J/ψ → K⁰ Σ⁺ p̅

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from __future__ import annotations

import logging
import os
import warnings
from typing import TYPE_CHECKING

import graphviz
import jax.numpy as jnp
import matplotlib.pyplot as plt
import qrules
import sympy as sp
from ampform.dynamics import BlattWeisskopfSquared, EnergyDependentWidth
from ampform.sympy import perform_cached_doit
from IPython.display import Latex, Markdown
from tensorwaves.data.transform import SympyDataTransformer
from tqdm.auto import tqdm

from ampform_dpd import DalitzPlotDecompositionBuilder
from ampform_dpd.adapter.qrules import normalize_state_ids, to_three_body_decay
from ampform_dpd.decay import State
from ampform_dpd.dynamics import FormFactor, RelativisticBreitWigner
from ampform_dpd.dynamics.builder import formulate_breit_wigner_with_form_factor
from ampform_dpd.io import (
    as_markdown_table,
    aslatex,
    perform_cached_lambdify,
    simplify_latex_rendering,
)

simplify_latex_rendering()
logging.getLogger("absl").setLevel(logging.ERROR)  # mute JAX
warnings.simplefilter("ignore", category=RuntimeWarning)

NO_TQDM = "EXECUTE_NB" in os.environ
if NO_TQDM:
    logging.getLogger("ampform_dpd.io").setLevel(logging.ERROR)
if TYPE_CHECKING:
    from tensorwaves.interface import DataSample, ParametrizedFunction

Decay definition

We follow this example, which was generated with QRules, and leave out the \(K\)-resonances and the resonances that lie far outside of phase space.

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REACTION = qrules.generate_transitions(
    initial_state="J/psi(1S)",
    final_state=["K0", "Sigma+", "p~"],
    allowed_interaction_types="strong",
    formalism="canonical-helicity",
    mass_conservation_factor=0.05,
)
REACTION = normalize_state_ids(REACTION)
dot = qrules.io.asdot(REACTION, collapse_graphs=True)
graphviz.Source(dot)

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DECAY = to_three_body_decay(REACTION.transitions, min_ls=True)
Markdown(as_markdown_table([DECAY.initial_state, *DECAY.final_state.values()]))

index	name	LaTeX	\(J^P\)	mass (MeV)	width (MeV)
0	J/psi(1S)	\(J/\psi(1S)\)	\(1^-\)	3,096	0
1	K0	\(K^{0}\)	\(0^-\)	497	0
2	Sigma+	\(\Sigma^{+}\)	\(\frac{1}{2}^+\)	1,189	0
3	p~	\(\overline{p}\)	\(\frac{1}{2}^-\)	938	0

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resonances = sorted(
    {t.resonance for t in DECAY.chains},
    key=lambda p: (p.name[0], p.mass),
)
resonance_names = [p.name for p in resonances]
Markdown(as_markdown_table(resonances))

name	LaTeX	\(J^P\)	mass (MeV)	width (MeV)
N(1710)+	\(N(1710)^{+}\)	\(\frac{1}{2}^+\)	1,710	140
N(1700)+	\(N(1700)^{+}\)	\(\frac{3}{2}^-\)	1,720	200
N(1720)+	\(N(1720)^{+}\)	\(\frac{3}{2}^+\)	1,720	250
Sigma(1660)~-	\(\overline{\Sigma}(1660)^{-}\)	\(\frac{1}{2}^-\)	1,660	200
Sigma(1670)~-	\(\overline{\Sigma}(1670)^{-}\)	\(\frac{3}{2}^+\)	1,675	70
Sigma(1750)~-	\(\overline{\Sigma}(1750)^{-}\)	\(\frac{1}{2}^+\)	1,750	150
Sigma(1775)~-	\(\overline{\Sigma}(1775)^{-}\)	\(\frac{5}{2}^+\)	1,775	120
Sigma(1910)~-	\(\overline{\Sigma}(1940)^{-}\)	\(\frac{3}{2}^+\)	1,910	220

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Latex(aslatex(DECAY, with_jp=True))

\[\begin{split}\begin{array}{c} J/\psi(1S) \xrightarrow[S=1]{L=1} \left(N(1700)^{+} \xrightarrow[S=1/2]{L=2} K^{0} \Sigma^{+}\right) \overline{p} \\ J/\psi(1S) \xrightarrow[S=1]{L=0} \left(N(1710)^{+} \xrightarrow[S=1/2]{L=1} K^{0} \Sigma^{+}\right) \overline{p} \\ J/\psi(1S) \xrightarrow[S=1]{L=0} \left(N(1720)^{+} \xrightarrow[S=1/2]{L=1} K^{0} \Sigma^{+}\right) \overline{p} \\ J/\psi(1S) \xrightarrow[S=1]{L=0} \left(\overline{\Sigma}(1660)^{-} \xrightarrow[S=1/2]{L=1} K^{0} \overline{p}\right) \Sigma^{+} \\ J/\psi(1S) \xrightarrow[S=1]{L=1} \left(\overline{\Sigma}(1670)^{-} \xrightarrow[S=1/2]{L=2} K^{0} \overline{p}\right) \Sigma^{+} \\ J/\psi(1S) \xrightarrow[S=0]{L=1} \left(\overline{\Sigma}(1750)^{-} \xrightarrow[S=1/2]{L=0} K^{0} \overline{p}\right) \Sigma^{+} \\ J/\psi(1S) \xrightarrow[S=2]{L=1} \left(\overline{\Sigma}(1775)^{-} \xrightarrow[S=1/2]{L=2} K^{0} \overline{p}\right) \Sigma^{+} \\ J/\psi(1S) \xrightarrow[S=1]{L=1} \left(\overline{\Sigma}(1940)^{-} \xrightarrow[S=1/2]{L=2} K^{0} \overline{p}\right) \Sigma^{+} \\ \end{array}\end{split}\]

Model formulation

The total, aligned intensity expression looks as follows:

model_builder = DalitzPlotDecompositionBuilder(DECAY, min_ls=False)
for chain in model_builder.decay.chains:
    model_builder.dynamics_choices.register_builder(
        chain, formulate_breit_wigner_with_form_factor
    )
model = model_builder.formulate(reference_subsystem=2)
model.intensity

\[\displaystyle \sum_{\lambda_{0}=-1}^{1} \sum_{\lambda_{1}=0} \sum_{\lambda_{2}=-1/2}^{1/2} \sum_{\lambda_{3}=-1/2}^{1/2}{\left|{\sum_{\lambda_0^{\prime}=-1}^{1} \sum_{\lambda_1^{\prime}=0} \sum_{\lambda_2^{\prime}=-1/2}^{1/2} \sum_{\lambda_3^{\prime}=-1/2}^{1/2}{A^{2}_{\lambda_0^{\prime}, \lambda_1^{\prime}, \lambda_2^{\prime}, \lambda_3^{\prime}} d^{\frac{1}{2}}_{\lambda_2^{\prime},\lambda_{2}}\left(\zeta^2_{2(2)}\right) d^{\frac{1}{2}}_{\lambda_3^{\prime},\lambda_{3}}\left(\zeta^3_{2(2)}\right) d^{1}_{\lambda_{0},\lambda_0^{\prime}}\left(\zeta^0_{2(2)}\right) + A^{3}_{\lambda_0^{\prime}, \lambda_1^{\prime}, \lambda_2^{\prime}, \lambda_3^{\prime}} d^{\frac{1}{2}}_{\lambda_2^{\prime},\lambda_{2}}\left(\zeta^2_{3(2)}\right) d^{\frac{1}{2}}_{\lambda_3^{\prime},\lambda_{3}}\left(\zeta^3_{3(2)}\right) d^{1}_{\lambda_{0},\lambda_0^{\prime}}\left(\zeta^0_{3(2)}\right)}}\right|^{2}}\]

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Latex(aslatex(model.variables))

\[\begin{split}\begin{array}{rcl} \theta_{31} &=& \operatorname{acos}{\left(\frac{2 \sigma_{2} \left(- m_{2}^{2} - m_{3}^{2} + \sigma_{1}\right) - \left(m_{0}^{2} - m_{2}^{2} - \sigma_{2}\right) \left(- m_{1}^{2} + m_{3}^{2} + \sigma_{2}\right)}{\sqrt{\lambda\left(m_{0}^{2}, m_{2}^{2}, \sigma_{2}\right)} \sqrt{\lambda\left(\sigma_{2}, m_{3}^{2}, m_{1}^{2}\right)}} \right)} \\ \theta_{12} &=& \operatorname{acos}{\left(\frac{2 \sigma_{3} \left(- m_{1}^{2} - m_{3}^{2} + \sigma_{2}\right) - \left(m_{0}^{2} - m_{3}^{2} - \sigma_{3}\right) \left(m_{1}^{2} - m_{2}^{2} + \sigma_{3}\right)}{\sqrt{\lambda\left(m_{0}^{2}, m_{3}^{2}, \sigma_{3}\right)} \sqrt{\lambda\left(\sigma_{3}, m_{1}^{2}, m_{2}^{2}\right)}} \right)} \\ \zeta^0_{2(2)} &=& 0 \\ \zeta^2_{2(2)} &=& 0 \\ \zeta^3_{2(2)} &=& 0 \\ \zeta^0_{3(2)} &=& - \operatorname{acos}{\left(\frac{- 2 m_{0}^{2} \left(- m_{2}^{2} - m_{3}^{2} + \sigma_{1}\right) + \left(m_{0}^{2} + m_{2}^{2} - \sigma_{2}\right) \left(m_{0}^{2} + m_{3}^{2} - \sigma_{3}\right)}{\sqrt{\lambda\left(m_{0}^{2}, m_{3}^{2}, \sigma_{3}\right)} \sqrt{\lambda\left(m_{0}^{2}, \sigma_{2}, m_{2}^{2}\right)}} \right)} \\ \zeta^2_{3(2)} &=& \operatorname{acos}{\left(\frac{2 m_{2}^{2} \left(- m_{0}^{2} - m_{1}^{2} + \sigma_{1}\right) + \left(m_{0}^{2} + m_{2}^{2} - \sigma_{2}\right) \left(- m_{1}^{2} - m_{2}^{2} + \sigma_{3}\right)}{\sqrt{\lambda\left(m_{0}^{2}, m_{2}^{2}, \sigma_{2}\right)} \sqrt{\lambda\left(\sigma_{3}, m_{2}^{2}, m_{1}^{2}\right)}} \right)} \\ \zeta^3_{3(2)} &=& \operatorname{acos}{\left(\frac{2 m_{3}^{2} \left(- m_{0}^{2} - m_{1}^{2} + \sigma_{1}\right) + \left(m_{0}^{2} + m_{3}^{2} - \sigma_{3}\right) \left(- m_{1}^{2} - m_{3}^{2} + \sigma_{2}\right)}{\sqrt{\lambda\left(m_{0}^{2}, m_{3}^{2}, \sigma_{3}\right)} \sqrt{\lambda\left(\sigma_{2}, m_{3}^{2}, m_{1}^{2}\right)}} \right)} \\ \end{array}\end{split}\]

Each unaligned amplitude is defined as follows:

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Latex(aslatex(model.amplitudes))

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\[\begin{split}\begin{array}{rcl} A^{2}_{-1, 0, - \frac{1}{2}, - \frac{1}{2}} &=& \sum_{\lambda_{R}=-1/2}^{1/2}{- \frac{\sqrt{2} \delta_{-1, \lambda_{R} + \frac{1}{2}} \left(C^{\frac{1}{2},\frac{1}{2}}_{0,0,\frac{1}{2},\frac{1}{2}}\right)^{2} C^{0,\lambda_{R} + \frac{1}{2}}_{\frac{1}{2},\lambda_{R},\frac{1}{2},\frac{1}{2}} C^{1,\lambda_{R} + \frac{1}{2}}_{1,0,0,\lambda_{R} + \frac{1}{2}} \mathcal{F}_{1}\left(m_{0}^{2}, m_{\overline{\Sigma}(1750)^{-}}, m_{\Sigma^{+}}\right) \mathcal{F}_{0}\left(\sigma_{2}^{2}, m_{K^{0}}, m_{\overline{p}}\right) \mathcal{H}^\mathrm{LS,decay}_{\overline{\Sigma}(1750)^{-}, 0, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{\Sigma}(1750)^{-}, 1, 0} \mathcal{R}_{0}\left(\sigma_{2}, m_{\overline{\Sigma}(1750)^{-}}, \Gamma_{\overline{\Sigma}(1750)^{-}}\right) d^{\frac{1}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)}{2}} + \sum_{\lambda_{R}=-1/2}^{1/2}{- \frac{\sqrt{2} \delta_{-1, \lambda_{R} + \frac{1}{2}} C^{\frac{1}{2},\frac{1}{2}}_{0,0,\frac{1}{2},\frac{1}{2}} C^{1,\lambda_{R} + \frac{1}{2}}_{0,0,1,\lambda_{R} + \frac{1}{2}} C^{1,\lambda_{R} + \frac{1}{2}}_{\frac{1}{2},\lambda_{R},\frac{1}{2},\frac{1}{2}} C^{\frac{1}{2},\frac{1}{2}}_{1,0,\frac{1}{2},\frac{1}{2}} \mathcal{F}_{0}\left(m_{0}^{2}, m_{\overline{\Sigma}(1660)^{-}}, m_{\Sigma^{+}}\right) \mathcal{F}_{1}\left(\sigma_{2}^{2}, m_{K^{0}}, m_{\overline{p}}\right) \mathcal{H}^\mathrm{LS,decay}_{\overline{\Sigma}(1660)^{-}, 1, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{\Sigma}(1660)^{-}, 0, 1} \mathcal{R}_{1}\left(\sigma_{2}, m_{\overline{\Sigma}(1660)^{-}}, \Gamma_{\overline{\Sigma}(1660)^{-}}\right) d^{\frac{1}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)}{2}} + \sum_{\lambda_{R}=-5/2}^{5/2}{- \frac{\sqrt{30} \delta_{-1, \lambda_{R} + \frac{1}{2}} C^{\frac{1}{2},\frac{1}{2}}_{0,0,\frac{1}{2},\frac{1}{2}} C^{1,\lambda_{R} + \frac{1}{2}}_{1,0,2,\lambda_{R} + \frac{1}{2}} C^{\frac{5}{2},\frac{1}{2}}_{2,0,\frac{1}{2},\frac{1}{2}} C^{2,\lambda_{R} + \frac{1}{2}}_{\frac{5}{2},\lambda_{R},\frac{1}{2},\frac{1}{2}} \mathcal{F}_{1}\left(m_{0}^{2}, m_{\overline{\Sigma}(1775)^{-}}, m_{\Sigma^{+}}\right) \mathcal{F}_{2}\left(\sigma_{2}^{2}, m_{K^{0}}, m_{\overline{p}}\right) \mathcal{H}^\mathrm{LS,decay}_{\overline{\Sigma}(1775)^{-}, 2, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{\Sigma}(1775)^{-}, 1, 2} \mathcal{R}_{2}\left(\sigma_{2}, m_{\overline{\Sigma}(1775)^{-}}, \Gamma_{\overline{\Sigma}(1775)^{-}}\right) d^{\frac{5}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)}{6}} + \sum_{\lambda_{R}=-3/2}^{3/2}{- \frac{\sqrt{5} \delta_{-1, \lambda_{R} + \frac{1}{2}} C^{\frac{1}{2},\frac{1}{2}}_{0,0,\frac{1}{2},\frac{1}{2}} C^{1,\lambda_{R} + \frac{1}{2}}_{1,0,1,\lambda_{R} + \frac{1}{2}} C^{1,\lambda_{R} + \frac{1}{2}}_{\frac{3}{2},\lambda_{R},\frac{1}{2},\frac{1}{2}} C^{\frac{3}{2},\frac{1}{2}}_{2,0,\frac{1}{2},\frac{1}{2}} \mathcal{F}_{1}\left(m_{0}^{2}, m_{\overline{\Sigma}(1670)^{-}}, m_{\Sigma^{+}}\right) \mathcal{F}_{2}\left(\sigma_{2}^{2}, m_{K^{0}}, m_{\overline{p}}\right) \mathcal{H}^\mathrm{LS,decay}_{\overline{\Sigma}(1670)^{-}, 2, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{\Sigma}(1670)^{-}, 1, 1} \mathcal{R}_{2}\left(\sigma_{2}, m_{\overline{\Sigma}(1670)^{-}}, \Gamma_{\overline{\Sigma}(1670)^{-}}\right) d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)}{2}} + \sum_{\lambda_{R}=-3/2}^{3/2}{- \frac{\sqrt{5} \delta_{-1, \lambda_{R} + \frac{1}{2}} C^{\frac{1}{2},\frac{1}{2}}_{0,0,\frac{1}{2},\frac{1}{2}} C^{1,\lambda_{R} + \frac{1}{2}}_{1,0,1,\lambda_{R} + \frac{1}{2}} C^{1,\lambda_{R} + \frac{1}{2}}_{\frac{3}{2},\lambda_{R},\frac{1}{2},\frac{1}{2}} C^{\frac{3}{2},\frac{1}{2}}_{2,0,\frac{1}{2},\frac{1}{2}} \mathcal{F}_{1}\left(m_{0}^{2}, m_{\overline{\Sigma}(1940)^{-}}, m_{\Sigma^{+}}\right) \mathcal{F}_{2}\left(\sigma_{2}^{2}, m_{K^{0}}, m_{\overline{p}}\right) \mathcal{H}^\mathrm{LS,decay}_{\overline{\Sigma}(1940)^{-}, 2, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{\Sigma}(1940)^{-}, 1, 1} \mathcal{R}_{2}\left(\sigma_{2}, m_{\overline{\Sigma}(1940)^{-}}, \Gamma_{\overline{\Sigma}(1940)^{-}}\right) d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)}{2}} \\ A^{3}_{-1, 0, - \frac{1}{2}, - \frac{1}{2}} &=& \sum_{\lambda_{R}=-3/2}^{3/2}{\frac{\delta_{-1, \lambda_{R} + \frac{1}{2}} C^{\frac{1}{2},\frac{1}{2}}_{0,0,\frac{1}{2},\frac{1}{2}} C^{1,\lambda_{R} + \frac{1}{2}}_{0,0,1,\lambda_{R} + \frac{1}{2}} C^{\frac{3}{2},\frac{1}{2}}_{1,0,\frac{1}{2},\frac{1}{2}} C^{1,\lambda_{R} + \frac{1}{2}}_{\frac{3}{2},\lambda_{R},\frac{1}{2},\frac{1}{2}} \mathcal{F}_{0}\left(m_{0}^{2}, m_{N(1720)^{+}}, m_{\overline{p}}\right) \mathcal{F}_{1}\left(\sigma_{3}^{2}, m_{K^{0}}, m_{\Sigma^{+}}\right) \mathcal{H}^\mathrm{LS,decay}_{N(1720)^{+}, 1, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{N(1720)^{+}, 0, 1} \mathcal{R}_{1}\left(\sigma_{3}, m_{N(1720)^{+}}, \Gamma_{N(1720)^{+}}\right) d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{12}\right)}{2}} + \sum_{\lambda_{R}=-1/2}^{1/2}{\frac{\sqrt{2} \delta_{-1, \lambda_{R} + \frac{1}{2}} C^{\frac{1}{2},\frac{1}{2}}_{0,0,\frac{1}{2},\frac{1}{2}} C^{1,\lambda_{R} + \frac{1}{2}}_{0,0,1,\lambda_{R} + \frac{1}{2}} C^{1,\lambda_{R} + \frac{1}{2}}_{\frac{1}{2},\lambda_{R},\frac{1}{2},\frac{1}{2}} C^{\frac{1}{2},\frac{1}{2}}_{1,0,\frac{1}{2},\frac{1}{2}} \mathcal{F}_{0}\left(m_{0}^{2}, m_{N(1710)^{+}}, m_{\overline{p}}\right) \mathcal{F}_{1}\left(\sigma_{3}^{2}, m_{K^{0}}, m_{\Sigma^{+}}\right) \mathcal{H}^\mathrm{LS,decay}_{N(1710)^{+}, 1, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{N(1710)^{+}, 0, 1} \mathcal{R}_{1}\left(\sigma_{3}, m_{N(1710)^{+}}, \Gamma_{N(1710)^{+}}\right) d^{\frac{1}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{12}\right)}{2}} + \sum_{\lambda_{R}=-3/2}^{3/2}{\frac{\sqrt{5} \delta_{-1, \lambda_{R} + \frac{1}{2}} C^{\frac{1}{2},\frac{1}{2}}_{0,0,\frac{1}{2},\frac{1}{2}} C^{1,\lambda_{R} + \frac{1}{2}}_{1,0,1,\lambda_{R} + \frac{1}{2}} C^{1,\lambda_{R} + \frac{1}{2}}_{\frac{3}{2},\lambda_{R},\frac{1}{2},\frac{1}{2}} C^{\frac{3}{2},\frac{1}{2}}_{2,0,\frac{1}{2},\frac{1}{2}} \mathcal{F}_{1}\left(m_{0}^{2}, m_{N(1700)^{+}}, m_{\overline{p}}\right) \mathcal{F}_{2}\left(\sigma_{3}^{2}, m_{K^{0}}, m_{\Sigma^{+}}\right) \mathcal{H}^\mathrm{LS,decay}_{N(1700)^{+}, 2, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{N(1700)^{+}, 1, 1} \mathcal{R}_{2}\left(\sigma_{3}, m_{N(1700)^{+}}, \Gamma_{N(1700)^{+}}\right) d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{12}\right)}{2}} \\ A^{2}_{-1, 0, - \frac{1}{2}, \frac{1}{2}} &=& \sum_{\lambda_{R}=-1/2}^{1/2}{- \frac{\sqrt{2} \delta_{-1, \lambda_{R} + \frac{1}{2}} \left(C^{\frac{1}{2},- \frac{1}{2}}_{0,0,\frac{1}{2},- \frac{1}{2}}\right)^{2} C^{0,\lambda_{R} + \frac{1}{2}}_{\frac{1}{2},\lambda_{R},\frac{1}{2},\frac{1}{2}} C^{1,\lambda_{R} + \frac{1}{2}}_{1,0,0,\lambda_{R} + \frac{1}{2}} \mathcal{F}_{1}\left(m_{0}^{2}, m_{\overline{\Sigma}(1750)^{-}}, m_{\Sigma^{+}}\right) \mathcal{F}_{0}\left(\sigma_{2}^{2}, m_{K^{0}}, m_{\overline{p}}\right) \mathcal{H}^\mathrm{LS,decay}_{\overline{\Sigma}(1750)^{-}, 0, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{\Sigma}(1750)^{-}, 1, 0} \mathcal{R}_{0}\left(\sigma_{2}, m_{\overline{\Sigma}(1750)^{-}}, \Gamma_{\overline{\Sigma}(1750)^{-}}\right) d^{\frac{1}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)}{2}} + \sum_{\lambda_{R}=-1/2}^{1/2}{- \frac{\sqrt{2} \delta_{-1, \lambda_{R} + \frac{1}{2}} C^{\frac{1}{2},- \frac{1}{2}}_{0,0,\frac{1}{2},- \frac{1}{2}} C^{1,\lambda_{R} + \frac{1}{2}}_{0,0,1,\lambda_{R} + \frac{1}{2}} C^{1,\lambda_{R} + \frac{1}{2}}_{\frac{1}{2},\lambda_{R},\frac{1}{2},\frac{1}{2}} C^{\frac{1}{2},- \frac{1}{2}}_{1,0,\frac{1}{2},- \frac{1}{2}} \mathcal{F}_{0}\left(m_{0}^{2}, m_{\overline{\Sigma}(1660)^{-}}, m_{\Sigma^{+}}\right) \mathcal{F}_{1}\left(\sigma_{2}^{2}, m_{K^{0}}, m_{\overline{p}}\right) \mathcal{H}^\mathrm{LS,decay}_{\overline{\Sigma}(1660)^{-}, 1, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{\Sigma}(1660)^{-}, 0, 1} \mathcal{R}_{1}\left(\sigma_{2}, m_{\overline{\Sigma}(1660)^{-}}, \Gamma_{\overline{\Sigma}(1660)^{-}}\right) d^{\frac{1}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)}{2}} + \sum_{\lambda_{R}=-5/2}^{5/2}{- \frac{\sqrt{30} \delta_{-1, \lambda_{R} + \frac{1}{2}} C^{\frac{1}{2},- \frac{1}{2}}_{0,0,\frac{1}{2},- \frac{1}{2}} C^{1,\lambda_{R} + \frac{1}{2}}_{1,0,2,\lambda_{R} + \frac{1}{2}} C^{\frac{5}{2},- \frac{1}{2}}_{2,0,\frac{1}{2},- \frac{1}{2}} C^{2,\lambda_{R} + \frac{1}{2}}_{\frac{5}{2},\lambda_{R},\frac{1}{2},\frac{1}{2}} \mathcal{F}_{1}\left(m_{0}^{2}, m_{\overline{\Sigma}(1775)^{-}}, m_{\Sigma^{+}}\right) \mathcal{F}_{2}\left(\sigma_{2}^{2}, m_{K^{0}}, m_{\overline{p}}\right) \mathcal{H}^\mathrm{LS,decay}_{\overline{\Sigma}(1775)^{-}, 2, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{\Sigma}(1775)^{-}, 1, 2} \mathcal{R}_{2}\left(\sigma_{2}, m_{\overline{\Sigma}(1775)^{-}}, \Gamma_{\overline{\Sigma}(1775)^{-}}\right) d^{\frac{5}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)}{6}} + \sum_{\lambda_{R}=-3/2}^{3/2}{- \frac{\sqrt{5} \delta_{-1, \lambda_{R} + \frac{1}{2}} C^{\frac{1}{2},- \frac{1}{2}}_{0,0,\frac{1}{2},- \frac{1}{2}} C^{1,\lambda_{R} + \frac{1}{2}}_{1,0,1,\lambda_{R} + \frac{1}{2}} C^{1,\lambda_{R} + \frac{1}{2}}_{\frac{3}{2},\lambda_{R},\frac{1}{2},\frac{1}{2}} C^{\frac{3}{2},- \frac{1}{2}}_{2,0,\frac{1}{2},- \frac{1}{2}} \mathcal{F}_{1}\left(m_{0}^{2}, m_{\overline{\Sigma}(1670)^{-}}, m_{\Sigma^{+}}\right) \mathcal{F}_{2}\left(\sigma_{2}^{2}, m_{K^{0}}, m_{\overline{p}}\right) \mathcal{H}^\mathrm{LS,decay}_{\overline{\Sigma}(1670)^{-}, 2, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{\Sigma}(1670)^{-}, 1, 1} \mathcal{R}_{2}\left(\sigma_{2}, m_{\overline{\Sigma}(1670)^{-}}, \Gamma_{\overline{\Sigma}(1670)^{-}}\right) d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)}{2}} + \sum_{\lambda_{R}=-3/2}^{3/2}{- \frac{\sqrt{5} \delta_{-1, \lambda_{R} + \frac{1}{2}} C^{\frac{1}{2},- \frac{1}{2}}_{0,0,\frac{1}{2},- \frac{1}{2}} C^{1,\lambda_{R} + \frac{1}{2}}_{1,0,1,\lambda_{R} + \frac{1}{2}} C^{1,\lambda_{R} + \frac{1}{2}}_{\frac{3}{2},\lambda_{R},\frac{1}{2},\frac{1}{2}} C^{\frac{3}{2},- \frac{1}{2}}_{2,0,\frac{1}{2},- \frac{1}{2}} \mathcal{F}_{1}\left(m_{0}^{2}, m_{\overline{\Sigma}(1940)^{-}}, m_{\Sigma^{+}}\right) \mathcal{F}_{2}\left(\sigma_{2}^{2}, m_{K^{0}}, m_{\overline{p}}\right) \mathcal{H}^\mathrm{LS,decay}_{\overline{\Sigma}(1940)^{-}, 2, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{\Sigma}(1940)^{-}, 1, 1} \mathcal{R}_{2}\left(\sigma_{2}, m_{\overline{\Sigma}(1940)^{-}}, \Gamma_{\overline{\Sigma}(1940)^{-}}\right) d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)}{2}} \\ A^{3}_{-1, 0, - \frac{1}{2}, \frac{1}{2}} &=& \sum_{\lambda_{R}=-3/2}^{3/2}{- \frac{\delta_{-1, \lambda_{R} - \frac{1}{2}} C^{\frac{1}{2},\frac{1}{2}}_{0,0,\frac{1}{2},\frac{1}{2}} C^{1,\lambda_{R} - \frac{1}{2}}_{0,0,1,\lambda_{R} - \frac{1}{2}} C^{\frac{3}{2},\frac{1}{2}}_{1,0,\frac{1}{2},\frac{1}{2}} C^{1,\lambda_{R} - \frac{1}{2}}_{\frac{3}{2},\lambda_{R},\frac{1}{2},- \frac{1}{2}} \mathcal{F}_{0}\left(m_{0}^{2}, m_{N(1720)^{+}}, m_{\overline{p}}\right) \mathcal{F}_{1}\left(\sigma_{3}^{2}, m_{K^{0}}, m_{\Sigma^{+}}\right) \mathcal{H}^\mathrm{LS,decay}_{N(1720)^{+}, 1, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{N(1720)^{+}, 0, 1} \mathcal{R}_{1}\left(\sigma_{3}, m_{N(1720)^{+}}, \Gamma_{N(1720)^{+}}\right) d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{12}\right)}{2}} + \sum_{\lambda_{R}=-1/2}^{1/2}{- \frac{\sqrt{2} \delta_{-1, \lambda_{R} - \frac{1}{2}} C^{\frac{1}{2},\frac{1}{2}}_{0,0,\frac{1}{2},\frac{1}{2}} C^{1,\lambda_{R} - \frac{1}{2}}_{0,0,1,\lambda_{R} - \frac{1}{2}} C^{1,\lambda_{R} - \frac{1}{2}}_{\frac{1}{2},\lambda_{R},\frac{1}{2},- \frac{1}{2}} C^{\frac{1}{2},\frac{1}{2}}_{1,0,\frac{1}{2},\frac{1}{2}} \mathcal{F}_{0}\left(m_{0}^{2}, m_{N(1710)^{+}}, m_{\overline{p}}\right) \mathcal{F}_{1}\left(\sigma_{3}^{2}, m_{K^{0}}, m_{\Sigma^{+}}\right) \mathcal{H}^\mathrm{LS,decay}_{N(1710)^{+}, 1, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{N(1710)^{+}, 0, 1} \mathcal{R}_{1}\left(\sigma_{3}, m_{N(1710)^{+}}, \Gamma_{N(1710)^{+}}\right) d^{\frac{1}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{12}\right)}{2}} + \sum_{\lambda_{R}=-3/2}^{3/2}{- \frac{\sqrt{5} \delta_{-1, \lambda_{R} - \frac{1}{2}} C^{\frac{1}{2},\frac{1}{2}}_{0,0,\frac{1}{2},\frac{1}{2}} C^{1,\lambda_{R} - \frac{1}{2}}_{1,0,1,\lambda_{R} - \frac{1}{2}} C^{1,\lambda_{R} - \frac{1}{2}}_{\frac{3}{2},\lambda_{R},\frac{1}{2},- \frac{1}{2}} C^{\frac{3}{2},\frac{1}{2}}_{2,0,\frac{1}{2},\frac{1}{2}} \mathcal{F}_{1}\left(m_{0}^{2}, m_{N(1700)^{+}}, m_{\overline{p}}\right) \mathcal{F}_{2}\left(\sigma_{3}^{2}, m_{K^{0}}, m_{\Sigma^{+}}\right) \mathcal{H}^\mathrm{LS,decay}_{N(1700)^{+}, 2, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{N(1700)^{+}, 1, 1} \mathcal{R}_{2}\left(\sigma_{3}, m_{N(1700)^{+}}, \Gamma_{N(1700)^{+}}\right) d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{12}\right)}{2}} \\ A^{2}_{-1, 0, \frac{1}{2}, - \frac{1}{2}} &=& \sum_{\lambda_{R}=-1/2}^{1/2}{\frac{\sqrt{2} \delta_{-1, \lambda_{R} - \frac{1}{2}} \left(C^{\frac{1}{2},\frac{1}{2}}_{0,0,\frac{1}{2},\frac{1}{2}}\right)^{2} C^{0,\lambda_{R} - \frac{1}{2}}_{\frac{1}{2},\lambda_{R},\frac{1}{2},- \frac{1}{2}} C^{1,\lambda_{R} - \frac{1}{2}}_{1,0,0,\lambda_{R} - \frac{1}{2}} \mathcal{F}_{1}\left(m_{0}^{2}, m_{\overline{\Sigma}(1750)^{-}}, m_{\Sigma^{+}}\right) \mathcal{F}_{0}\left(\sigma_{2}^{2}, m_{K^{0}}, m_{\overline{p}}\right) \mathcal{H}^\mathrm{LS,decay}_{\overline{\Sigma}(1750)^{-}, 0, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{\Sigma}(1750)^{-}, 1, 0} \mathcal{R}_{0}\left(\sigma_{2}, m_{\overline{\Sigma}(1750)^{-}}, \Gamma_{\overline{\Sigma}(1750)^{-}}\right) d^{\frac{1}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)}{2}} + \sum_{\lambda_{R}=-1/2}^{1/2}{\frac{\sqrt{2} \delta_{-1, \lambda_{R} - \frac{1}{2}} C^{\frac{1}{2},\frac{1}{2}}_{0,0,\frac{1}{2},\frac{1}{2}} C^{1,\lambda_{R} - \frac{1}{2}}_{0,0,1,\lambda_{R} - \frac{1}{2}} C^{1,\lambda_{R} - \frac{1}{2}}_{\frac{1}{2},\lambda_{R},\frac{1}{2},- \frac{1}{2}} C^{\frac{1}{2},\frac{1}{2}}_{1,0,\frac{1}{2},\frac{1}{2}} \mathcal{F}_{0}\left(m_{0}^{2}, m_{\overline{\Sigma}(1660)^{-}}, m_{\Sigma^{+}}\right) \mathcal{F}_{1}\left(\sigma_{2}^{2}, m_{K^{0}}, m_{\overline{p}}\right) \mathcal{H}^\mathrm{LS,decay}_{\overline{\Sigma}(1660)^{-}, 1, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{\Sigma}(1660)^{-}, 0, 1} \mathcal{R}_{1}\left(\sigma_{2}, m_{\overline{\Sigma}(1660)^{-}}, \Gamma_{\overline{\Sigma}(1660)^{-}}\right) d^{\frac{1}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)}{2}} + \sum_{\lambda_{R}=-5/2}^{5/2}{\frac{\sqrt{30} \delta_{-1, \lambda_{R} - \frac{1}{2}} C^{\frac{1}{2},\frac{1}{2}}_{0,0,\frac{1}{2},\frac{1}{2}} C^{1,\lambda_{R} - \frac{1}{2}}_{1,0,2,\lambda_{R} - \frac{1}{2}} C^{\frac{5}{2},\frac{1}{2}}_{2,0,\frac{1}{2},\frac{1}{2}} C^{2,\lambda_{R} - \frac{1}{2}}_{\frac{5}{2},\lambda_{R},\frac{1}{2},- \frac{1}{2}} \mathcal{F}_{1}\left(m_{0}^{2}, m_{\overline{\Sigma}(1775)^{-}}, m_{\Sigma^{+}}\right) \mathcal{F}_{2}\left(\sigma_{2}^{2}, m_{K^{0}}, m_{\overline{p}}\right) \mathcal{H}^\mathrm{LS,decay}_{\overline{\Sigma}(1775)^{-}, 2, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{\Sigma}(1775)^{-}, 1, 2} \mathcal{R}_{2}\left(\sigma_{2}, m_{\overline{\Sigma}(1775)^{-}}, \Gamma_{\overline{\Sigma}(1775)^{-}}\right) d^{\frac{5}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)}{6}} + \sum_{\lambda_{R}=-3/2}^{3/2}{\frac{\sqrt{5} \delta_{-1, \lambda_{R} - \frac{1}{2}} C^{\frac{1}{2},\frac{1}{2}}_{0,0,\frac{1}{2},\frac{1}{2}} C^{1,\lambda_{R} - \frac{1}{2}}_{1,0,1,\lambda_{R} - \frac{1}{2}} C^{1,\lambda_{R} - \frac{1}{2}}_{\frac{3}{2},\lambda_{R},\frac{1}{2},- \frac{1}{2}} C^{\frac{3}{2},\frac{1}{2}}_{2,0,\frac{1}{2},\frac{1}{2}} \mathcal{F}_{1}\left(m_{0}^{2}, m_{\overline{\Sigma}(1670)^{-}}, m_{\Sigma^{+}}\right) \mathcal{F}_{2}\left(\sigma_{2}^{2}, m_{K^{0}}, m_{\overline{p}}\right) \mathcal{H}^\mathrm{LS,decay}_{\overline{\Sigma}(1670)^{-}, 2, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{\Sigma}(1670)^{-}, 1, 1} \mathcal{R}_{2}\left(\sigma_{2}, m_{\overline{\Sigma}(1670)^{-}}, \Gamma_{\overline{\Sigma}(1670)^{-}}\right) d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)}{2}} + \sum_{\lambda_{R}=-3/2}^{3/2}{\frac{\sqrt{5} \delta_{-1, \lambda_{R} - \frac{1}{2}} C^{\frac{1}{2},\frac{1}{2}}_{0,0,\frac{1}{2},\frac{1}{2}} C^{1,\lambda_{R} - \frac{1}{2}}_{1,0,1,\lambda_{R} - \frac{1}{2}} C^{1,\lambda_{R} - \frac{1}{2}}_{\frac{3}{2},\lambda_{R},\frac{1}{2},- \frac{1}{2}} C^{\frac{3}{2},\frac{1}{2}}_{2,0,\frac{1}{2},\frac{1}{2}} \mathcal{F}_{1}\left(m_{0}^{2}, m_{\overline{\Sigma}(1940)^{-}}, m_{\Sigma^{+}}\right) \mathcal{F}_{2}\left(\sigma_{2}^{2}, m_{K^{0}}, m_{\overline{p}}\right) \mathcal{H}^\mathrm{LS,decay}_{\overline{\Sigma}(1940)^{-}, 2, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{\Sigma}(1940)^{-}, 1, 1} \mathcal{R}_{2}\left(\sigma_{2}, m_{\overline{\Sigma}(1940)^{-}}, \Gamma_{\overline{\Sigma}(1940)^{-}}\right) d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)}{2}} \\ A^{3}_{-1, 0, \frac{1}{2}, - \frac{1}{2}} &=& \sum_{\lambda_{R}=-3/2}^{3/2}{- \frac{\delta_{-1, \lambda_{R} + \frac{1}{2}} C^{\frac{1}{2},- \frac{1}{2}}_{0,0,\frac{1}{2},- \frac{1}{2}} C^{1,\lambda_{R} + \frac{1}{2}}_{0,0,1,\lambda_{R} + \frac{1}{2}} C^{\frac{3}{2},- \frac{1}{2}}_{1,0,\frac{1}{2},- \frac{1}{2}} C^{1,\lambda_{R} + \frac{1}{2}}_{\frac{3}{2},\lambda_{R},\frac{1}{2},\frac{1}{2}} \mathcal{F}_{0}\left(m_{0}^{2}, m_{N(1720)^{+}}, m_{\overline{p}}\right) \mathcal{F}_{1}\left(\sigma_{3}^{2}, m_{K^{0}}, m_{\Sigma^{+}}\right) \mathcal{H}^\mathrm{LS,decay}_{N(1720)^{+}, 1, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{N(1720)^{+}, 0, 1} \mathcal{R}_{1}\left(\sigma_{3}, m_{N(1720)^{+}}, \Gamma_{N(1720)^{+}}\right) d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{12}\right)}{2}} + \sum_{\lambda_{R}=-1/2}^{1/2}{- \frac{\sqrt{2} \delta_{-1, \lambda_{R} + \frac{1}{2}} C^{\frac{1}{2},- \frac{1}{2}}_{0,0,\frac{1}{2},- \frac{1}{2}} C^{1,\lambda_{R} + \frac{1}{2}}_{0,0,1,\lambda_{R} + \frac{1}{2}} C^{1,\lambda_{R} + \frac{1}{2}}_{\frac{1}{2},\lambda_{R},\frac{1}{2},\frac{1}{2}} C^{\frac{1}{2},- \frac{1}{2}}_{1,0,\frac{1}{2},- \frac{1}{2}} \mathcal{F}_{0}\left(m_{0}^{2}, m_{N(1710)^{+}}, m_{\overline{p}}\right) \mathcal{F}_{1}\left(\sigma_{3}^{2}, m_{K^{0}}, m_{\Sigma^{+}}\right) \mathcal{H}^\mathrm{LS,decay}_{N(1710)^{+}, 1, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{N(1710)^{+}, 0, 1} \mathcal{R}_{1}\left(\sigma_{3}, m_{N(1710)^{+}}, \Gamma_{N(1710)^{+}}\right) d^{\frac{1}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{12}\right)}{2}} + \sum_{\lambda_{R}=-3/2}^{3/2}{- \frac{\sqrt{5} \delta_{-1, \lambda_{R} + \frac{1}{2}} C^{\frac{1}{2},- \frac{1}{2}}_{0,0,\frac{1}{2},- \frac{1}{2}} C^{1,\lambda_{R} + \frac{1}{2}}_{1,0,1,\lambda_{R} + \frac{1}{2}} C^{1,\lambda_{R} + \frac{1}{2}}_{\frac{3}{2},\lambda_{R},\frac{1}{2},\frac{1}{2}} C^{\frac{3}{2},- \frac{1}{2}}_{2,0,\frac{1}{2},- \frac{1}{2}} \mathcal{F}_{1}\left(m_{0}^{2}, m_{N(1700)^{+}}, m_{\overline{p}}\right) \mathcal{F}_{2}\left(\sigma_{3}^{2}, m_{K^{0}}, m_{\Sigma^{+}}\right) \mathcal{H}^\mathrm{LS,decay}_{N(1700)^{+}, 2, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{N(1700)^{+}, 1, 1} \mathcal{R}_{2}\left(\sigma_{3}, m_{N(1700)^{+}}, \Gamma_{N(1700)^{+}}\right) d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{12}\right)}{2}} \\ A^{2}_{-1, 0, \frac{1}{2}, \frac{1}{2}} &=& \sum_{\lambda_{R}=-1/2}^{1/2}{\frac{\sqrt{2} \delta_{-1, \lambda_{R} - \frac{1}{2}} \left(C^{\frac{1}{2},- \frac{1}{2}}_{0,0,\frac{1}{2},- \frac{1}{2}}\right)^{2} C^{0,\lambda_{R} - \frac{1}{2}}_{\frac{1}{2},\lambda_{R},\frac{1}{2},- \frac{1}{2}} C^{1,\lambda_{R} - \frac{1}{2}}_{1,0,0,\lambda_{R} - \frac{1}{2}} \mathcal{F}_{1}\left(m_{0}^{2}, m_{\overline{\Sigma}(1750)^{-}}, m_{\Sigma^{+}}\right) \mathcal{F}_{0}\left(\sigma_{2}^{2}, m_{K^{0}}, m_{\overline{p}}\right) \mathcal{H}^\mathrm{LS,decay}_{\overline{\Sigma}(1750)^{-}, 0, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{\Sigma}(1750)^{-}, 1, 0} \mathcal{R}_{0}\left(\sigma_{2}, m_{\overline{\Sigma}(1750)^{-}}, \Gamma_{\overline{\Sigma}(1750)^{-}}\right) d^{\frac{1}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)}{2}} + \sum_{\lambda_{R}=-1/2}^{1/2}{\frac{\sqrt{2} \delta_{-1, \lambda_{R} - \frac{1}{2}} C^{\frac{1}{2},- \frac{1}{2}}_{0,0,\frac{1}{2},- \frac{1}{2}} C^{1,\lambda_{R} - \frac{1}{2}}_{0,0,1,\lambda_{R} - \frac{1}{2}} C^{1,\lambda_{R} - \frac{1}{2}}_{\frac{1}{2},\lambda_{R},\frac{1}{2},- \frac{1}{2}} C^{\frac{1}{2},- \frac{1}{2}}_{1,0,\frac{1}{2},- \frac{1}{2}} \mathcal{F}_{0}\left(m_{0}^{2}, m_{\overline{\Sigma}(1660)^{-}}, m_{\Sigma^{+}}\right) \mathcal{F}_{1}\left(\sigma_{2}^{2}, m_{K^{0}}, m_{\overline{p}}\right) \mathcal{H}^\mathrm{LS,decay}_{\overline{\Sigma}(1660)^{-}, 1, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{\Sigma}(1660)^{-}, 0, 1} \mathcal{R}_{1}\left(\sigma_{2}, m_{\overline{\Sigma}(1660)^{-}}, \Gamma_{\overline{\Sigma}(1660)^{-}}\right) d^{\frac{1}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)}{2}} + \sum_{\lambda_{R}=-5/2}^{5/2}{\frac{\sqrt{30} \delta_{-1, \lambda_{R} - \frac{1}{2}} C^{\frac{1}{2},- \frac{1}{2}}_{0,0,\frac{1}{2},- \frac{1}{2}} C^{1,\lambda_{R} - \frac{1}{2}}_{1,0,2,\lambda_{R} - \frac{1}{2}} C^{\frac{5}{2},- \frac{1}{2}}_{2,0,\frac{1}{2},- \frac{1}{2}} C^{2,\lambda_{R} - \frac{1}{2}}_{\frac{5}{2},\lambda_{R},\frac{1}{2},- \frac{1}{2}} \mathcal{F}_{1}\left(m_{0}^{2}, m_{\overline{\Sigma}(1775)^{-}}, m_{\Sigma^{+}}\right) \mathcal{F}_{2}\left(\sigma_{2}^{2}, m_{K^{0}}, m_{\overline{p}}\right) \mathcal{H}^\mathrm{LS,decay}_{\overline{\Sigma}(1775)^{-}, 2, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{\Sigma}(1775)^{-}, 1, 2} \mathcal{R}_{2}\left(\sigma_{2}, m_{\overline{\Sigma}(1775)^{-}}, \Gamma_{\overline{\Sigma}(1775)^{-}}\right) d^{\frac{5}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)}{6}} + \sum_{\lambda_{R}=-3/2}^{3/2}{\frac{\sqrt{5} \delta_{-1, \lambda_{R} - \frac{1}{2}} C^{\frac{1}{2},- \frac{1}{2}}_{0,0,\frac{1}{2},- \frac{1}{2}} C^{1,\lambda_{R} - \frac{1}{2}}_{1,0,1,\lambda_{R} - \frac{1}{2}} C^{1,\lambda_{R} - \frac{1}{2}}_{\frac{3}{2},\lambda_{R},\frac{1}{2},- \frac{1}{2}} C^{\frac{3}{2},- \frac{1}{2}}_{2,0,\frac{1}{2},- \frac{1}{2}} \mathcal{F}_{1}\left(m_{0}^{2}, m_{\overline{\Sigma}(1670)^{-}}, m_{\Sigma^{+}}\right) \mathcal{F}_{2}\left(\sigma_{2}^{2}, m_{K^{0}}, m_{\overline{p}}\right) \mathcal{H}^\mathrm{LS,decay}_{\overline{\Sigma}(1670)^{-}, 2, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{\Sigma}(1670)^{-}, 1, 1} \mathcal{R}_{2}\left(\sigma_{2}, m_{\overline{\Sigma}(1670)^{-}}, \Gamma_{\overline{\Sigma}(1670)^{-}}\right) d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)}{2}} + \sum_{\lambda_{R}=-3/2}^{3/2}{\frac{\sqrt{5} \delta_{-1, \lambda_{R} - \frac{1}{2}} C^{\frac{1}{2},- \frac{1}{2}}_{0,0,\frac{1}{2},- \frac{1}{2}} C^{1,\lambda_{R} - \frac{1}{2}}_{1,0,1,\lambda_{R} - \frac{1}{2}} C^{1,\lambda_{R} - \frac{1}{2}}_{\frac{3}{2},\lambda_{R},\frac{1}{2},- \frac{1}{2}} C^{\frac{3}{2},- \frac{1}{2}}_{2,0,\frac{1}{2},- \frac{1}{2}} \mathcal{F}_{1}\left(m_{0}^{2}, m_{\overline{\Sigma}(1940)^{-}}, m_{\Sigma^{+}}\right) \mathcal{F}_{2}\left(\sigma_{2}^{2}, m_{K^{0}}, m_{\overline{p}}\right) \mathcal{H}^\mathrm{LS,decay}_{\overline{\Sigma}(1940)^{-}, 2, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{\Sigma}(1940)^{-}, 1, 1} \mathcal{R}_{2}\left(\sigma_{2}, m_{\overline{\Sigma}(1940)^{-}}, \Gamma_{\overline{\Sigma}(1940)^{-}}\right) d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)}{2}} \\ A^{3}_{-1, 0, \frac{1}{2}, \frac{1}{2}} &=& \sum_{\lambda_{R}=-3/2}^{3/2}{\frac{\delta_{-1, \lambda_{R} - \frac{1}{2}} C^{\frac{1}{2},- \frac{1}{2}}_{0,0,\frac{1}{2},- \frac{1}{2}} C^{1,\lambda_{R} - \frac{1}{2}}_{0,0,1,\lambda_{R} - \frac{1}{2}} C^{\frac{3}{2},- \frac{1}{2}}_{1,0,\frac{1}{2},- \frac{1}{2}} C^{1,\lambda_{R} - \frac{1}{2}}_{\frac{3}{2},\lambda_{R},\frac{1}{2},- \frac{1}{2}} \mathcal{F}_{0}\left(m_{0}^{2}, m_{N(1720)^{+}}, m_{\overline{p}}\right) \mathcal{F}_{1}\left(\sigma_{3}^{2}, m_{K^{0}}, m_{\Sigma^{+}}\right) \mathcal{H}^\mathrm{LS,decay}_{N(1720)^{+}, 1, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{N(1720)^{+}, 0, 1} \mathcal{R}_{1}\left(\sigma_{3}, m_{N(1720)^{+}}, \Gamma_{N(1720)^{+}}\right) d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{12}\right)}{2}} + \sum_{\lambda_{R}=-1/2}^{1/2}{\frac{\sqrt{2} \delta_{-1, \lambda_{R} - \frac{1}{2}} C^{\frac{1}{2},- \frac{1}{2}}_{0,0,\frac{1}{2},- \frac{1}{2}} C^{1,\lambda_{R} - \frac{1}{2}}_{0,0,1,\lambda_{R} - \frac{1}{2}} C^{1,\lambda_{R} - \frac{1}{2}}_{\frac{1}{2},\lambda_{R},\frac{1}{2},- \frac{1}{2}} C^{\frac{1}{2},- \frac{1}{2}}_{1,0,\frac{1}{2},- \frac{1}{2}} \mathcal{F}_{0}\left(m_{0}^{2}, m_{N(1710)^{+}}, m_{\overline{p}}\right) \mathcal{F}_{1}\left(\sigma_{3}^{2}, m_{K^{0}}, m_{\Sigma^{+}}\right) \mathcal{H}^\mathrm{LS,decay}_{N(1710)^{+}, 1, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{N(1710)^{+}, 0, 1} \mathcal{R}_{1}\left(\sigma_{3}, m_{N(1710)^{+}}, \Gamma_{N(1710)^{+}}\right) d^{\frac{1}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{12}\right)}{2}} + \sum_{\lambda_{R}=-3/2}^{3/2}{\frac{\sqrt{5} \delta_{-1, \lambda_{R} - \frac{1}{2}} C^{\frac{1}{2},- \frac{1}{2}}_{0,0,\frac{1}{2},- \frac{1}{2}} C^{1,\lambda_{R} - \frac{1}{2}}_{1,0,1,\lambda_{R} - \frac{1}{2}} C^{1,\lambda_{R} - \frac{1}{2}}_{\frac{3}{2},\lambda_{R},\frac{1}{2},- \frac{1}{2}} C^{\frac{3}{2},- \frac{1}{2}}_{2,0,\frac{1}{2},- \frac{1}{2}} \mathcal{F}_{1}\left(m_{0}^{2}, m_{N(1700)^{+}}, m_{\overline{p}}\right) \mathcal{F}_{2}\left(\sigma_{3}^{2}, m_{K^{0}}, m_{\Sigma^{+}}\right) \mathcal{H}^\mathrm{LS,decay}_{N(1700)^{+}, 2, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{N(1700)^{+}, 1, 1} \mathcal{R}_{2}\left(\sigma_{3}, m_{N(1700)^{+}}, \Gamma_{N(1700)^{+}}\right) d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{12}\right)}{2}} \\ A^{2}_{0, 0, - \frac{1}{2}, - \frac{1}{2}} &=& \sum_{\lambda_{R}=-1/2}^{1/2}{- \frac{\sqrt{2} \delta_{0, \lambda_{R} + \frac{1}{2}} \left(C^{\frac{1}{2},\frac{1}{2}}_{0,0,\frac{1}{2},\frac{1}{2}}\right)^{2} C^{0,\lambda_{R} + \frac{1}{2}}_{\frac{1}{2},\lambda_{R},\frac{1}{2},\frac{1}{2}} C^{1,\lambda_{R} + \frac{1}{2}}_{1,0,0,\lambda_{R} + \frac{1}{2}} \mathcal{F}_{1}\left(m_{0}^{2}, m_{\overline{\Sigma}(1750)^{-}}, m_{\Sigma^{+}}\right) \mathcal{F}_{0}\left(\sigma_{2}^{2}, m_{K^{0}}, m_{\overline{p}}\right) \mathcal{H}^\mathrm{LS,decay}_{\overline{\Sigma}(1750)^{-}, 0, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{\Sigma}(1750)^{-}, 1, 0} \mathcal{R}_{0}\left(\sigma_{2}, m_{\overline{\Sigma}(1750)^{-}}, \Gamma_{\overline{\Sigma}(1750)^{-}}\right) d^{\frac{1}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)}{2}} + \sum_{\lambda_{R}=-1/2}^{1/2}{- \frac{\sqrt{2} \delta_{0, \lambda_{R} + \frac{1}{2}} C^{\frac{1}{2},\frac{1}{2}}_{0,0,\frac{1}{2},\frac{1}{2}} C^{1,\lambda_{R} + \frac{1}{2}}_{0,0,1,\lambda_{R} + \frac{1}{2}} C^{1,\lambda_{R} + \frac{1}{2}}_{\frac{1}{2},\lambda_{R},\frac{1}{2},\frac{1}{2}} C^{\frac{1}{2},\frac{1}{2}}_{1,0,\frac{1}{2},\frac{1}{2}} \mathcal{F}_{0}\left(m_{0}^{2}, m_{\overline{\Sigma}(1660)^{-}}, m_{\Sigma^{+}}\right) \mathcal{F}_{1}\left(\sigma_{2}^{2}, m_{K^{0}}, m_{\overline{p}}\right) \mathcal{H}^\mathrm{LS,decay}_{\overline{\Sigma}(1660)^{-}, 1, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{\Sigma}(1660)^{-}, 0, 1} \mathcal{R}_{1}\left(\sigma_{2}, m_{\overline{\Sigma}(1660)^{-}}, \Gamma_{\overline{\Sigma}(1660)^{-}}\right) d^{\frac{1}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)}{2}} + \sum_{\lambda_{R}=-5/2}^{5/2}{- \frac{\sqrt{30} \delta_{0, \lambda_{R} + \frac{1}{2}} C^{\frac{1}{2},\frac{1}{2}}_{0,0,\frac{1}{2},\frac{1}{2}} C^{1,\lambda_{R} + \frac{1}{2}}_{1,0,2,\lambda_{R} + \frac{1}{2}} C^{\frac{5}{2},\frac{1}{2}}_{2,0,\frac{1}{2},\frac{1}{2}} C^{2,\lambda_{R} + \frac{1}{2}}_{\frac{5}{2},\lambda_{R},\frac{1}{2},\frac{1}{2}} \mathcal{F}_{1}\left(m_{0}^{2}, m_{\overline{\Sigma}(1775)^{-}}, m_{\Sigma^{+}}\right) \mathcal{F}_{2}\left(\sigma_{2}^{2}, m_{K^{0}}, m_{\overline{p}}\right) \mathcal{H}^\mathrm{LS,decay}_{\overline{\Sigma}(1775)^{-}, 2, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{\Sigma}(1775)^{-}, 1, 2} \mathcal{R}_{2}\left(\sigma_{2}, m_{\overline{\Sigma}(1775)^{-}}, \Gamma_{\overline{\Sigma}(1775)^{-}}\right) d^{\frac{5}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)}{6}} + \sum_{\lambda_{R}=-3/2}^{3/2}{- \frac{\sqrt{5} \delta_{0, \lambda_{R} + \frac{1}{2}} C^{\frac{1}{2},\frac{1}{2}}_{0,0,\frac{1}{2},\frac{1}{2}} C^{1,\lambda_{R} + \frac{1}{2}}_{1,0,1,\lambda_{R} + \frac{1}{2}} C^{1,\lambda_{R} + \frac{1}{2}}_{\frac{3}{2},\lambda_{R},\frac{1}{2},\frac{1}{2}} C^{\frac{3}{2},\frac{1}{2}}_{2,0,\frac{1}{2},\frac{1}{2}} \mathcal{F}_{1}\left(m_{0}^{2}, m_{\overline{\Sigma}(1670)^{-}}, m_{\Sigma^{+}}\right) \mathcal{F}_{2}\left(\sigma_{2}^{2}, m_{K^{0}}, m_{\overline{p}}\right) \mathcal{H}^\mathrm{LS,decay}_{\overline{\Sigma}(1670)^{-}, 2, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{\Sigma}(1670)^{-}, 1, 1} \mathcal{R}_{2}\left(\sigma_{2}, m_{\overline{\Sigma}(1670)^{-}}, \Gamma_{\overline{\Sigma}(1670)^{-}}\right) d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)}{2}} + \sum_{\lambda_{R}=-3/2}^{3/2}{- \frac{\sqrt{5} \delta_{0, \lambda_{R} + \frac{1}{2}} C^{\frac{1}{2},\frac{1}{2}}_{0,0,\frac{1}{2},\frac{1}{2}} C^{1,\lambda_{R} + \frac{1}{2}}_{1,0,1,\lambda_{R} + \frac{1}{2}} C^{1,\lambda_{R} + \frac{1}{2}}_{\frac{3}{2},\lambda_{R},\frac{1}{2},\frac{1}{2}} C^{\frac{3}{2},\frac{1}{2}}_{2,0,\frac{1}{2},\frac{1}{2}} \mathcal{F}_{1}\left(m_{0}^{2}, m_{\overline{\Sigma}(1940)^{-}}, m_{\Sigma^{+}}\right) \mathcal{F}_{2}\left(\sigma_{2}^{2}, m_{K^{0}}, m_{\overline{p}}\right) \mathcal{H}^\mathrm{LS,decay}_{\overline{\Sigma}(1940)^{-}, 2, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{\Sigma}(1940)^{-}, 1, 1} \mathcal{R}_{2}\left(\sigma_{2}, m_{\overline{\Sigma}(1940)^{-}}, \Gamma_{\overline{\Sigma}(1940)^{-}}\right) d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)}{2}} \\ A^{3}_{0, 0, - \frac{1}{2}, - \frac{1}{2}} &=& \sum_{\lambda_{R}=-3/2}^{3/2}{\frac{\delta_{0, \lambda_{R} + \frac{1}{2}} C^{\frac{1}{2},\frac{1}{2}}_{0,0,\frac{1}{2},\frac{1}{2}} C^{1,\lambda_{R} + \frac{1}{2}}_{0,0,1,\lambda_{R} + \frac{1}{2}} C^{\frac{3}{2},\frac{1}{2}}_{1,0,\frac{1}{2},\frac{1}{2}} C^{1,\lambda_{R} + \frac{1}{2}}_{\frac{3}{2},\lambda_{R},\frac{1}{2},\frac{1}{2}} \mathcal{F}_{0}\left(m_{0}^{2}, m_{N(1720)^{+}}, m_{\overline{p}}\right) \mathcal{F}_{1}\left(\sigma_{3}^{2}, m_{K^{0}}, m_{\Sigma^{+}}\right) \mathcal{H}^\mathrm{LS,decay}_{N(1720)^{+}, 1, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{N(1720)^{+}, 0, 1} \mathcal{R}_{1}\left(\sigma_{3}, m_{N(1720)^{+}}, \Gamma_{N(1720)^{+}}\right) d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{12}\right)}{2}} + \sum_{\lambda_{R}=-1/2}^{1/2}{\frac{\sqrt{2} \delta_{0, \lambda_{R} + \frac{1}{2}} C^{\frac{1}{2},\frac{1}{2}}_{0,0,\frac{1}{2},\frac{1}{2}} C^{1,\lambda_{R} + \frac{1}{2}}_{0,0,1,\lambda_{R} + \frac{1}{2}} C^{1,\lambda_{R} + \frac{1}{2}}_{\frac{1}{2},\lambda_{R},\frac{1}{2},\frac{1}{2}} C^{\frac{1}{2},\frac{1}{2}}_{1,0,\frac{1}{2},\frac{1}{2}} \mathcal{F}_{0}\left(m_{0}^{2}, m_{N(1710)^{+}}, m_{\overline{p}}\right) \mathcal{F}_{1}\left(\sigma_{3}^{2}, m_{K^{0}}, m_{\Sigma^{+}}\right) \mathcal{H}^\mathrm{LS,decay}_{N(1710)^{+}, 1, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{N(1710)^{+}, 0, 1} \mathcal{R}_{1}\left(\sigma_{3}, m_{N(1710)^{+}}, \Gamma_{N(1710)^{+}}\right) d^{\frac{1}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{12}\right)}{2}} + \sum_{\lambda_{R}=-3/2}^{3/2}{\frac{\sqrt{5} \delta_{0, \lambda_{R} + \frac{1}{2}} C^{\frac{1}{2},\frac{1}{2}}_{0,0,\frac{1}{2},\frac{1}{2}} C^{1,\lambda_{R} + \frac{1}{2}}_{1,0,1,\lambda_{R} + \frac{1}{2}} C^{1,\lambda_{R} + \frac{1}{2}}_{\frac{3}{2},\lambda_{R},\frac{1}{2},\frac{1}{2}} C^{\frac{3}{2},\frac{1}{2}}_{2,0,\frac{1}{2},\frac{1}{2}} \mathcal{F}_{1}\left(m_{0}^{2}, m_{N(1700)^{+}}, m_{\overline{p}}\right) \mathcal{F}_{2}\left(\sigma_{3}^{2}, m_{K^{0}}, m_{\Sigma^{+}}\right) \mathcal{H}^\mathrm{LS,decay}_{N(1700)^{+}, 2, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{N(1700)^{+}, 1, 1} \mathcal{R}_{2}\left(\sigma_{3}, m_{N(1700)^{+}}, \Gamma_{N(1700)^{+}}\right) d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{12}\right)}{2}} \\ A^{2}_{0, 0, - \frac{1}{2}, \frac{1}{2}} &=& \sum_{\lambda_{R}=-1/2}^{1/2}{- \frac{\sqrt{2} \delta_{0, \lambda_{R} + \frac{1}{2}} \left(C^{\frac{1}{2},- \frac{1}{2}}_{0,0,\frac{1}{2},- \frac{1}{2}}\right)^{2} C^{0,\lambda_{R} + \frac{1}{2}}_{\frac{1}{2},\lambda_{R},\frac{1}{2},\frac{1}{2}} C^{1,\lambda_{R} + \frac{1}{2}}_{1,0,0,\lambda_{R} + \frac{1}{2}} \mathcal{F}_{1}\left(m_{0}^{2}, m_{\overline{\Sigma}(1750)^{-}}, m_{\Sigma^{+}}\right) \mathcal{F}_{0}\left(\sigma_{2}^{2}, m_{K^{0}}, m_{\overline{p}}\right) \mathcal{H}^\mathrm{LS,decay}_{\overline{\Sigma}(1750)^{-}, 0, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{\Sigma}(1750)^{-}, 1, 0} \mathcal{R}_{0}\left(\sigma_{2}, m_{\overline{\Sigma}(1750)^{-}}, \Gamma_{\overline{\Sigma}(1750)^{-}}\right) d^{\frac{1}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)}{2}} + \sum_{\lambda_{R}=-1/2}^{1/2}{- \frac{\sqrt{2} \delta_{0, \lambda_{R} + \frac{1}{2}} C^{\frac{1}{2},- \frac{1}{2}}_{0,0,\frac{1}{2},- \frac{1}{2}} C^{1,\lambda_{R} + \frac{1}{2}}_{0,0,1,\lambda_{R} + \frac{1}{2}} C^{1,\lambda_{R} + \frac{1}{2}}_{\frac{1}{2},\lambda_{R},\frac{1}{2},\frac{1}{2}} C^{\frac{1}{2},- \frac{1}{2}}_{1,0,\frac{1}{2},- \frac{1}{2}} \mathcal{F}_{0}\left(m_{0}^{2}, m_{\overline{\Sigma}(1660)^{-}}, m_{\Sigma^{+}}\right) \mathcal{F}_{1}\left(\sigma_{2}^{2}, m_{K^{0}}, m_{\overline{p}}\right) \mathcal{H}^\mathrm{LS,decay}_{\overline{\Sigma}(1660)^{-}, 1, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{\Sigma}(1660)^{-}, 0, 1} \mathcal{R}_{1}\left(\sigma_{2}, m_{\overline{\Sigma}(1660)^{-}}, \Gamma_{\overline{\Sigma}(1660)^{-}}\right) d^{\frac{1}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)}{2}} + \sum_{\lambda_{R}=-5/2}^{5/2}{- \frac{\sqrt{30} \delta_{0, \lambda_{R} + \frac{1}{2}} C^{\frac{1}{2},- \frac{1}{2}}_{0,0,\frac{1}{2},- \frac{1}{2}} C^{1,\lambda_{R} + \frac{1}{2}}_{1,0,2,\lambda_{R} + \frac{1}{2}} C^{\frac{5}{2},- \frac{1}{2}}_{2,0,\frac{1}{2},- \frac{1}{2}} C^{2,\lambda_{R} + \frac{1}{2}}_{\frac{5}{2},\lambda_{R},\frac{1}{2},\frac{1}{2}} \mathcal{F}_{1}\left(m_{0}^{2}, m_{\overline{\Sigma}(1775)^{-}}, m_{\Sigma^{+}}\right) \mathcal{F}_{2}\left(\sigma_{2}^{2}, m_{K^{0}}, m_{\overline{p}}\right) \mathcal{H}^\mathrm{LS,decay}_{\overline{\Sigma}(1775)^{-}, 2, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{\Sigma}(1775)^{-}, 1, 2} \mathcal{R}_{2}\left(\sigma_{2}, m_{\overline{\Sigma}(1775)^{-}}, \Gamma_{\overline{\Sigma}(1775)^{-}}\right) d^{\frac{5}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)}{6}} + \sum_{\lambda_{R}=-3/2}^{3/2}{- \frac{\sqrt{5} \delta_{0, \lambda_{R} + \frac{1}{2}} C^{\frac{1}{2},- \frac{1}{2}}_{0,0,\frac{1}{2},- \frac{1}{2}} C^{1,\lambda_{R} + \frac{1}{2}}_{1,0,1,\lambda_{R} + \frac{1}{2}} C^{1,\lambda_{R} + \frac{1}{2}}_{\frac{3}{2},\lambda_{R},\frac{1}{2},\frac{1}{2}} C^{\frac{3}{2},- \frac{1}{2}}_{2,0,\frac{1}{2},- \frac{1}{2}} \mathcal{F}_{1}\left(m_{0}^{2}, m_{\overline{\Sigma}(1670)^{-}}, m_{\Sigma^{+}}\right) \mathcal{F}_{2}\left(\sigma_{2}^{2}, m_{K^{0}}, m_{\overline{p}}\right) \mathcal{H}^\mathrm{LS,decay}_{\overline{\Sigma}(1670)^{-}, 2, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{\Sigma}(1670)^{-}, 1, 1} \mathcal{R}_{2}\left(\sigma_{2}, m_{\overline{\Sigma}(1670)^{-}}, \Gamma_{\overline{\Sigma}(1670)^{-}}\right) d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)}{2}} + \sum_{\lambda_{R}=-3/2}^{3/2}{- \frac{\sqrt{5} \delta_{0, \lambda_{R} + \frac{1}{2}} C^{\frac{1}{2},- \frac{1}{2}}_{0,0,\frac{1}{2},- \frac{1}{2}} C^{1,\lambda_{R} + \frac{1}{2}}_{1,0,1,\lambda_{R} + \frac{1}{2}} C^{1,\lambda_{R} + \frac{1}{2}}_{\frac{3}{2},\lambda_{R},\frac{1}{2},\frac{1}{2}} C^{\frac{3}{2},- \frac{1}{2}}_{2,0,\frac{1}{2},- \frac{1}{2}} \mathcal{F}_{1}\left(m_{0}^{2}, m_{\overline{\Sigma}(1940)^{-}}, m_{\Sigma^{+}}\right) \mathcal{F}_{2}\left(\sigma_{2}^{2}, m_{K^{0}}, m_{\overline{p}}\right) \mathcal{H}^\mathrm{LS,decay}_{\overline{\Sigma}(1940)^{-}, 2, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{\Sigma}(1940)^{-}, 1, 1} \mathcal{R}_{2}\left(\sigma_{2}, m_{\overline{\Sigma}(1940)^{-}}, \Gamma_{\overline{\Sigma}(1940)^{-}}\right) d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)}{2}} \\ A^{3}_{0, 0, - \frac{1}{2}, \frac{1}{2}} &=& \sum_{\lambda_{R}=-3/2}^{3/2}{- \frac{\delta_{0, \lambda_{R} - \frac{1}{2}} C^{\frac{1}{2},\frac{1}{2}}_{0,0,\frac{1}{2},\frac{1}{2}} C^{1,\lambda_{R} - \frac{1}{2}}_{0,0,1,\lambda_{R} - \frac{1}{2}} C^{\frac{3}{2},\frac{1}{2}}_{1,0,\frac{1}{2},\frac{1}{2}} C^{1,\lambda_{R} - \frac{1}{2}}_{\frac{3}{2},\lambda_{R},\frac{1}{2},- \frac{1}{2}} \mathcal{F}_{0}\left(m_{0}^{2}, m_{N(1720)^{+}}, m_{\overline{p}}\right) \mathcal{F}_{1}\left(\sigma_{3}^{2}, m_{K^{0}}, m_{\Sigma^{+}}\right) \mathcal{H}^\mathrm{LS,decay}_{N(1720)^{+}, 1, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{N(1720)^{+}, 0, 1} \mathcal{R}_{1}\left(\sigma_{3}, m_{N(1720)^{+}}, \Gamma_{N(1720)^{+}}\right) d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{12}\right)}{2}} + \sum_{\lambda_{R}=-1/2}^{1/2}{- \frac{\sqrt{2} \delta_{0, \lambda_{R} - \frac{1}{2}} C^{\frac{1}{2},\frac{1}{2}}_{0,0,\frac{1}{2},\frac{1}{2}} C^{1,\lambda_{R} - \frac{1}{2}}_{0,0,1,\lambda_{R} - \frac{1}{2}} C^{1,\lambda_{R} - \frac{1}{2}}_{\frac{1}{2},\lambda_{R},\frac{1}{2},- \frac{1}{2}} C^{\frac{1}{2},\frac{1}{2}}_{1,0,\frac{1}{2},\frac{1}{2}} \mathcal{F}_{0}\left(m_{0}^{2}, m_{N(1710)^{+}}, m_{\overline{p}}\right) \mathcal{F}_{1}\left(\sigma_{3}^{2}, m_{K^{0}}, m_{\Sigma^{+}}\right) \mathcal{H}^\mathrm{LS,decay}_{N(1710)^{+}, 1, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{N(1710)^{+}, 0, 1} \mathcal{R}_{1}\left(\sigma_{3}, m_{N(1710)^{+}}, \Gamma_{N(1710)^{+}}\right) d^{\frac{1}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{12}\right)}{2}} + \sum_{\lambda_{R}=-3/2}^{3/2}{- \frac{\sqrt{5} \delta_{0, \lambda_{R} - \frac{1}{2}} C^{\frac{1}{2},\frac{1}{2}}_{0,0,\frac{1}{2},\frac{1}{2}} C^{1,\lambda_{R} - \frac{1}{2}}_{1,0,1,\lambda_{R} - \frac{1}{2}} C^{1,\lambda_{R} - \frac{1}{2}}_{\frac{3}{2},\lambda_{R},\frac{1}{2},- \frac{1}{2}} C^{\frac{3}{2},\frac{1}{2}}_{2,0,\frac{1}{2},\frac{1}{2}} \mathcal{F}_{1}\left(m_{0}^{2}, m_{N(1700)^{+}}, m_{\overline{p}}\right) \mathcal{F}_{2}\left(\sigma_{3}^{2}, m_{K^{0}}, m_{\Sigma^{+}}\right) \mathcal{H}^\mathrm{LS,decay}_{N(1700)^{+}, 2, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{N(1700)^{+}, 1, 1} \mathcal{R}_{2}\left(\sigma_{3}, m_{N(1700)^{+}}, \Gamma_{N(1700)^{+}}\right) d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{12}\right)}{2}} \\ A^{2}_{0, 0, \frac{1}{2}, - \frac{1}{2}} &=& \sum_{\lambda_{R}=-1/2}^{1/2}{\frac{\sqrt{2} \delta_{0, \lambda_{R} - \frac{1}{2}} \left(C^{\frac{1}{2},\frac{1}{2}}_{0,0,\frac{1}{2},\frac{1}{2}}\right)^{2} C^{0,\lambda_{R} - \frac{1}{2}}_{\frac{1}{2},\lambda_{R},\frac{1}{2},- \frac{1}{2}} C^{1,\lambda_{R} - \frac{1}{2}}_{1,0,0,\lambda_{R} - \frac{1}{2}} \mathcal{F}_{1}\left(m_{0}^{2}, m_{\overline{\Sigma}(1750)^{-}}, m_{\Sigma^{+}}\right) \mathcal{F}_{0}\left(\sigma_{2}^{2}, m_{K^{0}}, m_{\overline{p}}\right) \mathcal{H}^\mathrm{LS,decay}_{\overline{\Sigma}(1750)^{-}, 0, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{\Sigma}(1750)^{-}, 1, 0} \mathcal{R}_{0}\left(\sigma_{2}, m_{\overline{\Sigma}(1750)^{-}}, \Gamma_{\overline{\Sigma}(1750)^{-}}\right) d^{\frac{1}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)}{2}} + \sum_{\lambda_{R}=-1/2}^{1/2}{\frac{\sqrt{2} \delta_{0, \lambda_{R} - \frac{1}{2}} C^{\frac{1}{2},\frac{1}{2}}_{0,0,\frac{1}{2},\frac{1}{2}} C^{1,\lambda_{R} - \frac{1}{2}}_{0,0,1,\lambda_{R} - \frac{1}{2}} C^{1,\lambda_{R} - \frac{1}{2}}_{\frac{1}{2},\lambda_{R},\frac{1}{2},- \frac{1}{2}} C^{\frac{1}{2},\frac{1}{2}}_{1,0,\frac{1}{2},\frac{1}{2}} \mathcal{F}_{0}\left(m_{0}^{2}, m_{\overline{\Sigma}(1660)^{-}}, m_{\Sigma^{+}}\right) \mathcal{F}_{1}\left(\sigma_{2}^{2}, m_{K^{0}}, m_{\overline{p}}\right) \mathcal{H}^\mathrm{LS,decay}_{\overline{\Sigma}(1660)^{-}, 1, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{\Sigma}(1660)^{-}, 0, 1} \mathcal{R}_{1}\left(\sigma_{2}, m_{\overline{\Sigma}(1660)^{-}}, \Gamma_{\overline{\Sigma}(1660)^{-}}\right) d^{\frac{1}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)}{2}} + \sum_{\lambda_{R}=-5/2}^{5/2}{\frac{\sqrt{30} \delta_{0, \lambda_{R} - \frac{1}{2}} C^{\frac{1}{2},\frac{1}{2}}_{0,0,\frac{1}{2},\frac{1}{2}} C^{1,\lambda_{R} - \frac{1}{2}}_{1,0,2,\lambda_{R} - \frac{1}{2}} C^{\frac{5}{2},\frac{1}{2}}_{2,0,\frac{1}{2},\frac{1}{2}} C^{2,\lambda_{R} - \frac{1}{2}}_{\frac{5}{2},\lambda_{R},\frac{1}{2},- \frac{1}{2}} \mathcal{F}_{1}\left(m_{0}^{2}, m_{\overline{\Sigma}(1775)^{-}}, m_{\Sigma^{+}}\right) \mathcal{F}_{2}\left(\sigma_{2}^{2}, m_{K^{0}}, m_{\overline{p}}\right) \mathcal{H}^\mathrm{LS,decay}_{\overline{\Sigma}(1775)^{-}, 2, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{\Sigma}(1775)^{-}, 1, 2} \mathcal{R}_{2}\left(\sigma_{2}, m_{\overline{\Sigma}(1775)^{-}}, \Gamma_{\overline{\Sigma}(1775)^{-}}\right) d^{\frac{5}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)}{6}} + \sum_{\lambda_{R}=-3/2}^{3/2}{\frac{\sqrt{5} \delta_{0, \lambda_{R} - \frac{1}{2}} C^{\frac{1}{2},\frac{1}{2}}_{0,0,\frac{1}{2},\frac{1}{2}} C^{1,\lambda_{R} - \frac{1}{2}}_{1,0,1,\lambda_{R} - \frac{1}{2}} C^{1,\lambda_{R} - \frac{1}{2}}_{\frac{3}{2},\lambda_{R},\frac{1}{2},- \frac{1}{2}} C^{\frac{3}{2},\frac{1}{2}}_{2,0,\frac{1}{2},\frac{1}{2}} \mathcal{F}_{1}\left(m_{0}^{2}, m_{\overline{\Sigma}(1670)^{-}}, m_{\Sigma^{+}}\right) \mathcal{F}_{2}\left(\sigma_{2}^{2}, m_{K^{0}}, m_{\overline{p}}\right) \mathcal{H}^\mathrm{LS,decay}_{\overline{\Sigma}(1670)^{-}, 2, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{\Sigma}(1670)^{-}, 1, 1} \mathcal{R}_{2}\left(\sigma_{2}, m_{\overline{\Sigma}(1670)^{-}}, \Gamma_{\overline{\Sigma}(1670)^{-}}\right) d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)}{2}} + \sum_{\lambda_{R}=-3/2}^{3/2}{\frac{\sqrt{5} \delta_{0, \lambda_{R} - \frac{1}{2}} C^{\frac{1}{2},\frac{1}{2}}_{0,0,\frac{1}{2},\frac{1}{2}} C^{1,\lambda_{R} - \frac{1}{2}}_{1,0,1,\lambda_{R} - \frac{1}{2}} C^{1,\lambda_{R} - \frac{1}{2}}_{\frac{3}{2},\lambda_{R},\frac{1}{2},- \frac{1}{2}} C^{\frac{3}{2},\frac{1}{2}}_{2,0,\frac{1}{2},\frac{1}{2}} \mathcal{F}_{1}\left(m_{0}^{2}, m_{\overline{\Sigma}(1940)^{-}}, m_{\Sigma^{+}}\right) \mathcal{F}_{2}\left(\sigma_{2}^{2}, m_{K^{0}}, m_{\overline{p}}\right) \mathcal{H}^\mathrm{LS,decay}_{\overline{\Sigma}(1940)^{-}, 2, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{\Sigma}(1940)^{-}, 1, 1} \mathcal{R}_{2}\left(\sigma_{2}, m_{\overline{\Sigma}(1940)^{-}}, \Gamma_{\overline{\Sigma}(1940)^{-}}\right) d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)}{2}} \\ A^{3}_{0, 0, \frac{1}{2}, - \frac{1}{2}} &=& \sum_{\lambda_{R}=-3/2}^{3/2}{- \frac{\delta_{0, \lambda_{R} + \frac{1}{2}} C^{\frac{1}{2},- \frac{1}{2}}_{0,0,\frac{1}{2},- \frac{1}{2}} C^{1,\lambda_{R} + \frac{1}{2}}_{0,0,1,\lambda_{R} + \frac{1}{2}} C^{\frac{3}{2},- \frac{1}{2}}_{1,0,\frac{1}{2},- \frac{1}{2}} C^{1,\lambda_{R} + \frac{1}{2}}_{\frac{3}{2},\lambda_{R},\frac{1}{2},\frac{1}{2}} \mathcal{F}_{0}\left(m_{0}^{2}, m_{N(1720)^{+}}, m_{\overline{p}}\right) \mathcal{F}_{1}\left(\sigma_{3}^{2}, m_{K^{0}}, m_{\Sigma^{+}}\right) \mathcal{H}^\mathrm{LS,decay}_{N(1720)^{+}, 1, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{N(1720)^{+}, 0, 1} \mathcal{R}_{1}\left(\sigma_{3}, m_{N(1720)^{+}}, \Gamma_{N(1720)^{+}}\right) d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{12}\right)}{2}} + \sum_{\lambda_{R}=-1/2}^{1/2}{- \frac{\sqrt{2} \delta_{0, \lambda_{R} + \frac{1}{2}} C^{\frac{1}{2},- \frac{1}{2}}_{0,0,\frac{1}{2},- \frac{1}{2}} C^{1,\lambda_{R} + \frac{1}{2}}_{0,0,1,\lambda_{R} + \frac{1}{2}} C^{1,\lambda_{R} + \frac{1}{2}}_{\frac{1}{2},\lambda_{R},\frac{1}{2},\frac{1}{2}} C^{\frac{1}{2},- \frac{1}{2}}_{1,0,\frac{1}{2},- \frac{1}{2}} \mathcal{F}_{0}\left(m_{0}^{2}, m_{N(1710)^{+}}, m_{\overline{p}}\right) \mathcal{F}_{1}\left(\sigma_{3}^{2}, m_{K^{0}}, m_{\Sigma^{+}}\right) \mathcal{H}^\mathrm{LS,decay}_{N(1710)^{+}, 1, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{N(1710)^{+}, 0, 1} \mathcal{R}_{1}\left(\sigma_{3}, m_{N(1710)^{+}}, \Gamma_{N(1710)^{+}}\right) d^{\frac{1}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{12}\right)}{2}} + \sum_{\lambda_{R}=-3/2}^{3/2}{- \frac{\sqrt{5} \delta_{0, \lambda_{R} + \frac{1}{2}} C^{\frac{1}{2},- \frac{1}{2}}_{0,0,\frac{1}{2},- \frac{1}{2}} C^{1,\lambda_{R} + \frac{1}{2}}_{1,0,1,\lambda_{R} + \frac{1}{2}} C^{1,\lambda_{R} + \frac{1}{2}}_{\frac{3}{2},\lambda_{R},\frac{1}{2},\frac{1}{2}} C^{\frac{3}{2},- \frac{1}{2}}_{2,0,\frac{1}{2},- \frac{1}{2}} \mathcal{F}_{1}\left(m_{0}^{2}, m_{N(1700)^{+}}, m_{\overline{p}}\right) \mathcal{F}_{2}\left(\sigma_{3}^{2}, m_{K^{0}}, m_{\Sigma^{+}}\right) \mathcal{H}^\mathrm{LS,decay}_{N(1700)^{+}, 2, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{N(1700)^{+}, 1, 1} \mathcal{R}_{2}\left(\sigma_{3}, m_{N(1700)^{+}}, \Gamma_{N(1700)^{+}}\right) d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{12}\right)}{2}} \\ A^{2}_{0, 0, \frac{1}{2}, \frac{1}{2}} &=& \sum_{\lambda_{R}=-1/2}^{1/2}{\frac{\sqrt{2} \delta_{0, \lambda_{R} - \frac{1}{2}} \left(C^{\frac{1}{2},- \frac{1}{2}}_{0,0,\frac{1}{2},- \frac{1}{2}}\right)^{2} C^{0,\lambda_{R} - \frac{1}{2}}_{\frac{1}{2},\lambda_{R},\frac{1}{2},- \frac{1}{2}} C^{1,\lambda_{R} - \frac{1}{2}}_{1,0,0,\lambda_{R} - \frac{1}{2}} \mathcal{F}_{1}\left(m_{0}^{2}, m_{\overline{\Sigma}(1750)^{-}}, m_{\Sigma^{+}}\right) \mathcal{F}_{0}\left(\sigma_{2}^{2}, m_{K^{0}}, m_{\overline{p}}\right) \mathcal{H}^\mathrm{LS,decay}_{\overline{\Sigma}(1750)^{-}, 0, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{\Sigma}(1750)^{-}, 1, 0} \mathcal{R}_{0}\left(\sigma_{2}, m_{\overline{\Sigma}(1750)^{-}}, \Gamma_{\overline{\Sigma}(1750)^{-}}\right) d^{\frac{1}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)}{2}} + \sum_{\lambda_{R}=-1/2}^{1/2}{\frac{\sqrt{2} \delta_{0, \lambda_{R} - \frac{1}{2}} C^{\frac{1}{2},- \frac{1}{2}}_{0,0,\frac{1}{2},- \frac{1}{2}} C^{1,\lambda_{R} - \frac{1}{2}}_{0,0,1,\lambda_{R} - \frac{1}{2}} C^{1,\lambda_{R} - \frac{1}{2}}_{\frac{1}{2},\lambda_{R},\frac{1}{2},- \frac{1}{2}} C^{\frac{1}{2},- \frac{1}{2}}_{1,0,\frac{1}{2},- \frac{1}{2}} \mathcal{F}_{0}\left(m_{0}^{2}, m_{\overline{\Sigma}(1660)^{-}}, m_{\Sigma^{+}}\right) \mathcal{F}_{1}\left(\sigma_{2}^{2}, m_{K^{0}}, m_{\overline{p}}\right) \mathcal{H}^\mathrm{LS,decay}_{\overline{\Sigma}(1660)^{-}, 1, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{\Sigma}(1660)^{-}, 0, 1} \mathcal{R}_{1}\left(\sigma_{2}, m_{\overline{\Sigma}(1660)^{-}}, \Gamma_{\overline{\Sigma}(1660)^{-}}\right) d^{\frac{1}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)}{2}} + \sum_{\lambda_{R}=-5/2}^{5/2}{\frac{\sqrt{30} \delta_{0, \lambda_{R} - \frac{1}{2}} C^{\frac{1}{2},- \frac{1}{2}}_{0,0,\frac{1}{2},- \frac{1}{2}} C^{1,\lambda_{R} - \frac{1}{2}}_{1,0,2,\lambda_{R} - \frac{1}{2}} C^{\frac{5}{2},- \frac{1}{2}}_{2,0,\frac{1}{2},- \frac{1}{2}} C^{2,\lambda_{R} - \frac{1}{2}}_{\frac{5}{2},\lambda_{R},\frac{1}{2},- \frac{1}{2}} \mathcal{F}_{1}\left(m_{0}^{2}, m_{\overline{\Sigma}(1775)^{-}}, m_{\Sigma^{+}}\right) \mathcal{F}_{2}\left(\sigma_{2}^{2}, m_{K^{0}}, m_{\overline{p}}\right) \mathcal{H}^\mathrm{LS,decay}_{\overline{\Sigma}(1775)^{-}, 2, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{\Sigma}(1775)^{-}, 1, 2} \mathcal{R}_{2}\left(\sigma_{2}, m_{\overline{\Sigma}(1775)^{-}}, \Gamma_{\overline{\Sigma}(1775)^{-}}\right) d^{\frac{5}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)}{6}} + \sum_{\lambda_{R}=-3/2}^{3/2}{\frac{\sqrt{5} \delta_{0, \lambda_{R} - \frac{1}{2}} C^{\frac{1}{2},- \frac{1}{2}}_{0,0,\frac{1}{2},- \frac{1}{2}} C^{1,\lambda_{R} - \frac{1}{2}}_{1,0,1,\lambda_{R} - \frac{1}{2}} C^{1,\lambda_{R} - \frac{1}{2}}_{\frac{3}{2},\lambda_{R},\frac{1}{2},- \frac{1}{2}} C^{\frac{3}{2},- \frac{1}{2}}_{2,0,\frac{1}{2},- \frac{1}{2}} \mathcal{F}_{1}\left(m_{0}^{2}, m_{\overline{\Sigma}(1670)^{-}}, m_{\Sigma^{+}}\right) \mathcal{F}_{2}\left(\sigma_{2}^{2}, m_{K^{0}}, m_{\overline{p}}\right) \mathcal{H}^\mathrm{LS,decay}_{\overline{\Sigma}(1670)^{-}, 2, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{\Sigma}(1670)^{-}, 1, 1} \mathcal{R}_{2}\left(\sigma_{2}, m_{\overline{\Sigma}(1670)^{-}}, \Gamma_{\overline{\Sigma}(1670)^{-}}\right) d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)}{2}} + \sum_{\lambda_{R}=-3/2}^{3/2}{\frac{\sqrt{5} \delta_{0, \lambda_{R} - \frac{1}{2}} C^{\frac{1}{2},- \frac{1}{2}}_{0,0,\frac{1}{2},- \frac{1}{2}} C^{1,\lambda_{R} - \frac{1}{2}}_{1,0,1,\lambda_{R} - \frac{1}{2}} C^{1,\lambda_{R} - \frac{1}{2}}_{\frac{3}{2},\lambda_{R},\frac{1}{2},- \frac{1}{2}} C^{\frac{3}{2},- \frac{1}{2}}_{2,0,\frac{1}{2},- \frac{1}{2}} \mathcal{F}_{1}\left(m_{0}^{2}, m_{\overline{\Sigma}(1940)^{-}}, m_{\Sigma^{+}}\right) \mathcal{F}_{2}\left(\sigma_{2}^{2}, m_{K^{0}}, m_{\overline{p}}\right) \mathcal{H}^\mathrm{LS,decay}_{\overline{\Sigma}(1940)^{-}, 2, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{\Sigma}(1940)^{-}, 1, 1} \mathcal{R}_{2}\left(\sigma_{2}, m_{\overline{\Sigma}(1940)^{-}}, \Gamma_{\overline{\Sigma}(1940)^{-}}\right) d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)}{2}} \\ A^{3}_{0, 0, \frac{1}{2}, \frac{1}{2}} &=& \sum_{\lambda_{R}=-3/2}^{3/2}{\frac{\delta_{0, \lambda_{R} - \frac{1}{2}} C^{\frac{1}{2},- \frac{1}{2}}_{0,0,\frac{1}{2},- \frac{1}{2}} C^{1,\lambda_{R} - \frac{1}{2}}_{0,0,1,\lambda_{R} - \frac{1}{2}} C^{\frac{3}{2},- \frac{1}{2}}_{1,0,\frac{1}{2},- \frac{1}{2}} C^{1,\lambda_{R} - \frac{1}{2}}_{\frac{3}{2},\lambda_{R},\frac{1}{2},- \frac{1}{2}} \mathcal{F}_{0}\left(m_{0}^{2}, m_{N(1720)^{+}}, m_{\overline{p}}\right) \mathcal{F}_{1}\left(\sigma_{3}^{2}, m_{K^{0}}, m_{\Sigma^{+}}\right) \mathcal{H}^\mathrm{LS,decay}_{N(1720)^{+}, 1, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{N(1720)^{+}, 0, 1} \mathcal{R}_{1}\left(\sigma_{3}, m_{N(1720)^{+}}, \Gamma_{N(1720)^{+}}\right) d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{12}\right)}{2}} + \sum_{\lambda_{R}=-1/2}^{1/2}{\frac{\sqrt{2} \delta_{0, \lambda_{R} - \frac{1}{2}} C^{\frac{1}{2},- \frac{1}{2}}_{0,0,\frac{1}{2},- \frac{1}{2}} C^{1,\lambda_{R} - \frac{1}{2}}_{0,0,1,\lambda_{R} - \frac{1}{2}} C^{1,\lambda_{R} - \frac{1}{2}}_{\frac{1}{2},\lambda_{R},\frac{1}{2},- \frac{1}{2}} C^{\frac{1}{2},- \frac{1}{2}}_{1,0,\frac{1}{2},- \frac{1}{2}} \mathcal{F}_{0}\left(m_{0}^{2}, m_{N(1710)^{+}}, m_{\overline{p}}\right) \mathcal{F}_{1}\left(\sigma_{3}^{2}, m_{K^{0}}, m_{\Sigma^{+}}\right) \mathcal{H}^\mathrm{LS,decay}_{N(1710)^{+}, 1, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{N(1710)^{+}, 0, 1} \mathcal{R}_{1}\left(\sigma_{3}, m_{N(1710)^{+}}, \Gamma_{N(1710)^{+}}\right) d^{\frac{1}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{12}\right)}{2}} + \sum_{\lambda_{R}=-3/2}^{3/2}{\frac{\sqrt{5} \delta_{0, \lambda_{R} - \frac{1}{2}} C^{\frac{1}{2},- \frac{1}{2}}_{0,0,\frac{1}{2},- \frac{1}{2}} C^{1,\lambda_{R} - \frac{1}{2}}_{1,0,1,\lambda_{R} - \frac{1}{2}} C^{1,\lambda_{R} - \frac{1}{2}}_{\frac{3}{2},\lambda_{R},\frac{1}{2},- \frac{1}{2}} C^{\frac{3}{2},- \frac{1}{2}}_{2,0,\frac{1}{2},- \frac{1}{2}} \mathcal{F}_{1}\left(m_{0}^{2}, m_{N(1700)^{+}}, m_{\overline{p}}\right) \mathcal{F}_{2}\left(\sigma_{3}^{2}, m_{K^{0}}, m_{\Sigma^{+}}\right) \mathcal{H}^\mathrm{LS,decay}_{N(1700)^{+}, 2, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{N(1700)^{+}, 1, 1} \mathcal{R}_{2}\left(\sigma_{3}, m_{N(1700)^{+}}, \Gamma_{N(1700)^{+}}\right) d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{12}\right)}{2}} \\ A^{2}_{1, 0, - \frac{1}{2}, - \frac{1}{2}} &=& \sum_{\lambda_{R}=-1/2}^{1/2}{- \frac{\sqrt{2} \delta_{1, \lambda_{R} + \frac{1}{2}} \left(C^{\frac{1}{2},\frac{1}{2}}_{0,0,\frac{1}{2},\frac{1}{2}}\right)^{2} C^{0,\lambda_{R} + \frac{1}{2}}_{\frac{1}{2},\lambda_{R},\frac{1}{2},\frac{1}{2}} C^{1,\lambda_{R} + \frac{1}{2}}_{1,0,0,\lambda_{R} + \frac{1}{2}} \mathcal{F}_{1}\left(m_{0}^{2}, m_{\overline{\Sigma}(1750)^{-}}, m_{\Sigma^{+}}\right) \mathcal{F}_{0}\left(\sigma_{2}^{2}, m_{K^{0}}, m_{\overline{p}}\right) \mathcal{H}^\mathrm{LS,decay}_{\overline{\Sigma}(1750)^{-}, 0, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{\Sigma}(1750)^{-}, 1, 0} \mathcal{R}_{0}\left(\sigma_{2}, m_{\overline{\Sigma}(1750)^{-}}, \Gamma_{\overline{\Sigma}(1750)^{-}}\right) d^{\frac{1}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)}{2}} + \sum_{\lambda_{R}=-1/2}^{1/2}{- \frac{\sqrt{2} \delta_{1, \lambda_{R} + \frac{1}{2}} C^{\frac{1}{2},\frac{1}{2}}_{0,0,\frac{1}{2},\frac{1}{2}} C^{1,\lambda_{R} + \frac{1}{2}}_{0,0,1,\lambda_{R} + \frac{1}{2}} C^{1,\lambda_{R} + \frac{1}{2}}_{\frac{1}{2},\lambda_{R},\frac{1}{2},\frac{1}{2}} C^{\frac{1}{2},\frac{1}{2}}_{1,0,\frac{1}{2},\frac{1}{2}} \mathcal{F}_{0}\left(m_{0}^{2}, m_{\overline{\Sigma}(1660)^{-}}, m_{\Sigma^{+}}\right) \mathcal{F}_{1}\left(\sigma_{2}^{2}, m_{K^{0}}, m_{\overline{p}}\right) \mathcal{H}^\mathrm{LS,decay}_{\overline{\Sigma}(1660)^{-}, 1, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{\Sigma}(1660)^{-}, 0, 1} \mathcal{R}_{1}\left(\sigma_{2}, m_{\overline{\Sigma}(1660)^{-}}, \Gamma_{\overline{\Sigma}(1660)^{-}}\right) d^{\frac{1}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)}{2}} + \sum_{\lambda_{R}=-5/2}^{5/2}{- \frac{\sqrt{30} \delta_{1, \lambda_{R} + \frac{1}{2}} C^{\frac{1}{2},\frac{1}{2}}_{0,0,\frac{1}{2},\frac{1}{2}} C^{1,\lambda_{R} + \frac{1}{2}}_{1,0,2,\lambda_{R} + \frac{1}{2}} C^{\frac{5}{2},\frac{1}{2}}_{2,0,\frac{1}{2},\frac{1}{2}} C^{2,\lambda_{R} + \frac{1}{2}}_{\frac{5}{2},\lambda_{R},\frac{1}{2},\frac{1}{2}} \mathcal{F}_{1}\left(m_{0}^{2}, m_{\overline{\Sigma}(1775)^{-}}, m_{\Sigma^{+}}\right) \mathcal{F}_{2}\left(\sigma_{2}^{2}, m_{K^{0}}, m_{\overline{p}}\right) \mathcal{H}^\mathrm{LS,decay}_{\overline{\Sigma}(1775)^{-}, 2, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{\Sigma}(1775)^{-}, 1, 2} \mathcal{R}_{2}\left(\sigma_{2}, m_{\overline{\Sigma}(1775)^{-}}, \Gamma_{\overline{\Sigma}(1775)^{-}}\right) d^{\frac{5}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)}{6}} + \sum_{\lambda_{R}=-3/2}^{3/2}{- \frac{\sqrt{5} \delta_{1, \lambda_{R} + \frac{1}{2}} C^{\frac{1}{2},\frac{1}{2}}_{0,0,\frac{1}{2},\frac{1}{2}} C^{1,\lambda_{R} + \frac{1}{2}}_{1,0,1,\lambda_{R} + \frac{1}{2}} C^{1,\lambda_{R} + \frac{1}{2}}_{\frac{3}{2},\lambda_{R},\frac{1}{2},\frac{1}{2}} C^{\frac{3}{2},\frac{1}{2}}_{2,0,\frac{1}{2},\frac{1}{2}} \mathcal{F}_{1}\left(m_{0}^{2}, m_{\overline{\Sigma}(1670)^{-}}, m_{\Sigma^{+}}\right) \mathcal{F}_{2}\left(\sigma_{2}^{2}, m_{K^{0}}, m_{\overline{p}}\right) \mathcal{H}^\mathrm{LS,decay}_{\overline{\Sigma}(1670)^{-}, 2, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{\Sigma}(1670)^{-}, 1, 1} \mathcal{R}_{2}\left(\sigma_{2}, m_{\overline{\Sigma}(1670)^{-}}, \Gamma_{\overline{\Sigma}(1670)^{-}}\right) d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)}{2}} + \sum_{\lambda_{R}=-3/2}^{3/2}{- \frac{\sqrt{5} \delta_{1, \lambda_{R} + \frac{1}{2}} C^{\frac{1}{2},\frac{1}{2}}_{0,0,\frac{1}{2},\frac{1}{2}} C^{1,\lambda_{R} + \frac{1}{2}}_{1,0,1,\lambda_{R} + \frac{1}{2}} C^{1,\lambda_{R} + \frac{1}{2}}_{\frac{3}{2},\lambda_{R},\frac{1}{2},\frac{1}{2}} C^{\frac{3}{2},\frac{1}{2}}_{2,0,\frac{1}{2},\frac{1}{2}} \mathcal{F}_{1}\left(m_{0}^{2}, m_{\overline{\Sigma}(1940)^{-}}, m_{\Sigma^{+}}\right) \mathcal{F}_{2}\left(\sigma_{2}^{2}, m_{K^{0}}, m_{\overline{p}}\right) \mathcal{H}^\mathrm{LS,decay}_{\overline{\Sigma}(1940)^{-}, 2, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{\Sigma}(1940)^{-}, 1, 1} \mathcal{R}_{2}\left(\sigma_{2}, m_{\overline{\Sigma}(1940)^{-}}, \Gamma_{\overline{\Sigma}(1940)^{-}}\right) d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)}{2}} \\ A^{3}_{1, 0, - \frac{1}{2}, - \frac{1}{2}} &=& \sum_{\lambda_{R}=-3/2}^{3/2}{\frac{\delta_{1, \lambda_{R} + \frac{1}{2}} C^{\frac{1}{2},\frac{1}{2}}_{0,0,\frac{1}{2},\frac{1}{2}} C^{1,\lambda_{R} + \frac{1}{2}}_{0,0,1,\lambda_{R} + \frac{1}{2}} C^{\frac{3}{2},\frac{1}{2}}_{1,0,\frac{1}{2},\frac{1}{2}} C^{1,\lambda_{R} + \frac{1}{2}}_{\frac{3}{2},\lambda_{R},\frac{1}{2},\frac{1}{2}} \mathcal{F}_{0}\left(m_{0}^{2}, m_{N(1720)^{+}}, m_{\overline{p}}\right) \mathcal{F}_{1}\left(\sigma_{3}^{2}, m_{K^{0}}, m_{\Sigma^{+}}\right) \mathcal{H}^\mathrm{LS,decay}_{N(1720)^{+}, 1, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{N(1720)^{+}, 0, 1} \mathcal{R}_{1}\left(\sigma_{3}, m_{N(1720)^{+}}, \Gamma_{N(1720)^{+}}\right) d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{12}\right)}{2}} + \sum_{\lambda_{R}=-1/2}^{1/2}{\frac{\sqrt{2} \delta_{1, \lambda_{R} + \frac{1}{2}} C^{\frac{1}{2},\frac{1}{2}}_{0,0,\frac{1}{2},\frac{1}{2}} C^{1,\lambda_{R} + \frac{1}{2}}_{0,0,1,\lambda_{R} + \frac{1}{2}} C^{1,\lambda_{R} + \frac{1}{2}}_{\frac{1}{2},\lambda_{R},\frac{1}{2},\frac{1}{2}} C^{\frac{1}{2},\frac{1}{2}}_{1,0,\frac{1}{2},\frac{1}{2}} \mathcal{F}_{0}\left(m_{0}^{2}, m_{N(1710)^{+}}, m_{\overline{p}}\right) \mathcal{F}_{1}\left(\sigma_{3}^{2}, m_{K^{0}}, m_{\Sigma^{+}}\right) \mathcal{H}^\mathrm{LS,decay}_{N(1710)^{+}, 1, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{N(1710)^{+}, 0, 1} \mathcal{R}_{1}\left(\sigma_{3}, m_{N(1710)^{+}}, \Gamma_{N(1710)^{+}}\right) d^{\frac{1}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{12}\right)}{2}} + \sum_{\lambda_{R}=-3/2}^{3/2}{\frac{\sqrt{5} \delta_{1, \lambda_{R} + \frac{1}{2}} C^{\frac{1}{2},\frac{1}{2}}_{0,0,\frac{1}{2},\frac{1}{2}} C^{1,\lambda_{R} + \frac{1}{2}}_{1,0,1,\lambda_{R} + \frac{1}{2}} C^{1,\lambda_{R} + \frac{1}{2}}_{\frac{3}{2},\lambda_{R},\frac{1}{2},\frac{1}{2}} C^{\frac{3}{2},\frac{1}{2}}_{2,0,\frac{1}{2},\frac{1}{2}} \mathcal{F}_{1}\left(m_{0}^{2}, m_{N(1700)^{+}}, m_{\overline{p}}\right) \mathcal{F}_{2}\left(\sigma_{3}^{2}, m_{K^{0}}, m_{\Sigma^{+}}\right) \mathcal{H}^\mathrm{LS,decay}_{N(1700)^{+}, 2, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{N(1700)^{+}, 1, 1} \mathcal{R}_{2}\left(\sigma_{3}, m_{N(1700)^{+}}, \Gamma_{N(1700)^{+}}\right) d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{12}\right)}{2}} \\ A^{2}_{1, 0, - \frac{1}{2}, \frac{1}{2}} &=& \sum_{\lambda_{R}=-1/2}^{1/2}{- \frac{\sqrt{2} \delta_{1, \lambda_{R} + \frac{1}{2}} \left(C^{\frac{1}{2},- \frac{1}{2}}_{0,0,\frac{1}{2},- \frac{1}{2}}\right)^{2} C^{0,\lambda_{R} + \frac{1}{2}}_{\frac{1}{2},\lambda_{R},\frac{1}{2},\frac{1}{2}} C^{1,\lambda_{R} + \frac{1}{2}}_{1,0,0,\lambda_{R} + \frac{1}{2}} \mathcal{F}_{1}\left(m_{0}^{2}, m_{\overline{\Sigma}(1750)^{-}}, m_{\Sigma^{+}}\right) \mathcal{F}_{0}\left(\sigma_{2}^{2}, m_{K^{0}}, m_{\overline{p}}\right) \mathcal{H}^\mathrm{LS,decay}_{\overline{\Sigma}(1750)^{-}, 0, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{\Sigma}(1750)^{-}, 1, 0} \mathcal{R}_{0}\left(\sigma_{2}, m_{\overline{\Sigma}(1750)^{-}}, \Gamma_{\overline{\Sigma}(1750)^{-}}\right) d^{\frac{1}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)}{2}} + \sum_{\lambda_{R}=-1/2}^{1/2}{- \frac{\sqrt{2} \delta_{1, \lambda_{R} + \frac{1}{2}} C^{\frac{1}{2},- \frac{1}{2}}_{0,0,\frac{1}{2},- \frac{1}{2}} C^{1,\lambda_{R} + \frac{1}{2}}_{0,0,1,\lambda_{R} + \frac{1}{2}} C^{1,\lambda_{R} + \frac{1}{2}}_{\frac{1}{2},\lambda_{R},\frac{1}{2},\frac{1}{2}} C^{\frac{1}{2},- \frac{1}{2}}_{1,0,\frac{1}{2},- \frac{1}{2}} \mathcal{F}_{0}\left(m_{0}^{2}, m_{\overline{\Sigma}(1660)^{-}}, m_{\Sigma^{+}}\right) \mathcal{F}_{1}\left(\sigma_{2}^{2}, m_{K^{0}}, m_{\overline{p}}\right) \mathcal{H}^\mathrm{LS,decay}_{\overline{\Sigma}(1660)^{-}, 1, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{\Sigma}(1660)^{-}, 0, 1} \mathcal{R}_{1}\left(\sigma_{2}, m_{\overline{\Sigma}(1660)^{-}}, \Gamma_{\overline{\Sigma}(1660)^{-}}\right) d^{\frac{1}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)}{2}} + \sum_{\lambda_{R}=-5/2}^{5/2}{- \frac{\sqrt{30} \delta_{1, \lambda_{R} + \frac{1}{2}} C^{\frac{1}{2},- \frac{1}{2}}_{0,0,\frac{1}{2},- \frac{1}{2}} C^{1,\lambda_{R} + \frac{1}{2}}_{1,0,2,\lambda_{R} + \frac{1}{2}} C^{\frac{5}{2},- \frac{1}{2}}_{2,0,\frac{1}{2},- \frac{1}{2}} C^{2,\lambda_{R} + \frac{1}{2}}_{\frac{5}{2},\lambda_{R},\frac{1}{2},\frac{1}{2}} \mathcal{F}_{1}\left(m_{0}^{2}, m_{\overline{\Sigma}(1775)^{-}}, m_{\Sigma^{+}}\right) \mathcal{F}_{2}\left(\sigma_{2}^{2}, m_{K^{0}}, m_{\overline{p}}\right) \mathcal{H}^\mathrm{LS,decay}_{\overline{\Sigma}(1775)^{-}, 2, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{\Sigma}(1775)^{-}, 1, 2} \mathcal{R}_{2}\left(\sigma_{2}, m_{\overline{\Sigma}(1775)^{-}}, \Gamma_{\overline{\Sigma}(1775)^{-}}\right) d^{\frac{5}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)}{6}} + \sum_{\lambda_{R}=-3/2}^{3/2}{- \frac{\sqrt{5} \delta_{1, \lambda_{R} + \frac{1}{2}} C^{\frac{1}{2},- \frac{1}{2}}_{0,0,\frac{1}{2},- \frac{1}{2}} C^{1,\lambda_{R} + \frac{1}{2}}_{1,0,1,\lambda_{R} + \frac{1}{2}} C^{1,\lambda_{R} + \frac{1}{2}}_{\frac{3}{2},\lambda_{R},\frac{1}{2},\frac{1}{2}} C^{\frac{3}{2},- \frac{1}{2}}_{2,0,\frac{1}{2},- \frac{1}{2}} \mathcal{F}_{1}\left(m_{0}^{2}, m_{\overline{\Sigma}(1670)^{-}}, m_{\Sigma^{+}}\right) \mathcal{F}_{2}\left(\sigma_{2}^{2}, m_{K^{0}}, m_{\overline{p}}\right) \mathcal{H}^\mathrm{LS,decay}_{\overline{\Sigma}(1670)^{-}, 2, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{\Sigma}(1670)^{-}, 1, 1} \mathcal{R}_{2}\left(\sigma_{2}, m_{\overline{\Sigma}(1670)^{-}}, \Gamma_{\overline{\Sigma}(1670)^{-}}\right) d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)}{2}} + \sum_{\lambda_{R}=-3/2}^{3/2}{- \frac{\sqrt{5} \delta_{1, \lambda_{R} + \frac{1}{2}} C^{\frac{1}{2},- \frac{1}{2}}_{0,0,\frac{1}{2},- \frac{1}{2}} C^{1,\lambda_{R} + \frac{1}{2}}_{1,0,1,\lambda_{R} + \frac{1}{2}} C^{1,\lambda_{R} + \frac{1}{2}}_{\frac{3}{2},\lambda_{R},\frac{1}{2},\frac{1}{2}} C^{\frac{3}{2},- \frac{1}{2}}_{2,0,\frac{1}{2},- \frac{1}{2}} \mathcal{F}_{1}\left(m_{0}^{2}, m_{\overline{\Sigma}(1940)^{-}}, m_{\Sigma^{+}}\right) \mathcal{F}_{2}\left(\sigma_{2}^{2}, m_{K^{0}}, m_{\overline{p}}\right) \mathcal{H}^\mathrm{LS,decay}_{\overline{\Sigma}(1940)^{-}, 2, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{\Sigma}(1940)^{-}, 1, 1} \mathcal{R}_{2}\left(\sigma_{2}, m_{\overline{\Sigma}(1940)^{-}}, \Gamma_{\overline{\Sigma}(1940)^{-}}\right) d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)}{2}} \\ A^{3}_{1, 0, - \frac{1}{2}, \frac{1}{2}} &=& \sum_{\lambda_{R}=-3/2}^{3/2}{- \frac{\delta_{1, \lambda_{R} - \frac{1}{2}} C^{\frac{1}{2},\frac{1}{2}}_{0,0,\frac{1}{2},\frac{1}{2}} C^{1,\lambda_{R} - \frac{1}{2}}_{0,0,1,\lambda_{R} - \frac{1}{2}} C^{\frac{3}{2},\frac{1}{2}}_{1,0,\frac{1}{2},\frac{1}{2}} C^{1,\lambda_{R} - \frac{1}{2}}_{\frac{3}{2},\lambda_{R},\frac{1}{2},- \frac{1}{2}} \mathcal{F}_{0}\left(m_{0}^{2}, m_{N(1720)^{+}}, m_{\overline{p}}\right) \mathcal{F}_{1}\left(\sigma_{3}^{2}, m_{K^{0}}, m_{\Sigma^{+}}\right) \mathcal{H}^\mathrm{LS,decay}_{N(1720)^{+}, 1, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{N(1720)^{+}, 0, 1} \mathcal{R}_{1}\left(\sigma_{3}, m_{N(1720)^{+}}, \Gamma_{N(1720)^{+}}\right) d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{12}\right)}{2}} + \sum_{\lambda_{R}=-1/2}^{1/2}{- \frac{\sqrt{2} \delta_{1, \lambda_{R} - \frac{1}{2}} C^{\frac{1}{2},\frac{1}{2}}_{0,0,\frac{1}{2},\frac{1}{2}} C^{1,\lambda_{R} - \frac{1}{2}}_{0,0,1,\lambda_{R} - \frac{1}{2}} C^{1,\lambda_{R} - \frac{1}{2}}_{\frac{1}{2},\lambda_{R},\frac{1}{2},- \frac{1}{2}} C^{\frac{1}{2},\frac{1}{2}}_{1,0,\frac{1}{2},\frac{1}{2}} \mathcal{F}_{0}\left(m_{0}^{2}, m_{N(1710)^{+}}, m_{\overline{p}}\right) \mathcal{F}_{1}\left(\sigma_{3}^{2}, m_{K^{0}}, m_{\Sigma^{+}}\right) \mathcal{H}^\mathrm{LS,decay}_{N(1710)^{+}, 1, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{N(1710)^{+}, 0, 1} \mathcal{R}_{1}\left(\sigma_{3}, m_{N(1710)^{+}}, \Gamma_{N(1710)^{+}}\right) d^{\frac{1}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{12}\right)}{2}} + \sum_{\lambda_{R}=-3/2}^{3/2}{- \frac{\sqrt{5} \delta_{1, \lambda_{R} - \frac{1}{2}} C^{\frac{1}{2},\frac{1}{2}}_{0,0,\frac{1}{2},\frac{1}{2}} C^{1,\lambda_{R} - \frac{1}{2}}_{1,0,1,\lambda_{R} - \frac{1}{2}} C^{1,\lambda_{R} - \frac{1}{2}}_{\frac{3}{2},\lambda_{R},\frac{1}{2},- \frac{1}{2}} C^{\frac{3}{2},\frac{1}{2}}_{2,0,\frac{1}{2},\frac{1}{2}} \mathcal{F}_{1}\left(m_{0}^{2}, m_{N(1700)^{+}}, m_{\overline{p}}\right) \mathcal{F}_{2}\left(\sigma_{3}^{2}, m_{K^{0}}, m_{\Sigma^{+}}\right) \mathcal{H}^\mathrm{LS,decay}_{N(1700)^{+}, 2, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{N(1700)^{+}, 1, 1} \mathcal{R}_{2}\left(\sigma_{3}, m_{N(1700)^{+}}, \Gamma_{N(1700)^{+}}\right) d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{12}\right)}{2}} \\ A^{2}_{1, 0, \frac{1}{2}, - \frac{1}{2}} &=& \sum_{\lambda_{R}=-1/2}^{1/2}{\frac{\sqrt{2} \delta_{1, \lambda_{R} - \frac{1}{2}} \left(C^{\frac{1}{2},\frac{1}{2}}_{0,0,\frac{1}{2},\frac{1}{2}}\right)^{2} C^{0,\lambda_{R} - \frac{1}{2}}_{\frac{1}{2},\lambda_{R},\frac{1}{2},- \frac{1}{2}} C^{1,\lambda_{R} - \frac{1}{2}}_{1,0,0,\lambda_{R} - \frac{1}{2}} \mathcal{F}_{1}\left(m_{0}^{2}, m_{\overline{\Sigma}(1750)^{-}}, m_{\Sigma^{+}}\right) \mathcal{F}_{0}\left(\sigma_{2}^{2}, m_{K^{0}}, m_{\overline{p}}\right) \mathcal{H}^\mathrm{LS,decay}_{\overline{\Sigma}(1750)^{-}, 0, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{\Sigma}(1750)^{-}, 1, 0} \mathcal{R}_{0}\left(\sigma_{2}, m_{\overline{\Sigma}(1750)^{-}}, \Gamma_{\overline{\Sigma}(1750)^{-}}\right) d^{\frac{1}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)}{2}} + \sum_{\lambda_{R}=-1/2}^{1/2}{\frac{\sqrt{2} \delta_{1, \lambda_{R} - \frac{1}{2}} C^{\frac{1}{2},\frac{1}{2}}_{0,0,\frac{1}{2},\frac{1}{2}} C^{1,\lambda_{R} - \frac{1}{2}}_{0,0,1,\lambda_{R} - \frac{1}{2}} C^{1,\lambda_{R} - \frac{1}{2}}_{\frac{1}{2},\lambda_{R},\frac{1}{2},- \frac{1}{2}} C^{\frac{1}{2},\frac{1}{2}}_{1,0,\frac{1}{2},\frac{1}{2}} \mathcal{F}_{0}\left(m_{0}^{2}, m_{\overline{\Sigma}(1660)^{-}}, m_{\Sigma^{+}}\right) \mathcal{F}_{1}\left(\sigma_{2}^{2}, m_{K^{0}}, m_{\overline{p}}\right) \mathcal{H}^\mathrm{LS,decay}_{\overline{\Sigma}(1660)^{-}, 1, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{\Sigma}(1660)^{-}, 0, 1} \mathcal{R}_{1}\left(\sigma_{2}, m_{\overline{\Sigma}(1660)^{-}}, \Gamma_{\overline{\Sigma}(1660)^{-}}\right) d^{\frac{1}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)}{2}} + \sum_{\lambda_{R}=-5/2}^{5/2}{\frac{\sqrt{30} \delta_{1, \lambda_{R} - \frac{1}{2}} C^{\frac{1}{2},\frac{1}{2}}_{0,0,\frac{1}{2},\frac{1}{2}} C^{1,\lambda_{R} - \frac{1}{2}}_{1,0,2,\lambda_{R} - \frac{1}{2}} C^{\frac{5}{2},\frac{1}{2}}_{2,0,\frac{1}{2},\frac{1}{2}} C^{2,\lambda_{R} - \frac{1}{2}}_{\frac{5}{2},\lambda_{R},\frac{1}{2},- \frac{1}{2}} \mathcal{F}_{1}\left(m_{0}^{2}, m_{\overline{\Sigma}(1775)^{-}}, m_{\Sigma^{+}}\right) \mathcal{F}_{2}\left(\sigma_{2}^{2}, m_{K^{0}}, m_{\overline{p}}\right) \mathcal{H}^\mathrm{LS,decay}_{\overline{\Sigma}(1775)^{-}, 2, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{\Sigma}(1775)^{-}, 1, 2} \mathcal{R}_{2}\left(\sigma_{2}, m_{\overline{\Sigma}(1775)^{-}}, \Gamma_{\overline{\Sigma}(1775)^{-}}\right) d^{\frac{5}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)}{6}} + \sum_{\lambda_{R}=-3/2}^{3/2}{\frac{\sqrt{5} \delta_{1, \lambda_{R} - \frac{1}{2}} C^{\frac{1}{2},\frac{1}{2}}_{0,0,\frac{1}{2},\frac{1}{2}} C^{1,\lambda_{R} - \frac{1}{2}}_{1,0,1,\lambda_{R} - \frac{1}{2}} C^{1,\lambda_{R} - \frac{1}{2}}_{\frac{3}{2},\lambda_{R},\frac{1}{2},- \frac{1}{2}} C^{\frac{3}{2},\frac{1}{2}}_{2,0,\frac{1}{2},\frac{1}{2}} \mathcal{F}_{1}\left(m_{0}^{2}, m_{\overline{\Sigma}(1670)^{-}}, m_{\Sigma^{+}}\right) \mathcal{F}_{2}\left(\sigma_{2}^{2}, m_{K^{0}}, m_{\overline{p}}\right) \mathcal{H}^\mathrm{LS,decay}_{\overline{\Sigma}(1670)^{-}, 2, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{\Sigma}(1670)^{-}, 1, 1} \mathcal{R}_{2}\left(\sigma_{2}, m_{\overline{\Sigma}(1670)^{-}}, \Gamma_{\overline{\Sigma}(1670)^{-}}\right) d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)}{2}} + \sum_{\lambda_{R}=-3/2}^{3/2}{\frac{\sqrt{5} \delta_{1, \lambda_{R} - \frac{1}{2}} C^{\frac{1}{2},\frac{1}{2}}_{0,0,\frac{1}{2},\frac{1}{2}} C^{1,\lambda_{R} - \frac{1}{2}}_{1,0,1,\lambda_{R} - \frac{1}{2}} C^{1,\lambda_{R} - \frac{1}{2}}_{\frac{3}{2},\lambda_{R},\frac{1}{2},- \frac{1}{2}} C^{\frac{3}{2},\frac{1}{2}}_{2,0,\frac{1}{2},\frac{1}{2}} \mathcal{F}_{1}\left(m_{0}^{2}, m_{\overline{\Sigma}(1940)^{-}}, m_{\Sigma^{+}}\right) \mathcal{F}_{2}\left(\sigma_{2}^{2}, m_{K^{0}}, m_{\overline{p}}\right) \mathcal{H}^\mathrm{LS,decay}_{\overline{\Sigma}(1940)^{-}, 2, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{\Sigma}(1940)^{-}, 1, 1} \mathcal{R}_{2}\left(\sigma_{2}, m_{\overline{\Sigma}(1940)^{-}}, \Gamma_{\overline{\Sigma}(1940)^{-}}\right) d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)}{2}} \\ A^{3}_{1, 0, \frac{1}{2}, - \frac{1}{2}} &=& \sum_{\lambda_{R}=-3/2}^{3/2}{- \frac{\delta_{1, \lambda_{R} + \frac{1}{2}} C^{\frac{1}{2},- \frac{1}{2}}_{0,0,\frac{1}{2},- \frac{1}{2}} C^{1,\lambda_{R} + \frac{1}{2}}_{0,0,1,\lambda_{R} + \frac{1}{2}} C^{\frac{3}{2},- \frac{1}{2}}_{1,0,\frac{1}{2},- \frac{1}{2}} C^{1,\lambda_{R} + \frac{1}{2}}_{\frac{3}{2},\lambda_{R},\frac{1}{2},\frac{1}{2}} \mathcal{F}_{0}\left(m_{0}^{2}, m_{N(1720)^{+}}, m_{\overline{p}}\right) \mathcal{F}_{1}\left(\sigma_{3}^{2}, m_{K^{0}}, m_{\Sigma^{+}}\right) \mathcal{H}^\mathrm{LS,decay}_{N(1720)^{+}, 1, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{N(1720)^{+}, 0, 1} \mathcal{R}_{1}\left(\sigma_{3}, m_{N(1720)^{+}}, \Gamma_{N(1720)^{+}}\right) d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{12}\right)}{2}} + \sum_{\lambda_{R}=-1/2}^{1/2}{- \frac{\sqrt{2} \delta_{1, \lambda_{R} + \frac{1}{2}} C^{\frac{1}{2},- \frac{1}{2}}_{0,0,\frac{1}{2},- \frac{1}{2}} C^{1,\lambda_{R} + \frac{1}{2}}_{0,0,1,\lambda_{R} + \frac{1}{2}} C^{1,\lambda_{R} + \frac{1}{2}}_{\frac{1}{2},\lambda_{R},\frac{1}{2},\frac{1}{2}} C^{\frac{1}{2},- \frac{1}{2}}_{1,0,\frac{1}{2},- \frac{1}{2}} \mathcal{F}_{0}\left(m_{0}^{2}, m_{N(1710)^{+}}, m_{\overline{p}}\right) \mathcal{F}_{1}\left(\sigma_{3}^{2}, m_{K^{0}}, m_{\Sigma^{+}}\right) \mathcal{H}^\mathrm{LS,decay}_{N(1710)^{+}, 1, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{N(1710)^{+}, 0, 1} \mathcal{R}_{1}\left(\sigma_{3}, m_{N(1710)^{+}}, \Gamma_{N(1710)^{+}}\right) d^{\frac{1}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{12}\right)}{2}} + \sum_{\lambda_{R}=-3/2}^{3/2}{- \frac{\sqrt{5} \delta_{1, \lambda_{R} + \frac{1}{2}} C^{\frac{1}{2},- \frac{1}{2}}_{0,0,\frac{1}{2},- \frac{1}{2}} C^{1,\lambda_{R} + \frac{1}{2}}_{1,0,1,\lambda_{R} + \frac{1}{2}} C^{1,\lambda_{R} + \frac{1}{2}}_{\frac{3}{2},\lambda_{R},\frac{1}{2},\frac{1}{2}} C^{\frac{3}{2},- \frac{1}{2}}_{2,0,\frac{1}{2},- \frac{1}{2}} \mathcal{F}_{1}\left(m_{0}^{2}, m_{N(1700)^{+}}, m_{\overline{p}}\right) \mathcal{F}_{2}\left(\sigma_{3}^{2}, m_{K^{0}}, m_{\Sigma^{+}}\right) \mathcal{H}^\mathrm{LS,decay}_{N(1700)^{+}, 2, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{N(1700)^{+}, 1, 1} \mathcal{R}_{2}\left(\sigma_{3}, m_{N(1700)^{+}}, \Gamma_{N(1700)^{+}}\right) d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{12}\right)}{2}} \\ A^{2}_{1, 0, \frac{1}{2}, \frac{1}{2}} &=& \sum_{\lambda_{R}=-1/2}^{1/2}{\frac{\sqrt{2} \delta_{1, \lambda_{R} - \frac{1}{2}} \left(C^{\frac{1}{2},- \frac{1}{2}}_{0,0,\frac{1}{2},- \frac{1}{2}}\right)^{2} C^{0,\lambda_{R} - \frac{1}{2}}_{\frac{1}{2},\lambda_{R},\frac{1}{2},- \frac{1}{2}} C^{1,\lambda_{R} - \frac{1}{2}}_{1,0,0,\lambda_{R} - \frac{1}{2}} \mathcal{F}_{1}\left(m_{0}^{2}, m_{\overline{\Sigma}(1750)^{-}}, m_{\Sigma^{+}}\right) \mathcal{F}_{0}\left(\sigma_{2}^{2}, m_{K^{0}}, m_{\overline{p}}\right) \mathcal{H}^\mathrm{LS,decay}_{\overline{\Sigma}(1750)^{-}, 0, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{\Sigma}(1750)^{-}, 1, 0} \mathcal{R}_{0}\left(\sigma_{2}, m_{\overline{\Sigma}(1750)^{-}}, \Gamma_{\overline{\Sigma}(1750)^{-}}\right) d^{\frac{1}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)}{2}} + \sum_{\lambda_{R}=-1/2}^{1/2}{\frac{\sqrt{2} \delta_{1, \lambda_{R} - \frac{1}{2}} C^{\frac{1}{2},- \frac{1}{2}}_{0,0,\frac{1}{2},- \frac{1}{2}} C^{1,\lambda_{R} - \frac{1}{2}}_{0,0,1,\lambda_{R} - \frac{1}{2}} C^{1,\lambda_{R} - \frac{1}{2}}_{\frac{1}{2},\lambda_{R},\frac{1}{2},- \frac{1}{2}} C^{\frac{1}{2},- \frac{1}{2}}_{1,0,\frac{1}{2},- \frac{1}{2}} \mathcal{F}_{0}\left(m_{0}^{2}, m_{\overline{\Sigma}(1660)^{-}}, m_{\Sigma^{+}}\right) \mathcal{F}_{1}\left(\sigma_{2}^{2}, m_{K^{0}}, m_{\overline{p}}\right) \mathcal{H}^\mathrm{LS,decay}_{\overline{\Sigma}(1660)^{-}, 1, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{\Sigma}(1660)^{-}, 0, 1} \mathcal{R}_{1}\left(\sigma_{2}, m_{\overline{\Sigma}(1660)^{-}}, \Gamma_{\overline{\Sigma}(1660)^{-}}\right) d^{\frac{1}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)}{2}} + \sum_{\lambda_{R}=-5/2}^{5/2}{\frac{\sqrt{30} \delta_{1, \lambda_{R} - \frac{1}{2}} C^{\frac{1}{2},- \frac{1}{2}}_{0,0,\frac{1}{2},- \frac{1}{2}} C^{1,\lambda_{R} - \frac{1}{2}}_{1,0,2,\lambda_{R} - \frac{1}{2}} C^{\frac{5}{2},- \frac{1}{2}}_{2,0,\frac{1}{2},- \frac{1}{2}} C^{2,\lambda_{R} - \frac{1}{2}}_{\frac{5}{2},\lambda_{R},\frac{1}{2},- \frac{1}{2}} \mathcal{F}_{1}\left(m_{0}^{2}, m_{\overline{\Sigma}(1775)^{-}}, m_{\Sigma^{+}}\right) \mathcal{F}_{2}\left(\sigma_{2}^{2}, m_{K^{0}}, m_{\overline{p}}\right) \mathcal{H}^\mathrm{LS,decay}_{\overline{\Sigma}(1775)^{-}, 2, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{\Sigma}(1775)^{-}, 1, 2} \mathcal{R}_{2}\left(\sigma_{2}, m_{\overline{\Sigma}(1775)^{-}}, \Gamma_{\overline{\Sigma}(1775)^{-}}\right) d^{\frac{5}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)}{6}} + \sum_{\lambda_{R}=-3/2}^{3/2}{\frac{\sqrt{5} \delta_{1, \lambda_{R} - \frac{1}{2}} C^{\frac{1}{2},- \frac{1}{2}}_{0,0,\frac{1}{2},- \frac{1}{2}} C^{1,\lambda_{R} - \frac{1}{2}}_{1,0,1,\lambda_{R} - \frac{1}{2}} C^{1,\lambda_{R} - \frac{1}{2}}_{\frac{3}{2},\lambda_{R},\frac{1}{2},- \frac{1}{2}} C^{\frac{3}{2},- \frac{1}{2}}_{2,0,\frac{1}{2},- \frac{1}{2}} \mathcal{F}_{1}\left(m_{0}^{2}, m_{\overline{\Sigma}(1670)^{-}}, m_{\Sigma^{+}}\right) \mathcal{F}_{2}\left(\sigma_{2}^{2}, m_{K^{0}}, m_{\overline{p}}\right) \mathcal{H}^\mathrm{LS,decay}_{\overline{\Sigma}(1670)^{-}, 2, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{\Sigma}(1670)^{-}, 1, 1} \mathcal{R}_{2}\left(\sigma_{2}, m_{\overline{\Sigma}(1670)^{-}}, \Gamma_{\overline{\Sigma}(1670)^{-}}\right) d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)}{2}} + \sum_{\lambda_{R}=-3/2}^{3/2}{\frac{\sqrt{5} \delta_{1, \lambda_{R} - \frac{1}{2}} C^{\frac{1}{2},- \frac{1}{2}}_{0,0,\frac{1}{2},- \frac{1}{2}} C^{1,\lambda_{R} - \frac{1}{2}}_{1,0,1,\lambda_{R} - \frac{1}{2}} C^{1,\lambda_{R} - \frac{1}{2}}_{\frac{3}{2},\lambda_{R},\frac{1}{2},- \frac{1}{2}} C^{\frac{3}{2},- \frac{1}{2}}_{2,0,\frac{1}{2},- \frac{1}{2}} \mathcal{F}_{1}\left(m_{0}^{2}, m_{\overline{\Sigma}(1940)^{-}}, m_{\Sigma^{+}}\right) \mathcal{F}_{2}\left(\sigma_{2}^{2}, m_{K^{0}}, m_{\overline{p}}\right) \mathcal{H}^\mathrm{LS,decay}_{\overline{\Sigma}(1940)^{-}, 2, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{\Sigma}(1940)^{-}, 1, 1} \mathcal{R}_{2}\left(\sigma_{2}, m_{\overline{\Sigma}(1940)^{-}}, \Gamma_{\overline{\Sigma}(1940)^{-}}\right) d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)}{2}} \\ A^{3}_{1, 0, \frac{1}{2}, \frac{1}{2}} &=& \sum_{\lambda_{R}=-3/2}^{3/2}{\frac{\delta_{1, \lambda_{R} - \frac{1}{2}} C^{\frac{1}{2},- \frac{1}{2}}_{0,0,\frac{1}{2},- \frac{1}{2}} C^{1,\lambda_{R} - \frac{1}{2}}_{0,0,1,\lambda_{R} - \frac{1}{2}} C^{\frac{3}{2},- \frac{1}{2}}_{1,0,\frac{1}{2},- \frac{1}{2}} C^{1,\lambda_{R} - \frac{1}{2}}_{\frac{3}{2},\lambda_{R},\frac{1}{2},- \frac{1}{2}} \mathcal{F}_{0}\left(m_{0}^{2}, m_{N(1720)^{+}}, m_{\overline{p}}\right) \mathcal{F}_{1}\left(\sigma_{3}^{2}, m_{K^{0}}, m_{\Sigma^{+}}\right) \mathcal{H}^\mathrm{LS,decay}_{N(1720)^{+}, 1, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{N(1720)^{+}, 0, 1} \mathcal{R}_{1}\left(\sigma_{3}, m_{N(1720)^{+}}, \Gamma_{N(1720)^{+}}\right) d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{12}\right)}{2}} + \sum_{\lambda_{R}=-1/2}^{1/2}{\frac{\sqrt{2} \delta_{1, \lambda_{R} - \frac{1}{2}} C^{\frac{1}{2},- \frac{1}{2}}_{0,0,\frac{1}{2},- \frac{1}{2}} C^{1,\lambda_{R} - \frac{1}{2}}_{0,0,1,\lambda_{R} - \frac{1}{2}} C^{1,\lambda_{R} - \frac{1}{2}}_{\frac{1}{2},\lambda_{R},\frac{1}{2},- \frac{1}{2}} C^{\frac{1}{2},- \frac{1}{2}}_{1,0,\frac{1}{2},- \frac{1}{2}} \mathcal{F}_{0}\left(m_{0}^{2}, m_{N(1710)^{+}}, m_{\overline{p}}\right) \mathcal{F}_{1}\left(\sigma_{3}^{2}, m_{K^{0}}, m_{\Sigma^{+}}\right) \mathcal{H}^\mathrm{LS,decay}_{N(1710)^{+}, 1, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{N(1710)^{+}, 0, 1} \mathcal{R}_{1}\left(\sigma_{3}, m_{N(1710)^{+}}, \Gamma_{N(1710)^{+}}\right) d^{\frac{1}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{12}\right)}{2}} + \sum_{\lambda_{R}=-3/2}^{3/2}{\frac{\sqrt{5} \delta_{1, \lambda_{R} - \frac{1}{2}} C^{\frac{1}{2},- \frac{1}{2}}_{0,0,\frac{1}{2},- \frac{1}{2}} C^{1,\lambda_{R} - \frac{1}{2}}_{1,0,1,\lambda_{R} - \frac{1}{2}} C^{1,\lambda_{R} - \frac{1}{2}}_{\frac{3}{2},\lambda_{R},\frac{1}{2},- \frac{1}{2}} C^{\frac{3}{2},- \frac{1}{2}}_{2,0,\frac{1}{2},- \frac{1}{2}} \mathcal{F}_{1}\left(m_{0}^{2}, m_{N(1700)^{+}}, m_{\overline{p}}\right) \mathcal{F}_{2}\left(\sigma_{3}^{2}, m_{K^{0}}, m_{\Sigma^{+}}\right) \mathcal{H}^\mathrm{LS,decay}_{N(1700)^{+}, 2, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{N(1700)^{+}, 1, 1} \mathcal{R}_{2}\left(\sigma_{3}, m_{N(1700)^{+}}, \Gamma_{N(1700)^{+}}\right) d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{12}\right)}{2}} \\ \end{array}\end{split}\]

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s, m0, w0, m1, m2, L, R, z = sp.symbols("s m0 Gamma0 m1 m2 L R z")
exprs = [
    RelativisticBreitWigner(s, m0, w0, m1, m2, L, R),
    EnergyDependentWidth(s, m0, w0, m1, m2, L, R),
    FormFactor(s, m1, m2, L, R),
    BlattWeisskopfSquared(z, L),
]
Latex(aslatex({e: e.doit(deep=False) for e in exprs}))

\[\begin{split}\begin{array}{rcl} \mathcal{R}_{L}\left(s, m_{0}, \Gamma_{0}\right) &=& \frac{\Gamma_{0} m_{0}}{m_{0}^{2} - i m_{0} \Gamma_{L}\left(s\right) - s} \\ \Gamma_{0}\left(s\right) &=& \frac{\Gamma_{0} B_{L}^2\left(R^{2} q^2\left(s\right)\right) \rho\left(s\right)}{B_{L}^2\left(R^{2} q^2_{0}\left(m_{0}^{2}\right)\right) \rho_{0}\left(m_{0}^{2}\right)} \\ \mathcal{F}_{L}\left(s, m_{1}, m_{2}\right) &=& \sqrt{B_{L}^2\left(R^{2} q^2\left(s\right)\right)} \\ B_{L}^2\left(z\right) &=& \begin{cases} 1 & \text{for}\: L = 0 \\\frac{2 z}{z + 1} & \text{for}\: L = 1 \\\frac{13 z^{2}}{9 z + \left(z - 3\right)^{2}} & \text{for}\: L = 2 \\\frac{277 z^{3}}{z \left(z - 15\right)^{2} + \left(2 z - 5\right) \left(18 z - 45\right)} & \text{for}\: L = 3 \\\frac{12746 z^{4}}{25 z \left(2 z - 21\right)^{2} + \left(z^{2} - 45 z + 105\right)^{2}} & \text{for}\: L = 4 \\\frac{998881 z^{5}}{z^{5} + 15 z^{4} + 315 z^{3} + 6300 z^{2} + 99225 z + 893025} & \text{for}\: L = 5 \\\frac{118394977 z^{6}}{z^{6} + 21 z^{5} + 630 z^{4} + 18900 z^{3} + 496125 z^{2} + 9823275 z + 108056025} & \text{for}\: L = 6 \\\frac{19727003738 z^{7}}{z^{7} + 28 z^{6} + 1134 z^{5} + 47250 z^{4} + 1819125 z^{3} + 58939650 z^{2} + 1404728325 z + 18261468225} & \text{for}\: L = 7 \\\frac{4392846440677 z^{8}}{z^{8} + 36 z^{7} + 1890 z^{6} + 103950 z^{5} + 5457375 z^{4} + 255405150 z^{3} + 9833098275 z^{2} + 273922023375 z + 4108830350625} & \text{for}\: L = 8 \end{cases} \\ \end{array}\end{split}\]

Preparing for input data

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i, j = (3, 2)
k, *_ = {1, 2, 3} - {i, j}
σk, σk_expr = list(model.invariants.items())[k - 1]
Latex(aslatex({σk: σk_expr}))

\[\begin{split}\begin{array}{rcl} \sigma_{1} &=& m_{0}^{2} + m_{1}^{2} + m_{2}^{2} + m_{3}^{2} - \sigma_{2} - \sigma_{3} \\ \end{array}\end{split}\]

Define meshgrid for Dalitz plot Hide code cell source

resolution = 1_000
m = sorted(model.masses, key=str)
x_min = float(((m[j] + m[k]) ** 2).xreplace(model.masses))
x_max = float(((m[0] - m[i]) ** 2).xreplace(model.masses))
y_min = float(((m[i] + m[k]) ** 2).xreplace(model.masses))
y_max = float(((m[0] - m[j]) ** 2).xreplace(model.masses))
x_diff = x_max - x_min
y_diff = y_max - y_min
x_min -= 0.05 * x_diff
x_max += 0.05 * x_diff
y_min -= 0.05 * y_diff
y_max += 0.05 * y_diff
X, Y = jnp.meshgrid(
    jnp.linspace(x_min, x_max, num=resolution),
    jnp.linspace(y_min, y_max, num=resolution),
)

Create data converter for Dalitz coordinates Hide code cell source

definitions = dict(model.variables)
definitions[σk] = σk_expr
definitions = {
    symbol: expr.xreplace(definitions).xreplace(model.masses)
    for symbol, expr in definitions.items()
}
data_transformer = SympyDataTransformer.from_sympy(definitions, backend="jax")
dalitz_data = {
    f"sigma{i}": X,
    f"sigma{j}": Y,
}
dalitz_data.update(data_transformer(dalitz_data))

Prepare parametrized numerical function Hide code cell source

unfolded_intensity_expr = perform_cached_doit(model.intensity)
full_intensity_expr = perform_cached_doit(
    unfolded_intensity_expr.xreplace(model.amplitudes)
)
free_parameters = {
    k: v
    for k, v in model.parameter_defaults.items()
    if isinstance(k, sp.Indexed)
    if "production" in str(k) or "decay" in str(k)
}
fixed_parameters = {
    k: v for k, v in model.parameter_defaults.items() if k not in free_parameters
}
intensity_func = perform_cached_lambdify(
    full_intensity_expr.xreplace(fixed_parameters),
    parameters=free_parameters,
    backend="jax",
)
intensities = intensity_func(dalitz_data)

Cached expression file /home/runner/.cache/ampform/sympy-v1.12/pythonhashseed-0+144204890906408906.pkl not found, performing doit()...

Cached expression file /home/runner/.cache/ampform/sympy-v1.12/pythonhashseed-0-6833387125911614667.pkl not found, performing doit()...

Dalitz plot

Show code cell source Hide code cell source

def get_decay_products(subsystem_id: int) -> tuple[State, State]:
    return tuple(s for s in DECAY.final_state.values() if s.index != subsystem_id)


plt.rc("font", size=18)
I_tot = jnp.nansum(intensities)
normalized_intensities = intensities / I_tot

fig, ax = plt.subplots(figsize=(14, 10))
mesh = ax.pcolormesh(X, Y, normalized_intensities)
ax.set_aspect("equal")
c_bar = plt.colorbar(mesh, ax=ax, pad=0.01)
c_bar.ax.set_ylabel("Normalized intensity (a.u.)")
sigma_labels = {
    i: Rf"$\sigma_{i} = M^2\left({' '.join(p.latex for p in get_decay_products(i))}\right)$"
    for i in (1, 2, 3)
}
ax.set_xlabel(sigma_labels[i])
ax.set_ylabel(sigma_labels[j])
plt.show()

Show code cell source Hide code cell source

def compute_sub_intensity(
    func: ParametrizedFunction, phsp: DataSample, resonance_latex: str
) -> jnp.ndarray:
    original_parameters = dict(func.parameters)
    zero_parameters = {
        k: 0
        for k, v in func.parameters.items()
        if R"\mathcal{H}" in k
        if resonance_latex not in k
    }
    func.update_parameters(zero_parameters)
    intensities = func(phsp)
    func.update_parameters(original_parameters)
    return intensities


plt.rc("font", size=16)
fig, axes = plt.subplots(figsize=(18, 6), ncols=2, sharey=True)
fig.subplots_adjust(wspace=0.02)
ax1, ax2 = axes
x = jnp.sqrt(X[0])
y = jnp.sqrt(Y[:, 0])
ax1.fill_between(x, jnp.nansum(normalized_intensities, axis=0), alpha=0.5)
ax2.fill_between(y, jnp.nansum(normalized_intensities, axis=1), alpha=0.5)
for ax in axes:
    _, y_max = ax.get_ylim()
    ax.set_ylim(0, y_max)
    ax.autoscale(enable=False, axis="x")
ax1.set_ylabel("Normalized intensity (a.u.)")
ax1.set_xlabel(sigma_labels[i])
ax2.set_xlabel(sigma_labels[j])
resonance_counter1, resonance_counter2 = 0, 0
for chain in tqdm(model.decay.chains, disable=NO_TQDM):
    resonance = chain.resonance
    if set(chain.decay_products) == set(get_decay_products(subsystem_id=i)):
        ax = ax1
        resonance_counter1 += 1
        color = f"C{resonance_counter1}"
        projection_axis = 0
        x_data = x
    elif set(chain.decay_products) == set(get_decay_products(subsystem_id=j)):
        ax = ax2
        resonance_counter2 += 1
        color = f"C{resonance_counter2}"
        projection_axis = 1
        x_data = y
    else:
        raise NotImplementedError
    sub_intensities = compute_sub_intensity(
        intensity_func, dalitz_data, resonance.latex
    )
    ax.plot(x_data, jnp.nansum(sub_intensities / I_tot, axis=projection_axis), c=color)
    ax.axvline(resonance.mass, label=f"${resonance.latex}$", c=color, ls="dashed")
for ax in axes:
    ax.legend(fontsize=12)
plt.show()