Ξ_b⁻ → p K⁻ K⁻

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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 EnergyDependentWidth
from ampform.dynamics.form_factor import BlattWeisskopfSquared, FormFactor
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 (
    load_particles,
    normalize_state_ids,
    permute_equal_final_states,
    to_three_body_decay,
)
from ampform_dpd.decay import State
from ampform_dpd.dynamics import 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

See DOI:10.1103/PhysRevD.104.052010 [pdf], Search for CP violation in \(\Xi_b^- \to p K^- K^-\) decays by LHCb. It found six asymmetry parameters, for \(\Lambda(1405)\), \(\Lambda(1520)\), \(\Lambda(1670)\), \(\Sigma(1385)\), \(\Sigma(1775)\), and \(\Sigma(1915)\).

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PARTICLES = load_particles()
excluded_particles = [
    "Lambda(1600)",
    "Lambda(1690)",
    "Lambda(1800)",
    "Lambda(1810)",
    "Lambda(1890)",
    "Lambda(2000)",
    "Sigma(1660)0",
    "Sigma(1670)0",
    "Sigma(1750)0",
    "Sigma(1910)0",
    "Sigma(c)(2455)0",
    "Sigma(c)(2520)0",
]
for name in excluded_particles:
    PARTICLES.remove(PARTICLES[name])

REACTION = qrules.generate_transitions(
    initial_state="Xi(b)-",
    final_state=["K-", "K-", "p"],
    formalism="canonical-helicity",
    allowed_intermediate_particles=["Lambda", "Sigma"],
    max_angular_momentum=2,
    particle_db=PARTICLES,
)
REACTION = normalize_state_ids(REACTION)
REACTION = permute_equal_final_states(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	Xi(b)-	\(\Xi_{b}^{-}\)	\(\frac{1}{2}^+\)	5,797	0
1	K-	\(K^{-}\)	\(0^-\)	493	0
2	K-	\(K^{-}\)	\(0^-\)	493	0
3	p	\(p\)	\(\frac{1}{2}^+\)	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)
Lambda(1405)	\(\Lambda(1405)\)	\(\frac{1}{2}^-\)	1,405	50
Lambda(1520)	\(\Lambda(1520)\)	\(\frac{3}{2}^-\)	1,519	16
Lambda(1670)	\(\Lambda(1670)\)	\(\frac{1}{2}^-\)	1,674	30
Sigma(1385)0	\(\Sigma(1385)^{0}\)	\(\frac{3}{2}^+\)	1,383	36

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def get_decay_identifier(decay):
    return (decay.resonance, *(particle.name for particle in decay.decay_products))


chains = {get_decay_identifier(c): c for c in DECAY.chains}
Latex(aslatex(chains.values(), with_jp=True))

\[\begin{split}\begin{array}{c} \Xi_{b}^{-}\left[\frac{1}{2}^+\right] \xrightarrow[S=1/2]{L=0} \left(\Lambda(1405)\left[\frac{1}{2}^-\right] \xrightarrow[S=1/2]{L=0} K^{-}\left[0^-\right] p\left[\frac{1}{2}^+\right]\right) K^{-}\left[0^-\right] \\ \Xi_{b}^{-}\left[\frac{1}{2}^+\right] \xrightarrow[S=3/2]{L=1} \left(\Lambda(1520)\left[\frac{3}{2}^-\right] \xrightarrow[S=1/2]{L=1} K^{-}\left[0^-\right] p\left[\frac{1}{2}^+\right]\right) K^{-}\left[0^-\right] \\ \Xi_{b}^{-}\left[\frac{1}{2}^+\right] \xrightarrow[S=1/2]{L=0} \left(\Lambda(1670)\left[\frac{1}{2}^-\right] \xrightarrow[S=1/2]{L=0} K^{-}\left[0^-\right] p\left[\frac{1}{2}^+\right]\right) K^{-}\left[0^-\right] \\ \Xi_{b}^{-}\left[\frac{1}{2}^+\right] \xrightarrow[S=3/2]{L=1} \left(\Sigma(1385)^{0}\left[\frac{3}{2}^+\right] \xrightarrow[S=1/2]{L=1} K^{-}\left[0^-\right] p\left[\frac{1}{2}^+\right]\right) K^{-}\left[0^-\right] \\ \end{array}\end{split}\]

Model formulation

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=1)
model.intensity

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

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Latex(aslatex(model.variables))

\[\begin{split}\begin{array}{rcl} \theta_{23} &=& \operatorname{acos}{\left(\frac{2 \sigma_{1} \left(- m_{1}^{2} - m_{2}^{2} + \sigma_{3}\right) - \left(m_{0}^{2} - m_{1}^{2} - \sigma_{1}\right) \left(m_{2}^{2} - m_{3}^{2} + \sigma_{1}\right)}{\sqrt{\lambda\left(m_{0}^{2}, m_{1}^{2}, \sigma_{1}\right)} \sqrt{\lambda\left(\sigma_{1}, m_{2}^{2}, m_{3}^{2}\right)}} \right)} \\ \theta_{31} &=& \operatorname{acos}{\left(\frac{2 \sigma_{2} \left(- m_{2}^{2} - m_{3}^{2} + \sigma_{1}\right) - \left(m_{0}^{2} - m_{2}^{2} - \sigma_{2}\right) \left(- m_{1}^{2} + m_{3}^{2} + \sigma_{2}\right)}{\sqrt{\lambda\left(m_{0}^{2}, m_{2}^{2}, \sigma_{2}\right)} \sqrt{\lambda\left(\sigma_{2}, m_{3}^{2}, m_{1}^{2}\right)}} \right)} \\ \zeta^0_{1(1)} &=& 0 \\ \zeta^3_{1(1)} &=& 0 \\ \zeta^0_{2(1)} &=& - \operatorname{acos}{\left(\frac{- 2 m_{0}^{2} \left(- m_{1}^{2} - m_{2}^{2} + \sigma_{3}\right) + \left(m_{0}^{2} + m_{1}^{2} - \sigma_{1}\right) \left(m_{0}^{2} + m_{2}^{2} - \sigma_{2}\right)}{\sqrt{\lambda\left(m_{0}^{2}, m_{2}^{2}, \sigma_{2}\right)} \sqrt{\lambda\left(m_{0}^{2}, \sigma_{1}, m_{1}^{2}\right)}} \right)} \\ \zeta^3_{2(1)} &=& - \operatorname{acos}{\left(\frac{2 m_{3}^{2} \left(m_{1}^{2} + m_{2}^{2} - \sigma_{3}\right) + \left(- m_{1}^{2} - m_{3}^{2} + \sigma_{2}\right) \left(- m_{2}^{2} - m_{3}^{2} + \sigma_{1}\right)}{\sqrt{\lambda\left(\sigma_{1}, m_{2}^{2}, m_{3}^{2}\right)} \sqrt{\lambda\left(\sigma_{2}, m_{3}^{2}, m_{1}^{2}\right)}} \right)} \\ \end{array}\end{split}\]

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^{1}_{- \frac{1}{2}, 0, 0, - \frac{1}{2}} &=& \sum_{\lambda_{R}=-1/2}^{1/2}{- \frac{\delta_{- \frac{1}{2} \lambda_{R}} \left(C^{\frac{1}{2},\frac{1}{2}}_{0,0,\frac{1}{2},\frac{1}{2}}\right)^{2} C^{\frac{1}{2},\lambda_{R}}_{0,0,\frac{1}{2},\lambda_{R}} C^{\frac{1}{2},\lambda_{R}}_{\frac{1}{2},\lambda_{R},0,0} \mathcal{F}_{0}\left(m_{0}^{2}, m_{\Lambda(1405)}, m_{K^{-}}\right) \mathcal{F}_{0}\left(\sigma_{1}^{2}, m_{K^{-}}, m_{p}\right) \mathcal{H}^\mathrm{LS,decay}_{\Lambda(1405), 0, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\Lambda(1405), 0, \frac{1}{2}} \mathcal{R}_{0}\left(\sigma_{1}, m_{\Lambda(1405)}, \Gamma_{\Lambda(1405)}\right) d^{\frac{1}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{23}\right)}{2}} + \sum_{\lambda_{R}=-1/2}^{1/2}{- \frac{\delta_{- \frac{1}{2} \lambda_{R}} \left(C^{\frac{1}{2},\frac{1}{2}}_{0,0,\frac{1}{2},\frac{1}{2}}\right)^{2} C^{\frac{1}{2},\lambda_{R}}_{0,0,\frac{1}{2},\lambda_{R}} C^{\frac{1}{2},\lambda_{R}}_{\frac{1}{2},\lambda_{R},0,0} \mathcal{F}_{0}\left(m_{0}^{2}, m_{\Lambda(1670)}, m_{K^{-}}\right) \mathcal{F}_{0}\left(\sigma_{1}^{2}, m_{K^{-}}, m_{p}\right) \mathcal{H}^\mathrm{LS,decay}_{\Lambda(1670), 0, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\Lambda(1670), 0, \frac{1}{2}} \mathcal{R}_{0}\left(\sigma_{1}, m_{\Lambda(1670)}, \Gamma_{\Lambda(1670)}\right) d^{\frac{1}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{23}\right)}{2}} + \sum_{\lambda_{R}=-3/2}^{3/2}{- \frac{3 \sqrt{2} \delta_{- \frac{1}{2} \lambda_{R}} C^{\frac{1}{2},\frac{1}{2}}_{0,0,\frac{1}{2},\frac{1}{2}} C^{\frac{3}{2},\frac{1}{2}}_{1,0,\frac{1}{2},\frac{1}{2}} C^{\frac{1}{2},\lambda_{R}}_{1,0,\frac{3}{2},\lambda_{R}} C^{\frac{3}{2},\lambda_{R}}_{\frac{3}{2},\lambda_{R},0,0} \mathcal{F}_{1}\left(m_{0}^{2}, m_{\Lambda(1520)}, m_{K^{-}}\right) \mathcal{F}_{1}\left(\sigma_{1}^{2}, m_{K^{-}}, m_{p}\right) \mathcal{H}^\mathrm{LS,decay}_{\Lambda(1520), 1, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\Lambda(1520), 1, \frac{3}{2}} \mathcal{R}_{1}\left(\sigma_{1}, m_{\Lambda(1520)}, \Gamma_{\Lambda(1520)}\right) d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{23}\right)}{4}} + \sum_{\lambda_{R}=-3/2}^{3/2}{- \frac{3 \sqrt{2} \delta_{- \frac{1}{2} \lambda_{R}} C^{\frac{1}{2},\frac{1}{2}}_{0,0,\frac{1}{2},\frac{1}{2}} C^{\frac{3}{2},\frac{1}{2}}_{1,0,\frac{1}{2},\frac{1}{2}} C^{\frac{1}{2},\lambda_{R}}_{1,0,\frac{3}{2},\lambda_{R}} C^{\frac{3}{2},\lambda_{R}}_{\frac{3}{2},\lambda_{R},0,0} \mathcal{F}_{1}\left(m_{0}^{2}, m_{\Sigma(1385)^{0}}, m_{K^{-}}\right) \mathcal{F}_{1}\left(\sigma_{1}^{2}, m_{K^{-}}, m_{p}\right) \mathcal{H}^\mathrm{LS,decay}_{\Sigma(1385)^{0}, 1, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\Sigma(1385)^{0}, 1, \frac{3}{2}} \mathcal{R}_{1}\left(\sigma_{1}, m_{\Sigma(1385)^{0}}, \Gamma_{\Sigma(1385)^{0}}\right) d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{23}\right)}{4}} \\ A^{2}_{- \frac{1}{2}, 0, 0, - \frac{1}{2}} &=& \sum_{\lambda_{R}=-1/2}^{1/2}{\frac{\delta_{- \frac{1}{2} \lambda_{R}} \left(C^{\frac{1}{2},\frac{1}{2}}_{0,0,\frac{1}{2},\frac{1}{2}}\right)^{2} C^{\frac{1}{2},\lambda_{R}}_{0,0,\frac{1}{2},\lambda_{R}} C^{\frac{1}{2},\lambda_{R}}_{\frac{1}{2},\lambda_{R},0,0} \mathcal{F}_{0}\left(m_{0}^{2}, m_{\Lambda(1405)}, m_{K^{-}}\right) \mathcal{F}_{0}\left(\sigma_{2}^{2}, m_{K^{-}}, m_{p}\right) \mathcal{H}^\mathrm{LS,decay}_{\Lambda(1405), 0, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\Lambda(1405), 0, \frac{1}{2}} \mathcal{R}_{0}\left(\sigma_{2}, m_{\Lambda(1405)}, \Gamma_{\Lambda(1405)}\right) d^{\frac{1}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)}{2}} + \sum_{\lambda_{R}=-1/2}^{1/2}{\frac{\delta_{- \frac{1}{2} \lambda_{R}} \left(C^{\frac{1}{2},\frac{1}{2}}_{0,0,\frac{1}{2},\frac{1}{2}}\right)^{2} C^{\frac{1}{2},\lambda_{R}}_{0,0,\frac{1}{2},\lambda_{R}} C^{\frac{1}{2},\lambda_{R}}_{\frac{1}{2},\lambda_{R},0,0} \mathcal{F}_{0}\left(m_{0}^{2}, m_{\Lambda(1670)}, m_{K^{-}}\right) \mathcal{F}_{0}\left(\sigma_{2}^{2}, m_{K^{-}}, m_{p}\right) \mathcal{H}^\mathrm{LS,decay}_{\Lambda(1670), 0, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\Lambda(1670), 0, \frac{1}{2}} \mathcal{R}_{0}\left(\sigma_{2}, m_{\Lambda(1670)}, \Gamma_{\Lambda(1670)}\right) d^{\frac{1}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)}{2}} + \sum_{\lambda_{R}=-3/2}^{3/2}{\frac{3 \sqrt{2} \delta_{- \frac{1}{2} \lambda_{R}} C^{\frac{1}{2},\frac{1}{2}}_{0,0,\frac{1}{2},\frac{1}{2}} C^{\frac{3}{2},\frac{1}{2}}_{1,0,\frac{1}{2},\frac{1}{2}} C^{\frac{1}{2},\lambda_{R}}_{1,0,\frac{3}{2},\lambda_{R}} C^{\frac{3}{2},\lambda_{R}}_{\frac{3}{2},\lambda_{R},0,0} \mathcal{F}_{1}\left(m_{0}^{2}, m_{\Lambda(1520)}, m_{K^{-}}\right) \mathcal{F}_{1}\left(\sigma_{2}^{2}, m_{K^{-}}, m_{p}\right) \mathcal{H}^\mathrm{LS,decay}_{\Lambda(1520), 1, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\Lambda(1520), 1, \frac{3}{2}} \mathcal{R}_{1}\left(\sigma_{2}, m_{\Lambda(1520)}, \Gamma_{\Lambda(1520)}\right) d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)}{4}} + \sum_{\lambda_{R}=-3/2}^{3/2}{\frac{3 \sqrt{2} \delta_{- \frac{1}{2} \lambda_{R}} C^{\frac{1}{2},\frac{1}{2}}_{0,0,\frac{1}{2},\frac{1}{2}} C^{\frac{3}{2},\frac{1}{2}}_{1,0,\frac{1}{2},\frac{1}{2}} C^{\frac{1}{2},\lambda_{R}}_{1,0,\frac{3}{2},\lambda_{R}} C^{\frac{3}{2},\lambda_{R}}_{\frac{3}{2},\lambda_{R},0,0} \mathcal{F}_{1}\left(m_{0}^{2}, m_{\Sigma(1385)^{0}}, m_{K^{-}}\right) \mathcal{F}_{1}\left(\sigma_{2}^{2}, m_{K^{-}}, m_{p}\right) \mathcal{H}^\mathrm{LS,decay}_{\Sigma(1385)^{0}, 1, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\Sigma(1385)^{0}, 1, \frac{3}{2}} \mathcal{R}_{1}\left(\sigma_{2}, m_{\Sigma(1385)^{0}}, \Gamma_{\Sigma(1385)^{0}}\right) d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)}{4}} \\ A^{1}_{- \frac{1}{2}, 0, 0, \frac{1}{2}} &=& \sum_{\lambda_{R}=-1/2}^{1/2}{\frac{\delta_{- \frac{1}{2} \lambda_{R}} \left(C^{\frac{1}{2},- \frac{1}{2}}_{0,0,\frac{1}{2},- \frac{1}{2}}\right)^{2} C^{\frac{1}{2},\lambda_{R}}_{0,0,\frac{1}{2},\lambda_{R}} C^{\frac{1}{2},\lambda_{R}}_{\frac{1}{2},\lambda_{R},0,0} \mathcal{F}_{0}\left(m_{0}^{2}, m_{\Lambda(1405)}, m_{K^{-}}\right) \mathcal{F}_{0}\left(\sigma_{1}^{2}, m_{K^{-}}, m_{p}\right) \mathcal{H}^\mathrm{LS,decay}_{\Lambda(1405), 0, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\Lambda(1405), 0, \frac{1}{2}} \mathcal{R}_{0}\left(\sigma_{1}, m_{\Lambda(1405)}, \Gamma_{\Lambda(1405)}\right) d^{\frac{1}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{23}\right)}{2}} + \sum_{\lambda_{R}=-1/2}^{1/2}{\frac{\delta_{- \frac{1}{2} \lambda_{R}} \left(C^{\frac{1}{2},- \frac{1}{2}}_{0,0,\frac{1}{2},- \frac{1}{2}}\right)^{2} C^{\frac{1}{2},\lambda_{R}}_{0,0,\frac{1}{2},\lambda_{R}} C^{\frac{1}{2},\lambda_{R}}_{\frac{1}{2},\lambda_{R},0,0} \mathcal{F}_{0}\left(m_{0}^{2}, m_{\Lambda(1670)}, m_{K^{-}}\right) \mathcal{F}_{0}\left(\sigma_{1}^{2}, m_{K^{-}}, m_{p}\right) \mathcal{H}^\mathrm{LS,decay}_{\Lambda(1670), 0, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\Lambda(1670), 0, \frac{1}{2}} \mathcal{R}_{0}\left(\sigma_{1}, m_{\Lambda(1670)}, \Gamma_{\Lambda(1670)}\right) d^{\frac{1}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{23}\right)}{2}} + \sum_{\lambda_{R}=-3/2}^{3/2}{\frac{3 \sqrt{2} \delta_{- \frac{1}{2} \lambda_{R}} C^{\frac{1}{2},- \frac{1}{2}}_{0,0,\frac{1}{2},- \frac{1}{2}} C^{\frac{3}{2},- \frac{1}{2}}_{1,0,\frac{1}{2},- \frac{1}{2}} C^{\frac{1}{2},\lambda_{R}}_{1,0,\frac{3}{2},\lambda_{R}} C^{\frac{3}{2},\lambda_{R}}_{\frac{3}{2},\lambda_{R},0,0} \mathcal{F}_{1}\left(m_{0}^{2}, m_{\Lambda(1520)}, m_{K^{-}}\right) \mathcal{F}_{1}\left(\sigma_{1}^{2}, m_{K^{-}}, m_{p}\right) \mathcal{H}^\mathrm{LS,decay}_{\Lambda(1520), 1, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\Lambda(1520), 1, \frac{3}{2}} \mathcal{R}_{1}\left(\sigma_{1}, m_{\Lambda(1520)}, \Gamma_{\Lambda(1520)}\right) d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{23}\right)}{4}} + \sum_{\lambda_{R}=-3/2}^{3/2}{\frac{3 \sqrt{2} \delta_{- \frac{1}{2} \lambda_{R}} C^{\frac{1}{2},- \frac{1}{2}}_{0,0,\frac{1}{2},- \frac{1}{2}} C^{\frac{3}{2},- \frac{1}{2}}_{1,0,\frac{1}{2},- \frac{1}{2}} C^{\frac{1}{2},\lambda_{R}}_{1,0,\frac{3}{2},\lambda_{R}} C^{\frac{3}{2},\lambda_{R}}_{\frac{3}{2},\lambda_{R},0,0} \mathcal{F}_{1}\left(m_{0}^{2}, m_{\Sigma(1385)^{0}}, m_{K^{-}}\right) \mathcal{F}_{1}\left(\sigma_{1}^{2}, m_{K^{-}}, m_{p}\right) \mathcal{H}^\mathrm{LS,decay}_{\Sigma(1385)^{0}, 1, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\Sigma(1385)^{0}, 1, \frac{3}{2}} \mathcal{R}_{1}\left(\sigma_{1}, m_{\Sigma(1385)^{0}}, \Gamma_{\Sigma(1385)^{0}}\right) d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{23}\right)}{4}} \\ A^{2}_{- \frac{1}{2}, 0, 0, \frac{1}{2}} &=& \sum_{\lambda_{R}=-1/2}^{1/2}{\frac{\delta_{- \frac{1}{2} \lambda_{R}} \left(C^{\frac{1}{2},- \frac{1}{2}}_{0,0,\frac{1}{2},- \frac{1}{2}}\right)^{2} C^{\frac{1}{2},\lambda_{R}}_{0,0,\frac{1}{2},\lambda_{R}} C^{\frac{1}{2},\lambda_{R}}_{\frac{1}{2},\lambda_{R},0,0} \mathcal{F}_{0}\left(m_{0}^{2}, m_{\Lambda(1405)}, m_{K^{-}}\right) \mathcal{F}_{0}\left(\sigma_{2}^{2}, m_{K^{-}}, m_{p}\right) \mathcal{H}^\mathrm{LS,decay}_{\Lambda(1405), 0, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\Lambda(1405), 0, \frac{1}{2}} \mathcal{R}_{0}\left(\sigma_{2}, m_{\Lambda(1405)}, \Gamma_{\Lambda(1405)}\right) d^{\frac{1}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)}{2}} + \sum_{\lambda_{R}=-1/2}^{1/2}{\frac{\delta_{- \frac{1}{2} \lambda_{R}} \left(C^{\frac{1}{2},- \frac{1}{2}}_{0,0,\frac{1}{2},- \frac{1}{2}}\right)^{2} C^{\frac{1}{2},\lambda_{R}}_{0,0,\frac{1}{2},\lambda_{R}} C^{\frac{1}{2},\lambda_{R}}_{\frac{1}{2},\lambda_{R},0,0} \mathcal{F}_{0}\left(m_{0}^{2}, m_{\Lambda(1670)}, m_{K^{-}}\right) \mathcal{F}_{0}\left(\sigma_{2}^{2}, m_{K^{-}}, m_{p}\right) \mathcal{H}^\mathrm{LS,decay}_{\Lambda(1670), 0, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\Lambda(1670), 0, \frac{1}{2}} \mathcal{R}_{0}\left(\sigma_{2}, m_{\Lambda(1670)}, \Gamma_{\Lambda(1670)}\right) d^{\frac{1}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)}{2}} + \sum_{\lambda_{R}=-3/2}^{3/2}{\frac{3 \sqrt{2} \delta_{- \frac{1}{2} \lambda_{R}} C^{\frac{1}{2},- \frac{1}{2}}_{0,0,\frac{1}{2},- \frac{1}{2}} C^{\frac{3}{2},- \frac{1}{2}}_{1,0,\frac{1}{2},- \frac{1}{2}} C^{\frac{1}{2},\lambda_{R}}_{1,0,\frac{3}{2},\lambda_{R}} C^{\frac{3}{2},\lambda_{R}}_{\frac{3}{2},\lambda_{R},0,0} \mathcal{F}_{1}\left(m_{0}^{2}, m_{\Lambda(1520)}, m_{K^{-}}\right) \mathcal{F}_{1}\left(\sigma_{2}^{2}, m_{K^{-}}, m_{p}\right) \mathcal{H}^\mathrm{LS,decay}_{\Lambda(1520), 1, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\Lambda(1520), 1, \frac{3}{2}} \mathcal{R}_{1}\left(\sigma_{2}, m_{\Lambda(1520)}, \Gamma_{\Lambda(1520)}\right) d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)}{4}} + \sum_{\lambda_{R}=-3/2}^{3/2}{\frac{3 \sqrt{2} \delta_{- \frac{1}{2} \lambda_{R}} C^{\frac{1}{2},- \frac{1}{2}}_{0,0,\frac{1}{2},- \frac{1}{2}} C^{\frac{3}{2},- \frac{1}{2}}_{1,0,\frac{1}{2},- \frac{1}{2}} C^{\frac{1}{2},\lambda_{R}}_{1,0,\frac{3}{2},\lambda_{R}} C^{\frac{3}{2},\lambda_{R}}_{\frac{3}{2},\lambda_{R},0,0} \mathcal{F}_{1}\left(m_{0}^{2}, m_{\Sigma(1385)^{0}}, m_{K^{-}}\right) \mathcal{F}_{1}\left(\sigma_{2}^{2}, m_{K^{-}}, m_{p}\right) \mathcal{H}^\mathrm{LS,decay}_{\Sigma(1385)^{0}, 1, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\Sigma(1385)^{0}, 1, \frac{3}{2}} \mathcal{R}_{1}\left(\sigma_{2}, m_{\Sigma(1385)^{0}}, \Gamma_{\Sigma(1385)^{0}}\right) d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)}{4}} \\ A^{1}_{\frac{1}{2}, 0, 0, - \frac{1}{2}} &=& \sum_{\lambda_{R}=-1/2}^{1/2}{- \frac{\delta_{\frac{1}{2} \lambda_{R}} \left(C^{\frac{1}{2},\frac{1}{2}}_{0,0,\frac{1}{2},\frac{1}{2}}\right)^{2} C^{\frac{1}{2},\lambda_{R}}_{0,0,\frac{1}{2},\lambda_{R}} C^{\frac{1}{2},\lambda_{R}}_{\frac{1}{2},\lambda_{R},0,0} \mathcal{F}_{0}\left(m_{0}^{2}, m_{\Lambda(1405)}, m_{K^{-}}\right) \mathcal{F}_{0}\left(\sigma_{1}^{2}, m_{K^{-}}, m_{p}\right) \mathcal{H}^\mathrm{LS,decay}_{\Lambda(1405), 0, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\Lambda(1405), 0, \frac{1}{2}} \mathcal{R}_{0}\left(\sigma_{1}, m_{\Lambda(1405)}, \Gamma_{\Lambda(1405)}\right) d^{\frac{1}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{23}\right)}{2}} + \sum_{\lambda_{R}=-1/2}^{1/2}{- \frac{\delta_{\frac{1}{2} \lambda_{R}} \left(C^{\frac{1}{2},\frac{1}{2}}_{0,0,\frac{1}{2},\frac{1}{2}}\right)^{2} C^{\frac{1}{2},\lambda_{R}}_{0,0,\frac{1}{2},\lambda_{R}} C^{\frac{1}{2},\lambda_{R}}_{\frac{1}{2},\lambda_{R},0,0} \mathcal{F}_{0}\left(m_{0}^{2}, m_{\Lambda(1670)}, m_{K^{-}}\right) \mathcal{F}_{0}\left(\sigma_{1}^{2}, m_{K^{-}}, m_{p}\right) \mathcal{H}^\mathrm{LS,decay}_{\Lambda(1670), 0, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\Lambda(1670), 0, \frac{1}{2}} \mathcal{R}_{0}\left(\sigma_{1}, m_{\Lambda(1670)}, \Gamma_{\Lambda(1670)}\right) d^{\frac{1}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{23}\right)}{2}} + \sum_{\lambda_{R}=-3/2}^{3/2}{- \frac{3 \sqrt{2} \delta_{\frac{1}{2} \lambda_{R}} C^{\frac{1}{2},\frac{1}{2}}_{0,0,\frac{1}{2},\frac{1}{2}} C^{\frac{3}{2},\frac{1}{2}}_{1,0,\frac{1}{2},\frac{1}{2}} C^{\frac{1}{2},\lambda_{R}}_{1,0,\frac{3}{2},\lambda_{R}} C^{\frac{3}{2},\lambda_{R}}_{\frac{3}{2},\lambda_{R},0,0} \mathcal{F}_{1}\left(m_{0}^{2}, m_{\Lambda(1520)}, m_{K^{-}}\right) \mathcal{F}_{1}\left(\sigma_{1}^{2}, m_{K^{-}}, m_{p}\right) \mathcal{H}^\mathrm{LS,decay}_{\Lambda(1520), 1, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\Lambda(1520), 1, \frac{3}{2}} \mathcal{R}_{1}\left(\sigma_{1}, m_{\Lambda(1520)}, \Gamma_{\Lambda(1520)}\right) d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{23}\right)}{4}} + \sum_{\lambda_{R}=-3/2}^{3/2}{- \frac{3 \sqrt{2} \delta_{\frac{1}{2} \lambda_{R}} C^{\frac{1}{2},\frac{1}{2}}_{0,0,\frac{1}{2},\frac{1}{2}} C^{\frac{3}{2},\frac{1}{2}}_{1,0,\frac{1}{2},\frac{1}{2}} C^{\frac{1}{2},\lambda_{R}}_{1,0,\frac{3}{2},\lambda_{R}} C^{\frac{3}{2},\lambda_{R}}_{\frac{3}{2},\lambda_{R},0,0} \mathcal{F}_{1}\left(m_{0}^{2}, m_{\Sigma(1385)^{0}}, m_{K^{-}}\right) \mathcal{F}_{1}\left(\sigma_{1}^{2}, m_{K^{-}}, m_{p}\right) \mathcal{H}^\mathrm{LS,decay}_{\Sigma(1385)^{0}, 1, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\Sigma(1385)^{0}, 1, \frac{3}{2}} \mathcal{R}_{1}\left(\sigma_{1}, m_{\Sigma(1385)^{0}}, \Gamma_{\Sigma(1385)^{0}}\right) d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{23}\right)}{4}} \\ A^{2}_{\frac{1}{2}, 0, 0, - \frac{1}{2}} &=& \sum_{\lambda_{R}=-1/2}^{1/2}{\frac{\delta_{\frac{1}{2} \lambda_{R}} \left(C^{\frac{1}{2},\frac{1}{2}}_{0,0,\frac{1}{2},\frac{1}{2}}\right)^{2} C^{\frac{1}{2},\lambda_{R}}_{0,0,\frac{1}{2},\lambda_{R}} C^{\frac{1}{2},\lambda_{R}}_{\frac{1}{2},\lambda_{R},0,0} \mathcal{F}_{0}\left(m_{0}^{2}, m_{\Lambda(1405)}, m_{K^{-}}\right) \mathcal{F}_{0}\left(\sigma_{2}^{2}, m_{K^{-}}, m_{p}\right) \mathcal{H}^\mathrm{LS,decay}_{\Lambda(1405), 0, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\Lambda(1405), 0, \frac{1}{2}} \mathcal{R}_{0}\left(\sigma_{2}, m_{\Lambda(1405)}, \Gamma_{\Lambda(1405)}\right) d^{\frac{1}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)}{2}} + \sum_{\lambda_{R}=-1/2}^{1/2}{\frac{\delta_{\frac{1}{2} \lambda_{R}} \left(C^{\frac{1}{2},\frac{1}{2}}_{0,0,\frac{1}{2},\frac{1}{2}}\right)^{2} C^{\frac{1}{2},\lambda_{R}}_{0,0,\frac{1}{2},\lambda_{R}} C^{\frac{1}{2},\lambda_{R}}_{\frac{1}{2},\lambda_{R},0,0} \mathcal{F}_{0}\left(m_{0}^{2}, m_{\Lambda(1670)}, m_{K^{-}}\right) \mathcal{F}_{0}\left(\sigma_{2}^{2}, m_{K^{-}}, m_{p}\right) \mathcal{H}^\mathrm{LS,decay}_{\Lambda(1670), 0, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\Lambda(1670), 0, \frac{1}{2}} \mathcal{R}_{0}\left(\sigma_{2}, m_{\Lambda(1670)}, \Gamma_{\Lambda(1670)}\right) d^{\frac{1}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)}{2}} + \sum_{\lambda_{R}=-3/2}^{3/2}{\frac{3 \sqrt{2} \delta_{\frac{1}{2} \lambda_{R}} C^{\frac{1}{2},\frac{1}{2}}_{0,0,\frac{1}{2},\frac{1}{2}} C^{\frac{3}{2},\frac{1}{2}}_{1,0,\frac{1}{2},\frac{1}{2}} C^{\frac{1}{2},\lambda_{R}}_{1,0,\frac{3}{2},\lambda_{R}} C^{\frac{3}{2},\lambda_{R}}_{\frac{3}{2},\lambda_{R},0,0} \mathcal{F}_{1}\left(m_{0}^{2}, m_{\Lambda(1520)}, m_{K^{-}}\right) \mathcal{F}_{1}\left(\sigma_{2}^{2}, m_{K^{-}}, m_{p}\right) \mathcal{H}^\mathrm{LS,decay}_{\Lambda(1520), 1, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\Lambda(1520), 1, \frac{3}{2}} \mathcal{R}_{1}\left(\sigma_{2}, m_{\Lambda(1520)}, \Gamma_{\Lambda(1520)}\right) d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)}{4}} + \sum_{\lambda_{R}=-3/2}^{3/2}{\frac{3 \sqrt{2} \delta_{\frac{1}{2} \lambda_{R}} C^{\frac{1}{2},\frac{1}{2}}_{0,0,\frac{1}{2},\frac{1}{2}} C^{\frac{3}{2},\frac{1}{2}}_{1,0,\frac{1}{2},\frac{1}{2}} C^{\frac{1}{2},\lambda_{R}}_{1,0,\frac{3}{2},\lambda_{R}} C^{\frac{3}{2},\lambda_{R}}_{\frac{3}{2},\lambda_{R},0,0} \mathcal{F}_{1}\left(m_{0}^{2}, m_{\Sigma(1385)^{0}}, m_{K^{-}}\right) \mathcal{F}_{1}\left(\sigma_{2}^{2}, m_{K^{-}}, m_{p}\right) \mathcal{H}^\mathrm{LS,decay}_{\Sigma(1385)^{0}, 1, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\Sigma(1385)^{0}, 1, \frac{3}{2}} \mathcal{R}_{1}\left(\sigma_{2}, m_{\Sigma(1385)^{0}}, \Gamma_{\Sigma(1385)^{0}}\right) d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)}{4}} \\ A^{1}_{\frac{1}{2}, 0, 0, \frac{1}{2}} &=& \sum_{\lambda_{R}=-1/2}^{1/2}{\frac{\delta_{\frac{1}{2} \lambda_{R}} \left(C^{\frac{1}{2},- \frac{1}{2}}_{0,0,\frac{1}{2},- \frac{1}{2}}\right)^{2} C^{\frac{1}{2},\lambda_{R}}_{0,0,\frac{1}{2},\lambda_{R}} C^{\frac{1}{2},\lambda_{R}}_{\frac{1}{2},\lambda_{R},0,0} \mathcal{F}_{0}\left(m_{0}^{2}, m_{\Lambda(1405)}, m_{K^{-}}\right) \mathcal{F}_{0}\left(\sigma_{1}^{2}, m_{K^{-}}, m_{p}\right) \mathcal{H}^\mathrm{LS,decay}_{\Lambda(1405), 0, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\Lambda(1405), 0, \frac{1}{2}} \mathcal{R}_{0}\left(\sigma_{1}, m_{\Lambda(1405)}, \Gamma_{\Lambda(1405)}\right) d^{\frac{1}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{23}\right)}{2}} + \sum_{\lambda_{R}=-1/2}^{1/2}{\frac{\delta_{\frac{1}{2} \lambda_{R}} \left(C^{\frac{1}{2},- \frac{1}{2}}_{0,0,\frac{1}{2},- \frac{1}{2}}\right)^{2} C^{\frac{1}{2},\lambda_{R}}_{0,0,\frac{1}{2},\lambda_{R}} C^{\frac{1}{2},\lambda_{R}}_{\frac{1}{2},\lambda_{R},0,0} \mathcal{F}_{0}\left(m_{0}^{2}, m_{\Lambda(1670)}, m_{K^{-}}\right) \mathcal{F}_{0}\left(\sigma_{1}^{2}, m_{K^{-}}, m_{p}\right) \mathcal{H}^\mathrm{LS,decay}_{\Lambda(1670), 0, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\Lambda(1670), 0, \frac{1}{2}} \mathcal{R}_{0}\left(\sigma_{1}, m_{\Lambda(1670)}, \Gamma_{\Lambda(1670)}\right) d^{\frac{1}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{23}\right)}{2}} + \sum_{\lambda_{R}=-3/2}^{3/2}{\frac{3 \sqrt{2} \delta_{\frac{1}{2} \lambda_{R}} C^{\frac{1}{2},- \frac{1}{2}}_{0,0,\frac{1}{2},- \frac{1}{2}} C^{\frac{3}{2},- \frac{1}{2}}_{1,0,\frac{1}{2},- \frac{1}{2}} C^{\frac{1}{2},\lambda_{R}}_{1,0,\frac{3}{2},\lambda_{R}} C^{\frac{3}{2},\lambda_{R}}_{\frac{3}{2},\lambda_{R},0,0} \mathcal{F}_{1}\left(m_{0}^{2}, m_{\Lambda(1520)}, m_{K^{-}}\right) \mathcal{F}_{1}\left(\sigma_{1}^{2}, m_{K^{-}}, m_{p}\right) \mathcal{H}^\mathrm{LS,decay}_{\Lambda(1520), 1, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\Lambda(1520), 1, \frac{3}{2}} \mathcal{R}_{1}\left(\sigma_{1}, m_{\Lambda(1520)}, \Gamma_{\Lambda(1520)}\right) d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{23}\right)}{4}} + \sum_{\lambda_{R}=-3/2}^{3/2}{\frac{3 \sqrt{2} \delta_{\frac{1}{2} \lambda_{R}} C^{\frac{1}{2},- \frac{1}{2}}_{0,0,\frac{1}{2},- \frac{1}{2}} C^{\frac{3}{2},- \frac{1}{2}}_{1,0,\frac{1}{2},- \frac{1}{2}} C^{\frac{1}{2},\lambda_{R}}_{1,0,\frac{3}{2},\lambda_{R}} C^{\frac{3}{2},\lambda_{R}}_{\frac{3}{2},\lambda_{R},0,0} \mathcal{F}_{1}\left(m_{0}^{2}, m_{\Sigma(1385)^{0}}, m_{K^{-}}\right) \mathcal{F}_{1}\left(\sigma_{1}^{2}, m_{K^{-}}, m_{p}\right) \mathcal{H}^\mathrm{LS,decay}_{\Sigma(1385)^{0}, 1, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\Sigma(1385)^{0}, 1, \frac{3}{2}} \mathcal{R}_{1}\left(\sigma_{1}, m_{\Sigma(1385)^{0}}, \Gamma_{\Sigma(1385)^{0}}\right) d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{23}\right)}{4}} \\ A^{2}_{\frac{1}{2}, 0, 0, \frac{1}{2}} &=& \sum_{\lambda_{R}=-1/2}^{1/2}{\frac{\delta_{\frac{1}{2} \lambda_{R}} \left(C^{\frac{1}{2},- \frac{1}{2}}_{0,0,\frac{1}{2},- \frac{1}{2}}\right)^{2} C^{\frac{1}{2},\lambda_{R}}_{0,0,\frac{1}{2},\lambda_{R}} C^{\frac{1}{2},\lambda_{R}}_{\frac{1}{2},\lambda_{R},0,0} \mathcal{F}_{0}\left(m_{0}^{2}, m_{\Lambda(1405)}, m_{K^{-}}\right) \mathcal{F}_{0}\left(\sigma_{2}^{2}, m_{K^{-}}, m_{p}\right) \mathcal{H}^\mathrm{LS,decay}_{\Lambda(1405), 0, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\Lambda(1405), 0, \frac{1}{2}} \mathcal{R}_{0}\left(\sigma_{2}, m_{\Lambda(1405)}, \Gamma_{\Lambda(1405)}\right) d^{\frac{1}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)}{2}} + \sum_{\lambda_{R}=-1/2}^{1/2}{\frac{\delta_{\frac{1}{2} \lambda_{R}} \left(C^{\frac{1}{2},- \frac{1}{2}}_{0,0,\frac{1}{2},- \frac{1}{2}}\right)^{2} C^{\frac{1}{2},\lambda_{R}}_{0,0,\frac{1}{2},\lambda_{R}} C^{\frac{1}{2},\lambda_{R}}_{\frac{1}{2},\lambda_{R},0,0} \mathcal{F}_{0}\left(m_{0}^{2}, m_{\Lambda(1670)}, m_{K^{-}}\right) \mathcal{F}_{0}\left(\sigma_{2}^{2}, m_{K^{-}}, m_{p}\right) \mathcal{H}^\mathrm{LS,decay}_{\Lambda(1670), 0, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\Lambda(1670), 0, \frac{1}{2}} \mathcal{R}_{0}\left(\sigma_{2}, m_{\Lambda(1670)}, \Gamma_{\Lambda(1670)}\right) d^{\frac{1}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)}{2}} + \sum_{\lambda_{R}=-3/2}^{3/2}{\frac{3 \sqrt{2} \delta_{\frac{1}{2} \lambda_{R}} C^{\frac{1}{2},- \frac{1}{2}}_{0,0,\frac{1}{2},- \frac{1}{2}} C^{\frac{3}{2},- \frac{1}{2}}_{1,0,\frac{1}{2},- \frac{1}{2}} C^{\frac{1}{2},\lambda_{R}}_{1,0,\frac{3}{2},\lambda_{R}} C^{\frac{3}{2},\lambda_{R}}_{\frac{3}{2},\lambda_{R},0,0} \mathcal{F}_{1}\left(m_{0}^{2}, m_{\Lambda(1520)}, m_{K^{-}}\right) \mathcal{F}_{1}\left(\sigma_{2}^{2}, m_{K^{-}}, m_{p}\right) \mathcal{H}^\mathrm{LS,decay}_{\Lambda(1520), 1, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\Lambda(1520), 1, \frac{3}{2}} \mathcal{R}_{1}\left(\sigma_{2}, m_{\Lambda(1520)}, \Gamma_{\Lambda(1520)}\right) d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)}{4}} + \sum_{\lambda_{R}=-3/2}^{3/2}{\frac{3 \sqrt{2} \delta_{\frac{1}{2} \lambda_{R}} C^{\frac{1}{2},- \frac{1}{2}}_{0,0,\frac{1}{2},- \frac{1}{2}} C^{\frac{3}{2},- \frac{1}{2}}_{1,0,\frac{1}{2},- \frac{1}{2}} C^{\frac{1}{2},\lambda_{R}}_{1,0,\frac{3}{2},\lambda_{R}} C^{\frac{3}{2},\lambda_{R}}_{\frac{3}{2},\lambda_{R},0,0} \mathcal{F}_{1}\left(m_{0}^{2}, m_{\Sigma(1385)^{0}}, m_{K^{-}}\right) \mathcal{F}_{1}\left(\sigma_{2}^{2}, m_{K^{-}}, m_{p}\right) \mathcal{H}^\mathrm{LS,decay}_{\Sigma(1385)^{0}, 1, \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\Sigma(1385)^{0}, 1, \frac{3}{2}} \mathcal{R}_{1}\left(\sigma_{2}, m_{\Sigma(1385)^{0}}, \Gamma_{\Sigma(1385)^{0}}\right) d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)}{4}} \\ \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", nonnegative=True)
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}))

\[\begin{split}\begin{array}{rcl} \mathcal{R}_{L}\left(s, m_{0}, \Gamma_{0}\right) &=& \frac{\Gamma_{0} m_{0}}{m_{0}^{2} - i m_{0} \Gamma_{L}\left(s\right) - s} \\ \Gamma_{0}\left(s\right) &=& \frac{\Gamma_{0} \mathcal{F}_{L}\left(s, m_{1}, m_{2}\right)^{2} \rho\left(s\right)}{\mathcal{F}_{L}\left(m_{0}^{2}, m_{1}, m_{2}\right)^{2} \rho_{0}\left(m_{0}^{2}\right)} \\ \mathcal{F}_{L}\left(s, m_{1}, m_{2}\right) &=& \sqrt{B_{L}^2\left(R^{2} q^2\left(s\right)\right)} \\ B_{L}^2\left(z\right) &=& \frac{\left|{h_{L}^{(1)}\left(1\right)}\right|^{2}}{z \left|{h_{L}^{(1)}\left(\sqrt{z}\right)}\right|^{2}} \\ \end{array}\end{split}\]

Preparing for input data

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

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

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-v1.13.3/61ccbc544c6ec8f3a259d89b0a71e1f5.pkl not found, performing doit()...

Cached expression file /home/runner/.cache/ampform/sympy-v1.13.3/7155d53897e56d88190213da5683758c.pkl not found, performing doit()...

Dalitz plot

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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()

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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, ax = plt.subplots(figsize=(10, 6), sharey=True)
fig.subplots_adjust(wspace=0.02)
x = jnp.sqrt(X[0])
y = jnp.sqrt(Y[:, 0])
ax.fill_between(x, jnp.nansum(normalized_intensities, axis=0), alpha=0.5)
_, y_max = ax.get_ylim()
ax.set_ylim(0, y_max)
ax.autoscale(enable=False, axis="x")
ax.set_ylabel("Normalized intensity (a.u.)")
ax.set_xlabel(sigma_labels[i])
resonance_counter = 0
for chain in tqdm(model.decay.chains, disable=NO_TQDM):
    if {p.index for p in chain.decay_products} != {1, 3}:
        continue
    resonance = chain.resonance
    sub_intensities = compute_sub_intensity(
        intensity_func, dalitz_data, resonance.latex
    )
    color = f"C{resonance_counter}"
    ax.plot(x, jnp.nansum(sub_intensities / I_tot, axis=0), c=color)
    ax.axvline(resonance.mass, label=f"${resonance.latex}$", c=color, ls="dashed")
    resonance_counter += 1
ax.legend(fontsize=12)
plt.show()