# Nucleon and $\Delta$ resonances in $\gamma n\to K^{+}\Sigma^{-}$
photoproduction

Neng-Chang Wei  School of Nuclear Science and Technology, University of
Chinese Academy of Sciences, Beijing 101408, China  Ai-Chao Wang  School of
Nuclear Science and Technology, University of Chinese Academy of Sciences,
Beijing 101408, China  Fei Huang  Corresponding author. Email:
<EMAIL_ADDRESS>School of Nuclear Science and Technology, University of
Chinese Academy of Sciences, Beijing 101408, China  Kanzo Nakayama  Department
of Physics and Astronomy, University of Georgia, Athens, Georgia 30602, USA

###### Abstract

The most recent data on beam asymmetries, $\Sigma$, and beam-target
asymmetries, $E$, from the CLAS Collaboration, together with the previous data
on differential cross sections and beam asymmetries from the CLAS and LEPS
Collaborations, for the $\gamma n\to K^{+}\Sigma^{-}$ reaction are studied
based on an effective Lagrangian approach in the tree-level Born
approximation. The $t$-channel $K$ and $K^{\ast}(892)$ exchanges, the
$u$-channel $\Sigma$ exchange, the interaction current, and the exchanges of
$N$, $\Delta$, and their excited states in the $s$ channel are considered in
constructing the reaction amplitudes to describe the available experimental
data. The reaction mechanisms of $\gamma n\to K^{+}\Sigma^{-}$ are analyzed,
and the associated resonances’ parameters are extracted. The numerical results
show that the $\Delta$ exchange and the $N(1710)1/2^{+}$, $N(1880)1/2^{+}$,
$N(1900)3/2^{+}$, and $\Delta(1920)3/2^{+}$ resonance exchanges in the
$s$-channel dominate the $\gamma n\to K^{+}\Sigma^{-}$ reaction in the lower
energy region, and the $t$-channel $K^{\ast}(892)$ exchange plays a crucial
role at forward angles in the higher energy region.

$K^{+}\Sigma^{-})$ photoproduction, effective Lagrangian approach, photo-beam
asymmetries, beam-target asymmetries

###### pacs:

25.20.Lj, 13.60.Le, 14.20.Gk, 13.75.Jz

##  I Introduction

The study of the properties of excited baryons such as their masses, widths,
and transition form factors provides essential information in help understand
the nonperturbative regime of quantum chromodynamics (QCD) and, in particular,
the internal structure of baryons. Today, however, it is still a challenge to
match the observed baryon spectrum with predictions from QCD-inspired
phenomenological models Isgur:1977ef ; Koniuk:1979vy  and/or lattice QCD
calculations Edwards:2013 ; Lang:2017 ; Kiratidis:2017 ; Andersen:2018 ,
since, theoretically, more resonances are predicted than what have been found
in experiments as listed in the most recent Review of Particle Physics (RPP)
Zyla:2020zbs . This is known as the missing resonance problem in baryon
spectroscopy.

Most of our earlier knowledge of baryon excitations have been gained from $\pi
N$ scattering experiments or single-pion photoproduction experiments. As such,
those “missing resonances” might have been escaped from detections due to
their weak couplings to the $\pi N$ channel. In this regard, concerted efforts
are being made at electron accelerator facilities worldwide – such as the
Jlab, MAMI, ELSA, and Spring8 – to measure the production reactions of mesons
other than the pion. In this respect, in the past few years, the $KY$
($Y=\Sigma,\Lambda$) photoproduction reaction has been receiving increasing
attention, both experimentally and theoretically.

Experimentally, as has been reviewed in Refs. Mart:2019fau ; Ireland:2019uwn ,
a large number of data on differential cross sections and polarization
observables have been measured in the past 20 years for the four elementary
$K\Sigma$ photoproduction reactions, namely, $\gamma p\to K^{0}\Sigma^{+}$,
$\gamma p\to K^{+}\Sigma^{0}$, $\gamma n\to K^{0}\Sigma^{0}$, and $\gamma n\to
K^{+}\Sigma^{-}$ Glander:2003jw ; McNabb:2003nf ; Zegers:2003ux ;
Sumihama:2005er ; Kohri:2006yx ; Bradford:2005pt ; Lleres:2007tx ;
Bradford:2006ba ; Dey:2010hh ; Jude:2013jzs ; Paterson:2016vmc ; Lawall:2005np
; Castelijns:2007qt ; Aguar-Bartolome:2013fqw ; Akondi:2018shh ;
AnefalosPereira:2009zw . A large body of these data have been measured in the
proton channel, while the measurements of the neutron channel are
comparatively scarce. Measurements of $KY$ photoproduction off neutron can
offer additional constraints to learn more about the underlying reaction
dynamics and, in particular, to determine the $N^{\ast}$ and $\Delta$
resonances that are excited in this reaction. Also, there are studies of Kaon
photoproduction reactions involving excited hyperon in the final states, both
experimentally Kohri:2009xe ; Wieland:2010cq ; Moriya:2013hwg ; CLAS:2021osv
and theoretically Oh:2007jd ; Wang:2014jxb ; Wang:2016dtb ; He:2013ksa ;
Wang:2020mdn ; Wei:2021qnc ; Zhang:2021iez ; Kim:2017nxg ; Yu:2017kng .

Very recently, the CLAS Collaboration has just released the data on beam
asymmetries $\Sigma$ CLAS-beam  and beam-target asymmetries $E$
Zachariou:2020kkb  for the $\gamma n\to K^{+}\Sigma^{-}$ reaction. Their
measurements of $E$ are the world’s first measurement of this observable for
$\gamma n\to K^{+}\Sigma^{-}$, and their measurements of $\Sigma$, compared
with the very limited previous data from the LEPS Collaboration Kohri:2006yx ,
cover broader ranges of kaon emission angle and photon energy. These new data
on $\Sigma$ and $E$ are expected to help provide further constraints in the
determination of the reaction dynamics in the $\gamma n\to K^{+}\Sigma^{-}$
reaction.

Theoretically, several works Vancraeyveld:2009qt ; Mart:2019fau ;
Clymton:2021wof ; Luthfiyah:2021yqe  have already been devoted to analyze the
data for $K^{+}\Sigma^{-}$ photoproduction published before 2010 in Refs.
Kohri:2006yx ; AnefalosPereira:2009zw . In Ref. Vancraeyveld:2009qt , a Regge-
plus-resonance model was adapted to analyze the LEPS data Kohri:2006yx  with
amplitudes being constructed by $K$ and $K^{*}$ Regge-trajectory exchanges
supplemented with a selection of $s$-channel resonance diagrams. In a series
works of Refs. Mart:2019fau ; Clymton:2021wof ; Luthfiyah:2021yqe , the data
from Refs. Kohri:2006yx ; AnefalosPereira:2009zw  for $K^{+}\Sigma^{-}$
photoproduction were analyzed together with the data for $K^{0}\Sigma^{0}$,
$K^{+}\Sigma^{0}$, $K^{0}\Sigma^{+}$ photoproduction reactions. In Ref.
Mart:2019fau , an isobar model was constructed from the Feynman diagrammatic
approach for the background terms and multipoles formulation for the resonance
terms to analyze the data, and all the resonances rated with two or more stars
in RPP with spins ranging from 1/2 to 13/2 were included in the fit. In Ref.
Clymton:2021wof , an updated isobar model with resonance terms also being
constructed from Feynman diagrammatic approach was developed for $K\Sigma$
photoproduction. In Ref. Luthfiyah:2021yqe , the effects of the nucleon and
$\Delta$ resonances with spin 11/2, 13/2, and 15/2 in $K\Lambda$ and $K\Sigma$
photoproduction reactions were investigated.

Stimulated by the recently released data on $\Sigma$ CLAS-beam  and $E$
Zachariou:2020kkb , in the present work we perform an analysis of all the
available data for $\gamma n\to K^{+}\Sigma^{-}$ Kohri:2006yx ;
AnefalosPereira:2009zw ; CLAS-beam ; Zachariou:2020kkb  based on an effective
Lagrangian approach in the Bonn approximation. We consider the $t$-channel $K$
and $K^{\ast}(892)$ exchanges, the $u$-channel $\Sigma$ exchange, the
interaction current, and the $s$-channel $N$, $\Delta$, and their excited
states ($N^{\ast}$s and $\Delta^{\ast}$s) exchanges in constructing the
reaction amplitude. It is shown that all the available data for $\gamma n\to
K^{+}\Sigma^{-}$ can be well reproduced in our model. The reaction mechanisms
of $\gamma n\to K^{+}\Sigma^{-}$ are analyzed and the parameters of the
relevant $N$ and $\Delta$ resonances are extracted and compared with those
quoted by RPP Zyla:2020zbs .

We mention that, in principle, a coupled-channels approach that properly
accounts for the unitarity and analyticity of the reaction amplitude should be
employed to analyze the data. This is beyond the scope of the present work,
which may be considered as a first step toward performing such a more complete
model calculation.

The paper is organized as follows. In Sec. II, we introduce the framework of
our theoretical model by explicitly giving the effective interaction
Lagrangians, the resonance propagators, and the phenomenological form factors
employed in this work. In Sec. III, our numerical results are shown and
discussed. Finally, a brief summary and conclusion of the present work is
given in Sec. IV.

##  II Formalism

(a) $s$ channel (b) $t$ channel

(c) $u$ channel (d) Interaction current

Figure 1: Generic structure of the amplitude for $\gamma N\to K\Sigma$. Time
proceeds from left to right.

In the present work, we analyze the available data for $\gamma n\to
K^{+}\Sigma^{-}$ within a tree-level effective Lagrangian approach. As
diagrammatically depicted in Fig. 1, the following diagrams are taken into
account in constructing the reaction amplitude: (a) the $s$-channel $N$,
$N^{\ast}$, $\Delta$, and $\Delta^{\ast}$ exchanges, (b) the $t$-channel $K$
and $K^{\ast}(892)$ exchanges, (c) the $u$-channel $\Sigma$ exchange, and (d)
the interaction current. We mention that the $u$-channel $\Sigma^{*}$ exchange
and $t$-channel $K_{1}$ exchange are not considered in the present work as we
don’t have experimental information on their radiative decays to constrain the
corresponding coupling constants, and moreover, as can be seen in Fig. 2, the
cross-section data for the considered reaction at high-energy backward angles
and forward angles where the $\Sigma^{*}$ and $K_{1}$ exchanges contribute
dominantly are scarce to fix the strengths of the exchanges of these two
hadrons. The full photoproduction amplitude of $\gamma n\to K^{+}\Sigma^{-}$
can be written as Haberzettl:1997 ; Haberzettl:2006bn ; Huang:2012 ;
Huang:2012xj

$M^{\mu}\equiv M^{\mu}_{s}+M^{\mu}_{t}+M^{\mu}_{u}+M^{\mu}_{\rm int},$ (1)

with $\mu$ denoting the Lorentz index of the incoming photon $\gamma$. The
first three terms in Eq. (1) represent the amplitudes arising from the $s$-,
$t$-, and $u$-channel diagrams, respectively, and they can be
straightforwardly calculated from the effective Lagrangians, propagators and
form factors which will be given in detail in the following part of this
section. The last term in Eq. (1), $M^{\mu}_{\rm int}$, denotes the
interaction current which contains contributions from other diagrams that do
not have $s$-, $t$-, or $u$-channel poles. An exact calculation of
$M^{\mu}_{\rm int}$ is impractical as it includes very complicated non-linear
interaction dynamics. In practice, we follow Refs. Haberzettl:2006bn ;
Huang:2012xj  to model the interaction current $M^{\mu}_{\rm int}$ by a
generalized contact current

$M^{\mu}_{\rm int}=\Gamma_{\Sigma NK}(q)C^{\mu}+M^{\mu}_{\rm KR}f_{t},$ (2)

with $\Gamma_{\Sigma NK}(q)$ being the vertex function of $\Sigma NK$
interaction obtained from the Lagrangian of Eq. (10),

$\Gamma_{\Sigma NK}(q)=g_{\Sigma
NK}\gamma_{5}\left(\lambda+\frac{1-\lambda}{2M_{N}}{{q\\!\\!\\!/}}\right).$
(3)

Here $q$ is the four-momentum of the outgoing $K$ meson, and $\lambda$ is the
mixing parameter for pseudoscalar ($\lambda=1$) and pseudovector ($\lambda=0$)
$\Sigma NK$ couplings. $M^{\mu}_{\rm KR}$ in Eq. (2) is the Kroll-Ruderman
term obtained from the Lagrangian of Eq. (17),

$M^{\mu}_{KR}=-g_{\Sigma NK}\frac{1-\lambda}{2M_{N}}\gamma_{5}\gamma^{\mu}\tau
Q_{K},$ (4)

with $\tau$ denoting the isospin factor and $Q_{K}$ representing the electric
charge of $K$ meson. $f_{t}$ in Eq. (2) is the phenomenological form factor
attaching to the $\Sigma NK$ vertex in $t$-channel $K$ exchange amplitude, and
its form will be given in Eq. (36). $C^{\mu}$ in Eq. (2) is an auxiliary
current introduced to ensure that the full photoproduction amplitude given in
Eq. (1) satisfies the generalized Ward-Takahashi identity and thus is fully
gauge invariant. Following Refs. Haberzettl:1997 ; Haberzettl:2006bn ;
Huang:2012 ; Huang:2012xj , for the $\gamma n\to K^{+}\Sigma^{-}$ reaction, we
choose the following prescription for $C^{\mu}$:

$C^{\mu}=-Q_{K}\tau\frac{f_{t}-\hat{F}}{t-q^{2}}(2q-k)^{\mu}-Q_{\Sigma}\tau\frac{f_{u}-\hat{F}}{u-p^{\prime
2}}(2p^{\prime}-k)^{\mu},$ (5)

with

$\hat{F}=1-\hat{h}\left(1-f_{t}\right)\left(1-f_{u}\right),$ (6)

where $k$ and $p^{\prime}$ are the four-momenta for the incoming photon
$\gamma$ and outgoing $\Sigma$, respectively; $Q_{\Sigma}$ is the electric
charge of $\Sigma$; $f_{u}$ is the phenomenological form factor attaching to
the $\Sigma NK$ vertex in $u$-channel $\Sigma$ exchange amplitude and will be
given in Eq. (37); $\hat{h}=1$ is used for simplicity as usual Wang:2017tpe ;
Wang:2018vlv ; Wei:2019 .

###  II.1 Effective Lagrangians

In this section, we present the explicit expressions of the Lagrangians
employed in the present work. For the sake of brevity, we define the following
operators

$\Gamma^{(+)}=\gamma_{5}\qquad{\rm and}\qquad\Gamma^{(-)}=1,$ (7)

and the field-strength tensor of the photon field $A^{\mu}$

$F^{\mu\nu}=\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu}.$ (8)

The following Lagrangians are used for non-resonant production amplitudes:

$\displaystyle{\cal L}_{\gamma KK}$ $\displaystyle=$ $\displaystyle
ie\\!\left[K^{+}\left(\partial_{\mu}K^{-}\right)-K^{-}\left(\partial_{\mu}K^{+}\right)\right]\\!A^{\mu},$
(9) $\displaystyle{\cal L}_{\Sigma NK}$ $\displaystyle=$
$\displaystyle-\,g_{\Sigma
NK}\bar{\Sigma}\gamma_{5}\left[\left(i\lambda+\frac{1-\lambda}{2M_{N}}{\partial\\!\\!\\!/}\right)K\right]N$
(10) $\displaystyle+\,\text{H.\,c.},$ $\displaystyle{\cal L}_{\gamma
K{K^{\ast}}}$ $\displaystyle=$ $\displaystyle e\frac{g_{\gamma
K{K^{\ast}}}}{M_{K}}\varepsilon^{\alpha\mu\lambda\nu}\left(\partial_{\alpha}A_{\mu}\right)\left(\partial_{\lambda}K\right)K^{\ast}_{\nu},$
(11) $\displaystyle{\cal L}_{\Sigma N{K^{\ast}}}$ $\displaystyle=$
$\displaystyle-\,g_{\Sigma
N{K^{\ast}}}\bar{\Sigma}\left[\left(\gamma^{\mu}-\frac{\kappa_{\Sigma
N{K^{\ast}}}}{2M_{N}}\sigma^{\mu\nu}\partial_{\nu}\right)K^{\ast}_{\mu}\right]N$
(12) $\displaystyle+\,\text{H.\,c.},$ $\displaystyle{\cal
L}_{\Sigma\Sigma\gamma}$ $\displaystyle=$
$\displaystyle-\,e\bar{\Sigma}\left[\left(\hat{e}\gamma^{\mu}-\frac{\hat{\kappa}_{\Sigma}}{2M_{N}}\sigma^{\mu\nu}\partial_{\nu}\right)A_{\mu}\right]\Sigma,$
(13) $\displaystyle{\cal L}_{\gamma NN}$ $\displaystyle=$
$\displaystyle-\,e\bar{N}\\!\left[\\!\left(\hat{e}\gamma^{\mu}-\frac{\hat{\kappa}_{N}}{2M_{N}}\sigma^{\mu\nu}\partial_{\nu}\\!\right)\\!A_{\mu}\right]\\!N,$
(14) $\displaystyle{\cal L}_{\Delta\Sigma K}$ $\displaystyle=$
$\displaystyle\frac{g_{\Delta\Sigma
K}}{M_{K}}\bar{\Sigma}\left(\partial^{\mu}K\right)\Delta_{\mu}+\text{H.\,c.},$
(15) $\displaystyle{\cal L}_{\Delta N\gamma}$ $\displaystyle=$ $\displaystyle-
ie\frac{g^{(1)}_{\Delta
N\gamma}}{2M_{N}}\bar{\Delta}_{\mu}\gamma_{\nu}\gamma_{5}F^{\mu\nu}N$ (16)
$\displaystyle+\,e\frac{g^{(2)}_{\Delta
N\gamma}}{(2M_{N})^{2}}\bar{\Delta}_{\mu}\gamma_{5}F^{\mu\nu}\partial_{\nu}N+\text{H.\,c.},$
$\displaystyle L_{\Sigma NK\gamma}$ $\displaystyle=$ $\displaystyle ig_{\Sigma
NK}\frac{1-\lambda}{2M_{N}}\bar{\Sigma}\gamma_{5}\gamma^{\mu}A_{\mu}\hat{Q}_{K}K\tau
N.$ (17)

Here, $e$ stands for the elementary charge unit and $\hat{e}$ stands for the
charge operator acting on the $N$ field or the $\Sigma$ field. $\hat{Q}_{K}$
is the charge operator acting on the $K$ field.
$\hat{\kappa}_{N}\equiv\kappa_{p}\hat{e}+\kappa_{n}(1-\hat{e})$ with
$\kappa_{p}=1.793$ being the anomalous magnetic moment of proton and
$\kappa_{n}=-1.913$ the anomalous magnetic moment of neutron.
$\hat{\kappa}_{\Sigma}\equiv\kappa_{\Sigma^{+}}(1+\hat{e})/2+\kappa_{\Sigma^{-}}(1-\hat{e})/2$
with $\kappa_{\Sigma^{+}}=1.458$ being the anomalous magnetic moment of
$\Sigma^{+}$ and $\kappa_{\Sigma^{-}}=-0.16$ the anomalous magnetic moment of
$\Sigma^{-}$. $M_{K}$ and $M_{N}$ denote the masses of $K$ and $N$,
respectively. $\varepsilon^{\alpha\mu\lambda\nu}$ is the totally antisymmetric
Levi-Civita tensor with $\varepsilon^{0123}=+1$. In Eq. (10) $\lambda$ is the
pseudoscalar-pseudovector mixing parameter for $\Sigma NK$ coupling and we use
the pure pseudovector coupling, namely, $\lambda=0$ in our calculations. The
coupling constant $g_{\gamma K^{\pm}K^{\ast\pm}}=0.413$ is obtained by using
the radiative decay width of $K^{\ast}\to K\gamma$ reported by RPP
Zyla:2020zbs  with the sign determined by the flavor SU(3) symmetry
Garcilazo:1993av . The coupling constants $g^{(1)}_{\Delta N\gamma}=-4.18$ and
$g^{(2)}_{\Delta N\gamma}=4.327$ are determined by the helicity amplitudes of
$\Delta\to N\gamma$ from RPP Zyla:2020zbs . The strong interaction coupling
parameters are obtained via the flavor SU(3) symmetry Wang:2017tpe ;
Wang:2018vlv ; Swart:1963 ; Ronchen:2013 :

$\displaystyle g_{\Sigma NK}$ $\displaystyle=$
$\displaystyle\frac{1}{5}g_{NN\pi}=2.692,$ (18) $\displaystyle g_{\Sigma
NK^{\ast}}$ $\displaystyle=$
$\displaystyle\dfrac{1}{2}g_{NN\rho}-\dfrac{1}{2}g_{NN\omega}=-4.26,$ (19)
$\displaystyle\kappa_{\Sigma N{K^{\ast}}}$ $\displaystyle=$
$\displaystyle\dfrac{f_{\Sigma NK^{\ast}}}{g_{\Sigma
NK^{\ast}}}=\dfrac{f_{NN\rho}-f_{NN\omega}}{g_{NN\rho}-g_{NN\omega}}=-2.33,$
(20) $\displaystyle g_{\Delta\Sigma K}$ $\displaystyle=$
$\displaystyle\frac{M_{K}}{M_{\pi}}g_{\Delta N\pi}=7.89,$ (21)

where the empirical values $g_{NN\pi}=13.46$, $g_{NN\rho}=3.25$,
$g_{NN\omega}=11.76$, $\kappa_{NN\rho}=f_{NN\rho}/g_{NN\rho}=6.10$,
$f_{NN\omega}=0$, and $g_{\Delta N\pi}=2.23$ are used. We mention that the
SU(3) relation is not “exact” but a rough approximation. Its applicability
needs to be tested by the capability of the model to describe the data.

For the $s$-channel $N^{\ast}$ and $\Delta^{\ast}$ exchanges, we use the
following Lagrangians for the electromagnetic interactions

$\displaystyle{\cal L}_{RN\gamma}^{1/2\pm}$ $\displaystyle=$ $\displaystyle
e\frac{g_{RN\gamma}^{(1)}}{2M_{N}}\bar{R}\Gamma^{(\mp)}\sigma_{\mu\nu}\left(\partial^{\nu}A^{\mu}\right)N+\text{H.\,c.},$
(22) $\displaystyle{\cal L}_{RN\gamma}^{3/2\pm}$ $\displaystyle=$
$\displaystyle-\,ie\frac{g_{RN\gamma}^{(1)}}{2M_{N}}\bar{R}_{\mu}\gamma_{\nu}\Gamma^{(\pm)}F^{\mu\nu}N$
(23)
$\displaystyle+\,e\frac{g_{RN\gamma}^{(2)}}{\left(2M_{N}\right)^{2}}\bar{R}_{\mu}\Gamma^{(\pm)}F^{\mu\nu}\partial_{\nu}N+\text{H.\,c.},$
$\displaystyle{\cal L}_{RN\gamma}^{5/2\pm}$ $\displaystyle=$ $\displaystyle
e\frac{g_{RN\gamma}^{(1)}}{\left(2M_{N}\right)^{2}}\bar{R}_{\mu\alpha}\gamma_{\nu}\Gamma^{(\mp)}\left(\partial^{\alpha}F^{\mu\nu}\right)N$
(24)
$\displaystyle\pm\,ie\frac{g_{RN\gamma}^{(2)}}{\left(2M_{N}\right)^{3}}\bar{R}_{\mu\alpha}\Gamma^{(\mp)}\left(\partial^{\alpha}F^{\mu\nu}\right)\partial_{\nu}N$
$\displaystyle+\,\text{H.\,c.},$ $\displaystyle{\cal L}_{RN\gamma}^{7/2\pm}$
$\displaystyle=$ $\displaystyle
ie\frac{g_{RN\gamma}^{(1)}}{\left(2M_{N}\right)^{3}}\bar{R}_{\mu\alpha\beta}\gamma_{\nu}\Gamma^{(\pm)}\left(\partial^{\alpha}\partial^{\beta}F^{\mu\nu}\right)N$
(25)
$\displaystyle-\,e\frac{g_{RN\gamma}^{(2)}}{\left(2M_{N}\right)^{4}}\bar{R}_{\mu\alpha\beta}\Gamma^{(\pm)}\left(\partial^{\alpha}\partial^{\beta}F^{\mu\nu}\right)\partial_{\nu}N$
$\displaystyle+\,\text{H.\,c.},$

and use the following Lagrangians for the hadronic interactions

$\displaystyle{\cal L}_{R\Sigma K}^{1/2\pm}$ $\displaystyle=$ $\displaystyle-
g_{R\Sigma
K}\bar{N}\Gamma^{(\pm)}\left[\left(i\lambda+\frac{1-\lambda}{M_{R}\pm
M_{\Sigma}}{{\partial\\!\\!\\!/}}\right)K\right]R$ (26)
$\displaystyle+\,\text{H.\,c.},$ $\displaystyle{\cal L}_{R\Sigma K}^{3/2\pm}$
$\displaystyle=$ $\displaystyle\pm\frac{g_{R\Sigma
K}}{M_{K}}\bar{N}\Gamma^{(\mp)}\left(\partial^{\alpha}K\right)R_{\alpha}+\text{H.\,c.},$
(27) $\displaystyle{\cal L}_{R\Sigma K}^{5/2\pm}$ $\displaystyle=$
$\displaystyle i\frac{g_{R\Sigma
K}}{M_{K}^{2}}\bar{N}\Gamma^{(\pm)}\left(\partial^{\alpha}\partial^{\beta}K\right)R_{\alpha\beta}+\text{H.\,c.},$
(28) $\displaystyle{\cal L}_{R\Sigma K}^{7/2\pm}$ $\displaystyle=$
$\displaystyle\mp\frac{g_{R\Sigma
K}}{M_{K}^{3}}\bar{N}\Gamma^{(\mp)}\left(\partial^{\alpha}\partial^{\beta}\partial^{\gamma}K\right)R_{\alpha\beta\gamma}$
(29) $\displaystyle+\,\text{H.\,c.},$

where $R$ denotes the $N^{\ast}$ or $\Delta^{\ast}$ resonance and the
superscript of ${\cal L}$ denotes the spin and parity of the resonance $R$.
The pseudoscalar-pseudovector mixing parameter $\lambda=0$ is employed in Eq.
(26). When the resonance helicity amplitudes of $R\to N\gamma$ are available
in RPP Zyla:2020zbs , they will be used to fix the resonance electromagnetic
couplings. Otherwise, the products of the hadronic coupling constant
$g_{R\Sigma K}$ and the electromagnetic coupling constants
$g_{RN\gamma}^{(i)}$ ($i=1,2$), which are relevant to the photoproduction
amplitudes in the present work, will be determined by a fit to the data for
$\gamma n\to K^{+}\Sigma^{-}$.

###  II.2 Resonance propagators

The following prescriptions for propagators of the resonances with spin $1/2$,
$3/2$, $5/2$, and $7/2$ are employed in the present work Wang:2017tpe :

$\displaystyle S_{1/2}(p)$ $\displaystyle=$
$\displaystyle\frac{i}{{p\\!\\!\\!/}-M_{R}+i\Gamma_{R}/2},$ (30)
$\displaystyle S_{3/2}(p)$ $\displaystyle=$
$\displaystyle\frac{i}{{p\\!\\!\\!/}-M_{R}+i\Gamma_{R}/2}\left(\tilde{g}_{\mu\nu}+\frac{1}{3}\tilde{\gamma}_{\mu}\tilde{\gamma}_{\nu}\right),$
(31) $\displaystyle S_{5/2}(p)$ $\displaystyle=$
$\displaystyle\frac{i}{{p\\!\\!\\!/}-M_{R}+i\Gamma_{R}/2}\,\bigg{[}\,\frac{1}{2}\big{(}\tilde{g}_{\mu\alpha}\tilde{g}_{\nu\beta}+\tilde{g}_{\mu\beta}\tilde{g}_{\nu\alpha}\big{)}$
(32)
$\displaystyle-\,\frac{1}{5}\tilde{g}_{\mu\nu}\tilde{g}_{\alpha\beta}+\frac{1}{10}\big{(}\tilde{g}_{\mu\alpha}\tilde{\gamma}_{\nu}\tilde{\gamma}_{\beta}+\tilde{g}_{\mu\beta}\tilde{\gamma}_{\nu}\tilde{\gamma}_{\alpha}$
$\displaystyle+\,\tilde{g}_{\nu\alpha}\tilde{\gamma}_{\mu}\tilde{\gamma}_{\beta}+\tilde{g}_{\nu\beta}\tilde{\gamma}_{\mu}\tilde{\gamma}_{\alpha}\big{)}\bigg{]},$
$\displaystyle S_{7/2}(p)$ $\displaystyle=$
$\displaystyle\frac{i}{{p\\!\\!\\!/}-M_{R}+i\Gamma_{R}/2}\,\frac{1}{36}\sum_{P_{\mu}P_{\nu}}\bigg{(}\tilde{g}_{\mu_{1}\nu_{1}}\tilde{g}_{\mu_{2}\nu_{2}}\tilde{g}_{\mu_{3}\nu_{3}}$
(33)
$\displaystyle-\,\frac{3}{7}\tilde{g}_{\mu_{1}\mu_{2}}\tilde{g}_{\nu_{1}\nu_{2}}\tilde{g}_{\mu_{3}\nu_{3}}+\frac{3}{7}\tilde{\gamma}_{\mu_{1}}\tilde{\gamma}_{\nu_{1}}\tilde{g}_{\mu_{2}\nu_{2}}\tilde{g}_{\mu_{3}\nu_{3}}$
$\displaystyle-\,\frac{3}{35}\tilde{\gamma}_{\mu_{1}}\tilde{\gamma}_{\nu_{1}}\tilde{g}_{\mu_{2}\mu_{3}}\tilde{g}_{\nu_{2}\nu_{3}}\bigg{)},$

with $\tilde{g}_{\mu\nu}$ and $\tilde{\gamma}_{\mu}$ being defined as

$\displaystyle\tilde{g}_{\mu\nu}$ $\displaystyle=$
$\displaystyle-\,g_{\mu\nu}+\frac{p_{\mu}p_{\nu}}{M_{R}^{2}},$ (34)
$\displaystyle\tilde{\gamma}_{\mu}$ $\displaystyle=$
$\displaystyle-\,\gamma_{\mu}+\frac{p_{\mu}{p\\!\\!\\!/}}{M_{R}^{2}}.$ (35)

Here $M_{R}$ and $\Gamma_{R}$ are the mass and width for the resonance $R$
with four-momentum $p$, respectively. The summation over $P_{\mu}$
$\left(P_{\nu}\right)$ in Eq. (33) goes over all the $3!=6$ possible
permutations of the indices $\mu_{1}\mu_{2}\mu_{3}$
$\left(\nu_{1}\nu_{2}\nu_{3}\right)$.

###  II.3 Form factors

In the present work, each hadronic vertex obtained from the Lagrangians given
in Sec. II.1 is accompanied with a phenomenological form factor. For
$t$-channel meson exchanges, the following form factor is employed
Huang:2012xj ; Wang:2017tpe ; Wang:2018vlv :

$\displaystyle
f_{M}(q^{2}_{M})=\left(\frac{\Lambda_{M}^{2}-M_{M}^{2}}{\Lambda_{M}^{2}-q^{2}_{M}}\right)^{2},$
(36)

where $M_{M}$ and $q_{M}$ denote the mass and four-momentum of the
intermediate meson, respectively, and $\Lambda_{M}$ is the cut off parameter
for meson exchange diagrams. For $s$-channel and $u$-channel baryon exchanges,
we use the following form factor Huang:2012xj ; Wang:2017tpe ; Wang:2018vlv :

$\displaystyle
f_{B}(p^{2}_{x})=\left(\frac{\Lambda_{B}^{4}}{\Lambda_{B}^{4}+\left(p_{x}^{2}-M_{B}^{2}\right)^{2}}\right)^{2},$
(37)

where $M_{B}$ and $p_{x}$ ($x=s,u$) represent the mass and four-momentum of
the exchanged baryon in $s$ or $u$ channel, respectively, and $\Lambda_{B}$ is
the cut off parameter for baryon exchange diagrams.

##  III Results and discussion

Table 1: Fitted values of resonant parameters and the extracted resonance decay branching ratios. The asterisks below the resonance names denote the overall rating of each resonance evaluated by RPP Zyla:2020zbs . The values in the brackets blow resonances’ masses, widths, and branching ratios are corresponding values advocated by RPP Zyla:2020zbs .  |  $M_{R}$ [MeV] |  $\Gamma_{R}$ [MeV] | $g^{(1)}_{RN\gamma}g_{R\Sigma K}$ | $g^{(2)}_{RN\gamma}g_{R\Sigma K}$ |  $\beta_{N\gamma}$ [%] |  $\beta_{K\Sigma}$ [%]
---|---|---|---|---|---|---
$N(1710)1/2^{+}$ | $1692\pm 1$ | $161\pm 1$ | $1.661\pm 0.012$ |  | $\bm{0.01}$ | $0.008$
$\ast\\!\ast\\!\ast\ast$ | $[1680\sim 1740]$ | $[80\sim 200]$ |  |  | $[0.0\sim 0.02]$ | [seen]
$N(1880)1/2^{+}$ | $1893\pm 1$ | $400\pm 2$ | $1.252\pm 0.001$ |  | $\bm{0.05}$ | $18.531$
$\ast\\!\ast\\!\ast$ | $[1830\sim 1930]$ | $[200\sim 400]$ |  |  | $[0.002\sim 0.63]$ | $[10\sim 24]$
$N(1895)1/2^{-}$ | $1870\pm 1$ | $200\pm 1$ | $0.054\pm 0.001$ |  | $\bm{0.027}$ | $8.795$
$\ast\\!\ast\\!\ast\ast$ | $[1870\sim 1920]$ | $[80\sim 200]$ |  |  | $[0.003\sim 0.05]$ | $[6\sim 20]$
$N(1900)3/2^{+}$ | $1942\pm 1$ | $122\pm 2$ | $-0.116\pm 0.001$ | $0.024\pm 0.004$ | $\bm{0.04}$ | $6.999$
$\ast\\!\ast\\!\ast\ast$ | $[1890\sim 1950]$ | $[100\sim 320]$ |  |  | $[\textless 0.04]$ | $[3\sim 7]$
$N(2060)5/2^{-}$ | $2030\pm 2$ | $450\pm 2$ | $-0.648\pm 0.023$ | $0.254\pm 0.034$ | $\bm{0.036}$ | $1.024$
$\ast\\!\ast\\!\ast$ | $[2030\sim 2200]$ | $[300\sim 450]$ |  |  | $[0.003\sim 0.07]$ | $[1\sim 5]$
$\Delta(1910)1/2^{+}$ | $1913\pm 2$ | $200\pm 1$ | $0.541\pm 0.006$ |  | $\bm{0.01}$ | $13.980$
$\ast\\!\ast\\!\ast\ast$ | $[1850\sim 1950]$ | $[200\sim 400]$ |  |  | $[0.0\sim 0.02]$ | $[4\sim 14]$
$\Delta(1920)3/2^{+}$ | $1870\pm 1$ | $360\pm 1$ | $-1.204\pm 0.003$ | $-0.103\pm 0.034$ | $\bm{0.2}$ | $3.019$
$\ast\\!\ast\\!\ast$ | $[1870\sim 1970]$ | $[240\sim 360]$ |  |  |  | $[2\sim 6]$
$\Delta(1950)7/2^{+}$ | $1915\pm 1$ | $235\pm 5$ | $7.416\pm 0.220$ | $-7.261\pm 0.215$ | $\bm{0.001}$ | $0.300$
$\ast\\!\ast\\!\ast\ast$ | $[1915\sim 1950]$ | $[235\sim 335]$ |  |  |  | $[0.3\sim 0.5]$
Table 2: Fitted values of cutoff parameters (in MeV). $\Lambda_{N(1710)}$ | $\Lambda_{N(1880)}$ | $\Lambda_{N(1895)}$ | $\Lambda_{N(1900)}$ | $\Lambda_{N(2060)}$ | $\Lambda_{\Delta(1910)}$ | $\Lambda_{\Delta(1920)}$ | $\Lambda_{\Delta(1950)}$ | $\Lambda_{K^{\ast}(K)}$ | $\Lambda_{\Delta(N)}$ | $\Lambda_{\Sigma}$
---|---|---|---|---|---|---|---|---|---|---
$1932\pm 9$ | $980\pm 5$ | $982\pm 16$ | $2000\pm 8$ | $1620\pm 8$ | $1222\pm 25$ | $1317\pm 6$ | $800\pm 1$ | $584\pm 1$ | $1140\pm 2$ | $1092\pm 8$
Figure 2: Differential cross sections for $\gamma n\to K^{+}\Sigma^{-}$. The
solid (black), dotted (red), and long-dashed (green) lines represent the
results from the full calculation, the $t$-channel amplitudes ($K$ and
$K^{\ast}(892)$ exchanges), and $s$-channel amplitudes ($N$, $N^{\ast}$,
$\Delta$, and $\Delta^{\ast}$ exchanges), respectively. Data with red circles
and blue triangles are taken from the CLAS Collaboration
AnefalosPereira:2009zw  and the LEPS Collaboration Kohri:2006yx ,
respectively. The numbers in parentheses denote the incident energies (left
number) and the corresponding center-of-mass energies of the system (right
number), in MeV.  Figure 3: Photo-beam asymmetries $\Sigma$ for $\gamma n\to
K^{+}\Sigma^{-}$. The solid (black) lines represent the results from the full
calculation. The dotted (red), dashed (blue), long-dashed (green), and dot-
dashed (magenta) lines represent the results obtained by switching off the
$t$-channel amplitudes ($K$ and $K^{\ast}(892)$ exchanges), $u$-channel
amplitude ($\Sigma$ exchange), $s$-channel amplitudes ($N$, $N^{\ast}$,
$\Delta$, and $\Delta^{\ast}$ exchanges), and interaction current,
respectively. Data with red circles and blue triangles are taken from the CLAS
Collaboration CLAS-beam  and the LEPS Collaboration Kohri:2006yx ,
respectively. The numbers in parentheses denote the incident energies (left
number) and the corresponding center-of-mass energies of the system (right
number), in MeV. Figure 4: Beam-target asymmetries $E$ for $\gamma n\to
K^{+}\Sigma^{-}$. Notations for lines are the same as Fig. 3. Data with red
circles are taken from the CLAS Collaboration Zachariou:2020kkb . $E_{\gamma}$
and $W$ denote the incident energies and the corresponding center-of-mass
energies of the system, respectively.

Figure 5: Predicted total cross sections for $\gamma n\to K^{+}\Sigma^{-}$
with contributions from individual terms.

The main purpose of the present work is to perform a combined analysis of all
the available data, especially the very recently released photo-beam asymmetry
data $\Sigma$ and the beam-target asymmetry data $E$, for the $\gamma n\to
K^{+}\Sigma^{-}$ reaction in an effective Lagrangian approach as introduced in
Sec. II. We consider the $s$-channel $N$, $N^{\ast}$, $\Delta$, and
$\Delta^{\ast}$ exchanges, the $t$-channel $K$ and $K^{\ast}(892)$ exchanges,
the $u$-channel $\Sigma$ exchange, and the interaction current in constructing
the reaction amplitudes.

As can be seen from Fig. 2, the differential cross-section data for $\gamma
n\to K^{+}\Sigma^{-}$ from the CLAS Collaboration AnefalosPereira:2009zw
indicate possible contributions from the $s$-channel resonance exchanges in
the energy range from the $\Sigma K$ threshold up to $E_{\gamma}\sim 1850$ MeV
($W\sim 2088$ MeV). In the most recent RPP Zyla:2020zbs , there are in total
seven $N$ and $\Delta$ resonances with masses $M\leq 2100$ MeV that have
considerable $K\Sigma$ branching ratios, namely, the $N(1880)1/2^{+}$,
$N(1895)1/2^{-}$, $N(1900)3/2^{+}$, $N(2060)5/2^{-}$, $\Delta(1910)1/2^{+}$,
$\Delta(1920)3/2^{+}$, and $\Delta(1950)7/2^{+}$ resonances. Among these seven
resonances, the $N(1895)1/2^{-}$, $N(1900)3/2^{+}$, $\Delta(1910)1/2^{+}$, and
$\Delta(1950)7/2^{+}$ are rated as four-star resonances, and the other three
as three-star resonances in RPP Zyla:2020zbs . As a first step, we try to see
if the available experimental data for $K^{+}\Sigma^{-}$ photoproduction can
be well described by considering in the $s$ channel the contributions from
these seven resonances. Their masses and widths will be varied within the
ranges advocated by RPP Zyla:2020zbs  to fit the data for $\gamma n\to
K^{+}\Sigma^{-}$.

We use MINUIT to fit the model parameters, and numerous different starting
values of the model parameters have been tested to achieve the best converged
fitted results. It turns out, however, that with only the seven resonances
mentioned above being taken into account in the calculation, we were unable to
obtain an acceptable description of the available data for $\gamma n\to
K^{+}\Sigma^{-}$. In particular, the near-threshold angular distribution data
cannot be well reproduced. This hints for contribution of additional
resonance(s) located near this reaction threshold.

Actually, in addition to the above-mentioned seven resonances which have
considerable $K\Sigma$ branching ratios, in the $K\Sigma$ near threshold
region, there is another four-star resonance, i.e. the $N(1710)1/2^{+}$, that
is marked as “seen” in its decay branching ratio to the $K\Sigma$ channel in
RPP Zyla:2020zbs . We then try to refit the model parameters by further
including this resonance in our model. It turns out that a satisfactory
description of all the available data for $\gamma n\to K^{+}\Sigma^{-}$ can be
obtained, as we shall show and discuss in detail in the following.

As a result, we include five nucleon resonances and three $\Delta$ resonances
with 38 fitting parameters for resonances and background contributions in our
model. These parameters are determined by a fit to 780 available data for
$K^{+}\Sigma^{-}$ photoproduction within the energy range from threshold up to
$E_{\gamma}=3550$ MeV. The resulted $\chi^{2}$ per degree of freedom is
$\chi^{2}/N=3.11$. In Table 1 we list the fitted values of resonance
parameters and the extracted resonance branching ratios. There, the number of
asterisks below the resonance names denote the overall status of each
resonance evaluated by RPP Zyla:2020zbs . The values in the brackets below the
resonance masses and widths denote the ranges of the corresponding values
given by RPP Zyla:2020zbs . As for the resonance couplings, since the reaction
amplitudes are only sensitive to the products of the electromagnetic and
hadronic coupling constants, we show in Table 1 these products instead of
individual electromagnetic and hadronic coupling constants. For the four-star
resonances $\Delta(1910)1/2^{+}$ and $\Delta(1950)7/2^{+}$, their helicity
amplitudes are available in RPP, which can be used to fix the resonance
electromagnetic coupling constants. The results give
$g_{RN\gamma}^{(1)}=-0.103$ for the $\Delta(1910)1/2^{+}$ resonance, and
$g_{RN\gamma}^{(1)}=-12.92$, $g_{RN\gamma}^{(2)}=12.65$ for the
$\Delta(1950)7/2^{+}$ resonance. Hence, the hadronic coupling constants
$g_{R\Sigma K}=-5.252$ for $\Delta(1910)1/2^{+}$ and $g_{R\Sigma K}=-0.574$
for $\Delta(1950)7/2^{+}$ can be extracted from the products of hadronic and
electromagnetic couplings listed in Table 1. For other resonances, the
helicity amplitudes are not available in RPP. The last two columns in Table 1
show the extracted decay branching rations of $R\to N\gamma$ and $R\to
K\Sigma$, respectively. The values in brackets denote the ranges of the
corresponding branching ratios available in RPP Zyla:2020zbs . The
electromagnetic branching ratios in bold fonts are fixed and the hadronic
branching ratios are then extracted from the products of the resonance
electromagnetic and hadronic coupling constants. For the $\Delta(1910)1/2^{+}$
and $\Delta(1950)7/2^{+}$ resonances, the electromagnetic partial decay widths
are calculated from their electromagnetic coupling constants, and for the
$N(1710)1/2^{+}$, $N(1895)1/2^{-}$, and $N(2060)5/2^{-}$ resonances, the
electromagnetic branching ratios are fixed to be the centroid values of the
corresponding ranges given by RPP. For the $N(1880)1/2^{+}$ resonance, a value
$\beta_{N\gamma}=0.05\%$ which is smaller than the centroid RPP value of
$0.316\%$ but still within the range quote by RRP is used to avoid that the
corresponding hadronic branching ratio to fall well outside its range given in
RPP. For the same reason, a value $\beta_{N\gamma}=0.04\%$ is adopted for the
$N(1900)3/2^{+}$ resonance. For the $\Delta(1920)3/2^{+}$ resonance, there is
no information about its electromagnetic branching ratio from RPP and a value
$\beta_{N\gamma}=0.2\%$ is adopted to extract the hadronic branching ratio.
One can see from Table 1 that, for all the resonances, the electromagnetic
branching ratios and hadronic branching ratios are all consistent to those
advocated by RPP Zyla:2020zbs . The fitted values of cut-off parameters for
resonances and background diagrams are given in Table 2. Note that, based on
the facts that the contributions of $K$ and $N$ exchanges are rather small and
the results are insensitive to the values of $\Lambda_{K}$ and $\Lambda_{N}$
when both of these two cutoffs are treated as free parameters, we simply set
$\Lambda_{K}=\Lambda_{K^{*}}$ and $\Lambda_{N}=\Lambda_{\Delta}$ to reduce the
number of fitting parameters. The values of the cut-off parameters for
resonances, which are determined by fitting the data, vary in a wide range. As
will be shown in Fig. 5, a small cut-off means the s-channel resonance
contributes in a narrow energy range, while a big value of the cut-off allows
contributions at large energies.

We remark that a proper extraction of the resonances parameters should be
done, in principle, by searching for the poles of the reaction amplitude in
the complex energy plane Doring2009PLB ; Doring2009NPA . In particular, the
resonance coupling constants are related to the residues at those poles. This
is beyond the scope of the present work which is a first step toward such a
more complete analysis.

In Fig. 2, our numerical results for differential cross sections (black-solid
lines) are shown and compared to the data from the CLAS AnefalosPereira:2009zw
Collaboration (red circles) and LEPS Kohri:2006yx  Collaboration (blue
triangle). Here the numbers in parentheses denote the incident energies (left
number) and the corresponding center-of-mass energies of the system (right
number), in MeV. Note that the CLAS data are measured for $K^{+}\Sigma^{-}$
photoproduction off quasi-free neutron. We have checked and found that the
Fermi motion effect here is very tiny and thus is ignored in our calculations.
One sees that, overall, the model results agree with the data quite well in
the whole energy region considered. The contributions from individual terms
are also shown. The red-dotted and green-long-dashed lines represent the
results from the $t$-channel amplitudes ($K$ and $K^{\ast}(892)$ exchanges)
and $s$-channel amplitudes ($N$, $N^{\ast}$, $\Delta$, and $\Delta^{\ast}$
exchanges), respectively. It is seen that in the lower energy region, the
contributions from the $s$-channel mechanisms dominate the differential cross
sections of the $K^{+}\Sigma^{-}$ photoproduction. As the energy increases,
the contributions from the $t$-channel mechanism become more and more
prominent and are essential in describing the data at forward angles.
Furthermore, the $K^{\ast}$ exchange is largely responsible for the forward-
angle-peaked behavior exhibited at higher energies. This explains that
$\Lambda_{K^{\ast}(K)}$, which is the only parameter in the $t$-channel
amplitudes, is well determined to be $\Lambda_{K^{\ast}(K)}=584\pm 1$ MeV as
shown in Table 2. Note that the data seem to show a backward-angle enhancement
in the energy region of $E_{\gamma}=2250-2450$ MeV, although they have large
error bars. The model results seem to underestimate these data, but they are
still within the experimental error range. The contributions of $u$-channel
mechanism and the interaction current are negligible to the differential cross
sections and they are not plotted out in this figure. We mention that, in
principle, a stronger contribution at backward angles could be obtained if one
includes the exchange of one more $\Sigma^{\ast}$ resonance in the
$u$-channel. But in this case, one needs to introduce extra parameters as both
the theoretical and experimental information on $\Sigma^{\ast}N\gamma$
interaction are quite scarce. These parameters cannot be well determined at
present due to the large error bars of the backward-angle differential cross-
section data. We postpone the inclusion of more $u$-channel resonances till
more precision data for this reaction become accessible.

Our numerical results for beam asymmetries $\Sigma$ and beam-target
asymmetries $E$ are shown and compared to the experimental data Kohri:2006yx ;
CLAS-beam ; Zachariou:2020kkb  in Figs. 3 and 4, respectively. Here, the
black-solid lines represent the results of the full reaction amplitude. One
can see that the overall agreement of the theoretical predictions with the
data is rather reasonable. The red-dotted, blue-dashed, green-long-dashed, and
magenta-dot-dashed lines represent the results obtained by switching off the
$t$-channel amplitudes ($K$ and $K^{\ast}(892)$ exchanges), $u$-channel
amplitude ($\Sigma$ exchange), $s$-channel amplitudes ($N$, $N^{\ast}$,
$\Delta$, and $\Delta^{\ast}$ exchanges), and the interaction current,
respectively. For photo-beam asymmetries $\Sigma$ in Fig. 3, one sees that the
$s$-and the $t$-channel contributions (green-long-dashed lines and red-dotted
lines, respectively) have significant effects on the photo-beam asymmetries.
Noticeable influence of the $u$-channel contribution (blue-dashed lines) can
be also seen at higher energies and backward angles. The contribution from the
interaction current is negligible, similar to the situation encountered in the
angular distributions as shown in Fig. 2. For the beam-target asymmetries $E$
in Fig. 4, one sees that the dominant contribution is coming from the
$s$-channel resonance exchanges. The $t$\- and $u$-channels also contribute
significantly. The contribution from the interaction current is negligible,
same as the situation seen in the differential cross sections and photo-beam
asymmetries $\Sigma$ as shown in Figs. 2 and 3, respectively.

The prediction of the total cross section for $\gamma n\to K^{+}\Sigma^{-}$ is
shown in Fig. 5 (left panel), where its dynamical content is also exhibited.
It is clearly seen that the contributions from the $s$-channel resonance
exchanges dominate the total cross sections in the lower energy region and are
responsible for the bump structure exhibited around $W\sim 1900$ MeV. As
displayed in Fig. 5 (right panel), the resonance contributions are dominated
by the $N(1710)1/2^{+}$ resonance, followed by the $N(1880)1/2^{+}$,
$\Delta(1920)3/2^{+}$, and $N(1900)3/2^{+}$ resonances. The $N(2060)5/2^{-}$
resonance also contributes considerably at energies above $W\sim 2100$ MeV,
while the contributions from other resonances are rather small. Note that the
$N(1710)1/2^{+}$ resonance contributes in a much wider energy range with a
stronger strength than the $N(1880)1/2^{+}$ resonance. This is because that
the $N(1710)1/2^{+}$ has a much bigger cutoff and a bigger coupling constant
than the $N(1880)1/2^{+}$, as listed in Tables 1 and 2. Nevertheless, the
$N(1710)1/2^{+}$ resonance has a much smaller branching ratio into $K\Sigma$
than the $N(1880)1/2^{+}$ resonance. The reason is that the fitted mass of
$N(1710)1/2^{+}$ is rather close to the $K\Sigma$ threshold, resulting in a
much small phase space which suppresses the resonance decay width to $K\Sigma$
channel. Getting back to Fig. 5 (left panel), the $s$-channel $\Delta$
exchange provides a significant contribution in the low energy region, while
the $t$-channel $K^{\ast}$ exchange provides significant contribution at
higher energies. The contributions from other non-resonant diagrams to the
total cross sections are negligible, and they are not plotted in Fig. 5.

In Ref. Byd2021 , the data for $\gamma n\to K^{+}\Sigma^{-}$ were analyzed
within an isobar model. There are two sets of fit results, i.e. fit M and fit
L, which were obtained by using, respectively, the standard MINUIT and the so-
called LASSO method for fitting the data. The latter method, together with an
information criterium, allows a blindfold determination of the relevant baryon
resonances by the data from a given set of resonances considered within a
specified model. Compared with our present work, in fit M of Ref. Byd2021 , an
additional $K_{1}(1270)$ meson exchange was considered in the $t$ channel, and
six more nucleon resonances in the $s$ channel, namely, the $N(1535)1/2^{-}$,
$N(1650)1/2^{-}$, $N(1675)5/2^{-}$, $N(1720)3/2^{+}$, $N(1875)3/2^{-}$, and
$N(2120)3/2^{-}$ resonances. For $\Delta$ resonances, the
$\Delta(1900)1/2^{-}$ exchange was considered, while the
$\Delta(1910)1/2^{+}$, $\Delta(1920)3/2^{+}$, and $\Delta(1950)7/2^{+}$
resonances which are included in our present work were not considered in Ref.
Byd2021 . In fit L of Ref. Byd2021 , the $K_{1}(1270)$ meson exchange in the
$t$ channel was not considered. Apart from the three nucleon resonances,
$N(1895)1/2^{-}$, $N(1900)3/2^{+}$ and $N(2060)5/2^{-}$, also found in the
present work, the fit L also finds the $N(1650)1/2^{-}$, $N(1675)5/2^{-}$,
$N(1720)3/2^{+}$, $N(1875)3/2^{-}$, and $N(2120)3/2^{-}$ resonances to be
relevant. In contrast, in the present work we include the $N(1710)1/2^{+}$,
$N(1880)1/2^{+}$, $\Delta(1910)1/2^{+}$, $\Delta(1920)3/2^{+}$, and
$\Delta(1950)7/2^{+}$ resonances, in addition to those three nucleon
resonances mentioned above which are also found in fit L of Ref. Byd2021 .
Note that in our present work, the resonances are selected in such a way that
only the resonances that have considerable branching ratios to $K\Sigma$
channel in RPP Zyla:2020zbs  are considered, while in Ref. Byd2021  the
resonances were determined by a fit to the data for the $\gamma n\to
K^{+}\Sigma^{-}$ reaction. The differential cross sections and the photo-beam
asymmetries $\Sigma$ for $\gamma n\to K^{+}\Sigma^{-}$ were well described in
Ref. Byd2021 , but their description of the data on beam-target asymmetries
$E$ were worse than the description in the present work. We mention that the
data on beam-target asymmetries $E$ have not been included in the fits of Ref.
Byd2021 . It was concluded in Ref. Byd2021  that the $s$-channel
$N(1720)3/2^{+}$ resonance exchange is very important to describe the data.
This conclusion was reached based on the fact that the $N(1720)3/2^{+}$
resonance exchange had big influence in reproducing the photo-beam asymmetry
data (see Fig. 7 and Fig. 8 of Ref. Byd2021 ).

The fact that the two independent analyses of Ref. Byd2021  and the present
work have led to different resonance contents in describing the same data and
have drawn different conclusions about the reaction mechanisms, indicates that
the currently available data for $\gamma n\to K^{+}\Sigma^{-}$ are
insufficient to uniquely determine the reaction amplitude in theoretical
models. More data are needed to further learn about the resonance content in
this reaction. In fact, we mention that a pseudoscalar-meson photoproduction
process requires at least eight carefully chosen independent observables to
determine the corresponding reaction amplitude up to an overall phase
Chiang:1996em ; Nakayama:2019cys ; Nakayama:2018yzw .

##  IV Summary and conclusion

Quite recently, the CLAS Collaboration has released the data on beam
asymmetries $\Sigma$ and beam-target asymmetries $E$ for the $\gamma n\to
K^{+}\Sigma^{-}$ reaction CLAS-beam ; Zachariou:2020kkb . Stimulated by these
new measurements, in the present work, we have performed an analysis of all
the available data, including the differential cross sections
AnefalosPereira:2009zw ; Kohri:2006yx , the beam asymmetries $\Sigma$ CLAS-
beam ; Kohri:2006yx , and the beam-target asymmetries $E$ Zachariou:2020kkb ,
for the reaction $\gamma n\to K^{+}\Sigma^{-}$ in an effective Lagrangian
approach. The purpose is to understand the reaction mechanisms and to extract
the information on the $N$ and $\Delta$ resonances involved.

We have constructed the reaction amplitude for $\gamma n\to K^{+}\Sigma^{-}$
by considering the $t$-channel $K$ and $K^{\ast}(892)$ exchanges, the
$u$-channel $\Sigma$ exchange, the $s$-channel $N$, $N^{\ast}$, $\Delta$, and
$\Delta^{\ast}$ exchanges, and the interaction current. The gauge invariance
of the production amplitude, dictated by the generalized Ward-Takahashi
identity, has been strictly implemented. The $N(1710)1/2^{+}$,
$N(1880)1/2^{+}$, $N(1895)1/2^{-}$, $N(1900)3/2^{+}$, $N(2060)5/2^{-}$,
$\Delta(1910)1/2^{+}$, $\Delta(1920)3/2^{+}$, and $\Delta(1950)7/2^{+}$
resonances are considered in the $s$ channel with their masses and widths been
varied within the ranges advocated by the most recent RPP Zyla:2020zbs  to fit
the data.

All the available data for $\gamma n\to K^{+}\Sigma^{-}$ are well reproduced.
The present analysis reveals that the contributions from the $s$-channel
mechanisms dominate the $\gamma n\to K^{+}\Sigma^{-}$ reaction in the lower
energy region. The most prominent resonance contributions come from the
$N(1710)1/2^{+}$, $N(1880)1/2^{+}$, $N(1900)3/2^{+}$, and
$\Delta(1920)3/2^{+}$ resonances. The $N(2060)5/2^{-}$ resonance contributes
considerably at energies above $W\sim 2100$ MeV. The contributions from the
$t$-channel $K^{\ast}$ exchange are crucial to reproduce the differential
cross-section data at forward angles in the higher energy region and are also
significant to the photo-beam asymmetries $\Sigma$ and beam-target asymmetries
$E$. The interaction current turns out to contribute slightly for all the
observables considered, namely, the cross sections, beam asymmetries $\Sigma$,
and beam-target asymmetries $E$ of the $\gamma n\to K^{+}\Sigma^{-}$ reaction.
However, the resonance content in this reaction is far from being understood.
In this regard, more independent data on this reaction are required.

A combined analysis of all the available data for $\gamma p\to
K^{0}\Sigma^{+}$, $\gamma p\to K^{+}\Sigma^{0}$, $\gamma n\to
K^{0}\Sigma^{0}$, and $\gamma n\to K^{+}\Sigma^{-}$ will provide more
constraints on theoretical models, especially, in help disentangling the
hadronic and electromagnetic resonance couplings. Further work in this
direction is under consideration and will be presented elsewhere. This will
help prepare for a more complete analysis within a dynamical coupled channels
approach to better understand the reaction dynamics of the $K\Sigma$
photoproduction reactions and to extract the resonance parameters from the
poles of the reaction amplitude in the complex energy plane.

###### Acknowledgements.

This work is partially supported by the National Natural Science Foundation of
China under Grants No. 12175240, No. 12147153, and No. 11635009, the
Fundamental Research Funds for the Central Universities, and the China
Postdoctoral Science Foundation under Grants No. 2021M693141 and No.
2021M693142.

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# CALM: Convolution As Local Mixture

Lifan Liang

###### Abstract

In this paper, we showed that the feature map of a convolution layer is
equivalent to the unnormalized log posterior of a special kind of Gaussian
mixture for image modeling. Then we expanded the model to drive diverse
features and proposed a corresponding EM algorithm to learn the model.
Learning convolution weights using this approach is efficient, guaranteed to
converge, and does not need supervised information. Code is available at:
https://github.com/LifanLiang/CALM.

Machine Learning, Generative model,

##  1 Introduction

Unsupervised learning of image representations has been the focus of computer
vision research for decades. In recent years, the discriminative approach has
shown great promise in a series of benchmarking tasks (Chen et al., 2020;
Donahue et al., 2014; Doersch et al., 2015; Caron et al., 2020; Van Gansbeke
et al., 2020). However, the generality of such learned representation would be
limited by the scope of the discriminative tasks. Meanwhile, generative
representation learning on pixel level is computationally expensive (Lee et
al., 2009).

In this paper, we deviated from the current trend, traced back to the
generative approach to learning latent features, and proposed a patch-level
generative model for s single convolution layer. We reinterpreted the
convolution layer as computing posterior membership for our generative model.
The corresponding training algorithm only relies on the operation of
convolution without gradient computation and is guaranteed to converge. Our
work provides both theoretical and practical insights for future research on
unsupervised representation learning related to convolution neural network.

##  2 Related Work

To our knowledge, existing works have not yet proposed to learn convolution
weights by using the corresponding feature map. Restricted Boltzmann Machine
(RBM) has a similar learning approach since the latent features can be learned
from the output activation (Hinton, 2002).

In terms of the generative model we developed, the most similar existing
method is using Kmeans clustering on every patch that can be extracted from
images. This method has performed favorably compared to other more
sophisticated clustering methods such as RBM (Coates et al., 2011). This
benchmark observation cannot be well explained. Our model formulation suggests
that Kmeans extracts features invariant to original pixel intensities when
images are normalized with variance as one.

Another important difference is that Kmeans clustering on all the image
patches is computationally expensive in both time complexity and space
complexity, while our training algorithm can be easily implemented in existing
tools (i.e. pytorch, tensorflow) that enable GPU parallelization. In addition,
the batch size will not affect model convergence.

A more recent development is Convolutional Kmeans (Dundar et al., 2015), which
encouraged Kmeans to avoid redundant centroids by increasing the diversity of
the image patches. For our model, weight penalty combined with normalization
of feature map across filters is sufficient to drive diverse convolution
weights.

##  3 Model Formulation

###  3.1 The generative model with a single patch

An image with height $H$, width $W$, and $C$ channels is a patch embedded in
random background. Let $(Z_{1},Z_{2})$ denote the top left coordinate of the
patch. We assign a Multinomial distribution to $Z$ such that the patch is
equally likely to be anywhere in an image. Assuming the patch is square size
with length $L$, the patch area would be

$U=\\{i,j|Z_{1}\leq i\leq Z_{1}+L-1,Z_{2}\leq j\leq Z_{2}+L-1\\}$

Pixel intensities in the background area are sampled from the standard Normal
distribution, namely:

$X_{\\{i,j,c|(i,j)\notin U\\}}\sim Normal(0,1)$

where $X_{ijc}$ is pixel intensity located at $i$th row, $j$th column, and
$c$th channel of the image. And the pixel intensities in the patch area follow
a Multivariate Normal:

$X_{\\{i,j,c|(i,j)\in U\\}}\sim MVN(\mu,\mathcal{I})$

where U is the set of pixel locations within the patch area, $\mu$ is a vector
of size $L\times L\times C$, representing the mean for every patch pixel in
every channel, $\mathcal{I}$ is the identify matrix for covariance. We set all
variance to be one for model simplicity. Unit variance also makes it easier to
see the connection between our image modeling and convolution.

With the generative model stated above, the likelihood function regarding $U$
and $\mu$ is:

$P(X|\mu,U)\propto
exp(-0.5\sum_{U}^{i,j}\mu_{i,j}^{2}+\sum_{U}^{i,j}\mu_{i,j}X_{i,j})$ (1)

The log posterior of patch location, $U$ can be computed as:

$\displaystyle\log P(U|X,\mu)$
$\displaystyle=\log[\dfrac{P(X|\mu,U)P(U)}{\sum_{U}{P(X|\mu,U)P(U)}}]$ (2)
$\displaystyle=\sum_{U}^{i,j}\mu_{i,j}X_{i,j}-C$

where $C$ is the log total sum of $\sum_{U}^{i,j}\mu_{i,j}X_{i,j}$ across
patch locations. Please note that the term $\sum_{U}^{i,j}\mu_{i,j}X_{i,j}$ is
identical to applying a convolution filter to location $U$ of an image.
Therefore, the feature map generated from a single convolutional layer is
effectively the unnormalized log posterior of patch location for this model.

###  3.2 The generative model with multiple patches

Although the single patch model works well with supervised or self-supervised
learning, it will result in poor performance in unsupervised learning. This is
because CNN naturally deploys multiple convolution filters. Many of them will
converge to the same local optima without supervised information or drop out,
resulting in many redundant features.

To address this issue, the model needs to consider multiple patches
altogether. Similar to a conventional Gaussian mixture, a mixture of patches
reaches higher likelihood when each patch captures different edge features of
an image. Suppose the model is a mixture of $A$ patches and there are $B$
possible locations of $U$, then the number of components for multiple patch
model is $A\times B$. Conditioned on patch location $U$ and the patch $V$, the
two models have the same likelihood function as Equation (1). But the
posterior is different:

$logP(U,V|X,\mu)=\sum_{U^{V}}^{i,j}\mu^{V}_{i,j}X_{i,j}-0.5\sum_{U^{V}}^{i,j}{\mu^{V}_{i,j}}^{2}-const$
(3)

where $V$ is one patch in the mixture, $U^{V}$ is the image area covered by
the patch $V$, $\mu^{V}$ is the parameters of the patch $V$, and $const$
denotes the log sum of likelihood for all possible $Z$ and $V$:

$const=log\sum_{U,V}(\sum_{U^{V}}^{i,j}\mu^{V}_{i,j}X_{i,j}-0.5\sum_{U^{V}}^{i,j}{\mu^{V}_{i,j}}^{2})$

Please note that the term $0.5\sum_{U^{V}}^{i,j}{\mu^{V}_{i,j}}^{2}$ cannot be
canceled out in the multiple patch model. This is consistent with previous
observations that unsupervised learning of images requires sparsity
regularization.

Figure 1: Graphical illustration of the single patch model and multiple patch
model. Pixels in gray area are sampled from standard normal. Patches in
different colors are sampled from different $MVN$

##  4 Learning Algorithm

###  4.1 Expectional maximization (EM)

Similar to the Gaussian mixture, our model can be estimated via EM algorithm,
which always increases the model likelihood and guarantees convergence to zero
gradients. For the expectation step (E-step), we computed the joint posterior
of $U$ and $V$, as in Equation 3. This can be implemented as conventional
convolution with the term $-0.5\sum_{U^{V}}^{i,j}{\mu^{V}_{i,j}}^{2}$ set to
the bias.

In the maximization step (M-step), we need to estimate the mean vector for
each patch. This is computed as the average across pixel intensities weighted
by the joint posterior from E-step. For one parameter in a patch, the maximum
likelihood estimate is:

$\hat{\mu}_{ncijk}=\dfrac{\sum_{n,i^{\prime},j^{\prime}}\hat{\gamma}_{nci^{\prime}j^{\prime}}\cdot
X_{n,i^{\prime}+i-1,j^{\prime}+j-1,k}}{\sum_{n,i^{\prime},j^{\prime}}\hat{\gamma}_{nci^{\prime},j^{\prime}}}$
(4)

where $\hat{\gamma}$ is the estimated joint posterior, $n$ is the index for
samples, $c$ is the index for patches, $ijk$ are indices within the height,
width, and input channel ranges respectively, while $i^{\prime}j^{\prime}$
denotes the coordinates in $\hat{\gamma}$. When we move to estimate the
element next to $\mu_{ncijk}$, we essentially apply dot product between the
same $\hat{\gamma}$ and the next cover area of images. Hence the nominator in
Equation 4 can be computed by applying convolution using $\hat{\gamma}$ as the
filter.

However, such convolution is different from conventional convolution in two
ways: (1) each image has a unique convolution filter because different images
yield different $\hat{\gamma}$; (2) $\hat{\gamma}$ was applied to each color
channel separately to estimate different parameters. To address the first
difference with the existing deep learning framework, we took advantage of the
grouped convolution method. The batch dimension was set to the input channel
dimension and channels within one image make one group. As for the second
difference, we used 3D convolution instead of 2D, treating the color channel
as an extra dimension of the images.

Figure 2: Simplified illustration of the EM algorithm developed in this paper.
E-step is a conventional convolution; M-step uses the feature map as
convolution filter.

###  4.2 Convergence of the algorithm

Treating $U$ and $V$ as the missing data and $\mu$ as the parameter, previous
analysis of EM algorithm (Neal & Hinton, 1998; Wu, 1983) is applicable to our
model. Essentially, we are directly optimizing the function $F$:

$F(U,V,\mu)=\sum_{U,V}q(U,V)\log P(X,U,V|\mu)$

where $q(U,V)$ is the distribution set to $P(U,V|X,\mu)$ in E-step. The
marginal likelihood, $P(X|\mu)$ will also increase as $F$ increases.

Alternatively, our algorithm is similar to performing coordinate ascent over
the function $Q$:

$Q(U,V,\mu)=E_{q}[\log P(X,U,V|\mu)]+E_{q}\log q(U,V)$

###  4.3 Batch EM

The size of $\hat{\gamma}$ does not scale well with sample size. Fortunately,
we are more interested in learning convolution weights. For this purpose,
running batch EM is much better. For each batch, we only need to retain the
nominator and denominator of Equation 4. After one epoch, we summed up
retained values across all batches and updated patch parameters according to
Equation 4. Algorithm 1 describes the procedure.

Algorithm 1 Batch EM

Input: data $X$, sample size $N$, number of patches $K$, patch size $L$

Initialize $batches=SequentialBatch(X)$.

Initialize $\mu=Array[K,3,L,L]$.

repeat

Initialize $\mu^{*}=0,\beta^{*}=0$

for $b$ in $batches$ do

$\gamma_{b}=Estep(X[b],\mu)$

$\mu_{b},\beta_{b}=BatchMstep(X[b],\gamma_{b})$

$\mu^{*}=\mu^{*}+\mu_{b};\beta^{*}=\beta^{*}+\beta_{b}$

end for

$\mu=UpdateParameters(\mu^{*},\beta^{*})$

until $\mu$ does not change

###  4.4 Image-wide pooling

Feature maps learned from our model still contain position information. In
principle, image classification should only be concerned with the activation
of features, regardless of the location of activation. We summarize the
feature map by integrating out the position variable from the feature map.
More specifically, we performed log-sum-exp across the entire feature map for
each channel.

##  5 Result

###  5.1 Application to MNIST

We applied our model to the MNIST dataset for handwritten digit recognition.
Pixel intensities were divided by 255 so that they were between 0 and 1. We
used 64 convolution filters with the size $20\times 20$. Shown in Figure 3 are
the 64 features learned from images. We further applied image-wide pooling on
the feature map, such that every image yields a 64-dimension vector indicating
feature activation. We trained a Softmax regression with this feature
activation vector and image labels within the training set.

For comparison, we trained a supervised model with the same number of
parameters. The supervised model also has one convolution layer with 64
convolution filters of size $20\times 20$, followed by image-wide pooling,
Relu activation, and Softmax output. As shown in Table 1, our model performed
slightly worse than the supervised counterpart. This is probably because
supervised learning can identify image features more related to the task.

Figure 3: Visualization of the 64 features extracted from the MNIST images.

Figure 4: Unnormalized marginal likelihood along with epochs on the MNIST
dataset. EM algorithm converged after the second epoch. Table 1:
Classification accuracies for a supervised and unsupervised convolution layer.
The column named ”unlabeled” means that our model utilized the unlabeled data
during feature extraction.
Data set | Supervised | Unsupervised | Unlabeled
---|---|---|---
MNIST | 96.06 | 94.82 | –
STL10 | 10.00 | 40.03 | 53.54

###  5.2 Application to STL-10

We evaluated our model performance on STL-10 compared to a supervised
counterpart. We used 200 convolution kernels of size $15\times 15$. The rest
is similar to MNIST. Figure 6 showed that our learning algorithm converged
after 5 epochs. Most features extracted are edges (Figure 5).

Figure 5: Visualization of the first 64 features extracted from the STL10
images (labeled training data and unlabeled data).

Figure 6: Unnormalized marginal likelihood along with epochs on the STL10
dataset (labeled training data and unlabeled data).

As shown in the second row of Table 1, the supervised model failed to learn
meaningful features from the 5000 labeled images, while our model still
managed to extract some features related to the labels. When labeled training
data is combined with unlabeled data, our model performance has increased more
than 10%.

##  6 Conclusion

In this work, we reinterpreted a convolution layer as a special type of
Gaussian mixture. The corresponding EM algorithm can be implemented as
alternating the kernels for convolution, which is very simple and fast. The
major limitation is that our model only allows training of a single
convolution layer. A straightforward solution would be stacking these layers
together to construct deep neural networks. But computational cost increased
as the number of kernels increased. We leave the issue of constructing a
hierarchical model to future research.

Our training algorithm performs convolution with feature map instead of
backward propagation. Although running speed may not be much faster, this
approach has several advantages: (1) It enables training without any labels or
self-supervision; (2) There is no need for gradient computation; (3) It
strictly increases model likelihood after each epoch; (4) It does not require
any parameter tuning for optimization. In terms of model performance, our
experiments suggest supervised learning and unsupervised learning excel in
different scenarios. When there are sufficient labels or supervised
information, supervised learning is more effective because it focuses on
discriminative image features. When labels are scarce compared to the
complexity of images, supervised learning may struggle while unsupervised
learning can still extract effective features from images. In addition,
unsupervised learning can directly utilize unlabeled data to enhance
performance.

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