A2 Collaboration at MAMI

# Evaluation of the E2/M1 ratio in the $N\to\Delta(1232)$ transition from the
$\vec{\gamma}\vec{p}\to p\pi^{0}$ reaction

E. Mornacchi  Institut für Kernphysik, University of Mainz, D-55099 Mainz,
Germany  P. Pedroni  INFN Sezione di Pavia, I-27100 Pavia, Italy  F. Afzal  Y.
Wunderlich  Helmholtz-Institut für Strahlen- und Kernphysik, Universität Bonn,
D-53115 Bonn, Germany  S. Abt  Department für Physik, Universität Basel,
CH-4056 Basel, Switzerland  P. Achenbach  Institut für Kernphysik, University
of Mainz, D-55099 Mainz, Germany  J.R.M. Annand  SUPA School of Physics and
Astronomy, University of Glasgow, Glasgow G12 8QQ, United Kingdom  H.J. Arends
Institut für Kernphysik, University of Mainz, D-55099 Mainz, Germany  M.
Bashkanov  Department of Physics, University of York, Heslington, York, Y010
5DD, UK  M. Biroth  Institut für Kernphysik, University of Mainz, D-55099
Mainz, Germany  R. Beck  Helmholtz-Institut für Strahlen- und Kernphysik,
Universität Bonn, D-53115 Bonn, Germany  N. Borisov  Joint Institute for
Nuclear Research, 141980 Dubna, Russia  A. Braghieri  INFN Sezione di Pavia,
I-27100 Pavia, Italy  W.J. Briscoe  The George Washington University,
Washington, DC 20052-0001, USA  F. Cividini  Institut für Kernphysik,
University of Mainz, D-55099 Mainz, Germany  C. Collicott  Institut für
Kernphysik, University of Mainz, D-55099 Mainz, Germany  A. Denig  Institut
für Kernphysik, University of Mainz, D-55099 Mainz, Germany  A. S. Dolzhikov
Joint Institute for Nuclear Research, 141980 Dubna, Russia  E. Downie  The
George Washington University, Washington, DC 20052-0001, USA  S. Fegan
Department of Physics, University of York, Heslington, York, Y010 5DD, UK  A.
Fix  Tomsk Polytechnic University, 634034 Tomsk, Russia  D. Ghosal  Department
für Physik, Universität Basel, CH-4056 Basel, Switzerland  I. Gorodnov  Joint
Institute for Nuclear Research, 141980 Dubna, Russia  W. Gradl  Institut für
Kernphysik, University of Mainz, D-55099 Mainz, Germany  G. G. Gurevich
Institute for Nuclear Research, 125047 Moscow, Russia  L. Heijkenskjöld
Institut für Kernphysik, University of Mainz, D-55099 Mainz, Germany  D.
Hornidge  Mount Allison University, Sackville, New Brunswick E4L 1E6, Canada
G.M. Huber  University of Regina, Regina, Saskatchewan S4S 0A2, Canada  V.L.
Kashevarov  Institut für Kernphysik, University of Mainz, D-55099 Mainz,
Germany  S.J.D. Kay  Department of Physics, University of York, Heslington,
York, Y010 5DD, UK  M. Korolija  Rudjer Boskovic Institute, HR-10000 Zagreb,
Croatia  A. Lazarev  Joint Institute for Nuclear Research, 141980 Dubna,
Russia  K. Livingston  SUPA School of Physics and Astronomy, University of
Glasgow, Glasgow G12 8QQ, United Kingdom  S. Lutterer  Department für Physik,
Universität Basel, CH-4056 Basel, Switzerland  I.J.D. MacGregor  SUPA School
of Physics and Astronomy, University of Glasgow, Glasgow G12 8QQ, United
Kingdom  D.M. Manley  Kent State University, Kent, Ohio 44242-0001, USA  P.P.
Martel  Institut für Kernphysik, University of Mainz, D-55099 Mainz, Germany
R. Miskimen  University of Massachusetts, Amherst, Massachusetts 01003, USA
M. Mocanu  Department of Physics, University of York, Heslington, York, Y010
5DD, UK  C. Mullen  SUPA School of Physics and Astronomy, University of
Glasgow, Glasgow G12 8QQ, United Kingdom  A. Neganov  Joint Institute for
Nuclear Research, 141980 Dubna, Russia  A. Neiser  Institut für Kernphysik,
University of Mainz, D-55099 Mainz, Germany  M. Oberle  Department für Physik,
Universität Basel, CH-4056 Basel, Switzerland  M. Ostrick  Institut für
Kernphysik, University of Mainz, D-55099 Mainz, Germany  P.B. Otte  Institut
für Kernphysik, University of Mainz, D-55099 Mainz, Germany  D. Paudyal
University of Regina, Regina, Saskatchewan S4S 0A2, Canada  A. Powell  SUPA
School of Physics and Astronomy, University of Glasgow, Glasgow G12 8QQ,
United Kingdom  T. Rostomyan  Department für Physik, Universität Basel,
CH-4056 Basel, Switzerland  V. Sokhoyan  Institut für Kernphysik, University
of Mainz, D-55099 Mainz, Germany  K. Spieker  Helmholtz-Institut für Strahlen-
und Kernphysik, Universität Bonn, D-53115 Bonn, Germany  O. Steffen  Institut
für Kernphysik, University of Mainz, D-55099 Mainz, Germany  I.I. Strakovsky
The George Washington University, Washington, DC 20052-0001, USA  M. Thiel
Institut für Kernphysik, University of Mainz, D-55099 Mainz, Germany  A.
Thomas  Institut für Kernphysik, University of Mainz, D-55099 Mainz, Germany
Yu.A. Usov  Joint Institute for Nuclear Research, 141980 Dubna, Russia  S.
Wagner  Institut für Kernphysik, University of Mainz, D-55099 Mainz, Germany
D.P. Watts  Department of Physics, University of York, Heslington, York, Y010
5DD, UK  D. Werthmüller  Department of Physics, University of York,
Heslington, York, Y010 5DD, UK  J. Wettig  Institut für Kernphysik, University
of Mainz, D-55099 Mainz, Germany  L. Witthauer  Department für Physik,
Universität Basel, CH-4056 Basel, Switzerland  M. Wolfes  Institut für
Kernphysik, University of Mainz, D-55099 Mainz, Germany  N. Zachariou
Department of Physics, University of York, Heslington, York, Y010 5DD, UK

###### Abstract

A new data set for the helicity-dependent differential cross section of the
single-meson photoproduction reaction $\gamma p\to p\pi^{0}$ was obtained for
the photon energy interval 150-400 MeV. The experiment was performed at the A2
tagged photon facility of the Mainz Microtron MAMI using a circularly
polarized photon beam and a longitudinally polarized proton target. The
reaction products were detected with the large acceptance Crystal Ball/TAPS
calorimeter covering 97% of the full solid angle. These new results, obtained
with a fine energy and polar angle binning, greatly increase both the existing
quantity and quality of the data available for this observable. A moment
analysis, based on a finite expansion in Legendre polynomials, was applied to
these data by using a bootstrap-based fitting method to correctly account for
their systematic uncertainties. From the resulting decomposition of the
differential cross sections, the $E2/M1$ ratio for the $N\to\Delta(1232)$
transition was determined to be $[-2.38\pm 0.16{\hbox{ (stat.+sys.)}}\pm
0.10{\hbox{ (model)}}]\%$. Combining this value with previous results also
allowed us to evaluate the most precise available estimate of the $E2/M1$
ratio to be used for all further reference and model comparisons.

##  I Introduction

Fundamental states of hadrons are characterized by complex quark-gluon and
meson cloud dynamics, which are governed by non-perturbative QCD and give rise
to non-spherical components in their hadronic wave-function. Precise
experimental determination of the shapes then gives deep insight into these
interactions and provides a fundamental precision benchmark for all types of
models describing the hadronic structure (see, for instance, Refs. Bernstein
and Papanicolas (2007); Alexandrou  _et al._ (2012) and references therein).

The most direct and reliable evidence of a shape deformation is provided by
the determination of the particles’ electric quadrupole moment. However, the
proton, the only stable hadron, has a vanishing spectroscopic quadrupole
moment in the laboratory frame due to its spin-1/2 nature. Therefore, one has
to study the transition to its lowest $J=3/2$ excited state, namely the
$\Delta(1232)$ resonance.

The $N\to\Delta(1232)$ electromagnetic transition is predominantly due to the
magnetic dipole component $M1$. In a very simple constituent quark model
framework, this process is described by a spin flip of a single quark in the
$s$-wave state (see, for instance, Ref. Krusche and Schadmand (2003) and
references therein). Using real photons, any $d$-wave mixture in the nucleon
and/or in the $\Delta(1232)$ wave functions allows for the electric quadrupole
transition $E2$. Therefore, by measuring the $\gamma p\to\Delta(1232)\to N\pi$
reactions, one can assess the presence of the $d$-wave components and thus
quantify to what extent the nucleon and/or the $\Delta(1232)$ resonance
deviate from the spherical shape.

The amplitudes in the $\gamma N\to N\pi$ final states are usually described by
the notation $E_{\ell\pm}^{I}$ and $M_{\ell\pm}^{I}$, where $E$ and $M$ are
the electric and magnetic multipoles, respectively, $\ell$ is the orbital
angular momentum of the photoproduced pion, the sign $\pm$ refers to the total
$N\pi$ angular momentum $J=\ell\pm 1/2$, and $I$ is the isospin of the $N\pi$
system.

A common practice (see again Ref. Krusche and Schadmand (2003)) is to measure
the resonant quadrupole strength relative to the resonant dipole amplitude via
the ratio

$R_{EM}=\frac{E2}{M1}\equiv\left.\frac{\hbox{Im}[E_{1+}^{3/2}]}{\hbox{Im}[M_{1+}^{3/2}]}\right\rvert_{M_{\Delta}}$
(1)

evaluated at the $\Delta(1232)$ mass value $M_{\Delta}$.

Early empirical quark and Skyrme models, as well as relativistic models
including two-body exchange currents, gave a variety of $R_{EM}$ values in the
range $-6\%<R_{EM}<0\%$, where the minus sign indicates an oblate shape
deformation (see e.g., Refs. Beck  _et al._ (1997); Blanpied  _et al._ (2001);
Tiator  _et al._ (2003) and references therein).

More recently, a similar range of values was predicted by EFT approaches
Pascalutsa and Vanderhaeghen (2006); Gail and Hemmert (2006), dynamical and
effective Lagrangian models Tiator  _et al._ (2003); Sato and Lee (2001);
Pascalutsa and Tjon (2004); Fernandez-Ramirez  _et al._ (2006), and the most
recent lattice calculations Alexandrou  _et al._ (2011).

From an experimental point of view, isolating the resonant $E_{1+}$ amplitude
is complicated by its rather small value compared to the dominant $M_{1+}$
multipole transition. To overcome these difficulties, high-intensity photon
beams of precisely known energy are required, along with the measurement of
selected polarization observables that highlight the role of the small
$E_{1+}$ multipole thanks to the presence of interference terms between
$E_{1+}$ and the dominant $M_{1+}$ multipole.

These conditions were met by the advent of a newer generation of tagged photon
facilities in the late 1990s. In the last 20 years, several precision
measurements have been performed at the photon tagging facility of the MAMI
Microtron in Mainz (Germany) Ahrens et al. (2004); Beck  _et al._ (2000,
2002); Galler et al. (2001) and at the laser back-scattering facility (LEGS)
of the Brookhaven National Laboratory (USA) Blanpied  _et al._ (2001), where
$R_{EM}$ was evaluated from both pion production and Compton scattering data.

The experimental apparatus of the LEGS Collaboration Blanpied  _et al._ (2001)
had a complex geometry with a small angular acceptance and was best suited for
photon detection. Measurements at the MAMI tagging photon facility have
instead been performed over the years with different different detector
setups: the large acceptance DAPHNE detector Beck  _et al._ (2000); Ahrens et
al. (2004), covering the polar angle range
$\theta_{\text{lab}}\in[21^{\circ},159^{\circ}]$ with full azimuthal
acceptance and good charged particle detection capabilites, the TAPS
calorimeter Beck  _et al._ (2000) and the LARA apparatus Galler et al. (2001),
both suited for the photon detection, which had a complex geometry and
somewhat limited geometrical acceptance, although they covered more extreme
forward and backward angles compared to the LEGS apparatus.

All of these experiments gave estimates of $R_{EM}$ that are compatible,
within their errors, with the interval given by the latest Particle Data Group
(PDG) estimate Workman (2022) of $-3\%\leq R_{EM}\leq-2\%$. A very similar
interval ($R_{EM}=-2.5\%\pm 0.4\%$) has also been obtained from an analysis of
the more recent $\gamma N\to N\pi$ data using the AMIAS methodology Markou
_et al._ (2018), a general-purpose algorithm applied for the analysis of
several different hadronic and nuclear physics data sets Stiliaris and
Papanicolas (2007); Alexandrou  _et al._ (2015).

In this paper, we present a new precise determination of the $R_{EM}$ ratio
from the measurement of the helicity-dependent differential cross section of
the $\gamma p\to p\pi^{0}$ reaction in the incident photon energy range from
150 to 400 MeV. This experiment was performed at the A2 tagged photon facility
of the MAMI electron accelerator in Mainz, Germany Kaiser et al. (2008), using
the experimental setup of the A2 Collaboration that combines all the strengths
of the previous experiments described above: a very large ($\sim 97\%$ of
$4\pi$) angular acceptance with a good energy and angular resolution for
photons and protons, as well as a high photon detection efficiency. The
measured double-polarization observable, accessed using a circularly polarized
photon beam and a longitudinally polarized proton target, can be defined as:

$\frac{d\Delta\sigma}{d\Omega}=\Bigg{(}\frac{d\sigma}{d\Omega}\Bigg{)}_{3/2}-\Bigg{(}\frac{d\sigma}{d\Omega}\Bigg{)}_{1/2}\
,$ (2)

where the subscripts $3/2$ and $1/2$ indicate the total helicity states of the
$\gamma p$ system corresponding to the relative parallel or antiparallel
photon-proton spin configurations, respectively.

Thanks to the very high statistics collected, it was possible to measure
$d\Delta\sigma/d\Omega$ in very fine beam photon energy and polar angle
binning, with a width of $\simeq 2$ MeV and $10^{\circ}$, respectively. This
precise mapping of the entire $\Delta(1232)$ resonance excitation region
greatly enhanced both the existing quantity and quality of the data available
for this observable and allowed the existing estimates of the $E2/M1$ ratio to
be improved using a Legendre-moment analysis.

The paper is organized as follows. In Secs. II and III, the experimental setup
and the offline analysis methods used to obtain the measured observable are
briefly described. In Sec. IV, the new results for $d\Delta\sigma/d\Omega$ are
presented and compared with the scarce existing database. Section V reports
the Legendre-moment analysis of the $d\Delta\sigma/d\Omega$ data, with the
novel bootstrap-based fitting method used in the current analysis described in
Subsection V.1. The determination of the $E2/M1$ ratio from the definitions of
the fitted Legendre-moments in terms of multipoles is reported in Sec. VI.
Finally, the summary and outlook are given in Sec. VII.

##  II Experimental setup

The helicity-dependent data used for this analysis were collected at the A2
tagged photon facility of the MAMI electron accelerator in Mainz, Germany
Kaiser et al. (2008).

Figure 1: Sketch of the experimental setup of the A2 Collaboration tagged
photon facility, including photon tagging apparatus and detectors Mornacchi
(2021). The figure is not to scale.

Figure 1 shows a sketch of the A2 experimental setup used for the measurement.
Since this setup has already been described in detail (see, for instance,
Refs. Witthauer et al. (A2 Collaboration at MAMI); Werthmüller et al. (A2
Collaboration at MAMI); Dieterle et al. (A2 Collaboration at MAMI); Witthauer
et al. (A2 collaboration at MAMI); Dieterle et al. (A2 collaboration at MAMI);
Cividini et al. (2022) and references therein), we will limit the discussion
to the features relevant to the present experiment.

###  II.1 The photon beam

The circularly polarized photons used for this measurement were produced via
Bremsstrahlung on an amorphous radiator of the 450-MeV longitudinally
polarized electron beam. To avoid polarization dependent photon flux values,
the helicity of the electron beam was flipped at a rate of 1 Hz.

The electron polarization degree, $P_{e}$, was regularly determined by Mott
scattering close to the electron source Tioukine  _et al._ (2011) and was
found to be more than 80% with a systematic uncertainty of $\pm 3\%$.

The recoil electrons from the Bremsstrahlung process were momentum-analyzed
using the Glasgow-Mainz spectrometer with an energy resolution of $\sim 1$
MeV, corresponding to the width of the detector channels C. McGeorge et al.
(2008). The resulting photon beam passed through a 2 mm-diameter lead
collimator, reaching the target and detection apparatus.

The degree of the energy-dependent circular photon polarization,
$P_{\odot}^{\gamma}$, was determined using the Olsen and Maximon equation
Olsen and Maximon (1959):

$\frac{P_{\odot}^{\gamma}}{P_{e}}=\frac{4x-x^{2}}{4-4x+3x^{2}}\ ,$

where $x=E_{e}/E_{\gamma}$, with $E_{e}$ and $E_{\gamma}$ being the energy of
the electron and the Bremsstrahlung photon, respectively.

The photon tagging efficiency was measured once a day using a Pb-Glass
Cherenkov detector in dedicated low flux runs. During the standard data taking
operation, fluctuations in the photon flux were monitored using a low-
efficiency pair spectrometer located in the photon beamline after the
collimator. An absolute systematic uncertainty in the photon flux of 4% was
estimated by comparing the data from these detectors obtained under a range of
different experimental conditions.

###  II.2 The target system

The longitudinally polarized proton target used in this experiment was the
Mainz-Dubna Frozen Spin Target Rohlof and Dutz (1999); Goertz et al. (2004).
The filling factor for the $\sim 2$-mm-diameter butanol spheres contained in
the 2-cm-long, 2-cm-diameter target container was estimated to be 60%, with a
systematic uncertainty of 2% Rohlof and Dutz (1999).

The target material (butanol) was polarized using the Dynamic Nuclear
Polarization effect Bradtke et al. (1999), which requires a high magnetic
field (about 2.5 T) and a temperature of about 25 mK. A small holding magnetic
field of 0.6 T, which replaced the polarizing magnet during the data taking
phase, allowed regular relaxation times of about 1000 h to be achieved.

The target polarization was measured with an NMR system before and after each
data taking period and then exponentially interpolated at intermediate times.
Corrections to the calculated polarization values were necessary due to ice
formation on the NMR coils. They were taken from Refs. Paudyal (2017); Paudyal
et al. (2020), which independently analyzed the same data set used in the
present work. Due to these corrections, as in Ref. Paudyal et al. (2020), a
conservative systematic uncertainty of 10% was applied to the target
polarization values.

###  II.3 The hadron detector

The photon-induced reaction products were detected by the Crystal Ball-TAPS
apparatus. The Crystal Ball (CB) calorimeter was placed around the target cell
and covered the full azimuthal ($\phi$) angle and a polar ($\theta$) angle
range from 21∘ to 159∘ Starostin  _et al._ (2001). It consisted of 672 NaI(Tl)
crystals and had a $\sim 100\%$ detection efficiency for photons coming from
the $\pi^{0}$ decay. Inside CB, from the inside out, there were a Particle
Identification Detector (PID), consisting of a barrel of 24 plastic
scintillators, and two Multi-Wire Proportional Chambers (MWPCs). The
combination of all these detectors provided a precise tracking and
identification of charged particles. TAPS was an hexagonal wall covering the
polar angle forward region outside the CB acceptance,
$1^{\circ}<\theta<20^{\circ}$, and was made of of 366 BaF2 and 72 PbWO4
crystals Novotny et al. (1991); A. Gabler et al. (1994). In front of each
crystal there was a 5-mm-thick plastic scintillator (VETO) that was used for
charged particle identification. The combination of the large acceptance CB
and TAPS calorimeters covered $\sim$ 97$\%$ of the full solid angle.

##  III Data Analysis

After offline energy and time calibration of all detectors, the data from the
butanol target were analyzed, and all the implemented algorithms were tested
and checked with simulation, to obtain an optimal identification of the
$\gamma p\to p\pi^{0}$ channel.

No unpolarized background had to be evaluated since the effect of the $C$ and
$O$ spinless nuclei present in the target vanishes in the
$d\Delta\sigma/d\Omega$ difference (see Eq. (1)).

###  III.1 $\pi^{0}$ reconstruction and identification

The algorithms used for the data selection were basically the same as those
described in Ref. Cividini et al. (2022). Therefore, only a summary of the
main analysis steps required to identify the $p\pi^{0}$ channel is given here.

The candidate events accepted for the differential cross section evaluation
were those with 2 neutral or 3 (with at least two neutral) clusters
reconstructed inside the detector. A neutral cluster is defined as an energy
deposition in one of the two calorimeters, without an associated hit in either
the PID or MWPC, or VETO for clusters in CB and TAPS, respectively.

Due to the relatively high tagged photon flux ($\approx~{}\\!10^{7}\
\gamma/s$), a time coincidence within $20$ ns was required between the trigger
in the calorimeters and the hits in the tagger focal plane detector. To remove
the random coincidences in the selected time window, a side-band subtraction
was also performed by selecting a background sample on each side of the prompt
peak.

The first offline analysis step was the evaluation of the two-photon invariant
mass (IM) using all neutral clusters of each event. For all events with more
than 2 neutral hits, all possible combinations were used to calculate the IM
and only the combination that gave the closest value to the nominal $\pi^{0}$
mass was retained for subsequent analysis steps.

Figure 2: Left panel: two photon invariant mass (IM) distribution from the
$\pi^{0}$ reconstruction procedure obtained for the difference of the
candidate events with the parallel and antiparallel spin configurations. Right
panel: same as before, but for the missing mass (MM) distribution after the
IM-based selection shown in the left panel. In both cases, the acceptance
region for the selection of a good $p\pi^{0}$ event is inside the vertical red
lines.

The overall IM distribution obtained for the difference of candidate events
with the parallel and antiparallel spin configurations is shown in the left
panel of Fig. 2, together with the cuts ($\hbox{IM}\in[85-165]$ MeV) applied
in the offline analysis that selected the events for further analyses.

The next step was to evaluate the missing mass (MM) of the event, where the
recoil nucleon of the reaction $\gamma N\rightarrow\pi^{0}N$ was considered as
a missing particle, even when it was detected. The missing mass was calculated
as follows:

${\rm
MM}=\sqrt{(E_{\gamma}+m_{p}-E_{\pi^{0}})^{2}-(\vec{p}_{\gamma}-\vec{p}_{\pi^{0}})^{2}}\
\ ,$ (3)

where $E_{\gamma}$ and $\vec{p}_{\gamma}$ are the energy and momentum of the
incoming photon in the lab frame, respectively, $m_{p}$ is the proton mass in
the initial state, $E_{\pi^{0}}$ and $\vec{p}_{\pi^{0}}$ are the reconstructed
$\pi^{0}$ total energy and momentum, respectively.

The MM distribution obtained for the difference of candidate events after the
previous analysis steps with the parallel and anti-parallel spin
configurations is shown in the right panel of Fig. 2. Events from the
$\pi^{0}\pi^{0}$ channel, coming from the high energy part of the photon
Bremsstrahlung spectrum, can be clearly seen as a small peak in the right part
of the distribution. They are completely rejected by the applied selection
cuts ($\hbox{MM}\in[870-1005]$ MeV) shown in the right panel of Fig. 2.

The detection and reconstruction efficiency of the $p\pi^{0}$ events was
evaluated using a GEANT4-based simulation Agostinelli et al. (2003) that
accurately modeled the geometry and composition of the detection apparatus and
considered the applied electronic thresholds. As an example, the simulated
$\pi^{0}$ reconstruction efficiency at the photon beam energy $E_{\gamma}=300$
MeV is shown in Fig. 3 as a function the polar $\pi^{0}$ emission angle in the
c.m. system ($\theta_{\pi^{0}}^{cm}$). This efficiency, over the full measured
photon energy interval, varies in the forward, central, and backward angular
regions within the ranges $[40\%-60\%]$, $[50\%-90\%]$, and $[25\%-50\%]$,
respectively.

The relative systematic uncertainty was evaluated by examining the different
cuts and selection conditions applied to both the experimental and the
simulated data and it was estimated to be 10% of the value of the applied
correction.

Figure 3: $\pi^{0}$ reconstruction efficiency obtained using Monte Carlo
simulated events at $E_{\gamma}=300$ MeV, as a function of the polar $\pi^{0}$
emission angle in the c.m. system $\theta_{\pi^{0}}^{\text{cm}}$.

###  III.2 Systematic uncertainties

The various sources of systematic uncertainties discussed previously are
summarized in Table 1.

Sources of common constant systematic uncertainties come from the photon flux
normalization, the beam and target polarization, and from the target surface
density. In contrast, the systematic uncertainty related to the $\pi^{0}$
reconstruction efficiency depends on both $E_{\gamma}$ and
$\theta^{\text{cm}}_{\pi^{0}}$, and ranges from $\sim$ 1% to $\sim$ 7% of the
absolute $d\Delta\sigma/d\Omega$ values.

Table 1: Relative systematic uncertainties given as total widths of uniformly-distributed values.  Source | Error
---|---
Tagging efficiency | $\pm 4\%$
Beam polarization | $\pm 3\%$
Target polarization | $\pm 10\%$
Target density | $\pm 2\%$
$\pi^{0}$reconstruction efficiency | $\pm 1\%-7\%$

##  IV Results

Figure 4:  Excitation functions of the $\vec{\gamma}\vec{p}\to p\pi^{0}$
reaction as a function of the photon beam energy for the measured forward
$\theta_{\pi^{0}}^{\text{cm}}$ bins. The experimental results in blue are
compared with the predictions given by the BnGa-2019 (dashed red lines) and
SAID-SM22 (dotted green lines) energy-dependent PW analyses, and the DMT-2001
dynamical model (solid orange lines). The total contribution of all the
systematic uncertainties, given in half width units (see Sec. III.2), is shown
as gray bars.

A compact representation of the differential cross section
$d\Delta\sigma/d\Omega$ data can be obtained by plotting them as excitation
functions at the measured $\theta_{\pi^{0}}^{\text{cm}}$ angles, as shown in
Figs. 4 and 5.

The different lines show the predictions for this observable given by two
energy-dependent partial wave (PW) analyses: BnGa-2019 V. Anisovich et al.
(2016) (dashed red lines), SAID-SM22 Briscoe  _et al._ (2023) (dotted green
lines), and by the DMT-2001 dynamical model Kamalov and Yang (1999); Kamalov
_et al._ (2001a, b); Tiator  _et al._ (2010) (solid orange lines). The
DMT-2001 model is a field-theoretical meson-exchange model for $\pi N$
scattering, optimized for the description of the single pion photoproduction
process below $400$ MeV. In contrast to the previous PW analyses, the energy
dependence of the DMT amplitudes is largely determined from theoretical
considerations even though there are free parameters, describing the non-
resonant part of the amplitudes, that are determined from the fit of $\gamma
N\to\pi N$ data.

Figure 5:  Same as in Fig. 4, but for for the measured backward
$\theta_{\pi^{0}}^{\text{cm}}$ bins.

In the comparison with our data, all the predictions show a rather good
agreement in all energy ranges, except for slight differences at the very
forward and backward angles.

Figure 6:  The new data on the differential helicity-dependent cross section
(blue circles) are compared with available published data from the GDH Ahrens
et al. (2004) (red squares) and LEGS Hoblit  _et al._ (2009) (green up
triangles) collaborations. The total contribution of the systematic
uncertainties of the present data, given in half width units (see Sec. III.2),
is shown as gray bars.  Figure 7:  The new data on the differential helicity-
dependent cross section (blue circles) are compared to a combination of
previously published A2 data Adlarson et al. (2015); Afzal et al. (2023) (red
down triangles). See text for details. The total contribution of the
systematic uncertainties of the present data, given in half width units (see
Sec. III.2), is shown as gray bars.

In Fig. 6 some of the new $d\Delta\sigma/d\Omega$ data are compared, at fixed
$E_{\gamma}$ values, to all available results for this observable, that were
published by the GDH Ahrens et al. (2004) (red squares) and LEGS Hoblit  _et
al._ (2009) (green up triangles) collaborations. As can be easily seen from
the previous figures, the present work, with 18 ($10^{\circ}$ wide)
$\theta_{\pi^{0}}^{\text{cm}}$ bins measured at 114 $E_{\gamma}$ ($\sim\\!2$
MeV wide) bins, provides a huge improvement in both quantity and quality
compared to the previous data in the $\Delta(1232)$ resonance region.

As a further check, Fig. 7 compares our results with a combination of
previously published data from the A2 Collaboration (red down triangles).
These combined values were obtained, for $E_{\gamma}<400$ MeV, by multiplying
the unpolarized differential cross section values $d\sigma_{0}/d\Omega$ from
Ref. Adlarson et al. (2015) with the recently published $E$ asymmetry values
from Ref. Afzal et al. (2023), according to the well-known identity (see, for
instance, Ref. Wunderlich  _et al._ (2017)):

$\frac{d\Delta\sigma}{d\Omega}=-2\cdot\frac{d\sigma_{0}}{d\Omega}E\ .$ (4)

The excellent agreement found with both the previously published GDH and A2
data gives a strong indication of the overall correctness of the offline
analysis procedures.

Figure 8: The new total helicity-dependent cross section $\Delta\sigma$ for
the $\gamma p\to p\pi^{0}$ channel (blue circles) is compared with the results
obtained by the GDH Ahrens et al. (2004) (red squares) and LEGS Hoblit  _et
al._ (2009) (green up triangles) collaborations. The different line styles
show the predictions of the BnGa-2019 V. Anisovich et al. (2016) (dashed red
line) and SAID-SM22 Briscoe  _et al._ (2023) (green dotted line) PW analyses,
and of the DMT-2001 dynamical model Kamalov and Yang (1999); Kamalov  _et al._
(2001a, b); Tiator  _et al._ (2010) (solid orange line). The total
contribution of the systematic uncertainties, given in half width units (see
Sec. III.2), is shown as gray bars.

The total helicity-dependent cross section $\Delta\sigma$ for the $\gamma p\to
p\pi^{0}$ reaction, obtained by integrating the differential cross section
$d\Delta\sigma/d\Omega$ over the full solid angle, is shown in Fig. 8 (blue
points) for the photon beam energy range from $E_{\gamma}$=150 MeV up to 400
MeV. It is again compared to the predictions given by the previous PW analyses
and model and to the avaliable data from the GDH Ahrens et al. (2004) (red
points) and LEGS Hoblit  _et al._ (2009) (green points) collaborations.

As already noticed for the excitation functions shown in Figs. 4 and 5, our
data are in excellent agreement with the BnGa-2019 and SAID-SM22 PW analyses,
the DMT-2001 model, and with the GDH data, while differing from the LEGS data
in the region around the $\Delta$ resonance mass. As can be clearly deduced
from the differential cross section data shown in Fig. 6, this discrepancy
comes from differences in the angular shape, especially present in the forward
$\theta_{\pi^{0}}^{\text{cm}}$ region, that can not be accommodated with a
single common scale shift.

##  V Legendre analysis of the $\bm{d\Delta\sigma/d\Omega}$ data

The multipole content of the reaction amplitude can be easily accessed by
expanding the measured $d\Delta\sigma/d\Omega$ angular distributions in a
(truncated) Legendre series.

The energy dependence of the expansion coefficients can reveal specific
correlations between individual resonance states of certain parities (see, for
instance, Ref. Wunderlich  _et al._ (2017) and references therein). This
method proved to be particularly effective for $E_{\gamma}<400$ MeV, where
only one well-known resonance, the $\Delta(1232)$, dominates the amplitude and
low-lying multipoles can be accessed via the interference terms with the
dominant $M_{1+}$ term.

The Legendre coefficients $a_{k}$ were then obtained by fitting a series of
associated Legendre polynomials $P_{k}$ to the angular distributions
$d\Delta\sigma/d\Omega$ with the function (see Ref. Wunderlich  _et al._
(2017)):

$\frac{d\Delta\sigma}{d\Omega}=-2\,\frac{q}{k}\cdot{\displaystyle\sum_{k=0}^{2\ell_{\text{max}}}(a_{\ell_{\text{max}}})_{k}({W})P_{k}(\cos\theta)\
,}$ (5)

where $q,k$, and $W$ are the photon and $\pi^{0}$ momenta, and the total
energy in the c.m. system, respectively. The notation
$(a_{\ell_{\text{max}}})_{k}$ means that in the fitting procedure only the
partial waves with the $\pi N$ relative angular momentum up to
$\ell=\ell_{\text{max}}$ were included in the fit. The multipoles contributing
to the fit for $\ell_{\text{max}}=1,2$ are listed in Table 2.

Table 2: The multipole amplitudes contributing to the fitted cross section reported in Eq. (5) for different choices of $\ell_{\text{max}}$.  $\mathbf{\ell_{\text{max}}}$ | Wave | M-poles
---|---|---
1 |  $s$-wave | $E_{0+}$
$p$-wave | $E_{1+},M_{1+},M_{1-}$
2 |  $d$-wave | $E_{2+},E_{2-},M_{2+},M_{2-}$

###  V.1 Bootstrap-based fitting procedure

As noted in Ref. Wunderlich  _et al._ (2017), an important issue when using
this type of algorithm is the proper handling of the systematic uncertainties
associated with the data being fit. A widely used method, when one single
common and Gaussian-distributed multiplicative systematic uncertainty is
present, is to introduce a modified $\chi^{2}$ function, with an additional
overall scale parameter to be fitted (see, for instance, Refs. D’Agostini
(1994); Pedroni and Sconfietti (2020)):

$\chi_{\text{mod}}^{2}(\bm{\vartheta})=\sum_{i}{\left(\frac{fy_{i}-T_{i}(\bm{\vartheta})}{f\sigma_{i}}\right)^{2}}+\left(\frac{f-1}{\sigma_{sys}}\right)^{2}.$
(6)

Here $y_{i}$ and $\sigma_{i}$ are the experimental values to be fitted and
their corresponding statistical uncertainties in root mean square units (rms),
respectively. $T_{i}$ are the theoretical predictions given by a model
depending on the set of unknown parameters $\bm{\vartheta}$ to be evaluated
from the data, $f$ and $\sigma_{sys}$ are the additional fit parameter and its
estimated uncertainty (in rms units), respectively.

However, Eq. (6) cannot be used to fit the new $d\Delta\sigma/d\Omega$ data,
since they have both uniform and angular-dependent systematic uncertainties,
as discussed in Sec. III.2, and these effects can not be properly accounted
for by the single factor $f$. Moreover, under these conditions, the goodness-
of-fit distribution is generally not given by the $\chi^{2}$ function and the
fit parameter errors are not a priori Gaussian, since the sum appearing in Eq.
(6) consists of correlated and non-Gaussian variables.

To overcome all these difficulties, the fit to the present data was performed
by using an innovative method Pedroni and Sconfietti (2020), based on the
parametric bootstrap technique, already successfully deployed for different
analyses of nuclear Compton scattering data (see Refs. Pasquini  _et al._
(2018, 2019); Mornacchi  _et al._ (2022)).

As an example, in the case of a single data set with Gaussian statistical
errors and uniform common multiplicative systematic uncertainties, this method
consists of randomly generating $N$ Monte Carlo replicas of the experimental
data, where each data point $y_{i}$ is replaced by:

$y_{i}\rightarrow y_{i}^{(b)}=(1+\delta_{b})(y_{i}+r_{i,b}\sigma_{i})\ .$

The indices $i$ and $b$ run over the number of data points and the bootstrap
replica, respectively; $r_{i,b}$ is a random number extracted from the normal
distribution $\mathcal{N}(0,1)$, and $\delta_{b}$ is a uniform random variable
distributed according to $\mathcal{U}[-\Delta,\Delta]$, where $\pm\Delta$ is
the quoted systematic uncertainty interval. A set of fitted parameters
(${\bm{\vartheta}}_{b}$) is extracted from each of the $N$ simulated replicas
and, at the end of the procedure, the best values
($\hat{\bm{\vartheta}}_{\text{boot}}$) and the errors of the fitted quantities
can be easily extracted from the empirical statistical parameters of the the
resulting distributions.

The goodness of fit (see again Ref. Pedroni and Sconfietti (2020)) is
evaluated by comparing the minimum $\hat{\chi}_{\text{boot}}^{2}$ value
obtained as

$\hat{\chi}_{\text{boot}}^{2}=\sum_{i}{\left(\frac{y_{i}-T_{i}(\hat{\bm{\vartheta}}_{\text{boot}})}{\sigma_{i}}\right)^{2}}\
,$ (7)

with the theoretical distribution empirically obtained by repeating the
bootstrapping algorithm with each experimental datum $y_{i}$ replaced by
$T_{i}(\hat{\bm{\vartheta}}_{\text{boot}})$. With this procedure, after a
suitably large number $k$ of bootstrap cycles, an estimate of the goodness-of-
fit probability function can be obtained from the set of minimum values

$\hat{\chi}_{th,\text{boot},1}^{2},\hat{\chi}_{th,\text{boot},2}^{2},\ldots,\hat{\chi}_{th,\text{boot},k}^{2}$

evaluated at the end of each cycle. This ensures that the correct $p$-value is
then always provided by the present fitting procedure.

###  V.2 Fit results

A total of $N=10^{4}$ bootstrap samples were generated for each $W$ bin, and
the minimization was performed at the end of each iteration.

All the different systematic uncertainties were assumed to follow a uniform
distribution over the interval defined by the bounds given in Table 1. In the
case of the angular-dependent source, the same common fraction of the full
variation interval was randomly generated, for all angular bins, at each
bootstrap cycle. The final total uncertainty factor entered into the bootstrap
procedure is given by the product of all the random uniform variables
generated by the previous procedure.

Figure 9: The cumulative density function (CDF) of the goodness-of-fit
distribution divided by the number of degrees of freedom ($\nu$), as obtained
from the bootstrap procedure (red solid line) at $E_{\gamma}=350$ MeV, is
compared with the CDF of the reduced $\chi^{2}$ distribution (black dash-
dotted line).  Figure 10: Reduced $\hat{\chi^{2}}$ values obtained from the
bootstrap procedure, as a function of the center-of-mass energy $W$ at
$\ell_{\text{max}}=1$ (blue circles) and $\ell_{\text{max}}=2$ (red squares).

First, we evaluated the expected goodness-of-fit distribution, from which the
$p$-value associated with the minimum $\hat{\chi}_{\text{boot}}^{2}$ values
obtained by the bootstrap procedure has to be computed. Figure 9 shows the
obtained cumulative density function (CDF) divided by the number of degrees of
freedom ($\nu$), obtained at $W=1240$ MeV ($E_{\gamma}=350$ MeV) and with
$\ell_{\text{max}}=1$, compared with the CDF of the reduced $\chi^{2}$
distribution. These two distributions were found to basically coincide, and
the same result was obtained for all $W$ bins as well as for
$\ell_{\text{max}}=2$.

In the second step of this procedure, we evaluated the minimum value of the
$\ell_{\text{max}}$ parameter that could reproduce sufficiently well the
behavior of the experimental data. This was done by comparing, at each $W$
bin, the $\hat{\chi}^{2}_{\text{boot}}$ values obtained for both
$\ell_{\text{max}}=1$ and $\ell_{\text{max}}=2$ after a simultaneous fit of
all the involved coefficients, as shown in in Fig. 10.

Except for a few points in the center-of-mass energy region $W\sim 1200$ MeV,
the fit with $\ell_{max}=2$ gives, in general, no significant improvement in
the $\hat{\chi}^{2}_{\text{boot}}$ values. In several cases where the
difference is significant, the $\hat{\chi}^{2}_{\text{boot}}$ values obtained
with $\ell_{max}=2$ have suspiciously low values which may indicate
overfitting.

In addition, for $\ell_{max}=1$, we have that, after averaging over all $W$
bins, $\left<\hat{\chi}_{\text{boot}}^{2}/\nu\right>=1.14$, which corresponds
to a $p-$value of about $31\%$. All previous indications lead us to conclude
that the fit with $\ell_{\text{max}}=1$ reproduces our data sufficiently well.
Such a result was to be expected since, in the $\Delta(1232)$ region, the
production of $s$ and $p$-wave pions gives, by far, the largest contribution
to the $\pi^{0}$ channel, while $d$-waves only contribute thanks to very small
interference terms with the dominant $p$-waves. Their effect can be quantified
by the value of the $(a_{2})_{3}$ Legendre coefficient calculated with the
previous PW analyses and model. In the $\Delta(1232)$ region, the absolute
value of this coefficient was found to be $\lesssim 1\mu b/sr$, more than one
order of magnitude smaller than the absolute contributions given by the
$(a_{1})_{0}$ and $(a_{1})_{2}$ coefficients, dominated by the $p$-waves
multipoles, and also at least 2 times lower than the $(a_{1})_{1}$
coefficient, which quantifies the small $s$-$p$ interference contribution (see
Fig. 12). Contributions due to $d$-wave pions can then be safely neglected in
the present context.

An example of the probability distributions for the fit parameters
$(a_{1})^{b}_{0}$, $(a_{1})^{b}_{1}$, and $(a_{1})^{b}_{2}$, computed at the
end of each bootstrap replica, is given in Fig. 11, where these densities are
plotted at $W=1240$ MeV ($E_{\gamma}=350$ MeV).

Figure 11: Probability distributions for the fit parameters $(a_{1})^{b}_{0}$,
$(a_{1})^{b}_{1}$, and $(a_{1})^{b}_{2}$ as determined at the end of each
bootstrap replica at $W=1240$ MeV ($E_{\gamma}=350$ MeV). These distributions
are compared with the best-fit Gaussian curves (red lines), and the
corresponding $\hat{\chi}^{2}/\nu$ values are also given at the bottom of each
plot.

For the reasons discussed earlier, these distributions, especially for the
$(a_{1})^{b}_{0}$ and $(a_{1})^{b}_{2}$ parameters, show significant
deviations from the pure Gaussian shape, as indicated by the comparison with
the best-fit Gaussian red curves. Very similar density functions were found
for all other $W$ bins.

Figure 12: The fitted Legendre coefficients $(a_{1})_{0}$, $(a_{1})_{1}$, and
$(a_{1})_{2}$ are compared with the predictions of the BnGa-2019 V. Anisovich
et al. (2016) (dashed red lines) and SAID-SM22 Briscoe  _et al._ (2023)
(dotted green lines) PW analyses, and of the DMT-2001 dynamical model Kamalov
and Yang (1999); Kamalov  _et al._ (2001a, b); Tiator  _et al._ (2010) (solid
orange lines).

The values of the Legendre coefficients $(a_{1})_{0}$, $(a_{1})_{1}$, and
$(a_{1})_{2}$, determined as the mean of distributions obtained by the
bootstrap-based fit procedure at each $W$ bin, are plotted in Fig. 12. The
quoted fit errors are the 68% confidence level (CL) determined using the
quantiles of the bootstrapped parameter distributions, and include the
contribution of both the statistical and systematic uncertainties of the
experimental data.

The different curves represent the corresponding coefficients evaluated with
the BnGa-2019 and SAID-SM22 PW analyses with the DMT-2001 dynamical model. A
quite good agreement is found with all the previous predictions for the
$(a_{1})_{0}$ and $(a_{1})_{2}$ coefficients, whose value is mostly determined
by the dominant $p$-wave contributions (see later Eqs. 8-9). On the contrary,
these predictions give sizeable differences in the value of the $(a_{1})_{1}$
coefficient, which quantifies the effects of the $s-p$ interference terms. Our
new data will allow to resolve these discrepancies and improve our
understanding of the role of the low-lying multipoles for the $\pi N$ process
in the $\Delta(1232)$ energy region.

##  VI Derivation of the E2/M1 ratio

In the $\ell_{max}=1$ approximation (i.e., only $s$ and $p$-waves are
considered - see Table 2), the Legendre coefficients of the helicity-dependent
differential cross section can be decomposed in terms of multipoles as
Wunderlich  _et al._ (2017); Wunderlich (2019):

$\displaystyle(a_{1})_{0}$
$\displaystyle=|E_{0+}|^{2}+|M_{1-}|^{2}+3E^{*}_{1+}(3E_{1+}+M_{1+})+M^{*}_{1+}(3E_{1+}-M_{1+})\
;$ (8) $\displaystyle(a_{1})_{1}$
$\displaystyle{}=E^{*}_{0+}(3E_{1+}-M_{1-}+M_{1+})+E_{0+}(3E^{*}_{1+}-M^{*}_{1-}+M^{*}_{1+})\
;$ $\displaystyle(a_{1})_{2}$
$\displaystyle=-M^{*}_{1-}(3E_{1+}+M_{1+})+E^{*}_{1+}(6E_{1+}-3M_{1-})-M^{*}_{1+}(M_{1-}-2M_{1+})\
.$

To evaluate $R_{EM}$ (see Eq. (1)) from these equations, the following two
approximations were sequentially applied:

  * (i)

all terms involving only the $E_{0+}$, $E_{1+}$, and $M_{1-}$ multipoles were
dropped as, in the region of the $\Delta(1232)$ resonance mass, they can be
assumed to be negligible compared to the terms where $M_{1+}$ contributes;

  * (ii)

a perfect Breit-Wigner form was assumed for the $M_{1+}$ multipole, which
gives, by far, the largest contribution to the $\Delta(1232)$ resonance
excitation. Under this assumption, the real part of the $M_{1+}$ multipole
vanishes exactly at the $M_{\Delta}$ value. We can therefore set:

$M_{1+}\simeq i\hbox{Im}[M_{1+}].$

The first step in deriving $R_{EM}$ was to apply approximation (i) to the
coefficients given in Eq. (8):

$\displaystyle(a_{1})_{0}$
$\displaystyle\simeq-|M_{1+}|^{2}+3E^{*}_{1+}M_{1+}+3M^{*}_{1+}E_{1+}=-|M_{1+}|^{2}+6\hbox{Re}[E^{*}_{1+}M_{1+}]\
;$ (9) $\displaystyle(a_{1})_{1}$ $\displaystyle\simeq
2\hbox{Re}[E^{*}_{0+}M_{1+}]\ ;$ $\displaystyle(a_{1})_{2}$
$\displaystyle\simeq-2\hbox{Re}[M^{*}_{1-}M_{1+}]+2|M_{1+}|^{2}\ .$

It can be immediately noticed that the coefficient $(a_{1})_{1}$ is irrelevant
for further considerations and will be ignored from now on. Furthermore,
applying approximation (ii) resulted in:

$\displaystyle(a_{1})_{0}$
$\displaystyle\simeq-\hbox{Im}[M_{1+}]^{2}+6\hbox{Im}[E_{1+}]\hbox{Im}[M_{1+}]\
;$ (10) $\displaystyle(a_{1})_{2}$ $\displaystyle\simeq
2\hbox{Im}[M_{1+}]^{2}-2\hbox{Im}[M_{1-}]\hbox{Im}[M_{1+}]\simeq
2\hbox{Im}[M_{1+}]^{2}\ ,$

where, in the last step of the previous equation, it was considered that
$\hbox{Im}[M_{1-}]\ll\hbox{Im}[M_{1+}]$ around the $M_{\Delta}$ value.

Using Eq. (10), and recalling Eq. (1), the ratio $(a_{1})_{0}/(a_{1})_{2}$ can
now be calculated as:

$\frac{(a_{1})_{0}}{(a_{1})_{2}}\simeq-\frac{-\hbox{Im}[M_{1+}]^{2}}{2\hbox{Im}[M_{1+}]^{2}}+\frac{6\hbox{Im}[E_{1+}]\hbox{Im}[M_{1+}]}{2\hbox{Im}[M_{1+}]^{2}}=3R_{EM}-\frac{1}{2}\
.$ (11)

The following quantity can then be defined as a suitable approximation for
$R_{EM}$:

$R_{L}=\frac{1}{3}\frac{(a_{1})_{0}}{(a_{1})_{2}}+\frac{1}{6}\simeq R_{EM}\ .$
(12)

This allows for the extraction of the parameter $R_{L}$ from the ratio of only
two Legendre coefficients fitted from the angular distribution of only one
observable, $d\Delta\sigma/d\Omega$. This procedure has two main advantages:
i) it reduces by one the number of experimental observables to be measured
compared to some of the previous extractions (e.g., Ref. Beck  _et al._
(2002)); ii) the experimental uncertainty of $R_{L}$ is independent of all
common scaling systematic uncertainties affecting the experimental data, as
previously noted in Ref. Beck  _et al._ (2000).

Figure 13:  Left panel: probability distribution for the $R_{b}$ parameter
(see Eq. (12)), as determined by the bootstrap procedure at $W=1240$ MeV
($E_{\gamma}=350$ MeV). This distribution is compared to the best-fit Gaussian
curves (red lines), and the corresponding $\hat{\chi}^{2}/\nu$ value is also
given in the canvas. Right panel: The $W$ (total c.m. energy) dependence of
$R_{L}$ (see eq. 12) for the upper part of the measured photon energy interval
is compared to the $R_{W}$ ratio (see eq. 13) predicted by BOGA-2019 (dashed
red line), SAID-SM22 (dotted green line) and DMT-2001 (solid orange line). The
vertical dotted black lines define the $W$ region $1232\pm 10$ MeV selected to
evaluate $R_{EM}$, while the horizontal black segment is drawn at $R=-0.0238$.

The probability distribution for the parameter $R_{L}^{b}$ obtained by the
bootstrap procedure at $W=1240$ MeV ($E_{\gamma}=350$ MeV) is shown in the
left panel of Fig. 13. As can be seen from the result of the fit, this
distribution can be considered approximately Gaussian, even though it is given
by the ratio of two non-Gaussian variables.

In the right panel of Fig. 13, $R_{L}$ is plotted as a function of the center-
of-mass energy $W$ for the upper part of the measured photon energy interval.
The quoted fit errors are the 68% CL, and include the contribution of both the
statistical and angular-dependent systematic uncertainties of the experimental
data.

In the same figure, the $W$ dependence of the ratio:

$R_{W}=\frac{\hbox{Im}[E_{1+}^{3/2}]}{\hbox{Im}[M_{1+}^{3/2}]}\ ,$ (13)

as predicted by BOGA-2019, SAID-SM22 and DMT-2001, is also shown. For all the
different predictions, $R_{W}$ is strongly dependent on the $W$ value, while
the experimental $R_{L}$ values show quite small variations around the
$M_{\Delta}$ value.

In order to give an estimate of $R_{EM}$ from the distribution of $R_{L}$, a
small center-of-mass energy interval $W=1232\pm 10$ MeV was then chosen, and
the weighted average of all $R$ values included in this interval was
evaluated. This interval is centered around the $M_{\Delta}$ value, where
$R_{L}$ can be considered constant and the approximated Eq. (11) holds. The
model systematic uncertainty associated with the approximations that led to
Eq. (12) was evaluated, using the multipoles from both the previous PW
analyses and the DMT-2001 model, by comparing the exact $R_{EM}$ value
obtained with Eq. (1) with the average of the $R$ values calculated with the
approximated formula of Eq. (12) in the interval $W=1232\pm 10$ MeV. The
maximum relative deviation between these two estimates was found to be 4%.

A 30% variation in the width of the selected interval resulted in a $\sim 4\%$
change in both the evaluated $R_{EM}$ value and its model systematic
uncertainty. We then conservatively assume that the overall relative
systematic uncertainty associated with both the model approximation and the
interval choice is $\pm 4\%$ in rms units.

Finally, our new estimate of the parameter $R_{EM}$ is then:

$R_{EM}=[-2.38\pm 0.16\hbox{(stat.+sys.)}\pm 0.10\hbox{(model)}]\%\ .$ (14)

Figure 14: The $R_{EM}$ value obtained in this work (blue circle) is compared
to the most recent evaluations from Ref. Markou  _et al._ (2018) (AMIAS -
olive yellow star), Ref. Ahrens et al. (2004) (GDH - red square), Ref. Beck
_et al._ (2002) (TAPS - orange circle), Ref. Arndt  _et al._ (2002) (SAID-SM02
- gray down triangle), Ref. Blanpied  _et al._ (2001) (LEGS - green up
triangle), Ref. Galler et al. (2001) (LARA - pink cross), Ref. Beck  _et al._
(2000) (DAPHNE - cyan diamond). Except for the result from Ref. Arndt  _et
al._ (2002), all quoted errors are the sum of both statistical, systematic,
and model-dependent uncertainties. The vertical yellow band indicates the 100%
CL interval estimated by the PDG Workman (2022). The black square below the
dotted line is the newly calculated weighted average $R_{EM}^{AV}$, using some
of the previous results (see text for details).

In Fig. 14, the newly evaluated $R_{EM}$ value is compared to other recently
published extractions given by the DAPHNE Beck  _et al._ (2000), LARA Galler
et al. (2001), LEGS Blanpied  _et al._ (2001), TAPS Beck  _et al._ (2002), and
GDH Ahrens et al. (2004) collaborations, by the SAID-SM02 PW analysis Arndt
_et al._ (2002), and by the AMIAS Markou  _et al._ (2018) calculation. In this
last approach, the $\gamma N\to N\pi$ reactions were described using the the
multipole expansions of the Chew Goldberger Low Nambu (CGLN) amplitudes (Chew
_et al._ (1957)). A random set of multipoles coefficients was then generated
by taking into account the allowed physical limits and the required
constraints. The predictions for the different observables given by each set
of coefficients were finally compared to the selected $\gamma N\to N\pi$
experimental data set (see Ref.Markou  _et al._ (2018)) and the final result
and its uncertainty were evaluated from the obtained $\chi^{2}$ probability
values.

The present work provides an estimate with uncertainties that are not only
more favorable, but also more rigorously calculated compared to previous work.

Within the quoted errors, there is a good agreement between all the different
evaluations. For this reason, it is possible to combine them to obtain the
most accurate extraction of $R_{EM}$ available, which should be used for any
further reference and for model comparison.

For this estimate, we did not include data from the LEGS Blanpied  _et al._
(2001) collaboration because, as the authors themselves stated in their
original publication, a part of the data set from the $\gamma p\to N\pi$
reactions used for their evaluation is not consistent with the data sets
obtained at MAMI (see also Refs. Beck  _et al._ (1997, 2000, 2002); Ahrens et
al. (2004)). Similarly, we have also not included the $R_{EM}$ value from the
LARA collaboration Galler et al. (2001) since their proton Compton scattering
data give inconsistent results when compared to all other available data for
this reaction (see Ref. Mornacchi  _et al._ (2022)).

Our new evaluation was then calculated by combining the present value with
those from Refs. Ahrens et al. (2004); Beck  _et al._ (2000, 2002), all based
on independent data sets. The weighted average of the selected values is:

$R_{EM}^{AV}=[-2.47\pm 0.12\hbox{(stat.+sys.+mod.)}]\%\ ,$ (15)

where the quoted error is the sum of the statistical, systematic, and model-
dependent errors.

##  VII Summary and conclusions

New precise data on the helicity-dependent differential cross section of
single $\pi^{0}$ photoproduction on the proton have been obtained. Compared to
the existing data, this new measurement covers a much larger energy and polar
angular interval with a significantly improved precision. These new data
improve our understanding of the $p\pi^{0}$ photoproduction process in the
$\Delta(1232)$ energy region, and, in particular, of the role of the low-lying
multipoles, whose predicted contribution differs significantly between the
available PW analyses and models.

From a Legendre moment analysis of the obtained angular distributions, the
ratio $R_{EM}=(-2.38\pm 0.16\pm 0.10)\%$ at the $\Delta(1232)$ resonance mass
value was obtained. This is the most accurate estimate from one single
experiment ever published until now.

By combining some of the available estimates, the most accurate experimental
$R_{EM}$ value has been determined to be $R_{EM}^{AV}=(-2.47\pm 0.12)\%$,
where the quoted error is the sum of statistical, systematic, and model-
dependent errors. This value should be used for all further reference and
model comparisons.

##  VIII Acknowledgments

The authors wish to acknowledge the excellent support of the accelerator group
of MAMI. This work was supported by SchweizerischerNationalfonds
(200020-156983,132799, 121781,117601), DeutscheForschungsgemeinschaft
(SFB443,SFB1044,SFB/TR16,FOR5327), the INFN-Italy, the European Community-
Research Infrastructure Activity under FP7 programme (HadronPhysics,
grantagreement No.227431), the U.K.Science and Technology Facilities Council
grants ST/J000175/1, ST/G008604/1, ST/G008582/1,ST/J00006X/1,
ST/V001035/1ST/V002570/1, ST/P004385/2, ST/T002077/1, and ST/L00478X/2, the
Natural Sciences and Engineering Research Council (NSERC,
FRN:SAPPJ-2015-00023), Canada. This material is based upon work also supported
by the U.S.Department of Energy, Office of Science, Office of Nuclear Physics
Research Division, under Award Numbers DE-FG02-99ER41110, DE-FG02-88ER40415,
DE-FG02-01-ER41194, DE-SC0014323andDE-SC0016583 and by the National Science
Foundation, under Grant Nos.PHY-1039130 and IIA-1358175.

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