∎

e1e-mail<EMAIL_ADDRESS>
11institutetext:  SUPA, School of Physics and Astronomy, University of
Glasgow, Glasgow G12 8QQ, UK 22institutetext: SUPA, School of Physics and
Astronomy, University of Edinburgh, Edinburgh EH9 3FD, UK 33institutetext:
Department of Physics, University of Regina, Regina, SK S4S 0A2, Canada
44institutetext: Department of Physics, Institute for Nuclear Studies, The
George Washington University, Washington, DC 20052, USA 55institutetext:
Institut für Physik, University of Basel, Basel CH-4056, Switzerland
66institutetext: Institut für Kernphysik, Johanes Gutenberg-University Mainz,
D-55099 Mainz, Germany 77institutetext: Helmholtz–Institut für Strahlen- und
Kernphysik, University of Bonn, Bonn D-53115, Germany 88institutetext: INFN
Sezione di Pavia, I-27100 Pavia, Italy 99institutetext: Kent State University,
Kent, OH 44242, USA JINR, 141980 Dubna, Russia 1010institutetext: University
of Massachusetts, Amherst, MA 01003, USA 1111institutetext: Institute for
Nuclear Research, 125047 Moscow, Russia 1212institutetext: Racah Institute of
Physics, Hebrew University of Jerusalem, Israel 1313institutetext: Mount
Allison University, Sackville, NB E4L3B5, Canada 1414institutetext: Department
of Physics, University of York, Heslington, York Y010 5DD, UK
1515institutetext: Rudjer Boskovic Institute, Zagreb HR-10000, Croatia
1616institutetext: NRC ”Kurchatov Institute” - ITEP, Moscow 117218, Russia
1717institutetext: Institut für Experimentalphysik, Ruhr-University of Bochum,
Bochum D-44801, Germany

# Single $\pi^{0}$ Production Off Neutrons Bound in Deuteron with Linearly
Polarized Photons

C. Mullenaddr2  S. Gardneraddr2  D. I. Glaziere1,addr2  S. J. D.
Kayaddr4,addr3  K. Livingstonaddr2  I. I. Strakovskyaddr5  R. L. Workmanaddr5
S. Abtaddr6  P. Achenbachaddr7  F. Afzaladdr8  Z. Ahmedaddr3  C. S.
Akondiaddr10  J. R. M. Annandaddr2  M. Bashkanovaddr17  R. Beckaddr11  M.
Birothaddr7  N. S. Borisovaddr11  A. Braghieriaddr9  W. J. Briscoeaddr5  F.
Cividiniaddr7  C. Collicottaddr7  S. Costanzaaddr9  A. Denigaddr7  M.
Dieterleaddr6  E. J. Downieaddr5  P. Drexleraddr7  S. Feganaddr17  M. I.
Ferretti-Bondyaddr7  D. Ghosal addr6  I. Gorodnovaddr11  W. Gradladdr7  M.
Güntheraddr6  G. Gurevicaddr13  L. Heijkenskjöldaddr7  D. Hornidgeaddr16  G.
M. Huberaddr3  N. Jermannaddr6  A. Kaeseraddr6  M. Korolijaaddr18  V. L
Kashevarovaddr7  B. Kruscheaddr6  V. V. Kulikovaddr19  A. Lazarevaddr8  S.
Luttereraddr6  I. J. D. MacGregoraddr2  D. M. Manleyaddr10  P. P. Marteladdr7
M. A. Martemianovaddr19  C. Meieraddr6  R. Miskimenaddr12  M. Mocanuaddr17  E.
Mornacchiaddr7  A. Neganovaddr11  M. Oberleaddr6  M. Ostrickaddr7  P.
Otteaddr7  D. Paudyaladdr3  P. Pedroniaddr9  A. Powelladdr2  S. N.
Prakhovaddr7  G. Reicherzaddr20  G. Ronaddr14  T. Rostomyan addr6  C.
Sfientiaddr7  V. Sokhoyanaddr7  K. Spiekeraddr8  O. Steffenaddr7  Th.
Strubaddr6  I. Supekaddr18  A. Thieladdr8  M. Thieladdr7  A. Thomasaddr7  M.
Unverzagtaddr7  Yu. A. Usovaddr11  S. Wagneraddr7  N. K. Walfordaddr6  D. P.
Wattsaddr17  D. Werthmülleraddr2  J. Wettigaddr7  L. Witthaueraddr6  M.
Wolfesaddr7  N. Zachariouaddr17
(A2 Collaboration at MAMI)

(Received: date / Accepted: date)

###### Abstract

The quasifree $\overrightarrow{\gamma}d\to\pi^{0}n(p)$ photon beam asymmetry,
$\Sigma$, has been measured at photon energies, $E_{\gamma}$, from 390 to 610
MeV, corresponding to center of mass energy from 1.271 to 1.424 GeV, for the
first time. The data were collected in the A2 hall of the MAMI electron beam
facility with the Crystal Ball and TAPS calorimeters covering pion center-of-
mass angles from 49 to 148$\degree$. In this kinematic region, polarization
observables are sensitive to contributions from the $\Delta(1232)$ and
$N(1440)$ resonances. The extracted values of $\Sigma$ have been compared to
predictions based on partial-wave analyses (PWAs) of the existing pion
photoproduction database. Our comparison includes the SAID, MAID, and Bonn-
Gatchina analyses; while a revised SAID fit, including the new $\Sigma$
measurements, has also been performed. In addition, isospin symmetry is
examined as a way to predict $\pi^{0}n$ photoproduction observables, based on
fits to published data in the channels $\pi^{0}p$, $\pi^{+}n$, and $\pi^{-}p$.

###### pacs:

13.60.Le Meson production 13.88.+e Polarization in interactions and scattering

††journal: Eur. Phys. J. A

##  1 Introduction

Knowledge of the $N^{\ast}$ and $\Delta^{\ast}$ resonance decay couplings to
nucleons and photons is largely restricted to charged states. Increasing the
body of neutron-target measurements will allow a more highly constrained study
of neutral states and their $n\gamma$ couplings. The four charge channels
($\pi^{0}p$, $\pi^{+}n$, $\pi^{-}p$, and $\pi^{0}n$) of pion photoproduction
can be described in terms of three isospin amplitudes. This gives the
possibility of predicting properties of one channel based on sufficiently
detailed measurements of the other three. The $\pi^{0}n$ channel is the least-
studied and was the subject of this experiment.

Most existing $\gamma n$ data are unpolarized and provide around 1900
$\pi^{0}n$ photoproduction data points, spanning the full nucleon resonance
region Ireland:2019uwn  (Table 1). The $\overrightarrow{\gamma}n\to\pi^{0}n$
beam asymmetry, $\Sigma$, was previously measured by the GRAAL Collaboration
DiSalvo:2009zz . The beam asymmetry measures the relative strength of the
production with respect to the plane of photon linear polarisation. Their
measurements covered beam energy, $E_{\gamma}$, from 703 to 1475 MeV,
corresponding to a centre-of-mass energy, $W$, range from 1.484 to 1.912 GeV,
just above the current results.

Recently, the A2 Collaboration at MAMI published high-quality unpolarized
measurements for $\pi^{0}$ photoproduction off a neutron below $E_{\gamma}$ =
813 MeV Briscoe:2019cyo . The present data extend the range of the previous
GRAAL polarized measurements DiSalvo:2009zz , for $\pi^{0}n$ photoproduction.
Further A2 Collaboration measurements of the $\pi^{0}n$ $E$ asymmetry, with
longitudinal polarized target and circularly polarized photons, for
$E_{\gamma}$ = 216 – 1606 MeV Costanza:2020wnx , extend previous A2 $\pi^{0}n$
$E$ measurements Dieterle:2017myg . These data will provide the basis for
better-constrained $\gamma n$ decay amplitudes in the near future.

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. In several papers,
the GWU-ITEP group have shown that the FSI corrections on unpolarized cross
sections are less than 20% (see, for instance, Refs. Tarasov:2011ec ;
Mattione:2017fxc ; Tarasov:2015sta ). As polarization asymmetries measure
ratios of cross sections, FSI effects are expected to have a considerably
smaller effect on these, including $\Sigma$, and will be comparable, or less
than our quoted systematic uncertainties from experimental sources. In this
publication, $\Sigma$ for the neutron bound in a deuteron is presented
uncorrected for potential FSI effects so as not to add any model dependence to
the results.

Table 1: Published data for $\gamma n\to\pi^{0}n$ reaction as given in the SAID database Briscoe:2020zzz : 1st column is the observable, 2nd column is the number of energy bins, 3rd column is the number of data points. Observable | Nexp | Ndata | Eγ(min) | Eγ(max) |  $\theta$(min) |  $\theta$(max) | Laboratory | Ref
---|---|---|---|---|---|---|---|---
|  |  | (MeV) | (MeV) | ($\degree$) | ($\degree$) |  |
$d\sigma/d\Omega$ | 9 | 9 | 208 | 373 | 147 | 148 | MAMI | Kossert:2003zf
| 27 | 492 | 290 | 813 | 18 | 162 | MAMI | Briscoe:2019cyo
| 40 | 43 | 299 | 889 | 70 | 130 | Tokyo | Ando:1977aa
| 49 | 931 | 446 | 1427 | 32 | 162 | MAMI | Dieterle:2018adj
| 42 | 42 | 455 | 905 | 45 | 143 | Tokyo | Hemmi:1973ii
| 35 | 35 | 462 | 784 | 60 | 135 | Frascati | Bacci:1972bh
| 3 | 28 | 911 | 1390 | 3 | 91 | SLAC | Clinesmith:1967zn
$\Sigma$ | 27 | 216 | 703 | 1475 | 53 | 164 | GRAAL | DiSalvo:2009zz
$E$ | 17 | 151 | 446 | 1427 | 46 | 154 | MAMI | Dieterle:2017myg

The organization for this paper is as follows. In Section 2, details of the A2
experiment and detectors are given; Section 3 outlines the event selection;
Section 4 reviews the background subtraction; Section 5 covers the
determination of the photon asymmetry; and Section 6 outlines the dominant
sources of systematic uncertainty; Section 7 outlines the PWA methods used in
the fits and predictions compared to data. Finally, Section 8 presents the
results and interpretation of the present A2 $\Sigma$ data.

##  2 Experiment

The reaction $\overrightarrow{\gamma}$d $\rightarrow\pi^{0}$n(p) was measured
at the Mainzer Microtron (MAMI) electron accelerator facility, in August 2016.
The 1.5 GeV MAMI electron beam, incident on an aligned diamond radiator,
produced a photon beam via coherent bremsstrahlung, with significant linear
polarization up to photon energies of $610$ MeV. The energy of the photon beam
was measured using the Glasgow-Mainz Tagged Photon Spectrometer with a
resolution of around $4$ MeV. This spectrometer measured the position of the
degraded post-bremsstrahlung electron on a plastic scintillator focal plane
consisting of 353 elements after traversing a 1.8 T magnetic dipole field
McGeorge:2007tg . The energy of the detected electron, and therefore also the
energy of the photon, was deduced from this position.

The photon beam interacted in a 10 cm long liquid deuterium target ($LD_{2}$).
The reaction products were detected in two calorimeters: the Crystal Ball
(CB), a highly segmented array of 672 NaI(Tl) crystals arranged in a sphere
centered on the target cell Starostin:2001zz ; and the TAPS calorimeter, a
forward wall of 366 BaF2 and 72 PbWO4 crystals arranged 1.5 m downstream from
the CB center Gabler:1994ay  (Fig. 1). The CB covers lab. frame angles
21$\degree<\theta_{L}<159\degree$ and TAPS approximately
$2\degree<\theta_{L}<20\degree$. The $LD_{2}$ target cell was surrounded by a
Particle Identification Detector (PID) consisting of 24, 30 cm long plastic
scintillators arranged in a cylindrical formation. This allowed separation of
reactions with a scattered neutron from those with a proton. A Multi-Wire
Proportional Chamber barrel (MWPC) provided tracking information for charged
particles which were not used for the all-neutral final state investigated
here, however here they provided additional proton rejection. Charge particle
identification was provided in the case of the TAPS detector by a thin plastic
veto layer in front of each crystal. In addition, a 2.6 cm thick graphite
cylinder was situated between the PID and the MWPC to be used as the analysing
material for a nucleon polarimeter Bashkanov:2019mpj , and was not required
for this analysis Mullen:2020fgd .

All simulations used in this analysis were performed with a full detector
model using the Geant4 GEANT4  toolkit.

Figure 1: Set-up of the A2 experiment. CB shows a NaI(Tl) calorimeter, TAPS
shows a BaF2 and PbWO4 calorimeter, PID shows a plastic scintillator detector
for particle identification, MWPC are two cylindrical multiwire proportional
chambers, and target shos the liquid deuterium target $LD_{2}$ (see text for
details).

The linear polarization of the photons was produced by coherent bremsstrahlung
Lohmann:1994vz ; Timm:1969mf , with the electron beam scattered coherently
from an aligned radiator. A thin diamond crystal (30 $\mu$m) with low mosaic-
structure was used to minimize the energy smearing of the coherent spectrum
arising from electron multiple scattering effects and crystal defects in the
lattice Kellie:2005wy . The alignment of the diamond was carried out using the
Stonehenge technique Livingston:2008hv  with two orthogonal polarization plane
orientations chosen to be at azimuthal angles of $\pm 45\degree$ with respect
to the equatorial plane of the CB detector. To increase the degree of linear
polarization a 2 mm diameter Pb-collimator was installed 2.5 m downstream of
the radiator, enhancing the ratio of coherent to incoherently scattered
photons that reached the $LD_{2}$. The degree of linear polarization was
determined by calibrating against the linearly polarized photon beam asymmetry
for $\pi^{0}$ production off the proton. This was measured for each photon
energy bin and compared to a recent SAID PWA solution including recent high
statistics measurements in the same energy range Gardner:2016irh . The ratio
of the measured asymmetry to the SAID values gave the photon polarization for
each energy bin. The analysis of the proton asymmetry was performed in the
same manner as the neutron asymmetry described here. The resulting photon
polarization ranged from $15\%$ at $E_{\gamma}$ = 390 MeV to a maximum of
$55\%$ at 610 MeV Mullen:2020fgd .

##  3 Data Analysis

The photon asymmetry, $\Sigma$, for the reaction $\overrightarrow{\gamma}$d
$\rightarrow\pi^{0}$ n(p) has been measured for beam energies in the range 390
– 610 MeV and a center-of-mass (c.m.) production angle, $\theta$, range of
49$\degree$ – 148$\degree$. The semi-inclusive final state of interest
included the recoiling neutron and $\pi^{0}$, and omitted the spectator
proton. It was identified by detecting three neutral particles, two $\gamma$s
stemming from the decay of the $\pi^{0}$, and a neutron as the third. The
energy of the three particles was measured by the CB and TAPS calorimeters
which, in coincidence with a tagged photon, allowed the reaction to be
reconstructed. The classification of a neutral state was made if there were no
hits in the PID or MWPC detectors for the CB or the TAPS veto layer. The
spectator proton was not considered as it typically did not have sufficient
energy to reach the calorimeters. The $\pi^{0}$ was reconstructed from the
combination of two of the three particles detected. All combinations were
considered, and any incorrect 2$\gamma$ combinations were removed, either by
subsequent cuts or by background subtraction.

Table 2: A summary of the loose cuts applied to the data before the sPlots fits. Variable | Cut Range | Units
---|---|---
Tagged Time |  $-80$ < t${}_{\pi^{0}}$ < 20 | ns
Coplanarity | -50 < $\Delta\phi$ < 50 | degree
Missing Mass | 1850 < Mmiss < 2300 | MeV/c2
Cone Angle | 0 < $\theta_{\mathrm{Cone}}$ < 0.5 | radian
Invariant Mass | 80 < Minv < 200 | MeV/c2
Spectator Momentum | 0 < $\left|\mathrm{P}_{\mathrm{spec}}\right|$ < 200 | MeV/$c$

Preliminary cuts were placed on a number of variables. These were guided by
simulated signal and background channels to ensure no actual signal was lost
and are given in Table 2. The missing mass was constructed using the mass of
the missing 4-momentum defined as:

$P_{\mathrm{miss}}=P_{\mathrm{beam}}+P_{\mathrm{d}}-P_{\pi^{0}},$ (1)

where Pbeam is the 4-momentum for the beam photon, Pd is the 4-momentum of the
stationary deuterium target, and P${}_{\pi^{0}}$ is the 4-momentum of the
detected $\pi^{0}$. The mass of this missing 4-momentum gave a peak
distributed around the summed mass of the two nucleons with some extra
smearing from initial Fermi motion within the deuteron. When this initial
momentum was low, as was generally the case, the resulting spectator proton
momentum was also low and the reaction was approximately two-body with the
$\pi^{0}$ and participant nucleon being nearly coplanar in $\phi$:

$\Delta\phi=\phi_{\pi^{0}}-\phi_{n-180\degree}~{}\sim 0\degree,$ (2)

where $\phi_{\pi^{0}}$ is the reconstructed azimuthal angle of the $\pi^{0}$
and $\phi_{n-180\degree}$ the azimuthal angle of the detected neutron after
rotation by $180\degree$ around the z-axis.

The detection of the $\pi^{0}$ and the neutron allowed the construction of the
difference between the detected nucleon polar angle and the nucleon polar
angle reconstructed from the $\pi^{0}$ assuming a stationary initial state
neutron. This gives the definition of the “Cone Angle”,
$\theta_{\mathrm{Cone}}$.

The momentum of the spectator proton was also used to distinguish the quasi-
free final state. It was calculated via:

$P_{\mathrm{spec}}=P_{\mathrm{beam}}+P_{\mathrm{d}}-P_{\pi^{0}}-P_{\mathrm{n}},$
(3)

where $P_{\mathrm{n}}$ is the 4-momentum of the detected participant nucleon.
The magnitude of momentum for the participant nucleon was calculated using
conservation of momentum and energy in the three-body final state, using the
measured $\pi^{0}$ momentum and the neutron direction, as given by its cluster
hit position in the CB.

##  4 Background Subtractions

To further isolate the true $\pi^{0}$n final state, several sources of
background had to be subtracted from the selected events. The main sources of
these backgrounds were: random electrons in the photon tagger; background to
the two photon combination giving the $\pi^{0}$; and other reactions producing
the same detected particles as the $\pi^{0}$n(p) reaction. The sPlot technique
was used to remove these background events using a separate discriminatory
variable for each source to produce event-by-event weights termed sWeights,
for full details see Pivk:2004ty . The sWeights are normalized using the
relative yields and covariance matrix of the signal and background derived
from the fits. Weights corresponding to regions of high background are
negative and effectively subtract off this contribution to the distribution.
This is similar to how a “sideband subtraction” method works, but is more
generally applicable. Consecutive fits were performed applying the sWeights
from the previous fit. An sPlot fit to the discriminatory variables using
appropriate probability density functions (PDFs) derived from simulated event
samples, determined the yields of the different event species. The covariance
matrix of this yield fit was then used to calculate the sWeights associated
with each event in the fitted sample. Including these sWeights in the
subsequent observable fits allowed determination of the photon asymmetry for
our signal. The fits to the discriminatory variables are described in the
following sections.

###  4.1 Random Tagged Photons

Random coincidences with background electrons in the photon tagger were
removed via the coincidence time between the $\pi^{0}$ and the tagged beam
photon as given by:

$t_{\mathrm{coin}}=\frac{t_{\gamma 1}+t_{\gamma 2}}{2}-t_{\mathrm{tagger}},$
(4)

with the time of the electron in the tagger, $t_{\mathrm{tagger}}$ and the
time in the calorimeters of the 2 photons $t_{\gamma 1,2}$. This resulted in a
timing distribution strongly peaked at zero with a flat random background, as
shown in Fig. 2 for the bin $E_{\gamma}=610$ MeV and $\cos\theta_{CM}=0.05$.
In this case, a Gaussian PDF was used for the signal with a uniform background
function.

###  4.2 Background in $2\gamma$ Invariant Mass

Background to real $\pi^{0}$ decays in the two $\gamma$ invariant mass
distribution can arise from a wrong combination of the three neutral clusters,
or multiple clusters created by one actual particle. These background sources
will not give a peaking structure in the invariant mass distribution and were
thus subtracted using the sPlot technique. The $\pi^{0}$ signal PDF was taken
from a histogram template of simulated events, while the background was
modelled by a third order Chebychev polynomial. An example fit is shown for
the bin $E_{\gamma}=610$ MeV and $\cos\theta=0.05$ in Fig. 3. In this mass
range, the $\pi^{0}$ signal was typically around $90\%$ of the total events.

###  4.3 Background to the $\pi^{0}$n(p) final state

Background to the final state may come, for example, from events in which more
than one pion is produced. This background is reduced with the loose cuts
given in Table 2. To determine the sWeights for subtracting the residual
background, the coplanarity given in Eq. 2 was used. The signal PDF shape was
given by simulated data and the background by a second order Chebychev
polynomial. The resulting fit, for the bin $E_{\gamma}=610$ MeV and
$\cos\theta=0.05$, is shown in Fig. 4.

Figure 2: The timing coincidence spectra between the photon beam tagger and
the calorimeters. Black points are data; red solid line is full fit result;
dashed black is signal Gaussian function; dashed red is flat background
function. This fit was used to produce weights to subtract the random
background events.

Figure 3: The invariant mass of the two detected $\gamma$s. Black points are
data; red solid line is full fit result; dashed black is simulated signal
function; dashed red is third degree Chebychev polynomial. This fit was used
to produce weights to subtract background events that did not have a
$\pi^{0}$.

Figure 4: The coplanarity between the $\pi^{0}$ and detected neutron. Black
points are data; red solid line is full fit result; dashed black is simulated
signal function; dashed red is second degree Chebychev polynomial. This fit
was used to produce weights to subtract events that did not originate from the
$\overrightarrow{\gamma}$d $\rightarrow\pi^{0}$ n(p) final state.

Figure 5:  Example of fitted asymmetry used to extract $\Sigma$. Blue points
are the data and blue solid line is the result of the maximum likelihood fit.

##  5 Determination of the Photon Asymmetry

The photon asymmetry, $\Sigma$, quantifies the effect of the polarization of
the beam on the excitation of the neutron and its subsequent decay to a pion
and nucleon. With a linearly polarized photon beam the differential cross
section is

$\frac{d\sigma}{d\Omega}=\left(\frac{d\sigma}{d\Omega}\right)_{0},(1+P_{L}\Sigma\cos
2\phi),$ (5)

where $\phi$ is the azimuthal angle of the meson production plane relative to
the plane of linear polarization and $P_{L}$ is the degree of linear
polarization. Rotating the orientation of the diamond radiator allowed the
plane of linear polarization to flip between $-\frac{\pi}{4}$ and
$\frac{\pi}{4}$. Shifting the plane by $\frac{\pi}{2}$ effectively flips the
sign of the asymmetry giving two polarization states $P_{S}=\pm 1$.

To extract $\Sigma$ from the measured $\phi$ distributions unbinned extended
maximum likelihood fits were performed. The fit function was given by

$F(\Sigma:\phi,P_{S},P_{L})=1+\Sigma P_{S}P_{L}\cos(2\phi+\phi_{0}),$ (6)

with $\phi_{0}$ determined from fits to be $95.8\degree$.

The negative log likelihood function given by

$-\ln{L}=-\sum^{N}_{i}w_{i}\ln{F(\Sigma:\phi_{i},P_{S,i},P_{L,i})}+B(\Sigma),$
(7)

was minimized using Minuit as part of the ROOFIT Verkerke:2003ir  library.
Here, $N$ is the number of data events in the $E_{\gamma}$ and $\theta$ bin,
while subscript $i$ refers to the value of the variables for a given event. In
particular, $w_{i}$ represents the value of the sWeight used to subtract
background events from the likelihood summation.

The PDF normalization integral term $B(\Sigma)$ was determined by Monte-Carlo
integration using simulated data. For this, $P_{S}$ and $P_{L}$ values were
randomly chosen to match the fluxes and polarization degrees of the real data.
This effectively corrected for second order systematic effects due to
differences in polarization state luminosity and degree of polarization.

An example of the resulting fitted asymmetry in polarization state compared to
the background subtracted data asymmetry, where the plotted asymmetry is
calculated as,

$A(\phi)=\frac{F(\phi:P_{S}=+1)-F(\phi:P_{S}=-1)}{F(\phi,P_{S}=+1)+F(\phi,P_{S}=-1)}$
(8)

is shown in Fig. 5, for the bin $E_{\gamma}=610$ MeV and $\cos\theta=0.05$, as
an illustration. The fits were performed in bins of 20 MeV for $E_{\gamma}$
and 0.1 in $\cos\theta$. Results are shown in Fig. 6 alongside solutions of
various PWAs described in Section 7.

Figure 6: $\Sigma$ for $\overrightarrow{\gamma}n\to\pi^{0}n$ vs. pion
production angle $\theta$ in c.m. frame: A2 (blue filled circles); fit: SAID
MU22 (red solid curves), SAID MA19 Briscoe:2019cyo  (blue dash-dotted curves),
Bonn-Gatchina BG2014-02 Gutz:2014wit  (magenta dotted curves), and MAID2007
Drechsel:2007if  (black dashed curves). Only angle-dependent uncertainties are
shown for all data. Each plot corresponds to a 20 MeV wide bin in Eγ with the
central value given in each.

##  6 Systematic Uncertainties

The dominant sources of systematic uncertainty in the results came from the
linear polarization calibration and the background subtraction. The fractional
difference between a simple cuts-based analysis and the sPlot background
subtraction methods gave an estimate of the systematic uncertainty in
background subtraction method for each point. For the cuts-based analysis a
similar procedure was performed but, rather than use a sPlot subtraction,
tighter cuts were placed on the discriminatory variables to identify a cleaner
sample of $n\pi^{0}$ events with residual backgrounds estimated by simulations
to be around 3.6$\%$. These cuts are summarised in Table 3.

Table 3: A summary of the tight cuts applied to the data to produce a low background event sample. Variable | Cut Range | Units
---|---|---
Tagged Time | -5 < t${}_{\pi^{0}}$ < 5 | ns
Coplanarity | -30 < $\Delta\phi$ < 30 | degree
Missing Mass | 1850 < Mmiss < 2100 | MeV/c2
Cone Angle | 0 < $\theta_{\mathrm{Cone}}$ < 0.3 | radian
Invariant Mass | 110 < Minv < 160 | MeV/c2
Spectator Momentum | 0 < $\left|\mathrm{P}_{\mathrm{spec}}\right|$ < 200 | MeV/$c$

While a cuts-only analysis retains some small amount of background, as
illustrated in Figs. 3 and 4, the sPlot method is expected to remove all the
background. Any error in the subtraction of the background by the weights-
based method is expected to be less than the effect of not subtracting the
background. Hence, the difference between $\Sigma$ extracted from the cuts and
sPlot subtracted results is used as a conservative estimate of this systematic
uncertainty Mullen:2020fgd . Values for this systematic uncertainty were
calculated for each $E_{\gamma}$ and $\theta$ bin with a mean value of $3\%$.
For the systematic uncertainty in the degree of linear polarization there are
two factors, first the uncertainty in the SAID solution for $\Sigma$ on
$p\pi^{0}$, which was used to determine the polarization, and was estimated to
be $2\%$. Second, the uncertainty on our extraction of the $p\pi^{0}$
asymmetries which had a main contribution from the background subtraction
which was estimated in a similar manner to the $n\pi^{0}$ background
subtraction and found to be $4\%$. Adding these two factors in quadrature
gives an overall $4.5\%$ systematic uncertainty in our $\Sigma$ results due to
the linear polarization. Other uncertainties were found to be much smaller
than these sources: acceptance effects, such as the neutron detection
efficiency, cancel due to the polarization flip; polarization degree and
luminosity asymmetries were incorporated into the likelihood fit; and unbinned
fits were used, removing binning artifacts from the results.

##  7 Multipole Analysis

The SAID Workman:2012jf , MAID Drechsel:2007if , and Bonn-Gatchina
Anisovich:2009zy  analyses use different fit formalisms to extract the
partial-wave (multipole) amplitudes underlying different data-sets. Comparing
the different resulting amplitudes gives an estimate of the systematic errors
inherent in the process.

For the MAID analysis, which was completed in 2007, the most recent data-sets
have not been included and this must be considered when making comparisons
Drechsel:2007if . The MAID unitary isobar approach applies a Breit-Wigner
resonance plus background model, guaranteeing unitarity up to the two-pion
production threshold.

The Bonn-Gatchina method fits a wider range of reactions utilizing elements of
the K-matrix and P-vector approaches Anisovich:2009zy . Both pion and photo-
induced reactions are included in a multi-channel fit. Reactions with three-
body final states are included using an event-based likelihood fit. The
elastic pion-nucleon reaction is fitted based on existing amplitudes. The
various data types are fitted with the possibility of renormalization and
weighting.

The SAID method is an extension of the Chew-Mandelstam K-matrix approach used
to fit pion-nucleon elastic scattering and $\eta N$ production data. The
resonance spectrum is fixed from this fit Arndt:2006bf  and only the photo-
couplings are allowed to vary. This differs from the MAID and Bonn-Gatchina
analyses, which can add new resonances to improve the agreement with data. The
formalism has built-in cuts associated with the $\pi\Delta$, $\rho N$, and
$\eta N$ thresholds but only single-pion photoproduction data are fitted. Data
have been weighted and renormalized in previous fits. No weighting and only
renormalization at the one percent level was utilized in fitting the present
set of $\Sigma$ data.

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

$\chi^{2}=\sum_{i}\left(\frac{X\eta_{i}-\eta_{i}^{exp}}{\epsilon_{i}}\right)^{2}+\left(\frac{X-1}{\epsilon_{X}}\right)^{2},$
(9)

where the subscript $i$ labels the data points within the distribution,
$\eta_{i}^{exp}$ is an individual measurement, $\eta_{i}$ is the corresponding
calculated value, and $\epsilon_{i}$ represents the angular-dependent
statistical uncertainty. The total $\chi^{2}$ is then found by summing over
all measurements. This re-normalization freedom is often important in
obtaining the best SAID fit results. For other data analyzed in the fit, such
as the total cross sections and excitation data, the statistical and
systematic uncertainties were combined in quadrature and no re-normalization
was allowed.

In fitting the present set of $\Sigma$ data, an overall angle-independent
systematic uncertainty of $1\%$ was used for $\epsilon_{X}$ in Eq. (9). The
resulting values for $X$ remained within $\epsilon_{X}$ of unity on average.

Figure 7: Comparison of $\chi^{2}$ per data point for the previous SAID
solution MA19 Briscoe:2019cyo  and MAID2007 Drechsel:2007if  applied to the
present A2 data with blue open circles and black open triangles, respectively.
The new SAID solution MU22 (MUXX) with red full circles and magenta full
squares obtained after adding the present A2 data (MUXX does not have any
$\gamma n\to\pi^{0}n$ data). Also shown are the fit $\chi^{2}$ per data point
values averaged over each energy bin $E_{\gamma}$, where the horizontal dashed
lines are for the MU22 (red), MUXX (magenta), MA19 (blue), and MAID2007
(black) solutions. The solid lines connecting the points are included only to
guide the eye.

A revised SAID multipole analysis has been completed, including the present
set of $\overrightarrow{\gamma}n\to\pi^{0}n$ $\Sigma$ data. This new global
energy-dependent solution has been labeled as MU22. The overall fit quality of
the present MU22 and previous MA19 Briscoe:2019cyo  SAID fits is compared with
the MAID2007 Drechsel:2007if  solution in Tables 4 and 5. The inclusion of the
present A2 data set provides a fit with significantly improved
$\chi^{2}/data$, specifically at higher energies, in comparisons between the
$\pi^{0}n$ fits and data ($\chi^{2}/data$ for MA19 = 3.93 and $\chi^{2}/data$
for MU22 = 1.44) as shown in Fig. 7 and Table 4. This demonstrates the
influence of these asymmetry measurements with their small uncertainties. The
overall comparison of the MA19 and MU22 solutions shows that the fit
$\chi^{2}/data$ values are essentially unchanged for $\pi^{0}p$ and $\pi^{+}n$
channels. The $\chi^{2}$ per data point including all available data and the
present A2 data for MA19 and MU22 (with MAID2007) is given in Table 5.

Table 4: $\chi^{2}$ per data point for new A2 data. Predictions are from MAID2007 Drechsel:2007if  and the SAID fit MUXX (no world $\pi^{0}n$ data fitted), an older SAID fit to existing data (MA19 Briscoe:2019cyo ), and the SAID fit including present data (MU22). Solution |  $\chi^{2}/(\pi^{0}n$ data)
---|---
MU22 | 275/189=1.46
MUXX | 624/189=3.30
MA19 | 743/189=3.93
MAID2007 | 1151/189=6.09

Additionally, an alternative MUXX solution was generated excluding all world
$\gamma n\to\pi^{0}n$ data and show results in Fig. 8 and Table 4. The
excellent comparison of the isospin-predicted $\Sigma$ to the data strongly
suggests the systematics in the new data are well under control. The
comparisons with MAID are interesting as both solutions use isospin symmetry
to predict the $\pi^{0}n$ observables based on the available data from the
other three charge channels in 2007 and 2020.

Table 5: $\chi^{2}$ per data point values for all charge channels covering the energy range from 155 MeV to 1000 MeV. Fits as described in Table 4. Solution |  $\chi^{2}/(\pi^{0}p$ data) |  $\chi^{2}/(\pi^{+}n$ data)
---|---|---
MU22 | 13274/9534=1.39 | 7454/4039=1.85
MUXX | 13171/9534=1.38 | 7259/4039=1.80
MA19 | 12565/9534=1.32 | 7461/4039=1.85
MAID2007 | 73638/9534=7.72 | 14599/4039=3.61
Solution |  $\chi^{2}/(\pi^{0}n$ data) |  $\chi^{2}/(\pi^{-}p$ data)
MU22 | 2345/ 798=2.94 | 5879/3456=1.70
MUXX | 7639/ 798=9.57 | 5384/3456=1.56
MA19 | 2649/ 798=3.32 | 5999/3456=1.74
MAID2007 | 4846/ 798=6.07 | 15365/3456=4.45

##  8 Results and Interpretation

A comprehensive set of $\Sigma$ data for
$\overrightarrow{\gamma}d\to\pi^{0}n(p)$ at 12 photon energies has been
determined with the CB and TAPS spectrometers using a tagged photon beam at
incident photon energies from 390 to 610 MeV. The present $\Sigma$ data cover
the resonance region from above the maximum of the $\Delta$-isobar to the
Roper resonance.

Figure 8: $d\sigma/d\Omega$, $\Sigma$, and $E$ observables for
$\overrightarrow{\gamma}n\to\pi^{0}n$ (blue full circles). All data were
produced by the A2 Collaboration at MAMI: $d\sigma/d\Omega$ at
$E_{\gamma}=610$ MeV (top) are from Ref. Briscoe:2019cyo , $\Sigma$ at
$E_{\gamma}=610$ MeV (middle) are present measurement, and $E$ at
$E_{\gamma}=603$ MeV (bottom) are from Ref. Dieterle:2017myg . New SAID
solutions MU22 (MUXX) are shown by red solid (magenta dotted) curves and
MAID2007 Drechsel:2007if  by black dashed curves.

The SAID MA19 Briscoe:2019cyo , Bonn-Gatchina BG2014-02 Gutz:2014wit , and
MAID2007 Drechsel:2007if  curves shown in Fig. 6 did not include the present
A2 data in their fits. In addition, the MAID2007 fit does not include
measurements after 2007. MU22 includes all previous measurement and includes
the A2 data. All fits and predictions agree well for the lowest energy where
the $\Delta$ resonance dominates. The angular distribution retains the shape
of SAID MA19 up until the highest energies where some larger deviations become
apparent.

Exploring the effectiveness of isospin symmetry to predict $\pi^{0}n$
observables, in Fig. 8 the fit (MU22) is compared to predictions from MUXX and
MAID2007 for $d\sigma/d\Omega$, $\Sigma$, and the double-polarization
asymmetry $E$. At this energy, the qualitative features are generally
reproduced, particularly for the current data. In the fit, cross sections have
larger angle-independent systematic uncertainties and the renormalization
factor from Eq. (9) improves the description, but is not included in the plot.
While the new data cover parts of the $\Delta$ and Roper resonance regions,
selected isospin multipoles are compared up to a photon energy of 1 GeV. The
isospin $3/2$ multipoles are taken as determined by the much larger proton-
target database and only the isospin $1/2$ neutron multipoles are shown in
Fig. 9. Comparing MA19 to MA22, there are no significant changes seen in the
$M_{1-}^{1/2}$ multipole. In addition, comparing imaginary parts of multipoles
connected to the nearby $N(1535)$ ($E_{0+}^{1/2}$) and $N(1520)$
($E_{2-}^{1/2}$ and $M_{2-}^{1/2}$), there is also good agreement between the
SAID and Bonn-Gatchina plots. The resonance couplings for the $N(1535)$,
$N(1440)$, and $N(1520)$ are expected to be in agreement with those reported
in Ref. Briscoe:2019cyo .

As a final comment on the predictive ability of our fit, excluding all
$\pi^{0}n$ data, a comparison of Tables 4 and 5 shows that fit MUXX is much
less successful between the upper energy limit of the present experiment and 1
GeV in the photon energy. This change is due mainly to poor compatibility with
GRAAL $\Sigma$ data DiSalvo:2009zz .

Figure 9: Selected neutron multipole $I$ = 1/2 amplitude from threshold to $W$
= 1.68 GeV ($E_{\gamma}$ = 1030 MeV) for the states $0^{+}$, $1^{-}$, and
$2^{-}$. The blue dash-dotted (red solid) curves correspond to the SAID MA19
Briscoe:2019cyo  (new MU22 including present A2 data) solution. The magenta
dotted (black dashed) curves give the Bonn-Gatchina BG2014-02 Gutz:2014wit
(MAID2007 Drechsel:2007if ). The vertical black arrows indicate Breit-Wigner
(BW) mass ($W_{R}$), and horizontal bars show full ($\Gamma$) and partial
($\Gamma_{\pi N}$) widths of resonances extracted by the BW fit of the $\pi N$
data associated with the SAID solution SP06 Arndt:2006bf . Vertical red arrows
show the $\eta$ meson production threshold.

Our results for $\Sigma$ for $\overrightarrow{\gamma}d\to\pi^{0}n(p)$ consist
of 189 experimental points and are available from the SAID database
Briscoe:2020zzz , where systematic uncertainties for each bin have been added
in quadrature.

###### Acknowledgements.

This work was supported in part by the UK Science and Technology Facilities
Council (STFC Grants No. 57071/1 and No. 50727/1), the U. S. Department of
Energy, Office of Science, Office of Nuclear Physics, under Awards No.
DE–FG02–01ER41194, No. DE–-SC0016583, No. DE-–SC0016582, and No.
DE-–SC0014323. This work was supported by Schweizerischer Nationalfonds
(Grants No. 200020–132799, No. 121781, No. 117601, and No. 113511),
Deutsche Forschungsgemeinschaft (SFB Grant No. 443,
No. SFB/TR 16, No. SFB 1044), DFG–RFBR (Grant No. 05–-02-–04014), European
Community Research Infrastructure Activity (FP6), the U. S. DOE, U. S. NSF,
and NSERC (Grant No. SAPPJ–-2018-–00020) Canada. This publication is part of a
project that has received funding from the European Union’s Horizon 2020
research and innovation programme under grant agreement STRONG – 2020 - No
824093. We would like to thank all the technical and nontechnical staff of
MAMI for their support.

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# Reply to the comment on the letter “Geometric Origin of the Tennis Racket
Effect”

P. Mardešić111Institut de Mathématiques de Bourgogne - UMR 5584 CNRS,
Université de Bourgogne, 9 avenue Alain Savary, BP 47870, 21078 DIJON, FRANCE,
<EMAIL_ADDRESS>G. J. Gutierrez Guillen222Institut de
Mathématiques de Bourgogne - UMR 5584 CNRS, Université de Bourgogne, 9 avenue
Alain Savary, BP 47870, 21078 DIJON, FRANCE, D. Sugny333Laboratoire
Interdisciplinaire Carnot de Bourgogne (ICB), UMR 6303 CNRS-Université
Bourgogne, 9 Av. A. Savary, BP 47 870, F-21078 Dijon Cedex, France,
<EMAIL_ADDRESS>
###### Abstract

The author of the comment [arXiv:2302.04190] criticizes our published results
in Phys. Rev. Lett. 125, 064301 (2020) about the Tennis Racket Effect (TRE).
The TRE is a geometric effect which occurs in the free rotation of any
asymmetric rigid body. We explain why the criticism of this comment is not
valid.

The author of the comment [1] and of the work [2] criticizes our published
results in [3] and [4] about the Tennis Racket Effect (TRE). Our studies were
based on a preliminary analysis by Cushman and co-workers [6, 7]. Even if it
is not described in standard textbooks of classical mechanics [8, 9, 10], TRE
is a very well-known phenomenon with many videos illustrating this effect in
different rigid bodies. The TRE is a geometric effect which occurs in the free
rotation of any asymmetric rigid body. This effect is manifested in a tennis
racket by an almost $\displaystyle\pi$-twist of the head of the racket when
the handle performs a full rotation. We show in [3, 4] that a perfect TRE can
be observed in the limit of an ideal asymmetric rigid body when the moments of
inertia are far from each other. We recently investigate the dynamical role of
physical constraints on the moments of inertia and we find in which conditions
a perfect TRE can be performed within such constraints [5].

In the comment [1], the author mentions that (verbatim only adapting the
numbering of the references): “In the recent work [3], authors introduced and
discussed a first-order differential equation that relates two of the Euler
angles obeying the Euler-Poisson equations under very special initial
conditions, see Eq. (1) in [3]. Since it was obtained from the equations, that
can be used to study the dynamics of a free asymmetric rigid body, authors
assumed that their equation is suitable for describing some effects in the
theory of a rigid body, including the Tennis Racket, Dzhanibekov and Monster
Flip effects. However, this assumption is not justified, and the relationship
of their solution $\displaystyle\psi(\varphi)$ to Eq. (1) in [3] with the
motions of a rigid body is not clear.”

If we understand properly the comment [1], the core of the criticism is based
on the choice of the moving frame, where the _preserved_ angular momentum
$\displaystyle\mathbf{m}$ is taken in the direction of the $\displaystyle
Z$-axis of the space-fixed frame (laboratory frame). The free rotation of the
rigid body is described by the _relative_ position of two frames: the fixed
laboratory frame and the moving frame attached to the rigid body. This
position is described by three Euler angles whose dynamics can be derived from
Euler equations. The use of such equations in studying the rotation of the
rigid body is by no means our contribution to the subject, but goes back to
Euler and appears in all standard textbooks of classical mechanics [8, 10]. In
our analysis, we consider the convention where the fixed frame is the one
attached to the body and the moving frame is the laboratory one. The TRE is
then described in our papers by considering specific trajectories of the
angular momentum close to the separatrix between the rotating and the
oscillating trajectories. Our main contribution was to express the TRE in
terms of elliptic integrals of the first and third kinds [4]. A geometric
analysis is proposed in [3] to show that the geometric origin of TRE is a pole
of a Riemann surface defined in a complex phase space. We also confirm the TRE
from a numerical integration of the differential system.

The author of [1] further claims that solutions to the Euler equations with
the convention of the angular momentum $\displaystyle m$ in the direction of
the $\displaystyle Z$-axis do not present the TRE. However, this statement is
not true. The proof is based on the wrong assumption that the initial
condition for the angular momentum is of the form $\displaystyle(0,0,m_{3})$
in the body-fixed frame, whereas this condition is only valid in the
laboratory frame. This leads to an erroneous counter-example in [1] which in
no way proves the absence of TRE.

## References

  * [1] A. A. Deriglazov, _Comment on the Letter ”Geometric Origin of the Tennis Racket Effect” by P. Mardesic, et al, Phys. Rev. Lett. 125, 064301 (2020)._ , arXiv:2302.04190
  * [2] A. A. Deriglazov, _Lagrangian and Hamiltonian formulations of asymmetric rigid body, considered as a constrained system_ , arXiv:2301.10741
  * [3] P. Mardesic, G. J. Gutierrez Guillen, L. Van Damme and D. Sugny, _Geometric origin of the Tennis Racket Effect_ , Phys. Rev. Lett. 125, 064301 (2020)
  * [4] L. Van Damme, P. Mardešić and D. Sugny, _The tennis racket effect in a three-dimensional rigid body_ , Physica D 338, 17 (2017)
  * [5] G. J. Gutierrez Guillen, E. Aldo Arroyo, P. Mardesic, D. Sugny, _Signatures of physical constraints in rotating rigid bodies_ , arXiv:2302.01758
  * [6] M. S. Ashbaugh, C. C. Chicone and R. H. Cushman, _The twisting tennis racket_ , J. Dyn. Diff. Eq. 3, 67 (1991).
  * [7] R. H. Cushman and L. Bates, _Global Aspects of Classical Integrable Systems_ (Birkhauser, Basel, 1997).
  * [8] H. Goldstein, _Classical Mechanics_ (Addison-Wesley, Reading, MA, 1950).
  * [9] V. I. Arnol’d, _Mathematical Methods of Classical Mechanics_ (Springer-Verlag, New York, 1989).
  * [10] Landau L.D. and Lifshitz E.M. (1976) Mechanics, 3rd. ed., Pergamon Press.