††thanks: Deceased

Current address:]Ohio University, Athens, Ohio 45701 Current address:]Idaho
State University, Pocatello, Idaho 83209

CLAS Collaboration

# Beam-spin asymmetry $\bm{\Sigma}$ for $\Sigma^{-}$ hyperon photoproduction
off the neutron

N. Zachariou<EMAIL_ADDRESS>University of York, York YO10 5DD, United
Kingdom  E. Munevar  Universidad Distrital Francisco José de Caldas, Bogotá,
Colombia  B.L. Berman  The George Washington University, Washington, DC 20052
P. Bydžovský  Nuclear Physics Institute of the Czech Academy of Sciences, 250
68 Řež, Czechia  A. Cieplý  Nuclear Physics Institute of the Czech Academy of
Sciences, 250 68 Řež, Czechia  G. Feldman  The George Washington University,
Washington, DC 20052  Y. Ilieva  University of South Carolina, Columbia, South
Carolina 29208  P. Nadel-Turonski  CFNS Stony Brook University, Stony Brook,
New York 11794  D. Skoupil  Nuclear Physics Institute of the Czech Academy of
Sciences, 250 68 Řež, Czechia  A.V. Sarantsev  Helmholtz-Institut fuer
Strahlen- und Kernphysik, Universität Bonn, 53115 Bonn, Germany  D. P. Watts
University of York, York YO10 5DD, United Kingdom  M.J. Amaryan  Old Dominion
University, Norfolk, Virginia 23529  G. Angelini  The George Washington
University, Washington, DC 20052  W.R. Armstrong  Argonne National Laboratory,
Argonne, Illinois 60439  H. Atac  Temple University, Philadelphia,
Pennsylvania 19122  H. Avakian  Thomas Jefferson National Accelerator
Facility, Newport News, Virginia 23606  L. Barion  INFN, Sezione di Ferrara,
44100 Ferrara, Italy  M. Bashkanov  University of York, York YO10 5DD, United
Kingdom  M. Battaglieri  Thomas Jefferson National Accelerator Facility,
Newport News, Virginia 23606  INFN, Sezione di Genova, 16146 Genova, Italy  I.
Bedlinskiy  National Research Centre Kurchatov Institute - ITEP, Moscow,
117259, Russia  F. Benmokhtar  Duquesne University, 600 Forbes Avenue,
Pittsburgh, Pennsylvania 15282  A. Bianconi  Università degli Studi di
Brescia, 25123 Brescia, Italy  INFN, Sezione di Pavia, 27100 Pavia, Italy  L.
Biondo  INFN, Sezione di Genova, 16146 Genova, Italy  INFN, Sezione di
Catania, 95123 Catania, Italy  Università degli Studi di Messina, 98166
Messina, Italy  A.S. Biselli  Fairfield University, Fairfield, Connecticut
06824  M. Bondi  INFN, Sezione di Genova, 16146 Genova, Italy  F. Bossù  IRFU,
CEA, Université Paris-Saclay, F-91191 Gif-sur-Yvette, France  S. Boiarinov
Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606
W.J. Briscoe  The George Washington University, Washington, DC 20052  W.K.
Brooks  Universidad Técnica Federico Santa María, Casilla 110-V Valparaíso,
Chile  Thomas Jefferson National Accelerator Facility, Newport News, Virginia
23606  D. Bulumulla  Old Dominion University, Norfolk, Virginia 23529  V.D.
Burkert  Thomas Jefferson National Accelerator Facility, Newport News,
Virginia 23606  D.S. Carman  Thomas Jefferson National Accelerator Facility,
Newport News, Virginia 23606  J.C. Carvajal  Florida International University,
Miami, Florida 33199  A. Celentano  INFN, Sezione di Genova, 16146 Genova,
Italy  P. Chatagnon  Universit’e Paris-Saclay, CNRS/IN2P3, IJCLab, 91405
Orsay, France  T. Chetry  Mississippi State University, Mississippi State, MS
39762-5167  G. Ciullo  INFN, Sezione di Ferrara, 44100 Ferrara, Italy
Universita’ di Ferrara , 44121 Ferrara, Italy  L. Clark  University of
Glasgow, Glasgow G12 8QQ, United Kingdom  P.L. Cole  Lamar University, 4400
MLK Blvd, PO Box 10046, Beaumont, Texas 77710  M. Contalbrigo  INFN, Sezione
di Ferrara, 44100 Ferrara, Italy  G. Costantini  Università degli Studi di
Brescia, 25123 Brescia, Italy  INFN, Sezione di Pavia, 27100 Pavia, Italy  V.
Crede  Florida State University, Tallahassee, Florida 32306  A. D’Angelo
INFN, Sezione di Roma Tor Vergata, 00133 Rome, Italy  Università di Roma Tor
Vergata, 00133 Rome, Italy  N. Dashyan  Yerevan Physics Institute, 375036
Yerevan, Armenia  R. De Vita  INFN, Sezione di Genova, 16146 Genova, Italy  M.
Defurne  IRFU, CEA, Université Paris-Saclay, F-91191 Gif-sur-Yvette, France
A. Deur  Thomas Jefferson National Accelerator Facility, Newport News,
Virginia 23606  S. Diehl  II Physikalisches Institut der Universitaet Giessen,
35392 Giessen, Germany  University of Connecticut, Storrs, Connecticut 06269
C. Djalali  Ohio University, Athens, Ohio 45701  University of South Carolina,
Columbia, South Carolina 29208  R. Dupre  Universit’e Paris-Saclay,
CNRS/IN2P3, IJCLab, 91405 Orsay, France  M. Dugger  Arizona State University,
Tempe, Arizona 85287-1504  H. Egiyan  Thomas Jefferson National Accelerator
Facility, Newport News, Virginia 23606  University of New Hampshire, Durham,
New Hampshire 03824-3568  M. Ehrhart  Argonne National Laboratory, Argonne,
Illinois 60439  A. El Alaoui  Universidad Técnica Federico Santa María,
Casilla 110-V Valparaíso, Chile  L. El Fassi  Mississippi State University,
Mississippi State, MS 39762-5167  Argonne National Laboratory, Argonne,
Illinois 60439  P. Eugenio  Florida State University, Tallahassee, Florida
32306  G. Fedotov  [  Skobeltsyn Institute of Nuclear Physics, Lomonosov
Moscow State University, 119234 Moscow, Russia  S. Fegan  University of York,
York YO10 5DD, United Kingdom  A. Filippi  INFN, Sezione di Torino, 10125
Torino, Italy  A. Fradi  University of Gabes, 6072-Gabes, Tunisia  G. Gavalian
Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606
Old Dominion University, Norfolk, Virginia 23529  G.P. Gilfoyle  University of
Richmond, Richmond, Virginia 23173  F.X. Girod  Thomas Jefferson National
Accelerator Facility, Newport News, Virginia 23606  IRFU, CEA, Université
Paris-Saclay, F-91191 Gif-sur-Yvette, France  C. Gleason  Union College,
Schenectady, New York, 12308  A.A. Golubenko  Skobeltsyn Institute of Nuclear
Physics, Lomonosov Moscow State University, 119234 Moscow, Russia  R.W. Gothe
University of South Carolina, Columbia, South Carolina 29208  K.A. Griffioen
College of William and Mary, Williamsburg, Virginia 23187-8795  M. Guidal
Universit’e Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France  K. Hafidi
Argonne National Laboratory, Argonne, Illinois 60439  H. Hakobyan  Universidad
Técnica Federico Santa María, Casilla 110-V Valparaíso, Chile  Yerevan Physics
Institute, 375036 Yerevan, Armenia  M. Hattawy  Old Dominion University,
Norfolk, Virginia 23529  T.B. Hayward  University of Connecticut, Storrs,
Connecticut 06269  D. Heddle  Christopher Newport University, Newport News,
Virginia 23606  Thomas Jefferson National Accelerator Facility, Newport News,
Virginia 23606  K. Hicks  Ohio University, Athens, Ohio 45701  A. Hobart
Universit’e Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France  M. Holtrop
University of New Hampshire, Durham, New Hampshire 03824-3568  D.G. Ireland
University of Glasgow, Glasgow G12 8QQ, United Kingdom  E.L. Isupov
Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University,
119234 Moscow, Russia  D. Jenkins  Virginia Tech, Blacksburg, Virginia
24061-0435  H.S. Jo  Kyungpook National University, Daegu 41566, Republic of
Korea  Universit’e Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France  K.
Joo  University of Connecticut, Storrs, Connecticut 06269  D. Keller
University of Virginia, Charlottesville, Virginia 22901  A. Khanal  Florida
International University, Miami, Florida 33199  M. Khandaker  [  Norfolk State
University, Norfolk, Virginia 23504  A. Kim  University of Connecticut,
Storrs, Connecticut 06269  F.J. Klein  Catholic University of America,
Washington, D.C. 20064  A. Kripko  II Physikalisches Institut der Universitaet
Giessen, 35392 Giessen, Germany  V. Kubarovsky  Thomas Jefferson National
Accelerator Facility, Newport News, Virginia 23606  Rensselaer Polytechnic
Institute, Troy, New York 12180-3590  L. Lanza  INFN, Sezione di Roma Tor
Vergata, 00133 Rome, Italy  M. Leali  Università degli Studi di Brescia, 25123
Brescia, Italy  INFN, Sezione di Pavia, 27100 Pavia, Italy  K. Livingston
University of Glasgow, Glasgow G12 8QQ, United Kingdom  I.J.D. MacGregor
University of Glasgow, Glasgow G12 8QQ, United Kingdom  D. Marchand
Universit’e Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France  N. Markov
Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606
University of Connecticut, Storrs, Connecticut 06269  L. Marsicano  INFN,
Sezione di Genova, 16146 Genova, Italy  V. Mascagna  Università degli Studi
dell’Insubria, 22100 Como, Italy  INFN, Sezione di Pavia, 27100 Pavia, Italy
B. McKinnon  University of Glasgow, Glasgow G12 8QQ, United Kingdom  S.
Migliorati  Università degli Studi di Brescia, 25123 Brescia, Italy  INFN,
Sezione di Pavia, 27100 Pavia, Italy  T. Mineeva  Universidad Técnica Federico
Santa María, Casilla 110-V Valparaíso, Chile  M. Mirazita  INFN, Laboratori
Nazionali di Frascati, 00044 Frascati, Italy  V. Mokeev  Thomas Jefferson
National Accelerator Facility, Newport News, Virginia 23606  Skobeltsyn
Institute of Nuclear Physics, Lomonosov Moscow State University, 119234
Moscow, Russia  C. Munoz Camacho  Universit’e Paris-Saclay, CNRS/IN2P3,
IJCLab, 91405 Orsay, France  P. Nadel-Turonski  Thomas Jefferson National
Accelerator Facility, Newport News, Virginia 23606  Catholic University of
America, Washington, D.C. 20064  K. Neupane  University of South Carolina,
Columbia, South Carolina 29208  S. Niccolai  Universit’e Paris-Saclay,
CNRS/IN2P3, IJCLab, 91405 Orsay, France  G. Niculescu  James Madison
University, Harrisonburg, Virginia 22807  T.R. O’Connell  University of
Connecticut, Storrs, Connecticut 06269  M. Osipenko  INFN, Sezione di Genova,
16146 Genova, Italy  A.I. Ostrovidov  Florida State University, Tallahassee,
Florida 32306  P. Pandey  Old Dominion University, Norfolk, Virginia 23529  M.
Paolone  New Mexico State University, PO Box 30001, Las Cruces, New Mexico
88003, USA  L.L. Pappalardo  INFN, Sezione di Ferrara, 44100 Ferrara, Italy
Universita’ di Ferrara , 44121 Ferrara, Italy  R. Paremuzyan  Thomas Jefferson
National Accelerator Facility, Newport News, Virginia 23606  E. Pasyuk  Thomas
Jefferson National Accelerator Facility, Newport News, Virginia 23606  W.
Phelps  Christopher Newport University, Newport News, Virginia 23606  O.
Pogorelko  National Research Centre Kurchatov Institute - ITEP, Moscow,
117259, Russia  J.W. Price  California State University, Dominguez Hills,
Carson, California 90747  Y. Prok  Old Dominion University, Norfolk, Virginia
23529  University of Virginia, Charlottesville, Virginia 22901  B.A. Raue
Florida International University, Miami, Florida 33199  M. Ripani  INFN,
Sezione di Genova, 16146 Genova, Italy  J. Ritman  Institute fur Kernphysik
(Juelich), Juelich, Germany  A. Rizzo  INFN, Sezione di Roma Tor Vergata,
00133 Rome, Italy  Università di Roma Tor Vergata, 00133 Rome, Italy  G.
Rosner  University of Glasgow, Glasgow G12 8QQ, United Kingdom  J. Rowley
Ohio University, Athens, Ohio 45701  F. Sabatie  IRFU, CEA, Université Paris-
Saclay, F-91191 Gif-sur-Yvette, France  C. Salgado  Norfolk State University,
Norfolk, Virginia 23504  A. Schmidt  The George Washington University,
Washington, DC 20052  R.A. Schumacher  Carnegie Mellon University, Pittsburgh,
Pennsylvania 15213  Y.G. Sharabian  Thomas Jefferson National Accelerator
Facility, Newport News, Virginia 23606  E.V. Shirokov  Skobeltsyn Institute of
Nuclear Physics, Lomonosov Moscow State University, 119234 Moscow, Russia  U.
Shrestha  University of Connecticut, Storrs, Connecticut 06269  D. Sokhan
University of Glasgow, Glasgow G12 8QQ, United Kingdom  O. Soto  INFN,
Laboratori Nazionali di Frascati, 00044 Frascati, Italy  N. Sparveris  Temple
University, Philadelphia, Pennsylvania 19122  S. Stepanyan  Thomas Jefferson
National Accelerator Facility, Newport News, Virginia 23606  P. Stoler
University of Connecticut, Storrs, Connecticut 06269  Rensselaer Polytechnic
Institute, Troy, New York 12180-3590  I.I. Strakovsky  The George Washington
University, Washington, DC 20052  S. Strauch  University of South Carolina,
Columbia, South Carolina 29208  R. Tyson  University of Glasgow, Glasgow G12
8QQ, United Kingdom  M. Ungaro  Thomas Jefferson National Accelerator
Facility, Newport News, Virginia 23606  University of Connecticut, Storrs,
Connecticut 06269  L. Venturelli  Università degli Studi di Brescia, 25123
Brescia, Italy  INFN, Sezione di Pavia, 27100 Pavia, Italy  H. Voskanyan
Yerevan Physics Institute, 375036 Yerevan, Armenia  A. Vossen  Duke
University, Durham, North Carolina 27708-0305  Thomas Jefferson National
Accelerator Facility, Newport News, Virginia 23606  E. Voutier  Universit’e
Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France  K. Wei  University of
Connecticut, Storrs, Connecticut 06269  X. Wei  Thomas Jefferson National
Accelerator Facility, Newport News, Virginia 23606  R. Wishart  University of
Glasgow, Glasgow G12 8QQ, United Kingdom  M.H. Wood  Canisius College,
Buffalo, NY  University of South Carolina, Columbia, South Carolina 29208  B.
Yale  College of William and Mary, Williamsburg, Virginia 23187-8795  J. Zhang
University of Virginia, Charlottesville, Virginia 22901  Old Dominion
University, Norfolk, Virginia 23529  Z.W. Zhao  Duke University, Durham, North
Carolina 27708-0305  University of South Carolina, Columbia, South Carolina
29208

###### Abstract

We report a new measurement of the beam-spin asymmetry $\bm{\Sigma}$ for the
$\vec{\gamma}n\rightarrow K^{+}\Sigma^{-}$ reaction using quasi-free neutrons
in a liquid-deuterium target. The new dataset includes data at previously
unmeasured photon energy and angular ranges, thereby providing new constraints
on partial wave analyses used to extract properties of the excited nucleon
states. The experimental data were obtained using the CEBAF Large Acceptance
Spectrometer (CLAS), housed in Hall B of the Thomas Jefferson National
Accelerator Facility (JLab). The CLAS detector measured reaction products from
a liquid-deuterium target produced by an energy-tagged, linearly polarised
photon beam with energies in the range 1.1 to 2.3 GeV. Predictions from an
isobar model indicate strong sensitivity to $N(1720)3/2^{+}$,
$\Delta(1900)1/2^{-}$, and $N(1895)1/2^{-}$, with the latter being a state not
considered in previous photoproduction analyses. When our data are
incorporated in the fits of partial-wave analyses, one observes significant
changes in $\gamma$-$n$ couplings of the resonances which have small branching
ratios to the $\pi N$ channel.

##  1  Introduction

The excitation spectrum of the nucleon provides fundamental information on the
dynamics and interactions of its constituents, the quarks and gluons, and is
an important tool to achieve a more detailed understanding of the nature of
Quantum Chromodynamics (QCD) in the non-perturbative regime. Phenomenological
constituent quark models Capstick and Roberts (2000, 1998a); Capstick and
Isgur (1986); Löring  _et al._ (2001); Glozman  _et al._ (1998); Giannini  _et
al._ (2001) and lattice QCD Edwards  _et al._ (2011); Dudek and Edwards
(2012); Edwards  _et al._ (2013) predict a plethora of excited states of the
nucleon that have yet to be experimentally determined. Alternative
interpretations of nucleon structure that result in a reduced number of
excited states (and therefore fewer “missing” resonances) have also been
proposed Anselmino  _et al._ (1993); Brodsky (2007); Kolomeitsev and Lutz
(2004); Afonin (2007). Experimentally establishing the existence, or absence,
of these missing nucleon resonances in nature has thus the potential to
provide important insights into fundamental nucleon structure. As a result,
the investigation continues to be a major focus at the world’s leading
electromagnetic beam facilities. Here we report a new precise measurement of
the single polarisation observable, $\bm{\Sigma}$, and we discuss the effect
this dataset has on partial wave analyses and models that aim at understanding
the excited spectrum of nucleons.

The clean extraction of the nucleon excitation spectrum from experiment is
complicated by the fact that the excited states are short-lived (broad) and
overlapping. This complicates the extraction of their fundamental properties
(photocouplings, lifetimes, spins, parities, decay branches, and even
existence), with the difficulties exacerbated for states that produce weak
signals in the decay channel under study. In the photoproduction of a
pseudoscalar meson off the nucleon, the excited nucleon states contribute
through their initial photoexcitation from the nucleon followed by the strong
decay of the state. Values of the four complex amplitudes may be extracted up
to an arbitrary phase, given data from a suitable combination of polarization
measurements of sufficient accuracy, which would therefore provide a maximum
constraint on subsequent partial-wave analyses Chiang and Tabakin (1997). It
has been recently argued that a reduced requirement on the number of measured
observables may still allow convergence to a unique set of multipole
amplitudes Workman  _et al._ (2017).

It is clear that eliminating the ambiguities in partial-wave analysis
extraction of the excited nucleon states requires a precise and complete set
of measurements of single- and double-polarisation observables, involving
polarised beams, targets, and recoiling baryon polarimetry Chiang and Tabakin
(1997); Nys  _et al._ (2016); Sandorfi  _et al._ (2011); Fasano  _et al._
(1992); Keaton and Workman (1996). Furthermore, measurements on both proton
and (more challenging) neutron targets are indispensable, since resonances can
have isospin dependent photocouplings Sandorfi and Hoblit (2013); Mart  _et
al._ (1995). Additionally, the predicted differences in the preferred decay
branches of individual states Capstick and Roberts (1998a); Chiang and Tabakin
(1997); Capstick and Roberts (1998b), mean that measurement of a wide range of
pseudoscalar meson photoproduction final states, including $N\pi$, $K\Lambda$,
$K\Sigma$ are crucial, and even data on vector meson (e.g., $N\omega$) or
multiple meson decays (e.g., $N\pi\pi$) could be necessary. A recent review of
the available results on non-strange baryon spectroscopy is given in Ref.
Ireland  _et al._ (2020).

The relative importance of decay channels to strange quark containing
particles (e.g., $K\Lambda$, $K\Sigma$) for missing or poorly established
states has been emphasized by constituent quark model calculations Capstick
and Roberts (1998a). Recent measurements of exclusive photoproduction of
$K\Lambda$ and $K\Sigma$ from proton targets Anisovich et al. (2017); Paterson
et al. (2016) was key to achieve sensitivity to the newly discovered states
reported in the PDG 2020 Zyla et al. (2020). However, the corresponding data
from neutron targets are much more limited. Although the differential cross
section for $K^{+}\Sigma^{-}$ Kohri et al. (2006); Pereira et al. (2010) and
$K^{0}\Lambda$ Pereira et al. (2010) reactions are measured with good
precision, only one single polarisation measurement exists. The beam-spin
asymmetry, $\bm{\Sigma}$, was originally obtained at LEPS Kohri et al. (2006),
having kinematical coverage only at very forward kaon angles. The few double-
polarisation measurements, for $K^{+}\Sigma^{-}$ Zachariou et al. (2020),
$K^{0}\Lambda$, and $K^{0}\Sigma^{0}$ Ho et al. (2018), have more complete
kinematic coverage but modest statistical accuracy, limiting definitive
interpretations about contributing resonant states in partial wave analyses.
It was highlighted in the most recent work on the beam-target helicity
asymmetry in $K^{+}\Sigma^{-}$ photoproduction Zachariou et al. (2020) that
the $\bm{\Sigma}$ at backward kaon angles showed an enhanced sensitivity to
the contribution of the $N(2120){3/2^{-}}$ ($D_{13}$) excited state, which was
found to improve the interpretation of the beam-target helicity asymmetry data
Zachariou et al. (2020).

In this work, we provide new experimental data on the beam-spin asymmetry
$\bm{\Sigma}$ for the reaction $\vec{\gamma}n\rightarrow K^{+}\Sigma^{-}$, for
the first time covering a wide range of kinematics and in previously
unexplored mass ranges for contributing states. The experiment used a
linearly-polarised tagged-photon beam incident on a (bound) quasi-free neutron
target (liquid deuterium). The paper is ordered as follows: Sec. 2 provides a
brief description of the experimental setup, Sec. 3 describes the method we
employed to determine the observable $\bm{\Sigma}$, and details of the data
analysis procedure and a description of the systematic uncertainties are
discussed in Secs. 4 and 5, respectively. The results and a discussion of
their implications are presented in Sec. 6.

##  2  Experimental setup

The data for this work were collected during the E06-103 experiment Nadel-
Turonski et al. (2006), which was conducted at the Thomas Jefferson National
Accelerator Facility (JLab) utilising the Continuous Electron Beam Accelerator
Facility (CEBAF) and the CEBAF Large Acceptance Spectrometer (CLAS) Mecking et
al. (2003) housed in Hall B. The CLAS detector was comprised of a drift-
chamber (DC) tracking system, a time-of-flight (ToF) system, and a calorimeter
system that allowed particle identification and four-vector determination for
charged and neutral particles. The charged particles’ momenta were determined
by tracing them as they traversed a toroidal magnetic field, providing
momentum resolution of $\sigma_{p}/p\sim 1\%$. A start counter (ST) Sharabian
et al. (2006) that surrounded the target cell provided the event start-time
information in photoproduction experiments. The ST, in conjunction with the
ToF system, was used to determine the speed of charged particles. The E06-103
experiment utilised a 40-cm long liquid-deuterium target, centered 20 cm
upstream of the nominal CLAS center to maximize acceptance for hyperon decays.
Overall, the CLAS detector provided an efficient detection of charged
particles over a large fraction of the full solid angle (between 8∘ and 142∘
in polar angles with $\sim$83% azimuthal coverage).

Hall B also housed the Tagger Facility Sober et al. (2000), which enabled the
selection and characterisation of the photons that initiated the photo-induced
reactions detected within the CLAS detector on an event-by-event basis. The
real photon beam was produced via the bremsstrahlung technique, by impinging a
monochromatic electron beam on a thin radiator. The post-bremsstrahlung
electrons were momentum analysed in a magnetic spectrometer that provided
energy and timing information of the incident photon beam. With an energy
resolution of $\sim$0.2%, this system permitted the tagging of photons with
energies between 20% and 95% of the incident electron beam energy. The
production of linearly polarized photons was based on the coherent
bremsstrahlung radiation technique Überall (1962) utilising a 50-$\mu$m thick
diamond radiator. With the use of a precise goniometer, data for two
orientations of the photon polarisation were collected: one parallel (Para)
and one perpendicular (Perp) to the Hall-B floor. Data were also obtained
using an amorphous carbon radiator that enabled the determination of the
degree of photon polarisation as discussed in the next section. For a fixed
electron energy, the position of the coherent edge 111The coherent edge refers
to the sharp falling edge in the enhancement spectrum as indicated in Fig. 1.
was selected by appropriately orienting the diamond radiator. Data were
obtained for different electron beam energies, varying from 3.3 to 5.2 GeV, to
enhance the degree of photon polarisation in six coherent peak positions, in
steps of 200 MeV between 1.1 and 2.3 GeV.

##  3  Beam-spin asymmetry

The differential cross section for meson photoproduction off an unpolarised
target with a linearly polarised photon beam is given by Barker and Donnachie
(1975):

$\displaystyle\frac{d\sigma}{d\Omega}$ $\displaystyle=$
$\displaystyle\left(\frac{d\sigma}{d\Omega}\right)_{0}[1-P_{lin}\bm{\Sigma}\cos(2\eta)],$
(1)

where $P_{lin}$ is the magnitude of the beam polarisation vector at an angle
$\eta$ to the reaction plane222The reaction plane is defined by the cross
product of the incoming photon and the outgoing meson.. The above equation is
obtained by integrating over the angular distribution of the hyperon decay
products 333The full cross section equation as shown in Ref. Barker and
Donnachie (1975) depends on additional double polarisation observables
accessible by studying the angular dependence of the hyperon decay products.
Integrating over the nucleon angle in the hyperon rest frame would eliminate
such contributions only when the detector acceptance is uniform in these
kinematics. In principle, the detector acceptance might affect the implied
integration over the angles of the hyperon decay products. However, as the
self-analyticity of the $\Sigma^{-}$ is very small ($\alpha=0.068$), any
double polarisation observables contributions/effects to the beam-spin
asymmetry are negligible compared to the quoted systematics. ($\Sigma^{-}\to
n\pi^{-}$ with a 99.85% branching ratio). The determination of $\bm{\Sigma}$
was done using a maximum likelihood approach. The likelihood function for a
given event, $i$, taken from the cross-section Eq. (1) is

$\displaystyle L_{i}=c\left[1-P_{lin}^{i}\bm{\Sigma}\cos(2\eta_{i})\right]A,$
(2)

where $c$ is a normalisation coefficient and $A$ is the detector acceptance.
In the construction of the log-likelihood function an approximation was made
concerning the detector acceptance. Specifically, an acceptance that is
largely independent of the kinematic variable $\eta$ was assumed, which
resulted in a normalisation coefficient that is independent of the value of
the polarisation observable. This approximation significantly simplified the
extraction of the observable, but could potentially result in systematic
biases. Extensive studies of such systematic effects showed that any residual
effects on the polarisation observable are negligible.

The log-likelihood function that was maximized to obtain the polarisation
observables is thus given by

$\displaystyle\log L$ $\displaystyle=$ $\displaystyle
b+\sum_{i}\log\left[1-P_{lin}^{i}\bm{\Sigma}\cos(2\eta_{i})\right],$ (3)

where the constant $b$ is the observable-independent constant that absorbs the
normalisation coefficient (associated with the photon flux) and the detector
acceptance. The summation is over all events within a given kinematic bin. A
transformation from the reaction frame (where the $y$ axis is perpendicular to
the reaction plane) to the lab frame (where the $y$ axis is vertical to the
Hall B floor) was done using the following equations for the two orthogonal
orientations of the photon polarisation (Para and Perp)

$\displaystyle\eta^{Para}$ $\displaystyle=$ $\displaystyle-(\phi-\phi_{0})$
$\displaystyle\eta^{Perp}$ $\displaystyle=$
$\displaystyle\frac{\pi}{2}-(\phi-\phi_{0}),$

where $\phi$ is the meson azimuthal angle as measured in the lab frame, and
$\phi_{0}$ is the offset of the photon polarisation with respect to the lab
$x$ (for Para) or $y$ (for Perp) axis. Using the above two equations, Eq. (3)
can be written as

$\displaystyle\log L$ $\displaystyle=$ $\displaystyle
b+\sum_{i}\log\left[1-\mathbb{P}_{lin}^{i}\bm{\Sigma}\cos(2\phi_{i}-2\phi_{0})\right],$
(4)

where $\mathbb{P}_{lin}^{i}=P_{lin}^{i}$ for Para events, and
$\mathbb{P}_{lin}^{i}=-P_{lin}^{i}$ for Perp events444The sign of
$\mathbb{P}_{\gamma}$ absorbs the sign from the trigonometric function when
translating $\eta$ to $\phi$, since $\cos(180-2\phi)=-\cos(2\phi)$.. This
likelihood function was maximized using MINUIT James and Winkler (2004) to
obtain the value of the observable $\bm{\Sigma}$ and its uncertainty. The
$\phi_{0}$ offset was determined using a high-statistics channel from the same
dataset (single pion photoproduction), found to be consistent with zero.

The determination of the beam-spin asymmetry requires a precise knowledge of
the degree of photon polarisation, $P_{lin}^{i}$. The determination of
$P_{lin}^{i}$ involved using the coherent and incoherent bremsstrahlung
spectra to obtain an enhancement distribution that was then fit by a spectrum
obtained from theoretical bremsstrahlung calculations (referred to as the
Analytical Bremsstrahlung Calculation – ANB). Specifically, the enhancement
distribution was obtained by taking the ratio of the photon energy spectrum
from the diamond radiator to one obtained using the amorphous radiator, and
was used to constrain the relative contribution of the coherent and incoherent
bremsstrahlung to the total photon yield. This ratio also removed Tagger
channel efficiency fluctuations allowing a precise determination of the degree
of photon polarisation. Subsequently, the enhancement plot was fit with the
theoretical spectrum from ANB. More details on the procedure can be found in
Refs. Überall (1962); Livingston (2011); Zachariou (2012).

The ANB calculation takes into account 17 experimental parameters
characterizing the geometry of the radiator, collimator, and photon beam.
Several of these parameters were measured experimentally (such as the photon
beam energy and beam spot size), whereas others (such as electron beam
divergence on the radiator) were varied until a good agreement was obtained
between the enhancement plot and the ANB calculation. These parameters were
then used to calculate the degree of polarisation as a function of photon
energy. An example of a fit to an enhancement spectrum with the ANB
calculation is shown in Fig. 1 along with the calculated photon polarisation
(dashed line). This procedure was done for the various coherent-edge
positions, allowing the determination of the photon polarisation on an event-
by-event basis. The degree of photon polarisation throughout the experiment
was on average 72%.

Figure 1: Example of an enhancement distribution (blue points) fit with the
ANB calculation (red histogram) to determine the photon polarisation (dashed
line). The arrow indicates the coherent edge position.

##  4  Data analysis

The reaction of interest was reconstructed by selecting events with exactly
one negative pion and one positive kaon, identifying the photon that initiated
it, and applying the missing-mass technique under the assumption that the
target was a nucleon at rest. Particle identification was done by following
the standard procedures adopted for E06-103 analyses, by comparing the
particle’s speed calculated from two independent measurements: time-of-flight
and momentum, with the latter requiring an assumption about the particle’s
rest mass. The photon that initiated the reaction detected in CLAS was
identified by timing coincidence at the event vertex between the tracks in
CLAS and photons, with the latter being reconstructed using information from
the Tagger spectrometer. The 2-ns beam bunch structure of the delivered
electron beam allowed an unambiguous identification of the photon that
initiated the reaction for $\sim 90\%$ of the events. The remaining $10\%$ of
events were associated with two or more photons with coincidence times within
$\pm 1$ ns, and such events were discarded from further analysis.

Figure 2: Mass of the missing state $X$ of $\gamma n\rightarrow K^{+}X$ vs.
$\gamma n\rightarrow K^{+}\pi^{-}X$ indicating the different physics channels
that contribute to the event sample.

A fraction of positive pions from the reactions $\gamma N\to\pi^{+}\pi^{-}X$
(where $N$ can be either a proton or a neutron) were misidentified as kaons.
Contributions from these events were eliminated by applying a cut on the mass
of the missing state $X$ in $\gamma N\to\pi^{+}\pi^{-}X$ (assuming the
reconstructed kaon was a misidentified pion).

The reaction of interest was identified by further exploiting the missing-mass
technique. Specifically, the correlation in the missing mass, $m_{X}$,
distribution of $\gamma n\to K^{+}X$ ($MM_{\gamma n\to K^{+}X}$) and the
$m_{X}$ distribution of $\gamma n\to K^{+}\pi^{-}X$ ($MM_{\gamma n\to
K^{+}\pi^{-}X}$) allows a clean identification of the reaction of interest.
This correlation is shown Fig. 2 along with the elliptical (two-dimensional)
cut employed to select the events of interest. The parameters of the
elliptical cut were optimised using simulated data (processed through a
realistic detector simulation). This approach resulted in an event sample
where background contributions were minimised, while retaining a large
fraction of good events. With the parameters of the adopted cut, the average
background contributions were found to be below 2% (with such contributions
accounted for in the systematic uncertainty, as listed in Table 1).

##  5  Systematic Uncertainties

An extensive investigation of potential sources of systematic uncertainty was
carried out with estimates summarised in Table 1. Most sources have negligible
contributions compared to the statistical uncertainty of the data. The largest
contribution originates from uncertainties of the degree of photon
polarisation, and the second largest is due to the dilution of the measured
observable stemming from having a bound rather than a free neutron target.

Figure 3: Beam spin asymmetry $\bm{\Sigma}$ as a function of kaon angle in the
$c.m.$ for four photon energy bins as indicated in the panels. Experimental
data from this work are shown with solid blue circles, whereas magenta points
show the previously published results from LEPS Kohri et al. (2006).
Statistical uncertainties are indicated with the errors bars, whereas the
total systematic uncertainties of the CLAS results are shown by the shaded bar
chart. The left set of panels shows the new Bonn-Gatchina solution that was
fit to our data (red solid lines), as well as previously published Bonn
Gatchina solutions with (green dash dotted lines) and without (black dashed
lines) contributions from the $D_{13}$ resonance (see Ref. Zachariou et al.
(2020) for a detailed discussion). The right set of panels show the full
solution of the isobar model (red solid lines), as well as the solution
without the $N(1720)\>3/2^{+}$ (black dashed lines), and the
$\Delta(1900)\>1/2^{-}$ (green dashed dotted lines) resonance.

The latter arises from effects of the Fermi motion of the target neutron and
Final State Interactions (FSI) of the outgoing reaction products with the
deuterium remnants. Such dilution effects were investigated in detail using a
smaller subset of the data sample in which the final-state neutron was
detected in addition to the $K^{+}$ and $\pi^{-}$ Munevar (2014), as well as
through simulations. Only a weak dependence of $\bm{\Sigma}$ on the momentum
of the target neutron was discovered. The asymmetric (positive) systematic
uncertainties reflect the fact that FSI effects only dilute and do not enhance
the measured $\bm{\Sigma}$. Additional sources, including background
contributions and misidentified kaon events, contributed to a much smaller
degree as summarised in Table 1. The uncertainties are split in two
categories: an absolute uncertainty that is the same for all kinematics, and a
relative uncertainty (associated with the photon polarisation) with its
magnitude determined for each point.

Source | $\sigma^{sys}$
---|---
Maximum Likelihood | negligible
Kaon PID | $\pm 0.008$
Pion PID | $\pm 0.004$
Photon selection | $\pm 0.002$
Misidentified kaons | $\pm 0.003$
Kaon Decay in flight | $\pm 0.0064$
$\Sigma^{*}$ background contribution | $\pm 0.007$
$\Lambda$ and $\Sigma^{0}$ contributions | $\pm 0.008$
Fiducial cut | $\pm 0.002$
FSI | +0.024
Total Absolute Systematic | $\mathbf{{}^{+0.029}_{-0.016}}$
Photon polarisation | 8%
Table 1: Summary of systematic uncertainties of $\bm{\Sigma}$.

##  6  Results and Discussion

The extracted $\bm{\Sigma}$ data are shown by the blue solid circles in Figs.
3 and 4, binned in 50-MeV wide photon-energy bins (from 1.1 to 2.3 GeV) and in
10 bins of kaon production angle in the center-of-momentum ($c.m.$)
frame555Commonly known as the center-of-mass frame., $\cos\theta_{K^{+}}$.
Figure 3 shows how the new precise results in four photon-energy bins compare
with previous and current Bonn Gatchina solutions (left) or with an isobar
model predictions that focuses on contributions from specific resonance states
(right). Figure 4 compares the two new solutions for all available kinematic
bins as discussed in detailed later on. The angular bins are contiguous but
with varying widths to accommodate the angular variation of the reaction yield
as to keep the statistics per bin rather constant. The statistical
uncertainties are shown by the error bars for each point, and the systematic
uncertainties are shown by the grey bands. For all photon energies, the
measured $\bm{\Sigma}$ is large, positive, and for forward-central kaon angles
rather uniform. The data exhibit a fall off at backward kaon angles, with
$\bm{\Sigma}$ typically larger in the forward angle region. As $\bm{\Sigma}$
must have a value of 0 at $\cos\theta_{K^{+}}=\pm 1$, the observable values
outside of our acceptance range (i.e., between $\cos\theta_{K^{+}}=0.75$ and
$\cos\theta_{K^{+}}=1.0$ for forward angles) must vary rapidly to reach 0. At
backward angles the transition to zero exhibits a more gradual trend.

Figure 4: Beam spin asymmetry $\bm{\Sigma}$ as a function of kaon angle in the
$c.m.$ The different panels show bins in photon energy. Experimental data from
this work are shown with solid blue circles, whereas magenta points show the
previously published results from LEPS Kohri et al. (2006). Statistical
uncertainties are indicated with the errors bars, whereas the total systematic
uncertainties of the CLAS results are shown by the shaded bar chart. The black
dashed line indicates the full solution of the isobar model as described in
the text and the red solid line indicates the new Bonn-Gatchina solution that
was fit to our data.

The published results for $\bm{\Sigma}$ from LEPS Kohri et al. (2006) are also
shown by the magenta solid squares in Figs. 3 and 4. The LEPS data are limited
to very forward kaon angles and have larger statistical and systematic
uncertainties than our data. Nevertheless, the results from CLAS are in good
agreement with these previously published data (note the LEPS data were
obtained in 100-MeV wide photon energy bins). The improvement in the quality
and range of available data with this new measurement is apparent in Fig. 4.

The solutions of the Bonn-Gatchina group BG2016 Müller et al. (2020) (not
shown here) predicted the beam asymmetry for the $\gamma n\to K^{+}\Sigma^{-}$
reaction above 1850 MeV to be negative at the backward and the central angular
region. The most recent predictions from the Bonn-Gatchina model Zachariou et
al. (2020) with (without) the proposed $D_{13}$ resonance are shown by the
dash-dotted green (dashed black) lines in the left panels of Fig. 3 (only four
representative photon energy bins are shown). These predictions were fit to
the current world dataset in meson photoproduction, including the unpolarised
differential cross section, the data on the beam-target helicity asymmetry
measured by the CLAS Collaboration, and the LEPS data on the beam asymmetry,
but excluding data from this work. The $\bm{\Sigma}$ data from LEPS were
measured only in the very forward angular region and mostly were defined by
the contribution from the $t$-channel exchange amplitudes. Moreover, the CLAS
data on the unpolarized cross section and beam-target helicity asymmetry did
not cover the very backward angular region, which allowed ambiguous solutions.
It is clear that neither of these solutions can reproduce the angular
dependence of $\bm{\Sigma}$, with discrepancies especially apparent at photon
energies above 1.3 GeV. Clearly, the new data have the potential to impact
this partial-wave analysis and, therefore, the excited nucleon spectrum
therein.

The inclusion of the new beam asymmetry data in the full combined analysis led
to significant changes in the $\gamma$-$n$ couplings of the resonances that
have small branching ratios to the $\pi N$ channel. The largest changes were
found in the $D_{13}$ and $P_{13}$ partial waves: here the states in the
region above 1850 MeV were mostly seen in the reactions with open strangeness.
The newly obtained solution is indicated by the red solid curves in Figs. 3
and 4. The detailed and systematic analysis of this solution will be presented
in a separate paper, which will follow the present publication.

The full predictions from an isobar model Skoupil and Bydžovský (2016) for
$\bm{\Sigma}$ are shown by the black dashed line in Fig. 4. These are based on
an effective Lagrangian in a tree-level approximation. The non-resonant part
of the amplitude consists of the Born terms and exchanges of resonances in the
$t$\- ($K^{*}$ and $K_{1}$) and $u$-channels ($\Sigma^{*}$). The main coupling
constant $g_{K^{+}\Sigma^{-}n}=\sqrt{2}g_{K^{+}\Sigma^{0}p}=1.568$, which
determines the strength of the Born terms, was taken from the $K^{+}\Lambda$
channel Skoupil and Bydžovský (2016) and kept unchanged in the present fit.
The resonant part is modeled by $s$-channel exchanges of nucleon and $\Delta$
resonances with masses below about 2 GeV. Hadronic form factors included in
the strong vertices account for hadron structure and regularize the amplitude
at large energies. The form factors are introduced in the way that keeps gauge
invariance intact, in analogy with the method used in Refs. Skoupil and
Bydžovský (2016) and Skoupil and Bydžovský (2018). The solution presented in
Fig. 4 was fit to the current CLAS (and LEPS) $\bm{\Sigma}$ data, as well as
the differential cross section of $\gamma n\to K^{+}\Sigma^{-}$ from CLAS
Pereira et al. (2010). In total, 24 free parameters (22 couplings and 2 hadron
form factor cut-offs) were used to fit 332 cross section data points and 284
asymmetries, all of them are restricted to energies up to $E_{\gamma}=2.6$
GeV. The fit parameters of the isobar model were extracted adopting the
procedure outlined in Refs. Skoupil and Bydžovský (2016, 2018) for the
$K^{+}\Lambda$ channel. More details are provided in Ref. Bydžovský  _et al._
(2021).

The considered set of nucleon resonances in the isobar model was motivated by
previous analyses of $K^{+}\Lambda$ Skoupil and Bydžovský (2016, 2018) and
$K\Sigma$ photoproduction David  _et al._ (1996).

Table 2: Characteristics of included resonances with their masses and widths taken as the PDG Breit-Wigner averages. The available branching ratios to the $K\Lambda$ and $K\Sigma$ channels are also taken from the PDG Zyla et al. (2020). For the nucleon and Delta resonances, the values $g_{1}$ and $g_{2}$ show the baryon-$K\Sigma$ scalar and tensor couplings obtained in our fit, while for the $K^{*}$ and $K_{1}$ states they represent the vector and tensor couplings, respectively.  Resonance | Mass | Width | Branching ratios | Couplings
---|---|---|---|---
| (MeV) | (MeV) | $\Lambda K$ | $\Sigma K$ | $g_{1}$ | $g_{2}$
$N(1535)\;1/2^{-}$ | 1530 | 150 | — | — | -0.709 | —
$N(1650)\;1/2^{-}$ | 1650 | 125 | 0.07 | 0.00 | 0.314 | —
$N(1675)\;5/2^{-}$ | 1675 | 145 | — | — | -0.013 | 0.022
$N(1710)\;1/2^{+}$ | 1710 | 140 | 0.15 | 0.01 | -0.940 | —
$N(1720)\;3/2^{+}$ | 1720 | 250 | 0.05 | 0.00 | -0.098 | -0.082
$N(1875)\;3/2^{-}$ | 1875 | 200 | 0.01 | 0.01 | -0.220 | -0.223
$N(1880)\;1/2^{+}$ | 1880 | 300 | 0.16 | 0.14 | -0.050 | —
$N(1895)\;1/2^{-}$ | 1895 | 120 | 0.18 | 0.13 | -0.063 | —
$N(1900)\;3/2^{+}$ | 1920 | 200 | 0.11 | 0.05 | -0.051 | -0.004
$N(2060)\;5/2^{-}$ | 2100 | 400 | 0.01 | 0.03 | -0.00001 | 0.003
$N(2120)\;3/2^{-}$ | 2120 | 300 | — | — | -0.034 | -0.010
$\Delta(1900)\;1/2^{-}$ | 1860 | 250 | — | 0.01 | 0.298 | —
$K^{*}(892)$ | 891.7 | 50.8 | — | — | 0.366 | 1.103
$K_{1}(1270)$ | 1270 | 90 | — | — | -1.448 | 0.473

Some additional $N^{*}$ resonances predicted to strongly couple the $K\Sigma$
channel were also investigated in the analysis. The variant with the smallest
$\chi^{2}/ndf$ and reasonable values of the parameters was selected. The
complete set of resonances from this best fit is provided in Table 2 666Note
that only the statistical uncertainties of the fit data were used in the
computation of the $\chi^{2}$, which results in a relatively large value
$\chi^{2}/ndf=2.39$ for the selected solution. This approach was chosen due to
missing systematic uncertainties in some data sets. When systematics are taken
into account, the $\chi^{2}$ value typically drops without changing the
quality of results.. The solution indicates contribution from two kaon
resonances, multiple nucleon resonances, one $\Delta$ resonance, and no
hyperon resonances 777The obtained couplings $g_{1}$ and $g_{2}$ from the fit
listed in Table 2 are all reasonable, as are the hadronic form factor cut-offs
$\Lambda_{\rm bgr}=0.874$ GeV and $\Lambda_{N}=1.451$ GeV (see Ref. Skoupil
and Bydžovský (2018) for a description of these parameters).. The combined
asymmetry and cross section data show a strong sensitivity to
$N(1720)\>3/2^{+}$, whose omission significantly diminishes both observables
(see right panels of Fig. 3). Sensitivity was also observed from the
$\Delta(1900)1/2^{-}$ resonance, specifically at central angles, as indicated
by the green dashed dotted lines in the right panels of Fig. 3. A significant
contribution of the $N(1895)\>1/2^{-}$ state was obtained, a state not
considered in previous photoproduction analyses, but with a relatively large
$K\Sigma$ branching ratio. The role of hyperon resonances appears small,
giving negligible effects on the predicted observables.

##  7  Summary

We present the first precise measurement of the beam-spin asymmetry
$\bm{\Sigma}$, employing a linearly polarised photon beam, for
$\vec{\gamma}n\rightarrow K^{+}\Sigma^{-}$ up to photon energies of
$E_{\gamma}=2.3$ GeV. The new data obtained using a deuterium target agree
well with previously published data from LEPS (limited only to forward
angles), while significantly extending the available kinematic coverage for
$\vec{\gamma}n\rightarrow K^{+}\Sigma^{-}$ down to photon energies of
$E_{\gamma}=1.1$ GeV and cover a large angular range. The new $\bm{\Sigma}$
data are an important addition to the world database and have a large effect
on the determined $\gamma$-$n$ couplings of resonances that have small
branching ratios to the $\pi N$ channels. The largest changes were found in
the $D_{13}$ and $P_{13}$ partial waves. A more detailed analysis in the Bonn-
Gatchina framework will be presented in a planned joint publication. The new
data were also fit using an isobar model based on an effective Lagrangian in a
tree-level approximation, with results indicating contributions from two kaon
resonances, multiple nucleon resonances (with significant contributions from
the $N(1720)3/2^{+}$ and $N(1895)1/2^{-}$), one $\Delta$ resonance, and no
hyperon resonances. Details on the isobar model will also be presented in a
longer planned joint publication.

##  8  Acknowledgments

This work has been supported by the U.K. Science and Technology Facilities
Council (ST/P004385/2, ST/T002077/1, and ST/L00478X/2) grants, as well as by
the Czech Science Foundation GACR grant 19-19640S. We also acknowledge the
outstanding efforts of the staff of the Accelerator and Physics Divisions at
Jefferson Lab that made this experiment possible. The Southeastern
Universities Research Association (SURA) operated the Thomas Jefferson
National Accelerator Facility for the United States Department of Energy under
contract DE-AC05-06OR23177. Further support was provided by the National
Science Foundation, the Italian Istituto Nazionale di Fisica Nucleare, the
Chilean Comisión Nacional de Investigación Científica y Tecnológica (CONICYT),
the French Centre National de la Recherche Scientifique, the French
Commissariat à l’Energie Atomique, and the National Research Foundation of
Korea.

## References

  * Capstick and Roberts (2000) S. Capstick and W. Roberts, Prog. Part. Nucl. Phys. 45, 241 (2000).
  * Capstick and Roberts (1998a) S. Capstick and W. Roberts, Phys. Rev. D 58, 074011 (1998a).
  * Capstick and Isgur (1986) S. Capstick and N. Isgur, Phys. Rev. D 34, 2809 (1986).
  * Löring  _et al._ (2001) U. Löring, B.Ch. Metsch, and H.R. Petry, Eur. Phys. J. A 10, 395 (2001).
  * Glozman  _et al._ (1998) L. Ya. Glozman, W. Plessas, K. Varga, and R. F. Wagenbrunn, Phys. Rev. D 58, 094030 (1998).
  * Giannini  _et al._ (2001) M. M. Giannini, E. Santopinto, and A. Vassallo, Eur. Phys. J. A 12, 447 (2001).
  * Edwards  _et al._ (2011) R. G. Edwards, J. J. Dudek, D. G. Richards, and S. J. Wallace, Phys. Rev. D 84, 074508 (2011).
  * Dudek and Edwards (2012) J. J. Dudek and R.G. Edwards, Phys. Rev. D 85, 054016 (2012).
  * Edwards  _et al._ (2013) R. G. Edwards, N. Mathur, D. G. Richards, and S. J. Wallace, Phys. Rev. D 87, 054506 (2013).
  * Anselmino  _et al._ (1993) M. Anselmino, E. Predazzi, S. Ekelin, S. Fredriksson, and D. B. Lichtenberg, Rev. Mod. Phys. 65, 1199 (1993).
  * Brodsky (2007) S. J. Brodsky, Eur. Phys. J. A 31, 638 (2007).
  * Kolomeitsev and Lutz (2004) E. E. Kolomeitsev and M. F. M. Lutz, Phys. Lett. B 585, 243 (2004).
  * Afonin (2007) S. S. Afonin, Int. J. Mod. Phys. A 22, 4537 (2007).
  * Chiang and Tabakin (1997) W-T. Chiang and F. Tabakin, Phys. Rev. C 55, 2054 (1997).
  * Workman  _et al._ (2017) R. L. Workman, L. Tiator, Y. Wunderlich, M. Döring, and H. Haberzettl, Phys. Rev. C 95, 015206 (2017).
  * Nys  _et al._ (2016) J. Nys, J. Ryckebusch, D. G. Ireland, and D. I. Glazier, Phys. Lett. B 759, 260 (2016).
  * Sandorfi  _et al._ (2011) A. M. Sandorfi, S. Hoblit, H. Kamano, and T. S. H. Lee, J. Phys. G 38, 053001 (2011).
  * Fasano  _et al._ (1992) C. G. Fasano, F. Tabakin, and B. Saghai, Phys. Rev. C 46, 2430 (1992).
  * Keaton and Workman (1996) G. Keaton and R. Workman, Phys. Rev. C 54, 1437 (1996).
  * Sandorfi and Hoblit (2013) A. M. Sandorfi and S. Hoblit, Nucl. Phys. A 914, 538 (2013).
  * Mart  _et al._ (1995) T. Mart, C. Bennhold, and C. E. Hyde-Wright, Phys. Rev. C 51, 1074(R) (1995).
  * Capstick and Roberts (1998b) S. Capstick and W. Roberts, Phys. Rev. D 57, 4301 (1998b).
  * Ireland  _et al._ (2020) D. G. Ireland, E. Pasyuk, and I. Strakovsky, Prog. Part. Nucl. Phys. 111, 103752 (2020).
  * Anisovich et al. (2017) A. V. Anisovich et al., Phys. Rev. Lett. 119, 062004 (2017).
  * Paterson et al. (2016) C. A. Paterson et al. (CLAS Collaboration), Phys. Rev. C 93, 065201 (2016).
  * Zyla et al. (2020) P.A. Zyla et al. (Particle Data Group), Progress of Theoretical and Experimental Physics 2020, 083C01 (2020).
  * Kohri et al. (2006) H. Kohri et al. (LEPS Collaboration), Phys. Rev. Lett. 97, 082003 (2006).
  * Pereira et al. (2010) S. A. Pereira et al. (CLAS Collaboration), Phys. Lett. B 688, 289 (2010).
  * Zachariou et al. (2020) N. Zachariou et al. (CLAS Collaboration), Phys. Lett. B 808, 135662 (2020).
  * Ho et al. (2018) D. Ho et al. (CLAS Collaboration), Phys. Rev. C 98, 045205 (2018).
  * Nadel-Turonski et al. (2006) P. Nadel-Turonski et al., Thomas Jefferson Lab, CLAS Approved Experiment E06-103 (2006), https://www.jlab.org/exp_prog/proposals/06/PR-06-103.pdf.
  * Mecking et al. (2003) B. A. Mecking et al., Nucl. Instrum. Meth. A 503, 513 (2003).
  * Sharabian et al. (2006) Y. G. Sharabian et al., Nucl. Instrum. Meth. A 556, 246 (2006).
  * Sober et al. (2000) D. I. Sober et al., Nucl. Instrum. Meth. A 440, 263 (2000).
  * Überall (1962) H. Überall, Zeitschrift für Naturforschung A 17, 332 (1962).
  * Barker and Donnachie (1975) I. S. Barker and A. Donnachie, Nucl. Phys. B 95, 347 (1975).
  * James and Winkler (2004) F. James and M. Winkler, “MINUIT User’s Guide,” (2004).
  * Livingston (2011) K. Livingston, Jefferson Laboratory, CLAS Note 2011-020 (2011), https://misportal.jlab.org/ul/Physics/Hall-B/clas/viewFile.cfm/2011-020.pdf?documentId=656.
  * Zachariou (2012) N. Zachariou, Ph.D. Thesis George Washington University (2012), http://www.jlab.org/Hall-B/general/thesis/Zachariou_thesis.pdf.
  * Munevar (2014) E. Munevar, Ph.D. Thesis George Washington University (2014), https://www.jlab.org/Hall-B/general/thesis/Munevar_thesis.pdf.
  * Müller et al. (2020) J. Müller et al., Phys. Lett. B 803, 135323 (2020).
  * Skoupil and Bydžovský (2016) D. Skoupil and P. Bydžovský, Phys. Rev. C 93, 025204 (2016).
  * Skoupil and Bydžovský (2018) D. Skoupil and P. Bydžovský, Phys. Rev. C 97, 025202 (2018).
  * Bydžovský  _et al._ (2021) P. Bydžovský, A. Cieplý, D. Petrellis, and D. Skoupil, prepared for publication (2021).
  * David  _et al._ (1996) J. C. David, C. Fayard, G. H. Lamot, and B. Saghai, Phys. Rev. C 53, 2613 (1996).