# Combined analysis of the $\gamma n\to K^{0}\Sigma^{0}$ and $\gamma n\to
K^{+}\Sigma^{-}$ reactions

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

###### Abstract

The recently released data on differential cross sections for $\gamma n\to
K^{0}\Sigma^{0}$ from the A2 and BGOOD Collaborations are used to examine the
theoretical model constructed in our previous work [Phys. Rev. D 105, 094017
(2022)] for $\gamma n\to K^{+}\Sigma^{-}$, and it is found that the model
predictions are able to qualitatively reproduce the A2 data but fail to
describe the BGOOD data. Then, a combined analysis of the $\gamma n\to
K^{0}\Sigma^{0}$ and $\gamma n\to K^{+}\Sigma^{-}$ reactions is performed to
revise the theoretical model. Due to the inconsistency problem, the A2 and
BGOOD data are included in fits separately. In the case of including the A2
data, both the data for $\gamma n\to K^{0}\Sigma^{0}$ and $\gamma n\to
K^{+}\Sigma^{-}$ can be fairly well described, and the contributions from the
$N(1710)1/2^{+}$, $N(1880)1/2^{+}$, $N(1900)3/2^{+}$, and
$\Delta(1920)3/2^{+}$ resonances are found to dominate the reactions in the
lower energy region. While in the case of including the BGOOD data, although
most of the data for the $\gamma n\to K^{+}\Sigma^{-}$ reaction can be
described with the exception of some noticeable discrepancies on beam
asymmetries at lower energies, the BGOOD data for $\gamma n\to
K^{0}\Sigma^{0}$ can be only qualitatively described, and the contributions
from the $N(1710)1/2^{+}$, $N(1900)3/2^{+}$, and $\Delta(1910)1/2^{+}$
resonances are found to dominate the reactions in the lower energy region. In
both cases, the $t$-channel $K^{\ast}(892)$ exchange is found to play a
crucial role at forward angles in the higher energy region. Further precise
measurements of data for $\gamma n\to K^{0}\Sigma^{0}$ are called on to
disentangle the discrepancies between the data sets from the A2 and BGOOD
Collaborations.

$K^{+}\Sigma^{-})$ 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

Studies of the spectrum and structure of the excited states of the nucleon and
$\Delta$ ($N^{\ast}$s and $\Delta^{\ast}$s) can provide us with essential
information about the dynamics of the strong interaction in the
nonperturbative regime of quantum chromodynamics (QCD). Extracting information
on resonances from the data of the $\pi N$ scattering experiments and single-
pion photoproduction experiments represents the most important part of the
preceding exploration of the $N^{\ast}$s and $\Delta^{\ast}$s. However, both
the QCD-inspired phenomenological models Isgur:1977ef ; Koniuk:1979vy  and
lattice QCD calculations Edwards:2013 ; Lang:2017 ; Kiratidis:2017 ;
Andersen:2018  have predicted many more $N^{\ast}$s and $\Delta^{\ast}$s than
those have been unraveled in experiments as listed in the Review of Particle
Physics (RPP) Zyla:2020zbs , which is known as the “missing resonance problem”
in baryon spectroscopy.

Since the $\pi N$ channels are sensitive to resonances with sizable
substantial decays to the $\pi N$ final states, those “missing resonances” may
have escaped from observation in the $\pi N$ scattering experiments and/or
single-pion photoproduction experiments due to their small couplings to the
$\pi N$ final states. In this regards, the hadrons photo- and
electroproduction processes can offer opportunity to study resonances with
sizable decays not only to $\pi N$ final states but also to other final states
such as $\eta N$, $\omega N$, $\rho N$, $K\Lambda$, $K\Sigma$, et al..
Consequently, studying the spectrum and structure of $N^{\ast}$s and
$\Delta^{\ast}$s through probes of real and virtual photons has been a key
objective of the hadron physics community in the past 20 years Mokeev:2022afv
; Ireland:2019uwn ; Laget:2019tou ; Aznauryan:2011qj . Actually, the
accumulated high-quality data of various exclusive meson photo- and
electroproduction experiments have helped to identify several long-awaited new
resonances known previously as the “missing resonances” Mokeev:2022afv .

Due to the broad and overlapping nature of light-flavor resonances, the
identification of individual resonance contributions requires analyzing the
data of different exclusive meson photo- and electroproduction reactions that
have selective sensitivity to different resonances. Stimulated by the recently
released data on beam asymmetries $\Sigma$ CLAS-beam  and beam-target
asymmetries $E$ Zachariou:2020kkb  for the $\gamma n\to K^{+}\Sigma^{-}$
reaction, in our previous work Wei:2022nqp  we have performed a timely
theoretical analysis of all the available data Kohri:2006yx ;
AnefalosPereira:2009zw ; CLAS-beam ; Zachariou:2020kkb  for the $\gamma n\to
K^{+}\Sigma^{-}$ reaction based on an effective Lagrangian approach in the
Bonn approximation. There, all the available data for the $\gamma n\to
K^{+}\Sigma^{-}$ reaction were well described and explained by the model
results. The reaction mechanisms were analyzed, and the parameters of the
relevant resonances were extracted and compared to those quoted by RPP
Zyla:2020zbs .

Meanwhile, in Ref. Byd2021  the same data for the $\gamma n\to
K^{+}\Sigma^{-}$ reaction were also analyzed within an isobar model with 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 Least Absolute Shrinkage
and Selection Operator method for fitting the data. The data for the $\gamma
n\to K^{+}\Sigma^{-}$ reaction can be also well described with the two sets of
fit results in Ref. Byd2021 , but different conclusions about the reaction
mechanisms, in particular about the resonance contents and major resonance
contributions, were drew.

The fact that the two independent analyses of Ref. Wei:2022nqp  and Ref.
Byd2021  have led to quite different results indicates that the currently
available data for $\gamma n\to K^{+}\Sigma^{-}$ are insufficient to uniquely
determine the reaction amplitudes in theoretical models. In fact, even a
mathematically complete experiment for a pseudoscalar-meson photoproduction
process would require data on at least eight carefully chosen independent
observables to determine the corresponding reaction amplitudes at each center-
of-mass energy and meson polar angle Chiang:1996em ; Nakayama:2019cys ;
Nakayama:2018yzw . In this sense, further experimental constraints are needed
in fully determining the reaction mechanisms of $\gamma n\to K^{+}\Sigma^{-}$
by testing and distinguishing the phenomenological models for the $\gamma n\to
K^{+}\Sigma^{-}$ reaction.

In 2019, the A2 Collaboration released the world first data on differential
cross sections for the $\gamma n\to K^{0}\Sigma^{0}$ reaction from threshold
up to center-of-mass energy $W=1855$ MeV A2:2018doh . Very recently, the BGOOD
Collaboration reported the differential cross sections for this reaction from
threshold up to $W=2400$ MeV BGOOD:2021oxp . Since the hadronic and
electromagnetic vertices are exactly the same in both the $\gamma n\to
K^{+}\Sigma^{-}$ and $\gamma n\to K^{0}\Sigma^{0}$ reactions except for some
possible isospin factors, the differential cross-section data from the A2 and
BGOOD Collaborations for $\gamma n\to K^{0}\Sigma^{0}$ can be, in principle,
used to test the theoretical models for $\gamma n\to K^{+}\Sigma^{-}$.
Furthermore, it is expected that a combined analysis of the data for these two
reactions would provide more experimental constraints on theoretical models to
disentangle the resonance hadronic and electromagnetic couplings, and to
achieve a better understanding of the reaction dynamics of the $K\Sigma$
photoproduction off neutron.

It is worthy to mention that except for the two most recent works of Refs.
Wei:2022nqp ; Byd2021  which have been devoted to analyze all the available
data Kohri:2006yx ; AnefalosPereira:2009zw ; CLAS-beam ; Zachariou:2020kkb
for $K^{+}\Sigma^{-}$ photoproduction, in literature several works
Vancraeyveld:2009qt ; Mart:2019fau ; Clymton:2021wof ; Luthfiyah:2021yqe  have
been devoted to analyze the LEPS data published in 2006 Kohri:2006yx  and CLAS
data published in 2010 AnefalosPereira:2009zw  for $K^{+}\Sigma^{-}$
photoproduction and the A2 data published in 2019 A2:2018doh  for
$K^{0}\Sigma^{0}$ photoproduction. In Ref. Vancraeyveld:2009qt , the LEPS data
Kohri:2006yx  for $K^{+}\Sigma^{-}$ photoproduction were analyzed within a
Regge-plus-resonance model. The data from Refs. Kohri:2006yx ;
AnefalosPereira:2009zw  for $K^{+}\Sigma^{-}$ photoproduction and the data
from Ref. A2:2018doh  for $K^{0}\Sigma^{0}$ photoproduction together with the
data for $K^{+}\Sigma^{0}$ and $K^{0}\Sigma^{+}$ photoproduction reactions
were analyzed in isobar models in a series works of Refs. Mart:2019fau ;
Clymton:2021wof ; Luthfiyah:2021yqe , where the contributions from resonances
with spin from $1/2$ up to $15/2$ were considered.

In the present paper, we perform a combined analysis of the $\gamma n\to
K^{0}\Sigma^{0}$ and $\gamma n\to K^{+}\Sigma^{-}$ reactions within the model
constructed in our pervious work Wei:2022nqp  by further taking into account
the differential cross-section data from the A2 A2:2018doh  and BGOOD
BGOOD:2021oxp  Collaborations for $\gamma n\to K^{0}\Sigma^{0}$. The purpose
is to implement more experimental constraints on the theoretical model and
thus to achieve a better understanding of the reaction dynamics of the
$K\Sigma$ photoproduction off neutron, especially, to learn more about the
resonances that couple significantly to these two reactions.

The paper is organized as follows. In Sec. II, we briefly review the framework
of the theoretical model. The results are shown and discussed in Sec. III. The
summary and conclusions are 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. Table 1: Fitted values of resonant parameters and the extracted resonance decay branching ratios obtained by fit A. 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] | $g^{(1)}_{RN\gamma}g_{R\Sigma K}$ | $g^{(2)}_{RN\gamma}g_{R\Sigma K}$ |  $\beta_{N\gamma}$ [%] |  $\beta_{K\Sigma}$ [%]
---|---|---|---|---|---|---
$N(1710)1/2^{+}$ | $1692\pm 1$ | $94\pm 1$ | $1.475\pm 0.010$ |  | $\bm{0.01}$ | $0.018$
$\ast\\!\ast\\!\ast\ast$ | $[1680\sim 1740]$ | $[80\sim 200]$ |  |  | $[0.0\sim 0.02]$ | [seen]
$N(1880)1/2^{+}$ | $1860\pm 1$ | $400\pm 2$ | $1.605\pm 0.001$ |  | $\bm{0.05}$ | $21.363$
$\ast\\!\ast\\!\ast$ | $[1830\sim 1930]$ | $[200\sim 400]$ |  |  | $[0.002\sim 0.63]$ | $[10\sim 24]$
$N(1895)1/2^{-}$ | $1920\pm 1$ | $113\pm 10$ | $0.015\pm 0.001$ |  | $\bm{0.005}$ | $14.085$
$\ast\\!\ast\\!\ast\ast$ | $[1870\sim 1920]$ | $[80\sim 200]$ |  |  | $[0.003\sim 0.05]$ | $[6\sim 20]$
$N(1900)3/2^{+}$ | $1931\pm 1$ | $141\pm 2$ | $-0.151\pm 0.001$ | $0.083\pm 0.005$ | $\bm{0.04}$ | $7.000$
$\ast\\!\ast\\!\ast\ast$ | $[1890\sim 1950]$ | $[100\sim 320]$ |  |  | $[\textless 0.04]$ | $[3\sim 7]$
$N(2060)5/2^{-}$ | $2051\pm 2$ | $450\pm 1$ | $-1.025\pm 0.023$ | $-0.942\pm 0.043$ | $\bm{0.037}$ | $1.948$
$\ast\\!\ast\\!\ast$ | $[2030\sim 2200]$ | $[300\sim 450]$ |  |  | $[0.003\sim 0.07]$ | $[1\sim 5]$
$\Delta(1910)1/2^{+}$ | $1895\pm 1$ | $200\pm 2$ | $0.589\pm 0.001$ |  | $\bm{0.02}$ | $14.000$
$\ast\\!\ast\\!\ast\ast$ | $[1850\sim 1950]$ | $[200\sim 400]$ |  |  | $[0.0\sim 0.02]$ | $[4\sim 14]$
$\Delta(1920)3/2^{+}$ | $1870\pm 1$ | $360\pm 1$ | $-1.715\pm 0.004$ | $0.365\pm 0.066$ | $\bm{0.2}$ | $5.771$
$\ast\\!\ast\\!\ast$ | $[1870\sim 1970]$ | $[240\sim 360]$ |  |  |  | $[2\sim 6]$
Table 2: Fitted values of cutoff parameters (in MeV) obtained by fit A. $\Lambda_{N(1710)}$ | $\Lambda_{N(1880)}$ | $\Lambda_{N(1895)}$ | $\Lambda_{N(1900)}$ | $\Lambda_{N(2060)}$ | $\Lambda_{\Delta(1910)}$ | $\Lambda_{\Delta(1920)}$ | $\Lambda_{K^{\ast}(K)}$ | $\Lambda_{\Delta(N)}$ | $\Lambda_{\Sigma}$
---|---|---|---|---|---|---|---|---|---
$1655\pm 7$ | $1082\pm 4$ | $800\pm 32$ | $2000\pm 8$ | $1579\pm 6$ | $1316\pm 30$ | $1248\pm 4$ | $589\pm 1$ | $984\pm 2$ | $1218\pm 7$
Table 3: Fitted values of resonant parameters and the extracted resonance decay branching ratios obtained by fit B. 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] | $g^{(1)}_{RN\gamma}g_{R\Sigma K}$ | $g^{(2)}_{RN\gamma}g_{R\Sigma K}$ |  $\beta_{N\gamma}$ [%] |  $\beta_{K\Sigma}$ [%]
---|---|---|---|---|---|---
$N(1710)1/2^{+}$ | $1692\pm 1$ | $200\pm 1$ | $2.677\pm 0.056$ |  | $\bm{0.01}$ | $0.020$
$\ast\\!\ast\\!\ast\ast$ | $[1680\sim 1740]$ | $[80\sim 200]$ |  |  | $[0.0\sim 0.02]$ | [seen]
$N(1880)1/2^{+}$ | $1830\pm 1$ | $200\pm 1$ | $0.306\pm 0.001$ |  | $\bm{0.005}$ | $21.275$
$\ast\\!\ast\\!\ast$ | $[1830\sim 1930]$ | $[200\sim 400]$ |  |  | $[0.002\sim 0.63]$ | $[10\sim 24]$
$N(1895)1/2^{-}$ | $1920\pm 1$ | $200\pm 1$ | $-0.046\pm 0.001$ |  | $\bm{0.027}$ | $8.462$
$\ast\\!\ast\\!\ast\ast$ | $[1870\sim 1920]$ | $[80\sim 200]$ |  |  | $[0.003\sim 0.05]$ | $[6\sim 20]$
$N(1900)3/2^{+}$ | $1890\pm 1$ | $235\pm 3$ | $0.438\pm 0.001$ | $-0.837\pm 0.007$ | $\bm{0.04}$ | $9.668$
$\ast\\!\ast\\!\ast\ast$ | $[1890\sim 1950]$ | $[100\sim 320]$ |  |  | $[\textless 0.04]$ | $[3\sim 7]$
$N(2060)5/2^{-}$ | $2200\pm 1$ | $450\pm 2$ | $-0.000\pm 0.001$ | $0.562\pm 0.001$ | $\bm{0.01}$ | $1.994$
$\ast\\!\ast\\!\ast$ | $[2030\sim 2200]$ | $[300\sim 450]$ |  |  | $[0.003\sim 0.07]$ | $[1\sim 5]$
$\Delta(1910)1/2^{+}$ | $1850\pm 1$ | $200\pm 1$ | $-0.854\pm 0.012$ |  | $\bm{0.02}$ | $17.939$
$\ast\\!\ast\\!\ast\ast$ | $[1850\sim 1950]$ | $[200\sim 400]$ |  |  | $[0.0\sim 0.02]$ | $[4\sim 14]$
$\Delta(1920)3/2^{+}$ | $1970\pm 1$ | $286\pm 16$ | $-0.318\pm 0.006$ | $0.220\pm 0.008$ | $\bm{0.04}$ | $3.262$
$\ast\\!\ast\\!\ast$ | $[1870\sim 1970]$ | $[240\sim 360]$ |  |  |  | $[2\sim 6]$
Table 4: Fitted values of cutoff parameters (in MeV) obtained by fit B. $\Lambda_{N(1710)}$ | $\Lambda_{N(1880)}$ | $\Lambda_{N(1895)}$ | $\Lambda_{N(1900)}$ | $\Lambda_{N(2060)}$ | $\Lambda_{\Delta(1910)}$ | $\Lambda_{\Delta(1920)}$ | $\Lambda_{K^{\ast}(K)}$ | $\Lambda_{\Delta(N)}$ | $\Lambda_{\Sigma}$
---|---|---|---|---|---|---|---|---|---
$876\pm 4$ | $2066\pm 3$ | $2039\pm 46$ | $2100\pm 2$ | $2000\pm 4$ | $1436\pm 25$ | $750\pm 15$ | $567\pm 1$ | $907\pm 4$ | $1188\pm 30$

The tree-level effective Lagrangian model constructed for the $\gamma N\to
K\Sigma$ reaction is diagrammatically depicted in Fig. 1. One sees from this
figure that the following reaction mechanisms are taken into account in
constructing the $K\Sigma$ photoproduction amplitudes: Fig. 1(a) the
$s$-channel $N$, $N^{\ast}$, $\Delta$, and $\Delta^{\ast}$ exchanges, Fig.
1(b) the $t$-channel $K$ and $K^{\ast}(892)$ exchanges, Fig. 1(c) the
$u$-channel $\Sigma$ exchange, and Fig. 1(d) the interaction current which
stands for other diagrams that do not have $s$-, $t$-, or $u$-channel poles
and ensures the gauge invariance of the full photoproduction amplitudes. The
gauge-invariant amputated photoproduction amplitudes 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)

where the letter $\mu$ denotes the Lorentz index of the incoming photon
$\gamma$. The terms $M^{\mu}_{s}$, $M^{\mu}_{t}$, $M^{\mu}_{u}$ and
$M^{\mu}_{\rm int}$ stand for the amplitudes calculated from the $s$-channel
mechanism, $t$-channel mechanism, $u$-channel mechanism and the interaction
current, respectively. According to Refs. Haberzettl:2006bn ; Huang:2012xj ,
the interaction current $M^{\mu}_{\rm int}$ can be modeled by a generalized
contact current as

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

with $q$ being the four-momentum of the outgoing $K$ meson and $f_{t}$ being
the phenomenological form factor attaching to the amplitude of the $t$-channel
$K$ exchange. Here, $\Gamma_{\Sigma NK}(q)$ is the vertex function of the
$\Sigma NK$ interaction and reads

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

which is calculated from the following Lagrangian for the $\Sigma NK$
interaction,

${\cal L}_{\Sigma NK}=-g_{\Sigma
NK}\bar{\Sigma}\gamma_{5}\left[\left(i\lambda+\frac{1-\lambda}{2M_{N}}{\partial\\!\\!\\!/}\right)K\right]N+\text{H.\,c.},$
(4)

with $\lambda$ being the mixing parameter for the pseudoscalar ($\lambda=1$)
and pseudovector ($\lambda=0$) $\Sigma NK$ couplings. The traditional Kroll-
Ruderman term $M^{\mu}_{\rm KR}$ in Eq. (2) reads

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

which is obtained from the following Lagrangian for the $\Sigma NK\gamma$
interaction,

$\displaystyle L_{\Sigma NK\gamma}$ $\displaystyle=$ $\displaystyle ig_{\Sigma
NK}\frac{1-\lambda}{2M_{N}}\bar{\Sigma}\gamma_{5}\gamma^{\mu}A_{\mu}Q_{K}K\tau
N,$ (6)

with $\tau$ being the isospin factor and $Q_{K}$ the electric charge of
outgoing $K$ meson. The $C^{\mu}$ 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. According to Refs. Haberzettl:1997 ; Haberzettl:2006bn ; Huang:2012
; Huang:2012xj , for the $\gamma N\to K\Sigma$ reaction, the general
prescription for $C^{\mu}$ is chosen as

$\displaystyle C^{\mu}$ $\displaystyle=$ $\displaystyle-
Q_{K}\tau\frac{f_{t}-\hat{F}}{t-q^{2}}(2q-k)^{\mu}-Q_{\Sigma}\tau\frac{f_{u}-\hat{F}}{u-p^{\prime
2}}(2p^{\prime}-k)^{\mu}$ (7) $\displaystyle-\tau
Q_{N}\frac{f_{s}-\hat{F}}{s-p^{\prime 2}}(2p+k)^{\mu},$

with

$\hat{F}=1-\hat{h}\left(1-\delta_{t}f_{t}\right)\left(1-\delta_{u}f_{u}\right)\left(1-\delta_{s}f_{s}\right),$
(8)

where $k$, $p$, and $p^{\prime}$ are the four-momenta for the incoming photon
$\gamma$, incoming nucleon $N$, and outgoing $\Sigma$, respectively; $Q_{N}$
and $Q_{\Sigma}$ are the electric charges of incoming nucleon $N$ and outgoing
$\Sigma$, respectively; $f_{u}$ and $f_{s}$ denote the phenomenological form
factors attaching to the $u$-channel $\Sigma$ exchange and $s$-channel $N$
exchange, respectively; the constant $\delta_{x}=1$ for non-zero charges
$Q_{x}$, and $\delta_{x}=0$ for zero charges $Q_{x}$; $\hat{h}=1$ is used for
simplicity as usual Wang:2017tpe ; Wang:2018vlv ; Wei:2019 .

Note that the $t$-channel $K$ exchange, $u$-channel $\Sigma$ exchange as well
as the interaction current contribute to the $\gamma n\to K^{+}\Sigma^{-}$
reaction, but vanish in the $\gamma n\to K^{0}\Sigma^{0}$ reaction due to the
neutral charges of $K^{0}$ and $\Sigma^{0}$.

The amplitudes $M^{\mu}_{s}$, $M^{\mu}_{t}$, and $M^{\mu}_{u}$ can be
straightforwardly obtained with the effective Lagrangians, resonance
propagators, and phenomenological form factors that have been explicitly given
in Sec. II of Ref. Wei:2022nqp . We do not repeat these materials here for the
sake of brevity.

Figure 2: Differential cross sections for $\gamma n\to K^{0}\Sigma^{0}$ (black
solid line) predicted from the model of Ref. Wei:2022nqp . The red inverted
triangles and black circles denote the corresponding BGOOD data obtained by
the fitting methods of RD and PS, respectively BGOOD:2021oxp . The blue
diamonds and squares denote the A2 data measured at $\cos\theta=-0.125$ and
$\cos\theta=-0.375$, respectively A2:2018doh .  Figure 3: Differential cross
sections for $\gamma n\to K^{+}\Sigma^{-}$ obtained by fit A. 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 4: Photo-beam
asymmetries $\Sigma$ for $\gamma n\to K^{+}\Sigma^{-}$ obtained by fit A. 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 numbers) and the corresponding
center-of-mass energies of the system (right numbers), in MeV. Figure 5: Beam-
target asymmetries $E$ for $\gamma n\to K^{+}\Sigma^{-}$ obtained by fit A.
Notations for lines are the same as Fig. 4. 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 6: Differential cross sections for $\gamma n\to
K^{0}\Sigma^{0}$ obtained by fit A. The solid (black), dotted (red), and long-
dashed (green) lines represent the results from the full calculation, the
$t$-channel amplitude ($K^{\ast}(892)$ exchanges), and $s$-channel amplitudes
($N$, $N^{\ast}$, $\Delta$, and $\Delta^{\ast}$ exchanges), respectively. Data
with blue squares are taken from the A2 Collaboration A2:2018doh . The numbers
in each subfigure denote center-of-mass energies (in MeV) of the system.
Figure 7: Differential cross sections for $\gamma n\to K^{0}\Sigma^{0}$
obtained by fit A. The solid (black), dotted (red), and long-dashed (green)
lines represent the results from the full calculation, the $t$-channel
amplitude ($K^{\ast}(892)$ exchanges), and $s$-channel amplitudes ($N$,
$N^{\ast}$, $\Delta$, and $\Delta^{\ast}$ exchanges), respectively. Data with
red inverted triangles and black circles are taken from BGOOD Collaboration
obtained by the fitting methods of RD and PS, respectively BGOOD:2021oxp .
Data with blue squares are taken from the A2 Collaboration A2:2018doh .

Figure 8: Predicted total cross sections obtained by fit A for $\gamma n\to
K^{+}\Sigma^{-}$ with contributions from individual terms.

Figure 9: Predicted total cross sections obtained by fit A for $\gamma n\to
K^{0}\Sigma^{0}$ with contributions from individual terms. Data with blue
squares are taken from the A2 Collaboration A2:2018doh .

##  III Results and discussion

Figure 10: Differential cross sections for $\gamma n\to K^{+}\Sigma^{-}$
obtained by fit B. Notations for lines and data are the same as Fig. 3. Figure
11: Photo-beam asymmetries $\Sigma$ for $\gamma n\to K^{+}\Sigma^{-}$ obtained
by fit B. Notations for lines and data are the same as Fig. 4. Figure 12:
Beam-target asymmetries $E$ for $\gamma n\to K^{+}\Sigma^{-}$ obtained by fit
B. Notations for lines and data are the same as Fig. 5. Figure 13:
Differential cross sections for $\gamma n\to K^{0}\Sigma^{0}$ obtained by fit
B. Notations for lines and data are the same as Fig. 7.

Figure 14: Predicted total cross sections obtained by fit B for $\gamma n\to
K^{+}\Sigma^{-}$ with contributions from individual terms. The black dashed
line in the panel (a) represents the results from the full calculation of fit
A.

Figure 15: Predicted total cross sections obtained by fit B for $\gamma n\to
K^{0}\Sigma^{0}$ with contributions from individual terms. The black dashed
line in the panel (a) represents the results from the full calculation of fit
A. Data with blue squares are taken from the A2 Collaboration A2:2018doh .
Figure 16: Predictions of the photo-beam asymmetries $\Sigma$ obtained by Fit
A (red solid lines) and Fit B (blue dashed lines) for $\gamma n\to
K^{0}\Sigma^{0}$. Figure 17: Predictions of the beam-target asymmetries $E$
obtained by Fit A (red solid lines) and Fit B (blue dashed lines) for $\gamma
n\to K^{0}\Sigma^{0}$.

The main purpose of the present work is to perform a combined analysis of the
$\gamma n\to K^{0}\Sigma^{0}$ and $\gamma n\to K^{+}\Sigma^{-}$ reactions
within the model constructed in our pervious work Wei:2022nqp  by further
considering the differential cross-section data from the A2 and BGOOD
Collaborations for the $\gamma n\to K^{0}\Sigma^{0}$ reaction A2:2018doh ;
BGOOD:2021oxp . No extra model parameter is needed to be introduced in
constructing the amplitudes for the $\gamma n\to K^{0}\Sigma^{0}$ reaction as
the hadronic and electromagnetic vertices are exactly the same in both the
$\gamma n\to K^{+}\Sigma^{-}$ and $\gamma n\to K^{0}\Sigma^{0}$ reactions
except for some possible isospin factors. Therefore, the data for the $\gamma
n\to K^{0}\Sigma^{0}$ reaction can be used to test the theoretical model for
the $\gamma n\to K^{+}\Sigma^{-}$ reaction. Moreover, a combined analysis of
the data for both the $\gamma n\to K^{0}\Sigma^{0}$ and $\gamma n\to
K^{+}\Sigma^{-}$ reactions will implement more experimental constraints on the
theoretical model to determine better the reaction amplitudes and to result in
more reliable theoretical results.

Firstly we want to see how well the model of Ref. Wei:2022nqp  can describe
the data from the A2 and BGOOD Collaborations A2:2018doh ; BGOOD:2021oxp  for
$\gamma n\to K^{0}\Sigma^{0}$. As an illustration, part of the model
predictions of the differential cross sections for $\gamma n\to
K^{0}\Sigma^{0}$ is shown in Fig. 2 with a comparison to the corresponding
data from the A2 and BGOOD Collaborations A2:2018doh ; BGOOD:2021oxp . Here,
the prediction of the model results (black solid line) is calculated at
$\cos\theta=-0.25$. The red inverted triangles and black circles denote the
corresponding BGOOD data at $-0.40<\cos\theta<-0.10$ obtained by using the
fitting methods of, respectively, RD and PS, which denote two different
methods used to describe and subtract background BGOOD:2021oxp . The blue
diamonds and squares denote the A2 data measured at $\cos\theta=-0.125$ and
$\cos\theta=-0.375$, respectively A2:2018doh , which are located at the range
of the angle measured by the BGOOD Collaboration. One can see from Fig. 2 that
the model results of Ref. Wei:2022nqp  are able to qualitatively reproduce the
data from the A2 Collaboration A2:2018doh , but fail to describe the data from
the BGOOD Collaboration BGOOD:2021oxp . Furthermore, obvious discrepancies
between the data sets are shown: the A2 data have an upward tendency as the
center-of-mass energy increases with a maximum value $\approx 0.14$ ${\rm\mu
b/sr}$, while the BGOOD data appear almost flat with values remaining blow
$0.4$ ${\rm\mu b/sr}$.

Even though the BGOOD data are far from being satisfactorily described, the
qualitative description of the A2 data is, to some extent, an advocate of the
validity and feasibility of the theoretical model constructed in Ref.
Wei:2022nqp . No matter what, a simultaneous analysis of the data for both the
$\gamma n\to K^{0}\Sigma^{0}$ and $\gamma n\to K^{+}\Sigma^{-}$ reactions is
anticipated to implement more experimental constraints on the theoretical
model and thus to result in more reliable results. However, we have checked
and found that, due to the discrepancies between the data sets of the A2 and
BGOOD Collaborations, there is no way to obtain a unique fit if we consider
both of them simultaneously. As we have no clear reason to discard one data
set in favor of the other, we are therefore forced to consider them
separately. In one case, we have considered the data from the A2 Collaboration
A2:2018doh  for $\gamma n\to K^{0}\Sigma^{0}$ and all the available data
Kohri:2006yx ; AnefalosPereira:2009zw ; CLAS-beam ; Zachariou:2020kkb  for
$\gamma n\to K^{+}\Sigma^{-}$, which is denoted as fit A. In the other case,
we have considered the data from the BGOOD Collaboration BGOOD:2021oxp  for
$\gamma n\to K^{0}\Sigma^{0}$ and all the available data Kohri:2006yx ;
AnefalosPereira:2009zw ; CLAS-beam ; Zachariou:2020kkb  for $\gamma n\to
K^{+}\Sigma^{-}$, which is denoted as fit B.

In Ref. Wei:2022nqp  contributions from the $N(1880)1/2^{+}$,
$N(1895)1/2^{-}$, $N(1900)3/2^{+}$, $N(2060)5/2^{-}$, $\Delta(1910)1/2^{+}$,
$\Delta(1920)3/2^{+}$, $\Delta(1950)7/2^{+}$, and $N(1710)1/2^{+}$ resonances
have been taken into account to reproduce the available data for $\gamma n\to
K^{+}\Sigma^{-}$. Apart from the $N(1710)1/2^{+}$ which is marked as “seen” in
its decay branching ratio to the $K\Sigma$ channel, all the other considered
resonances have sizable $K\Sigma$ branching ratios in RPP Zyla:2020zbs .
Nevertheless, the results of Ref. Wei:2022nqp  have shown that the
contributions from the high partial wave resonance $\Delta(1950)7/2^{+}$ are
rather small. In the present combined analysis of $\gamma n\to
K^{0}\Sigma^{0}$ and $\gamma n\to K^{+}\Sigma^{-}$, we have checked and found
that the contributions from this resonance are even smaller. We thus do not
consider this resonance in the present work.

The results obtained by fit A are shown in Fig. 3-9. The resulted $\chi^{2}$
per degree of freedom, $\chi^{2}/N$, is $3.21$ with $N=828$. The fitted values
of resonant parameters as well as the extracted resonance decay branching
ratios are listed in Table 1. Here, the number of asterisks below each
resonance names denote the overall status of this resonance evaluated by RPP
Zyla:2020zbs . The masses and widths of resonances have been varied within the
ranges (values in the brackets below the resonance masses and widths)
advocated by RPP Zyla:2020zbs  to fit the data for $\gamma n\to
K^{0}\Sigma^{0}$ and $\gamma n\to K^{+}\Sigma^{-}$. In the present tree-level
calculations, the calculated results are only sensitive to the products of the
electromagnetic and hadronic coupling constants, and we, therefore, list the
values of these products instead of individual electromagnetic and hadronic
coupling constants. The last two columns of Table 1 show the extracted decay
branching rations of $R\to N\gamma$ and $R\to K\Sigma$, respectively. Here,
the values in brackets denote the ranges of the corresponding branching ratios
available in RPP Zyla:2020zbs . For each resonance, its electromagnetic
branching ratio is fixed at a selected value (in bold font) within the range
advocated by RPP Zyla:2020zbs  and its hadronic branching ratio is then
extracted from the corresponding products of the resonance electromagnetic and
hadronic coupling constants. One can see from Table 1 that, for all the
resonances, the electromagnetic branching ratios and the extracted hadronic
branching ratios are all consistent to those advocated by RPP Zyla:2020zbs .
The corresponding fitted values of cutoff parameters are listed in Table 2.

The differential cross sections for $\gamma n\to K^{+}\Sigma^{-}$ obtained by
fit A (black-solid lines) are shown in Fig. 3 and compared to the data from
the CLAS AnefalosPereira:2009zw  Collaboration (red circles) and LEPS
Kohri:2006yx  Collaboration (blue triangle). Individual contributions from
various reaction dynamics are also shown. The red-dotted and green-long-dashed
lines represent the results from the $t$-channel mechanisms ($K$ and
$K^{\ast}(892)$ exchanges) and $s$-channel mechanisms ($N$, $N^{\ast}$,
$\Delta$, and $\Delta^{\ast}$ exchanges), respectively. Note that,
contributions from the interaction current and the $u$-channel mechanism are
too small to be clearly seen with the scale used, and thus they are not
plotted in this figure. The numbers in the parentheses denote the
corresponding incident energies (left numbers) and the corresponding center-
of-mass energies of the system (right numbers), in MeV. One can see that, the
differential cross section data for $\gamma n\to K^{+}\Sigma^{-}$ can be well
described with the model results. In the lower energy region, contributions
from the $s$-channel mechanisms dominate the reaction, while contributions
from the $t$-channel mechanisms become more and more prominent as the center-
of-mass energy increases and finally dominate the reaction in the higher
energy region. In particular, contributions from the $t$-channel
$K^{\ast}(892)$ exchange account for the forward-angle-peaked behavior
exhibited at higher energies.

The beam asymmetries $\Sigma$ (black-solid lines) and beam-target asymmetries
$E$ (black-solid lines) for $\gamma n\to K^{+}\Sigma^{-}$ obtained by fit A
are shown and compared to the data Kohri:2006yx ; CLAS-beam ;
Zachariou:2020kkb  in Fig. 4 and Fig. 5, respectively. Here, the red-dotted,
blue-dashed, green-long-dashed, and magenta-dot-dashed lines denote 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, in the full
photoproduction amplitudes. Overall, the agreement of the model results with
the data is reasonable. For the photo-beam asymmetries $\Sigma$ in Fig. 4,
contributions from the $s$\- and $t$-channel mechanisms have significant
effects. Contributions from the $u$-channel mechanism have noticeable effects
at higher energies and backward angles, while the effects from the interaction
current is quite small. For the beam-target asymmetries $E$ in Fig. 5,
contributions from the $s$-channel mechanisms have dominant effects. The
effects from the $t$-and the $u$-channel mechanisms are significant, while the
effects from the interaction current are found to be negligible, just as the
situation seen in the differential cross sections and photo-beam asymmetries
$\Sigma$.

The results for the differential cross sections for $\gamma n\to
K^{0}\Sigma^{0}$ obtained by fit A (black-solid lines) are shown in Fig. 6 and
compared to the data from the A2 Collaboration A2:2018doh . One can see that,
the experimental data can be well described and the contributions from the
$s$-channel mechanisms dominate this reaction.

In Fig. 7, we also compare the differential cross sections for $\gamma n\to
K^{0}\Sigma^{0}$ obtained by fit A (black-solid lines) to the data from the
BGOOD Collaboration BGOOD:2021oxp . The data from the A2 Collaboration
A2:2018doh  measured at the same ranges of the meson polar angle are also
shown for comparison. One can see that, the model results of fit A can well
reproduce the data from the A2 Collaboration A2:2018doh  but fail to describe
the data from the BGOOD Collaboration BGOOD:2021oxp .

The predictions of the total cross sections for $\gamma n\to K^{+}\Sigma^{-}$
and $\gamma n\to K^{0}\Sigma^{0}$ obtained by fit A are shown in Fig. 8 and
Fig. 9, respectively. Clearly, one can see that in both reactions, the
$s$-channel resonance exchanges dominate the reactions in the lower energy
region, and the $t$-channel $K^{\ast}(892)$ exchange become more and more
prominent as the center-of-mass energy increases and finally dominates the
reactions in the higher energy region. In the lower energy region, the
dominant $s$-channel contributions in both reactions are from the
$N(1710)1/2^{+}$, $N(1880)1/2^{+}$, $N(1900)3/2^{+}$, and
$\Delta(1920)3/2^{+}$ resonances. We remark that, even though the dominant
resonances in $\gamma n\to K^{+}\Sigma^{-}$ found here are the same as those
in Ref. Wei:2022nqp , the individual contributions from each resonance are
noticeably different in these two works. In particular, the contributions from
the $N(1710){1/2}^{+}$ resonance are much narrower and located at lower energy
region in the present work due to the much smaller fit value of the cutoff
parameter as listed in Table 2. The contributions from both the
$N(1880)1/2^{+}$ and $\Delta(1920)3/2^{+}$ resonances are much stronger in the
present work mainly due to the larger branching ratios derived from the fitted
resonance couplings as listed in Table 1. The contributions from the
$N(1900)3/2^{+}$ resonance in the present work are similar to those in Ref.
Wei:2022nqp . The $N(2060){5/2}^{-}$ resonance contributes significantly at
higher energy region in both works, while its contributions are a little bit
more stronger in the present work due to the larger fit value of the cutoff
parameter and bigger branching ratio derived from the fitted coupling
constants. Nevertheless, the coherent sum of all resonance contributions in
the $\gamma n\to K^{+}\Sigma^{-}$ reaction in fit A of the present work is
similar to that in the fit of Ref. Wei:2022nqp . Note that the resulted
$\chi^{2}/N$ for fit A in the present work is $3.37$, bigger than the value
$3.11$ in Ref. Wei:2022nqp , indicating a little bit worse fitting quality of
fit A in the present work than the fit in Ref. Wei:2022nqp , which is
reasonable as the data from the A2 Collaboration A2:2018doh  implement
additional constraints on the theoretical model.

We now come to the discussions of the results obtained by fit B. The resulted
$\chi^{2}$ per degree of freedom is $\chi^{2}/N=15.81$ with $N=819$, which is
significantly larger than that of fit A. Note that, for the BGOOD data, the
two methods (RD and PS) used to describe and subtract back ground show a good
agreement BGOOD:2021oxp , and we have included the data obtained by RD in our
fits. The results obtained by fit B are shown in Fig. 10-15. The fitted values
of resonant parameters as well as the extracted resonance decay branching
ratios are listed in Table 3. One sees from Table 3 that the fitted values of
the model parameters have changed significantly compared with those shown in
Table 1. The extracted electromagnetic branching ratios and hadronic branching
ratios are consistent to the ranges advocated by RPP Zyla:2020zbs , except for
those of the $N(1900)3/2^{+}$ and $\Delta(1910)1/2^{+}$ resonances, which are
a little bigger than the corresponding values advocated by RPP Zyla:2020zbs .
The fitted values of cutoff parameters are listed in Table 4.

The differential cross sections for $\gamma n\to K^{+}\Sigma^{-}$ obtained by
fit B (black-solid lines) are shown in Fig. 10 and compared to the data from
the CLAS AnefalosPereira:2009zw  Collaboration (red circles) and LEPS
Kohri:2006yx  Collaboration (blue triangle). One can see that the differential
cross-section data can be also well described by the results of fit B. Similar
to the situation of fit A, contributions from the $s$\- and $t$-channel
mechanisms dominate the reaction in the lower and higher energy regions,
respectively.

The results for the beam asymmetries $\Sigma$ (black-solid lines) and beam-
target asymmetries $E$ (black-solid lines) for $\gamma n\to K^{+}\Sigma^{-}$
obtained by fit B are shown and compared to the data Kohri:2006yx ; CLAS-beam
; Zachariou:2020kkb  in Fig. 11 and Fig. 12, respectively. For the photo-beam
asymmetries $\Sigma$ in Fig. 11, noticeable discrepancies between the data and
model results are observed at lower energies, resulting in relative large
$\chi^{2}/N$ as the measured error bars in the data are quite small.
Contributions from the $s$-channel and $t$-channel mechanisms have significant
effects on photo-beam asymmetries in the lower energy region and higher energy
region, respectively, while the effects from the $u$-channel mechanism and the
interaction current are found to be negligible. For the beam-target
asymmetries $E$ in Fig. 12, the agreement of the model results with the data
is reasonable. Contributions from the $s$-channel mechanisms have the most
strong effects in the whole energy region. In the higher energy region, the
effects from $t$-channel mechanisms are significant at forward angles, and
noticeable effects from the $u$-channel mechanism are also shown. Again, the
effects from the interaction current are found to be negligible.

The differential cross sections for $\gamma n\to K^{0}\Sigma^{0}$ obtained by
fit B (black-solid lines) are shown in Fig. 13 and compared to the data from
the BGOOD Collaboration BGOOD:2021oxp . The data from the A2 Collaboration
A2:2018doh  measured at the same ranges of the meson polar angle are also
shown for comparison. One sees that, the experimental data can be only
qualitatively described. The bump structure shown at $W\approx 1900$ MeV and
$-0.10<\cos\theta<0.20$ can not be described by the model results.

The predictions of the total cross section for $\gamma n\to K^{+}\Sigma^{-}$
and $\gamma n\to K^{0}\Sigma^{0}$ obtained by fit B are shown in Fig. 14 and
Fig. 15, respectively. One can see that in both reactions, contributions from
the $s$-channel mechanisms dominate the reactions in the lower energy region,
while the contributions from the $t$-channel mechanisms dominate the reactions
in the higher energy region. Although the $t$-channel interactions provide
similar contributions in both $\gamma n\to K^{+}\Sigma^{-}$ and $\gamma n\to
K^{0}\Sigma^{0}$, the contributions from the $s$-channel resonance diagrams in
$\gamma n\to K^{0}\Sigma^{0}$ are much smaller than those in $\gamma n\to
K^{+}\Sigma^{-}$. Compared with fit A where the contributions from $s$-channel
diagrams are mainly coming from the $N(1710)1/2^{+}$, $N(1880)1/2^{+}$,
$N(1900)3/2^{+}$, $\Delta(1920)3/2^{+}$, and $N(2060)5/2^{-}$ resonances, the
dominant $s$-channel contributions in fit B are from the $N(1710)1/2^{+}$,
$N(1900)3/2^{+}$, and $\Delta(1910)1/2^{+}$ resonances. For the $\gamma n\to
K^{+}\Sigma^{-}$ reaction, the coherent sum of all resonance contributions
from fit B is similar to that from fit A. On the contrary, for the $\gamma
n\to K^{0}\Sigma^{0}$ reaction, the coherent sum of all resonance
contributions from fit B is less than $1/6$ of that from fit A. One also sees
that the predicted total cross sections for $\gamma n\to K^{+}\Sigma^{-}$ are
similar in fits A and B, while for $\gamma n\to K^{0}\Sigma^{0}$ they are
quite different.

We remark that the predictions of the differential cross sections and various
polarization observables obtained by Kaon-Maid for $\gamma n\to
K^{+}\Sigma^{-}$ and $\gamma n\to K^{0}\Sigma^{0}$ are available online Kaon-
Maid . Actually, the Kaon-Maid’s predictions of the differential cross
sections and beam asymmetries $\Sigma$ for $\gamma n\to K^{+}\Sigma^{-}$ have
been published in Ref. Clymton:2021wof , and those of the beam-target
asymmetries $E$ for $\gamma n\to K^{+}\Sigma^{-}$ have been published in Ref.
Zachariou:2020kkb . One notes that the predictions of Kaon-Maid for the
differential cross sections of $\gamma n\to K^{0}\Sigma^{0}$ agree well with
the BGOOD data. Nevertheless, as shown in Refs. Zachariou:2020kkb ;
Clymton:2021wof , noticeable discrepancies can be seen between the data and
the predictions of Kaon-Maid for the results of the differential cross
sections, beam asymmetries $\Sigma$, and beam-target asymmetries $E$ for
$\gamma n\to K^{+}\Sigma^{-}$.

In brief, by considering different data sets from A2 and BGOOD Collaborations
A2:2018doh ; BGOOD:2021oxp  for $\gamma n\to K^{0}\Sigma^{0}$, the model
results of fit A and fit B have led to different conclusions about the
reaction mechanisms. In fit A where the data from A2 Collaboration are
considered, all data can be well described by the theoretical model, while in
fit B where the data from BGOOD Collaboration are considered, noticeable
discrepancies between the data and model results are seen. At present, we have
no clear reason to discard on data set in favor of the other. Thus, further
precise measurements of data for $\gamma n\to K^{0}\Sigma^{0}$ are called on
to disentangle the discrepancies between the data sets from the A2 and BGOOD
Collaborations A2:2018doh ; BGOOD:2021oxp , and to further determine the
reaction mechanisms for $K\Sigma$ photoproduction reactions. Since the
polarization observables are more sensitive to the details of the reaction
dynamics in general and thus are preferred by experimentalist, in Figs. 16 and
17 the predictions of the photo-beam asymmetries $\Sigma$ and beam-target
asymmetries $E$ obtained by Fit A and Fit B for the $\gamma n\to
K^{0}\Sigma^{0}$ reaction at $W=1800$ and $W=1900$ MeV are given, which can be
examined by the future experiments.

##  IV Summary and conclusion

The available data for the $\gamma n\to K^{+}\Sigma^{-}$ reaction have been
analyzed in our previous work Wei:2022nqp  and in Ref. Byd2021 . However,
these two independent analyses have extracted different resonance contents and
different reaction mechanisms for this reaction even they described the same
sets of data.

Recently, the A2 A2:2018doh  and BGOOD BGOOD:2021oxp  Collaborations released
the data on differential cross sections for the $\gamma n\to K^{0}\Sigma^{0}$
reaction. Since the hadronic and electromagnetic vertices are exactly the same
in the $\gamma n\to K^{+}\Sigma^{-}$ and $\gamma n\to K^{0}\Sigma^{0}$
reactions except for some possible isospin factors, the data for $\gamma n\to
K^{0}\Sigma^{0}$ can be used to test the theoretical models of the $\gamma
n\to K^{+}\Sigma^{-}$ reaction. A combined analysis of the data for the
$\gamma n\to K^{+}\Sigma^{-}$ and $\gamma n\to K^{0}\Sigma^{0}$ reactions
would provide more experimental constraints on the theoretical models and,
therefore, allow us perform a better analysis of the reaction mechanism and
extract more reliably the resonance contents and parameters.

In the present paper, we have examined the theoretical model constructed in
our previous work Wei:2022nqp  for $\gamma n\to K^{+}\Sigma^{-}$ by comparing
the predicted differential cross sections for $\gamma n\to K^{0}\Sigma^{0}$
with the corresponding data, and it was found that only the A2 data can be
qualitatively reproduced. We have then performed a combined analysis of the
data for both the $\gamma n\to K^{0}\Sigma^{0}$ and $\gamma n\to
K^{+}\Sigma^{-}$ reactions Kohri:2006yx ; AnefalosPereira:2009zw ; CLAS-beam ;
Zachariou:2020kkb ; A2:2018doh ; BGOOD:2021oxp . Due to the inconsistencies of
data, we have included the data from the A2 and BGOOD Collaborations
separately in the fits.

In the case of including the A2 data, all the data for the $\gamma n\to
K^{0}\Sigma^{0}$ and $\gamma n\to K^{+}\Sigma^{-}$ reactions can be well
described. Contributions from the $s$\- and $t$-channel mechanisms were found
to dominate the reactions in the lower energy region and higher energy region,
respectively. The dominant $s$-channel contributions in the lower energy
region were found to be from the $N(1710)1/2^{+}$, $N(1880)1/2^{+}$,
$N(1900)3/2^{+}$, and $\Delta(1920)3/2^{+}$ resonances.

In the case of including the BGOOD data, the data for the $\gamma n\to
K^{+}\Sigma^{-}$ reaction can be also reproduced, with the exception of some
noticeable discrepancies in the descriptions of the beam asymmetries. The data
on the differential cross sections from the BGOOD BGOOD:2021oxp  Collaboration
for the $\gamma n\to K^{0}\Sigma^{0}$ reaction can be only qualitatively
described. The $s$\- and $t$-channel mechanisms were also found to dominate
the reactions in the lower energy region and higher energy region,
respectively. The dominant $s$-channel contributions in the lower energy
region were found to be from the $N(1710)1/2^{+}$, $N(1900)3/2^{+}$, and
$\Delta(1910)1/2^{+}$ resonances. Further precise measurements of data for
$\gamma n\to K^{0}\Sigma^{0}$ were called on to disentangle the discrepancies
between the data sets from the A2 and BGOOD Collaborations.

###### Acknowledgements.

The author N. C. Wei thanks K. Kohl, who kindly provided him with the
experimental data for the present work. 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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††thanks: These authors contributed equally††thanks: These authors contributed
equally

# Reentrant delocalization transition in one-dimensional photonic
quasicrystals

Sachin Vaidya  Department of Physics, The Pennsylvania State University,
University Park, Pennsylvania 16802, USA  Christina Jörg  Department of
Physics, The Pennsylvania State University, University Park, Pennsylvania
16802, USA  Kyle Linn  Department of Physics, The Pennsylvania State
University, University Park, Pennsylvania 16802, USA  Megan Goh  Department of
Physics, Amherst College,Amherst, MA 01002, USA  Mikael C. Rechtsman
Department of Physics, The Pennsylvania State University, University Park,
Pennsylvania 16802, USA

###### Abstract

Waves propagating in certain one-dimensional quasiperiodic lattices are known
to exhibit a sharp localization transition. We theoretically predict and
experimentally observe that the localization of light in one-dimensional
photonic quasicrystals may be followed by a second delocalization transition
for some states on increasing quasiperiodic modulation strength - an example
of a reentrant transition. We further propose that this phenomenon can be
qualitatively captured by a dimerized tight-binding model with long-range
couplings.

Anderson localization is a generic phenomenon of wave localization in randomly
disordered media Anderson_localization_original . The presence of localized
states implies the cessation of all wave transport in the thermodynamic limit
and thus the Anderson model has provided deep insights into the nature of
metal to insulator transitions for electrons in disordered solids
fifty_years_of_anderson  as well as for light propagating in disordered
photonic structures Anderson_localization_of_light . Specifically in
photonics, localization has been proposed and observed in photonic crystals
(PhCs) and waveguide arrays, both in truly random john1987_PhC ; Maret_Loc ;
genack_loc ; lahini2008anderson_loc ; Anderson_localization_of_light ;
Anderson_PhC_waveguide  and quasicrystalline cases AA_waveguides1 ;
PhC_aubry_Andre1 ; PhC_aubry_Andre2 . Furthermore, this localization
phenomenon can be employed for various photonic applications, such as for
random nanolasing PhC_anderson1 , formation of photonic pseudogaps
Anderson_PhC_bandgap , formation of high $Q/V$ nanocavities PhC_anderson2 ;
PhC_anderson3 ; PhC_aubry_Andre1 ; PhC_aubry_Andre2  and for reducing the
crosstalk between waveguides in fiber arrays for endoscopy and
telecommunications guglielmon2019inducing .

It can be shown that in one and two dimensions, an infinitesimal amount of
random disorder causes wave localization but in three dimensions, a sharp
transition occurs between extended and localized regimes at a finite value of
disorder strength Anderson_localization_1D ; Anderson_localization_2D . Such a
sharp transition between localized and extended regimes can also occur in one
dimension when the random disorder of lattice potentials is replaced by
quasiperiodicity. A model first proposed by Aubry and André consists of a one-
dimensional lattice with quasiperiodic on-site-energy modulation and nearest-
neighbor couplings that exhibits a sharp localization transition AA_original .
Specifically, the on-site potential for the $n$-th site in a chain of atoms is
modulated according to $E_{n}=E_{0}+\xi\cos(2\pi\beta n)$, where $E_{0}$ is
the unperturbed on-site energy, $\beta$ is an irrational number and $\xi$ is
the strength of the quasiperiodic modulation. For this simple model, the
localization transition occurs for the entire spectrum at a single value of
$\xi$ due to a duality between the extended and localized regimes AA_original
.

Extensions of the Aubry-André model with long-range couplings AA_longrange1 ;
AA_longrange2 ; AA_generic_critical  and non-Hermiticity AA_NH1 ; reentrant4
were investigated theoretically and found to possess single-particle mobility
edges and consequently intermediate regimes, where both extended and localized
states co-exist. Moreover, some dimerized tight-binding models were recently
found to exhibit a second reentrant transition of some states back to the same
localization regime reentrant1 ; reentrant2 ; reentrant3 ; reentrant4 . Simple
two-component one-dimensional PhCs can be thought of as naturally dimer-like
due to the patterning of their different dielectrics and may exhibit non-
Hermiticity from gain or radiative loss. They are therefore a potentially
useful platform for exploring the rich localization physics in complex models.

In this work, we experimentally demonstrate a surprising localization
phenomenon in multi-layer structures with quasiperiodic thickness modulation,
i.e., one-dimensional photonic quasicrystals (PhQCs). In particular, we
observe that in addition to the complete inhibition of transmission
corresponding to a sharp localization transition, there is a second transition
to an extended regime upon increasing the quasiperiodic modulation strength.
The experimental signature of this is the complete recovery of transmission
through the structure as the quasiperiodic modulation increases beyond the
localized regime. This reentrant delocalization transition is not known to
occur in random potentials and is a unique feature of quasicrystalline
systems. To further explore the reentrant transition, we develop a tight-
binding model inspired by our PhQCs, that captures the physics of localization
and delocalization in our system.

Figure 1:  Schematic of multi-layer photonic structures made out of Si and
SiO2 layers. The structures have increasing quasiperiodic modulation of layer
thicknesses (from left to right); the leftmost structure is a perfect one-
dimensional photonic crystal and the rest are photonic quasicrystals.

The system considered here is shown in Fig. 1 and consists of a set of multi-
layer structures made out of two materials, silicon and silica (SiO2), with
refractive indices $n_{\text{Si}}=3.5$ and $n_{\text{SiO}_{2}}=1.5$,
respectively. These layers are stacked along the $z$-direction and define the
dielectric function, $\epsilon(z)$. When propagation purely along the
$z$-direction is considered, this system is described by the following Maxwell
eigenvalue problem for a single scalar field $\mathcal{H}(z)$
photoniccrystalsbook ; photoniccrystalsbook2 :

$\displaystyle-\partial_{z}\left(\frac{1}{\epsilon(z)}\partial_{z}\right)\mathcal{H}(z)=$
$\displaystyle\left(\frac{\omega}{c}\right)^{2}\mathcal{H}(z),$ (1)

where $\mathbf{H}_{\text{TE}}=\mathcal{H}(z)\mathbf{\hat{x}}$ and
$\mathbf{H}_{\text{TM}}=\mathcal{H}(z)\mathbf{\hat{y}}$ are the degenerate TE-
and TM-polarized magnetic field solutions respectively, with frequency
eigenvalue $\omega$.

Motivated by the Aubry-André model, we modulate the thicknesses of each layer
in a unit cell, defined as a pair of neighboring Si and SiO2 layers, according
to

$\displaystyle t_{n}=t_{0}[1+A\cos(2\pi\beta n)],$ (2)

where $n\in\\{1,2,...,N\\}$ identifies a pair of layers, $2N$ is the total
number of layers, $A$ is the strength of the spatial modulation and $\beta$ is
the closest Diophantine (rational) approximation to the golden mean,
$\phi=(1+\sqrt{5})/2$, for a given value of system size, $N$.

When $A=0$, all layers have the same thickness $t_{0}$ and the system is a 1D
PhC with a lattice constant of $a=2t_{0}$, whereas for non-zero values of $A$,
the integer sampling frequency of the cosine term and the irrational
modulation frequency $\beta$ provide two competing and incommensurate periods
that result in a 1D PhQC. In the latter case, the average lattice constant
$\langle a\rangle=2t_{0}$ provides a convenient length scale. We note that
since $A$ modulates the thicknesses of layers, it is a bounded parameter with
$|A|\leq 1$.

Figure 2:  (a) Eigenvalue spectrum of the PhQC states and their corresponding
IPR as a function of $A$ for $N=89$. (b) The transmission spectrum as a
function of $A$ for $N=89$. Localization of various states corresponds to
sharp drops in transmission (white arrows). Some states undergo a second
delocalization transition around $A=0.8$, which results in a sharp recovery of
transmission (blue arrow). (c) $\mathcal{H}(z)$-field profiles of the state
marked with the black arrow in (a), for various values of $A$.

We obtain the states of our PhQCs using the plane-wave expansion method, as
implemented in the open source software package MIT Photonic Bands (MPB) MPB ,
and calculate their inverse participation ratios (IPR) given by

$\displaystyle\text{IPR}_{p}=\frac{\int|\mathcal{H}_{p}(z)|^{4}dz}{\left[\int|\mathcal{H}_{p}(z)|^{2}dz\right]^{2}},$
(3)

where $\mathcal{H}_{p}$ is the scalar field in (1), corresponding to the
$p$-th state and the integral is taken over the entire finite system. IPR is a
measure of localization of states, where small (large) values of IPR indicate
extended (localized) states.

The results for a system size of $N=89$ are shown in Fig. 2. Figure 2 (a)
shows a plot of the eigenvalue spectrum of the PhQC states as a function of
$A$ and their corresponding IPR. In this plot, we focus on states
corresponding to the second band in the PhC limit (i.e., at $A=0$), and
convert their corresponding frequency eigenvalues to dimensionless wavelength.
For small values of $A$ ($A<0.3$), the states are extended since the structure
may be thought of as being crystalline with a small quasicrystalline
perturbation. For larger values of $A$, the states undergo transitions to a
localized regime, as indicated by a sharp increase in their IPR. However, as
seen from Fig. 2 (a), these transitions do not all occur at the same value of
$A$. Moreover, for some states around $\lambda/\langle a\rangle=3.2$ and
$A=0.8$, we observe a sharp reduction in IPR on further increasing $A$,
marking a reentrant transition to a second extended regime for these states.
In Fig. 2 (c), we also examine the $\mathcal{H}(z)$-field profile for one such
state that undergoes a reentrant transition, marked by the arrow in Fig. 2
(a). The field profiles show the transition from extended to localized and
back to extended as $A$ is increased.

Our system thus exhibits some crucial distinctions from the simple Aubry-André
model. Each pair of layers that forms a unit cell in our PhQCs is not well
approximated as a resonator or atomic potential that is evanescently coupled
only to its nearest neighbors. If the PhQC corresponds to a tight-binding
model at all, it must be thought of as possessing long-range couplings that
can be accurately computed using Wannier-function methods PhC_wannier1 . The
presence of these effectively long-range couplings creates single-particle
mobility edges that result in intermediate regimes where both extended and
localized states co-exist AA_longrange1 . In fact, we find that due to the
bounded nature of the quasiperiodic modulation strength via the parameter $A$,
a large part of the spectrum of the PhQC is in an intermediate regime SupplMat
; AA_generic_critical . Moreover, the states corresponding to the lowest band
of the PhC limit never localize for any value of $A$ up to its bounds SupplMat
. This is because PhQCs act as effectively homogeneous dielectric media at
long wavelengths. Finally, the presence of a reentrant transition suggests the
breakdown of the duality between the localized and extended regimes that
exists in the simple Aubry-André model.

Since the localization of states causes the cessation of wave transport, we
explore its consequences in the transmission spectrum of the PhQCs. Fig. 2 (b)
shows a plot of the transmission spectrum of the PhQCs as a function of $A$,
calculated using a transfer matrix approach. We see that the localization
transitions from Fig. 2 (a) correspond to the vanishing of transmission
through the structures. This occurs because the system size is larger than the
localization length of the localized states and as such these states are
unable to form a transmission channel across the structures. Furthermore, we
also see a recovery of transmission around $\lambda/\langle a\rangle\sim 3.2$,
which corresponds to the reentrant transition of some of the states to an
extended regime. This observation is not a finite size effect and persists for
much larger system sizes SupplMat . PhQCs and their transmission spectra
therefore provide an accessible experimental setting in which to explore
localization phenomena in one-dimensional systems.

Figure 3:  (a) Scanning electron microscope (SEM) image of a cut through a
typical one-dimensional photonic quasicrystal fabricated by PECVD. Both layers
in a pair of neighboring Si and SiO2 layers have identical thicknesses. The
thickness values of each such pair are modulated according to Eq. (2). (b)
Experimentally measured transmission spectrum as a function of $A$ for $N=13$.
(c) Simulated transmission spectrum as a function of $A$ for $N=13$. In (b)
and (c), the localization transitions are marked with white arrows and the
reentrant delocalization transition is marked with a blue arrow.

For the experiment, we fabricate the PhQCs using plasma-enhanced chemical
vapor deposition (PECVD), alternating between Si and SiO2 deposition on a
glass substrate. The deposition times control the thicknesses of each layer
and are determined from (2). We fabricate a total of ten samples with $\langle
a\rangle=$ $0.25\text{\,}\mathrm{\SIUnitSymbolMicro m}$, $t_{0}=$
$0.125\text{\,}\mathrm{\SIUnitSymbolMicro m}$, $N=13$, $\beta=21/13$ and
varying values of $A$. A scanning electron microscope (SEM) image of a typical
sample is shown in Fig. 3 (a). To characterize our samples, we measure the
transmission spectrum of each sample as a function of wavelength, normalized
to the transmission through the bare glass substrate. This is performed using
a supercontinuum laser in combination with a filter that allows for wavelength
selection in the range of $690\text{\,}\mathrm{nm}$ to
$1100\text{\,}\mathrm{nm}$. The transmitted power is measured with a
photodiode power sensor.

The measured transmission spectrum is shown in Fig. 3 (b), along with the
simulation results for comparison in Fig. 3 (c). We find that despite the
relatively small system size, the localization transitions for states near
$\lambda\sim$ $0.85\text{\,}\mathrm{a}$nd
$0.75\text{\,}\mathrm{\SIUnitSymbolMicro m}$ are clearly observed as the sharp
inhibition of transmission. Furthermore, the second transition to an extended
regime near $\lambda\sim$ $0.78\text{\,}\mathrm{\SIUnitSymbolMicro m}$ is
observed as a sharp recovery of transmission for $A>0.8$. The states near
$\lambda\sim$ $1\text{\,}\mathrm{\SIUnitSymbolMicro m}$ are localized for
$A>0.5$, however the system size in the experiment is smaller than the
localization length of these states and as such we measure finite transmission
around this wavelength SupplMat . Therefore it would be possible in principle
to extract the localization length of states directly from the transmission
spectrum of PhQCs by varying the system size.

To further explore the observed localization features, we develop a tight-
binding model that qualitatively captures the physics of localization in our
PhQCs. In particular, we consider a 1D quasiperiodic model with nearest and
next-nearest neighbor couplings given by the Hamiltonian

$H=\sum_{j=A,B}\sum_{i=1}^{N}E_{i,j}\left[1+\alpha\cos(2\pi\beta
i)\right]n_{i,j}\\\
-t_{\text{NN}}\sum_{i=1}^{N}\left(c^{{\dagger}}_{i,A}c_{i,B}+\text{h.c.}\right)-t_{\text{NN}}\sum_{i=1}^{N-1}\left(c^{{\dagger}}_{i,B}c_{{i+1},A}+\text{h.c.}\right)\\\
-t_{\text{NNN}}\sum_{j=A,B}\sum_{i=1}^{N-1}\left(c^{{\dagger}}_{i,j}c_{i+1,j}+\text{h.c.}\right),$
(4)

where $c^{{\dagger}}_{i,j}$, $c_{i,j}$ and $n_{i,j}$ are respectively the
creation, annihilation and number operators on site $i$ of sublattice $j=A,B$.
$t_{\text{NN}}$, $t_{\text{NNN}}$ are the nearest- and next-nearest-neighbor
couplings respectively and $E_{i,j}$ are the unperturbed on-site energies of
site $i$ of sublattice $j$. This lattice of $2N$ sites is shown schematically
in Fig. 4 (a). We choose $E_{i,A}=1$ as the energy scale and set
$\beta=(1+\sqrt{5})/2$. $\alpha$ is an unbounded parameter that governs the
strength of the quasiperiodic modulation of the on-site energies and we choose
$E_{i,B}\neq E_{i,A}$ to introduce dimerization, akin to the two different
layers in the PhQCs.

We plot the IPR of the states of this model for $N=89$ in Fig. 4 (b), where we
observe some important qualitative similarities with our PhQCs. The states of
the lower band stay extended until a much larger value of $\alpha$, compared
to the upper band, similar to the lowest two bands in the PhQCs. We also
observe that the states of the upper band exhibit mobility edges and an
intermediate regime, before undergoing a transition to a completely localized
regime at $\alpha\sim 0.25$. Moreover, some states of this band undergo a
second transition at $\alpha\sim 0.6$ and remain extended for a range of
$\alpha$ values. Eventually all states of this model become localized for a
large enough value of $\alpha$ ($\alpha>2$). We find that these features are
generic and persist for a range of parameters of the model.

We also calculate the average IPR and average normalized participation ratio
(NPR) for a set of $M$ states, given by

$\displaystyle\langle\text{IPR}\rangle=$
$\displaystyle\frac{1}{M}\sum_{n=1}^{M}\sum_{i=1}^{2N}|\psi_{n,i}|^{4},$ (5)
$\displaystyle\langle\text{NPR}\rangle=$
$\displaystyle\frac{1}{M}\sum_{n=1}^{M}\left(2N\sum_{i=1}^{2N}|\psi_{n,i}|^{4}\right)^{-1},$
(6)

where $|\psi_{n,i}\rangle$ is the normalized $n$-th eigenstate of $H$ and $i$
labels the sites. The extended regime is characterized by near-zero
$\langle\text{IPR}\rangle$ and non-zero $\langle\text{NPR}\rangle$ and vice
versa for the localized regime. A non-zero value for both
$\langle\text{IPR}\rangle$ and $\langle\text{NPR}\rangle$ implies the presence
of an intermediate regime in the spectrum. The plot of
$\langle\text{IPR}\rangle$ and $\langle\text{NPR}\rangle$ for the states of
the upper band is shown in Fig. 4 (c). Comparing this plot with Fig. 4 (b), we
can see that the first intermediate regime arises due to a moblity edge and
the second intermediate regime arises due to a reentrant delocalization
transition for the lowest lying states of the band.

Through this model, we see that a combination of staggered potentials and
long-range couplings, competing with quasiperiodicity, can cause the
delocalization of previously localized states for a range of parameter values.
These findings are consistent with other models with similar qualities that
are known to host reentrant transitions reentrant1 ; reentrant3 .

Figure 4:  (a) Schematic of the tight binding model. The dimerized unit cell
for the periodic system ($\alpha=0$) is highlighted. The solid (dotted) lines
represent nearest-neighbor (next-nearest-neighbor) couplings. (b) The energy
spectrum and IPR of the corresponding states of the model for $E_{i,A}=1$,
$E_{i,B}=2$, $t_{\text{NN}}=0.7$, $t_{\text{NNN}}=0.35$ and $N=89$. The states
of the second band exhibit a mobility edge and are localized for
$0.25<\alpha<0.6$. Some states of this band undergo a reentrant delocalization
transition at $\alpha\sim 0.6$ (c) A plot of the $\langle\text{IPR}\rangle$
and $\langle\text{NPR}\rangle$ for the states of the second band. The
highlighted areas indicate intermediate regimes, where both
$\langle\text{IPR}\rangle$ and $\langle\text{NPR}\rangle$ are non-zero, and
localized and extended states co-exist.

In conclusion, we have observed a reentrant delocalization transition – a
feature that is not present in the standard Aubry-André model – in 1D PhQCs
with an Aubry-André-type quasiperiodic modulation by measuring their
transmission spectra. The PhQCs and their transmission spectra thus provide a
means to experimentally explore more complex models with richer localization
physics, as compared with simple nearest-neighbor tight-binding models.
Inspired by the PhQCs, we have also explored the localization features of our
system in a tight-binding setting, in order to lend physical insight into the
nature of the transition. In the future, it will be interesting to explore
localization in passive non-Hermitian 1D PhQCs enabled via a patterning of
lossy dielectric materials. Furthermore, examining localized states in higher
dimensional realizations of PhQCs is warranted since it could lead to better-
performing photonic nanocavities with lower index materials.

We thank Zeyu Zhang for SEM images of the PhQCs and Bill Mahoney for technical
help with the PECVD process. C.J. acknowledges funding from the Alexander von
Humboldt Foundation within the Feodor-Lynen Fellowship program. M.C.R. and
S.V. acknowledge the support of the Charles E. Kaufman Foundation under Grant
No. KA2020-114794, the U.S. Office of Naval Research Multidisciplinary
University Research Initiative (MURI) under Grant No. N00014-20-1-2325 as well
as the U.S. Army Research Office MURI under Grant No. W911NF-22-2-0103. K. L.
and M. G. acknowledge support from the Pennsylvania State University REU
program.

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