# Role of the high-spin nucleon and delta resonances in the $K\Lambda$ and
$K\Sigma$ photoproduction off the nucleon

N. H. Luthfiyah  T. Mart<EMAIL_ADDRESS>Departemen Fisika, FMIPA,
Universitas Indonesia, Depok 16424, Indonesia

###### Abstract

We have investigated the effect of nucleon and delta resonances with spins
11/2, 13/2, and 15/2 in the kaon photoproduction process $\gamma+N\to K+Y$ by
using two covariant isobar models. The formalism for high-spin propagators and
interaction Lagrangians were adopted from the works of Pascalutsa and Vrancx
et al. The calculated scattering amplitudes were decomposed into six Lorentz-
and gauge-invariant matrices, from which we calculated the cross sections and
polarization observables. The unknown parameters in the amplitudes, i.e., the
coupling constants and hadronic form factor cutoffs, were obtained by fitting
the calculated observables to experimental data. In the $K\Lambda$ channels
the inclusion of $N(2600)I_{1,11}$ and $N(2700)K_{1,13}$ resonances improves
the agreement between model calculations and experimental data significantly
and reduces the dominance of resonances in the model by increasing the
hadronic form factor cutoff of the Born terms. Furthermore, the inclusion of
these resonances reduces the number of resonance structures in cross sections,
including the structure in the $K^{0}\Lambda$ differential cross section at
$W\approx 1650$ MeV, which could become a hint of the narrow resonance. In the
$K\Sigma$ channels the inclusion of $N(2600)I_{1,11}$, $N(2700)K_{1,13}$,
$\Delta(2420)H_{3,11}$, $\Delta(2750)I_{3,13}$, and $\Delta(2950)K_{3,15}$
states also significantly improves the model and increases the hadronic form
factor cutoff of the Born terms. However, different from the $K\Lambda$
channels, the inclusion of these high-spin resonances leads to more resonance
structures in the $K^{+}\Sigma^{0}$ differential cross section. This
investigation reveals that the second and third peaks in the $K^{+}\Sigma^{0}$
differential cross section originate from the $\Delta(2000)F_{35}$ and
$N(2290)G_{19}$ resonances, respectively. We have also evaluated the resonance
properties at the pole positions and using the Breit-Wigner method. In both
$K\Lambda$ and $K\Sigma$ channels the inclusion of the high-spin baryon
resonances was found to improve the agreement between the resonance properties
obtained in this study and those listed by the Particle Data Group.

###### pacs:

13.60.Le, 14.20.Gk, 25.20.Lj

##  I INTRODUCTION

Recently, the effect of spins-7/2 and -9/2 nucleon resonances on kaon
photoproduction processes has been phenomenologically investigated by using a
covariant isobar model Clymton:2017nvp , in which the scattering amplitude was
calculated by using the appropriate Feynman diagrams depicted in Fig. 1. The
analytical calculation performed in this study made use of the consistent
interaction Lagrangians proposed by Pascalutsa Pascalutsa:2000kd . The
calculated observables were fitted to nearly 7400 experimental data points.
The result of the fitting process showed that the inclusion of spins-7/2 and
9/2 nucleon resonances could improve the agreement between the model
calculation and the experimental data. The model was later extended to
describe both $\gamma p\to K^{+}\Lambda$ and $\gamma n\to K^{0}\Lambda$
processes, simultaneously Mart:2019mtq . In the latter, the model was fitted
to nearly 9400 data points, including recent data from the CLAS and MAMI
collaborations. The extended model yielded a nice agreement between the
calculated observables and experimental data in both isospin channels.

Despite the success of the model, it has not yet consider the resonances with
spins-11/2, 13/2 and 15/2, which are tabulated by the Particle Data Group
(PDG) listing pdg  (see Table 1). Given the fact that the inclusion of
spins-7/2 and 9/2 resonances in Ref. Clymton:2017nvp  significantly improves
the model, we could also expect that a similar phenomenon would be obtained
with the inclusion of the resonances listed in Table 1. Furthermore, we can
also extend the model to include the four $K\Sigma$ isospin channels, i.e.
$K^{+}\Sigma^{0}$, $K^{0}\Sigma^{+}$, $K^{+}\Sigma^{-}$ and $K^{0}\Sigma^{0}$
channels. Since the total isospin of these channels is 3/2, the $\Delta$
resonances are allowed as the intermediate states. An isobar model for
$K\Sigma$ reaction can be constructed from the well-known $K\Lambda$ model
based on our previous studies, e.g., in Ref. Clymton:2019nrs . For the
$K\Lambda$ final states, the isospin symmetry couples the $K^{+}\Lambda$ and
$K^{0}\Lambda$ channels, whereas for the $K\Sigma$ final states, the
$K^{+}\Sigma^{0}$, $K^{0}\Sigma^{+}$, $K^{+}\Sigma^{-}$, and $K^{0}\Sigma^{0}$
production processes are coupled by isospin symmetry to one model. As
mentioned above, there are nearly 9400 data points available for the
$K\Lambda$ channels, and nearly 8000 data points for $K\Sigma$ ones. These
data sets will be fitted to the $K\Lambda$ and $K\Sigma$ models separately. In
principle, all six isospin channels can also be coupled, since all $K\Lambda$
and $K\Sigma$ isospin channels utilize the same leading Born coupling
constants, i.e., the $g_{K\Lambda N}$ and $g_{K\Sigma N}$. However, since we
have fixed these coupling constants to the SU(3) values Adelseck:1990ch , the
$K\Lambda$ and $K\Sigma$ channels are naturally decoupled.

Table 1: Nucleon and delta resonances with spins from 11/2 to 15/2 used in our study and tabulated by the PDG in their particle listing pdg . Resonance | $J^{P}$ | Status | Mass (MeV) | Width (MeV)
---|---|---|---|---
$N(2600)I_{1,11}$ | $11/2^{-}$ | *** | $2600\pm 50$ | $650\pm 150$
$N(2700)K_{1,13}$ | $13/2^{+}$ | ** | $2612\pm 45$ | $350\pm 50$
$\Delta(2420)H_{3,11}$ | $11/2^{+}$ | **** | $2450\pm 150$ | $500\pm 200$
$\Delta(2750)I_{3,13}$ | $13/2^{-}$ | ** | $2794\pm 80$ | $350\pm 100$
$\Delta(2950)K_{3,15}$ | $15/2^{+}$ | ** | $2990\pm 100$ | $330\pm 100$

To the best of our knowledge, there are limited studies investigating the
contribution of high-spin nucleon and delta resonances in kaon photoproduction
with the field-theoretic model. Presumably, this is due to the complicated
formulations of propagator and vertex factors along with the problem of lower-
spin background that plagued the formulation of high-spin ($J>1/2$) resonance
propagator. Therefore, the first purpose of this paper is to set forth the
formulation of higher-spin resonances amplitude. After that we can study their
effect on the six isospin channels of kaon photoproduction (see Table 2) by
means of a covariant isobar model.

Table 2: Six possible isospin channels of kaon photoproduction on the nucleon along with their threshold energies in terms of photon lab energy $k^{\rm thr.}$ and total c.m. energy $W^{\rm thr.}$. No. | Reactions |  $k^{\rm{thr.}}$ (MeV) |  $W^{\rm{thr.}}$ (MeV)
---|---|---|---
1) | $\gamma+p\to K^{+}+\Lambda$ | 911 | 1609
2) | $\gamma+n\to K^{0}+\Lambda$ | 915 | 1613
3) | $\gamma+p\to K^{+}+\Sigma^{0}$ | 1046 | 1686
4) | $\gamma+p\to K^{0}+\Sigma^{+}$ | 1048 | 1687
5) | $\gamma+n\to K^{+}+\Sigma^{-}$ | 1052 | 1691
6) | $\gamma+n\to K^{0}+\Sigma^{0}$ | 1051 | 1690
Figure 1: Feynman diagrams of the kaon photoproduction $\gamma(k)+N(p)\to
K(q)+Y(p_{Y})$ for the (a) $s$-channel, (b) $u$-channel, and (c) $t$-channel
intermediate states.

We have organized this paper as follows. In Sec. II we present the formalism
used in our study. In Sec. III we present the numerical result and discuss the
comparison between model calculations and experimental data. Finally, in Sec.
IV we summarize and conclude our work. The extracted form functions used to
calculate the observables for the fitting process are given in Appendix A.

##  II Formalism

As mentioned above we adopt the formalism of the nucleon propagators and the
interaction Lagrangian developed by Pascalutsa Pascalutsa:2000kd  and Vrancx
et al. Vrancx:2011qv . In our previous work, we explained this formalism in
details and constructed the reaction amplitude for the nucleon resonances with
spins 7/2 and 9/2 Clymton:2017nvp . To facilitate the reader, in this section
we briefly discuss this formalism and derive the construction of the
spins-11/2, -13/2, and -15/2 resonance propagators along with their
interaction Lagrangians. A preliminary result for the analytical form of the
production amplitudes involving nucleon resonances with spins up to 13/2 has
been reported in a conference Luthfiyah:2020ogb . The notation of the four-
momentum of photon, nucleon, kaon, and hyperon used in the following
discussion is given in the caption of Fig. 1, with $p_{R}=p+k=p_{Y}+q$.

###  II.1 Consistent Interaction Theory

A consistent interaction Lagrangian is required to eliminate the appearance of
the lower-spin background amplitude, which is known as an intrinsic problem in
the Rarita-Schwinger (R-S) formulation of the spin-3/2 (or higher) propagator.
A number of solutions have been put forward to solve this problem in the last
decades. Among them, those of Pascalutsa Pascalutsa:2000kd  and Vrancx et al.
Vrancx:2011qv  are relevant to our present work. They constructed the
interaction Lagrangians which automatically cancel out the lower spin
contributions to the scattering amplitude. Pascalutsa proposed a gauge-
invariant interaction structure for spin-3/2 particles by introducing a local
symmetry to the R-S field $\psi_{\mu_{1}...\mu_{n}}$. The local symmetry reads
Vrancx:2011qv

$\displaystyle\psi_{\mu_{1}...\mu_{n}}\rightarrow\psi_{\mu_{1}...\mu_{n}}\,+\frac{1}{n(n-1)!}\,\sum_{P(\mu)}\,\partial_{\mu_{1}}\,\xi_{\mu_{2}...\mu_{n}},$
(1)

where the totally symmetric and space-time dependent tensor-spinor field
$\xi_{\mu_{2}...\mu_{n}}$ fulfills

$\displaystyle\gamma^{\mu_{1}}\,\xi_{\mu_{1}\mu_{2}...\mu_{n-1}}=0~{}.$ (2)

To construct a consistent interaction Lagrangian for particles with spin-3/2
we need a gauge-invariant field $G_{\mu\nu}$, taken from Ref.
Pascalutsa:2000kd , and the interaction operator
$O^{(3/2)}_{(\mu,\nu)\lambda}$, adopted from Ref. Vrancx:2011qv . The
invariant field and the interaction operator is written in the form of

$\displaystyle G_{\mu\nu}$ $\displaystyle=$
$\displaystyle\partial_{\mu}\psi_{\nu}\,-\,\partial_{\nu}\psi_{\mu}$ (3)
$\displaystyle=$
$\displaystyle(\partial_{\mu}g_{\nu\lambda}-\partial_{\nu}g_{\mu\lambda})\psi^{\lambda}$
$\displaystyle=$ $\displaystyle O^{(3/2)}_{(\mu,\nu)\lambda}\psi^{\lambda},$

with

$O^{(3/2)}_{(\mu,\nu)\lambda}=(\partial_{\mu}g_{\nu\lambda}-\partial_{\nu}g_{\mu\lambda}),$
(4)

where $\psi_{\mu}$ is the massive R-S field that obeys the R-S equation and
its constraints. The interaction operator fulfills the following property,

$\displaystyle\partial^{\lambda}\,O^{3/2}_{(\mu,\nu)\lambda}\,(\partial)=\,O^{3/2}_{(\mu,\nu)\lambda}\,(\partial)\,\partial^{\lambda},$
(5)

which maintains the invariance of $G_{\mu\nu}$ under the uR-$\rm{S_{5/2}}$
gauge as stated in Eq. (1). With the invariant field and spin-3/2 interaction
operator, Vrancx et al. constructed a consistent interaction theory for
spin-5/2 field. This formulation is later expanded for general high-spin
field. Vrancx et al. defined a general interaction operator as Vrancx:2011qv

$\displaystyle
O^{(n+1/2)}_{(\mu_{1}...\mu_{n},\nu_{1}...\nu_{n})\lambda_{1}...\lambda_{n}}\,(\partial)=$
$\displaystyle{\displaystyle\frac{1}{(n!)^{2}}\sum_{P(\nu)}\sum_{P(\lambda)}}O^{(3/2)}_{(\mu_{1},\nu_{1})\lambda_{1}}\,(\partial)\,...\,O^{(3/2)}_{(\mu_{n},\nu_{n})\lambda_{n}}\,(\partial)\,,$
(6)

with $P(\mu)$$P(\lambda)$ indicate all possible permutations for all $\mu$ and
$\lambda$. Vrancx reduced the indices of the $G_{\mu\nu}$ field introduced by
Pascalutsa in Eq. (3), because the $G_{\mu\nu}$ field contains too many
indices compared to the original field $\psi_{\mu_{1}..\mu_{n}}$. A new field
is introduced by Vrancx as
$\Psi_{\mu}=O^{(3/2)}_{(\mu,\nu)\lambda}\,\psi^{\lambda}\,\gamma^{\nu}$. For
particles with spin-$(n+1/2)$ the gauge-invariant field can be written as

$\displaystyle\Psi_{\mu_{1}...\mu_{n}}=O^{(n+1/2)}_{(\mu_{1}...\mu_{n},\nu_{1}...\nu_{n})\lambda_{1}...\lambda_{n}}\,(\partial)\,\psi^{\lambda_{1}...\lambda_{n}}\,\gamma^{\nu_{1}...\nu_{n}}.$
(7)

The interaction operator for the above field reads

$\displaystyle\mathcal{O}^{(n+1/2)}_{(\mu_{1}...\mu_{n})\lambda_{1}...\lambda_{n}}\,(\partial)=\gamma^{\nu_{1}..\nu_{n}}O^{(n+1/2)}_{(\mu_{1}...\mu_{n},\nu_{1}...\nu_{n})\lambda_{1}...\lambda_{n}}\,(\partial)$
$\displaystyle={\displaystyle\frac{1}{(n!)^{2}}\sum_{P(\lambda)}}\mathcal{O}^{(3/2)}_{(\mu_{1})\lambda_{1}}\,(\partial)\,...\,\mathcal{O}^{(3/2)}_{(\mu_{n})\lambda_{n}}\,(\partial)\,,$
(8)

where

$\displaystyle\mathcal{O}^{(3/2)}_{(\mu_{n})\lambda_{n}}\,(\partial)=\gamma^{\nu}\,O^{(3/2)}_{(\mu,\nu)\lambda}\,(\partial)\,.$
(9)

With this consistent interaction we are ready to construct the Lagrangians for
high-spin interactions.

###  II.2 Interaction Lagrangians

The basic Lagrangian for the kaon-hyperon-nucleon interaction is

$\displaystyle\mathcal{L}_{\mathrm{had}}=g_{KYN}\,\bar{\Psi}_{Y}\,\gamma_{5}\,\Psi_{N}\,\Phi_{K}.$
(10)

Following this basic Lagrangian, the standard interaction Lagrangian for the
interaction with spin-1/2 resonances is written as

$\displaystyle\mathcal{L}_{\mathrm{had}}=g_{KYR}\,\bar{\Psi}_{Y}\,\gamma_{5}\,\Psi_{R}\,\Phi_{K}+{\rm
H.c.}$ (11)

According to Pascalutsa the hadronic interaction Lagrangian for spin-3/2
resonance with mass $m_{R}$ reads Pascalutsa-PRD1998

$\displaystyle\mathcal{L}_{\mathrm{had}}=\frac{g_{KYR}}{m_{R}^{2}}\,\epsilon^{\mu\nu\alpha\beta}\,\bar{\Psi}_{Y}\,\partial_{\beta}\phi^{*}\,\gamma_{5}\,\gamma_{\alpha}\,(\partial_{\mu}\psi_{\nu})+\mathrm{H.c},$
(12)

where $\bar{\Psi}_{Y}$ is the spinor field of the hyperon, $\phi$ is the
pseudoscalar field of the kaon, and $\psi_{\nu}$ is the massive R-S field of
the nucleon or resonance. However, the above Lagrangian is inconsistent for
higher-spin resonances. With the substitution proposed by Vrancx,
$\psi_{\mu}\,\rightarrow\,\bar{\Psi}_{\mu}/m_{R}$, the Lagrangian for a
resonance with spin 3/2 can be written as

$\displaystyle\mathcal{L}_{\mathrm{had}}$ $\displaystyle=$
$\displaystyle\frac{g_{KYR}}{m_{R}^{3}}\,\epsilon^{\mu\nu\alpha\beta}\,\bar{\Psi}_{Y}\,\partial_{\beta}\phi^{*}\,\gamma_{5}\,\gamma_{\alpha}\,\partial_{\mu}\Psi_{\nu}+\mathrm{H.c.},~{}~{}$
(13)

whereas the Lagrangian for hadronic interaction with spin-$(n+1/2)$ resonances
has the form of

$\displaystyle\mathcal{L}_{\mathrm{had}}$ $\displaystyle=$
$\displaystyle\frac{g_{KYR}}{m_{R}^{2n+1}}\,\epsilon^{\mu\nu_{n}\alpha\beta}\,\partial^{\nu_{1}}\,...\partial^{\nu_{n-1}}\,\bar{\Psi}_{Y}\,\partial_{\beta}\phi^{*}$
(14)
$\displaystyle\times\gamma_{5}\,\gamma_{\alpha}\,(\partial_{\mu}\Psi_{\nu_{1}...\nu_{n}})+\mathrm{H.c}$
$\displaystyle=$
$\displaystyle\frac{g_{KYR}}{m_{R}^{2n+1}}\,\epsilon^{\mu\nu\alpha\beta}\,\bar{\Psi}_{Y}\,\partial_{\beta}\phi^{*}\,\gamma_{5}\,\gamma_{\alpha}$
$\displaystyle\times\partial_{\mu}O^{(n+1/2)}_{(\mu_{1}...\mu_{n},\nu_{1}...\nu_{n})\lambda_{1}...\lambda_{n}}\,(\partial)$
$\displaystyle\times\gamma^{\alpha_{1}}\,...\gamma^{\alpha_{n}}\,\psi^{\lambda_{1}...\lambda_{n}}+{\rm
H.c}.$

In addition to the hadronic Lagrangian, Pascalutsa has also constructed the
Lagrangian for the electromagnetic interaction, which is written as
Pascalutsa-PRC1999

$\displaystyle\mathcal{L}_{\rm em}$ $\displaystyle=$
$\displaystyle\frac{e}{m_{R}^{3}}\bar{\Psi}^{\beta}\Bigl{\\{}(g_{1}\,\epsilon_{\mu\nu\alpha\beta}\partial^{\alpha}\Psi+g_{2}\,\gamma_{5}\,g_{\beta\nu}\partial_{\mu}\Psi$
(15)
$\displaystyle+g_{3}\,\gamma_{\mu}\,\gamma^{\rho}\,\epsilon_{\rho\nu\alpha\beta}\partial^{\alpha}\Psi+g_{4}\,\gamma_{5}\,\gamma_{\mu}\,\gamma^{\rho}$
$\displaystyle\times(\partial_{\rho}g_{\nu\beta}-\partial_{\nu}g_{\rho\beta})\Psi\Bigr{\\}}F^{\mu\nu}+\mathrm{H.c}\,,$

where $F^{\mu\nu}$ is the conventional electromagnetic field strength tensor.
To correctly model the interaction in discussion, a consistent interaction for
the electromagnetic interaction Lagrangian is needed. The electromagnetic
interaction Lagrangian for the resonance particles with spin-$(n+1/2)$ can be
written as

$\displaystyle\mathcal{L}_{\rm em}$ $\displaystyle=$
$\displaystyle\frac{e}{m_{R}^{2n+1}}\bar{\Psi}^{\beta_{1}\cdots\beta_{n}}\bigl{\\{}(g_{1}\,\epsilon_{\mu\nu\alpha\beta_{n}}\partial^{\alpha}\Psi$
(16)
$\displaystyle+g_{2}\,\gamma_{5}\,g_{\beta_{n}\nu}\partial_{\mu}\Psi+g_{3}\,\gamma_{\mu}\,\gamma^{\rho}\,\epsilon_{\rho\nu\alpha\beta_{n}}\partial^{\alpha}\Psi$
$\displaystyle+g_{4}\,\gamma_{5}\,\gamma_{\mu}\,\gamma^{\rho}\,(\partial_{\rho}g_{\nu\beta_{n}}-\partial_{\nu}g_{\rho\beta_{n}})\Psi\bigr{\\}}$
$\displaystyle\times\partial_{\beta_{1}}\cdots\partial_{\beta_{n-1}}F^{\mu\nu}+\mathrm{H.c.}$

The consistency of the interaction is guaranteed by the operator interaction
which fulfills:

$\displaystyle
p_{R}^{\lambda_{i}}\,O^{(n+1/2)}_{(\mu_{1}...\mu_{n},\nu_{1}...\nu_{n})\lambda_{1}...\lambda_{n}}(p_{R})=0,$
(17)

with $i=1,2,...,n$ and $p_{R}^{\lambda_{i}}$ is the four-momenta of the
resonance particles with spin-$(n+1/2)$.

The electromagnetic and hadronic vertices, which are required to calculate the
scattering amplitude, can be obtained from the interaction Lagrangians
constructed in the previous discussion. As the result the hadronic vertex can
be written as

$\displaystyle\Gamma^{\mathrm{had}}_{\mu_{1}...\mu_{2}}$ $\displaystyle=$
$\displaystyle\frac{g_{KYR}}{m_{R}^{2n+1}}\,\epsilon^{\mu\nu_{n}\alpha\beta}\,p_{\Lambda}^{\nu_{1}}\,...p_{\Lambda}^{\nu_{n-1}}\,q_{\beta}\,\gamma_{5}\,\gamma_{\alpha}\,{p_{R}}_{\mu}$
(18) $\displaystyle\times
O^{(n+1/2)}_{(\nu_{1}...\nu_{n},\alpha_{1}...\alpha_{n})\mu_{1}...\mu_{n}}\,(p_{R})\gamma^{\alpha_{1}}\,...\gamma^{\alpha_{n}}.~{}~{}~{}$

and the electromagnetic vertex reads

$\displaystyle\Gamma^{\rm em}_{\nu_{1}\cdots\nu_{n}}$ $\displaystyle=$
$\displaystyle\frac{e}{m_{R}^{2n+1}}\,O_{n+1/2}^{(\beta_{1}\cdots\beta_{n},\alpha_{1}\cdots\alpha_{n})\nu_{1}\cdots\nu_{n}}(p_{R})\gamma_{\alpha_{1}}\cdots\gamma_{\alpha_{n}}$
(19)
$\displaystyle\times\,\Bigl{\\{}g_{1}\,\epsilon_{\mu\nu\alpha\beta_{n}}p^{\alpha}+g_{2}\,\gamma_{5}\,g_{\beta_{n}\nu}p_{\mu}$
$\displaystyle+g_{3}\,\gamma_{\mu}\,\gamma^{\rho}\,\epsilon_{\rho\nu\alpha\beta_{n}}p^{\alpha}$
$\displaystyle+g_{4}\,\gamma_{5}\,\gamma_{\mu}\,\gamma^{\rho}\,(p_{\rho}g_{\nu\beta_{n}}-p_{\nu}g_{\rho\beta_{n}})\Bigr{\\}}$
$\displaystyle\times k_{\beta_{1}}\cdots
k_{\beta_{n-1}}\,(k^{\mu}\epsilon^{\nu}-k^{\nu}\epsilon^{\mu})+\mathrm{H.c.}$

The vertex factors can be simplified to

$\displaystyle\Gamma^{\mathrm{had}}_{\mu_{1}...\mu_{2}}$ $\displaystyle=$
$\displaystyle\frac{1}{m_{R}^{n}}\tilde{\Gamma}_{\mathrm{had}}^{\nu_{1}...\nu_{n}}\,\tilde{O}^{(n+1/2)}_{(\nu_{1}...\nu_{n})\mu_{1}...\mu_{n}}\,(p_{R})$
(20) $\displaystyle\Gamma_{\mathrm{em}}^{\mu_{1}...\mu_{2}}$ $\displaystyle=$
$\displaystyle\frac{1}{m_{R}^{n}}\,\tilde{O}^{(n+1/2)}_{(\nu_{1}...\nu_{n})\mu_{1}...\mu_{n}}\,(p_{R})\,\tilde{\Gamma}^{\mathrm{em}}_{\nu_{1}...\nu_{n}}$
(21)

where $\tilde{O}^{(n+1/2)}_{(\nu_{1}...\nu_{n})\mu_{1}...\mu_{n}}\,(p_{R})$ is
the interaction operator defined by

$\displaystyle\tilde{O}^{(n+1/2)}_{(\nu_{1}...\nu_{n})\mu_{1}...\mu_{n}}\,(p_{R})=$
$\displaystyle
O^{(n+1/2)}_{(\nu_{1}...\nu_{n},\alpha_{1}...\alpha_{n})\mu_{1}...\mu_{n}}\,(p_{R})\gamma^{\alpha_{1}}\,...\gamma^{\alpha_{n}}~{}.$
(22)

In the present work we use the propagator

$\displaystyle P_{\mu_{1}..\mu_{n};\nu_{1}..\nu_{n}}\,(p_{R})=$
$\displaystyle{\displaystyle\frac{\not{p}_{R}+m_{R}}{p_{R}^{2}-m_{R}^{2}+im_{R}\Gamma}}\,\tilde{\mathcal{P}}^{n+1/2}_{\mu_{1}...\mu_{n};\nu_{1}...\nu_{n}}(p_{R}),$
(23)

where
$\tilde{\mathcal{P}}^{n+1/2}_{\mu_{1}...\mu_{n};\nu_{1}...\nu_{n}}(p_{R})$ is
the on-shell projection operator. The complicated form of the projection
operator will be discussed later.

The production amplitude is obtained by sandwiching the propagator between the
two vertices, i.e.,

$\displaystyle\mathcal{M}_{\mathrm{res}}^{\mathrm{n+1/2}}=\bar{u}_{\Lambda}\Gamma^{\rm
had}_{\mu_{1}...\mu_{n}}P^{\mu_{1}...\mu_{n},\nu_{1}...\nu_{n}}_{(n+1/2)}\,(p_{R})\,\Gamma^{\rm
em}_{\nu_{1}...\nu_{n}}\,u_{p}.~{}$ (24)

By inserting Eqs. (20), (21), and (II.2) into Eq. (24) we obtain

$\displaystyle\mathcal{M}_{\mathrm{res}}^{\mathrm{n+1/2}}$ $\displaystyle=$
$\displaystyle\bar{u}_{\Lambda}\tilde{\Gamma}^{\alpha_{1}...\alpha_{n}}_{\rm
had}\tilde{O}^{n+1/2}_{(\alpha_{1}...\alpha_{n})\mu_{1}...\mu_{n}}\,(p_{R})$
(25)
$\displaystyle\times\frac{1}{m_{R}^{2n}}\,P^{\mu_{1}...\mu_{n},\nu_{1}...\nu_{n}}_{(n+1/2)}\,(p_{R})$
$\displaystyle\times\tilde{O}_{n+1/2}^{(\beta_{1}...\beta_{n})\nu_{1}...\nu_{n}}\,(p_{R})\,\tilde{\Gamma}_{\rm
em}^{\beta_{1}...\beta_{n}}\,u_{p},$ $\displaystyle=$
$\displaystyle\bar{u}_{\Lambda}\tilde{\Gamma}^{\alpha_{1}...\alpha_{n}}_{\rm
had}\tilde{O}^{n+1/2}_{(\alpha_{1}...\alpha_{n})\mu_{1}...\mu_{n}}\,(p_{R})$
$\displaystyle\times\frac{1}{m_{R}^{2n}}\,\frac{\not{p}_{R}+m_{R}}{p_{R}^{2}-m_{R}^{2}+im_{R}\Gamma}\;$
$\displaystyle\times\mathcal{P}^{\mu_{1}...\mu_{n},\nu_{1}...\nu_{n}}_{(n+1/2)}\,(p_{R})\tilde{O}_{n+1/2}^{(\beta_{1}...\beta_{n})\nu_{1}...\nu_{n}}\,(p_{R})$
$\displaystyle\times\tilde{\Gamma}_{\rm
em}^{\beta_{1}...\beta_{n}}\,u_{p}~{}.$

The above formulation is simplified by considering the orthogonalities of the
projection operator,

$\displaystyle\gamma_{\mu_{i}}\,\mathcal{P}^{\mu_{1}..\mu_{n},\nu_{1}..\nu_{n}}_{(n+1/2)}\,(p_{R})$
$\displaystyle=$
$\displaystyle\mathcal{P}^{\mu_{1}..\mu_{n},\nu_{1}..\nu_{n}}_{(n+1/2)}\,(p_{R})\,\gamma_{\nu_{i}}$
(26) $\displaystyle=$ $\displaystyle 0,$

and

$\displaystyle{p_{R}}_{\mu_{i}}\,\mathcal{P}^{\mu_{1}..\mu_{n},\nu_{1}..\nu_{n}}_{(n+1/2)}\,(p_{R})$
$\displaystyle=$
$\displaystyle{p_{R}}_{\nu_{i}}\,\mathcal{P}^{\mu_{1}..\mu_{n},\nu_{1}..\nu_{n}}_{(n+1/2)}\,(p_{R})$
(27) $\displaystyle=$ $\displaystyle 0,$

with $i=1,2,..n$.

Thus, the scattering amplitude can be written as

$\displaystyle\mathcal{M}_{\mathrm{res}}^{\mathrm{n+1/2}}$ $\displaystyle=$
$\displaystyle\bar{u}_{\Lambda}\,\prod_{i=1}^{n}\not{p}_{R}\,g_{\alpha_{i}\,\mu_{i}}\tilde{\Gamma}_{\rm
had}^{\alpha_{1}...\alpha_{n}}\,\frac{1}{m_{R}^{2n}}\,\frac{\not{p}_{R}+m_{R}}{p_{R}^{2}-m_{R}^{2}+im_{R}\Gamma}\mathcal{P}^{\mu_{1}...\mu_{n},\nu_{1}...\nu_{n}}_{(n+1/2)}\,(p_{R})\,\prod_{i=1}^{n}\not{p}_{R}\,g_{\beta_{j}\,\nu_{j}}\tilde{\Gamma}_{\rm
em}^{\beta_{1}..\beta_{n}}\,u_{p}$ $\displaystyle=$
$\displaystyle\bar{u}_{\Lambda}\,\tilde{\Gamma}^{\rm
had}_{\mu_{1}...\mu_{n}}\,\frac{p_{R}^{2n}}{m_{R}^{2n}}\,\frac{\not{p}_{R}+m_{R}}{p_{R}^{2}-m_{R}^{2}+im_{R}\Gamma}\,\mathcal{P}^{\mu_{1}...\mu_{n},\nu_{1}...\nu_{n}}_{(n+1/2)}\,(p_{R})\,\tilde{\Gamma}_{\rm
em}^{\beta_{1}..\beta_{n}}\,u_{p}.$

The above equation shows how a consistent interaction structure constructed
intuitively.

###  II.3 Propagators

The propagators used in this model are constructed from the corresponding
particle projection operators. The generalized projection operator has been
explored by Huang, et al. in Ref. Huang:2004we . The projection operator for
spin-11/2 can be written as

$\displaystyle\mathcal{P}^{11/2}_{\begin{subarray}{c}\mu\mu_{1}\mu_{2}\mu_{3}\mu_{4}\\\
\nu\nu_{1}\nu_{2}\nu_{3}\nu_{4}\end{subarray}}$ $\displaystyle=$
$\displaystyle\frac{1}{1440}\sum_{\mathrm{P}(\mu),\mathrm{P}(\nu)}\Bigl{\\{}P_{\mu\nu}P_{\mu_{1}\nu_{1}}P_{\mu_{2}\nu_{2}}P_{\mu_{3}\nu_{3}}P_{\mu_{4}\nu_{4}}-\textstyle{\frac{10}{11}}P_{\mu\mu_{1}}P_{\nu\nu_{1}}P_{\mu_{2}\nu_{2}}P_{\mu_{3}\nu_{3}}P_{\mu_{4}\nu_{4}}$
(29)
$\displaystyle+\textstyle\frac{5}{33}P_{\mu\mu_{1}}P_{\nu\nu_{1}}P_{\mu_{2}\mu_{3}}P_{\nu_{2}\nu_{3}}P_{\mu_{4}\nu_{4}}+\textstyle{\frac{5}{11}}\gamma^{\,\rho}\gamma^{\sigma}P_{\mu\rho}P_{\nu\sigma}P_{\mu_{1}\nu_{1}}P_{\mu_{2}\nu_{2}}P_{\mu_{3}\nu_{3}}P_{\mu_{4}\nu_{4}}$
$\displaystyle-\textstyle{\frac{10}{33}}\gamma^{\,\rho}\gamma^{\sigma}P_{\mu\rho}P_{\nu\sigma}P_{\mu_{1}\mu_{2}}P_{\nu_{1}\nu_{2}}P_{\mu_{3}\nu_{3}}P_{\mu_{4}\nu_{4}}+\textstyle{\frac{5}{231}}\gamma^{\,\rho}\gamma^{\sigma}P_{\mu\rho}P_{\nu\sigma}P_{\mu_{1}\mu_{2}}P_{\nu_{1}\nu_{2}}P_{\mu_{3}\mu_{4}}P_{\nu_{3}\nu_{4}}\Bigr{\\}}~{},$

and for spin-13/2 particles the projection operator is

$\displaystyle\mathcal{P}^{13/2}_{\begin{subarray}{c}\mu\mu_{1}\mu_{2}\mu_{3}\mu_{4}\mu_{5}\\\
\nu\nu_{1}\nu_{2}\nu_{3}\nu_{4}\nu_{5}\end{subarray}}$ $\displaystyle=$
$\displaystyle\frac{1}{(6!)^{2}}\sum_{\mathrm{P}(\mu),\mathrm{P}(\nu)}\Bigl{\\{}P_{\mu\nu}P_{\mu_{1}\nu_{1}}P_{\mu_{2}\nu_{2}}P_{\mu_{3}\nu_{3}}P_{\mu_{4}\nu_{4}}P_{\mu_{5}\nu_{5}}-\textstyle{\frac{15}{13}}P_{\mu\mu_{1}}P_{\nu\nu_{1}}P_{\mu_{2}\nu_{2}}P_{\mu_{3}\nu_{3}}P_{\mu_{4}\nu_{4}}P_{\mu_{5}\nu_{5}}$
(30)
$\displaystyle+\textstyle\frac{45}{143}P_{\mu\mu_{1}}P_{\nu\nu_{1}}P_{\mu_{2}\mu_{3}}P_{\nu_{2}\nu_{3}}P_{\mu_{4}\nu_{4}}P_{\mu_{5}\nu_{5}}-\textstyle\frac{5}{429}P_{\mu\mu_{1}}P_{\nu\nu_{1}}P_{\mu_{2}\mu_{3}}P_{\nu_{2}\nu_{3}}P_{\mu_{4}\mu_{5}}P_{\nu_{4}\nu_{5}}$
$\displaystyle+\textstyle{\frac{6}{13}}\gamma^{\,\rho}\gamma^{\sigma}P_{\mu\rho}P_{\nu\sigma}P_{\mu_{1}\nu_{1}}P_{\mu_{2}\nu_{2}}P_{\mu_{3}\nu_{3}}P_{\mu_{4}\nu_{4}}P_{\mu_{5}\nu_{5}}-\textstyle{\frac{60}{143}}\gamma^{\,\rho}\gamma^{\sigma}P_{\mu\rho}P_{\nu\sigma}P_{\mu_{1}\mu_{2}}P_{\nu_{1}\nu_{2}}P_{\mu_{3}\nu_{3}}P_{\mu_{4}\nu_{4}}P_{\mu_{5}\nu_{5}}$
$\displaystyle+\textstyle{\frac{30}{429}}\gamma^{\,\rho}\gamma^{\sigma}P_{\mu\rho}P_{\nu\sigma}P_{\mu_{1}\mu_{2}}P_{\nu_{1}\nu_{2}}P_{\mu_{3}\mu_{4}}P_{\nu_{3}\nu_{4}}P_{\mu_{5}\nu_{5}}\Bigr{\\}}~{},$

while the projection operator for spin-15/2 particles reads

$\displaystyle\mathcal{P}^{15/2}_{\begin{subarray}{c}\mu\mu_{1}\mu_{2}\mu_{3}\mu_{4}\mu_{5}\mu_{6}\\\
\nu\nu_{1}\nu_{2}\nu_{3}\nu_{4}\nu_{5}\nu_{6}\end{subarray}}=\frac{1}{(7!)^{2}}\sum_{\mathrm{P}(\mu),\mathrm{P}(\nu)}\Bigl{\\{}P_{\mu\nu}P_{\mu_{1}\nu_{1}}P_{\mu_{2}\nu_{2}}P_{\mu_{3}\nu_{3}}P_{\mu_{4}\nu_{4}}P_{\mu_{5}\nu_{5}}P_{\mu_{6}\nu_{6}}$
(31)
$\displaystyle-\textstyle{\frac{21}{15}}P_{\mu\mu_{1}}P_{\nu\nu_{1}}P_{\mu_{2}\nu_{2}}P_{\mu_{3}\nu_{3}}P_{\mu_{4}\nu_{4}}P_{\mu_{5}\nu_{5}}P_{\mu_{6}\nu_{6}}+\textstyle\frac{7}{13}P_{\mu\mu_{1}}P_{\nu\nu_{1}}P_{\mu_{2}\mu_{3}}P_{\nu_{2}\nu_{3}}P_{\mu_{4}\nu_{4}}P_{\mu_{5}\nu_{5}}P_{\mu_{6}\nu_{6}}$
$\displaystyle-\textstyle\frac{7}{143}P_{\mu\mu_{1}}P_{\nu\nu_{1}}P_{\mu_{2}\mu_{3}}P_{\nu_{2}\nu_{3}}P_{\mu_{4}\mu_{5}}P_{\nu_{4}\nu_{5}}P_{\mu_{6}\nu_{6}}+\textstyle{\frac{7}{15}}\gamma^{\,\rho}\gamma^{\sigma}P_{\mu\rho}P_{\nu\sigma}P_{\mu_{1}\nu_{1}}P_{\mu_{2}\nu_{2}}P_{\mu_{3}\nu_{3}}P_{\mu_{4}\nu_{4}}P_{\mu_{5}\nu_{5}}P_{\mu_{6}\nu_{6}}$
$\displaystyle-\textstyle{\frac{7}{13}}\gamma^{\,\rho}\gamma^{\sigma}P_{\mu\rho}P_{\nu\sigma}P_{\mu_{1}\mu_{2}}P_{\nu_{1}\nu_{2}}P_{\mu_{3}\nu_{3}}P_{\mu_{4}\nu_{4}}P_{\mu_{5}\nu_{5}}P_{\mu_{6}\nu_{6}}+\textstyle{\frac{21}{143}}\gamma^{\,\rho}\gamma^{\sigma}P_{\mu\rho}P_{\nu\sigma}P_{\mu_{1}\mu_{2}}P_{\nu_{1}\nu_{2}}P_{\mu_{3}\mu_{4}}P_{\nu_{3}\nu_{4}}P_{\mu_{5}\nu_{5}}P_{\mu_{6}\nu_{6}}$
$\displaystyle\textstyle{\frac{7}{1287}}\gamma^{\,\rho}\gamma^{\sigma}P_{\mu\rho}P_{\nu\sigma}P_{\mu_{1}\mu_{2}}P_{\nu_{1}\nu_{2}}P_{\mu_{3}\mu_{4}}P_{\nu_{3}\nu_{4}}P_{\mu_{5}\mu_{6}}P_{\nu_{5}\nu_{6}}\Bigr{\\}}~{},$

The projection operators written above fulfill the orthogonality condition
given by Eq.(26). According to Eq.(II.2), the propagator for spin-11/2
resonance can be written as

$\displaystyle P^{11/2}_{\begin{subarray}{c}\mu\mu_{1}\mu_{2}\mu_{3}\mu_{4}\\\
\nu\nu_{1}\nu_{2}\nu_{3}\nu_{4}\end{subarray}}$ $\displaystyle=$
$\displaystyle\frac{s^{5}}{m_{R}^{10}}\frac{(\not{p}+\not{k}+m_{R})}{(s-m_{R}^{2}+im_{R}\Gamma_{R})}\mathcal{P}^{11/2}_{\begin{subarray}{c}\mu\mu_{1}\mu_{2}\mu_{3}\mu_{4}\\\
\nu\nu_{1}\nu_{2}\nu_{3}\nu_{4}\end{subarray}}\,,$ (32)

whereas for spin-13/2 and spin-15/2 resonances the propagators read

$\displaystyle
P^{13/2}_{\begin{subarray}{c}\mu\mu_{1}\mu_{2}\mu_{3}\mu_{4}\mu_{5}\\\
\nu\nu_{1}\nu_{2}\nu_{3}\nu_{4}\nu_{5}\end{subarray}}$ $\displaystyle=$
$\displaystyle\frac{s^{6}}{m_{R}^{12}}\frac{(\not{p}+\not{k}+m_{R})}{(s-m_{R}^{2}+im_{R}\Gamma_{R})}\mathcal{P}^{13/2}_{\begin{subarray}{c}\mu\mu_{1}\mu_{2}\mu_{3}\mu_{4}\mu_{5}\\\
\nu\nu_{1}\nu_{2}\nu_{3}\nu_{4}\nu_{5}\end{subarray}}$ (33)

and

$\displaystyle{P}^{15/2}_{\begin{subarray}{c}\mu\mu_{1}\mu_{2}\mu_{3}\mu_{4}\mu_{5}\mu_{6}\\\
\nu\nu_{1}\nu_{2}\nu_{3}\nu_{4}\nu_{5}\nu_{6}\end{subarray}}$ $\displaystyle=$
$\displaystyle\frac{s^{7}}{m_{R}^{14}}\frac{(\not{p}+\not{k}+m_{R})}{(s-m_{R}^{2}+im_{R}\Gamma_{R})}\mathcal{P}^{15/2}_{\begin{subarray}{c}\mu\mu_{1}\mu_{2}\mu_{3}\mu_{4}\mu_{5}\mu_{6}\\\
\nu\nu_{1}\nu_{2}\nu_{3}\nu_{4}\nu_{5}\nu_{6}\end{subarray}}\,,$ (34)

respectively, with $s=p_{R}^{2}=(p+k)^{2}=W^{2}$. Notice that the factors
${s^{5}}/{m_{R}^{10}}$, ${s^{6}}/{m_{R}^{12}}$ and ${s^{7}}/{m_{R}^{14}}$
originate from the consequence of the consistent interaction.

###  II.4 Hadronic and Electromagnetic Vertices

By using the above prescription we obtain that the hadronic and
electromagnetic vertices for spin-11/2 reads as

$\displaystyle\Gamma_{\rm{had}}^{\mu\mu_{1}\mu_{2}\mu_{3}\mu_{4}\pm}$
$\displaystyle=$
$\displaystyle\frac{g_{KYR}}{m_{R}^{6}}\Gamma_{\mp}\bigl{[}(p_{\Lambda}\cdot
q-\not{p}_{\Lambda}\not{q})\gamma^{\mu}+\not{p}_{\Lambda}q^{\mu}-\not{q}p^{\mu}_{\Lambda})\bigr{]}p^{\mu_{1}}_{\Lambda}p^{\mu_{2}}_{\Lambda}p^{\mu_{3}}_{\Lambda}p^{\mu_{4}}_{\Lambda}$
(35)

and

$\displaystyle\Gamma_{\rm{em}}^{\nu\nu_{1}\nu_{2}\nu_{3}\nu_{4}\pm}$
$\displaystyle=$
$\displaystyle\frac{-i}{m_{R}^{6}}\biggl{[}g_{1}p^{\nu}(\not{k}\not{\epsilon}-\not{\epsilon}\not{k})+g_{2}(k^{\nu}p\cdot\epsilon-\epsilon^{\nu}p\cdot
k)+g_{3}(\epsilon^{\nu}\not{k}-k^{\nu}\not{\epsilon})\not{p}+g_{4}\gamma^{\nu}(\not{k}\not{\epsilon}-\not{\epsilon}\not{k})\not{p}$
(36) $\displaystyle+g_{5}\gamma^{\nu}(p\cdot\epsilon\not{k}-p\cdot
k\not{\epsilon})\biggr{]}k^{\nu_{1}}k^{\nu_{2}}k^{\nu_{3}}k^{\nu_{4}}\Gamma_{\pm}~{},$

respectively, whereas for the spin-13/2 resonance

$\displaystyle\Gamma_{\rm{had}}^{\mu\mu_{1}\mu_{2}\mu_{3}\mu_{4}\mu_{5}\pm}$
$\displaystyle=$
$\displaystyle\frac{g_{KYR}}{m_{R}^{7}}\Gamma_{\pm}\bigl{[}(p_{\Lambda}\cdot
q-\not{p}_{\Lambda}\not{q})\gamma^{\mu}+\not{p}_{\Lambda}q^{\mu}-\not{q}p^{\mu}_{\Lambda})\bigr{]}p^{\mu_{1}}_{\Lambda}p^{\mu_{2}}_{\Lambda}p^{\mu_{3}}_{\Lambda}p^{\mu_{4}}_{\Lambda}p^{\mu_{5}}_{\Lambda}~{},$
(37)

and

$\displaystyle\Gamma_{\rm{em}}^{\nu\nu_{1}\nu_{2}\nu_{3}\nu_{4}\nu_{5}\pm}$
$\displaystyle=$
$\displaystyle\frac{-i}{m_{R}^{7}}\biggl{[}g_{1}p^{\nu}(\not{k}\not{\epsilon}-\not{\epsilon}\not{k})+g_{2}(k^{\nu}p\cdot\epsilon-\epsilon^{\nu}p\cdot
k)+g_{3}(\epsilon^{\nu}\not{k}-k^{\nu}\not{\epsilon})\not{p}+g_{4}\gamma^{\nu}(\not{k}\not{\epsilon}-\not{\epsilon}\not{k})\not{p}$
(38) $\displaystyle+g_{5}\gamma^{\nu}(p\cdot\epsilon\not{k}-p\cdot
k\not{\epsilon})\biggr{]}k^{\nu_{1}}k^{\nu_{2}}k^{\nu_{3}}k^{\nu_{4}}k^{\nu_{5}}\Gamma_{\mp}~{},$

respectively. The above result indicates that the number of momentum
dependence increases with the number of spin. This conclusion was previously
made by Ref. Huang:2004we .

For the spin-15/2 resonance the hadronic and electromagnetic vertices are
given by

$\displaystyle\Gamma_{\rm{had}}^{\mu\mu_{1}\mu_{2}\mu_{3}\mu_{4}\mu_{5}\mu_{6}\pm}$
$\displaystyle=$
$\displaystyle\frac{g_{KYR}}{m_{R}^{8}}\Gamma_{\pm}\bigl{[}(p_{\Lambda}\cdot
q-\not{p}_{\Lambda}\not{q})\gamma^{\mu}+\not{p}_{\Lambda}q^{\mu}-\not{q}p^{\mu}_{\Lambda})\bigr{]}p^{\mu_{1}}_{\Lambda}p^{\mu_{2}}_{\Lambda}p^{\mu_{3}}_{\Lambda}p^{\mu_{4}}_{\Lambda}p^{\mu_{5}}_{\Lambda}p^{\mu_{6}}_{\Lambda}~{},$
(39)

and

$\displaystyle\Gamma_{\rm{em}}^{\nu\nu_{1}\nu_{2}\nu_{3}\nu_{4}\nu_{5}\nu_{6}\pm}$
$\displaystyle=$
$\displaystyle\frac{-i}{m_{R}^{8}}\biggl{[}g_{1}p^{\nu}(\not{k}\not{\epsilon}-\not{\epsilon}\not{k})+g_{2}(k^{\nu}p\cdot\epsilon-\epsilon^{\nu}p\cdot
k)+g_{3}(\epsilon^{\nu}\not{k}-k^{\nu}\not{\epsilon})\not{p}+g_{4}\gamma^{\nu}(\not{k}\not{\epsilon}-\not{\epsilon}\not{k})\not{p}$
(40) $\displaystyle+g_{5}\gamma^{\nu}(p\cdot\epsilon\not{k}-p\cdot
k\not{\epsilon})\biggr{]}k^{\nu_{1}}k^{\nu_{2}}k^{\nu_{3}}k^{\nu_{4}}k^{\nu_{5}}k^{\nu_{6}}\Gamma_{\mp}~{},$

respectively. The above formalism is valid for both positive and negative
parities, for which the parity factors are denoted by $\Gamma_{+}=i\gamma_{5}$
and $\Gamma_{-}=1$, respectively.

###  II.5 Production Amplitudes

As stated above the production amplitudes for spin-11/2, -13/2, and -15/2
resonances are obtained by sandwiching the propagators given by Eqs. (32),
(33), and (34) between the corresponding hadronic vertex factors given by Eqs.
(35), (37), and (39), and electromagnetic vertex factors given by Eqs. (36),
(38), and (40), respectively. As a result we obtain the amplitude for
spin-11/2

$\displaystyle\mathcal{M}^{\pm}_{11/2}$ $\displaystyle=$
$\displaystyle\bar{u}_{\Lambda}\gamma_{5}\\{-s\pm
m_{R}(\not{p}+\not{k})\\}\Bigl{[}7(33c_{1}^{4}+18c_{1}^{2}c_{2}c_{3}+7c_{2}^{2}c_{3}^{2})\times\bigl{\\{}-p_{\Lambda\nu}+{\frac{1}{s}}c_{\Lambda}(p+k)_{\nu}\bigr{\\}}-14c_{1}c_{2}(6c_{1}^{2}+2c_{2}c_{3})$
(41)
$\displaystyle\times\bigl{\\{}-k_{\nu}+{\frac{1}{s}}c_{k}(p+k)_{\nu}\bigr{\\}}+(21c_{1}^{4}+14c_{1}^{2}c_{2}c_{3}+c_{2}^{2}c_{3}^{2})\bigl{\\{}-\not{p}_{\Lambda}+{\frac{1}{s}}c_{\Lambda}(\not{p}+\not{k})\bigr{\\}}\bigl{\\{}-\gamma_{\nu}+{\frac{1}{s}}(\not{p}+\not{k})(p+k)_{\nu}\bigr{\\}}$
$\displaystyle-14c_{1}(6c_{1}^{2}+2c_{2}c_{3})\bigl{\\{}-\not{p}_{\Lambda}+{\frac{1}{s}}c_{\Lambda}(\not{p}+\not{k})\bigr{\\}}\bigl{\\{}-\not{k}+{\frac{1}{s}}c_{k}(\not{p}+\not{k})\bigr{\\}}\bigl{\\{}-p_{\Lambda\nu}+{\frac{1}{s}}c_{\Lambda}(p+k)_{\nu}\bigr{\\}}+4c_{2}(7c_{1}^{2}+c_{2}c_{3})$
$\displaystyle\times\bigl{\\{}-\not{p}_{\Lambda}+{\frac{1}{s}}c_{\Lambda}(\not{p}+\not{k})\bigr{\\}}\bigl{\\{}-\not{k}+{\frac{1}{s}}c_{k}(\not{p}+\not{k})\bigr{\\}}\bigl{\\{}-k_{\nu}+{\frac{1}{s}}c_{k}(p+k)_{\nu}\bigr{\\}}\Bigr{]}$
$\displaystyle\times\Bigl{[}G_{1a}p^{\nu}(\not{k}\not{\epsilon}-\not{\epsilon}\not{k})+G_{2a}(k^{\nu}p\cdot\epsilon-\epsilon^{\nu}p\cdot
k)+G_{3a}(\epsilon^{\nu}\not{k}-k^{\nu}\not{\epsilon})\not{p}\Bigr{]}u_{p}\,,$

with:

$\displaystyle G_{1a}$ $\displaystyle=$
$\displaystyle\frac{s^{5}g_{KYR}g_{1}}{231m^{22}_{R}(s-m^{2}_{R}+im_{R}\Gamma_{R})}\,,$
(42) $\displaystyle G_{2a}$ $\displaystyle=$
$\displaystyle\frac{s^{5}g_{KYR}g_{2}}{231m^{22}_{R}(s-m^{2}_{R}+im_{R}\Gamma_{R})}\,,$
(43) $\displaystyle G_{3a}$ $\displaystyle=$
$\displaystyle\frac{s^{5}g_{KYR}g_{3}}{231m^{22}_{R}(s-m^{2}_{R}+im_{R}\Gamma_{R})}\,.$
(44)

The production amplitude for spin-13/2 particle is given by

$\displaystyle\mathcal{M}^{\pm}_{13/2}$ $\displaystyle=$
$\displaystyle\bar{u}_{\Lambda}\gamma_{5}\\{s\pm
m_{R}(\not{p}+\not{k})\\}\Bigl{[}3c_{1}(143c_{1}^{4}+110c_{1}^{2}c_{2}c_{3}+15c_{2}^{2}c_{3}^{2})\bigl{\\{}-p_{\Lambda\nu}+{\frac{1}{s}}c_{\Lambda}(p+k)_{\nu}\bigr{\\}}-5c_{2}(33c_{1}^{4}$
(45)
$\displaystyle+18c_{1}^{2}c_{2}c_{3}+c_{2}^{2}c_{3}^{2})\bigl{\\{}-k_{\nu}+{\frac{1}{s}}c_{k}(p+k)_{\nu}\bigr{\\}}+c_{1}(33c_{1}^{4}+120c_{1}^{2}c_{2}c_{3}+5c_{2}^{2}c_{3}^{2})\bigl{\\{}-\not{p}_{\Lambda}+{\frac{1}{s}}c_{\Lambda}(\not{p}+\not{k})\bigr{\\}}$
$\displaystyle\times\bigl{\\{}-\gamma_{\nu}+{\frac{1}{s}}(\not{p}+\not{k})(p+k)_{\nu}\bigr{\\}}-5(33c_{1}^{4}+18c_{1}^{2}c_{2}c_{3}+c_{2}^{2}c_{3}^{2})\bigl{\\{}-\not{p}_{\Lambda}+{\frac{1}{s}}c_{\Lambda}(\not{p}+\not{k})\bigr{\\}}\bigl{\\{}-\not{k}+{\frac{1}{s}}c_{k}(\not{p}+\not{k})\bigr{\\}}$
$\displaystyle\times\bigl{\\{}-p_{\Lambda\nu}+{\frac{1}{s}}c_{\Lambda}(p+k)_{\nu}\bigr{\\}}+20c_{1}c_{2}(3c_{1}^{2}+c_{2}c_{3})\bigl{\\{}-\not{p}_{\Lambda}+{\frac{1}{s}}c_{\Lambda}(\not{p}+\not{k})\bigr{\\}}\bigl{\\{}-\not{k}+{\frac{1}{s}}c_{k}(\not{p}+\not{k})\bigr{\\}}$
$\displaystyle\times\bigl{\\{}-k_{\nu}+{\frac{1}{s}}c_{k}(p+k)_{\nu}\bigr{\\}}\Bigr{]}\Bigl{[}G_{1b}p^{\nu}(\not{k}\not{\epsilon}-\not{\epsilon}\not{k})+G_{2b}(k^{\nu}p\cdot\epsilon-\epsilon^{\nu}p\cdot
k)+G_{3b}(\epsilon^{\nu}\not{k}-k^{\nu}\not{\epsilon})\not{p}\Bigr{]}u_{p}\,,$

with

$\displaystyle G_{1b}$ $\displaystyle=$
$\displaystyle\frac{s^{6}g_{KYR}g_{1}}{429m^{26}_{R}(s-m^{2}_{R}+im_{R}\Gamma_{R})}\,,$
(46) $\displaystyle G_{2b}$ $\displaystyle=$
$\displaystyle\frac{s^{6}g_{KYR}g_{2}}{429m^{26}_{R}(s-m^{2}_{R}+im_{R}\Gamma_{R})}\,,$
(47) $\displaystyle G_{3b}$ $\displaystyle=$
$\displaystyle\frac{s^{6}g_{KYR}g_{3}}{429m^{26}_{R}(s-m^{2}_{R}+im_{R}\Gamma_{R})}\,.$
(48)

Finally, the amplitude for spin-15/2 resonance reads

$\displaystyle\mathcal{M}^{\pm}_{15/2}$ $\displaystyle=$
$\displaystyle\bar{u}_{\Lambda}\gamma_{5}\\{-s\pm
m_{R}(\not{p}+\not{k})\\}\Bigl{[}45(143c_{1}^{6}+143c_{1}^{4}c_{2}c_{3}+33c_{1}^{2}c_{2}^{2}c_{3}^{2}+c_{2}^{3}c_{3}^{3})\bigl{\\{}-p_{\Lambda\nu}+{\frac{1}{s}}c_{\Lambda}(p+k)_{\nu}\bigr{\\}}$
(49)
$\displaystyle-9c_{1}c_{2}(286c_{1}^{4}+220c_{1}^{2}c_{2}c_{3}+30c_{2}^{2}c_{3}^{2})\bigl{\\{}-k_{\nu}+{\frac{1}{s}}c_{k}(p+k)_{\nu}\bigr{\\}}+(429c_{1}^{6}+495c_{1}^{4}c_{2}c_{3}+135c_{1}^{2}c_{2}^{2}c_{3}^{2})$
$\displaystyle\times\bigl{\\{}-\not{p}_{\Lambda}+{\frac{1}{s}}c_{\Lambda}(\not{p}+\not{k})\bigr{\\}}\bigl{\\{}-\gamma_{\nu}+{\frac{1}{s}}(\not{p}+\not{k})(p+k)_{\nu}\bigr{\\}}-18c_{1}(143c_{1}^{4}+110c_{1}c_{2}c_{3}+15c_{2}^{2}c_{3}^{2})$
$\displaystyle\times\bigl{\\{}-\not{p}_{\Lambda}+{\frac{1}{s}}c_{\Lambda}(\not{p}+\not{k})\bigr{\\}}\bigl{\\{}-\not{k}+{\frac{1}{s}}c_{k}(\not{p}+\not{k})\bigr{\\}}\bigl{\\{}-p_{\Lambda\nu}+{\frac{1}{s}}c_{\Lambda}(p+k)_{\nu}\bigr{\\}}+30c_{2}(33c_{1}^{4}+18c_{1}c_{2}c_{3}^{2}+c_{2}^{2}c_{3}^{2})$
$\displaystyle\times\bigl{\\{}-\not{p}_{\Lambda}+{\frac{1}{s}}c_{\Lambda}(\not{p}+\not{k})\bigr{\\}}\bigl{\\{}-\not{k}+{\frac{1}{s}}c_{k}(\not{p}+\not{k})\bigr{\\}}\bigl{\\{}-k_{\nu}+{\frac{1}{s}}c_{k}(p+k)_{\nu}\bigr{\\}}\Bigr{]}$
$\displaystyle\times\Bigl{[}G_{1c}p^{\nu}(\not{k}\not{\epsilon}-\not{\epsilon}\not{k})+G_{2c}(k^{\nu}p\cdot\epsilon-\epsilon^{\nu}p\cdot
k)+G_{3c}(\epsilon^{\nu}\not{k}-k^{\nu}\not{\epsilon})\not{p}\Bigr{]}u_{p}\,,$

with

$\displaystyle G_{1c}$ $\displaystyle=$
$\displaystyle\frac{s^{7}g_{KYR}g_{1}}{6435m^{30}_{R}(s-m^{2}_{R}+im_{R}\Gamma_{R})}\,,$
(50) $\displaystyle G_{2c}$ $\displaystyle=$
$\displaystyle\frac{s^{7}g_{KYR}g_{2}}{6435m^{30}_{R}(s-m^{2}_{R}+im_{R}\Gamma_{R})}\,,$
(51) $\displaystyle G_{3c}$ $\displaystyle=$
$\displaystyle\frac{s^{7}g_{KYR}g_{3}}{6435m^{30}_{R}(s-m^{2}_{R}+im_{R}\Gamma_{R})}\,.$
(52)

Note that in the fitting process only the product of the hadronic and
electromagnetic couplings, i.e., $g_{KYR}g_{i}$ with $i=1,2,3$, are extracted
from the data. Furthermore, in the production amplitudes given by Eqs. (41),
(45), and (49) we have used the following definitions,

$\displaystyle c_{1}=b_{\Lambda}-c_{\Lambda}c_{k}/s\,,$ (53a) $\displaystyle
c_{2}=m_{\Lambda}^{2}-c_{\Lambda}^{2}/s\,,$ (53b) $\displaystyle
c_{3}=c_{k}^{2}/s-k^{2}\,,$ (53c) $\displaystyle c_{4}=2b_{p}+k^{2}\,,$ (53d)
$\displaystyle c_{5}=4b_{p}+k^{2}\,,$ (53e) $\displaystyle b_{p}=p\cdot k\,,$
(53f) $\displaystyle b_{\Lambda}=p_{\Lambda}\cdot k\,,$ (53g) $\displaystyle
b_{q}=q\cdot k\,,$ (53h) $\displaystyle c_{p}=(p+k)\cdot p\,,$ (53i)
$\displaystyle c_{\Lambda}=(p+k)\cdot p_{\Lambda}\,,$ (53j) $\displaystyle
c_{k}=(p+k)\cdot k\,,$ (53k) $\displaystyle c_{s}=1-c_{\Lambda}/s\,.$ (53l)

###  II.6 Calculation of the observables

The production amplitudes given by Eqs. (41), (45), and (49) can be decomposed
into six gauge and Lorentz invariant matrices $M_{i}$ through

$\displaystyle\mathcal{M}_{fi}=\bar{u}_{\Lambda}\sum^{6}_{i=1}A_{i}M_{i}\,u_{p}~{},$
(54)

where the gauge and Lorentz invariant matrices $M_{i}$ are given by Deo:1974ik
; Dennery:1961zz :

$\displaystyle M_{1}$ $\displaystyle=$
$\displaystyle\gamma_{5}\,\not{\epsilon}\not{k}~{},$ (55) $\displaystyle
M_{2}$ $\displaystyle=$ $\displaystyle 2\gamma_{5}\left(q\cdot\epsilon P\cdot
k-q\cdot k\,P\cdot\epsilon\right)~{},$ (56) $\displaystyle M_{3}$
$\displaystyle=$ $\displaystyle\gamma_{5}(q\cdot
k\not{\epsilon}-q\cdot\epsilon\not{k})~{},$ (57) $\displaystyle M_{4}$
$\displaystyle=$ $\displaystyle
i\varepsilon_{\mu\nu\rho\sigma}\gamma^{\mu}q^{\nu}\epsilon^{\rho}k^{\sigma}~{},$
(58) $\displaystyle M_{5}$ $\displaystyle=$
$\displaystyle\gamma_{5}(q\cdot\epsilon k^{2}-q\cdot kk\cdot\epsilon),$ (59)
$\displaystyle M_{6}$ $\displaystyle=$
$\displaystyle\gamma_{5}(k\cdot\epsilon\not{k}-k^{2}\not{\epsilon})~{},$ (60)

where $P=\frac{1}{2}(p+p_{\Lambda})$ and $\varepsilon_{\mu\nu\rho\sigma}$ is
the Levi-Civita antisymmetric tensor. All observables required for fitting the
experimental data can be calculated from the form function $A_{i}$ extracted
from Eq. (54), after adding the contributions from all involved intermediate
states. The form functions $A_{i}$ for baryon resonances with spins up to 9/2
are given in the previous works Clymton:2017nvp ; Mart:2015jof , whereas those
with spins 11/2, 13/2, and 15/2 considered in the present work are given in
Appendix A.

###  II.7 Pole Position

Besides the Breit-Wigner parameters, such as mass, width, and branching
ratios, the Particle Data Group has recently listed new information on the
resonance properties, i.e., the pole position. It is obvious that the Breit-
Wigner parameters extracted in each model depend highly on the background
terms of the model. Therefore, all resonance properties obtained by using such
parameterization is difficult to compare with those obtained from other
models. This problem does not appear in the case of pole position. Currently,
the pole position has been extensively used in the realm of hadronic physics.
In the Particle Data Book 2018 the pole positions of resonance are listed
before the Breit-Wigner parameters. The placement shows that the pole
positions is currently considered as the important properties of a resonance.

In principle, the pole position can be calculated by setting the denominator
of the scattering amplitude to zero. Approaching the pole position the
scattering amplitude of a resonance increases dramatically. Since the
resonance scattering amplitude becomes extremely larger than contributions
from other intermediate states, the resonance property calculated at the pole
position is insensitive to the contribution of background terms. As a result,
the evaluation of resonance properties at the pole position is practically
model independent.

In the present work, the pole position properties of a resonance are the
resonance mass and width. They are defined via

$\sqrt{s_{R}}=M_{\rm pole}-i\Gamma_{\rm pole}/2\,.$ (61)

As previously stated, this is obtained by setting the denominator of
scattering amplitude to zero, i.e.,

$s_{R}-m_{R}^{2}+im_{R}\Gamma(s_{R})=0\,.$ (62)

Notice that the above equation cannot be directly calculated, since in the
present study we use $\Gamma(s)$ that depends on the total c.m. energy.
Therefore, the solution of Eq. (62) must be obtained numerically.

##  III Results and Discussion

In the present work, the isobar model used to analyze the effect of spin-11/2
and -13/2 nucleon resonances in the $K\Lambda$ channels is based on our
previous model developed to describe all available data in these channels
Mart:2019mtq . Furthermore, in this study we also investigate the effect of
spin-11/2, -13/2, and -15/2 $\Delta$ resonances in the $K\Sigma$ reaction
channels. Along with the nucleon resonances used in the $K\Lambda$ channels,
these $\Delta$ resonances are listed in Table 1. In total, there are 23
nucleon resonances included in our analysis for the $K\Lambda$ and $K\Sigma$
channels and, in addition, 17 $\Delta$ resonances in the $K\Sigma$ channels
with spins up to spin-15/2. The result obtained in all channels will be
discussed in the following subsections.

###  III.1 $K\Lambda$ Channel

Table 3: Coupling constants and other driving parameters of the background terms for $K\Lambda$ channels obtained in the present work (Model A) and the previous one (Model B) Mart:2019mtq . Error bars were not reported in Model B. See Ref. Mart:2019mtq  for the explanation of the parameter notation. Parameter | Model A | Model B
---|---|---
$g_{K\Lambda N}/\sqrt{4\pi}$ | $-4.40\pm 0.03$ | $-3.00$
$g_{K\Sigma N}/\sqrt{4\pi}$ | $0.90\pm 0.04$ | $1.30$
$G^{V}_{K^{*}}/4\pi$ | $0.08\pm 0.00$ | $0.13$
$G^{T}_{K^{*}}/4\pi$ | $-0.07\pm 0.00$ | $0.17$
$G^{V}_{K_{1}}/4\pi$ | $0.12\pm 0.00$ | $0.13$
$G^{T}_{K_{1}}/4\pi$ | $2.43\pm 0.00$ | $3.89$
$r_{{K_{1}}{K_{\gamma}}}$ | $0.52\pm 0.01$ | $0.65$
$\Lambda_{\rm B}$ (GeV) | $0.89\pm 0.00$ | $0.70$
$\Lambda_{\rm R}$ (GeV) | $1.09\pm 0.00$ | $1.10$
$\theta_{\rm had}$ (deg) | $94.31\pm 0.64$ | $90.0$
$\phi_{\rm had}$ (deg) | $90.00\pm 4.09$ | $0.0$
$\chi^{2}$ | 13316 | 13867
$N_{\rm par}$ | 264 | 247
$N_{\rm data}$ | 9364 | 9364
$\chi^{2}/N_{\rm dof}$ | 1.46 | 1.52

In Table 3 we present the leading coupling constants and other background
parameters obtained from the previous work (Model B) Mart:2019mtq  and current
analysis (Model A). Note that in the present work we have omitted the
$K^{0}\Lambda$ photoproduction data obtained from MAMI collaboration mami2018
due to the problem of data discrepancy as discussed in Ref. Mart:2019mtq .
Furthermore, it was shown that by excluding these data from the database leads
to a better model that can nicely reproduce the $\gamma n\to K^{0}\Lambda$
helicity asymmetry $E$ Mart:2019mtq . It is important to note that Model B was
also obtained from fitting without these data.

From Table 3 we can conclude that there is no dramatic changes in the
background parameters after including the spin-11/2 and -13/2 nucleon
resonances in the model. Nevertheless, the increase of $g_{K\Lambda N}$
coupling and the Born hadronic cutoff $\Lambda_{\rm B}$ shows that the
inclusion of the two resonances increases the contribution of the background
terms. We note that in the case of Kaon-Maid, the Born cutoff is very soft,
i.e., $\Lambda_{\rm B}=0.637$ GeV kaon-maid . Clearly, the Kaon-Maid model is
dominated by the resonance terms, whereas the Born terms are strongly
suppressed. Such situation is completely different from the case of pion or
eta photoproduction and could raise a question, whether Kaon-Maid is a
realistic phenomenological model.

The listed $\chi^{2}$ values indicate that the agreement between model
calculation and experimental data is significantly improved after including
the two nucleon resonances, as clearly expected. Since the calculation
includes two isospin channels, i.e., $\gamma p\to K^{+}\Lambda$ and $\gamma
n\to K^{0}\Lambda$, in the followings we present comparison between model
calculations and experimental data in details.

####  III.1.1 $K^{+}\Lambda$ Channel

Comparison between calculated $\gamma p\to K^{+}\Lambda$ total cross sections
from Models A and B, Kaon-Maid, and experimental data is displayed in Fig. 2.
Note that the experimental data shown in this figure are only for visual
comparison. The data were not included in the fitting process, since
differential and total cross sections data come from the same experiment.

Figure 2: Calculated total cross section of the $\gamma p\rightarrow
K^{+}\Lambda$ channel obtained from previous Mart:2019mtq  and present works,
compared with the experimental data from the CLAS collaboration (solid squares
clas2006 ). Figure 3: Energy distributions of the $\gamma p\rightarrow
K^{+}\Lambda$ differential cross section obtained from Model A (solid red
curves) and Model B (dashed black curves) for different values of
$\cos\theta$. Experimental data shown in this figure are obtained from the
LEPS 2006 (solid triangles leps2006 ), CLAS 2006 (solid squares clas2006 ),
CLAS 2010 (solid circles clas2010 ), and Crystal Ball 2014 (open circles
crystalball ) collaborations.  Figure 4: As in Fig. 3, but for angular
distribution. The corresponding value of total c.m. energy $W$ in GeV is shown
in each panel.

Compared to the prediction of Kaon-Maid, both models A and B displayed in Fig.
2 show substantial improvement. However, since our main motivation in this
work is to investigate the effect of spin-11/2 and 13/2 nucleon resonance, we
will not compare our result with the prediction of Kaon-Maid in the following
discussion, except in the case of total cross section, in which recent
experimental data are in good agreement with Kaon-Maid for certain isospin
channel. Both current and previous models seem to have a great agreement, with
a tiny difference only at higher energy region, i.e., $W\geq 2.6$ GeV. The
difference originates from the use of the high spin nucleon resonances, as
obviously seen from their masses. However, since there are no available data
in this energy region, no conclusion can be drawn at this point. Future
experiments with 12 GeV electron source at JLab could be expected to reveal
more information in this energy regime.

Both peaks shown by the two models seem to agree with each other, with
minuscule difference at $W\approx 1.85$ GeV, where the second peak is
attributed to the $P_{13}(1990)$ state, as discussed in Ref. Mart:2012fa .

More information can be obtained from the differential cross sections shown in
Figs. 3 and 4. Figure 3 shows that the difference in the total cross sections
of models A and B originate from the forward and backward regions of the
differential cross section. Furthermore, as shown in Fig. 3 at backward angle
(cos $\theta=-0.70$), it is also apparent that the result of model A has a
better agreement than model B, except in the higher energy region, $W>2.4$
GeV. Another interesting result is that the second peak of Model A is slightly
shorter, but wider, than that of Model B. As a result, Model A yields a more
accurate explanation of experimental data, especially for the CLAS 2010
clas2010  and Crystal Ball crystalball  ones. However, in the forward region
Model A yields fewer peaks than Model B. Nevertheless, Model A seems to
produce more natural shape of the cross section at the very forward angle,
$\cos\theta=0.90$, where unfortunately, the available experimental data from
different collaborations produce uncertainty in differential cross section up
to nearly 40%.

The angular distribution of differential cross section displayed in Fig. 4
shows that both models are in good agreement with experimental data.
Furthermore, experimental data in higher energy region are better reproduced.
This figure again shows that the inclusion of the high-spin resonances with
higher masses does not influence the the cross section behavior in the lower
energy region, where experimental data exist.

Figure 5: Asymmetry of the single-polarization ($P$, $\Sigma$, and $T$), and
double-polarization ($O_{x}$, $O_{z}$, $C_{x}$ and $C_{z}$) observables for
the $\gamma p\to K^{+}\Lambda$ channel as a function of kaon angle for
different total c.m. energies $W$ shown in each panel. Dashed black curves are
obtained from the previous isobar model Mart:2019mtq , solid red curves are
obtained from the present calculation. Experimental data shown in this figure
are taken from the GRAAL 2009 (open squares graal2009 ), CLAS 2006 (solid
squares clas2006 ), CLAS 2007 (solid triangles clas2007 ), CLAS 2010 (open
circles clas2010 ), and CLAS 2016 (solid circles clas2016 ) collaborations.

Figure 5 shows the single- and double-polarization observables, for which
experimental data are abundantly available at present. In this case we do not
see a dramatic changes after including the spin-11/2 and 13/2 nucleon
resonance in the model, except improvement in the agreement between model
calculation and experimental data at forward direction and high energy region,
which is expected due to the higher masses of these resonances. Nevertheless,
we still see significant improvement in the beam-recoil double-polarization
observables $C_{x}$ and $C_{z}$ at higher energies, where experimental data
have large error bars. Note that only a small part of experimental data can be
displayed in Fig. 5. More data are available in the fitting database,
especially for the recoil polarization $P$, and are not shown in the figure
due to their different kinematics.

####  III.1.2 $K^{0}\Lambda$ Channel

The available data for the $K^{0}\Lambda$ channel are significantly fewer than
the $K^{+}\Lambda$ one, given that the experiment with neutron target is more
difficult to perform. The number of data included in this study is less than
1000, which will affect the accuracy between model calculation and
experimental data. The calculation of this channel is performed by using the
isospin relation of the hadronic coupling constants, as well as information on
the neutral kaon transition moment and neutron helicity photon coupling
obtained from PDG pdg , in the $K^{+}\Lambda$ model as discussed in Refs.
Mart:1995wu ; Mart_K0L:2011 . Thus, investigation of the $K^{0}\Lambda$
channel can be considered as a direct test of isospin symmetry in kaon
photoproduction.

Experimental data of this channel are already available from the CLAS g10 and
g13 collaboration clas2017  and MAMI 2018 collaboration mami2018 . In the
previous work Mart:2019mtq , both data sets were included in the analysis.
However, in the present work we exclude the data from the MAMI 2018
collaboration, since it was found that the data are more difficult to fit and
have a discrepancy problem with the CLAS g10 and g13 data. Furthermore, in the
present work our main motivation is to investigate the effect of the higher
spin nucleon resonances, in which we need an accurate isobar model.

Figure 6 shows the calculated total cross section of the $\gamma n\rightarrow
K^{0}\Lambda$ channel. Obviously, both models A and B yield similar cross
section trend, except in the lower energy region and especially near the
production threshold, where the cross section obtained from model B is steeper
than that of the present work. All models give the total cross sections within
the experimental error bars. As discussed in Ref. Mart_K0L:2011 , threshold
behavior of $K^{0}\Lambda$ photoproduction provides important information that
can shed more light on the difference between pseudovector and pseudoscalar
theories in the kaon photoproduction process. The absence of $K^{0}$ exchange
in this channel also reduces the number of unknown parameters in the model. As
a consequence, threshold properties of the $K^{0}\Lambda$ can be more
accurately investigated. Note also that the over prediction of Kaon-Maid model
is understandable, since it is pure prediction and the model was fitted to old
data.

Figure 6: Total cross section of the $\gamma n\rightarrow K^{0}\Lambda$
isospin channel calculated from different models. Experimental data are
obtained from the CLAS g10 and g13 collaboration (open circles and open
squares) clas2017  and MAMI 2018 collaboration (solid circles) mami2018 . Note
that the data shown in this figure were not used in the fitting process and
shown here only for comparison. Figure 7: Energy distribution of the $\gamma
n\rightarrow K^{0}\Lambda$ differential cross section for different values of
$\cos\theta$ shown in each panel. Notation of the curves and experimental data
is as in Fig. 6.

Figure 7 shows the differential cross section of the $\gamma n\rightarrow
K^{0}\Lambda$ channel. At a glance, both models seem to be similar, especially
at $W>1.8$ GeV region. However, the inclusion of the two high-spin nucleon
resonances leads to different differential cross section near the threshold.
The difference is more apparent at the forward angle, i.e., $\cos\theta=0.75$.
The angular distribution of differential cross section shown in Fig. 8
corroborates this result. In Fig. 8 we can see that the difference between the
two models is more obvious in the forward and backward regions.

In general, the $K^{0}\Lambda$ differential cross sections also show that the
inclusion of the two high-spin nucleon resonances improves the model. However,
there is an important phenomenon appears in the $K^{0}\Lambda$ channel. As in
the case of the $K^{+}\Lambda$ channel, the inclusion of these resonances
leads to fewer structures in differential cross section, especially at the
forward region (see Fig. 3). The same phenomenon is also displayed by the
$K^{0}\Lambda$ channel as shown in Fig. 7. In this channel the previous work
displays a clear structure at $W\approx 1.65$ GeV, which appears in a wide
range of angular distribution, but is more apparent near forward region. The
structure is eliminated by the inclusion of the two high-spin nucleon
resonances. We found that this structure is very interesting because it
originates from the $N(1650)$ resonance contribution and the corresponding
width is less than 50 MeV (see the dashed curves in Fig. 7, especially at
$\cos\theta=0.75$). In the previous work Mart_K0L:2011  it was concluded that
the structure could be a hint of the narrow resonance, which was found to have
the mass of 1650 MeV.

Figure 8: As in Fig. 7, but for angular distributions. The total c.m. energy
$W$ is shown in each panel in GeV.

###  III.2 $K\Sigma$ Channel

As in the $K\Lambda$ channels, the four available channels of $K\Sigma$
photoproduction can be also simultaneously analyzed by exploiting the isospin
symmetry and some information on the resonance properties from PDG pdg .
Recently, we have studied these channels by using a partial wave approach for
the resonance part, whereas the background part was still constructed from the
covariant Feynman diagram technique Mart:2019fau . The model was fitted to
nearly 8000 experimental data points available from all four channels, but
dominantly from the $K^{+}\Sigma^{0}$ one.

In addition, a fully covariant model to describe photoproduction of $K\Sigma$
has been also constructed by including nucleon resonances with spins up to 9/2
and the result has been submitted for publication samson_ksigma . In the
present work we add the nucleon and delta resonances that are not available in
this covariant model. They include the nucleon resonances with spins 11/2 and
13/2, as well as the delta resonances with spins 11/2, 13/2, and 15/2, listed
in Table 1. The experimental data used in this study were obtained from the
CLAS, Crystal Ball, GRAAL, SAPHIR, LEPS, and SPring8 collaborations. Thus, to
observe the effect of including these resonances, we will compare the result
of our present work to that of the covariant model reported in Ref.
samson_ksigma .

Table 4 lists the leading coupling constants and other background parameters
extracted from the present analysis. For the sake of discussion, the present
model and the model reported in Ref. samson_ksigma  will be referred to as
Model C and Model D, respectively. As seen from the values of $\chi^{2}/N_{\rm
dof}$ in Table 4, the agreement between model calculation and experimental
data is improved after the inclusion of the high-spin nucleon and delta
resonances. The result is clearly expected, because the addition of resonances
increases the free parameters in the model. Table 4 also shows that the
inclusion of the high-spin resonances helps to increase the hadronic form
factor cutoff of the Born terms as in the $K\Lambda$ case.

Table 4: Extracted coupling constants and other background parameters in the $K\Sigma$ channels obtained from the present work (Model C) and the previous one (Model D) samson_ksigma . Note that error bars were not reported in Model D. Parameter | Model C | Model D
---|---|---
$g_{K\Lambda N}/\sqrt{4\pi}$ | $-4.26\pm 0.01$ | $-3.00$
$g_{K\Sigma N}/\sqrt{4\pi}$ | $1.30\pm 0.25$ | 0.90
$G^{V}_{K^{*}}/4\pi$ | $-0.04\pm 0.00$ | $-0.15$
$G^{T}_{K^{*}}/4\pi$ | $-0.03\pm 0.00$ | $-0.21$
$G^{V}_{K_{1}}/4\pi$ | $-0.46\pm 0.00$ | 0.12
$G^{T}_{K_{1}}/4\pi$ | $0.07\pm 0.01$ | 4.37
$r_{{K_{1}}{K_{\gamma}}}$ | $-2.00\pm 0.36$ |  $\cdots$111Not reported
$\Lambda_{\rm B}$(GeV) | $0.84\pm 0.00$ | 0.72
$\Lambda_{\rm R}$(GeV) | $1.00\pm 0.00$ | 1.25
$\theta_{\rm had}$(deg) | $90.00\pm 10.11$ | 90.0
$\phi_{\rm had}$(deg) | $0.00\pm 20.05$ | 0.00
$\chi^{2}$ | 8729 | 9053
$N_{\rm par}$ | 379 | 341
$N_{\rm data}$ | 7784 | 7784
$\chi^{2}/N_{\rm dof}$ | 1.18 | 1.22

####  III.2.1 $K^{+}\Sigma^{0}$ Channel

Among all possible $K\Sigma$ isospin channels, the $\gamma p\to
K^{+}\Sigma^{0}$ channel has the most abundant experimental data. This is
understandable since, as the production of $K^{+}\Lambda$, the production of
$K^{+}\Sigma^{0}$ is relatively easier to measure due to stable proton target
and relatively simpler technique to measure the decay of $\Lambda$ or
$\Sigma^{0}$ hyperon in the final states. The experimental data mentioned here
include those obtained from the Crystal Ball (at MAMI) crystalball , CLAS
clas2004sig ; clas2006 ; biblap , GRAAL graal:2007tx , LEPS, SAPHIR and
SPring8 collaborations.

Figure 9 shows the comparison between the calculated total cross sections
before and after the inclusion of the high-spin resonances, where the
prediction of Kaon-Maid is also displayed to show the improvement made by the
current models. It is seen that the prediction of Model C (solid red curve) is
practically similar to that of Model D (dashed black curve). The difference
between both models is very subtle and can be seen only at $W\approx 2.0$ GeV
and $W\approx 2.25$ GeV. Overall, both models fit nicely the experimental
data, with an exception at $W\gtrsim 2.15$ GeV, where we observe that there is
a discrepancy problem in the existing experimental data of total and
differential cross sections. This problem will be clarified later when we
discuss the result for differential cross section.

Figure 9: Total cross section of the $\gamma p\to K^{+}\Sigma^{0}$ channel.
Notation of the curves and experimental is given in the figure. Data shown in
this figure were not used in the fitting process of the present model. Figure
10: Energy distributions of the $\gamma p\to K^{+}\Sigma^{0}$ differential
cross section for different values of $\cos\theta$. Shown in the figure are
the results obtained from Model C (solid red curve) and Model D (dashed black
curve). Experimental data shown are taken from the CLAS 2004 (open squares
clas2004sig ), SAPHIR 2004 (solid squares sigsaphir2004 ), CLAS 2006 (solid
diamonds clas2006 ), LEPS 2006 (solid triangles leps2006  and open inverted
triangles sigleps2006 ), CLAS 2007 (solid circles biblap ), and Crystal Ball
(open circles crystalball ) collaborations.

Comparison between differential cross sections obtained from the two models is
shown in Figs. 10 and 11. The energy distribution of differential cross
section shown in Fig. 10 reveals that the difference between the two models is
most obvious at $\cos\theta=0.95$. At this forward angle we can see that the
inclusion of the high-spin resonances yields at least two structures in
differential cross section at the $W\geq 2.6$ GeV, which can be traced back to
the resonance masses.

Furthermore, the second peak in differential cross section becomes more
apparent and much closer to experimental data after the inclusion of these
resonances. This peak still clearly appears at $\cos\theta=0.80$ and 0.65, and
quickly disappears as we move to larger kaon angles. We have investigated the
origin of this second peak and found that it is due to the
$\Delta(2000)F_{35}$ resonance.

In addition, we also observe a third peak at $W\approx 2.3$ GeV, which
originates from the $N(2290)G_{19}$ state with spin 9/2, has positive parity,
and earns a status of four-star in the Particle Data Book pdg . Interestingly,
this state appears in the $K^{+}\Sigma^{0}$ channel after the inclusion of
higher nucleon resonances. This peak also quickly disappears as we increase
the kaon angle, but appears again in the backward angles. In the latter, both
models are in agreement with each other.

The angular distributions of differential cross section shown in Fig. 11
support this finding. In general, the agreement with experimental data are
similar for the two models. In the case when the experimental data from
different collaborations are scattered, the models try to reproduce their
average.

Figure 11: As in Fig. 10, but for angular distribution. The corresponding
total c.m. energy $W$ is shown in each panel in GeV.

####  III.2.2 $K^{0}\Sigma^{+}$ Channel

Figure 12: Total cross section for the $\gamma p\to K^{0}\Sigma^{+}$ process
obtained from the present and previous models. Experimental data are obtained
from the SAPHIR 2005 saphir2005 , CBELSA 2008 cbelsa , MAMI A2 2013 mami2013 ,
and MAMI A2 2019 mami2018  collaborations. All data shown in this figure were
not used in the present analysis.

The $K^{0}\Sigma^{+}$ channel is the last channel measured for the proton
target. Although proton is stable, detection of the neutral kaon and
positively charged $\Sigma$ hyperon in the final state is more challenging.
Figure 12 shows the $K^{0}\Sigma^{+}$ total cross sections obtained from both
models, as well as Kaon-Maid for comparison. Obviously, the three models
display different shapes of total cross section, which originate from
different nucleon and delta resonances used in the models. We note that
experimental data for this channel are relatively scattered, especially in the
energy range $1.77\lesssim W\lesssim 1.90$ GeV, where the new SAPHIR data are
almost 50% smaller than the older SAPHIR data. Fortunately, near the
production threshold all data are in agreement with each other and,
interestingly, closer to the prediction of Kaon-Maid. Nevertheless, at this
point we still observe that the inclusion of higher-spin resonances improves
the model prediction. Furthermore, as shown in Fig. 12, very close to the
threshold the predictions of the three models are very different. While the
predicted cross section of Kaon-Maid is slowly increasing with energy, the
calculated cross section of the present model (Model C) is rising steeply and
reveals the contributions of resonances with masses near 1.7 GeV. We note that
there are four nucleon resonances and one delta resonance to this end. As
stated in Ref. ks_threshold , experimental data near the threshold region are
very crucial to understand the production mechanism and related
phenomenological applications Sakinah:2019zbd .

Despite the fact that there are limited data available for this reaction, the
agreement between model calculation and experimental data increases
significantly with the inclusion of high-spin resonances. As in the case of
previous channels, the inclusion of high-spin resonances leads also to a
number of structures in the total cross sections, as clearly shown in Fig. 12.

The limited number of experimental data for the $K^{0}\Sigma^{+}$
photoproduction clearly impose a strong constraint on the range of validity of
our present model. Figure 13 obviously shows that the available data are
relatively scattered, with apparent peak at $W\approx 1.95$ GeV. This peak is
more obvious in the forward regions. Furthermore, in this figure we can also
observe that the inclusion of high-spin resonances eliminates the small
structure at $W\approx 1.85$ GeV and emphasizes the contribution of resonances
with $m\approx 1.7$ GeV as in the case of total cross section. More
experimental data are strongly required, especially at these energy points, to
clarify the effects of the inclusion of high-spin resonances in the present
model.

Figure 13: Energy distribution of the $\gamma p\to K^{0}\Sigma^{+}$
differential cross section calculated from Models C and D for different values
of $\cos\theta$, compared with available experimental data. Notation of the
curves and experimental data is as in Fig. 12.

The angular distributions of differential cross section shown in Fig. 14 clear
up the difference between Models C and D, which is visible in the whole angles
covered by experimental data. The improvement of the model after including the
high-spin resonances is relatively unclear due to the scattered experimental
data. In general, the inclusion of the high-spin resonances slightly improves
the agreement between model calculation and experimental data. Both Figs. 13
and 14 indicate that the present model prefers the new MAMI A2 2019 data set
mami2018  in the energy range $1.8\lesssim W\lesssim 1.9$ GeV, where
experimental data from different collaborations are significantly scattered.

Figure 14: As in Fig. 13, but for angular distribution. The corresponding
total c.m. energy $W$ in GeV is shown in each panel.

####  III.2.3 $K^{+}\Sigma^{-}$ Channel

The $K^{+}\Sigma^{-}$ photoproduction channel has nearly 300 experimental data
points in the form of differential cross section sigleps2006 ; clas2010sig
and photon asymmetry sigleps2006 . These data were included in the fitting
process. Thus, the total cross sections shown in Fig. 15 are pure prediction
and cannot be compared with experimental measurement. Nevertheless, we still
can see the small effect of the higher-spin resonances near the
$K^{+}\Sigma^{-}$ threshold, as in the case of $K^{0}\Sigma^{+}$ channel, and
at high energy $W\gtrsim 2.1$ GeV where no experimental data are available to
constrain the model. Otherwise, both models C and D show similar trend.

Figure 15: Calculated otal cross section of $\gamma n\to K^{+}\Sigma^{-}$
process.

The calculated $K^{+}\Sigma^{-}$ differential cross sections obtained from
both models are compared with experimental data in Fig. 16. This figure
reveals that the origin of the structures shown in the total cross section.
Near the threshold the peak appears in the whole angular distribution, whereas
the difference between the two models at high energy originates from the
backward angles. It is interesting to note that at high energy the inclusion
of the high-spin resonances slightly increases the cross section, but in the
backward region this situation dramatically changes (see the panel with
$\cos\theta=-0.65$).

Figure 16: Energy distribution of the $\gamma n\to K^{+}\Sigma^{-}$
differential cross section. Notation of the curves is as in Fig. 15.
Experimental data are obtained from the LEPS 2006 sigleps2006  and CLAS 2010
clas2010sig  collaborations.

For completeness, we have also checked the angular distribution of
$K^{+}\Sigma^{-}$ differential cross section. Figure 17 shows the comparison
between both models and experimental data. Similar to Fig. 16, we can see that
the difference between models C and D is also small, except at high energy,
i.e, at $W=2.174$ GeV and backward angle.

Figure 17: As in Fig. 16, but for angular distribution.

####  III.2.4 $K^{0}\Sigma^{0}$ channel

The $K^{0}\Sigma^{0}$ channel has very limited experimental data. They were
obtained by the MAMI A2 2018 collaboration by measuring photoproduction of
neutral kaon on a deuteron target. Although the number is very limited, the
existence of experimental data in this channel significantly helps to
constrain the prediction of the present model. Figure 18 shows the comparison
between calculated total cross sections obtained from previous and present
models and experimental data. It can be seen that the inclusion of the high-
spin resonances improves the model, although the cross section trend is
relatively well reproduced by the three models. The importance of the nucleon
and delta resonances with $m\approx 1.7$ GeV is slightly shown by Model D near
the production threshold.

As shown in Fig. 18 the total cross sections increase monotonically with
increasing energy. In fact, the predicted total cross section of the present
model is more than 3 $\mu$b at $W\approx 2$ GeV, which seems to be unrealistic
if we compare it with those of the neutral kaon productions shown in Figs. 6
and 12. Thus, total cross section data up to 2.5 GeV are very important to
this end.

Figure 18: Total cross section of $\gamma n\to K^{0}\Sigma^{0}$ process.
Experimental data mami2018  shown in this figure were not included in the
fitting process of the present model.

Figures 19 and 20 compare the calculated differential cross sections obtained
from the previous and present models with experimental data. Figure 19 shows
that models C and D start to differ at $W\approx 1.86$ GeV, where no
experimental data are available to constrain them. The inclusion of the high-
spin resonances in this channel improves the cross section divergence, which
is urgently required in forward regions. Interestingly, the two models predict
a resonance structure above this energy point, albeit with different
positions. Certainly, experimental data in the energy range $1.8\lesssim
W\lesssim 2.2$ GeV are very important to determine which resonance is
responsible for this structure.

Figure 19: Energy distribution of the $\gamma n\to K^{0}\Sigma^{0}$
differential cross section. Notation for the curves and experimental data is
as in Fig. 18.

The angular distributions of differential cross section shown in Fig. 20
reveals that the cross section of the $K^{0}\Sigma^{0}$ photoproduction has
backward-peaking behavior. This indicates the dominance of $u$-channel in this
process, which is easily understood from the fact that this process does not
have a $t$-channel in the Born terms since a neutral kaon cannot interact with
real photon.

Figure 20: As in Fig. 19, but for angular distribution.

###  III.3 Extracted Resonance Properties

Having investigated the effect of high-spin resonances on our models, we are
ready to discuss the resonance properties, i.e., their masses and widths, at
their pole positions, before and after including the high-spin resonances.
Note that during the fitting process we allowed the resonance masses and
widths to vary within the estimated error bars of PDG. In Table 5 we show the
resonance masses and widths evaluated at their pole positions extracted from
the present and previous works, compared with those listed by PDG pdg . Note
that for the $N(2600)I_{1,11}$, $N(2700)K_{1,13}$, $\Delta(2750)I_{3,13}$ and
$\Delta(2950)K_{3,15}$ resonances, the PDG does not have any information yet.
Therefore, in this case, the result shown in Table 5 provides the first
estimate for these resonances.

Table 5: Masses and widths of nucleon and $\Delta$ resonances evaluated at the pole position in MeV, obtained from the present work (Models A and C), previous works (Models B Mart:2019mtq  and D samson_ksigma ) and PDG pdg . The status of resonances is due to the PDG pdg .  Resonances | Status | PDG | Model A | Model B | Model C | Model D
---|---|---|---|---|---|---
|  | $m_{\rm{pole}}$ | $\Gamma_{\rm{pole}}$ | $m_{\rm{pole}}$ | $\Gamma_{\rm{pole}}$ | $m_{\rm{pole}}$ | $\Gamma_{\rm{pole}}$ | $m_{\rm{pole}}$ | $\Gamma_{\rm{pole}}$ | $m_{\rm{pole}}$ | $\Gamma_{\rm{pole}}$
$N(1440)P_{11}$ | **** | 1370 $\pm$ 10 | 175 $\pm$ 15 | 1305 $\pm$ 27.9 | 173 $\pm$ 46.0 | 1305 | 173 | 1355 $\pm$ 5.2 | 208 $\pm$ 7.2 | 1324 | 188
$N(1520)D_{13}$ | **** | 1510 $\pm$ 5 | 110 ${}^{+10}_{-5}$ | 1488 $\pm$ 3.5 | 112 $\pm$ 10.8 | 1495 | 101 | 1487 $\pm$ 4.4 | 112 $\pm$ 11.7 | 1489 | 100
$N(1535)S_{11}$ | **** | 1510 $\pm$ 10 | 130 $\pm$ 20 | 1508 $\pm$ 15.0 | 169 $\pm$ 26.0 | 1475 | 162 | 1474 $\pm$ 15.2 | 225 $\pm$ 17.3 | 1530 | 129
$N(1650)S_{11}$ | **** | 1655 $\pm$ 15 | 135 $\pm$ 35 | 1599 $\pm$ 1.5 | 230 $\pm$ 5.0 | 1612 | 232 | 1600 $\pm$ 18.1 | 230 $\pm$ 4.7 | 1664 | 176
$N(1675)D_{15}$ | **** | 1660 $\pm$ 5 | 135 ${}^{+15}_{-10}$ | 1639 $\pm$ 0.0 | 147 $\pm$ 0.2 | 1640 | 147 | 1629 $\pm$ 2.7 | 164 $\pm$ 14.1 | 1643 | 136
$N(1680)F_{15}$ | **** | 1675 ${}^{+5}_{-10}$ | 120 ${}^{+15}_{-10}$ | 1644 $\pm$ 0.9 | 119 $\pm$ 0.6 | 1651 | 123 | 1653 $\pm$ 2.9 | 121 $\pm$ 7.6 | 1667 | 98
$N(1700)D_{13}$ | *** | 1700 $\pm$ 50 | 200 $\pm$ 100 | 1668 $\pm$ 0.3 | 155 $\pm$ 0.2 | 1692 | 158 | 1637 $\pm$ 0.6 | 183 $\pm$ 0.6 | 1630 | 111
$N(1710)P_{11}$ | **** | 1700 $\pm$ 20 | 120 $\pm$ 40 | 1658 $\pm$ 0.9 | 120 $\pm$ 2.9 | 1657 | 175 | 1676 $\pm$ 0.8 | 71 $\pm$ 3.0 | 1705 | 47
$N(1720)P_{13}$ | **** | 1675 $\pm$ 15 | 250 ${}^{+150}_{-100}$ | 1632 $\pm$ 0.1 | 220 $\pm$ 0.4 | 1648 | 193 | 1578 $\pm$ 0.7 | 242 $\pm$ 1.5 | 1665 | 300
$N(1860)F_{15}$ | ** | 1830 ${}^{+120}_{-60}$ | 250 ${}^{+150}_{-50}$ | 1874 $\pm$ 0.2 | 226 $\pm$ 0.4 | 1862 | 217 | 1836 $\pm$ 0.4 | 241 $\pm$ 0.7 | 1787 | 156
$N(1875)D_{13}$ | *** | 1900 $\pm$ 50 | 160 $\pm$ 60 | 1840 $\pm$ 0.2 | 230 $\pm$ 1.0 | 1815 | 222 | 1765 $\pm$ 4.9 | 223 $\pm$ 0.7 | 1757 | 219
$N(1880)P_{11}$ | *** | 1860 $\pm$ 40 | 230 $\pm$ 50 | 1753 $\pm$ 1.7 | 325 $\pm$ 0.7 | 1785 | 298 | 1836 $\pm$ 4.8 | 278 $\pm$ 3.2 | 1831 | 166
$N(1895)S_{11}$ | **** | 1910 $\pm$ 20 | 110 $\pm$ 30 | 1874 $\pm$ 0.2 | 299 $\pm$ 0.4 | 1876 | 224 | 1848 $\pm$ 0.1 | 320 $\pm$ 0.4 | 1893 | 90
$N(1900)P_{13}$ | **** | 1920 $\pm$ 20 | 150 $\pm$ 50 | 1846 $\pm$ 0.2 | 265 $\pm$ 0.4 | 1865 | 256 | 1874 $\pm$ 0.7 | 186 $\pm$ 0.8 | 1899 | 239
$N(1990)F_{17}$ | ** | 2030 $\pm$ 65 | 240 $\pm$ 60 | 1935 $\pm$ 0.8 | 249 $\pm$ 1.3 | 1867 | 240 | 1916 $\pm$ 0.5 | 225 $\pm$ 1.2 | 2044 | 273
$N(2000)F_{15}$ | ** | 2030 $\pm$ 40 | 380 $\pm$ 60 | 1858 $\pm$ 0.2 | 264 $\pm$ 0.5 | 1932 | 273 | 1898 $\pm$ 1.2 | 260 $\pm$ 2.6 | 1978 | 232
$N(2060)D_{15}$ | *** | 2070 ${}^{+60}_{-50}$ | 400 ${}^{+30}_{-50}$ | 1950 $\pm$ 0.7 | 400 $\pm$ 0.3 | 1856 | 341 | 1951 $\pm$ 1.2 | 401 $\pm$ 1.1 | 1968 | 334
$N(2120)D_{13}$ | *** | 2100 $\pm$ 50 | 280 $\pm$ 60 | 1963 $\pm$ 0.7 | 354 $\pm$ 0.1 | 1884 | 354 | 1957 $\pm$ 0.7 | 385 $\pm$ 1.5 | 2029 | 274
$N(2190)G_{17}$ | **** | 2100 $\pm$ 50 | 400 $\pm$ 100 | 2014 $\pm$ 0.6 | 241 $\pm$ 0.3 | 2025 | 244 | 2059 $\pm$ 30.7 | 256 $\pm$ 64.4 | 2142 | 211
$N(2220)H_{19}$ | **** | 2170 ${}^{+30}_{-40}$ | 400 ${}^{+80}_{-40}$ | 2047 $\pm$ 0.4 | 220 $\pm$ 2.0 | 2020 | 228 | 2114 $\pm$ 18.4 | 240 $\pm$ 53.0 | 2131 | 202
$N(2250)G_{19}$ | **** | 2200 $\pm$ 50 | 420 ${}^{+80}_{-80}$ | 2138 $\pm$ 0.5 | 284 $\pm$ 4.9 | 2085 | 265 | 2138 $\pm$ 2.2 | 285 $\pm$ 4.6 | 2193 | 219
$N(2600)I_{1,11}$ | *** | $\cdots$ | $\cdots$ | 2318 $\pm$ 1.9 | 272 $\pm$ 33.9 | $\cdots$ | $\cdots$ | 2389 $\pm$ 11.4 | 310 $\pm$ 34.9 | $\cdots$ | $\cdots$
$N(2700)K_{1,13}$ | ** | $\cdots$ | $\cdots$ | 2393 $\pm$ 3.4 | 247 $\pm$ 18.0 | $\cdots$ | $\cdots$ | 2457 $\pm$ 17.7 | 253 $\pm$ 41.8 | $\cdots$ | $\cdots$
$\Delta(1232)P_{33}$ | **** | 1210 $\pm$ 1 | 100 $\pm$ 2 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 1209 $\pm$ 1.7 | 82 $\pm$ 5.2 | 1205 | 82
$\Delta(1600)P_{33}$ | **** | 1510 $\pm$ 50 | 270 $\pm$ 70 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 1444 $\pm$ 64.3 | 170 $\pm$ 52.5 | 1457 | 168
$\Delta(1620)S_{31}$ | **** | 1600 $\pm$ 10 | 120 $\pm$ 20 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 1565 $\pm$ 15.6 | 140 $\pm$ 20.2 | 1598 | 152
$\Delta(1700)D_{33}$ | **** | 1665 $\pm$ 25 | 250 $\pm$ 50 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 1616 $\pm$ 3.6 | 221 $\pm$ 1.2 | 1646 | 161
$\Delta(1900)S_{31}$ | *** | 1865 $\pm$ 35 | 240 $\pm$ 60 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 1760 $\pm$ 0.3 | 375 $\pm$ 1.0 | 1938 | 330
$\Delta(1905)F_{35}$ | **** | 1800 $\pm$ 30 | 300 $\pm$ 40 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 1772 $\pm$ 4.7 | 228 $\pm$ 15.3 | 1797 | 212
$\Delta(1910)P_{31}$ | **** | 1860 $\pm$ 30 | 300 $\pm$ 100 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 1836 $\pm$ 0.4 | 351 $\pm$ 0.9 | 1859 | 317
$\Delta(1920)P_{33}$ | *** | 1900 $\pm$ 50 | 300 $\pm$ 100 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 1758 $\pm$ 4.0 | 281 $\pm$ 7.7 | 1893 | 193
$\Delta(1930)D_{35}$ | *** | 1880 $\pm$ 40 | 280 $\pm$ 50 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 1850 $\pm$ 0.2 | 314 $\pm$ 0.7 | 1933 | 199
$\Delta(1940)D_{33}$ | ** | 1950 $\pm$ 100 | 350 $\pm$ 150 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 1878 $\pm$ 0.2 | 334 $\pm$ 1.1 | 1880 | 349
$\Delta(1950)F_{37}$ | **** | 1880 $\pm$ 10 | 240 $\pm$ 20 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 1835 $\pm$ 2.3 | 208 $\pm$ 5.7 | 1848 | 208
$\Delta(2000)F_{35}$ | ** | 2150 $\pm$ 100 | 350 $\pm$ 100 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 1844 $\pm$ 16.8 | 275 $\pm$ 42.4 | 2081 | 328
$\Delta(2300)H_{39}$ | ** | 2370 $\pm$ 80 | 420 $\pm$ 160 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 2318 $\pm$ 4.7 | 288 $\pm$ 4.3 | 2341 | 200
$\Delta(2400)G_{39}$ | ** | 2260 $\pm$ 60 | 320 $\pm$ 160 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 2282 $\pm$ 1.2 | 386 $\pm$ 4.1 | 2409 | 329
$\Delta(2420)H_{3,11}$ | **** | 2400 $\pm$ 100 | 450 $\pm$ 100 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 2302 $\pm$ 60.4 | 288 $\pm$ 118.3 | $\cdots$ | $\cdots$
$\Delta(2750)I_{3,13}$ | ** | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 2454 $\pm$ 28.3 | 309 $\pm$ 64.3 | $\cdots$ | $\cdots$
$\Delta(2950)K_{3,15}$ | ** | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 2572 $\pm$ 33.6 | 297 $\pm$ 88.2 | $\cdots$ | $\cdots$
Table 6: Breit-Wigner mass and width of nucleon and $\Delta$ resonances in MeV from present work (Models A and C) previous works (Models B Mart:2019mtq  and D samson_ksigma ) and PDG pdg .  Resonances | PDG | Model A ($K\Lambda$) | Model B ($K\Lambda$) | Model C ($K\Sigma$) | Model D ($K\Sigma$)
---|---|---|---|---|---
| Mass | Width | Mass | Width | Mass | Width | Mass | Width | Mass | Width
$N(1440)P_{11}$ | 1440$\pm$30 | 350$\pm$100 | 1410 $\pm$ 49 | 450 $\pm$41.3 | 1410 | 450 | 1470 $\pm$ 8.0 | 450 $\pm$ 7.6 | 1420 | 450
$N(1520)D_{13}$ | 1515$\pm$5 | 110$\pm$10 | 1520 $\pm$ 2.1 | 120 $\pm$12.3 | 1520 | 100 | 1519 $\pm$ 0.9 | 120 $\pm$ 13.3 | 1510 | 125
$N(1535)S_{11}$ | 1530$\pm$15 | 150$\pm$25 | 1545 $\pm$ 3.7 | 125 $\pm$48.5 | 1545 | 175 | 1545 $\pm$ 27.7 | 175 $\pm$ 32.7 | 1545 | 125
$N(1650)S_{11}$ | 1650$\pm$15 | 125$\pm$25 | 1545 $\pm$ 2.5 | 150 $\pm$0.8 | 1645 | 159 | 1635 $\pm$ 29.4 | 150 $\pm$ 9.8 | 1670 | 170
$N(1675)D_{15}$ | 1675$\pm$15 | 145$\pm$15 | 1635 $\pm$ 0.1 | 130 $\pm$0.3 | 1680 | 130 | 1680 $\pm$ 9.6 | 160 $\pm$ 21.2 | 1670 | 165
$N(1680)F_{15}$ | 1685$\pm$5 | $120^{+10}_{-5}$ | 1680 $\pm$ 1.5 | 115 $\pm$0.7 | 1690 | 120 | 1690 $\pm$ 1.0 | 115 $\pm$ 9.9 | 1686 | 120
$N(1700)D_{13}$ | $1720^{+80}_{-70}$ | 200$\pm$100 | 1712 $\pm$ 0.5 | 134 $\pm$0.5 | 1731 | 102 | 1706 $\pm$ 1.0 | 201 $\pm$ 0.9 | 1650 | 129
$N(1710)P_{11}$ | 1710$\pm$30 | 140$\pm$60 | 1708 $\pm$ 0.5 | 182 $\pm$3.0 | 1733 | 1250 | 1697 $\pm$ 0.4 | 80 $\pm$ 2.8 | 1710 | 50
$N(1720)P_{13}$ | $1720^{+30}_{-40}$ | $250^{+150}_{-100}$ | 1703 $\pm$ 0.5 | 208 $\pm$0.6 | 1700 | 2189 | 1687 $\pm$ 0.0 | 340 $\pm$ 2.3 | 1750 | 400
$N(1860)F_{15}$ | 1928$\pm$21 | 376$\pm$58 | 1980 $\pm$ 0.7 | 235 $\pm$0.8 | 1960 | 220 | 1971 $\pm$ 1.0 | 337 $\pm$ 1.2 | 1829 | 220
$N(1875)D_{13}$ | $1875^{+45}_{-25}$ | $200^{+150}_{-80}$ | 1918 $\pm$ 1.4 | 177 $\pm$2.2 | 1858 | 180 | 1850 $\pm$ 8.7 | 213 $\pm$ 0.9 | 1820 | 320
$N(1880)P_{11}$ | 1880$\pm$50 | 300$\pm$100 | 1930 $\pm$ 4.4 | 400 $\pm$1.2 | 1915 | 280 | 1930 $\pm$ 9.6 | 200 $\pm$ 7.7 | 1856 | 180
$N(1895)S_{11}$ | 1895$\pm$25 | $120^{+80}_{-40}$ | 1903 $\pm$ 0.7 | 142 $\pm$1.2 | 1893 | 106 | 1884 $\pm$ 0.7 | 157 $\pm$ 1.4 | 1893 | 90
$N(1900)P_{13}$ | 1920$\pm$30 | $200^{+120}_{-100}$ | 1920 $\pm$ 0.6 | 169 $\pm$1.1 | 1930 | 151 | 1907 $\pm$ 1.6 | 100 $\pm$ 1.9 | 1930 | 250
$N(1990)F_{17}$ | $2020^{+80}_{-70}$ | 300$\pm$100 | 2057 $\pm$ 2.9 | 245 $\pm$2.7 | 1995 | 265 | 2013 $\pm$ 1.5 | 200 $\pm$ 2.5 | 2125 | 400
$N(2000)F_{15}$ | 2060$\pm$30 | 390$\pm$55 | 2030 $\pm$ 0.4 | 445 $\pm$0.9 | 2090 | 338 | 2046 $\pm$ 0.9 | 335 $\pm$ 4.8 | 2044 | 335
$N(2060)D_{15}$ | $2100^{+100}_{-70}$ | $400^{+50}_{-100}$ | 2200 $\pm$ 2.2 | 450 $\pm$0.8 | 2060 | 450 | 2200 $\pm$ 2.7 | 450 $\pm$ 2.7 | 2060 | 450
$N(2120)D_{13}$ | $2120^{+40}_{-60}$ | $300^{+60}_{-40}$ | 2126 $\pm$ 1.4 | 275 $\pm$0.3 | 2075 | 375 | 2160 $\pm$ 1.6 | 345 $\pm$ 4.2 | 2075 | 305
$N(2190)G_{17}$ | 2180$\pm$40 | 400$\pm$100 | 2159 $\pm$ 1.8 | 300 $\pm$0.5 | 2181 | 300 | 2216 $\pm$ 5.1 | 300 $\pm$ 127.8 | 2200 | 300
$N(2220)H_{19}$ | 2250$\pm$50 | $400^{+100}_{-50}$ | 2200 $\pm$ 2.5 | 350 $\pm$3.1 | 2200 | 500 | 2284 $\pm$ 24.2 | 350 $\pm$ 92.7 | 2204 | 369
$N(2250)G_{19}$ | $2280^{+40}_{-30}$ | $500^{+100}_{-200}$ | 2320 $\pm$ 1.8 | 300 $\pm$11.2 | 2283 | 300 | 2320 $\pm$ 2.2 | 300 $\pm$ 10.4 | 2250 | 300
$N(2600)I_{1,11}$ | $2600^{+150}_{-50}$ | 650$\pm$150 | 2573 $\pm$ 6.9 | 500 $\pm$61.8 | $\cdots$ | $\cdots$ | 2750 $\pm$ 7.8 | 800 $\pm$ 59.1 | $\cdots$ | $\cdots$
$N(2700)K_{1,13}$ | 2612$\pm$45 | 350$\pm$50 | 2621 $\pm$ 3.3 | 400 $\pm$17.8 | $\cdots$ | $\cdots$ | 2675 $\pm$ 9.6 | 300 $\pm$ 89.9 | $\cdots$ | $\cdots$
$\Delta(1232)P_{33}$ | 1232$\pm$2 | 117$\pm$3 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 1234 $\pm$ 2.8 | 114 $\pm$ 4.0 | 1230 | 120
$\Delta(1600)P_{33}$ | 1570$\pm$70 | 250$\pm$50 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 1500 $\pm$ 99 | 200 $\pm$ 68.6 | 1500 | 220
$\Delta(1620)S_{31}$ | 1610$\pm$20 | 130$\pm$20 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 1590 $\pm$ 28.8 | 110 $\pm$ 30.3 | 1600 | 150
$\Delta(1700)D_{33}$ | 1710$\pm$20 | 300$\pm$800 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 1730 $\pm$ 5.9 | 355 $\pm$ 1.2 | 1686 | 213
$\Delta(1900)S_{31}$ | $1860^{+80}_{-20}$ | 250$\pm$70 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 1920 $\pm$ 1.2 | 320 $\pm$ 2.7 | 1920 | 325
$\Delta(1905)F_{35}$ | $1880^{+30}_{-25}$ | $330^{+70}_{-60}$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 1910 $\pm$ 6.7 | 400 $\pm$ 21.9 | 1878 | 400
$\Delta(1910)P_{31}$ | 1900$\pm$50 | 300$\pm$100 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 1950 $\pm$ 1.0 | 242 $\pm$ 2.5 | 1910 | 340
$\Delta(1920)P_{33}$ | 1920$\pm$50 | 300$\pm$60 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 1970 $\pm$ 1.2 | 360 $\pm$ 14.5 | 1908 | 195
$\Delta(1930)D_{35}$ | 1950$\pm$50 | 300$\pm$100 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 2000 $\pm$ 0.6 | 313 $\pm$ 1.5 | 1963 | 220
$\Delta(1940)D_{33}$ | 2000$\pm$60 | 400$\pm$100 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 2044 $\pm$ 1.3 | 330 $\pm$ 2.7 | 1994 | 520
$\Delta(1950)F_{37}$ | 1930$\pm$15 | 285$\pm$50.0 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 1934 $\pm$ 2.2 | 235 $\pm$ 9.7 | 1915 | 335
$\Delta(2000)F_{35}$ | 2015$\pm$24 | 500$\pm$52 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 2039 $\pm$ 6.8 | 552 $\pm$ 65.6 | 2192 | 525
$\Delta(2300)H_{39}$ | 2400$\pm$125 | 425$\pm$150 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 2525 $\pm$ 12 | 275 $\pm$ 10.6 | 2393 | 275
$\Delta(2400)G_{39}$ | 2643$\pm$141 | 895$\pm$432 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 2641 $\pm$ 4.2 | 544 $\pm$ 10.1 | 2504 | 463
$\Delta(2420)H_{3,11}$ | 2300$\pm$150 | 500$\pm$200 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 2515 $\pm$ 12.5 | 300 $\pm$ 281.3 | $\cdots$ | $\cdots$
$\Delta(2750)I_{3,13}$ | 2794$\pm$80 | 350$\pm$100 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 2760 $\pm$ 2.3 | 450 $\pm$ 141.0 | $\cdots$ | $\cdots$
$\Delta(2950)K_{3,15}$ | 2990$\pm$100 | 330$\pm$100 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 2890 $\pm$ 129.2 | 430 $\pm$ 193.3 | $\cdots$ | $\cdots$

In the $K\Lambda$ photoproduction the agreement between the extracted
properties of nucleon resonances and those of PDG is in general from fair to
good. The same situation is also seen in the $K\Sigma$ one. Good agreement
with the PDG values is observed in the case of the $N(1520)D_{13}$ and
$\Delta(1232)P_{33}$ resonances. Certain resonances with low rating status,
e.g., the $N(1875)D_{13}$, $N(2000)F_{15}$ and $\Delta(1900)S_{31}$ states,
show notable deviation from the PDG value. This indicates that the less-
established resonances tend to produce the resonance properties that
significantly deviates from the PDG values. From Table 5 it is also apparent
that the nucleon properties obtained from Model A have a better agreement with
the PDG values, in contrast to Model B, especially for the resonances whose
masses are less than 2000 MeV, as well as for the $N(2220)H_{19}$ and
$N(2250)G_{19}$ states. Furthermore, we can also observe that Model D seems to
produce higher extracted masses than Model C. On the other hand, the
calculated widths obtained from Model C are found to be closer to the PDG
values, except for the $N(2190)G_{17}$, $N(2220)H_{19}$ and $N(2250)G_{19}$
resonances.

Figure 21: Energy distributions of the experimental data used in the fitting
process of Model A ($K\Lambda$ photoproduction) and model C ($K\Sigma$
photoproduction).

The difference between the mass of $N(2700)K_{1,13}$ resonance extracted from
the $K\Lambda$ and $K\Sigma$ channels seems to be large, i.e., nearly 100 MeV.
Furthermore, the extracted masses are smaller than the Breit-Wigner ones, as
can be seen in Table 5. Nevertheless, the masses and widths of the two
resonances extracted in this work provide the first prediction of their pole
properties.

In contrast to the $N(2600)I_{1,11}$ and $N(2700)K_{1,13}$ resonances, the
extracted properties of the $\Delta(2420)H_{3,11}$ resonance are in a good
agreement with the PDG values. For the other two high-spin $\Delta$
resonances, i.e., $\Delta(2750)I_{3,13}$ and $\Delta(2950)K_{3,15}$ states,
the PDG does not have data for comparison. For both $\Delta(2750)I_{3,13}$ and
$\Delta(2950)K_{3,15}$ resonances the extracted masses at the pole position
are much smaller than those of the Breit-Wigner. Since the resonances affect
the cross section and other polarization observables only at high energies,
i.e., $W\gtrsim 2.7$ GeV, a study devoted for high energy photoproduction
would be very relevant to this end. This would be also in line with the 12 GeV
JLab experiments that are currently in progress.

In Tables 5 and 6 we have also included the error bars both from PDG and the
present work. Note that the previous works did not report the uncertainties in
the extracted resonance masses and widths. The quoted error bars of the
present work originate from the CERN-MINUIT output used for the fitting
process, where we employed the MIGRAD minimizer that produces both estimated
error bars of the fitted parameters and error matrix James:1975dr .

The error bars obtained from the MINUIT indicate the flexibility of the model
to the variation of the resonance masses and widths to reproduce the data,
i.e., the larger the error bars the more flexible the model. As a consequence,
in the energy region where precise experimental data are abundantly available
the error bars are forced to be small. Thus, we might expect that the error
bars are large below and near the threshold region and for $W\gtrsim 2.4$ GeV
(see Fig. 21). This is proven in Table 5, where we can see that relatively
larger error bars are obtained in Model A ($K\Lambda$ photoproduction with
$W^{\rm thr.}\approx 1610$ MeV) for the $N(1440)P_{11}$, $N(1520)D_{13}$,
$N(1535)S_{11}$ resonances. In Model C ($K\Sigma$ photoproduction with $W^{\rm
thr.}\approx 1690$ MeV) three more resonances, i.e., the $N(1650)S_{11}$,
$N(1675)D_{15}$, and $N(1680)F_{15}$ states, also exhibit this phenomenon.
Only the $\Delta(1232)P_{33}$ resonance has very small error bars, since PDG
has estimated this resonance with very precise mass and width, whereas during
the fitting process we allowed these parameters to vary within the PDG
uncertainties. In the higher energy region we observe that all baryon
resonances with $m_{R}\gtrsim 2400$ MeV show the relatively larger error bars.

In conclusion, we have observed that the addition of the high-spin resonances,
with spins from 11/2 up to 15/2, in our isobar models improves the agreement
between their pole properties extracted in this work and those listed by PDG.

##  IV Summary and Conclusion

We have derived the spin-11/2, -13/2, and -15/2 resonance amplitudes for kaon
photoproduction off the nucleon by using the covariant Feynman diagrammatic
technique. For this purpose we made use of the consistent interaction
Lagrangians proposed by Pascalutsa and Vrancx et al., as well as the
formulation of spin-(n+1/2) resonance propagator put forwarded by Vrancx et
al. We have studied the effect of high-spin resonances in the kaon
photoproduction processes by including the $N(2600)I_{1,11}$ and
$N(2700)K_{1,13}$ states in our previous model for $K\Lambda$ photoproduction
and the $N(2600)I_{1,11}$, $N(2700)K_{1,13}$, $\Delta(2420)H_{3,11}$,
$\Delta(2750)I_{3,13}$, and $\Delta(2950)K_{3,15}$ states in our previous
model developed for $K\Sigma$ photoproduction. In general, the inclusion of
these high-spin resonances improves the agreement between model calculations
and experimental data, which is indicated by the smaller values of $\chi^{2}$
in all isospin channels. The inclusion of the high-spin resonances also helps
to overcome the problem of resonance-dominated model, since the inclusion
increases the hadronic form factor cutoff of the Born terms and, therefore,
increases the role of the Born terms in both $K\Lambda$ and $K\Sigma$ models.
Specifically, in the $K^{+}\Lambda$ channel the inclusion leads to fewer
resonance structures in the cross sections and polarization observables. The
effect is significant in the high energy differential cross section, near the
resonance masses and at forward direction. Different from the $K^{+}\Lambda$
channel, the effect in the $K^{0}\Lambda$ channel is more obvious and can be
observed in both low and high energy regions. Furthermore, in this channel the
effect is found to be large in both forward and backward angles. In the
$K^{+}\Sigma^{0}$ channel the effect is only significant in the forward
region, where a number of resonance structures appear after including the
high-spin resonances. Two of them are important to note here, i.e., the
$\Delta(2000)F_{35}$ and $N(2290)G_{19}$ resonances, which are responsible for
the second and third peaks in the $K^{+}\Sigma^{0}$ differential cross
section. In contrast to the $K^{+}\Sigma^{0}$ channel, the $K^{0}\Sigma^{+}$
channel is found to be sensitive to these high-spin resonances. In this case,
the effect can be observed in the whole energy range covered by experimental
data and the whole angular distribution. The effect is, however, not observed
in the $K^{+}\Sigma^{-}$ and $K^{0}\Sigma^{0}$ channels, at least in the whole
kinematics where experimental data are available. Finally, we found that the
addition of the high-spin resonances leads to a better agreement between the
extracted resonance properties and those listed by PDG.

## Acknowledgments

This work has been supported by the PUTI Q1 Grant of Universitas Indonesia,
under contract No. BA-1080/UN2.RST/PPM.00.03.01/2020.

##  Appendix A Form Functions $A_{i}$ for Baryon Resonances with Spins 11/2,
13/2 and 15/2

The extracted form functions $A_{i}$ given in Eq. (54) for baryon resonances
with spin 11/2 read

$\displaystyle A_{1}$ $\displaystyle=$
$\displaystyle\Bigl{[}\,14c_{1}D_{1}(s\pm
m_{N^{*}}m_{\Lambda})+10D_{3}\Bigl{\\{}(m_{\Lambda}^{2}c_{k}-b_{p}c_{\Lambda})\pm
m_{N^{*}}m_{\Lambda}(\textstyle{\frac{1}{s}}c_{k}c_{\Lambda}-b_{p})\Bigr{\\}}\,\Bigr{]}\,G_{1}$
(63)
$\displaystyle+\textstyle{\frac{1}{2}}(m_{p}+m_{\Lambda})\Bigl{[}\,\Bigl{\\{}(m_{p}c_{\Lambda}-m_{\Lambda}c_{p})\pm
m_{N^{*}}(m_{p}m_{\Lambda}-\textstyle{\frac{1}{s}}c_{\Lambda}c_{p})\Bigr{\\}}\Bigl{\\{}28c_{1}(c_{1}-b_{p}c_{s})D_{4}-\textstyle{\frac{4}{s}}k^{2}c_{2}c_{p}D_{5}+D_{3}\Bigr{\\}}$
$\displaystyle-
b_{p}(m_{\Lambda}\pm\textstyle{\frac{1}{s}}m_{N^{*}}c_{\Lambda})D_{3}\pm
7m_{N^{*}}\Bigl{\\{}(c_{1}-b_{p}c_{s})D_{2}-\textstyle{\frac{4}{s}}k^{2}c_{1}c_{2}c_{p}D_{4}\Bigr{\\}}\,\Bigr{]}\,G_{2}+\Bigl{[}\,\textstyle{\frac{1}{2}}(m_{p}+m_{\Lambda})\Bigl{\\{}(s\pm
m_{N^{*}}m_{p})$
$\displaystyle\times\bigl{\\{}-7c_{s}D_{2}-4c_{1}c_{2}D_{4}\textstyle{\frac{1}{s}}c_{k}\bigr{\\}}-\bigl{\\{}(m_{p}m_{\Lambda}+11c_{\Lambda})\pm
m_{N^{*}}(\textstyle{\frac{1}{s}}m_{p}c_{\Lambda}+11m_{\Lambda})\bigr{\\}}D_{3}$
$\displaystyle-\bigl{\\{}(b_{p}c_{\Lambda}-m_{p}m_{\Lambda}c_{k})\pm
m_{N^{*}}(\textstyle{\frac{1}{s}}m_{p}c_{\Lambda}c_{k}-m_{\Lambda}b_{p})\bigr{\\}}(-28c_{1}c_{s}D_{4}-4c_{2}D_{5}\textstyle{\frac{1}{s}}c_{k})\Bigr{\\}}$
$\displaystyle+\Bigl{\\{}(6m_{p}c_{\Lambda}+6m_{\Lambda}c_{p}+m_{\Lambda}c_{k})\pm
m_{N^{*}}(6m_{p}m_{\Lambda}+\textstyle{\frac{6}{s}}c_{p}c_{\Lambda}+\textstyle{\frac{1}{s}}c_{k}c_{\Lambda})\Bigr{\\}}D_{3}\mp
7m_{N^{*}}c_{1}D_{1}\,\Bigr{]}\,G_{3}\,,$ $\displaystyle A_{2}$
$\displaystyle=$
$\displaystyle{\frac{1}{t-m_{K}^{2}}}\Bigl{\\{}\Bigl{[}-20k^{2}D_{3}(c_{\Lambda}\pm
m_{N^{*}}m_{\Lambda})\Bigr{]}\,G_{1}+\Bigl{[}7(-s\pm
m_{N^{*}}m_{p})\Bigl{\\{}(b_{p}c_{s}-c_{1})D_{2}+\textstyle{\frac{4}{s}}k^{2}c_{1}c_{2}c_{p}D_{4}\Bigr{\\}}$
(64) $\displaystyle-
m_{p}k^{2}(m_{\Lambda}\pm\textstyle{\frac{1}{s}}m_{N^{*}}c_{\Lambda})D_{3}+4\Bigl{\\{}(m_{p}m_{\Lambda}c_{k}-b_{p}c_{\Lambda})\pm
m_{N^{*}}(\textstyle{\frac{1}{s}}m_{p}c_{\Lambda}c_{k}-m_{\Lambda}b_{p})\Bigr{\\}}\Bigl{\\{}7c_{1}D_{4}(b_{p}c_{s}-c_{1})$
$\displaystyle+\textstyle{\frac{1}{s}}k^{2}c_{2}c_{p}D_{5}\Bigr{\\}}\,\Bigr{]}\,G_{2}+m_{p}\Bigl{[}(m_{\Lambda}\pm\textstyle{\frac{1}{s}}m_{N^{*}}c_{\Lambda})\Bigl{\\{}D_{3}k^{2}+56c_{1}c_{k}D_{4}(c_{1}-b_{p}c_{s})-8c_{2}\textstyle{\frac{1}{s}}k^{2}c_{k}c_{p}D_{5}\Bigr{\\}}$
$\displaystyle+\Bigl{\\{}(m_{p}c_{\Lambda}+m_{\Lambda}c_{p})\pm
m_{N^{*}}(m_{p}m_{\Lambda}+\textstyle{\frac{1}{s}}c_{\Lambda}c_{p})\Bigr{\\}}4k^{2}(7c_{1}c_{s}D_{4}+\textstyle{\frac{1}{s}}c_{2}c_{k}D_{5})$
$\displaystyle\pm
7m_{N^{*}}\Bigl{\\{}D_{2}(k^{2}c_{s}-2b_{q})-4c_{1}c_{2}k^{2}D_{4}(1+\textstyle{\frac{1}{s}}c_{p})\Bigr{\\}}\,\Bigr{]}\,G_{3}\Bigr{\\}}\,,$
$\displaystyle A_{3}$ $\displaystyle=$
$\displaystyle\Bigl{[}\,10c_{k}D_{3}(m_{\Lambda}\pm\textstyle{\frac{1}{s}}m_{N^{*}}c_{\Lambda})\pm
14m_{N^{*}}c_{1}D_{1}\,\Bigr{]}\,G_{1}+\textstyle{\frac{1}{2}}\Bigl{[}\,\Bigl{\\{}(m_{p}c_{\Lambda}-m_{\Lambda}c_{p})\pm
m_{N^{*}}(m_{p}m_{\Lambda}-\textstyle{\frac{1}{s}}c_{\Lambda}c_{p})\Bigr{\\}}$
(65)
$\displaystyle\times\Bigl{\\{}28c_{1}D_{4}\bigl{(}c_{1}+b_{p}(1+\textstyle{\frac{1}{s}}c_{\Lambda})\bigr{)}-\textstyle{\frac{4}{s}}k^{2}c_{2}c_{p}D_{5}+D_{3}\Bigr{\\}}-b_{p}(m_{\Lambda}\pm\textstyle{\frac{1}{s}}m_{N^{*}}c_{\Lambda})D_{3}$
$\displaystyle\pm
7m_{N^{*}}\Bigl{\\{}\bigl{(}c_{1}+b_{p}(1+\textstyle{\frac{1}{s}}c_{\Lambda})\bigr{)}D_{2}-\textstyle{\frac{4}{s}}k^{2}c_{1}c_{2}c_{p}D_{4}\Bigr{\\}}\,\Bigr{]}\,G_{2}+\textstyle{\frac{1}{2}}\Bigl{[}\,7(s\pm
m_{N^{*}}m_{p})\Bigl{\\{}(1+\textstyle{\frac{1}{s}}c_{\Lambda})D_{2}-4c_{1}c_{2}\textstyle{\frac{1}{s}}c_{k}D_{4}\Bigr{\\}}$
$\displaystyle-\Bigl{\\{}(m_{p}m_{\Lambda}+11c_{\Lambda})\pm
m_{N^{*}}(\textstyle{\frac{1}{s}}m_{p}c_{\Lambda}+11m_{\Lambda})\Bigr{\\}}D_{3}-\Bigl{\\{}(b_{p}c_{\Lambda}-m_{p}m_{\Lambda}c_{k})\pm
m_{N^{*}}(\textstyle{\frac{1}{s}}m_{p}c_{\Lambda}c_{k}-m_{\Lambda}b_{p})\Bigr{\\}}$
$\displaystyle\times\Bigl{\\{}28c_{1}D_{4}(1+\textstyle{\frac{1}{s}}c_{\Lambda})-4c_{2}D_{5}\textstyle{\frac{1}{s}}c_{k}\Bigr{\\}}\,\Bigr{]}\,G_{3}\,,$
$\displaystyle A_{4}$ $\displaystyle=$
$\displaystyle\Bigl{[}\,10c_{k}D_{3}(m_{\Lambda}\pm\textstyle{\frac{1}{s}}m_{N^{*}}c_{\Lambda})\pm
14m_{N^{*}}c_{1}D_{2}\,\Bigr{]}\,G_{1}+\textstyle{\frac{1}{2}}\Bigl{[}\,\Bigl{\\{}(m_{p}c_{\Lambda}-m_{\Lambda}c_{p})\pm
m_{N^{*}}(m_{p}m_{\Lambda}-\textstyle{\frac{1}{s}}c_{\Lambda}c_{p})\Bigr{\\}}$
(66)
$\displaystyle\times\Bigl{\\{}28c_{1}D_{4}(c_{1}-b_{p}c_{s})-\textstyle{\frac{4}{s}}k^{2}c_{2}c_{p}D_{5}+D_{3}\Bigr{\\}}-b_{p}(m_{\Lambda}\pm\textstyle{\frac{1}{s}}m_{N^{*}}c_{\Lambda}D_{3}\pm
7m_{N^{*}}\Bigl{\\{}(c_{1}-b_{p}c_{s})D_{2}$
$\displaystyle-\textstyle{\frac{4}{s}}k^{2}c_{1}c_{2}c_{p}D_{4}\Bigr{\\}}\,\Bigr{]}\,G_{2}+\textstyle{\frac{1}{2}}\Bigl{[}\,7(s\pm
m_{N^{*}}m_{p})\Bigl{\\{}-c_{s}D_{2}-4c_{1}c_{2}D_{4}\textstyle{\frac{1}{s}}c_{k}\Bigr{\\}}-\Bigl{\\{}(m_{p}m_{\Lambda}+11c_{\Lambda}$
$\displaystyle\pm
m_{N^{*}}(\textstyle{\frac{1}{s}}m_{p}c_{\Lambda}+11m_{\Lambda})\Bigr{\\}}D_{3}-\Bigl{\\{}(b_{p}c_{\Lambda}-m_{p}m_{\Lambda}c_{k})\pm
m_{N^{*}}(\textstyle{\frac{1}{s}}m_{p}c_{\Lambda}c_{k}-m_{\Lambda}b_{p})\Bigr{\\}}$
$\displaystyle\times\Bigl{\\{}-28c_{1}c_{s}D_{4}-4c_{2}D_{5}\textstyle{\frac{1}{s}}c_{k}\Bigr{\\}}\,\Bigr{]}\,G_{3}\,,$
$\displaystyle A_{5}$ $\displaystyle=$
$\displaystyle{\frac{1}{t-m_{K}^{2}}}\Bigl{\\{}\Bigl{[}\,10c_{5}D_{3}(c_{\Lambda}\pm
m_{N^{*}}m_{\Lambda})\Bigr{]}\,G_{1}+\textstyle{\frac{1}{2}}\Bigl{[}7(-s\pm
m_{N^{*}}m_{p})\Bigl{\\{}D_{2}(b_{p}+b_{\Lambda}-\textstyle{\frac{1}{s}}c_{\Lambda}c_{5})-\textstyle{\frac{4}{s}}c_{1}c_{2}c_{5}c_{p}D_{4}\Bigr{\\}}$
(67)
$\displaystyle+m_{p}c_{5}(m_{\Lambda}\pm\textstyle{\frac{1}{s}}m_{N^{*}}c_{\Lambda})D_{3}+4\Bigl{\\{}(m_{p}m_{\Lambda}c_{k}-b_{p}c_{\Lambda})\pm
m_{N^{*}}(\textstyle{\frac{1}{s}}m_{p}c_{\Lambda}c_{k}-m_{\Lambda}b_{p})\Bigr{\\}}\Bigl{\\{}7c_{1}D_{4}(b_{p}+b_{\Lambda}-\textstyle{\frac{1}{s}}c_{\Lambda}c_{5})$
$\displaystyle-\textstyle{\frac{1}{s}}c_{2}c_{5}c_{p}D_{5}\Bigr{\\}}\Bigr{]}\,G_{2}+m_{p}\Bigl{[}\,\textstyle-{\frac{1}{2}}(m_{\Lambda}\pm\textstyle{\frac{1}{s}}m_{N^{*}}c_{\Lambda})\Bigl{\\{}c_{5}\Bigl{\\{}D_{3}-\textstyle{\frac{8}{s}}c_{2}c_{k}c_{p}D_{5}\Bigr{\\}}+56c_{1}c_{k}D_{4}(b_{p}+b_{\Lambda}-\textstyle{\frac{1}{s}}c_{\Lambda}c_{5})\Bigr{\\}}$
$\displaystyle-\Bigl{\\{}(m_{p}c_{\Lambda}+m_{\Lambda}c_{p})\pm
m_{N^{*}}(m_{p}m_{\Lambda}+\textstyle{\frac{1}{s}}c_{\Lambda}c_{p})\Bigr{\\}}\Bigl{\\{}28c_{1}D_{4}(4b_{\Lambda}-k^{2}-\textstyle{\frac{1}{s}}c_{\Lambda}c_{5})+\textstyle{\frac{4}{s}}c_{2}c_{5}c_{k}D_{5}\Bigr{\\}}$
$\displaystyle\pm\textstyle{\frac{7}{2}}m_{N^{*}}\Bigl{\\{}D_{2}(k^{2}-2b_{q}+\textstyle{\frac{1}{s}}c_{\Lambda}c_{5})+4c_{1}c_{2}c_{5}D_{4}(1+\textstyle{\frac{1}{s}}c_{p})\Bigr{\\}}\,\Bigr{]}\,G_{3}\Bigr{\\}}\,,$
$\displaystyle A_{6}$ $\displaystyle=$
$\displaystyle\Bigl{[}\,10\Bigl{\\{}(m_{\Lambda}s+m_{p}c_{\Lambda})\pm
m_{N^{*}}(c_{\Lambda}+m_{p}m_{\Lambda})\Bigr{\\}}D_{3}\,\Bigr{]}\,G_{1}\
+\textstyle{\frac{1}{2}}\Bigl{[}\Bigl{\\{}(m_{p}c_{\Lambda}-m_{\Lambda}c_{p})\pm
m_{N^{*}}(m_{p}m_{\Lambda}-\textstyle{\frac{1}{s}}c_{\Lambda}c_{p})\Bigr{\\}}$
(68)
$\displaystyle\times\Bigl{\\{}D_{3}+28c_{1}D_{4}(c_{1}+b_{p}c_{s})-\textstyle{\frac{4}{s}}c_{2}c_{4}c_{p}D_{5}\Bigr{\\}}+b_{p}(m_{\Lambda}\pm\textstyle{\frac{1}{s}}m_{N^{*}}c_{\Lambda})D_{3}$
$\displaystyle\pm
7m_{N^{*}}\Bigl{\\{}D_{2}(c_{1}+b_{p}c_{s})-\textstyle{\frac{4}{s}}c_{1}c_{2}c_{4}c_{p}D_{4}\Bigr{\\}}\,\Bigr{]}\,G_{2}+\textstyle{\frac{1}{2}}m_{p}\Bigl{[}7(s\pm
m_{N^{*}}m_{p})\Bigl{\\{}D_{2}c_{s}-4c_{1}c_{2}D_{4}(1+\textstyle{\frac{1}{s}}c_{p})\Bigr{\\}}$
$\displaystyle+D_{3}\Bigl{\\{}(m_{p}m_{\Lambda}-9c_{\Lambda})\pm
m_{N^{*}}(\textstyle{\frac{1}{s}}m_{p}c_{\Lambda}-9m_{\Lambda})\Bigr{\\}}+4\bigl{\\{}7c_{1}c_{s}D_{4}+c_{2}D_{5}(1+\textstyle{\frac{1}{s}}c_{p})\bigr{\\}}\Bigl{\\{}(m_{p}m_{\Lambda}c_{k}-b_{p}c_{\Lambda})$
$\displaystyle\pm
m_{N^{*}}(\textstyle{\frac{1}{s}}m_{p}c_{\Lambda}c_{k}-m_{\Lambda}b_{p})\Bigr{\\}}\,\Bigr{]}\,G_{3}\,,$

whereas the extracted functions $A_{i}$ for baryon resonances with spin 13/2
read

$\displaystyle A_{1}$ $\displaystyle=$ $\displaystyle\Bigl{[}\,2D_{6}(-s\pm
m_{N^{*}}m_{\Lambda})+12c_{1}D_{7}\Bigl{\\{}-(m_{\Lambda}^{2}c_{k}-b_{p}c_{\Lambda})\pm
m_{N^{*}}m_{\Lambda}(\textstyle{\frac{1}{s}}c_{k}c_{\Lambda}-b_{p})\Bigr{\\}}\,\Bigr{]}\,G_{1}$
(69)
$\displaystyle+\textstyle{\frac{1}{2}}(m_{p}+m_{\Lambda})\Bigl{[}\,\Bigl{\\{}-(m_{p}c_{\Lambda}-m_{\Lambda}c_{p})\pm
m_{N^{*}}(m_{p}m_{\Lambda}-\textstyle{\frac{1}{s}}c_{\Lambda}c_{p})\Bigr{\\}}\Bigl{\\{}5D_{2}(c_{1}-b_{p}c_{s})-\textstyle{\frac{20}{s}}k^{2}c_{1}c_{2}c_{p}D_{4}+c_{1}D_{8}\Bigr{\\}}$
$\displaystyle-
b_{p}(-m_{\Lambda}\pm\textstyle{\frac{1}{s}}m_{N^{*}}c_{\Lambda})c_{1}D_{8}\pm
m_{N^{*}}\Bigl{\\{}3c_{1}(c_{1}-b_{p}c_{s})D_{9}-\textstyle{\frac{5}{s}}k^{2}c_{2}c_{p}D_{2}\Bigr{\\}}\,\Bigr{]}\,G_{2}+\Bigl{[}\,\textstyle{\frac{1}{2}}(m_{p}+m_{\Lambda})$
$\displaystyle\times\Bigl{\\{}(-s\pm
m_{N^{*}}m_{p})\bigl{\\{}-3c_{s}c_{1}D_{9}-5\textstyle{\frac{1}{s}}c_{2}c_{k}D_{2}\bigr{\\}}-c_{1}\bigl{\\{}-(m_{p}m_{\Lambda}+13c_{\Lambda})\pm
m_{N^{*}}(\textstyle{\frac{1}{s}}m_{p}c_{\Lambda}+13m_{\Lambda})\bigr{\\}}D_{8}$
$\displaystyle+5\bigl{\\{}-(b_{p}c_{\Lambda}-m_{p}m_{\Lambda}c_{k})\pm
m_{N^{*}}(\textstyle{\frac{1}{s}}m_{p}c_{\Lambda}c_{k}-m_{\Lambda}b_{p})\bigr{\\}}\bigl{(}c_{s}c_{1}^{2}D_{2}+\textstyle{\frac{4}{s}}c_{1}c_{2}c_{k}D_{4}\bigr{)}\Bigr{\\}}$
$\displaystyle+c_{1}\Bigl{\\{}-(7m_{p}c_{\Lambda}+7m_{\Lambda}c_{p}+m_{\Lambda}c_{k})\pm
m_{N^{*}}(7m_{p}m_{\Lambda}+\textstyle{\frac{7}{s}}c_{p}c_{\Lambda}+\textstyle{\frac{1}{s}}c_{k}c_{\Lambda})\Bigr{\\}}D_{8}\mp
m_{N^{*}}D_{6}\,\Bigr{]}\,G_{3}\,,$ $\displaystyle A_{2}$ $\displaystyle=$
$\displaystyle{\frac{1}{t-m_{K}^{2}}}\Bigl{\\{}\Bigl{[}-24k^{2}c_{1}D_{7}(-c_{\Lambda}\pm
m_{N^{*}}m_{\Lambda})\Bigr{]}\,G_{1}+\Bigl{[}(s\pm
m_{N^{*}}m_{p})\Bigl{\\{}3c_{1}(b_{p}c_{s}-c_{1})D_{9}+\textstyle{\frac{5}{s}}k^{2}c_{2}c_{p}D_{2}\Bigr{\\}}$
(70) $\displaystyle-
m_{p}k^{2}c_{1}(-m_{\Lambda}\pm\textstyle{\frac{1}{s}}m_{N^{*}}c_{\Lambda})D_{8}+\Bigl{\\{}-(m_{p}m_{\Lambda}c_{k}-b_{p}c_{\Lambda})\pm
m_{N^{*}}(\textstyle{\frac{1}{s}}m_{p}c_{\Lambda}c_{k}-m_{\Lambda}b_{p})\Bigr{\\}}\Bigl{\\{}5D_{2}(b_{p}c_{s}-c_{1})$
$\displaystyle+\textstyle{\frac{20}{s}}k^{2}c_{1}c_{2}c_{p}D_{4}\Bigr{\\}}\,\Bigr{]}\,G_{2}+m_{p}\Bigl{[}(-m_{\Lambda}\pm\textstyle{\frac{1}{s}}m_{N^{*}}c_{\Lambda})\Bigl{\\{}k^{2}c_{1}D_{8}+10c_{k}(c_{1}-b_{p}c_{s})D_{2}-\textstyle{\frac{40}{s}}k^{2}c_{1}c_{2}c_{k}c_{p}D_{4}\Bigr{\\}}$
$\displaystyle+\Bigl{\\{}-(m_{p}c_{\Lambda}+m_{\Lambda}c_{p})\pm
m_{N^{*}}(m_{p}m_{\Lambda}+\textstyle{\frac{1}{s}}c_{\Lambda}c_{p})\Bigr{\\}}5k^{2}\bigl{(}c_{s}D_{2}+\textstyle{\frac{4}{s}}c_{1}c_{2}c_{k}D_{4}\bigr{)}$
$\displaystyle\pm
m_{N^{*}}\Bigl{\\{}3c_{1}D_{9}(k^{2}c_{s}-2b_{q})-5k^{2}c_{2}(1+\textstyle{\frac{1}{s}}c_{p})D_{2}\Bigr{\\}}\,\Bigr{]}\,G_{3}\Bigr{\\}}\,,$
$\displaystyle A_{3}$ $\displaystyle=$
$\displaystyle\Bigl{[}\,12c_{k}c_{1}D_{7}(-m_{\Lambda}\pm\textstyle{\frac{1}{s}}m_{N^{*}}c_{\Lambda})\pm
2m_{N^{*}}D_{6}\,\Bigr{]}\,G_{1}+\textstyle{\frac{1}{2}}\Bigl{[}\,\Bigl{\\{}-(m_{p}c_{\Lambda}-m_{\Lambda}c_{p})\pm
m_{N^{*}}(m_{p}m_{\Lambda}-\textstyle{\frac{1}{s}}c_{\Lambda}c_{p})\Bigr{\\}}$
(71)
$\displaystyle\times\Bigl{\\{}5D_{2}+c_{2}^{2}c_{3}^{2})\bigl{(}c_{1}+b_{p}(1+\textstyle{\frac{1}{s}}c_{\Lambda})\bigr{)}-\textstyle{\frac{20}{s}}k^{2}c_{1}c_{2}c_{p}D_{4}+c_{1}D_{2}\Bigr{\\}}-b_{p}c_{1}(-m_{\Lambda}\pm\textstyle{\frac{1}{s}}m_{N^{*}}c_{\Lambda})D_{8}$
$\displaystyle\pm
m_{N^{*}}\Bigl{\\{}3c_{1}\bigl{(}c_{1}+b_{p}(1+\textstyle{\frac{1}{s}}c_{\Lambda})\bigr{)}D_{9}-\textstyle{\frac{5}{s}}k^{2}c_{2}c_{p}D_{2}\Bigr{\\}}\,\Bigr{]}\,G_{2}+\textstyle{\frac{1}{2}}\Bigl{[}\,(-s\pm
m_{N^{*}}m_{p})\Bigl{\\{}3c_{1}(1+\textstyle{\frac{1}{s}}c_{\Lambda})D_{9}$
$\displaystyle-\textstyle{\frac{5}{s}}c_{2}c_{k}D_{2}\Bigr{\\}}-c_{1}\Bigl{\\{}-(m_{p}m_{\Lambda}+13c_{\Lambda})\pm
m_{N^{*}}(\textstyle{\frac{1}{s}}m_{p}c_{\Lambda}+13m_{\Lambda})\Bigr{\\}}D_{8}-5\Bigl{\\{}-(b_{p}c_{\Lambda}-m_{p}m_{\Lambda}c_{k})$
$\displaystyle\pm
m_{N^{*}}(\textstyle{\frac{1}{s}}m_{p}c_{\Lambda}c_{k}-m_{\Lambda}b_{p})\Bigr{\\}}\Bigl{\\{}(1+\textstyle{\frac{1}{s}}c_{\Lambda})D_{2}-\textstyle{\frac{4}{s}}c_{1}c_{2}c_{k}D_{4}\Bigr{\\}}\,\Bigr{]}\,G_{3}\,,$
$\displaystyle A_{4}$ $\displaystyle=$
$\displaystyle\Bigl{[}\,12c_{k}c_{1}D_{7}(-m_{\Lambda}\pm\textstyle{\frac{1}{s}}m_{N^{*}}c_{\Lambda})\pm
2m_{N^{*}}D_{6}\,\Bigr{]}\,G_{1}+\textstyle{\frac{1}{2}}\Bigl{[}\,\Bigl{\\{}-(m_{p}c_{\Lambda}-m_{\Lambda}c_{p})\pm
m_{N^{*}}(m_{p}m_{\Lambda}-\textstyle{\frac{1}{s}}c_{\Lambda}c_{p})\Bigr{\\}}$
(72)
$\displaystyle\times\Bigl{\\{}5D_{2}(c_{1}-b_{p}c_{s})-\textstyle{\frac{20}{s}}k^{2}c_{1}c_{2}c_{p}D_{4}+c_{1}D_{8}\Bigr{\\}}-b_{p}c_{1}(-m_{\Lambda}\pm\textstyle{\frac{1}{s}}m_{N^{*}}c_{\Lambda})D_{8}\pm
m_{N^{*}}\Bigl{\\{}3c_{1}(c_{1}-b_{p}c_{s})D_{9}$
$\displaystyle-\textstyle{\frac{5}{s}}k^{2}c_{2}c_{p}D_{2}\Bigr{\\}}\,\Bigr{]}\,G_{2}+\textstyle{\frac{1}{2}}\Bigl{[}\,(-s\pm
m_{N^{*}}m_{p})\bigl{(}-3c_{1}c_{s}D_{9}-\textstyle{\frac{5}{s}}c_{2}c_{k}D_{2}\bigr{)}-c_{1}\Bigl{\\{}-(m_{p}m_{\Lambda}+13c_{\Lambda})$
$\displaystyle\pm
m_{N^{*}}(\textstyle{\frac{1}{s}}m_{p}c_{\Lambda}13m_{\Lambda})\Bigr{\\}}D_{8}-5\Bigl{\\{}-(b_{p}c_{\Lambda}-m_{p}m_{\Lambda}c_{k})\pm
m_{N^{*}}(\textstyle{\frac{1}{s}}m_{p}c_{\Lambda}c_{k}-m_{\Lambda}b_{p})\Bigr{\\}}$
$\displaystyle\times\bigl{(}-c_{s}D_{2}-\textstyle{\frac{4}{s}}c_{1}c_{2}c_{k}D_{4}\bigr{)}\,\Bigr{]}\,G_{3}\,,$
$\displaystyle A_{5}$ $\displaystyle=$
$\displaystyle{\frac{1}{t-m_{K}^{2}}}\Bigl{\\{}\Bigl{[}\,12c_{5}c_{1}D_{7}\bigl{\\{}-c_{\Lambda}\pm
m_{N^{*}}m_{\Lambda}\bigr{\\}}\,\Bigr{]}\,G_{1}+\textstyle{\frac{1}{2}}\Bigl{[}(s\pm
m_{N^{*}}m_{p})\Bigl{\\{}3c_{1}(b_{p}+b_{\Lambda}-\textstyle{\frac{1}{s}}c_{\Lambda}c_{5})D_{9}-\textstyle{\frac{5}{s}}c_{2}c_{5}c_{p}D_{2}\Bigr{\\}}$
(73)
$\displaystyle+m_{p}c_{5}c_{1}(-m_{\Lambda}\pm\textstyle{\frac{1}{s}}m_{N^{*}}c_{\Lambda})D_{8}+\Bigl{\\{}-(m_{p}m_{\Lambda}c_{k}-b_{p}c_{\Lambda})\pm
m_{N^{*}}(\textstyle{\frac{1}{s}}m_{p}c_{\Lambda}c_{k}-m_{\Lambda}b_{p})\Bigr{\\}}$
$\displaystyle\times\Bigl{\\{}3c_{1}D_{9}(b_{p}+b_{\Lambda}-\textstyle{\frac{1}{s}}c_{\Lambda}c_{5})-\textstyle{\frac{20}{s}}c_{1}c_{2}c_{5}c_{p}D_{4}\Bigr{\\}}\,\Bigr{]}\,G_{2}+m_{p}\Bigl{[}(m_{\Lambda}\mp\textstyle{\frac{1}{s}}m_{N^{*}}c_{\Lambda})\Bigl{\\{}\textstyle{\frac{1}{2}}c_{1}c_{5}D_{8}$
$\displaystyle+10c_{k}D_{2}(b_{p}+b_{\Lambda}-\textstyle{\frac{1}{s}}c_{\Lambda}c_{5})+20c_{1}c_{2}c_{5}D_{4}\textstyle{\frac{1}{s}}c_{p}c_{k}\Bigr{\\}}+\textstyle{\frac{5}{2}}\Bigl{\\{}(m_{p}c_{\Lambda}+m_{\Lambda}c_{p})\mp
m_{N^{*}}(m_{p}m_{\Lambda}+\textstyle{\frac{1}{s}}c_{\Lambda}c_{p})\Bigr{\\}}$
$\displaystyle\times\bigl{\\{}(4b_{\Lambda}-k^{2}-\textstyle{\frac{1}{s}}c_{\Lambda}c_{5})D_{2}+4c_{1}c_{2}c_{5}\textstyle{\frac{1}{s}}c_{k}D_{4}\bigr{\\}}\pm\textstyle{\frac{1}{2}}m_{N^{*}}\Bigl{\\{}3c_{1}D_{9}(k^{2}-2b_{q}+\textstyle{\frac{1}{s}}c_{\Lambda}c_{5})$
$\displaystyle+5c_{2}c_{5}D_{2}(1+\textstyle{\frac{1}{s}}c_{p})\Bigr{\\}}\,\Bigr{]}\,G_{3}\Bigr{\\}}\,,$
$\displaystyle A_{6}$ $\displaystyle=$
$\displaystyle\Bigl{[}\,12c_{1}\bigl{\\{}-(m_{\Lambda}s+m_{p}c_{\Lambda})\pm
m_{N^{*}}(c_{\Lambda}+m_{p}m_{\Lambda})\bigr{\\}}D_{7}\,\Bigr{]}\,G_{1}+\textstyle{\frac{1}{2}}\Bigl{[}\Bigl{\\{}-(m_{p}c_{\Lambda}-m_{\Lambda}c_{p})\pm
m_{N^{*}}(m_{p}m_{\Lambda}-\textstyle{\frac{1}{s}}c_{\Lambda}c_{p})\Bigr{\\}}$
(74)
$\displaystyle\times\Bigl{\\{}c_{1}D_{8}+5D_{2}(c_{1}+b_{p}c_{s})-\textstyle{\frac{20}{s}}c_{1}c_{2}c_{4}c_{p}D_{4}\Bigr{\\}}+b_{p}(-m_{\Lambda}\pm\textstyle{\frac{1}{s}}m_{N^{*}}c_{\Lambda})c_{1}D_{8}\pm
m_{N^{*}}\Bigl{\\{}3c_{1}(c_{1}+b_{p}c_{s})D_{9}$
$\displaystyle-\textstyle{\frac{5}{s}}c_{2}c_{4}c_{p}D_{2}\Bigr{\\}}\Bigr{]}\,G_{2}+\textstyle{\frac{1}{2}}m_{p}\Bigl{[}(-s\pm
m_{N^{*}}m_{p})\Bigl{\\{}3c_{1}D_{9}c_{s}-5c_{2}D_{2}(1+\textstyle{\frac{1}{s}}c_{p})\Bigr{\\}}-c_{1}D_{8}\Bigl{\\{}(m_{p}m_{\Lambda}-11c_{\Lambda})$
$\displaystyle\mp
m_{N^{*}}(\textstyle{\frac{1}{s}}m_{p}c_{\Lambda}-11m_{\Lambda})\Bigr{\\}}+\bigl{\\{}5D_{2}c_{s}-20c_{1}c_{2}D_{4}(1+\textstyle{\frac{1}{s}}c_{p})\bigr{\\}}$
$\displaystyle\times\Bigl{\\{}(b_{p}c_{\Lambda}-m_{p}m_{\Lambda}c_{k})\pm
m_{N^{*}}(\textstyle{\frac{1}{s}}m_{p}c_{\Lambda}c_{k}-m_{\Lambda}b_{p})\Bigr{\\}}\Bigr{]}\,G_{3}\,,$

where we have defined

$\displaystyle D_{1}$ $\displaystyle=$ $\displaystyle
33c_{1}^{4}+30c_{1}^{2}c_{2}c_{3}+11c_{2}^{2}c_{3}^{2}\,,$ (75) $\displaystyle
D_{2}$ $\displaystyle=$ $\displaystyle
33c_{1}^{4}+18c_{1}^{2}c_{2}c_{3}+c_{2}^{2}c_{3}^{2}\,,$ (76) $\displaystyle
D_{3}$ $\displaystyle=$ $\displaystyle
21c_{1}^{4}+14c_{1}^{2}c_{2}c_{3}+c_{2}^{2}c_{3}^{2}\,,$ (77) $\displaystyle
D_{4}$ $\displaystyle=$ $\displaystyle 3c_{1}^{2}+c_{2}c_{3}\,,$ (78)
$\displaystyle D_{5}$ $\displaystyle=$ $\displaystyle
7c_{1}^{2}+c_{2}c_{3}\,,$ (79) $\displaystyle D_{6}$ $\displaystyle=$
$\displaystyle
429c_{1}^{6}+5(99c_{1}^{4}c_{2}c_{3}+27c_{1}^{2}c_{2}^{2}c_{3}^{2}+c_{2}^{3}c_{3}^{3})\,,$
(80) $\displaystyle D_{7}$ $\displaystyle=$ $\displaystyle
33c_{1}^{4}+45c_{1}^{2}c_{2}c_{3}+5c_{2}^{2}c_{3}^{2}\,,$ (81) $\displaystyle
D_{8}$ $\displaystyle=$ $\displaystyle
33c_{1}^{4}+120c_{1}^{2}c_{2}c_{3}+5c_{2}^{2}c_{3}^{2}\,,$ (82) $\displaystyle
D_{9}$ $\displaystyle=$ $\displaystyle
143c_{1}^{4}+110c_{1}^{2}c_{2}c_{3}+15c_{2}^{2}c_{3}^{2}\,,$ (83)

while $c_{1},c_{2}$, $c_{3}$ and other parameters are defined in Eq. (53).

Finally, the extracted functions $A_{i}$ for baryon resonances with spin 15/2
read

$\displaystyle A_{1}$ $\displaystyle=$
$\displaystyle\Bigl{[}\,\Bigl{(}12879c_{1}^{7}+126c_{1}F_{6}\Bigr{)}(s\pm
m_{N^{*}}m_{\Lambda})+\Bigl{(}14F_{3}+60c_{2}^{3}c_{3}^{3}\Bigr{)}\Bigl{\\{}(m_{\Lambda}^{2}c_{k}-b_{p}c_{\Lambda})\pm
m_{N^{*}}m_{\Lambda}(\textstyle{\frac{1}{s}}c_{k}c_{\Lambda}-b_{p})\Bigr{\\}}\,\Bigr{]}\,G_{1}$
(84)
$\displaystyle+\textstyle{\frac{1}{2}}(m_{p}+m_{\Lambda})\Bigl{[}\,\Bigl{\\{}(m_{p}c_{\Lambda}-m_{\Lambda}c_{p})\pm
m_{N^{*}}(m_{p}m_{\Lambda}-\textstyle{\frac{1}{s}}c_{\Lambda}c_{p})\Bigr{\\}}\Bigl{\\{}F_{4}(c_{1}-b_{p}c_{s})-\textstyle{\frac{1}{s}}k^{2}F_{5}c_{p}+F_{3}\Bigr{\\}}$
$\displaystyle-
b_{p}(m_{\Lambda}\pm\textstyle{\frac{1}{s}}m_{N^{*}}c_{\Lambda})F_{3}\pm
m_{N^{*}}\Bigl{\\{}(c_{1}-b_{p}c_{s})F_{1}-\textstyle{\frac{1}{s}}k^{2}F_{2}c_{p}\Bigr{\\}}\,\Bigr{]}\,G_{2}$
$\displaystyle+\Bigl{[}\,\textstyle{\frac{1}{2}}(m_{p}+m_{\Lambda})\Bigl{\\{}(s\pm
m_{N^{*}}m_{p})\bigl{\\{}-c_{s}F_{1}-\textstyle{\frac{1}{s}}c_{k}F_{2}\bigr{\\}}-\bigl{\\{}(m_{p}m_{\Lambda}+15c_{\Lambda})\pm
m_{N^{*}}(\textstyle{\frac{1}{s}}m_{p}c_{\Lambda}+15m_{\Lambda})\bigr{\\}}F_{3}$
$\displaystyle-\bigl{\\{}(b_{p}c_{\Lambda}-m_{p}m_{\Lambda}c_{k})\pm
m_{N^{*}}(\textstyle{\frac{1}{s}}m_{p}c_{\Lambda}c_{k}-m_{\Lambda}b_{p})\bigr{\\}}(-F_{4}c_{s}-\textstyle{\frac{1}{s}}c_{k}F_{5})\Bigr{\\}}$
$\displaystyle+\Bigl{\\{}(8m_{p}c_{\Lambda}+8m_{\Lambda}c_{p}+m_{\Lambda}c_{k})\pm
m_{N^{*}}(8m_{p}m_{\Lambda}+\textstyle{\frac{8}{s}}c_{p}c_{\Lambda}+\textstyle{\frac{1}{s}}c_{k}c_{\Lambda})\Bigr{\\}}F_{3}\mp
m_{N^{*}}c_{1}F_{1}\,\Bigr{]}\,G_{3}\,,$ $\displaystyle A_{2}$
$\displaystyle=$
$\displaystyle{\frac{1}{t-m_{K}^{2}}}\Bigl{\\{}\Bigl{[}-4k^{2}\Bigl{(}7F_{3}+30c_{2}^{3}c_{3}^{3}\Bigr{)}(c_{\Lambda}\pm
m_{N^{*}}m_{\Lambda})\Bigr{]}\,G_{1}+\Bigl{[}(-s\pm
m_{N^{*}}m_{p})\Bigl{\\{}(b_{p}c_{s}-c_{1})F_{1}+\textstyle{\frac{1}{s}}k^{2}c_{p}F_{2}\Bigr{\\}}$
(85) $\displaystyle-
m_{p}k^{2}(m_{\Lambda}\pm\textstyle{\frac{1}{s}}m_{N^{*}}c_{\Lambda})F_{3}+\Bigl{\\{}(m_{p}m_{\Lambda}c_{k}-b_{p}c_{\Lambda})\pm
m_{N^{*}}(\textstyle{\frac{1}{s}}m_{p}c_{\Lambda}c_{k}-m_{\Lambda}b_{p})\Bigr{\\}}$
$\displaystyle\times\Bigl{\\{}F_{4}(b_{p}c_{s}-c_{1})+\textstyle{\frac{1}{s}}k^{2}c_{p}F_{5}\Bigr{\\}}\,\Bigr{]}\,G_{2}+m_{p}\Bigl{[}(m_{\Lambda}\pm\textstyle{\frac{1}{s}}m_{N^{*}}c_{\Lambda})\Bigl{\\{}F_{3}k^{2}+2c_{k}F_{4}(c_{1}-b_{p}c_{s})$
$\displaystyle-\textstyle{\frac{1}{s}}k^{2}2c_{k}c_{p}F_{5}\Bigr{\\}}+\Bigl{\\{}(m_{p}c_{\Lambda}+m_{\Lambda}c_{p})\pm
m_{N^{*}}(m_{p}m_{\Lambda}+\textstyle{\frac{1}{s}}c_{\Lambda}c_{p})\Bigr{\\}}k^{2}(c_{s}F_{4}+\textstyle{\frac{1}{s}}c_{k}F_{5})$
$\displaystyle\pm
m_{N^{*}}\Bigl{\\{}F_{1}(k^{2}c_{s}-2b_{q})-F_{3}k^{2}(1+\textstyle{\frac{1}{s}}c_{p})\Bigr{\\}}\,\Bigr{]}\,G_{3}\Bigr{\\}}\,,$
$\displaystyle A_{3}$ $\displaystyle=$
$\displaystyle\Bigl{[}\,2c_{k}\Bigl{(}7F_{3}+30c_{2}^{3}c_{3}^{3}\Bigr{)}(m_{\Lambda}\pm\textstyle{\frac{1}{s}}m_{N^{*}}c_{\Lambda})\pm
2m_{N^{*}}\bigl{(}6435c_{1}^{7}+63c_{1}F_{6}\bigr{)}\,\Bigr{]}\,G_{1}$ (86)
$\displaystyle+\textstyle{\frac{1}{2}}\Bigl{[}\,\Bigl{\\{}(m_{p}c_{\Lambda}-m_{\Lambda}c_{p})\pm
m_{N^{*}}(m_{p}m_{\Lambda}-\textstyle{\frac{1}{s}}c_{\Lambda}c_{p})\Bigr{\\}}\Bigl{\\{}F_{4}\bigl{(}c_{1}+b_{p}(1+\textstyle{\frac{1}{s}}c_{\Lambda})\bigr{)}-\textstyle{\frac{1}{s}}k^{2}c_{p}F_{5}+F_{3}\Bigr{\\}}$
$\displaystyle-
b_{p}(m_{\Lambda}\pm\textstyle{\frac{1}{s}}m_{N^{*}}c_{\Lambda})F_{3}\pm
m_{N^{*}}\Bigl{\\{}\bigl{(}c_{1}+b_{p}(1+\textstyle{\frac{1}{s}}c_{\Lambda})\bigr{)}F_{1}-\textstyle{\frac{1}{s}}k^{2}c_{p}F_{2}\Bigr{\\}}\,\Bigr{]}\,G_{2}$
$\displaystyle+\textstyle{\frac{1}{2}}\Bigl{[}\,(s\pm
m_{N^{*}}m_{p})\Bigl{\\{}(1+\textstyle{\frac{1}{s}}c_{\Lambda})F_{1}-F_{2}\textstyle{\frac{1}{s}}c_{k}\Bigr{\\}}-\Bigl{\\{}(m_{p}m_{\Lambda}+15c_{\Lambda})\pm
m_{N^{*}}(\textstyle{\frac{1}{s}}m_{p}c_{\Lambda}+15m_{\Lambda})\Bigr{\\}}F_{3}$
$\displaystyle-\Bigl{\\{}(b_{p}c_{\Lambda}-m_{p}m_{\Lambda}c_{k})\pm
m_{N^{*}}(\textstyle{\frac{1}{s}}m_{p}c_{\Lambda}c_{k}-m_{\Lambda}b_{p})\Bigr{\\}}\Bigl{\\{}F_{4}(1+\textstyle{\frac{1}{s}}c_{\Lambda})-F_{5}\textstyle{\frac{1}{s}}c_{k}\Bigr{\\}}\,\Bigr{]}\,G_{3}\,,$
$\displaystyle A_{4}$ $\displaystyle=$
$\displaystyle\Bigl{[}\,2c_{k}\Bigl{(}7F_{3}+30c_{2}^{3}c_{3}^{3}\Bigr{)}(m_{\Lambda}\pm\textstyle{\frac{1}{s}}m_{N^{*}}c_{\Lambda})\pm
2m_{N^{*}}\bigl{(}6435c_{1}^{7}+63c_{1}F_{6}\bigr{)}\,\Bigr{]}\,G_{1}$ (87)
$\displaystyle+\textstyle{\frac{1}{2}}\Bigl{[}\,\Bigl{\\{}(m_{p}c_{\Lambda}-m_{\Lambda}c_{p})\pm
m_{N^{*}}(m_{p}m_{\Lambda}-\textstyle{\frac{1}{s}}c_{\Lambda}c_{p})\Bigr{\\}}\Bigl{\\{}F_{4}(c_{1}-b_{p}c_{s})-\textstyle{\frac{1}{s}}k^{2}c_{p}F_{5}+(5c_{1}^{2}+c_{2}c_{3})\Bigr{\\}}$
$\displaystyle-
b_{p}(m_{\Lambda}\pm\textstyle{\frac{1}{s}}m_{N^{*}}c_{\Lambda})F_{3}\pm
m_{N^{*}}\Bigl{\\{}(c_{1}-b_{p}c_{s})F_{1}-\textstyle{\frac{1}{s}}k^{2}c_{p}F_{2}\Bigr{\\}}\,\Bigr{]}\,G_{2}$
$\displaystyle+\textstyle{\frac{1}{2}}\Bigl{[}\,(s\pm
m_{N^{*}}m_{p})\bigl{(}-c_{s}F_{1}-F_{2}\textstyle{\frac{1}{s}}c_{k}\bigr{)}-\Bigl{\\{}(m_{p}m_{\Lambda}+15c_{\Lambda})\pm
m_{N^{*}}(\textstyle{\frac{1}{s}}m_{p}c_{\Lambda}+15m_{\Lambda})\Bigr{\\}}F_{3}$
$\displaystyle-\Bigl{\\{}(b_{p}c_{\Lambda}-m_{p}m_{\Lambda}c_{k})\pm
m_{N^{*}}(\textstyle{\frac{1}{s}}m_{p}c_{\Lambda}c_{k}-m_{\Lambda}b_{p})\Bigr{\\}}\bigl{(}-c_{s}F_{4}-F_{5}\textstyle{\frac{1}{s}}c_{k}\bigr{)}\,\Bigr{]}\,G_{3}\,,$
$\displaystyle A_{5}$ $\displaystyle=$
$\displaystyle{\frac{1}{t-m_{K}^{2}}}\Bigl{\\{}\Bigl{[}\,2c_{5}\bigl{(}7F_{3}+30c_{2}^{3}c_{3}^{3}\bigr{)}(c_{\Lambda}\pm
m_{N^{*}}m_{\Lambda})\Bigr{]}\,G_{1}+\textstyle{\frac{1}{2}}\Bigl{[}(-s\pm
m_{N^{*}}m_{p})\Bigl{\\{}F_{1}(b_{p}+b_{\Lambda}-\textstyle{\frac{1}{s}}c_{\Lambda}c_{5})-\textstyle{\frac{1}{s}}c_{5}c_{p}F_{2}\Bigr{\\}}$
(88)
$\displaystyle+m_{p}c_{5}(m_{\Lambda}\pm\textstyle{\frac{1}{s}}m_{N^{*}}c_{\Lambda})F_{3}+\Bigl{\\{}(m_{p}m_{\Lambda}c_{k}-b_{p}c_{\Lambda})\pm
m_{N^{*}}(\textstyle{\frac{1}{s}}m_{p}c_{\Lambda}c_{k}-m_{\Lambda}b_{p})\Bigr{\\}}\Bigl{\\{}F_{4}(b_{p}+b_{\Lambda}-\textstyle{\frac{1}{s}}c_{\Lambda}c_{5})$
$\displaystyle-\textstyle{\frac{1}{s}}c_{p}c_{5}F_{5}\Bigr{\\}}\Bigr{]}\,G_{2}+m_{p}\Bigl{[}\,\textstyle-{\frac{1}{2}}(m_{\Lambda}\pm\textstyle{\frac{1}{s}}m_{N^{*}}c_{\Lambda})\Bigl{\\{}c_{5}\bigl{(}F_{3}-\textstyle{\frac{1}{s}}c_{p}c_{k}F_{5}\bigr{)}+c_{k}F_{4}\bigl{(}b_{p}+b_{\Lambda}-\textstyle{\frac{1}{s}}c_{\Lambda}c_{5}\bigr{)}\Bigr{\\}}$
$\displaystyle-\Bigl{\\{}(m_{p}c_{\Lambda}+m_{\Lambda}c_{p})\pm
m_{N^{*}}(m_{p}m_{\Lambda}+\textstyle{\frac{1}{s}}c_{\Lambda}c_{p})\Bigr{\\}}\Bigl{\\{}F_{4}(4b_{\Lambda}-k^{2}-\textstyle{\frac{1}{s}}c_{\Lambda}c_{5})+\textstyle{\frac{1}{s}}c_{k}c_{5}F_{5}\Bigr{\\}}$
$\displaystyle\pm\textstyle{\frac{1}{2}}m_{N^{*}}\Bigl{\\{}F_{1}(k^{2}-2b_{q}+\textstyle{\frac{1}{s}}c_{\Lambda}c_{5})+c_{5}F_{2}(1+\textstyle{\frac{1}{s}}c_{p})\Bigr{\\}}\,\Bigr{]}\,G_{3}\Bigr{\\}}\,,$
$\displaystyle A_{6}$ $\displaystyle=$
$\displaystyle\Bigl{[}\,2\Bigl{\\{}(m_{\Lambda}s+m_{p}c_{\Lambda})\pm
m_{N^{*}}(c_{\Lambda}+m_{p}m_{\Lambda})\Bigr{\\}}\bigl{(}7F_{3}+30c_{2}^{3}c_{3}^{3}\bigr{)}\,\Bigr{]}\,G_{1}\
$ (89)
$\displaystyle+\textstyle{\frac{1}{2}}\Bigl{[}\Bigl{\\{}(m_{p}c_{\Lambda}-m_{\Lambda}c_{p})\pm
m_{N^{*}}(m_{p}m_{\Lambda}-\textstyle{\frac{1}{s}}c_{\Lambda}c_{p})\Bigr{\\}}\Bigl{\\{}F_{3}+F_{4}(c_{1}+b_{p}c_{s})-\textstyle{\frac{1}{s}}c_{p}c_{4}F_{5}\Bigr{\\}}$
$\displaystyle+b_{p}(m_{\Lambda}\pm\textstyle{\frac{1}{s}}m_{N^{*}}c_{\Lambda})F_{3}\pm
m_{N^{*}}\Bigl{\\{}F_{1}(c_{1}+b_{p}c_{s})-\textstyle{\frac{1}{s}}c_{4}c_{p}F_{2}\Bigr{\\}}\,\Bigr{]}\,G_{2}$
$\displaystyle+\textstyle{\frac{1}{2}}m_{p}\Bigl{[}(s\pm
m_{N^{*}}m_{p})\Bigl{\\{}F_{1}c_{s}-F_{2}(1+\textstyle{\frac{1}{s}}c_{p})\Bigr{\\}}+F_{3}\Bigl{\\{}(m_{p}m_{\Lambda}-11c_{\Lambda})\pm
m_{N^{*}}(\textstyle{\frac{1}{s}}m_{p}c_{\Lambda}-11m_{\Lambda})\Bigr{\\}}$
$\displaystyle+\bigl{\\{}c_{s}F_{4}+F_{5}(1+\textstyle{\frac{1}{s}}c_{p})\bigr{\\}}\Bigl{\\{}(m_{p}m_{\Lambda}c_{k}-b_{p}c_{\Lambda})\pm
m_{N^{*}}(\textstyle{\frac{1}{s}}m_{p}c_{\Lambda}c_{k}-m_{\Lambda}b_{p})\Bigr{\\}}\,\Bigr{]}\,G_{3}\,,$

where in terms of $c_{1},c_{2}$, and $c_{3}$ defined in Eq. (53) we have used

$\displaystyle F_{1}$ $\displaystyle=$ $\displaystyle
45(143c_{1}^{6}+143c_{1}^{4}c_{2}c_{3}+33c_{1}^{2}c_{2}^{2}c_{3}^{2}+c_{2}^{3}c_{3}^{3})\,,$
(90) $\displaystyle F_{2}$ $\displaystyle=$ $\displaystyle
9c_{1}c_{2}(286c_{1}^{4}+220c_{1}^{2}c_{2}c_{3}+30c_{2}^{2}c_{3}^{2})\,,$ (91)
$\displaystyle F_{3}$ $\displaystyle=$ $\displaystyle
429c_{1}^{6}+495c_{1}^{4}c_{2}c_{3}+135c_{1}^{2}c_{2}^{2}c_{3}^{2}\,,$ (92)
$\displaystyle F_{4}$ $\displaystyle=$ $\displaystyle
18c_{1}(143c_{1}^{4}+110c_{1}^{2}c_{2}c_{3}+15c_{2}^{2}c_{3}^{2})\,,$ (93)
$\displaystyle F_{5}$ $\displaystyle=$ $\displaystyle
30c_{2}(33c_{1}^{4}+18c_{1}^{2}c_{2}c_{3}+c_{2}^{2}c_{3}^{2})\,,$ (94)
$\displaystyle F_{6}$ $\displaystyle=$ $\displaystyle
143c_{1}^{4}c_{2}c_{3}+55c_{1}^{2}c_{2}^{2}c_{3}^{2}+5c_{2}^{3}c_{3}^{3}\,.$
(95)

## References

  * (1) S. Clymton and T. Mart, Phys. Rev. D 96, 054004 (2017).
  * (2) V. Pascalutsa, Phys. Lett. B 503, 85 (2001).
  * (3) T. Mart, Phys. Rev. D 100, 056008 (2019).
  * (4) P. A. Zyla et al. [Particle Data Group], Prog. Theor. Exp. Phys. 2020, 083C01 (2020).
  * (5) S. Clymton and T. Mart, JPS Conf. Proc. 26, 031009 (2019).
  * (6) R. A. Adelseck and B. Saghai, Phys. Rev. C 42, 108 (1990).
  * (7) T. Vrancx, L. De Cruz, J. Ryckebusch and P. Vancraeyveld, Phys. Rev. C 84, 045201 (2011).
  * (8) N. H. Luthfiyah and T. Mart, AIP Conf. Proc. 2234, 040015 (2020).
  * (9) V. Pascalutsa, Phys. Rev. D 58, 096002 (1998).
  * (10) V. Pascalutsa and R. Timmermans, Phys. Rev. C 60, 042201 (1999).
  * (11) S. Z. Huang, P. F. Zhang, T. N. Ruan, Y. C. Zhu and Z. P. Zheng, Commun. Theor. Phys. 41, 405 (2004).
  * (12) B. Deo and A. Bisoi, Phys. Rev. D 9, 288 (1974).
  * (13) P. Dennery, Phys. Rev. 124, 2000 (1961).
  * (14) T. Mart, S. Clymton, and A. J. Arifi, Phys. Rev. D 92, 094019 (2015).
  * (15) C. Akondi et al. [A2 Collaboration at MAMI], Eur. Phys. J. A 55, 202 (2019).
  * (16) R. Bradford et al. [CLAS Collaboration], Phys. Rev. C 73, 035202 (2006).
  * (17) T. Mart and C. Bennhold, Phys. Rev. C 61, 012201 (1999).
  * (18) T. Mart and M. Kholili, Phys. Rev. C 86, 022201 (2012).
  * (19) M. Sumihama et al. [LEPS Collaboration], Phys. Rev. C 73, 035214 (2006).
  * (20) M. McCracken et al. [CLAS Collaboration], Phys. Rev. C 81, 025201 (2010).
  * (21) T. Jude et al. [Crystal Ball Collaboration at MAMI], Phys. Lett. B 735, 112 (2014).
  * (22) T. Mart, C. Bennhold and C. E. Hyde, Phys. Rev. C 51, 1074 (1995).
  * (23) T. Mart, Phys. Rev. C 83, 048203 (2011).
  * (24) N. Compton et al. [CLAS Collaboration], Phys. Rev. C 96, 065201 (2017).
  * (25) A. Lleres et al. [GRAAL Collaboration], Eur. Phys. J. A 39, 149 (2009).
  * (26) R. Bradford et al. [CLAS Collaboration], Phys. Rev. C 75, 035205 (2007).
  * (27) C. Paterson et al. [CLAS Collaboration], Phys. Rev. C 93, 065201 (2016).
  * (28) T. Mart and M. Kholili, J. Phys. G 46, 105112 (2019).
  * (29) S. Clymton and T. Mart, arXiv:2104.10333 [hep-ph], Phys. Rev. D (accepted for publication 2021).
  * (30) J. McNabb et al. [CLAS Collaboration], Phys. Rev. C 69, 042201 (2004).
  * (31) B. Dey et al, Phys. Rev. C 82, 025202 (2010).
  * (32) A. Lleres et al. [GRAAL Collaboration], Eur. Phys. J. A 31, 79 (2007).
  * (33) K. Glander et al. [SAPHIR Collaboration], Eur. Phys. J. A 19, 251 (2004).
  * (34) H. Kohri et al. [LEPS Collaboration], Phys. Rev. Lett. 97, 082003 (2006).
  * (35) R. Lawall et al. [SAPHIR Collaboration], Eur. Phys. J. A 24, 275 (2005).
  * (36) R. Castelijns et al. [CBELSA/TAPS Collaboration], Eur. Phys. J. A 35, 39 (2008).
  * (37) P. Aguar-Bartolomé et al. [A2 Collaboration at MAMI], Phys. Rev. C 88, 044601 (2013).
  * (38) T. Mart, Phys. Rev. C 90, 065202 (2014).
  * (39) S. Sakinah, S. Clymton and T. Mart, Acta Phys. Polon. B 50, 1389 (2019).
  * (40) S. Pereira et al. [CLAS Collaboration], Phys. Lett. B 688, 289 (2010).
  * (41) F. James and M. Roos, Comput. Phys. Commun. 10, 343 (1975).