Λ_c⁺ → p π⁺ K⁻

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from __future__ import annotations

import graphviz
import qrules
import sympy as sp
from IPython.display import Latex, Markdown

from ampform_dpd import DalitzPlotDecompositionBuilder
from ampform_dpd.adapter.qrules import (
    load_particles,
    normalize_state_ids,
    to_three_body_decay,
)
from ampform_dpd.decay import ThreeBodyDecayChain
from ampform_dpd.dynamics import BreitWignerMinL
from ampform_dpd.dynamics.builder import create_mass_symbol, get_mandelstam_s
from ampform_dpd.io import as_markdown_table, aslatex, simplify_latex_rendering

simplify_latex_rendering()

Decay definition

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PARTICLES = load_particles()
for name in [
    "K*(1410)~0",
    "K(2)*(1430)~0",
    "K*(1680)~0",
    "Delta(1620)++",
    "Delta(1900)++",
    "Delta(1910)++",
    "Delta(1920)++",
    "Lambda(1800)",
    "Lambda(1810)",
    "Lambda(1890)",
]:
    PARTICLES.remove(PARTICLES[name])
STM = qrules.StateTransitionManager(
    initial_state=["Lambda(c)+"],
    final_state=["p", "K-", "pi+"],
    mass_conservation_factor=3,
    allowed_intermediate_particles=["K", "Delta", "Lambda"],
    particle_db=PARTICLES,
    max_angular_momentum=2,
    formalism="canonical-helicity",
)
STM.set_allowed_interaction_types([qrules.InteractionType.STRONG], node_id=1)
problem_sets = STM.create_problem_sets()
REACTION = STM.find_solutions(problem_sets)
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	Lambda(c)+	\(\Lambda_{c}^{+}\)	\(\frac{1}{2}^+\)	2,286	0
1	p	\(p\)	\(\frac{1}{2}^+\)	938	0
2	K-	\(K^{-}\)	\(0^-\)	493	0
3	pi+	\(\pi^{+}\)	\(0^-\)	139	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)
Delta(1232)++	\(\Delta(1232)^{++}\)	\(\frac{3}{2}^+\)	1,232	117
Delta(1600)++	\(\Delta(1600)^{++}\)	\(\frac{3}{2}^+\)	1,570	250
Delta(1700)++	\(\Delta(1700)^{++}\)	\(\frac{3}{2}^-\)	1,710	300
K(0)*(700)~0	\(\overline{K}_{0}^{*}(700)^{0}\)	\(0^+\)	845	468
K*(892)~0	\(\overline{K}^{*}(892)^{0}\)	\(1^-\)	895	47
K(0)*(1430)~0	\(\overline{K}_{0}^{*}(1430)^{0}\)	\(0^+\)	1,430	270
K(2)*(1980)~0	\(\overline{K}_{2}^{*}(1980)^{0}\)	\(2^+\)	1,990	348
Lambda(1405)	\(\Lambda(1405)\)	\(\frac{1}{2}^-\)	1,405	50
Lambda(1520)	\(\Lambda(1520)\)	\(\frac{3}{2}^-\)	1,519	16
Lambda(1600)	\(\Lambda(1600)\)	\(\frac{1}{2}^+\)	1,600	200
Lambda(1670)	\(\Lambda(1670)\)	\(\frac{1}{2}^-\)	1,674	30
Lambda(1690)	\(\Lambda(1690)\)	\(\frac{3}{2}^-\)	1,690	70
Lambda(2000)	\(\Lambda(2000)\)	\(\frac{1}{2}^-\)	2,000	210

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Latex(aslatex(DECAY, with_jp=True))

\[\begin{split}\begin{array}{c} \Lambda_{c}^{+}\left[\frac{1}{2}^+\right] \xrightarrow[S=3/2]{L=1} \left(\Delta(1232)^{++}\left[\frac{3}{2}^+\right] \xrightarrow[S=1/2]{L=1} p\left[\frac{1}{2}^+\right] \pi^{+}\left[0^-\right]\right) K^{-}\left[0^-\right] \\ \Lambda_{c}^{+}\left[\frac{1}{2}^+\right] \xrightarrow[S=3/2]{L=1} \left(\Delta(1600)^{++}\left[\frac{3}{2}^+\right] \xrightarrow[S=1/2]{L=1} p\left[\frac{1}{2}^+\right] \pi^{+}\left[0^-\right]\right) K^{-}\left[0^-\right] \\ \Lambda_{c}^{+}\left[\frac{1}{2}^+\right] \xrightarrow[S=3/2]{L=1} \left(\Delta(1700)^{++}\left[\frac{3}{2}^-\right] \xrightarrow[S=1/2]{L=2} p\left[\frac{1}{2}^+\right] \pi^{+}\left[0^-\right]\right) K^{-}\left[0^-\right] \\ \Lambda_{c}^{+}\left[\frac{1}{2}^+\right] \xrightarrow[S=1/2]{L=0} \left(\overline{K}_{0}^{*}(1430)^{0}\left[0^+\right] \xrightarrow[S=0]{L=0} K^{-}\left[0^-\right] \pi^{+}\left[0^-\right]\right) p\left[\frac{1}{2}^+\right] \\ \Lambda_{c}^{+}\left[\frac{1}{2}^+\right] \xrightarrow[S=1/2]{L=0} \left(\overline{K}_{0}^{*}(700)^{0}\left[0^+\right] \xrightarrow[S=0]{L=0} K^{-}\left[0^-\right] \pi^{+}\left[0^-\right]\right) p\left[\frac{1}{2}^+\right] \\ \Lambda_{c}^{+}\left[\frac{1}{2}^+\right] \xrightarrow[S=3/2]{L=1} \left(\overline{K}_{2}^{*}(1980)^{0}\left[2^+\right] \xrightarrow[S=0]{L=2} K^{-}\left[0^-\right] \pi^{+}\left[0^-\right]\right) p\left[\frac{1}{2}^+\right] \\ \Lambda_{c}^{+}\left[\frac{1}{2}^+\right] \xrightarrow[S=1/2]{L=0} \left(\overline{K}^{*}(892)^{0}\left[1^-\right] \xrightarrow[S=0]{L=1} K^{-}\left[0^-\right] \pi^{+}\left[0^-\right]\right) p\left[\frac{1}{2}^+\right] \\ \Lambda_{c}^{+}\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} p\left[\frac{1}{2}^+\right] K^{-}\left[0^-\right]\right) \pi^{+}\left[0^-\right] \\ \Lambda_{c}^{+}\left[\frac{1}{2}^+\right] \xrightarrow[S=3/2]{L=1} \left(\Lambda(1520)\left[\frac{3}{2}^-\right] \xrightarrow[S=1/2]{L=2} p\left[\frac{1}{2}^+\right] K^{-}\left[0^-\right]\right) \pi^{+}\left[0^-\right] \\ \Lambda_{c}^{+}\left[\frac{1}{2}^+\right] \xrightarrow[S=1/2]{L=0} \left(\Lambda(1600)\left[\frac{1}{2}^+\right] \xrightarrow[S=1/2]{L=1} p\left[\frac{1}{2}^+\right] K^{-}\left[0^-\right]\right) \pi^{+}\left[0^-\right] \\ \Lambda_{c}^{+}\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} p\left[\frac{1}{2}^+\right] K^{-}\left[0^-\right]\right) \pi^{+}\left[0^-\right] \\ \Lambda_{c}^{+}\left[\frac{1}{2}^+\right] \xrightarrow[S=3/2]{L=1} \left(\Lambda(1690)\left[\frac{3}{2}^-\right] \xrightarrow[S=1/2]{L=2} p\left[\frac{1}{2}^+\right] K^{-}\left[0^-\right]\right) \pi^{+}\left[0^-\right] \\ \Lambda_{c}^{+}\left[\frac{1}{2}^+\right] \xrightarrow[S=1/2]{L=0} \left(\Lambda(2000)\left[\frac{1}{2}^-\right] \xrightarrow[S=1/2]{L=0} p\left[\frac{1}{2}^+\right] K^{-}\left[0^-\right]\right) \pi^{+}\left[0^-\right] \\ \end{array}\end{split}\]

Lineshapes for dynamics

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s, m0, Γ0, m1, m2 = sp.symbols("s m0 Gamma0 m1 m2", nonnegative=True)
m_top, m_spec = sp.symbols(R"m_\mathrm{top} m_\mathrm{spectator}", nonnegative=True)
R_dec, R_prod = sp.symbols(R"R_\mathrm{res} R_{\Lambda_c}", nonnegative=True)
l_Λc, l_R = sp.symbols(R"l_{\Lambda_c} l_R", integer=True, nonnegative=True)
bw = BreitWignerMinL(s, m_top, m_spec, m0, Γ0, m1, m2, l_R, l_Λc, R_dec, R_prod)
Latex(aslatex({bw: bw.doit(deep=False)}))

\[\begin{split}\begin{array}{rcl} \mathcal{R}^\mathrm{BW}_{l_{R},l_{\Lambda_c}}\left(s\right) &=& \frac{\frac{F_{l_{R}}\left(R_\mathrm{res} p_{_{m_{1},m_{2}}}\left(s\right)\right)}{F_{l_{R}}\left(R_\mathrm{res} p_{_{m_{1},m_{2}}}\left(m_{0}^{2}\right)\right)} \frac{F_{l_{\Lambda_c}}\left(R_{\Lambda_c} q_{_{m_\mathrm{top},m_\mathrm{spectator}}}\left(s\right)\right)}{F_{l_{\Lambda_c}}\left(R_{\Lambda_c} q_{_{m_\mathrm{top},m_\mathrm{spectator}}}\left(m_{0}^{2}\right)\right)} \left(\frac{p_{_{m_{1},m_{2}}}\left(s\right)}{p_{_{m_{1},m_{2}}}\left(m_{0}^{2}\right)}\right)^{l_{R}} \left(\frac{q_{_{m_\mathrm{top},m_\mathrm{spectator}}}\left(s\right)}{q_{_{m_\mathrm{top},m_\mathrm{spectator}}}\left(m_{0}^{2}\right)}\right)^{l_{\Lambda_c}}}{m_{0}^{2} - i m_{0} \Gamma_{l_{R}}\left(s\right) - s} \\ \end{array}\end{split}\]

Dynamics builder function Hide code cell source

def formulate_breit_wigner(
    decay_chain: ThreeBodyDecayChain,
) -> tuple[BreitWignerMinL, dict[sp.Symbol, float]]:
    s = get_mandelstam_s(decay_chain.decay_node)
    child1_mass, child2_mass = map(create_mass_symbol, decay_chain.decay_products)
    l_dec = sp.Rational(decay_chain.outgoing_ls.L)
    l_prod = sp.Rational(decay_chain.incoming_ls.L)
    parent_mass = sp.Symbol(f"m_{{{decay_chain.parent.latex}}}", nonnegative=True)
    spectator_mass = sp.Symbol(f"m_{{{decay_chain.spectator.latex}}}", nonnegative=True)
    resonance_mass = sp.Symbol(f"m_{{{decay_chain.resonance.latex}}}", nonnegative=True)
    resonance_width = sp.Symbol(
        Rf"\Gamma_{{{decay_chain.resonance.latex}}}", nonnegative=True
    )
    R_dec = sp.Symbol(R"R_\mathrm{res}", nonnegative=True)
    R_prod = sp.Symbol(R"R_{\Lambda_c}", nonnegative=True)
    parameter_defaults = {
        parent_mass: decay_chain.parent.mass,
        spectator_mass: decay_chain.spectator.mass,
        resonance_mass: decay_chain.resonance.mass,
        resonance_width: decay_chain.resonance.width,
        child1_mass: decay_chain.decay_products[0].mass,
        child2_mass: decay_chain.decay_products[1].mass,
        # https://github.com/ComPWA/polarimetry/pull/11#issuecomment-1128784376
        R_dec: 1.5,
        R_prod: 5,
    }
    dynamics = BreitWignerMinL(
        s,
        parent_mass,
        spectator_mass,
        resonance_mass,
        resonance_width,
        child1_mass,
        child2_mass,
        l_dec,
        l_prod,
        R_dec,
        R_prod,
    )
    return dynamics, parameter_defaults

Model formulation

model_builder = DalitzPlotDecompositionBuilder(DECAY, min_ls=(False, True))
for chain in model_builder.decay.chains:
    model_builder.dynamics_choices.register_builder(chain, formulate_breit_wigner)
model = model_builder.formulate(reference_subsystem=1)
model.intensity

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

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Latex(aslatex(model.amplitudes).replace(R"\sum_", R"\sum\limits_"))

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\[\begin{split}\begin{array}{rcl} A^{1}_{- \frac{1}{2}, - \frac{1}{2}, 0, 0} &=& \sum\limits_{\lambda_{R}=-1}^{1}{- \frac{\sqrt{2} \delta_{- \frac{1}{2}, \lambda_{R} + \frac{1}{2}} \mathcal{R}^\mathrm{BW}_{1,0}\left(\sigma_{1}\right) C^{\frac{1}{2},\lambda_{R} + \frac{1}{2}}_{0,0,\frac{1}{2},\lambda_{R} + \frac{1}{2}} C^{\frac{1}{2},\lambda_{R} + \frac{1}{2}}_{1,\lambda_{R},\frac{1}{2},\frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{K}^{*}(892)^{0}, 0, \frac{1}{2}} \mathcal{H}^\mathrm{decay}_{\overline{K}^{*}(892)^{0}, 0, 0} d^{1}_{\lambda_{R},0}\left(\theta_{23}\right)}{2}} + \sum\limits_{\lambda_{R}=0}{- \frac{\sqrt{2} \delta_{- \frac{1}{2}, \lambda_{R} + \frac{1}{2}} \mathcal{R}^\mathrm{BW}_{0,0}\left(\sigma_{1}\right) C^{\frac{1}{2},\lambda_{R} + \frac{1}{2}}_{0,0,\frac{1}{2},\lambda_{R} + \frac{1}{2}} C^{\frac{1}{2},\lambda_{R} + \frac{1}{2}}_{0,\lambda_{R},\frac{1}{2},\frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{K}_{0}^{*}(1430)^{0}, 0, \frac{1}{2}} \mathcal{H}^\mathrm{decay}_{\overline{K}_{0}^{*}(1430)^{0}, 0, 0} d^{0}_{\lambda_{R},0}\left(\theta_{23}\right)}{2}} + \sum\limits_{\lambda_{R}=0}{- \frac{\sqrt{2} \delta_{- \frac{1}{2}, \lambda_{R} + \frac{1}{2}} \mathcal{R}^\mathrm{BW}_{0,0}\left(\sigma_{1}\right) C^{\frac{1}{2},\lambda_{R} + \frac{1}{2}}_{0,0,\frac{1}{2},\lambda_{R} + \frac{1}{2}} C^{\frac{1}{2},\lambda_{R} + \frac{1}{2}}_{0,\lambda_{R},\frac{1}{2},\frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{K}_{0}^{*}(700)^{0}, 0, \frac{1}{2}} \mathcal{H}^\mathrm{decay}_{\overline{K}_{0}^{*}(700)^{0}, 0, 0} d^{0}_{\lambda_{R},0}\left(\theta_{23}\right)}{2}} + \sum\limits_{\lambda_{R}=-2}^{2}{- \frac{\sqrt{6} \delta_{- \frac{1}{2}, \lambda_{R} + \frac{1}{2}} \mathcal{R}^\mathrm{BW}_{2,1}\left(\sigma_{1}\right) C^{\frac{1}{2},\lambda_{R} + \frac{1}{2}}_{1,0,\frac{3}{2},\lambda_{R} + \frac{1}{2}} C^{\frac{3}{2},\lambda_{R} + \frac{1}{2}}_{2,\lambda_{R},\frac{1}{2},\frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{K}_{2}^{*}(1980)^{0}, 1, \frac{3}{2}} \mathcal{H}^\mathrm{decay}_{\overline{K}_{2}^{*}(1980)^{0}, 0, 0} d^{2}_{\lambda_{R},0}\left(\theta_{23}\right)}{2}} \\ A^{2}_{- \frac{1}{2}, - \frac{1}{2}, 0, 0} &=& \sum\limits_{\lambda_{R}=-3/2}^{3/2}{- \frac{\sqrt{6} \delta_{- \frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{1,1}\left(\sigma_{2}\right) 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{H}^\mathrm{LS,production}_{\Delta(1232)^{++}, 1, \frac{3}{2}} \mathcal{H}^\mathrm{decay}_{\Delta(1232)^{++}, 0, - \frac{1}{2}} d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)}{2}} + \sum\limits_{\lambda_{R}=-3/2}^{3/2}{- \frac{\sqrt{6} \delta_{- \frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{1,1}\left(\sigma_{2}\right) 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{H}^\mathrm{LS,production}_{\Delta(1600)^{++}, 1, \frac{3}{2}} \mathcal{H}^\mathrm{decay}_{\Delta(1600)^{++}, 0, - \frac{1}{2}} d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)}{2}} + \sum\limits_{\lambda_{R}=-3/2}^{3/2}{- \frac{\sqrt{6} \delta_{- \frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{2,1}\left(\sigma_{2}\right) 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{H}^\mathrm{LS,production}_{\Delta(1700)^{++}, 1, \frac{3}{2}} \mathcal{H}^\mathrm{decay}_{\Delta(1700)^{++}, 0, - \frac{1}{2}} d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)}{2}} \\ A^{3}_{- \frac{1}{2}, - \frac{1}{2}, 0, 0} &=& \sum\limits_{\lambda_{R}=-1/2}^{1/2}{\frac{\sqrt{2} \delta_{- \frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{0,0}\left(\sigma_{3}\right) 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{H}^\mathrm{LS,production}_{\Lambda(1405), 0, \frac{1}{2}} \mathcal{H}^\mathrm{decay}_{\Lambda(1405), - \frac{1}{2}, 0} d^{\frac{1}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{12}\right)}{2}} + \sum\limits_{\lambda_{R}=-1/2}^{1/2}{\frac{\sqrt{2} \delta_{- \frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{1,0}\left(\sigma_{3}\right) 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{H}^\mathrm{LS,production}_{\Lambda(1600), 0, \frac{1}{2}} \mathcal{H}^\mathrm{decay}_{\Lambda(1600), - \frac{1}{2}, 0} d^{\frac{1}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{12}\right)}{2}} + \sum\limits_{\lambda_{R}=-1/2}^{1/2}{\frac{\sqrt{2} \delta_{- \frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{0,0}\left(\sigma_{3}\right) 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{H}^\mathrm{LS,production}_{\Lambda(1670), 0, \frac{1}{2}} \mathcal{H}^\mathrm{decay}_{\Lambda(1670), - \frac{1}{2}, 0} d^{\frac{1}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{12}\right)}{2}} + \sum\limits_{\lambda_{R}=-1/2}^{1/2}{\frac{\sqrt{2} \delta_{- \frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{0,0}\left(\sigma_{3}\right) 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{H}^\mathrm{LS,production}_{\Lambda(2000), 0, \frac{1}{2}} \mathcal{H}^\mathrm{decay}_{\Lambda(2000), - \frac{1}{2}, 0} d^{\frac{1}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{12}\right)}{2}} + \sum\limits_{\lambda_{R}=-3/2}^{3/2}{\frac{\sqrt{6} \delta_{- \frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{2,1}\left(\sigma_{3}\right) 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{H}^\mathrm{LS,production}_{\Lambda(1520), 1, \frac{3}{2}} \mathcal{H}^\mathrm{decay}_{\Lambda(1520), - \frac{1}{2}, 0} d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{12}\right)}{2}} + \sum\limits_{\lambda_{R}=-3/2}^{3/2}{\frac{\sqrt{6} \delta_{- \frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{2,1}\left(\sigma_{3}\right) 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{H}^\mathrm{LS,production}_{\Lambda(1690), 1, \frac{3}{2}} \mathcal{H}^\mathrm{decay}_{\Lambda(1690), - \frac{1}{2}, 0} d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{12}\right)}{2}} \\ A^{1}_{- \frac{1}{2}, \frac{1}{2}, 0, 0} &=& \sum\limits_{\lambda_{R}=-1}^{1}{\frac{\sqrt{2} \delta_{- \frac{1}{2}, \lambda_{R} - \frac{1}{2}} \mathcal{R}^\mathrm{BW}_{1,0}\left(\sigma_{1}\right) C^{\frac{1}{2},\lambda_{R} - \frac{1}{2}}_{0,0,\frac{1}{2},\lambda_{R} - \frac{1}{2}} C^{\frac{1}{2},\lambda_{R} - \frac{1}{2}}_{1,\lambda_{R},\frac{1}{2},- \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{K}^{*}(892)^{0}, 0, \frac{1}{2}} \mathcal{H}^\mathrm{decay}_{\overline{K}^{*}(892)^{0}, 0, 0} d^{1}_{\lambda_{R},0}\left(\theta_{23}\right)}{2}} + \sum\limits_{\lambda_{R}=0}{\frac{\sqrt{2} \delta_{- \frac{1}{2}, \lambda_{R} - \frac{1}{2}} \mathcal{R}^\mathrm{BW}_{0,0}\left(\sigma_{1}\right) C^{\frac{1}{2},\lambda_{R} - \frac{1}{2}}_{0,0,\frac{1}{2},\lambda_{R} - \frac{1}{2}} C^{\frac{1}{2},\lambda_{R} - \frac{1}{2}}_{0,\lambda_{R},\frac{1}{2},- \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{K}_{0}^{*}(1430)^{0}, 0, \frac{1}{2}} \mathcal{H}^\mathrm{decay}_{\overline{K}_{0}^{*}(1430)^{0}, 0, 0} d^{0}_{\lambda_{R},0}\left(\theta_{23}\right)}{2}} + \sum\limits_{\lambda_{R}=0}{\frac{\sqrt{2} \delta_{- \frac{1}{2}, \lambda_{R} - \frac{1}{2}} \mathcal{R}^\mathrm{BW}_{0,0}\left(\sigma_{1}\right) C^{\frac{1}{2},\lambda_{R} - \frac{1}{2}}_{0,0,\frac{1}{2},\lambda_{R} - \frac{1}{2}} C^{\frac{1}{2},\lambda_{R} - \frac{1}{2}}_{0,\lambda_{R},\frac{1}{2},- \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{K}_{0}^{*}(700)^{0}, 0, \frac{1}{2}} \mathcal{H}^\mathrm{decay}_{\overline{K}_{0}^{*}(700)^{0}, 0, 0} d^{0}_{\lambda_{R},0}\left(\theta_{23}\right)}{2}} + \sum\limits_{\lambda_{R}=-2}^{2}{\frac{\sqrt{6} \delta_{- \frac{1}{2}, \lambda_{R} - \frac{1}{2}} \mathcal{R}^\mathrm{BW}_{2,1}\left(\sigma_{1}\right) C^{\frac{1}{2},\lambda_{R} - \frac{1}{2}}_{1,0,\frac{3}{2},\lambda_{R} - \frac{1}{2}} C^{\frac{3}{2},\lambda_{R} - \frac{1}{2}}_{2,\lambda_{R},\frac{1}{2},- \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{K}_{2}^{*}(1980)^{0}, 1, \frac{3}{2}} \mathcal{H}^\mathrm{decay}_{\overline{K}_{2}^{*}(1980)^{0}, 0, 0} d^{2}_{\lambda_{R},0}\left(\theta_{23}\right)}{2}} \\ A^{2}_{- \frac{1}{2}, \frac{1}{2}, 0, 0} &=& \sum\limits_{\lambda_{R}=-3/2}^{3/2}{\frac{\sqrt{6} \delta_{- \frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{1,1}\left(\sigma_{2}\right) 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{H}^\mathrm{LS,production}_{\Delta(1232)^{++}, 1, \frac{3}{2}} \mathcal{H}^\mathrm{decay}_{\Delta(1232)^{++}, 0, \frac{1}{2}} d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)}{2}} + \sum\limits_{\lambda_{R}=-3/2}^{3/2}{\frac{\sqrt{6} \delta_{- \frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{1,1}\left(\sigma_{2}\right) 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{H}^\mathrm{LS,production}_{\Delta(1600)^{++}, 1, \frac{3}{2}} \mathcal{H}^\mathrm{decay}_{\Delta(1600)^{++}, 0, \frac{1}{2}} d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)}{2}} + \sum\limits_{\lambda_{R}=-3/2}^{3/2}{\frac{\sqrt{6} \delta_{- \frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{2,1}\left(\sigma_{2}\right) 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{H}^\mathrm{LS,production}_{\Delta(1700)^{++}, 1, \frac{3}{2}} \mathcal{H}^\mathrm{decay}_{\Delta(1700)^{++}, 0, \frac{1}{2}} d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)}{2}} \\ A^{3}_{- \frac{1}{2}, \frac{1}{2}, 0, 0} &=& \sum\limits_{\lambda_{R}=-1/2}^{1/2}{\frac{\sqrt{2} \delta_{- \frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{0,0}\left(\sigma_{3}\right) 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{H}^\mathrm{LS,production}_{\Lambda(1405), 0, \frac{1}{2}} \mathcal{H}^\mathrm{decay}_{\Lambda(1405), \frac{1}{2}, 0} d^{\frac{1}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{12}\right)}{2}} + \sum\limits_{\lambda_{R}=-1/2}^{1/2}{\frac{\sqrt{2} \delta_{- \frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{1,0}\left(\sigma_{3}\right) 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{H}^\mathrm{LS,production}_{\Lambda(1600), 0, \frac{1}{2}} \mathcal{H}^\mathrm{decay}_{\Lambda(1600), \frac{1}{2}, 0} d^{\frac{1}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{12}\right)}{2}} + \sum\limits_{\lambda_{R}=-1/2}^{1/2}{\frac{\sqrt{2} \delta_{- \frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{0,0}\left(\sigma_{3}\right) 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{H}^\mathrm{LS,production}_{\Lambda(1670), 0, \frac{1}{2}} \mathcal{H}^\mathrm{decay}_{\Lambda(1670), \frac{1}{2}, 0} d^{\frac{1}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{12}\right)}{2}} + \sum\limits_{\lambda_{R}=-1/2}^{1/2}{\frac{\sqrt{2} \delta_{- \frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{0,0}\left(\sigma_{3}\right) 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{H}^\mathrm{LS,production}_{\Lambda(2000), 0, \frac{1}{2}} \mathcal{H}^\mathrm{decay}_{\Lambda(2000), \frac{1}{2}, 0} d^{\frac{1}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{12}\right)}{2}} + \sum\limits_{\lambda_{R}=-3/2}^{3/2}{\frac{\sqrt{6} \delta_{- \frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{2,1}\left(\sigma_{3}\right) 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{H}^\mathrm{LS,production}_{\Lambda(1520), 1, \frac{3}{2}} \mathcal{H}^\mathrm{decay}_{\Lambda(1520), \frac{1}{2}, 0} d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{12}\right)}{2}} + \sum\limits_{\lambda_{R}=-3/2}^{3/2}{\frac{\sqrt{6} \delta_{- \frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{2,1}\left(\sigma_{3}\right) 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{H}^\mathrm{LS,production}_{\Lambda(1690), 1, \frac{3}{2}} \mathcal{H}^\mathrm{decay}_{\Lambda(1690), \frac{1}{2}, 0} d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{12}\right)}{2}} \\ A^{1}_{\frac{1}{2}, - \frac{1}{2}, 0, 0} &=& \sum\limits_{\lambda_{R}=-1}^{1}{- \frac{\sqrt{2} \delta_{\frac{1}{2}, \lambda_{R} + \frac{1}{2}} \mathcal{R}^\mathrm{BW}_{1,0}\left(\sigma_{1}\right) C^{\frac{1}{2},\lambda_{R} + \frac{1}{2}}_{0,0,\frac{1}{2},\lambda_{R} + \frac{1}{2}} C^{\frac{1}{2},\lambda_{R} + \frac{1}{2}}_{1,\lambda_{R},\frac{1}{2},\frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{K}^{*}(892)^{0}, 0, \frac{1}{2}} \mathcal{H}^\mathrm{decay}_{\overline{K}^{*}(892)^{0}, 0, 0} d^{1}_{\lambda_{R},0}\left(\theta_{23}\right)}{2}} + \sum\limits_{\lambda_{R}=0}{- \frac{\sqrt{2} \delta_{\frac{1}{2}, \lambda_{R} + \frac{1}{2}} \mathcal{R}^\mathrm{BW}_{0,0}\left(\sigma_{1}\right) C^{\frac{1}{2},\lambda_{R} + \frac{1}{2}}_{0,0,\frac{1}{2},\lambda_{R} + \frac{1}{2}} C^{\frac{1}{2},\lambda_{R} + \frac{1}{2}}_{0,\lambda_{R},\frac{1}{2},\frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{K}_{0}^{*}(1430)^{0}, 0, \frac{1}{2}} \mathcal{H}^\mathrm{decay}_{\overline{K}_{0}^{*}(1430)^{0}, 0, 0} d^{0}_{\lambda_{R},0}\left(\theta_{23}\right)}{2}} + \sum\limits_{\lambda_{R}=0}{- \frac{\sqrt{2} \delta_{\frac{1}{2}, \lambda_{R} + \frac{1}{2}} \mathcal{R}^\mathrm{BW}_{0,0}\left(\sigma_{1}\right) C^{\frac{1}{2},\lambda_{R} + \frac{1}{2}}_{0,0,\frac{1}{2},\lambda_{R} + \frac{1}{2}} C^{\frac{1}{2},\lambda_{R} + \frac{1}{2}}_{0,\lambda_{R},\frac{1}{2},\frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{K}_{0}^{*}(700)^{0}, 0, \frac{1}{2}} \mathcal{H}^\mathrm{decay}_{\overline{K}_{0}^{*}(700)^{0}, 0, 0} d^{0}_{\lambda_{R},0}\left(\theta_{23}\right)}{2}} + \sum\limits_{\lambda_{R}=-2}^{2}{- \frac{\sqrt{6} \delta_{\frac{1}{2}, \lambda_{R} + \frac{1}{2}} \mathcal{R}^\mathrm{BW}_{2,1}\left(\sigma_{1}\right) C^{\frac{1}{2},\lambda_{R} + \frac{1}{2}}_{1,0,\frac{3}{2},\lambda_{R} + \frac{1}{2}} C^{\frac{3}{2},\lambda_{R} + \frac{1}{2}}_{2,\lambda_{R},\frac{1}{2},\frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{K}_{2}^{*}(1980)^{0}, 1, \frac{3}{2}} \mathcal{H}^\mathrm{decay}_{\overline{K}_{2}^{*}(1980)^{0}, 0, 0} d^{2}_{\lambda_{R},0}\left(\theta_{23}\right)}{2}} \\ A^{2}_{\frac{1}{2}, - \frac{1}{2}, 0, 0} &=& \sum\limits_{\lambda_{R}=-3/2}^{3/2}{- \frac{\sqrt{6} \delta_{\frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{1,1}\left(\sigma_{2}\right) 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{H}^\mathrm{LS,production}_{\Delta(1232)^{++}, 1, \frac{3}{2}} \mathcal{H}^\mathrm{decay}_{\Delta(1232)^{++}, 0, - \frac{1}{2}} d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)}{2}} + \sum\limits_{\lambda_{R}=-3/2}^{3/2}{- \frac{\sqrt{6} \delta_{\frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{1,1}\left(\sigma_{2}\right) 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{H}^\mathrm{LS,production}_{\Delta(1600)^{++}, 1, \frac{3}{2}} \mathcal{H}^\mathrm{decay}_{\Delta(1600)^{++}, 0, - \frac{1}{2}} d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)}{2}} + \sum\limits_{\lambda_{R}=-3/2}^{3/2}{- \frac{\sqrt{6} \delta_{\frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{2,1}\left(\sigma_{2}\right) 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{H}^\mathrm{LS,production}_{\Delta(1700)^{++}, 1, \frac{3}{2}} \mathcal{H}^\mathrm{decay}_{\Delta(1700)^{++}, 0, - \frac{1}{2}} d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)}{2}} \\ A^{3}_{\frac{1}{2}, - \frac{1}{2}, 0, 0} &=& \sum\limits_{\lambda_{R}=-1/2}^{1/2}{\frac{\sqrt{2} \delta_{\frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{0,0}\left(\sigma_{3}\right) 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{H}^\mathrm{LS,production}_{\Lambda(1405), 0, \frac{1}{2}} \mathcal{H}^\mathrm{decay}_{\Lambda(1405), - \frac{1}{2}, 0} d^{\frac{1}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{12}\right)}{2}} + \sum\limits_{\lambda_{R}=-1/2}^{1/2}{\frac{\sqrt{2} \delta_{\frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{1,0}\left(\sigma_{3}\right) 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{H}^\mathrm{LS,production}_{\Lambda(1600), 0, \frac{1}{2}} \mathcal{H}^\mathrm{decay}_{\Lambda(1600), - \frac{1}{2}, 0} d^{\frac{1}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{12}\right)}{2}} + \sum\limits_{\lambda_{R}=-1/2}^{1/2}{\frac{\sqrt{2} \delta_{\frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{0,0}\left(\sigma_{3}\right) 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{H}^\mathrm{LS,production}_{\Lambda(1670), 0, \frac{1}{2}} \mathcal{H}^\mathrm{decay}_{\Lambda(1670), - \frac{1}{2}, 0} d^{\frac{1}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{12}\right)}{2}} + \sum\limits_{\lambda_{R}=-1/2}^{1/2}{\frac{\sqrt{2} \delta_{\frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{0,0}\left(\sigma_{3}\right) 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{H}^\mathrm{LS,production}_{\Lambda(2000), 0, \frac{1}{2}} \mathcal{H}^\mathrm{decay}_{\Lambda(2000), - \frac{1}{2}, 0} d^{\frac{1}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{12}\right)}{2}} + \sum\limits_{\lambda_{R}=-3/2}^{3/2}{\frac{\sqrt{6} \delta_{\frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{2,1}\left(\sigma_{3}\right) 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{H}^\mathrm{LS,production}_{\Lambda(1520), 1, \frac{3}{2}} \mathcal{H}^\mathrm{decay}_{\Lambda(1520), - \frac{1}{2}, 0} d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{12}\right)}{2}} + \sum\limits_{\lambda_{R}=-3/2}^{3/2}{\frac{\sqrt{6} \delta_{\frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{2,1}\left(\sigma_{3}\right) 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{H}^\mathrm{LS,production}_{\Lambda(1690), 1, \frac{3}{2}} \mathcal{H}^\mathrm{decay}_{\Lambda(1690), - \frac{1}{2}, 0} d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{12}\right)}{2}} \\ A^{1}_{\frac{1}{2}, \frac{1}{2}, 0, 0} &=& \sum\limits_{\lambda_{R}=-1}^{1}{\frac{\sqrt{2} \delta_{\frac{1}{2}, \lambda_{R} - \frac{1}{2}} \mathcal{R}^\mathrm{BW}_{1,0}\left(\sigma_{1}\right) C^{\frac{1}{2},\lambda_{R} - \frac{1}{2}}_{0,0,\frac{1}{2},\lambda_{R} - \frac{1}{2}} C^{\frac{1}{2},\lambda_{R} - \frac{1}{2}}_{1,\lambda_{R},\frac{1}{2},- \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{K}^{*}(892)^{0}, 0, \frac{1}{2}} \mathcal{H}^\mathrm{decay}_{\overline{K}^{*}(892)^{0}, 0, 0} d^{1}_{\lambda_{R},0}\left(\theta_{23}\right)}{2}} + \sum\limits_{\lambda_{R}=0}{\frac{\sqrt{2} \delta_{\frac{1}{2}, \lambda_{R} - \frac{1}{2}} \mathcal{R}^\mathrm{BW}_{0,0}\left(\sigma_{1}\right) C^{\frac{1}{2},\lambda_{R} - \frac{1}{2}}_{0,0,\frac{1}{2},\lambda_{R} - \frac{1}{2}} C^{\frac{1}{2},\lambda_{R} - \frac{1}{2}}_{0,\lambda_{R},\frac{1}{2},- \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{K}_{0}^{*}(1430)^{0}, 0, \frac{1}{2}} \mathcal{H}^\mathrm{decay}_{\overline{K}_{0}^{*}(1430)^{0}, 0, 0} d^{0}_{\lambda_{R},0}\left(\theta_{23}\right)}{2}} + \sum\limits_{\lambda_{R}=0}{\frac{\sqrt{2} \delta_{\frac{1}{2}, \lambda_{R} - \frac{1}{2}} \mathcal{R}^\mathrm{BW}_{0,0}\left(\sigma_{1}\right) C^{\frac{1}{2},\lambda_{R} - \frac{1}{2}}_{0,0,\frac{1}{2},\lambda_{R} - \frac{1}{2}} C^{\frac{1}{2},\lambda_{R} - \frac{1}{2}}_{0,\lambda_{R},\frac{1}{2},- \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{K}_{0}^{*}(700)^{0}, 0, \frac{1}{2}} \mathcal{H}^\mathrm{decay}_{\overline{K}_{0}^{*}(700)^{0}, 0, 0} d^{0}_{\lambda_{R},0}\left(\theta_{23}\right)}{2}} + \sum\limits_{\lambda_{R}=-2}^{2}{\frac{\sqrt{6} \delta_{\frac{1}{2}, \lambda_{R} - \frac{1}{2}} \mathcal{R}^\mathrm{BW}_{2,1}\left(\sigma_{1}\right) C^{\frac{1}{2},\lambda_{R} - \frac{1}{2}}_{1,0,\frac{3}{2},\lambda_{R} - \frac{1}{2}} C^{\frac{3}{2},\lambda_{R} - \frac{1}{2}}_{2,\lambda_{R},\frac{1}{2},- \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{K}_{2}^{*}(1980)^{0}, 1, \frac{3}{2}} \mathcal{H}^\mathrm{decay}_{\overline{K}_{2}^{*}(1980)^{0}, 0, 0} d^{2}_{\lambda_{R},0}\left(\theta_{23}\right)}{2}} \\ A^{2}_{\frac{1}{2}, \frac{1}{2}, 0, 0} &=& \sum\limits_{\lambda_{R}=-3/2}^{3/2}{\frac{\sqrt{6} \delta_{\frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{1,1}\left(\sigma_{2}\right) 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{H}^\mathrm{LS,production}_{\Delta(1232)^{++}, 1, \frac{3}{2}} \mathcal{H}^\mathrm{decay}_{\Delta(1232)^{++}, 0, \frac{1}{2}} d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)}{2}} + \sum\limits_{\lambda_{R}=-3/2}^{3/2}{\frac{\sqrt{6} \delta_{\frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{1,1}\left(\sigma_{2}\right) 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{H}^\mathrm{LS,production}_{\Delta(1600)^{++}, 1, \frac{3}{2}} \mathcal{H}^\mathrm{decay}_{\Delta(1600)^{++}, 0, \frac{1}{2}} d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)}{2}} + \sum\limits_{\lambda_{R}=-3/2}^{3/2}{\frac{\sqrt{6} \delta_{\frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{2,1}\left(\sigma_{2}\right) 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{H}^\mathrm{LS,production}_{\Delta(1700)^{++}, 1, \frac{3}{2}} \mathcal{H}^\mathrm{decay}_{\Delta(1700)^{++}, 0, \frac{1}{2}} d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)}{2}} \\ A^{3}_{\frac{1}{2}, \frac{1}{2}, 0, 0} &=& \sum\limits_{\lambda_{R}=-1/2}^{1/2}{\frac{\sqrt{2} \delta_{\frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{0,0}\left(\sigma_{3}\right) 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{H}^\mathrm{LS,production}_{\Lambda(1405), 0, \frac{1}{2}} \mathcal{H}^\mathrm{decay}_{\Lambda(1405), \frac{1}{2}, 0} d^{\frac{1}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{12}\right)}{2}} + \sum\limits_{\lambda_{R}=-1/2}^{1/2}{\frac{\sqrt{2} \delta_{\frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{1,0}\left(\sigma_{3}\right) 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{H}^\mathrm{LS,production}_{\Lambda(1600), 0, \frac{1}{2}} \mathcal{H}^\mathrm{decay}_{\Lambda(1600), \frac{1}{2}, 0} d^{\frac{1}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{12}\right)}{2}} + \sum\limits_{\lambda_{R}=-1/2}^{1/2}{\frac{\sqrt{2} \delta_{\frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{0,0}\left(\sigma_{3}\right) 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{H}^\mathrm{LS,production}_{\Lambda(1670), 0, \frac{1}{2}} \mathcal{H}^\mathrm{decay}_{\Lambda(1670), \frac{1}{2}, 0} d^{\frac{1}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{12}\right)}{2}} + \sum\limits_{\lambda_{R}=-1/2}^{1/2}{\frac{\sqrt{2} \delta_{\frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{0,0}\left(\sigma_{3}\right) 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{H}^\mathrm{LS,production}_{\Lambda(2000), 0, \frac{1}{2}} \mathcal{H}^\mathrm{decay}_{\Lambda(2000), \frac{1}{2}, 0} d^{\frac{1}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{12}\right)}{2}} + \sum\limits_{\lambda_{R}=-3/2}^{3/2}{\frac{\sqrt{6} \delta_{\frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{2,1}\left(\sigma_{3}\right) 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{H}^\mathrm{LS,production}_{\Lambda(1520), 1, \frac{3}{2}} \mathcal{H}^\mathrm{decay}_{\Lambda(1520), \frac{1}{2}, 0} d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{12}\right)}{2}} + \sum\limits_{\lambda_{R}=-3/2}^{3/2}{\frac{\sqrt{6} \delta_{\frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{2,1}\left(\sigma_{3}\right) 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{H}^\mathrm{LS,production}_{\Lambda(1690), 1, \frac{3}{2}} \mathcal{H}^\mathrm{decay}_{\Lambda(1690), \frac{1}{2}, 0} d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{12}\right)}{2}} \\ \end{array}\end{split}\]

By default, the aligned amplitudes are built up of Wigner-\(d\) functions, Clebsch–Gordan coefficients (\(C\)), a resonance parametrization (\(\mathcal{R}(\sigma)\)), and two coupling symbols \(\mathcal{H}^\text{prod}_\dots,\mathcal{H}^\text{dec}_\dots\). In some cases, you want to combine the couplings into one scaling factor. That can be done with the use_coefficients flag:

model = model_builder.formulate(reference_subsystem=1, use_coefficients=True)
(symbol, expr), *_ = model.amplitudes.items()
Latex(aslatex({symbol: expr}).replace(R"\sum_", R"\sum\limits_"))

\[\begin{split}\begin{array}{rcl} A^{1}_{- \frac{1}{2}, - \frac{1}{2}, 0, 0} &=& \sum\limits_{\lambda_{R}=-1}^{1}{- \frac{\sqrt{2} \delta_{- \frac{1}{2}, \lambda_{R} + \frac{1}{2}} \mathcal{R}^\mathrm{BW}_{1,0}\left(\sigma_{1}\right) C^{\frac{1}{2},\lambda_{R} + \frac{1}{2}}_{0,0,\frac{1}{2},\lambda_{R} + \frac{1}{2}} C^{\frac{1}{2},\lambda_{R} + \frac{1}{2}}_{1,\lambda_{R},\frac{1}{2},\frac{1}{2}} \mathcal{H}^\mathrm{\lambda,LS,\overline{K}^{*}(892)^{0}}_{0, \frac{1}{2}, 0, 0} d^{1}_{\lambda_{R},0}\left(\theta_{23}\right)}{2}} + \sum\limits_{\lambda_{R}=0}{- \frac{\sqrt{2} \delta_{- \frac{1}{2}, \lambda_{R} + \frac{1}{2}} \mathcal{R}^\mathrm{BW}_{0,0}\left(\sigma_{1}\right) C^{\frac{1}{2},\lambda_{R} + \frac{1}{2}}_{0,0,\frac{1}{2},\lambda_{R} + \frac{1}{2}} C^{\frac{1}{2},\lambda_{R} + \frac{1}{2}}_{0,\lambda_{R},\frac{1}{2},\frac{1}{2}} \mathcal{H}^\mathrm{\lambda,LS,\overline{K}_{0}^{*}(1430)^{0}}_{0, \frac{1}{2}, 0, 0} d^{0}_{\lambda_{R},0}\left(\theta_{23}\right)}{2}} + \sum\limits_{\lambda_{R}=0}{- \frac{\sqrt{2} \delta_{- \frac{1}{2}, \lambda_{R} + \frac{1}{2}} \mathcal{R}^\mathrm{BW}_{0,0}\left(\sigma_{1}\right) C^{\frac{1}{2},\lambda_{R} + \frac{1}{2}}_{0,0,\frac{1}{2},\lambda_{R} + \frac{1}{2}} C^{\frac{1}{2},\lambda_{R} + \frac{1}{2}}_{0,\lambda_{R},\frac{1}{2},\frac{1}{2}} \mathcal{H}^\mathrm{\lambda,LS,\overline{K}_{0}^{*}(700)^{0}}_{0, \frac{1}{2}, 0, 0} d^{0}_{\lambda_{R},0}\left(\theta_{23}\right)}{2}} + \sum\limits_{\lambda_{R}=-2}^{2}{- \frac{\sqrt{6} \delta_{- \frac{1}{2}, \lambda_{R} + \frac{1}{2}} \mathcal{R}^\mathrm{BW}_{2,1}\left(\sigma_{1}\right) C^{\frac{1}{2},\lambda_{R} + \frac{1}{2}}_{1,0,\frac{3}{2},\lambda_{R} + \frac{1}{2}} C^{\frac{3}{2},\lambda_{R} + \frac{1}{2}}_{2,\lambda_{R},\frac{1}{2},\frac{1}{2}} \mathcal{H}^\mathrm{\lambda,LS,\overline{K}_{2}^{*}(1980)^{0}}_{1, \frac{3}{2}, 0, 0} d^{2}_{\lambda_{R},0}\left(\theta_{23}\right)}{2}} \\ \end{array}\end{split}\]