Default amplitude model

1. Default amplitude model#

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import logging
import os
from textwrap import dedent

import sympy as sp
from ampform.sympy import cached
from ampform_dpd.io import as_markdown_table, aslatex, simplify_latex_rendering
from IPython.display import Markdown, display

from polarimetry.io import display_latex
from polarimetry.lhcb import load_model_builder, load_model_parameters
from polarimetry.lhcb.particle import Σ, K, Λc, load_particles, p, π

simplify_latex_rendering()

NO_LOG = "EXECUTE_NB" in os.environ
if NO_LOG:
    logging.disable(logging.CRITICAL)

1.1. Resonances and LS-scheme#

Particle definitions for \(\Lambda_c^+\) and \(p, \pi^+, K^-\) in the sequential order.

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decay_particles = [Λc, p, π, K, Σ]
Markdown(as_markdown_table(decay_particles))

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

pi+

\(\pi^+\)

\(0^-\)

139

0

3

K-

\(K^-\)

\(0^-\)

493

0

Sigma-

\(\Sigma^-\)

\(\frac{1}{2}^+\)

1,189

0

Particle definitions as defined in particle-definitions.yaml:

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particles = load_particles("../data/particle-definitions.yaml")
resonances = [p for p in particles.values() if p not in set(decay_particles)]
src = as_markdown_table(resonances)
Markdown(src)

name

LaTeX

\(J^P\)

mass (MeV)

width (MeV)

Lambda_c+

\(\Lambda_c^+\)

\(\frac{1}{2}^+\)

2,286

0

p

\(p\)

\(\frac{1}{2}^+\)

938

0

pi+

\(\pi^+\)

\(0^-\)

139

0

K-

\(K^-\)

\(0^-\)

493

0

L(1405)

\(\Lambda(1405)\)

\(\frac{1}{2}^-\)

1,405

50

L(1520)

\(\Lambda(1520)\)

\(\frac{3}{2}^-\)

1,519

15

L(1600)

\(\Lambda(1600)\)

\(\frac{1}{2}^+\)

1,630

250

L(1670)

\(\Lambda(1670)\)

\(\frac{1}{2}^-\)

1,670

30

L(1690)

\(\Lambda(1690)\)

\(\frac{3}{2}^-\)

1,690

70

L(1800)

\(\Lambda(1800)\)

\(\frac{1}{2}^-\)

1,800

300

L(1810)

\(\Lambda(1810)\)

\(\frac{1}{2}^+\)

1,810

150

L(2000)

\(\Lambda(2000)\)

\(\frac{1}{2}^-\)

2,000

210

D(1232)

\(\Delta(1232)\)

\(\frac{3}{2}^+\)

1,232

117

D(1600)

\(\Delta(1600)\)

\(\frac{3}{2}^+\)

1,640

300

D(1620)

\(\Delta(1620)\)

\(\frac{1}{2}^-\)

1,620

130

D(1700)

\(\Delta(1700)\)

\(\frac{3}{2}^-\)

1,690

380

K(700)

\(K(700)\)

\(0^+\)

824

478

K(892)

\(K(892)\)

\(1^-\)

895

47

K(1410)

\(K(1410)\)

\(1^-\)

1,421

236

K(1430)

\(K(1430)\)

\(0^+\)

1,375

190

See also

Amplitude model with LS-couplings

Most models work take the minimal \(L\)-value in each \(LS\)-coupling (only model 17 works in the full \(LS\)-basis. The generated \(LS\)-couplings look as follows:

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def sort_chains(chains):
    return sorted(
        chains,
        key=lambda c: (c.resonance.name[0], c.resonance.mass),
    )


def render_ls_table(with_jp: bool) -> None:
    all_ls_chains = load_model_builder(
        model_file="../data/model-definitions.yaml",
        particle_definitions=particles,
        model_id=17,
    ).decay.chains
    min_ls_chains = load_model_builder(
        model_file="../data/model-definitions.yaml",
        particle_definitions=particles,
        model_id=0,
    ).decay.chains

    all_ls_chains = sort_chains(all_ls_chains)
    min_ls_chains = sort_chains(min_ls_chains)

    n_all_ls = len(all_ls_chains)
    n_min_ls = len(min_ls_chains)

    src = Rf"""
    | Only minimum $LS$ ({n_min_ls}) | All $LS$-couplings ({n_all_ls}) |
    |:------------------------------:|:-------------------------------:|
    """
    src = dedent(src).strip() + "\n"
    min_ls_chain_iter = iter(min_ls_chains)
    min_ls_chain = None
    for all_ls_chain in all_ls_chains:
        min_ls_chain_latex = ""
        if (
            min_ls_chain is None
            or min_ls_chain.resonance.name != all_ls_chain.resonance.name
        ):
            try:
                min_ls_chain = next(min_ls_chain_iter)
                min_ls_chain_latex = f"${aslatex(min_ls_chain, with_jp=with_jp)}$"
            except StopIteration:
                pass
        all_ls_chain_latex = f"${aslatex(all_ls_chain, with_jp=with_jp)}$"
        src += f"| {min_ls_chain_latex} | {all_ls_chain_latex} |\n"
    display(Markdown(src))


render_ls_table(with_jp=False)

Only minimum \(LS\) (12)

All \(LS\)-couplings (26)

\(\Lambda_c^+ \xrightarrow[S=3/2]{L=1} \left(\Delta(1232) \xrightarrow[S=1/2]{L=1} p \pi^+\right) K^-\)

\(\Lambda_c^+ \xrightarrow[S=3/2]{L=1} \left(\Delta(1232) \xrightarrow[S=1/2]{L=1} p \pi^+\right) K^-\)

\(\Lambda_c^+ \xrightarrow[S=3/2]{L=2} \left(\Delta(1232) \xrightarrow[S=1/2]{L=1} p \pi^+\right) K^-\)

\(\Lambda_c^+ \xrightarrow[S=3/2]{L=1} \left(\Delta(1600) \xrightarrow[S=1/2]{L=1} p \pi^+\right) K^-\)

\(\Lambda_c^+ \xrightarrow[S=3/2]{L=1} \left(\Delta(1600) \xrightarrow[S=1/2]{L=1} p \pi^+\right) K^-\)

\(\Lambda_c^+ \xrightarrow[S=3/2]{L=2} \left(\Delta(1600) \xrightarrow[S=1/2]{L=1} p \pi^+\right) K^-\)

\(\Lambda_c^+ \xrightarrow[S=3/2]{L=1} \left(\Delta(1700) \xrightarrow[S=1/2]{L=2} p \pi^+\right) K^-\)

\(\Lambda_c^+ \xrightarrow[S=3/2]{L=1} \left(\Delta(1700) \xrightarrow[S=1/2]{L=2} p \pi^+\right) K^-\)

\(\Lambda_c^+ \xrightarrow[S=3/2]{L=2} \left(\Delta(1700) \xrightarrow[S=1/2]{L=2} p \pi^+\right) K^-\)

\(\Lambda_c^+ \xrightarrow[S=1/2]{L=0} \left(K(700) \xrightarrow[S=0]{L=0} \pi^+ K^-\right) p\)

\(\Lambda_c^+ \xrightarrow[S=1/2]{L=0} \left(K(700) \xrightarrow[S=0]{L=0} \pi^+ K^-\right) p\)

\(\Lambda_c^+ \xrightarrow[S=1/2]{L=1} \left(K(700) \xrightarrow[S=0]{L=0} \pi^+ K^-\right) p\)

\(\Lambda_c^+ \xrightarrow[S=1/2]{L=0} \left(K(892) \xrightarrow[S=0]{L=1} \pi^+ K^-\right) p\)

\(\Lambda_c^+ \xrightarrow[S=1/2]{L=0} \left(K(892) \xrightarrow[S=0]{L=1} \pi^+ K^-\right) p\)

\(\Lambda_c^+ \xrightarrow[S=1/2]{L=1} \left(K(892) \xrightarrow[S=0]{L=1} \pi^+ K^-\right) p\)

\(\Lambda_c^+ \xrightarrow[S=3/2]{L=1} \left(K(892) \xrightarrow[S=0]{L=1} \pi^+ K^-\right) p\)

\(\Lambda_c^+ \xrightarrow[S=3/2]{L=2} \left(K(892) \xrightarrow[S=0]{L=1} \pi^+ K^-\right) p\)

\(\Lambda_c^+ \xrightarrow[S=1/2]{L=0} \left(K(1430) \xrightarrow[S=0]{L=0} \pi^+ K^-\right) p\)

\(\Lambda_c^+ \xrightarrow[S=1/2]{L=0} \left(K(1430) \xrightarrow[S=0]{L=0} \pi^+ K^-\right) p\)

\(\Lambda_c^+ \xrightarrow[S=1/2]{L=1} \left(K(1430) \xrightarrow[S=0]{L=0} \pi^+ K^-\right) p\)

\(\Lambda_c^+ \xrightarrow[S=1/2]{L=0} \left(\Lambda(1405) \xrightarrow[S=1/2]{L=0} K^- p\right) \pi^+\)

\(\Lambda_c^+ \xrightarrow[S=1/2]{L=0} \left(\Lambda(1405) \xrightarrow[S=1/2]{L=0} K^- p\right) \pi^+\)

\(\Lambda_c^+ \xrightarrow[S=1/2]{L=1} \left(\Lambda(1405) \xrightarrow[S=1/2]{L=0} K^- p\right) \pi^+\)

\(\Lambda_c^+ \xrightarrow[S=3/2]{L=1} \left(\Lambda(1520) \xrightarrow[S=1/2]{L=2} K^- p\right) \pi^+\)

\(\Lambda_c^+ \xrightarrow[S=3/2]{L=1} \left(\Lambda(1520) \xrightarrow[S=1/2]{L=2} K^- p\right) \pi^+\)

\(\Lambda_c^+ \xrightarrow[S=3/2]{L=2} \left(\Lambda(1520) \xrightarrow[S=1/2]{L=2} K^- p\right) \pi^+\)

\(\Lambda_c^+ \xrightarrow[S=1/2]{L=0} \left(\Lambda(1600) \xrightarrow[S=1/2]{L=1} K^- p\right) \pi^+\)

\(\Lambda_c^+ \xrightarrow[S=1/2]{L=0} \left(\Lambda(1600) \xrightarrow[S=1/2]{L=1} K^- p\right) \pi^+\)

\(\Lambda_c^+ \xrightarrow[S=1/2]{L=1} \left(\Lambda(1600) \xrightarrow[S=1/2]{L=1} K^- p\right) \pi^+\)

\(\Lambda_c^+ \xrightarrow[S=1/2]{L=0} \left(\Lambda(1670) \xrightarrow[S=1/2]{L=0} K^- p\right) \pi^+\)

\(\Lambda_c^+ \xrightarrow[S=1/2]{L=0} \left(\Lambda(1670) \xrightarrow[S=1/2]{L=0} K^- p\right) \pi^+\)

\(\Lambda_c^+ \xrightarrow[S=1/2]{L=1} \left(\Lambda(1670) \xrightarrow[S=1/2]{L=0} K^- p\right) \pi^+\)

\(\Lambda_c^+ \xrightarrow[S=3/2]{L=1} \left(\Lambda(1690) \xrightarrow[S=1/2]{L=2} K^- p\right) \pi^+\)

\(\Lambda_c^+ \xrightarrow[S=3/2]{L=1} \left(\Lambda(1690) \xrightarrow[S=1/2]{L=2} K^- p\right) \pi^+\)

\(\Lambda_c^+ \xrightarrow[S=3/2]{L=2} \left(\Lambda(1690) \xrightarrow[S=1/2]{L=2} K^- p\right) \pi^+\)

\(\Lambda_c^+ \xrightarrow[S=1/2]{L=0} \left(\Lambda(2000) \xrightarrow[S=1/2]{L=0} K^- p\right) \pi^+\)

\(\Lambda_c^+ \xrightarrow[S=1/2]{L=0} \left(\Lambda(2000) \xrightarrow[S=1/2]{L=0} K^- p\right) \pi^+\)

\(\Lambda_c^+ \xrightarrow[S=1/2]{L=1} \left(\Lambda(2000) \xrightarrow[S=1/2]{L=0} K^- p\right) \pi^+\)

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render_ls_table(with_jp=True)

Only minimum \(LS\) (12)

All \(LS\)-couplings (26)

\(\Lambda_c^+\left[\frac{1}{2}^+\right] \xrightarrow[S=3/2]{L=1} \left(\Delta(1232)\left[\frac{3}{2}^+\right] \xrightarrow[S=1/2]{L=1} p\left[\frac{1}{2}^+\right] \pi^+\left[0^-\right]\right) K^-\left[0^-\right]\)

\(\Lambda_c^+\left[\frac{1}{2}^+\right] \xrightarrow[S=3/2]{L=1} \left(\Delta(1232)\left[\frac{3}{2}^+\right] \xrightarrow[S=1/2]{L=1} p\left[\frac{1}{2}^+\right] \pi^+\left[0^-\right]\right) K^-\left[0^-\right]\)

\(\Lambda_c^+\left[\frac{1}{2}^+\right] \xrightarrow[S=3/2]{L=2} \left(\Delta(1232)\left[\frac{3}{2}^+\right] \xrightarrow[S=1/2]{L=1} p\left[\frac{1}{2}^+\right] \pi^+\left[0^-\right]\right) K^-\left[0^-\right]\)

\(\Lambda_c^+\left[\frac{1}{2}^+\right] \xrightarrow[S=3/2]{L=1} \left(\Delta(1600)\left[\frac{3}{2}^+\right] \xrightarrow[S=1/2]{L=1} p\left[\frac{1}{2}^+\right] \pi^+\left[0^-\right]\right) K^-\left[0^-\right]\)

\(\Lambda_c^+\left[\frac{1}{2}^+\right] \xrightarrow[S=3/2]{L=1} \left(\Delta(1600)\left[\frac{3}{2}^+\right] \xrightarrow[S=1/2]{L=1} p\left[\frac{1}{2}^+\right] \pi^+\left[0^-\right]\right) K^-\left[0^-\right]\)

\(\Lambda_c^+\left[\frac{1}{2}^+\right] \xrightarrow[S=3/2]{L=2} \left(\Delta(1600)\left[\frac{3}{2}^+\right] \xrightarrow[S=1/2]{L=1} p\left[\frac{1}{2}^+\right] \pi^+\left[0^-\right]\right) K^-\left[0^-\right]\)

\(\Lambda_c^+\left[\frac{1}{2}^+\right] \xrightarrow[S=3/2]{L=1} \left(\Delta(1700)\left[\frac{3}{2}^-\right] \xrightarrow[S=1/2]{L=2} p\left[\frac{1}{2}^+\right] \pi^+\left[0^-\right]\right) K^-\left[0^-\right]\)

\(\Lambda_c^+\left[\frac{1}{2}^+\right] \xrightarrow[S=3/2]{L=1} \left(\Delta(1700)\left[\frac{3}{2}^-\right] \xrightarrow[S=1/2]{L=2} p\left[\frac{1}{2}^+\right] \pi^+\left[0^-\right]\right) K^-\left[0^-\right]\)

\(\Lambda_c^+\left[\frac{1}{2}^+\right] \xrightarrow[S=3/2]{L=2} \left(\Delta(1700)\left[\frac{3}{2}^-\right] \xrightarrow[S=1/2]{L=2} p\left[\frac{1}{2}^+\right] \pi^+\left[0^-\right]\right) K^-\left[0^-\right]\)

\(\Lambda_c^+\left[\frac{1}{2}^+\right] \xrightarrow[S=1/2]{L=0} \left(K(700)\left[0^+\right] \xrightarrow[S=0]{L=0} \pi^+\left[0^-\right] K^-\left[0^-\right]\right) p\left[\frac{1}{2}^+\right]\)

\(\Lambda_c^+\left[\frac{1}{2}^+\right] \xrightarrow[S=1/2]{L=0} \left(K(700)\left[0^+\right] \xrightarrow[S=0]{L=0} \pi^+\left[0^-\right] K^-\left[0^-\right]\right) p\left[\frac{1}{2}^+\right]\)

\(\Lambda_c^+\left[\frac{1}{2}^+\right] \xrightarrow[S=1/2]{L=1} \left(K(700)\left[0^+\right] \xrightarrow[S=0]{L=0} \pi^+\left[0^-\right] K^-\left[0^-\right]\right) p\left[\frac{1}{2}^+\right]\)

\(\Lambda_c^+\left[\frac{1}{2}^+\right] \xrightarrow[S=1/2]{L=0} \left(K(892)\left[1^-\right] \xrightarrow[S=0]{L=1} \pi^+\left[0^-\right] K^-\left[0^-\right]\right) p\left[\frac{1}{2}^+\right]\)

\(\Lambda_c^+\left[\frac{1}{2}^+\right] \xrightarrow[S=1/2]{L=0} \left(K(892)\left[1^-\right] \xrightarrow[S=0]{L=1} \pi^+\left[0^-\right] K^-\left[0^-\right]\right) p\left[\frac{1}{2}^+\right]\)

\(\Lambda_c^+\left[\frac{1}{2}^+\right] \xrightarrow[S=1/2]{L=1} \left(K(892)\left[1^-\right] \xrightarrow[S=0]{L=1} \pi^+\left[0^-\right] K^-\left[0^-\right]\right) p\left[\frac{1}{2}^+\right]\)

\(\Lambda_c^+\left[\frac{1}{2}^+\right] \xrightarrow[S=3/2]{L=1} \left(K(892)\left[1^-\right] \xrightarrow[S=0]{L=1} \pi^+\left[0^-\right] K^-\left[0^-\right]\right) p\left[\frac{1}{2}^+\right]\)

\(\Lambda_c^+\left[\frac{1}{2}^+\right] \xrightarrow[S=3/2]{L=2} \left(K(892)\left[1^-\right] \xrightarrow[S=0]{L=1} \pi^+\left[0^-\right] K^-\left[0^-\right]\right) p\left[\frac{1}{2}^+\right]\)

\(\Lambda_c^+\left[\frac{1}{2}^+\right] \xrightarrow[S=1/2]{L=0} \left(K(1430)\left[0^+\right] \xrightarrow[S=0]{L=0} \pi^+\left[0^-\right] K^-\left[0^-\right]\right) p\left[\frac{1}{2}^+\right]\)

\(\Lambda_c^+\left[\frac{1}{2}^+\right] \xrightarrow[S=1/2]{L=0} \left(K(1430)\left[0^+\right] \xrightarrow[S=0]{L=0} \pi^+\left[0^-\right] K^-\left[0^-\right]\right) p\left[\frac{1}{2}^+\right]\)

\(\Lambda_c^+\left[\frac{1}{2}^+\right] \xrightarrow[S=1/2]{L=1} \left(K(1430)\left[0^+\right] \xrightarrow[S=0]{L=0} \pi^+\left[0^-\right] K^-\left[0^-\right]\right) p\left[\frac{1}{2}^+\right]\)

\(\Lambda_c^+\left[\frac{1}{2}^+\right] \xrightarrow[S=1/2]{L=0} \left(\Lambda(1405)\left[\frac{1}{2}^-\right] \xrightarrow[S=1/2]{L=0} K^-\left[0^-\right] p\left[\frac{1}{2}^+\right]\right) \pi^+\left[0^-\right]\)

\(\Lambda_c^+\left[\frac{1}{2}^+\right] \xrightarrow[S=1/2]{L=0} \left(\Lambda(1405)\left[\frac{1}{2}^-\right] \xrightarrow[S=1/2]{L=0} K^-\left[0^-\right] p\left[\frac{1}{2}^+\right]\right) \pi^+\left[0^-\right]\)

\(\Lambda_c^+\left[\frac{1}{2}^+\right] \xrightarrow[S=1/2]{L=1} \left(\Lambda(1405)\left[\frac{1}{2}^-\right] \xrightarrow[S=1/2]{L=0} K^-\left[0^-\right] p\left[\frac{1}{2}^+\right]\right) \pi^+\left[0^-\right]\)

\(\Lambda_c^+\left[\frac{1}{2}^+\right] \xrightarrow[S=3/2]{L=1} \left(\Lambda(1520)\left[\frac{3}{2}^-\right] \xrightarrow[S=1/2]{L=2} K^-\left[0^-\right] p\left[\frac{1}{2}^+\right]\right) \pi^+\left[0^-\right]\)

\(\Lambda_c^+\left[\frac{1}{2}^+\right] \xrightarrow[S=3/2]{L=1} \left(\Lambda(1520)\left[\frac{3}{2}^-\right] \xrightarrow[S=1/2]{L=2} K^-\left[0^-\right] p\left[\frac{1}{2}^+\right]\right) \pi^+\left[0^-\right]\)

\(\Lambda_c^+\left[\frac{1}{2}^+\right] \xrightarrow[S=3/2]{L=2} \left(\Lambda(1520)\left[\frac{3}{2}^-\right] \xrightarrow[S=1/2]{L=2} K^-\left[0^-\right] p\left[\frac{1}{2}^+\right]\right) \pi^+\left[0^-\right]\)

\(\Lambda_c^+\left[\frac{1}{2}^+\right] \xrightarrow[S=1/2]{L=0} \left(\Lambda(1600)\left[\frac{1}{2}^+\right] \xrightarrow[S=1/2]{L=1} K^-\left[0^-\right] p\left[\frac{1}{2}^+\right]\right) \pi^+\left[0^-\right]\)

\(\Lambda_c^+\left[\frac{1}{2}^+\right] \xrightarrow[S=1/2]{L=0} \left(\Lambda(1600)\left[\frac{1}{2}^+\right] \xrightarrow[S=1/2]{L=1} K^-\left[0^-\right] p\left[\frac{1}{2}^+\right]\right) \pi^+\left[0^-\right]\)

\(\Lambda_c^+\left[\frac{1}{2}^+\right] \xrightarrow[S=1/2]{L=1} \left(\Lambda(1600)\left[\frac{1}{2}^+\right] \xrightarrow[S=1/2]{L=1} K^-\left[0^-\right] p\left[\frac{1}{2}^+\right]\right) \pi^+\left[0^-\right]\)

\(\Lambda_c^+\left[\frac{1}{2}^+\right] \xrightarrow[S=1/2]{L=0} \left(\Lambda(1670)\left[\frac{1}{2}^-\right] \xrightarrow[S=1/2]{L=0} K^-\left[0^-\right] p\left[\frac{1}{2}^+\right]\right) \pi^+\left[0^-\right]\)

\(\Lambda_c^+\left[\frac{1}{2}^+\right] \xrightarrow[S=1/2]{L=0} \left(\Lambda(1670)\left[\frac{1}{2}^-\right] \xrightarrow[S=1/2]{L=0} K^-\left[0^-\right] p\left[\frac{1}{2}^+\right]\right) \pi^+\left[0^-\right]\)

\(\Lambda_c^+\left[\frac{1}{2}^+\right] \xrightarrow[S=1/2]{L=1} \left(\Lambda(1670)\left[\frac{1}{2}^-\right] \xrightarrow[S=1/2]{L=0} K^-\left[0^-\right] p\left[\frac{1}{2}^+\right]\right) \pi^+\left[0^-\right]\)

\(\Lambda_c^+\left[\frac{1}{2}^+\right] \xrightarrow[S=3/2]{L=1} \left(\Lambda(1690)\left[\frac{3}{2}^-\right] \xrightarrow[S=1/2]{L=2} K^-\left[0^-\right] p\left[\frac{1}{2}^+\right]\right) \pi^+\left[0^-\right]\)

\(\Lambda_c^+\left[\frac{1}{2}^+\right] \xrightarrow[S=3/2]{L=1} \left(\Lambda(1690)\left[\frac{3}{2}^-\right] \xrightarrow[S=1/2]{L=2} K^-\left[0^-\right] p\left[\frac{1}{2}^+\right]\right) \pi^+\left[0^-\right]\)

\(\Lambda_c^+\left[\frac{1}{2}^+\right] \xrightarrow[S=3/2]{L=2} \left(\Lambda(1690)\left[\frac{3}{2}^-\right] \xrightarrow[S=1/2]{L=2} K^-\left[0^-\right] p\left[\frac{1}{2}^+\right]\right) \pi^+\left[0^-\right]\)

\(\Lambda_c^+\left[\frac{1}{2}^+\right] \xrightarrow[S=1/2]{L=0} \left(\Lambda(2000)\left[\frac{1}{2}^-\right] \xrightarrow[S=1/2]{L=0} K^-\left[0^-\right] p\left[\frac{1}{2}^+\right]\right) \pi^+\left[0^-\right]\)

\(\Lambda_c^+\left[\frac{1}{2}^+\right] \xrightarrow[S=1/2]{L=0} \left(\Lambda(2000)\left[\frac{1}{2}^-\right] \xrightarrow[S=1/2]{L=0} K^-\left[0^-\right] p\left[\frac{1}{2}^+\right]\right) \pi^+\left[0^-\right]\)

\(\Lambda_c^+\left[\frac{1}{2}^+\right] \xrightarrow[S=1/2]{L=1} \left(\Lambda(2000)\left[\frac{1}{2}^-\right] \xrightarrow[S=1/2]{L=0} K^-\left[0^-\right] p\left[\frac{1}{2}^+\right]\right) \pi^+\left[0^-\right]\)

1.2. Amplitude#

1.2.1. Spin-alignment amplitude#

The full intensity of the amplitude model is obtained by summing the following aligned amplitude over all helicity values \(\lambda_i\) in the initial state \(0\) and final states \(1, 2, 3\):

model_choice = 0
amplitude_builder = load_model_builder(
    model_file="../data/model-definitions.yaml",
    particle_definitions=particles,
    model_id=model_choice,
)
model = amplitude_builder.formulate()

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def simplify_notation(expr: sp.Expr) -> sp.Expr:
    def substitute_node(node):
        if isinstance(node, sp.Indexed) and node.indices[2:] == (0, 0):
            return sp.Indexed(node.base, *node.indices[:2])
        return node

    substitutions = {n: substitute_node(n) for n in sp.preorder_traversal(expr)}
    return expr.xreplace(substitutions)


display(simplify_notation(model.intensity.args[0].args[0].args[0].cleanup()))
\[\displaystyle \sum_{\lambda_0^{\prime}=-1/2}^{1/2} \sum_{\lambda_1^{\prime}=-1/2}^{1/2}{A^{1}_{\lambda_0^{\prime}, \lambda_1^{\prime}, 0, 0} d^{\frac{1}{2}}_{\lambda_1^{\prime},\lambda_{1}}\left(\zeta^1_{1(1)}\right) d^{\frac{1}{2}}_{\lambda_{0},\lambda_0^{\prime}}\left(\zeta^0_{1(1)}\right) + A^{2}_{\lambda_0^{\prime}, \lambda_1^{\prime}, 0, 0} d^{\frac{1}{2}}_{\lambda_1^{\prime},\lambda_{1}}\left(\zeta^1_{2(1)}\right) d^{\frac{1}{2}}_{\lambda_{0},\lambda_0^{\prime}}\left(\zeta^0_{2(1)}\right) + A^{3}_{\lambda_0^{\prime}, \lambda_1^{\prime}, 0, 0} d^{\frac{1}{2}}_{\lambda_1^{\prime},\lambda_{1}}\left(\zeta^1_{3(1)}\right) d^{\frac{1}{2}}_{\lambda_{0},\lambda_0^{\prime}}\left(\zeta^0_{3(1)}\right)}\]

Note that we simplified notation here: the amplitude indices for the spinless states are not rendered and their corresponding Wigner-\(d\) alignment functions are simply \(1\).

The relevant \(\zeta^i_{j(k)}\) angles are defined as:

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display_latex({k: v for k, v in model.variables.items() if "zeta" in str(k)})
\[\begin{split}\displaystyle \begin{aligned} \zeta^0_{1(1)} \;&=\; 0 \\ \zeta^1_{1(1)} \;&=\; 0 \\ \zeta^0_{2(1)} \;&=\; - \operatorname{acos}{\left(\frac{- 2 m_{0}^{2} \left(- m_{1}^{2} - m_{2}^{2} + \sigma_{3}\right) + \left(m_{0}^{2} + m_{1}^{2} - \sigma_{1}\right) \left(m_{0}^{2} + m_{2}^{2} - \sigma_{2}\right)}{\sqrt{\lambda\left(m_{0}^{2}, m_{2}^{2}, \sigma_{2}\right)} \sqrt{\lambda\left(m_{0}^{2}, \sigma_{1}, m_{1}^{2}\right)}} \right)} \\ \zeta^1_{2(1)} \;&=\; \operatorname{acos}{\left(\frac{2 m_{1}^{2} \left(- m_{0}^{2} - m_{3}^{2} + \sigma_{3}\right) + \left(m_{0}^{2} + m_{1}^{2} - \sigma_{1}\right) \left(- m_{1}^{2} - m_{3}^{2} + \sigma_{2}\right)}{\sqrt{\lambda\left(m_{0}^{2}, m_{1}^{2}, \sigma_{1}\right)} \sqrt{\lambda\left(\sigma_{2}, m_{1}^{2}, m_{3}^{2}\right)}} \right)} \\ \zeta^0_{3(1)} \;&=\; \operatorname{acos}{\left(\frac{- 2 m_{0}^{2} \left(- m_{1}^{2} - m_{3}^{2} + \sigma_{2}\right) + \left(m_{0}^{2} + m_{1}^{2} - \sigma_{1}\right) \left(m_{0}^{2} + m_{3}^{2} - \sigma_{3}\right)}{\sqrt{\lambda\left(m_{0}^{2}, m_{1}^{2}, \sigma_{1}\right)} \sqrt{\lambda\left(m_{0}^{2}, \sigma_{3}, m_{3}^{2}\right)}} \right)} \\ \zeta^1_{3(1)} \;&=\; - \operatorname{acos}{\left(\frac{2 m_{1}^{2} \left(- m_{0}^{2} - m_{2}^{2} + \sigma_{2}\right) + \left(m_{0}^{2} + m_{1}^{2} - \sigma_{1}\right) \left(- m_{1}^{2} - m_{2}^{2} + \sigma_{3}\right)}{\sqrt{\lambda\left(m_{0}^{2}, m_{1}^{2}, \sigma_{1}\right)} \sqrt{\lambda\left(\sigma_{3}, m_{1}^{2}, m_{2}^{2}\right)}} \right)} \\ \end{aligned}\end{split}\]

1.2.2. Sub-system amplitudes#

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display_latex({simplify_notation(k): v for k, v in model.amplitudes.items()}, wrap=True)
\[\begin{split}\displaystyle \begin{aligned} A^{1}_{- \frac{1}{2}, - \frac{1}{2}} \;&=\; \sum_{\lambda_{R}=-1}^{1}{- \delta_{- \frac{1}{2}, \lambda_{R} + \frac{1}{2}} \mathcal{R}^\mathrm{BW}_{1,0}\left(\sigma_{1}\right) \mathcal{H}^\mathrm{decay}_{K(892), 0, 0} \mathcal{H}^\mathrm{production}_{K(892), \lambda_{R}, - \frac{1}{2}} d^{1}_{\lambda_{R},0}\left(\theta_{23}\right)} \\ \;&+\; \sum_{\lambda_{R}=0}{- \delta_{- \frac{1}{2}, \lambda_{R} + \frac{1}{2}} \mathcal{R}^\mathrm{Bugg}\left(\sigma_{1}\right) \mathcal{H}^\mathrm{decay}_{K(1430), 0, 0} \mathcal{H}^\mathrm{production}_{K(1430), \lambda_{R}, - \frac{1}{2}} d^{0}_{\lambda_{R},0}\left(\theta_{23}\right)} \\ \;&+\; \sum_{\lambda_{R}=0}{- \delta_{- \frac{1}{2}, \lambda_{R} + \frac{1}{2}} \mathcal{R}^\mathrm{Bugg}\left(\sigma_{1}\right) \mathcal{H}^\mathrm{decay}_{K(700), 0, 0} \mathcal{H}^\mathrm{production}_{K(700), \lambda_{R}, - \frac{1}{2}} d^{0}_{\lambda_{R},0}\left(\theta_{23}\right)} \\ A^{2}_{- \frac{1}{2}, - \frac{1}{2}} \;&=\; \sum_{\lambda_{R}=-3/2}^{3/2}{- \delta_{- \frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{2,1}\left(\sigma_{2}\right) \mathcal{H}^\mathrm{decay}_{\Lambda(1520), 0, - \frac{1}{2}} \mathcal{H}^\mathrm{production}_{\Lambda(1520), \lambda_{R}, 0} d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)} \\ \;&+\; \sum_{\lambda_{R}=-1/2}^{1/2}{- \delta_{- \frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{1,0}\left(\sigma_{2}\right) \mathcal{H}^\mathrm{decay}_{\Lambda(1600), 0, - \frac{1}{2}} \mathcal{H}^\mathrm{production}_{\Lambda(1600), \lambda_{R}, 0} d^{\frac{1}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)} \\ \;&+\; \sum_{\lambda_{R}=-1/2}^{1/2}{- \delta_{- \frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{0,0}\left(\sigma_{2}\right) \mathcal{H}^\mathrm{decay}_{\Lambda(1670), 0, - \frac{1}{2}} \mathcal{H}^\mathrm{production}_{\Lambda(1670), \lambda_{R}, 0} d^{\frac{1}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)} \\ \;&+\; \sum_{\lambda_{R}=-3/2}^{3/2}{- \delta_{- \frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{2,1}\left(\sigma_{2}\right) \mathcal{H}^\mathrm{decay}_{\Lambda(1690), 0, - \frac{1}{2}} \mathcal{H}^\mathrm{production}_{\Lambda(1690), \lambda_{R}, 0} d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)} \\ \;&+\; \sum_{\lambda_{R}=-1/2}^{1/2}{- \delta_{- \frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{0,0}\left(\sigma_{2}\right) \mathcal{H}^\mathrm{decay}_{\Lambda(2000), 0, - \frac{1}{2}} \mathcal{H}^\mathrm{production}_{\Lambda(2000), \lambda_{R}, 0} d^{\frac{1}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)} \\ \;&+\; \sum_{\lambda_{R}=-1/2}^{1/2}{- \frac{\delta_{- \frac{1}{2} \lambda_{R}} F_{0}\left(R_{\Lambda_c} q_{0}\left(m_{0}^{2}\right)\right) \mathcal{R}^\mathrm{Flatté}\left(\sigma_{2}\right) \mathcal{H}^\mathrm{decay}_{\Lambda(1405), 0, - \frac{1}{2}} \mathcal{H}^\mathrm{production}_{\Lambda(1405), \lambda_{R}, 0} d^{\frac{1}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)}{F_{0}\left(R_{\Lambda_c} q_{0}\left(m_{0}^{2}\right)\right)}} \\ A^{3}_{- \frac{1}{2}, - \frac{1}{2}} \;&=\; \sum_{\lambda_{R}=-3/2}^{3/2}{\delta_{- \frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{1,1}\left(\sigma_{3}\right) \mathcal{H}^\mathrm{decay}_{\Delta(1232), - \frac{1}{2}, 0} \mathcal{H}^\mathrm{production}_{\Delta(1232), \lambda_{R}, 0} d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{12}\right)} \\ \;&+\; \sum_{\lambda_{R}=-3/2}^{3/2}{\delta_{- \frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{1,1}\left(\sigma_{3}\right) \mathcal{H}^\mathrm{decay}_{\Delta(1600), - \frac{1}{2}, 0} \mathcal{H}^\mathrm{production}_{\Delta(1600), \lambda_{R}, 0} d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{12}\right)} \\ \;&+\; \sum_{\lambda_{R}=-3/2}^{3/2}{\delta_{- \frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{2,1}\left(\sigma_{3}\right) \mathcal{H}^\mathrm{decay}_{\Delta(1700), - \frac{1}{2}, 0} \mathcal{H}^\mathrm{production}_{\Delta(1700), \lambda_{R}, 0} d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{12}\right)} \\ A^{1}_{- \frac{1}{2}, \frac{1}{2}} \;&=\; \sum_{\lambda_{R}=-1}^{1}{\delta_{- \frac{1}{2}, \lambda_{R} - \frac{1}{2}} \mathcal{R}^\mathrm{BW}_{1,0}\left(\sigma_{1}\right) \mathcal{H}^\mathrm{decay}_{K(892), 0, 0} \mathcal{H}^\mathrm{production}_{K(892), \lambda_{R}, \frac{1}{2}} d^{1}_{\lambda_{R},0}\left(\theta_{23}\right)} \\ \;&+\; \sum_{\lambda_{R}=0}{\delta_{- \frac{1}{2}, \lambda_{R} - \frac{1}{2}} \mathcal{R}^\mathrm{Bugg}\left(\sigma_{1}\right) \mathcal{H}^\mathrm{decay}_{K(1430), 0, 0} \mathcal{H}^\mathrm{production}_{K(1430), \lambda_{R}, \frac{1}{2}} d^{0}_{\lambda_{R},0}\left(\theta_{23}\right)} \\ \;&+\; \sum_{\lambda_{R}=0}{\delta_{- \frac{1}{2}, \lambda_{R} - \frac{1}{2}} \mathcal{R}^\mathrm{Bugg}\left(\sigma_{1}\right) \mathcal{H}^\mathrm{decay}_{K(700), 0, 0} \mathcal{H}^\mathrm{production}_{K(700), \lambda_{R}, \frac{1}{2}} d^{0}_{\lambda_{R},0}\left(\theta_{23}\right)} \\ A^{2}_{- \frac{1}{2}, \frac{1}{2}} \;&=\; \sum_{\lambda_{R}=-3/2}^{3/2}{\delta_{- \frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{2,1}\left(\sigma_{2}\right) \mathcal{H}^\mathrm{decay}_{\Lambda(1520), 0, \frac{1}{2}} \mathcal{H}^\mathrm{production}_{\Lambda(1520), \lambda_{R}, 0} d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)} \\ \;&+\; \sum_{\lambda_{R}=-1/2}^{1/2}{\delta_{- \frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{1,0}\left(\sigma_{2}\right) \mathcal{H}^\mathrm{decay}_{\Lambda(1600), 0, \frac{1}{2}} \mathcal{H}^\mathrm{production}_{\Lambda(1600), \lambda_{R}, 0} d^{\frac{1}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)} \\ \;&+\; \sum_{\lambda_{R}=-1/2}^{1/2}{\delta_{- \frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{0,0}\left(\sigma_{2}\right) \mathcal{H}^\mathrm{decay}_{\Lambda(1670), 0, \frac{1}{2}} \mathcal{H}^\mathrm{production}_{\Lambda(1670), \lambda_{R}, 0} d^{\frac{1}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)} \\ \;&+\; \sum_{\lambda_{R}=-3/2}^{3/2}{\delta_{- \frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{2,1}\left(\sigma_{2}\right) \mathcal{H}^\mathrm{decay}_{\Lambda(1690), 0, \frac{1}{2}} \mathcal{H}^\mathrm{production}_{\Lambda(1690), \lambda_{R}, 0} d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)} \\ \;&+\; \sum_{\lambda_{R}=-1/2}^{1/2}{\delta_{- \frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{0,0}\left(\sigma_{2}\right) \mathcal{H}^\mathrm{decay}_{\Lambda(2000), 0, \frac{1}{2}} \mathcal{H}^\mathrm{production}_{\Lambda(2000), \lambda_{R}, 0} d^{\frac{1}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)} \\ \;&+\; \sum_{\lambda_{R}=-1/2}^{1/2}{\frac{\delta_{- \frac{1}{2} \lambda_{R}} F_{0}\left(R_{\Lambda_c} q_{0}\left(m_{0}^{2}\right)\right) \mathcal{R}^\mathrm{Flatté}\left(\sigma_{2}\right) \mathcal{H}^\mathrm{decay}_{\Lambda(1405), 0, \frac{1}{2}} \mathcal{H}^\mathrm{production}_{\Lambda(1405), \lambda_{R}, 0} d^{\frac{1}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)}{F_{0}\left(R_{\Lambda_c} q_{0}\left(m_{0}^{2}\right)\right)}} \\ A^{3}_{- \frac{1}{2}, \frac{1}{2}} \;&=\; \sum_{\lambda_{R}=-3/2}^{3/2}{\delta_{- \frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{1,1}\left(\sigma_{3}\right) \mathcal{H}^\mathrm{decay}_{\Delta(1232), \frac{1}{2}, 0} \mathcal{H}^\mathrm{production}_{\Delta(1232), \lambda_{R}, 0} d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{12}\right)} \\ \;&+\; \sum_{\lambda_{R}=-3/2}^{3/2}{\delta_{- \frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{1,1}\left(\sigma_{3}\right) \mathcal{H}^\mathrm{decay}_{\Delta(1600), \frac{1}{2}, 0} \mathcal{H}^\mathrm{production}_{\Delta(1600), \lambda_{R}, 0} d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{12}\right)} \\ \;&+\; \sum_{\lambda_{R}=-3/2}^{3/2}{\delta_{- \frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{2,1}\left(\sigma_{3}\right) \mathcal{H}^\mathrm{decay}_{\Delta(1700), \frac{1}{2}, 0} \mathcal{H}^\mathrm{production}_{\Delta(1700), \lambda_{R}, 0} d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{12}\right)} \\ A^{1}_{\frac{1}{2}, - \frac{1}{2}} \;&=\; \sum_{\lambda_{R}=-1}^{1}{- \delta_{\frac{1}{2}, \lambda_{R} + \frac{1}{2}} \mathcal{R}^\mathrm{BW}_{1,0}\left(\sigma_{1}\right) \mathcal{H}^\mathrm{decay}_{K(892), 0, 0} \mathcal{H}^\mathrm{production}_{K(892), \lambda_{R}, - \frac{1}{2}} d^{1}_{\lambda_{R},0}\left(\theta_{23}\right)} \\ \;&+\; \sum_{\lambda_{R}=0}{- \delta_{\frac{1}{2}, \lambda_{R} + \frac{1}{2}} \mathcal{R}^\mathrm{Bugg}\left(\sigma_{1}\right) \mathcal{H}^\mathrm{decay}_{K(1430), 0, 0} \mathcal{H}^\mathrm{production}_{K(1430), \lambda_{R}, - \frac{1}{2}} d^{0}_{\lambda_{R},0}\left(\theta_{23}\right)} \\ \;&+\; \sum_{\lambda_{R}=0}{- \delta_{\frac{1}{2}, \lambda_{R} + \frac{1}{2}} \mathcal{R}^\mathrm{Bugg}\left(\sigma_{1}\right) \mathcal{H}^\mathrm{decay}_{K(700), 0, 0} \mathcal{H}^\mathrm{production}_{K(700), \lambda_{R}, - \frac{1}{2}} d^{0}_{\lambda_{R},0}\left(\theta_{23}\right)} \\ A^{2}_{\frac{1}{2}, - \frac{1}{2}} \;&=\; \sum_{\lambda_{R}=-3/2}^{3/2}{- \delta_{\frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{2,1}\left(\sigma_{2}\right) \mathcal{H}^\mathrm{decay}_{\Lambda(1520), 0, - \frac{1}{2}} \mathcal{H}^\mathrm{production}_{\Lambda(1520), \lambda_{R}, 0} d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)} \\ \;&+\; \sum_{\lambda_{R}=-1/2}^{1/2}{- \delta_{\frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{1,0}\left(\sigma_{2}\right) \mathcal{H}^\mathrm{decay}_{\Lambda(1600), 0, - \frac{1}{2}} \mathcal{H}^\mathrm{production}_{\Lambda(1600), \lambda_{R}, 0} d^{\frac{1}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)} \\ \;&+\; \sum_{\lambda_{R}=-1/2}^{1/2}{- \delta_{\frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{0,0}\left(\sigma_{2}\right) \mathcal{H}^\mathrm{decay}_{\Lambda(1670), 0, - \frac{1}{2}} \mathcal{H}^\mathrm{production}_{\Lambda(1670), \lambda_{R}, 0} d^{\frac{1}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)} \\ \;&+\; \sum_{\lambda_{R}=-3/2}^{3/2}{- \delta_{\frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{2,1}\left(\sigma_{2}\right) \mathcal{H}^\mathrm{decay}_{\Lambda(1690), 0, - \frac{1}{2}} \mathcal{H}^\mathrm{production}_{\Lambda(1690), \lambda_{R}, 0} d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)} \\ \;&+\; \sum_{\lambda_{R}=-1/2}^{1/2}{- \delta_{\frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{0,0}\left(\sigma_{2}\right) \mathcal{H}^\mathrm{decay}_{\Lambda(2000), 0, - \frac{1}{2}} \mathcal{H}^\mathrm{production}_{\Lambda(2000), \lambda_{R}, 0} d^{\frac{1}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)} \\ \;&+\; \sum_{\lambda_{R}=-1/2}^{1/2}{- \frac{\delta_{\frac{1}{2} \lambda_{R}} F_{0}\left(R_{\Lambda_c} q_{0}\left(m_{0}^{2}\right)\right) \mathcal{R}^\mathrm{Flatté}\left(\sigma_{2}\right) \mathcal{H}^\mathrm{decay}_{\Lambda(1405), 0, - \frac{1}{2}} \mathcal{H}^\mathrm{production}_{\Lambda(1405), \lambda_{R}, 0} d^{\frac{1}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)}{F_{0}\left(R_{\Lambda_c} q_{0}\left(m_{0}^{2}\right)\right)}} \\ A^{3}_{\frac{1}{2}, - \frac{1}{2}} \;&=\; \sum_{\lambda_{R}=-3/2}^{3/2}{\delta_{\frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{1,1}\left(\sigma_{3}\right) \mathcal{H}^\mathrm{decay}_{\Delta(1232), - \frac{1}{2}, 0} \mathcal{H}^\mathrm{production}_{\Delta(1232), \lambda_{R}, 0} d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{12}\right)} \\ \;&+\; \sum_{\lambda_{R}=-3/2}^{3/2}{\delta_{\frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{1,1}\left(\sigma_{3}\right) \mathcal{H}^\mathrm{decay}_{\Delta(1600), - \frac{1}{2}, 0} \mathcal{H}^\mathrm{production}_{\Delta(1600), \lambda_{R}, 0} d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{12}\right)} \\ \;&+\; \sum_{\lambda_{R}=-3/2}^{3/2}{\delta_{\frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{2,1}\left(\sigma_{3}\right) \mathcal{H}^\mathrm{decay}_{\Delta(1700), - \frac{1}{2}, 0} \mathcal{H}^\mathrm{production}_{\Delta(1700), \lambda_{R}, 0} d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{12}\right)} \\ A^{1}_{\frac{1}{2}, \frac{1}{2}} \;&=\; \sum_{\lambda_{R}=-1}^{1}{\delta_{\frac{1}{2}, \lambda_{R} - \frac{1}{2}} \mathcal{R}^\mathrm{BW}_{1,0}\left(\sigma_{1}\right) \mathcal{H}^\mathrm{decay}_{K(892), 0, 0} \mathcal{H}^\mathrm{production}_{K(892), \lambda_{R}, \frac{1}{2}} d^{1}_{\lambda_{R},0}\left(\theta_{23}\right)} \\ \;&+\; \sum_{\lambda_{R}=0}{\delta_{\frac{1}{2}, \lambda_{R} - \frac{1}{2}} \mathcal{R}^\mathrm{Bugg}\left(\sigma_{1}\right) \mathcal{H}^\mathrm{decay}_{K(1430), 0, 0} \mathcal{H}^\mathrm{production}_{K(1430), \lambda_{R}, \frac{1}{2}} d^{0}_{\lambda_{R},0}\left(\theta_{23}\right)} \\ \;&+\; \sum_{\lambda_{R}=0}{\delta_{\frac{1}{2}, \lambda_{R} - \frac{1}{2}} \mathcal{R}^\mathrm{Bugg}\left(\sigma_{1}\right) \mathcal{H}^\mathrm{decay}_{K(700), 0, 0} \mathcal{H}^\mathrm{production}_{K(700), \lambda_{R}, \frac{1}{2}} d^{0}_{\lambda_{R},0}\left(\theta_{23}\right)} \\ A^{2}_{\frac{1}{2}, \frac{1}{2}} \;&=\; \sum_{\lambda_{R}=-3/2}^{3/2}{\delta_{\frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{2,1}\left(\sigma_{2}\right) \mathcal{H}^\mathrm{decay}_{\Lambda(1520), 0, \frac{1}{2}} \mathcal{H}^\mathrm{production}_{\Lambda(1520), \lambda_{R}, 0} d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)} \\ \;&+\; \sum_{\lambda_{R}=-1/2}^{1/2}{\delta_{\frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{1,0}\left(\sigma_{2}\right) \mathcal{H}^\mathrm{decay}_{\Lambda(1600), 0, \frac{1}{2}} \mathcal{H}^\mathrm{production}_{\Lambda(1600), \lambda_{R}, 0} d^{\frac{1}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)} \\ \;&+\; \sum_{\lambda_{R}=-1/2}^{1/2}{\delta_{\frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{0,0}\left(\sigma_{2}\right) \mathcal{H}^\mathrm{decay}_{\Lambda(1670), 0, \frac{1}{2}} \mathcal{H}^\mathrm{production}_{\Lambda(1670), \lambda_{R}, 0} d^{\frac{1}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)} \\ \;&+\; \sum_{\lambda_{R}=-3/2}^{3/2}{\delta_{\frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{2,1}\left(\sigma_{2}\right) \mathcal{H}^\mathrm{decay}_{\Lambda(1690), 0, \frac{1}{2}} \mathcal{H}^\mathrm{production}_{\Lambda(1690), \lambda_{R}, 0} d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)} \\ \;&+\; \sum_{\lambda_{R}=-1/2}^{1/2}{\delta_{\frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{0,0}\left(\sigma_{2}\right) \mathcal{H}^\mathrm{decay}_{\Lambda(2000), 0, \frac{1}{2}} \mathcal{H}^\mathrm{production}_{\Lambda(2000), \lambda_{R}, 0} d^{\frac{1}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)} \\ \;&+\; \sum_{\lambda_{R}=-1/2}^{1/2}{\frac{\delta_{\frac{1}{2} \lambda_{R}} F_{0}\left(R_{\Lambda_c} q_{0}\left(m_{0}^{2}\right)\right) \mathcal{R}^\mathrm{Flatté}\left(\sigma_{2}\right) \mathcal{H}^\mathrm{decay}_{\Lambda(1405), 0, \frac{1}{2}} \mathcal{H}^\mathrm{production}_{\Lambda(1405), \lambda_{R}, 0} d^{\frac{1}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)}{F_{0}\left(R_{\Lambda_c} q_{0}\left(m_{0}^{2}\right)\right)}} \\ A^{3}_{\frac{1}{2}, \frac{1}{2}} \;&=\; \sum_{\lambda_{R}=-3/2}^{3/2}{\delta_{\frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{1,1}\left(\sigma_{3}\right) \mathcal{H}^\mathrm{decay}_{\Delta(1232), \frac{1}{2}, 0} \mathcal{H}^\mathrm{production}_{\Delta(1232), \lambda_{R}, 0} d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{12}\right)} \\ \;&+\; \sum_{\lambda_{R}=-3/2}^{3/2}{\delta_{\frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{1,1}\left(\sigma_{3}\right) \mathcal{H}^\mathrm{decay}_{\Delta(1600), \frac{1}{2}, 0} \mathcal{H}^\mathrm{production}_{\Delta(1600), \lambda_{R}, 0} d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{12}\right)} \\ \;&+\; \sum_{\lambda_{R}=-3/2}^{3/2}{\delta_{\frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{2,1}\left(\sigma_{3}\right) \mathcal{H}^\mathrm{decay}_{\Delta(1700), \frac{1}{2}, 0} \mathcal{H}^\mathrm{production}_{\Delta(1700), \lambda_{R}, 0} d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{12}\right)} \\ \end{aligned}\end{split}\]

The \(\theta_{ij}\) angles are defined as:

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display_latex({k: v for k, v in model.variables.items() if "theta" in str(k)})
\[\begin{split}\displaystyle \begin{aligned} \theta_{23} \;&=\; \operatorname{acos}{\left(\frac{2 \sigma_{1} \left(- m_{1}^{2} - m_{2}^{2} + \sigma_{3}\right) - \left(m_{0}^{2} - m_{1}^{2} - \sigma_{1}\right) \left(m_{2}^{2} - m_{3}^{2} + \sigma_{1}\right)}{\sqrt{\lambda\left(m_{0}^{2}, m_{1}^{2}, \sigma_{1}\right)} \sqrt{\lambda\left(\sigma_{1}, m_{2}^{2}, m_{3}^{2}\right)}} \right)} \\ \theta_{31} \;&=\; \operatorname{acos}{\left(\frac{2 \sigma_{2} \left(- m_{2}^{2} - m_{3}^{2} + \sigma_{1}\right) - \left(m_{0}^{2} - m_{2}^{2} - \sigma_{2}\right) \left(- m_{1}^{2} + m_{3}^{2} + \sigma_{2}\right)}{\sqrt{\lambda\left(m_{0}^{2}, m_{2}^{2}, \sigma_{2}\right)} \sqrt{\lambda\left(\sigma_{2}, m_{3}^{2}, m_{1}^{2}\right)}} \right)} \\ \theta_{12} \;&=\; \operatorname{acos}{\left(\frac{2 \sigma_{3} \left(- m_{1}^{2} - m_{3}^{2} + \sigma_{2}\right) - \left(m_{0}^{2} - m_{3}^{2} - \sigma_{3}\right) \left(m_{1}^{2} - m_{2}^{2} + \sigma_{3}\right)}{\sqrt{\lambda\left(m_{0}^{2}, m_{3}^{2}, \sigma_{3}\right)} \sqrt{\lambda\left(\sigma_{3}, m_{1}^{2}, m_{2}^{2}\right)}} \right)} \\ \end{aligned}\end{split}\]

Definitions for the \(\phi_{ij}\) angles can be found under DPD angles.

1.3. Parameter definitions#

Parameter values are provided in model-definitions.yaml, but the keys of the helicity couplings have to remapped to the helicity symbols that are used in this amplitude model. The function parameter_key_to_symbol() implements this remapping, following the supplementary material of [1]. It is asserted below that:

  1. the keys are mapped to symbols that exist in the default amplitude model

  2. all parameter symbols in the default amplitude model have a value assigned to them.

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imported_parameter_values = load_model_parameters(
    "../data/model-definitions.yaml",
    amplitude_builder.decay,
    model_choice,
    particle_definitions=particles,
)
unfolded_intensity_expr = cached.unfold(model)
model_symbols = unfolded_intensity_expr.free_symbols

non_existent = set(imported_parameter_values) - set(model_symbols)
error_message = "Imported symbols that don't exist in model:\n  "
error_message += "\n  ".join(map(str, sorted(non_existent, key=str)))
assert non_existent == set(), error_message

undefined = (
    set(model_symbols)
    - set(imported_parameter_values)
    - set(model.parameter_defaults)
    - set(model.variables)
    - set(sp.symbols("sigma1:4", nonnegative=True))
)
undefined = {
    s
    for s in undefined
    if not str(s).endswith("{decay}")
    if not str(s).endswith("production}")
}
error_message = "Symbols in model that don't have a definition:\n  "
error_message += "\n  ".join(map(str, sorted(undefined, key=str)))
assert undefined == set(), error_message
model.parameter_defaults.update(imported_parameter_values)

1.3.1. Helicity coupling values#

1.3.1.1. Production couplings#

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production_couplings = {
    key: value
    for key, value in model.parameter_defaults.items()
    if isinstance(key, sp.Indexed)
    if "production" in str(key.base)
    if str(value) != "1"
}
display_latex(production_couplings)
\[\begin{split}\displaystyle \begin{aligned} \mathcal{H}^\mathrm{production}_{K(892), -1, - \frac{1}{2}} \;&=\; 1.192614-1.025814i \\ \mathcal{H}^\mathrm{production}_{\Lambda(1405), - \frac{1}{2}, 0} \;&=\; -4.572486+3.190144i \\ \mathcal{H}^\mathrm{production}_{\Lambda(1520), - \frac{1}{2}, 0} \;&=\; 0.293998+0.044324i \\ \mathcal{H}^\mathrm{production}_{\Lambda(1600), - \frac{1}{2}, 0} \;&=\; -4.840649-3.082786i \\ \mathcal{H}^\mathrm{production}_{\Lambda(1670), - \frac{1}{2}, 0} \;&=\; -0.339585-0.144678i \\ \mathcal{H}^\mathrm{production}_{\Lambda(1690), - \frac{1}{2}, 0} \;&=\; -0.385772-0.110235i \\ \mathcal{H}^\mathrm{production}_{\Lambda(2000), - \frac{1}{2}, 0} \;&=\; -8.014857-7.614006i \\ \mathcal{H}^\mathrm{production}_{\Delta(1232), - \frac{1}{2}, 0} \;&=\; -6.778191+3.051805i \\ \mathcal{H}^\mathrm{production}_{\Delta(1600), - \frac{1}{2}, 0} \;&=\; 11.401585-3.125511i \\ \mathcal{H}^\mathrm{production}_{\Delta(1700), - \frac{1}{2}, 0} \;&=\; -10.37828-1.434872i \\ \mathcal{H}^\mathrm{production}_{K(700), 0, \frac{1}{2}} \;&=\; 0.068908+2.521444i \\ \mathcal{H}^\mathrm{production}_{K(892), 0, \frac{1}{2}} \;&=\; -0.727145-4.155027i \\ \mathcal{H}^\mathrm{production}_{K(1430), 0, \frac{1}{2}} \;&=\; -6.71516+10.479411i \\ \mathcal{H}^\mathrm{production}_{K(700), 0, - \frac{1}{2}} \;&=\; -2.68563+0.03849i \\ \mathcal{H}^\mathrm{production}_{K(892), 0, - \frac{1}{2}} \;&=\; 1+0i \\ \mathcal{H}^\mathrm{production}_{K(1430), 0, - \frac{1}{2}} \;&=\; 0.219754+8.741196i \\ \mathcal{H}^\mathrm{production}_{\Lambda(1405), \frac{1}{2}, 0} \;&=\; 10.44608+2.787441i \\ \mathcal{H}^\mathrm{production}_{\Lambda(1520), \frac{1}{2}, 0} \;&=\; -0.160687+1.498833i \\ \mathcal{H}^\mathrm{production}_{\Lambda(1600), \frac{1}{2}, 0} \;&=\; 6.971233-0.842435i \\ \mathcal{H}^\mathrm{production}_{\Lambda(1670), \frac{1}{2}, 0} \;&=\; -0.570978+1.011833i \\ \mathcal{H}^\mathrm{production}_{\Lambda(1690), \frac{1}{2}, 0} \;&=\; -2.730592-0.353613i \\ \mathcal{H}^\mathrm{production}_{\Lambda(2000), \frac{1}{2}, 0} \;&=\; -4.336255-3.796192i \\ \mathcal{H}^\mathrm{production}_{\Delta(1232), \frac{1}{2}, 0} \;&=\; -12.987193+4.528336i \\ \mathcal{H}^\mathrm{production}_{\Delta(1600), \frac{1}{2}, 0} \;&=\; 6.729211-1.002383i \\ \mathcal{H}^\mathrm{production}_{\Delta(1700), \frac{1}{2}, 0} \;&=\; -12.874102-2.10557i \\ \mathcal{H}^\mathrm{production}_{K(892), 1, \frac{1}{2}} \;&=\; -3.141446-3.29341i \\ \end{aligned}\end{split}\]

1.3.1.2. Decay couplings#

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decay_couplings = {
    key: value
    for key, value in model.parameter_defaults.items()
    if isinstance(key, sp.Indexed)
    if "decay" in str(key.base)
}
display_latex(decay_couplings)
\[\begin{split}\displaystyle \begin{aligned} \mathcal{H}^\mathrm{decay}_{K(892), 0, 0} \;&=\; 1 \\ \mathcal{H}^\mathrm{decay}_{\Lambda(1405), 0, - \frac{1}{2}} \;&=\; 1 \\ \mathcal{H}^\mathrm{decay}_{\Lambda(1520), 0, - \frac{1}{2}} \;&=\; -1 \\ \mathcal{H}^\mathrm{decay}_{\Lambda(1600), 0, - \frac{1}{2}} \;&=\; -1 \\ \mathcal{H}^\mathrm{decay}_{\Lambda(1670), 0, - \frac{1}{2}} \;&=\; 1 \\ \mathcal{H}^\mathrm{decay}_{\Lambda(1690), 0, - \frac{1}{2}} \;&=\; -1 \\ \mathcal{H}^\mathrm{decay}_{\Lambda(2000), 0, - \frac{1}{2}} \;&=\; 1 \\ \mathcal{H}^\mathrm{decay}_{\Delta(1232), - \frac{1}{2}, 0} \;&=\; 1 \\ \mathcal{H}^\mathrm{decay}_{\Delta(1600), - \frac{1}{2}, 0} \;&=\; 1 \\ \mathcal{H}^\mathrm{decay}_{\Delta(1700), - \frac{1}{2}, 0} \;&=\; -1 \\ \mathcal{H}^\mathrm{decay}_{K(700), 0, 0} \;&=\; 1 \\ \mathcal{H}^\mathrm{decay}_{K(1430), 0, 0} \;&=\; 1 \\ \mathcal{H}^\mathrm{decay}_{\Lambda(1405), 0, \frac{1}{2}} \;&=\; 1 \\ \mathcal{H}^\mathrm{decay}_{\Lambda(1520), 0, \frac{1}{2}} \;&=\; 1 \\ \mathcal{H}^\mathrm{decay}_{\Lambda(1600), 0, \frac{1}{2}} \;&=\; 1 \\ \mathcal{H}^\mathrm{decay}_{\Lambda(1670), 0, \frac{1}{2}} \;&=\; 1 \\ \mathcal{H}^\mathrm{decay}_{\Lambda(1690), 0, \frac{1}{2}} \;&=\; 1 \\ \mathcal{H}^\mathrm{decay}_{\Lambda(2000), 0, \frac{1}{2}} \;&=\; 1 \\ \mathcal{H}^\mathrm{decay}_{\Delta(1232), \frac{1}{2}, 0} \;&=\; 1 \\ \mathcal{H}^\mathrm{decay}_{\Delta(1600), \frac{1}{2}, 0} \;&=\; 1 \\ \mathcal{H}^\mathrm{decay}_{\Delta(1700), \frac{1}{2}, 0} \;&=\; 1 \\ \end{aligned}\end{split}\]

1.3.2. Non-coupling parameters#

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couplings = set(production_couplings) | set(decay_couplings)
non_coupling_parameters = {
    symbol: model.parameter_defaults[symbol]
    for symbol in sorted(model.parameter_defaults, key=str)
    if not isinstance(symbol, sp.Indexed)
}
display_latex(non_coupling_parameters)
\[\begin{split}\displaystyle \begin{aligned} R_\mathrm{res} \;&=\; 1.5 \\ R_{\Lambda_c} \;&=\; 5 \\ \Gamma_{D(1232)} \;&=\; 0.117 \\ \Gamma_{D(1600)} \;&=\; 0.3 \\ \Gamma_{D(1700)} \;&=\; 0.38 \\ \Gamma_{K(1430)} \;&=\; 0.19 \\ \Gamma_{K(700)} \;&=\; 0.47800000000000004 \\ \Gamma_{K(892)} \;&=\; 0.047299999999999995 \\ \Gamma_{L(1405) \to K^- p} \;&=\; 0.0505 \\ \Gamma_{L(1405) \to \pi^+ \Sigma^-} \;&=\; 0.0505 \\ \Gamma_{L(1520)} \;&=\; 0.015195 \\ \Gamma_{L(1600)} \;&=\; 0.25 \\ \Gamma_{L(1670)} \;&=\; 0.03 \\ \Gamma_{L(1690)} \;&=\; 0.07 \\ \Gamma_{L(2000)} \;&=\; 0.17926 \\ \gamma_{K(1430)} \;&=\; 0.020981 \\ \gamma_{K(700)} \;&=\; 0.94106 \\ m_{0} \;&=\; 2.28646 \\ m_{1} \;&=\; 0.938272046 \\ m_{2} \;&=\; 0.13957018 \\ m_{3} \;&=\; 0.49367700000000003 \\ m_{D(1232)} \;&=\; 1.232 \\ m_{D(1600)} \;&=\; 1.6400000000000001 \\ m_{D(1700)} \;&=\; 1.69 \\ m_{K(1430)} \;&=\; 1.375 \\ m_{K(700)} \;&=\; 0.8240000000000001 \\ m_{K(892)} \;&=\; 0.8955000000000001 \\ m_{L(1405)} \;&=\; 1.4051 \\ m_{L(1520)} \;&=\; 1.518467 \\ m_{L(1600)} \;&=\; 1.6300000000000001 \\ m_{L(1670)} \;&=\; 1.67 \\ m_{L(1690)} \;&=\; 1.69 \\ m_{L(2000)} \;&=\; 1.98819 \\ m_{\Sigma^-} \;&=\; 1.1893699999999998 \\ \end{aligned}\end{split}\]
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