legendre polynomial coefficients $A_{J}(W)$. These coefficients were very difficult to determine unambiguously with previously published $\pi^{-}$ photoproduction data of lower statistical accuracy. Because of the limited angular range of the g13 data, several sets of quasi-data were generated using the MA27 SAID solution (see Section X for details) in bins with width $\Delta\cos\theta_{\pi}^{c.m.}=0.05$ for the forward and backward directions to cover the full angular range.

It is important to note that the MA27 solution was constrained at the forward and backward angular ranges beyond the extent of the g13 data by the existing world data shown in Figs. 10 to 12. However, as the available data does not span the full $\cos\theta_{\pi}^{c.m.}$ range for the $W$ range of the g13 data, the MA27 quasi-data were conservatively assigned 10% uncertainties, which matches the largest of the experimental uncertainties reported within the g13 data, excepting a few regions with a problematic acceptance determination. Conservative assignment of uncertainties in these regions is important as these regions are quite sensitive to the highest partial waves.

As expected for such a fit using orthogonal polynomials, the Legendre coefficients $A_{J}(W)$ decrease markedly for large $J$. With the energy range and precision of the g13 data, a maximum value of $J=10$ was found to be sufficient to describe the data (similar to the analysis of the CLAS $\pi^{0}$ and $\pi^{+}$ $\Sigma$ beam asymmetry measurements ASU ). Thus, the infinite series is truncated as

[TABLE]

where the total cross section $\sigma^{\rm tot}=4\pi A_{0}(W)$.

In Fig. 15, the Legendre coefficients $A_{0}(W)$ to $A_{10}(W)$ are shown as a function of $W$ from the fit of the CLAS g13 $d\sigma/d\Omega$ data and the $d\sigma/d\Omega$ data generated from the MA27 predictions. The individual Legendre coefficients have been scaled by $n\times A_{J}+m$ to allow plotting on a common abscissa. The $n$ and $m$ scaling values are given on the subplots of Fig. 15.

The results of our fits yield unprecedented detail on the energy dependence of the Legendre coefficients $A_{J}(W)$, and should prove useful for performing a phase shift analysis of pion photoproduction data for the present energy range. As expected from the form of Eq. (28) of Ref. leg , resonance contributions from the second, third, and fourth resonance regions combine to produce clear peaks in the coefficient $A_{0}(W)$. It is interesting that all $A_{J}(W)$ coefficients show structure for the $W$ = 1.3 GeV to 1.8 GeV range, which was also seen in the MAMI A2 $\pi^{0}$ data pr15 . However, wide structures are also visible in the range $W$ = 1.8 GeV to 2.0 GeV, most likely attributable to contributions from one or more nucleon resonances known in this energy range with spin up to 7/2, as was seen in the recent CLAS g8 $\pi^{0}$ and $\pi^{+}$ $\Sigma$ beam asymmetry measurement Legendre analysis ASU .

The Legendre fit results shown in Fig. 15 do not include any assignment of model uncertainties associated with the extrapolations of the MA27 model beyond the range of the available data. Such assignments could be expected to be non-negligible for the higher Legendre moments shown here. However, our purpose in displaying the Legendre fit results is not to perform a quantitative amplitude analysis, but to showcase how the precision g13 cross section measurements can provide significant constraints on the resonance contributions over a broad range in $W$.

X Multipole Analysis

The SAID parameterization of the transition amplitude $T_{\alpha\beta}$ used in the hadronic fits to the $\pi N$ scattering data is given as

[TABLE]

where $\alpha$, $\beta$, and $\sigma$ are channel indices for the $\pi N$, $\pi\Delta$, $\rho N$, and $\eta N$ channels. Here $\overline{K}_{\alpha\beta}$ are the Chew-Mandelstam $K$-matrices, which are parameterized as polynomials in the scattering energy. $C_{\alpha}$ 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 ford .

In Ref. cm12 , it was shown that this form could be extended to $T_{\alpha\gamma}$ to include the electromagnetic channel as

[TABLE]

Here, the Chew-Mandelstam $K$-matrix elements associated with the hadronic channels are kept fixed from the previous SAID solution SP06 piN , 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 $\Delta^{*}$ states present in the SAID $\pi N$ scattering amplitudes are included in this multipole analysis.

For each angular distribution, a normalization constant $(X)$ and its uncertainty $(\epsilon_{X})$ were assigned. The quantity $\epsilon_{X}$ is generally associated with the normalization uncertainty (if known). The modified $\chi^{2}$ function to be minimized is given by

[TABLE]

where the subscript $i$ labels the data points within the distribution, $\theta_{i}^{exp}$ is an individual measurement, $\theta_{i}$ is the corresponding calculated value, and $\epsilon_{i}$ represents the total angle-dependent uncertainty. The total $\chi^{2}$ 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 renormalization was allowed.

In the previous fits to the $\gamma n\to\pi^{-}p$ differential cross sections of Ref. gb12 , the unrestricted best fit gave renormalization constants $X$ significantly different from unity. As can be seen from Eq. (11), if an angular distribution contains many measurements with small statistical uncertainties, a change in the renormalization may improve the fit with only a modest $\chi^{2}$ penalty. Here, however, the weight of the second term in Eq. (11) has been adjusted by the fit for each dataset to keep the renormalization constants approximately within $\epsilon_{X}$ of unity. This was possible without degrading the overall fit $\chi^{2}$, as can be seen in Fig. 16.

With the new high-precision $\gamma n\to\pi^{-}p$ cross sections from the CLAS g13 dataset, a new SAID multipole analysis has been completed. This new global energy-dependent solution has been labeled as MA27. The overall fit quality of the present MA27 and previous SAID PR15 solutions are compared in Tables 1 and 2. The inclusion of the g13 dataset shows significant improvement in the comparisons between the $\pi^{-}p$ fits and data ($\chi^{2}/d.p.$ for PR15 = 2.08 and $\chi^{2}/d.p.$ for MA27 = 1.10) as shown in Fig. 16 and Table 1. This demonstrates the power of these cross section measurements with their small uncertainties. The overall comparison of the PR15 and MA27 solutions in Table 2 shows the fit $\chi^{2}/d.p.$ values are essentially unchanged for the $\pi^{0}p$ and $\pi^{+}n$ channels but are notably worse for the $\pi^{0}n$ channel, which has very low statistics. The overall $\chi^{2}$ per data point including all available data and the new g13 data for PR15 is $\chi^{2}/d.p.=2.19$ (103747/47353) and for MA27 is $\chi^{2}/d.p.=1.99$ (94248/47353).

In Figs. 17 to 19, $I=1/2$ multipole amplitudes from the present and previous SAID fits are compared to predictions from the MAID and Bonn-Gatchina groups. The Bonn-Gatchina analysis has been regularly updated, whereas the MAID fit was published in 2007 and therefore does not include any results from the past decade, including the recent CLAS g10 cross section measurements of Ref. gb12 . The cross section requires $I=3/2$ multipoles as well, but these are highly constrained by proton-target measurements and have not changed significantly with the addition of neutron-target measurements (and therefore are not shown here). In the multipole plots, the subscript $n$ denotes a neutron target and $\ell\pm$ gives the value of $j=\ell\pm 1/2$, while the superscript gives the isospin index.

Changes in the multipole amplitudes can be seen in a comparison of the SAID curves in Figs. 17 to 19. Consistency among the analyses is visible in multipoles containing a dominant resonance, such as the ${}_{n}E_{2-}^{1/2}$ and ${}_{n}M_{2+}^{1/2}$ multipoles. However, the ${}_{n}E_{1+}^{1/2}$ and ${}_{n}M_{1+}^{1/2}$ multipoles differ even at the qualitative level. This discrepancy is evident in the proton-target multipoles as well.

The full world database of $\gamma n\to\pi^{-}p$ experiments above $E_{\gamma}=1.2$ GeV contains mainly differential cross sections, apart from some $\Sigma$ beam asymmetry measurements from Yerevan Yerevan , GRAAL GRAAL , and CEA CEA . Ultimately, more measurements of the polarization observables are needed in the $\pi^{-}p$ and $\pi^{0}n$ channels in order to fully constrain the underlying reaction amplitudes. New $\gamma n$ measurements from the CLAS g14 dataset sandorfiE will significantly add to the available polarization observable measurements.

Looking for significant changes in the imaginary parts of the multipoles (Figs. 17 to 19) in the energy region below the older set of CLAS g10 cross sections gb12 , several $N^{*}\to\gamma n$ photo-decay amplitudes have been extracted at their pole positions on the complex plane. This is the first determination of these amplitudes for the $N(1440)1/2^{+}$, $N(1535)1/2^{-}$, $N(1650)1/2^{-}$, and $N(1720)3/2^{+}$ states. A new approach has been applied to determine the pole positions and residues from the pion photoproduction multipoles Svarc . The method is based on a Laurent expansion of the multipoles, $M(W)$, with a Pietarinen series representing the regular (non-pole) part of the energy dependence as

[TABLE]

Here $W$, $a_{-1}^{(i)}$, and $W_{i}$ are complex numbers representing the c.m. energy, residues, and pole positions for the $i^{th}$ pole, respectively, and $B^{L}(W)$ is a regular function in the whole complex plane. A general unknown analytic function $B(W)$ can be expanded into a power series of Pietarinen functions as:

[TABLE]

where $c_{n},d_{n},e_{n}$ and $\alpha,\beta,\gamma$ are real numbers that represent tuning parameters and coefficients of the Pietarinen functions $X(W)$, $Y(W)$, and $Z(W)$, respectively. A variable number of series was used, depending on the structure of the non-pole part of each amplitude, and $x_{P}$, $x_{Q}$, and $x_{R}$ represent the branch points for each Pietarinen function. Once the pole position and residue were determined, the photo-decay amplitude at the pole could be constructed, as described in Ref. photo-pole . The residue of the corresponding $\pi N$ elastic scattering amplitude, required in this construction, was taken from the SAID analysis of elastic scattering data piN .

The $A_{1/2}(n)$ and $A_{3/2}(n)$ neutron helicity amplitudes for $N(1440)1/2^{+}$, $N(1535)1/2^{-}$, $N(1650)1/2^{-}$, and $N(1720)3/2^{+}$ for the new SAID MA27 solution are compared in Table 3 to the recent SAID GB12 gb12 and BG2013 BnGa13 solutions that were based on fits to all available data at the time, including the CLAS g10 dataset gb12 . From this table, the photo-decay amplitudes determined from the MA27 solution can be directly compared against the Breit-Wigner determinations. In addition, Table 3 includes a comparison to the older MAID2007 MAID07 solution, to the relativized quark model predictions of Ref. Capstick , and to the current PDG values PDG16 . The uncertainties on the modulus and phase quoted in Table 3 for the new MA27 solution were derived by comparing the global energy-dependent and energy-independent, single-energy amplitudes (see Ref. ed-se for a discussion on the two approaches). The comparison gave residues with uncertainties. Extracting the photo-decay amplitudes and considering the spread of possible values gave the MA27 uncertainties. These comparisons showed that the parameters from the MA27 solution are reasonably well under control.

The pole-valued and Breit-Wigner amplitudes from the fits are generally consistent in terms of the moduli. The comparisons are reasonable for the $N(1440)1/2^{+}$ and the $N(1535)1/2^{-}$. For the $N(1650)1/2^{-}$, the change from GB12 gb12 is significant and the result is in reasonable agreement with BG2013 BnGa13 , which used the FSI corrected g10 $\gamma n\to\pi^{-}p$ cross sections that were used for GB12. The $N(1650)1/2^{-}$ state has been difficult to describe as it is so close to the $N(1535)1/2^{-}$ and the $\eta N$ cusp, however, this pole-valued determination is believed to be more model-independent than the Breit-Wigner amplitude Svarc , which is reflected in the quoted uncertainties shown in Table 3. For the $N(1720)3/2^{+}$, the differences with respect to the BG2013 solution BnGa13 are significant and indicate that the CLAS g13 data provide tighter constraints in the coupled-channel model fits.

Comparing the new SAID MA27 solution with the relativized quark model predictions of Ref. Capstick , there are significant differences in the helicity amplitudes for the $N(1440)1/2^{+}$ and $N(1650)1/2^{-}$, while the helicity amplitudes for the $N(1535)1/2^{-}$ and $N(1720)3/2^{+}$ are in good agreement. With respect to the current PDG values PDG16 , Table 3 shows good correspondence with the MA27 solution for $N(1440)1/2^{+}$ and $N(1535)1/2^{-}$, but sizable disagreements for the higher-lying states.

A direct comparison of the quoted uncertainties on the neutron helicity amplitudes from the different solutions presented in Table 3 must be made with some caution. For the MA27, GB12, and BG2013 listings, the uncertainties do not take into account the significant model dependence in fitting the sparse database. In fact, the variance of the extracted results from different solutions fitting the same database would provide a reasonable estimate for this model dependence. However, this direct comparison is not possible given the different data sets employed for the different solutions shown in Table 3. Considering this issue, it is still meaningful that the overall quoted uncertainties for the helicity amplitudes from the MA27 solution are noticeably reduced relative to the BG2013 solution and to the GB12 solution (in particular for the $N(1650)1/2^{-}$) due to a combination of two factors. The first is the increased size of the database for MA27 that includes the new g13 $\gamma n$ cross sections and the second is the reduced model dependence of the pole fit approach employed for MA27.

XI Summary and Conclusions

A comprehensive set of $\gamma n\to\pi^{-}p$ differential cross sections at 157 photon energies has been determined with CLAS using a tagged-photon beam at incident photon energies from 0.445 GeV to 2.510 GeV. These data provide a factor of nearly three increase to the world’s data for this channel at these energies. To extract the $\gamma n$ cross section from the $\gamma d$ data, FSI corrections were included using a diagrammatic technique that takes into account a kinematic cut with momenta below (above) 200 MeV/c to select slow (fast) outgoing protons. In this analysis, the FSI correction factor depended on the photon energy and meson production angle, and was averaged over the rest of the variables in the region of the quasi-free process on the neutron.

The data collected in this CLAS g13 dataset spans a broad energy range, from just above the $\Delta$ isobar through the second, third, and fourth resonance regions. These data extend far into the poorly studied high-mass region above $W\sim 1.8$ GeV where many resonances are expected to exist but have not been firmly established. The precision of the data can be seen not only in the presented differential cross sections, but also through the uncertainties on the extracted Legendre coefficients. This approach of fitting the excitation functions with a Legendre series presents the data in a more compact and visual manner. These results will be useful for performing detailed phase shift analyses to better understand the resonant amplitudes.

On the experimental side, further improvements in the partial wave analyses await more precision data, specifically in the region above $E_{\gamma}=0.5$ GeV involving polarized photons and/or polarized targets. The data that are presently available are provided in Ref. SAID . Due to the closing of hadron facilities, new $\pi^{-}p\to\gamma n$ experiments are not planned, and only $\gamma n\to\pi^{-}p$ measurements are possible at electromagnetic facilities using deuterium targets. The agreement of these new $\gamma n\to\pi^{-}p$ cross section data with existing inverse $\pi^{-}$ photoproduction measurements indicates that these g13 measurements are reliable despite the use of deuterium as an effective neutron target.

As part of this new dataset for $\gamma n\to\pi^{-}p$, a new SAID multipole analysis called MA27 has been completed. This energy-dependent solution, which includes the CLAS g13 data, provides an improved understanding of the $N^{*}$ resonance parameters for several states, compared to the previous GB12 SAID solution that does not include the g13 CLAS data. In the MA27 solution, several photo-decay amplitudes $N^{*}\to\gamma n$ have been extracted at their pole positions on the complex plane with very small uncertainties. This is the first-ever determination of the excited neutron multipoles for the $N(1440)1/2^{+}$, $N(1535)1/2^{-}$, $N(1650)1/2^{-}$, and $N(1720)3/2^{+}$ resonances, contributing a crucial complement to the excited proton spectra. In addition, these new precision $\gamma n\to\pi^{-}p$ data will provide important and necessary constraints to advance coupled-channel analysis fits that are sorely lacking $\gamma n$ data over nearly the full nucleon resonance region.

Acknowledgements.

The authors acknowledge the outstanding efforts of the staff of the Accelerator and the Physics Divisions at Jefferson Lab that made this experiment possible. This work was supported by the US Department of Energy, the National Science Foundation, the Scottish Universities Physics Alliance (SUPA), the United Kingdom’s Science and Technology Facilities Council, the National Research Foundation of Korea, the Italian Instituto Nazionale di Fisica Nucleare, the French Centre National de la Recherche Scientifique, the French Commissariat à l’Energie Atomique, and the Deutsche Forschungsgemeinschaft (SFB 1044). This material is based upon work supported by the U. S. Department of Energy, Office of Science, Office of Nuclear Physics under Contract No. DE-AC05-06OR23177. The authors A.E.K. and V.E.T. also acknowledge the support of the grant RFBR 16-02-00767.