Amplitude model with ampform

PWA study on \(p \gamma \to \Lambda K^+ \pi^0\). We formulate the helicity amplitude model symbolically using AmpForm here.

Import Python libraries

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import logging
import os
import warnings
from collections import defaultdict

import ampform
import graphviz
import ipywidgets as w
import jax
import jax.numpy as jnp
import matplotlib.pyplot as plt
import numpy as np
import qrules
import sympy as sp
from ampform.dynamics.builder import RelativisticBreitWignerBuilder
from ampform.io import aslatex, improve_latex_rendering
from IPython.display import SVG, Image, Markdown, Math, display
from tensorwaves.data import (
    SympyDataTransformer,
    TFPhaseSpaceGenerator,
    TFUniformRealNumberGenerator,
)
from tensorwaves.function.sympy import create_parametrized_function

from gluex_nstar import create_pgamma, generate_markdown_table, load_particle_database

STATIC_PAGE = "EXECUTE_NB" in os.environ

os.environ["TF_CPP_MIN_LOG_LEVEL"] = "3"
logging.disable(logging.WARNING)
warnings.filterwarnings("ignore")

improve_latex_rendering()
particle_db = load_particle_database()

Decay definition

Particle definitions

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particles = ["Lambda", "K+", "pi0", "gamma", "p"]
src = generate_markdown_table(particle_db, particles)
Markdown(src)

Particle	Name	PID	\(J^{PC} (I^G)\)	\(I_3\)	\(M\)	\(\Gamma\)	\(Q\)	\(S\)	\(B\)
\(\Lambda\)	Lambda	3122	\(\frac{1}{2}^{+ } \; (0^{ })\)	\(0\)	1.12	2.515e-15	0	-1	1
\(K^{+}\)	K+	321	\(0^{- } \; (\frac{1}{2}^{ })\)	\(\frac{1}{2}\)	0.494	5.317e-17	1	1	0
\(\pi^{0}\)	pi0	111	\(0^{-+} \; (1^{-})\)	\(0\)	0.135	7.81e-09	0	0	0
\(\gamma\)	gamma	22	\(1^{--}\)	N/A	0	0	0	0	0
\(p\)	p	2212	\(\frac{1}{2}^{+ } \; (\frac{1}{2}^{ })\)	\(\frac{1}{2}\)	0.938	0	1	0	1

In the table above, PID is the PDG ID from PDG particle numbering scheme, \(J\) is the spin, \(P\) is the parity, \(C\) is the C parity, \(I\) is the isospin (magnitude), \(G\) is the G parity. \(I_3\) is the isospin projection (or the 3rd component), \(M\) is the mass, \(\Gamma\) is the width, \(Q\) is the charge, \(S\) is the strangeness number, and \(B\) is the baryon number.

Initial state definition

Mass for \(p \gamma\) system

E_lab_gamma = 8.5
m_proton = 0.938
m_0 = np.sqrt(2 * E_lab_gamma * m_proton + m_proton**2)
m_eta = 0.548
m_pi = 0.135
m_0

np.float64(4.101931740046389)

Add custom particle \(p \gamma\)

pgamma1, pgamma2 = create_pgamma(m_0)
particle_db.update([pgamma1, pgamma2])

Generate transitions

For simplicity, we use the initial state \(p \gamma\) (with spin-\(\frac{1}{2}\)), and set the allowed interaction type to be strong only, the formalism is selected to be helicity formalism instead of canonical.

See also

Helicity versus canonical

reaction = qrules.generate_transitions(
    initial_state=("pgamma1"),
    final_state=["Lambda", "K+", "pi0"],
    allowed_interaction_types=["strong"],
    formalism="helicity",
    particle_db=particle_db,
    max_angular_momentum=4,
    max_spin_magnitude=4,
    mass_conservation_factor=0,
)

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dot = qrules.io.asdot(reaction, collapse_graphs=True)
graphviz.Source(dot)

Resonance spectrum

Collect the resonances

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resonance_names = reaction.get_intermediate_particles().names
resonance_families = {
    R"$K^*$ resonances": "K(",
    R"$\Sigma^*$ resonances": "Sigma(",
    R"$N^*$ resonances": "N(",
}
tables = []
for title, prefix in resonance_families.items():
    names = sorted(
        (name for name in resonance_names if name.startswith(prefix)),
        key=lambda name: particle_db[name].mass,
    )
    table = generate_markdown_table(particle_db, names)
    tables.append(f"#### {title}\n\n{table}")
Markdown("\n".join(tables))

\(K^*\) resonances

Particle	Name	PID	\(J^{PC} (I^G)\)	\(I_3\)	\(M\)	\(\Gamma\)	\(Q\)	\(S\)	\(B\)
\(K_{0}^{*}(700)^{+}\)	K(0)*(700)+	9000321	\(0^{+ } \; (\frac{1}{2}^{ })\)	\(\frac{1}{2}\)	0.838	0.463	1	1	0
\(K_{2}^{*}(1430)^{+}\)	K(2)*(1430)+	325	\(2^{+ } \; (\frac{1}{2}^{ })\)	\(\frac{1}{2}\)	1.43	0.1	1	1	0
\(K_{0}^{*}(1430)^{+}\)	K(0)*(1430)+	10321	\(0^{+ } \; (\frac{1}{2}^{ })\)	\(\frac{1}{2}\)	1.43	0.27	1	1	0
\(K_{3}^{*}(1780)^{+}\)	K(3)*(1780)+	327	\(3^{- } \; (\frac{1}{2}^{ })\)	\(\frac{1}{2}\)	1.78	0.161	1	1	0
\(K_{0}^{*}(1950)^{+}\)	K(0)*(1950)+	9020321	\(0^{+ } \; (\frac{1}{2}^{ })\)	\(\frac{1}{2}\)	1.96	0.17	1	1	0
\(K_{2}^{*}(1980)^{+}\)	K(2)*(1980)+	9010325	\(2^{+ } \; (\frac{1}{2}^{ })\)	\(\frac{1}{2}\)	1.99	0.348	1	1	0
\(K_{4}^{*}(2045)^{+}\)	K(4)*(2045)+	329	\(4^{+ } \; (\frac{1}{2}^{ })\)	\(\frac{1}{2}\)	2.05	0.199	1	1	0

\(\Sigma^*\) resonances

Particle	Name	PID	\(J^{PC} (I^G)\)	\(I_3\)	\(M\)	\(\Gamma\)	\(Q\)	\(S\)	\(B\)
\(\Sigma(1385)^{0}\)	Sigma(1385)0	3214	\(\frac{3}{2}^{+ } \; (1^{ })\)	\(0\)	1.38	0.044	0	-1	1
\(\Sigma(1660)^{0}\)	Sigma(1660)0	13212	\(\frac{1}{2}^{+ } \; (1^{ })\)	\(0\)	1.66	0.2	0	-1	1
\(\Sigma(1670)^{0}\)	Sigma(1670)0	13214	\(\frac{3}{2}^{- } \; (1^{ })\)	\(0\)	1.68	0.07	0	-1	1
\(\Sigma(1750)^{0}\)	Sigma(1750)0	23212	\(\frac{1}{2}^{- } \; (1^{ })\)	\(0\)	1.75	0.15	0	-1	1
\(\Sigma(1775)^{0}\)	Sigma(1775)0	3216	\(\frac{5}{2}^{- } \; (1^{ })\)	\(0\)	1.77	0.12	0	-1	1
\(\Sigma(1940)^{0}\)	Sigma(1910)0	23214	\(\frac{3}{2}^{- } \; (1^{ })\)	\(0\)	1.91	0.22	0	-1	1
\(\Sigma(1915)^{0}\)	Sigma(1915)0	13216	\(\frac{5}{2}^{+ } \; (1^{ })\)	\(0\)	1.92	0.12	0	-1	1
\(\Sigma(2030)^{0}\)	Sigma(2030)0	3218	\(\frac{7}{2}^{+ } \; (1^{ })\)	\(0\)	2.03	0.18	0	-1	1

\(N^*\) resonances

Particle	Name	PID	\(J^{PC} (I^G)\)	\(I_3\)	\(M\)	\(\Gamma\)	\(Q\)	\(S\)	\(B\)
\(N(1650)^{+}\)	N(1650)+	32212	\(\frac{1}{2}^{- } \; (\frac{1}{2}^{ })\)	\(\frac{1}{2}\)	1.65	0.125	1	0	1
\(N(1675)^{+}\)	N(1675)+	2216	\(\frac{5}{2}^{- } \; (\frac{1}{2}^{ })\)	\(\frac{1}{2}\)	1.68	0.145	1	0	1
\(N(1680)^{+}\)	N(1680)+	12216	\(\frac{5}{2}^{+ } \; (\frac{1}{2}^{ })\)	\(\frac{1}{2}\)	1.69	0.12	1	0	1
\(N(1710)^{+}\)	N(1710)+	42212	\(\frac{1}{2}^{+ } \; (\frac{1}{2}^{ })\)	\(\frac{1}{2}\)	1.71	0.14	1	0	1
\(N(1720)^{+}\)	N(1720)+	32124	\(\frac{3}{2}^{+ } \; (\frac{1}{2}^{ })\)	\(\frac{1}{2}\)	1.72	0.25	1	0	1
\(N(1700)^{+}\)	N(1700)+	22124	\(\frac{3}{2}^{- } \; (\frac{1}{2}^{ })\)	\(\frac{1}{2}\)	1.72	0.2	1	0	1
\(N(1875)^{+}\)	N(1875)+	200002	\(\frac{3}{2}^{- } \; (\frac{1}{2}^{ })\)	\(\frac{1}{2}\)	1.88	0.2	1	0	1
\(N(1880)^{+}\)	N(1880)+	200000	\(\frac{1}{2}^{+ } \; (\frac{1}{2}^{ })\)	\(\frac{1}{2}\)	1.88	0.3	1	0	1
\(N(1895)^{+}\)	N(1895)+	200001	\(\frac{1}{2}^{- } \; (\frac{1}{2}^{ })\)	\(\frac{1}{2}\)	1.9	0.12	1	0	1
\(N(1900)^{+}\)	N(1900)+	200003	\(\frac{3}{2}^{+ } \; (\frac{1}{2}^{ })\)	\(\frac{1}{2}\)	1.92	0.2	1	0	1
\(N(2060)^{+}\)	N(2060)+	200004	\(\frac{5}{2}^{- } \; (\frac{1}{2}^{ })\)	\(\frac{1}{2}\)	2.1	0.4	1	0	1
\(N(2190)^{+}\)	N(2190)+	2128	\(\frac{7}{2}^{- } \; (\frac{1}{2}^{ })\)	\(\frac{1}{2}\)	2.18	0.4	1	0	1

Formulate amplitude model

model_builder = ampform.get_builder(reaction)
model_builder.config.scalar_initial_state_mass = True
model_builder.config.stable_final_state_ids = 0, 1, 2
bw_builder = RelativisticBreitWignerBuilder(
    energy_dependent_width=False,
    form_factor=False,
)
for name in reaction.get_intermediate_particles().names:
    model_builder.dynamics.assign(name, bw_builder)
model = model_builder.formulate()

model.intensity

\[\displaystyle \sum_{m_{A}=-1/2}^{1/2} \sum_{m_{0}=-1/2}^{1/2} \sum_{m_{1}=0} \sum_{m_{2}=0}{\left|{A^{01}_{m_{A}, m_{0}, m_{1}, m_{2}} + A^{02}_{m_{A}, m_{0}, m_{1}, m_{2}} + A^{12}_{m_{A}, m_{0}, m_{1}, m_{2}}}\right|^{2}}\]

The first term in the amplitude model:

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(symbol, expr), *_ = model.amplitudes.items()
Math(aslatex({symbol: expr}, terms_per_line=1))

\[\begin{split}\displaystyle \begin{aligned} A^{01}_{- \frac{1}{2}, - \frac{1}{2}, 0, 0} \;&=\; - \frac{C_{p\gamma (s1/2) \to N(1650)^{+}_{+1/2} \pi^{0}_{0}; N(1650)^{+} \to K^{+}_{0} \Lambda_{+1/2}} \Gamma_{N(1650)^{+}} m_{N(1650)^{+}} D^{\frac{1}{2}}_{- \frac{1}{2},- \frac{1}{2}}\left(- \phi_{01},\theta_{01},0\right) D^{\frac{1}{2}}_{- \frac{1}{2},- \frac{1}{2}}\left(- \phi^{01}_{0},\theta^{01}_{0},0\right)}{- i \Gamma_{N(1650)^{+}} m_{N(1650)^{+}} - m_{01}^{2} + \left(m_{N(1650)^{+}}\right)^{2}} \\ \;&+\; - \frac{C_{p\gamma (s1/2) \to N(1650)^{+}_{+1/2} \pi^{0}_{0}; N(1650)^{+} \to K^{+}_{0} \Lambda_{+1/2}} \Gamma_{N(1650)^{+}} m_{N(1650)^{+}} D^{\frac{1}{2}}_{- \frac{1}{2},\frac{1}{2}}\left(- \phi_{01},\theta_{01},0\right) D^{\frac{1}{2}}_{\frac{1}{2},- \frac{1}{2}}\left(- \phi^{01}_{0},\theta^{01}_{0},0\right)}{- i \Gamma_{N(1650)^{+}} m_{N(1650)^{+}} - m_{01}^{2} + \left(m_{N(1650)^{+}}\right)^{2}} \\ \;&+\; - \frac{C_{p\gamma (s1/2) \to N(1675)^{+}_{+1/2} \pi^{0}_{0}; N(1675)^{+} \to K^{+}_{0} \Lambda_{+1/2}} \Gamma_{N(1675)^{+}} m_{N(1675)^{+}} D^{\frac{1}{2}}_{- \frac{1}{2},- \frac{1}{2}}\left(- \phi_{01},\theta_{01},0\right) D^{\frac{5}{2}}_{- \frac{1}{2},- \frac{1}{2}}\left(- \phi^{01}_{0},\theta^{01}_{0},0\right)}{- i \Gamma_{N(1675)^{+}} m_{N(1675)^{+}} - m_{01}^{2} + \left(m_{N(1675)^{+}}\right)^{2}} \\ \;&+\; - \frac{C_{p\gamma (s1/2) \to N(1675)^{+}_{+1/2} \pi^{0}_{0}; N(1675)^{+} \to K^{+}_{0} \Lambda_{+1/2}} \Gamma_{N(1675)^{+}} m_{N(1675)^{+}} D^{\frac{1}{2}}_{- \frac{1}{2},\frac{1}{2}}\left(- \phi_{01},\theta_{01},0\right) D^{\frac{5}{2}}_{\frac{1}{2},- \frac{1}{2}}\left(- \phi^{01}_{0},\theta^{01}_{0},0\right)}{- i \Gamma_{N(1675)^{+}} m_{N(1675)^{+}} - m_{01}^{2} + \left(m_{N(1675)^{+}}\right)^{2}} \\ \;&+\; - \frac{C_{p\gamma (s1/2) \to N(1680)^{+}_{+1/2} \pi^{0}_{0}; N(1680)^{+} \to K^{+}_{0} \Lambda_{+1/2}} \Gamma_{N(1680)^{+}} m_{N(1680)^{+}} D^{\frac{1}{2}}_{- \frac{1}{2},- \frac{1}{2}}\left(- \phi_{01},\theta_{01},0\right) D^{\frac{5}{2}}_{- \frac{1}{2},- \frac{1}{2}}\left(- \phi^{01}_{0},\theta^{01}_{0},0\right)}{- i \Gamma_{N(1680)^{+}} m_{N(1680)^{+}} - m_{01}^{2} + \left(m_{N(1680)^{+}}\right)^{2}} \\ \;&+\; - \frac{C_{p\gamma (s1/2) \to N(1680)^{+}_{+1/2} \pi^{0}_{0}; N(1680)^{+} \to K^{+}_{0} \Lambda_{+1/2}} \Gamma_{N(1680)^{+}} m_{N(1680)^{+}} D^{\frac{1}{2}}_{- \frac{1}{2},\frac{1}{2}}\left(- \phi_{01},\theta_{01},0\right) D^{\frac{5}{2}}_{\frac{1}{2},- \frac{1}{2}}\left(- \phi^{01}_{0},\theta^{01}_{0},0\right)}{- i \Gamma_{N(1680)^{+}} m_{N(1680)^{+}} - m_{01}^{2} + \left(m_{N(1680)^{+}}\right)^{2}} \\ \;&+\; - \frac{C_{p\gamma (s1/2) \to N(1700)^{+}_{+1/2} \pi^{0}_{0}; N(1700)^{+} \to K^{+}_{0} \Lambda_{+1/2}} \Gamma_{N(1700)^{+}} m_{N(1700)^{+}} D^{\frac{1}{2}}_{- \frac{1}{2},- \frac{1}{2}}\left(- \phi_{01},\theta_{01},0\right) D^{\frac{3}{2}}_{- \frac{1}{2},- \frac{1}{2}}\left(- \phi^{01}_{0},\theta^{01}_{0},0\right)}{- i \Gamma_{N(1700)^{+}} m_{N(1700)^{+}} - m_{01}^{2} + \left(m_{N(1700)^{+}}\right)^{2}} \\ \;&+\; - \frac{C_{p\gamma (s1/2) \to N(1700)^{+}_{+1/2} \pi^{0}_{0}; N(1700)^{+} \to K^{+}_{0} \Lambda_{+1/2}} \Gamma_{N(1700)^{+}} m_{N(1700)^{+}} D^{\frac{1}{2}}_{- \frac{1}{2},\frac{1}{2}}\left(- \phi_{01},\theta_{01},0\right) D^{\frac{3}{2}}_{\frac{1}{2},- \frac{1}{2}}\left(- \phi^{01}_{0},\theta^{01}_{0},0\right)}{- i \Gamma_{N(1700)^{+}} m_{N(1700)^{+}} - m_{01}^{2} + \left(m_{N(1700)^{+}}\right)^{2}} \\ \;&+\; - \frac{C_{p\gamma (s1/2) \to N(1710)^{+}_{+1/2} \pi^{0}_{0}; N(1710)^{+} \to K^{+}_{0} \Lambda_{+1/2}} \Gamma_{N(1710)^{+}} m_{N(1710)^{+}} D^{\frac{1}{2}}_{- \frac{1}{2},- \frac{1}{2}}\left(- \phi_{01},\theta_{01},0\right) D^{\frac{1}{2}}_{- \frac{1}{2},- \frac{1}{2}}\left(- \phi^{01}_{0},\theta^{01}_{0},0\right)}{- i \Gamma_{N(1710)^{+}} m_{N(1710)^{+}} - m_{01}^{2} + \left(m_{N(1710)^{+}}\right)^{2}} \\ \;&+\; - \frac{C_{p\gamma (s1/2) \to N(1710)^{+}_{+1/2} \pi^{0}_{0}; N(1710)^{+} \to K^{+}_{0} \Lambda_{+1/2}} \Gamma_{N(1710)^{+}} m_{N(1710)^{+}} D^{\frac{1}{2}}_{- \frac{1}{2},\frac{1}{2}}\left(- \phi_{01},\theta_{01},0\right) D^{\frac{1}{2}}_{\frac{1}{2},- \frac{1}{2}}\left(- \phi^{01}_{0},\theta^{01}_{0},0\right)}{- i \Gamma_{N(1710)^{+}} m_{N(1710)^{+}} - m_{01}^{2} + \left(m_{N(1710)^{+}}\right)^{2}} \\ \;&+\; - \frac{C_{p\gamma (s1/2) \to N(1720)^{+}_{+1/2} \pi^{0}_{0}; N(1720)^{+} \to K^{+}_{0} \Lambda_{+1/2}} \Gamma_{N(1720)^{+}} m_{N(1720)^{+}} D^{\frac{1}{2}}_{- \frac{1}{2},- \frac{1}{2}}\left(- \phi_{01},\theta_{01},0\right) D^{\frac{3}{2}}_{- \frac{1}{2},- \frac{1}{2}}\left(- \phi^{01}_{0},\theta^{01}_{0},0\right)}{- i \Gamma_{N(1720)^{+}} m_{N(1720)^{+}} - m_{01}^{2} + \left(m_{N(1720)^{+}}\right)^{2}} \\ \;&+\; - \frac{C_{p\gamma (s1/2) \to N(1720)^{+}_{+1/2} \pi^{0}_{0}; N(1720)^{+} \to K^{+}_{0} \Lambda_{+1/2}} \Gamma_{N(1720)^{+}} m_{N(1720)^{+}} D^{\frac{1}{2}}_{- \frac{1}{2},\frac{1}{2}}\left(- \phi_{01},\theta_{01},0\right) D^{\frac{3}{2}}_{\frac{1}{2},- \frac{1}{2}}\left(- \phi^{01}_{0},\theta^{01}_{0},0\right)}{- i \Gamma_{N(1720)^{+}} m_{N(1720)^{+}} - m_{01}^{2} + \left(m_{N(1720)^{+}}\right)^{2}} \\ \;&+\; - \frac{C_{p\gamma (s1/2) \to N(1875)^{+}_{+1/2} \pi^{0}_{0}; N(1875)^{+} \to K^{+}_{0} \Lambda_{+1/2}} \Gamma_{N(1875)^{+}} m_{N(1875)^{+}} D^{\frac{1}{2}}_{- \frac{1}{2},- \frac{1}{2}}\left(- \phi_{01},\theta_{01},0\right) D^{\frac{3}{2}}_{- \frac{1}{2},- \frac{1}{2}}\left(- \phi^{01}_{0},\theta^{01}_{0},0\right)}{- i \Gamma_{N(1875)^{+}} m_{N(1875)^{+}} - m_{01}^{2} + \left(m_{N(1875)^{+}}\right)^{2}} \\ \;&+\; - \frac{C_{p\gamma (s1/2) \to N(1875)^{+}_{+1/2} \pi^{0}_{0}; N(1875)^{+} \to K^{+}_{0} \Lambda_{+1/2}} \Gamma_{N(1875)^{+}} m_{N(1875)^{+}} D^{\frac{1}{2}}_{- \frac{1}{2},\frac{1}{2}}\left(- \phi_{01},\theta_{01},0\right) D^{\frac{3}{2}}_{\frac{1}{2},- \frac{1}{2}}\left(- \phi^{01}_{0},\theta^{01}_{0},0\right)}{- i \Gamma_{N(1875)^{+}} m_{N(1875)^{+}} - m_{01}^{2} + \left(m_{N(1875)^{+}}\right)^{2}} \\ \;&+\; - \frac{C_{p\gamma (s1/2) \to N(1880)^{+}_{+1/2} \pi^{0}_{0}; N(1880)^{+} \to K^{+}_{0} \Lambda_{+1/2}} \Gamma_{N(1880)^{+}} m_{N(1880)^{+}} D^{\frac{1}{2}}_{- \frac{1}{2},- \frac{1}{2}}\left(- \phi_{01},\theta_{01},0\right) D^{\frac{1}{2}}_{- \frac{1}{2},- \frac{1}{2}}\left(- \phi^{01}_{0},\theta^{01}_{0},0\right)}{- i \Gamma_{N(1880)^{+}} m_{N(1880)^{+}} - m_{01}^{2} + \left(m_{N(1880)^{+}}\right)^{2}} \\ \;&+\; - \frac{C_{p\gamma (s1/2) \to N(1880)^{+}_{+1/2} \pi^{0}_{0}; N(1880)^{+} \to K^{+}_{0} \Lambda_{+1/2}} \Gamma_{N(1880)^{+}} m_{N(1880)^{+}} D^{\frac{1}{2}}_{- \frac{1}{2},\frac{1}{2}}\left(- \phi_{01},\theta_{01},0\right) D^{\frac{1}{2}}_{\frac{1}{2},- \frac{1}{2}}\left(- \phi^{01}_{0},\theta^{01}_{0},0\right)}{- i \Gamma_{N(1880)^{+}} m_{N(1880)^{+}} - m_{01}^{2} + \left(m_{N(1880)^{+}}\right)^{2}} \\ \;&+\; - \frac{C_{p\gamma (s1/2) \to N(1895)^{+}_{+1/2} \pi^{0}_{0}; N(1895)^{+} \to K^{+}_{0} \Lambda_{+1/2}} \Gamma_{N(1895)^{+}} m_{N(1895)^{+}} D^{\frac{1}{2}}_{- \frac{1}{2},- \frac{1}{2}}\left(- \phi_{01},\theta_{01},0\right) D^{\frac{1}{2}}_{- \frac{1}{2},- \frac{1}{2}}\left(- \phi^{01}_{0},\theta^{01}_{0},0\right)}{- i \Gamma_{N(1895)^{+}} m_{N(1895)^{+}} - m_{01}^{2} + \left(m_{N(1895)^{+}}\right)^{2}} \\ \;&+\; - \frac{C_{p\gamma (s1/2) \to N(1895)^{+}_{+1/2} \pi^{0}_{0}; N(1895)^{+} \to K^{+}_{0} \Lambda_{+1/2}} \Gamma_{N(1895)^{+}} m_{N(1895)^{+}} D^{\frac{1}{2}}_{- \frac{1}{2},\frac{1}{2}}\left(- \phi_{01},\theta_{01},0\right) D^{\frac{1}{2}}_{\frac{1}{2},- \frac{1}{2}}\left(- \phi^{01}_{0},\theta^{01}_{0},0\right)}{- i \Gamma_{N(1895)^{+}} m_{N(1895)^{+}} - m_{01}^{2} + \left(m_{N(1895)^{+}}\right)^{2}} \\ \;&+\; - \frac{C_{p\gamma (s1/2) \to N(1900)^{+}_{+1/2} \pi^{0}_{0}; N(1900)^{+} \to K^{+}_{0} \Lambda_{+1/2}} \Gamma_{N(1900)^{+}} m_{N(1900)^{+}} D^{\frac{1}{2}}_{- \frac{1}{2},- \frac{1}{2}}\left(- \phi_{01},\theta_{01},0\right) D^{\frac{3}{2}}_{- \frac{1}{2},- \frac{1}{2}}\left(- \phi^{01}_{0},\theta^{01}_{0},0\right)}{- i \Gamma_{N(1900)^{+}} m_{N(1900)^{+}} - m_{01}^{2} + \left(m_{N(1900)^{+}}\right)^{2}} \\ \;&+\; - \frac{C_{p\gamma (s1/2) \to N(1900)^{+}_{+1/2} \pi^{0}_{0}; N(1900)^{+} \to K^{+}_{0} \Lambda_{+1/2}} \Gamma_{N(1900)^{+}} m_{N(1900)^{+}} D^{\frac{1}{2}}_{- \frac{1}{2},\frac{1}{2}}\left(- \phi_{01},\theta_{01},0\right) D^{\frac{3}{2}}_{\frac{1}{2},- \frac{1}{2}}\left(- \phi^{01}_{0},\theta^{01}_{0},0\right)}{- i \Gamma_{N(1900)^{+}} m_{N(1900)^{+}} - m_{01}^{2} + \left(m_{N(1900)^{+}}\right)^{2}} \\ \;&+\; - \frac{C_{p\gamma (s1/2) \to N(2060)^{+}_{+1/2} \pi^{0}_{0}; N(2060)^{+} \to K^{+}_{0} \Lambda_{+1/2}} \Gamma_{N(2060)^{+}} m_{N(2060)^{+}} D^{\frac{1}{2}}_{- \frac{1}{2},- \frac{1}{2}}\left(- \phi_{01},\theta_{01},0\right) D^{\frac{5}{2}}_{- \frac{1}{2},- \frac{1}{2}}\left(- \phi^{01}_{0},\theta^{01}_{0},0\right)}{- i \Gamma_{N(2060)^{+}} m_{N(2060)^{+}} - m_{01}^{2} + \left(m_{N(2060)^{+}}\right)^{2}} \\ \;&+\; - \frac{C_{p\gamma (s1/2) \to N(2060)^{+}_{+1/2} \pi^{0}_{0}; N(2060)^{+} \to K^{+}_{0} \Lambda_{+1/2}} \Gamma_{N(2060)^{+}} m_{N(2060)^{+}} D^{\frac{1}{2}}_{- \frac{1}{2},\frac{1}{2}}\left(- \phi_{01},\theta_{01},0\right) D^{\frac{5}{2}}_{\frac{1}{2},- \frac{1}{2}}\left(- \phi^{01}_{0},\theta^{01}_{0},0\right)}{- i \Gamma_{N(2060)^{+}} m_{N(2060)^{+}} - m_{01}^{2} + \left(m_{N(2060)^{+}}\right)^{2}} \\ \;&+\; - \frac{C_{p\gamma (s1/2) \to N(2190)^{+}_{+1/2} \pi^{0}_{0}; N(2190)^{+} \to K^{+}_{0} \Lambda_{+1/2}} \Gamma_{N(2190)^{+}} m_{N(2190)^{+}} D^{\frac{1}{2}}_{- \frac{1}{2},- \frac{1}{2}}\left(- \phi_{01},\theta_{01},0\right) D^{\frac{7}{2}}_{- \frac{1}{2},- \frac{1}{2}}\left(- \phi^{01}_{0},\theta^{01}_{0},0\right)}{- i \Gamma_{N(2190)^{+}} m_{N(2190)^{+}} - m_{01}^{2} + \left(m_{N(2190)^{+}}\right)^{2}} \\ \;&+\; - \frac{C_{p\gamma (s1/2) \to N(2190)^{+}_{+1/2} \pi^{0}_{0}; N(2190)^{+} \to K^{+}_{0} \Lambda_{+1/2}} \Gamma_{N(2190)^{+}} m_{N(2190)^{+}} D^{\frac{1}{2}}_{- \frac{1}{2},\frac{1}{2}}\left(- \phi_{01},\theta_{01},0\right) D^{\frac{7}{2}}_{\frac{1}{2},- \frac{1}{2}}\left(- \phi^{01}_{0},\theta^{01}_{0},0\right)}{- i \Gamma_{N(2190)^{+}} m_{N(2190)^{+}} - m_{01}^{2} + \left(m_{N(2190)^{+}}\right)^{2}} \\ \end{aligned}\end{split}\]

Model parameters

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Math(aslatex(model.parameter_defaults))

\[\begin{split}\displaystyle \begin{aligned} m_{K_{0}^{*}(700)^{+}} \;&=\; 0.838 \\ \Gamma_{K_{0}^{*}(700)^{+}} \;&=\; 0.463 \\ C_{p\gamma (s1/2) \to {K_{0}^{*}(700)^{+}}_{0} \Lambda_{+1/2}; K_{0}^{*}(700)^{+} \to K^{+}_{0} \pi^{0}_{0}} \;&=\; 1+0i \\ m_{K^{*}(892)^{+}} \;&=\; 0.89188 \\ \Gamma_{K^{*}(892)^{+}} \;&=\; 0.0485 \\ C_{p\gamma (s1/2) \to K^{*}(892)^{+}_{+1} \Lambda_{+1/2}; K^{*}(892)^{+} \to K^{+}_{0} \pi^{0}_{0}} \;&=\; 1+0i \\ C_{p\gamma (s1/2) \to K^{*}(892)^{+}_{0} \Lambda_{+1/2}; K^{*}(892)^{+} \to K^{+}_{0} \pi^{0}_{0}} \;&=\; 1+0i \\ m_{K^{*}(1410)^{+}} \;&=\; 1.414 \\ \Gamma_{K^{*}(1410)^{+}} \;&=\; 0.232 \\ C_{p\gamma (s1/2) \to K^{*}(1410)^{+}_{+1} \Lambda_{+1/2}; K^{*}(1410)^{+} \to K^{+}_{0} \pi^{0}_{0}} \;&=\; 1+0i \\ C_{p\gamma (s1/2) \to K^{*}(1410)^{+}_{0} \Lambda_{+1/2}; K^{*}(1410)^{+} \to K^{+}_{0} \pi^{0}_{0}} \;&=\; 1+0i \\ m_{K_{2}^{*}(1430)^{+}} \;&=\; 1.4273 \\ \Gamma_{K_{2}^{*}(1430)^{+}} \;&=\; 0.1 \\ C_{p\gamma (s1/2) \to {K_{2}^{*}(1430)^{+}}_{+1} \Lambda_{+1/2}; K_{2}^{*}(1430)^{+} \to K^{+}_{0} \pi^{0}_{0}} \;&=\; 1+0i \\ C_{p\gamma (s1/2) \to {K_{2}^{*}(1430)^{+}}_{0} \Lambda_{+1/2}; K_{2}^{*}(1430)^{+} \to K^{+}_{0} \pi^{0}_{0}} \;&=\; 1+0i \\ m_{K_{0}^{*}(1430)^{+}} \;&=\; 1.43 \\ \Gamma_{K_{0}^{*}(1430)^{+}} \;&=\; 0.27 \\ C_{p\gamma (s1/2) \to {K_{0}^{*}(1430)^{+}}_{0} \Lambda_{+1/2}; K_{0}^{*}(1430)^{+} \to K^{+}_{0} \pi^{0}_{0}} \;&=\; 1+0i \\ m_{K^{*}(1680)^{+}} \;&=\; 1.718 \\ \Gamma_{K^{*}(1680)^{+}} \;&=\; 0.32 \\ C_{p\gamma (s1/2) \to K^{*}(1680)^{+}_{+1} \Lambda_{+1/2}; K^{*}(1680)^{+} \to K^{+}_{0} \pi^{0}_{0}} \;&=\; 1+0i \\ C_{p\gamma (s1/2) \to K^{*}(1680)^{+}_{0} \Lambda_{+1/2}; K^{*}(1680)^{+} \to K^{+}_{0} \pi^{0}_{0}} \;&=\; 1+0i \\ m_{K_{3}^{*}(1780)^{+}} \;&=\; 1.779 \\ \Gamma_{K_{3}^{*}(1780)^{+}} \;&=\; 0.161 \\ C_{p\gamma (s1/2) \to {K_{3}^{*}(1780)^{+}}_{+1} \Lambda_{+1/2}; K_{3}^{*}(1780)^{+} \to K^{+}_{0} \pi^{0}_{0}} \;&=\; 1+0i \\ C_{p\gamma (s1/2) \to {K_{3}^{*}(1780)^{+}}_{0} \Lambda_{+1/2}; K_{3}^{*}(1780)^{+} \to K^{+}_{0} \pi^{0}_{0}} \;&=\; 1+0i \\ m_{K_{0}^{*}(1950)^{+}} \;&=\; 1.957 \\ \Gamma_{K_{0}^{*}(1950)^{+}} \;&=\; 0.17 \\ C_{p\gamma (s1/2) \to {K_{0}^{*}(1950)^{+}}_{0} \Lambda_{+1/2}; K_{0}^{*}(1950)^{+} \to K^{+}_{0} \pi^{0}_{0}} \;&=\; 1+0i \\ m_{K_{2}^{*}(1980)^{+}} \;&=\; 1.99 \\ \Gamma_{K_{2}^{*}(1980)^{+}} \;&=\; 0.348 \\ C_{p\gamma (s1/2) \to {K_{2}^{*}(1980)^{+}}_{+1} \Lambda_{+1/2}; K_{2}^{*}(1980)^{+} \to K^{+}_{0} \pi^{0}_{0}} \;&=\; 1+0i \\ C_{p\gamma (s1/2) \to {K_{2}^{*}(1980)^{+}}_{0} \Lambda_{+1/2}; K_{2}^{*}(1980)^{+} \to K^{+}_{0} \pi^{0}_{0}} \;&=\; 1+0i \\ m_{K_{4}^{*}(2045)^{+}} \;&=\; 2.048 \\ \Gamma_{K_{4}^{*}(2045)^{+}} \;&=\; 0.199 \\ C_{p\gamma (s1/2) \to {K_{4}^{*}(2045)^{+}}_{+1} \Lambda_{+1/2}; K_{4}^{*}(2045)^{+} \to K^{+}_{0} \pi^{0}_{0}} \;&=\; 1+0i \\ C_{p\gamma (s1/2) \to {K_{4}^{*}(2045)^{+}}_{0} \Lambda_{+1/2}; K_{4}^{*}(2045)^{+} \to K^{+}_{0} \pi^{0}_{0}} \;&=\; 1+0i \\ m_{\Sigma(1385)^{0}} \;&=\; 1.3838 \\ \Gamma_{\Sigma(1385)^{0}} \;&=\; 0.044 \\ C_{p\gamma (s1/2) \to K^{+}_{0} \Sigma(1385)^{0}_{+1/2}; \Sigma(1385)^{0} \to \Lambda_{+1/2} \pi^{0}_{0}} \;&=\; 1+0i \\ m_{\Sigma(1660)^{0}} \;&=\; 1.66 \\ \Gamma_{\Sigma(1660)^{0}} \;&=\; 0.2 \\ C_{p\gamma (s1/2) \to K^{+}_{0} \Sigma(1660)^{0}_{+1/2}; \Sigma(1660)^{0} \to \Lambda_{+1/2} \pi^{0}_{0}} \;&=\; 1+0i \\ m_{\Sigma(1670)^{0}} \;&=\; 1.675 \\ \Gamma_{\Sigma(1670)^{0}} \;&=\; 0.07 \\ C_{p\gamma (s1/2) \to K^{+}_{0} \Sigma(1670)^{0}_{+1/2}; \Sigma(1670)^{0} \to \Lambda_{+1/2} \pi^{0}_{0}} \;&=\; 1+0i \\ m_{\Sigma(1750)^{0}} \;&=\; 1.75 \\ \Gamma_{\Sigma(1750)^{0}} \;&=\; 0.15 \\ C_{p\gamma (s1/2) \to K^{+}_{0} \Sigma(1750)^{0}_{+1/2}; \Sigma(1750)^{0} \to \Lambda_{+1/2} \pi^{0}_{0}} \;&=\; 1+0i \\ m_{\Sigma(1775)^{0}} \;&=\; 1.775 \\ \Gamma_{\Sigma(1775)^{0}} \;&=\; 0.12 \\ C_{p\gamma (s1/2) \to K^{+}_{0} \Sigma(1775)^{0}_{+1/2}; \Sigma(1775)^{0} \to \Lambda_{+1/2} \pi^{0}_{0}} \;&=\; 1+0i \\ m_{\Sigma(1940)^{0}} \;&=\; 1.91 \\ \Gamma_{\Sigma(1940)^{0}} \;&=\; 0.22 \\ C_{p\gamma (s1/2) \to K^{+}_{0} \Sigma(1940)^{0}_{+1/2}; \Sigma(1940)^{0} \to \Lambda_{+1/2} \pi^{0}_{0}} \;&=\; 1+0i \\ m_{\Sigma(1915)^{0}} \;&=\; 1.915 \\ \Gamma_{\Sigma(1915)^{0}} \;&=\; 0.12 \\ C_{p\gamma (s1/2) \to K^{+}_{0} \Sigma(1915)^{0}_{+1/2}; \Sigma(1915)^{0} \to \Lambda_{+1/2} \pi^{0}_{0}} \;&=\; 1+0i \\ m_{\Sigma(2030)^{0}} \;&=\; 2.03 \\ \Gamma_{\Sigma(2030)^{0}} \;&=\; 0.18 \\ C_{p\gamma (s1/2) \to K^{+}_{0} \Sigma(2030)^{0}_{+1/2}; \Sigma(2030)^{0} \to \Lambda_{+1/2} \pi^{0}_{0}} \;&=\; 1+0i \\ m_{N(1650)^{+}} \;&=\; 1.65 \\ \Gamma_{N(1650)^{+}} \;&=\; 0.125 \\ C_{p\gamma (s1/2) \to N(1650)^{+}_{+1/2} \pi^{0}_{0}; N(1650)^{+} \to K^{+}_{0} \Lambda_{+1/2}} \;&=\; 1+0i \\ m_{N(1675)^{+}} \;&=\; 1.675 \\ \Gamma_{N(1675)^{+}} \;&=\; 0.145 \\ C_{p\gamma (s1/2) \to N(1675)^{+}_{+1/2} \pi^{0}_{0}; N(1675)^{+} \to K^{+}_{0} \Lambda_{+1/2}} \;&=\; 1+0i \\ m_{N(1680)^{+}} \;&=\; 1.685 \\ \Gamma_{N(1680)^{+}} \;&=\; 0.12 \\ C_{p\gamma (s1/2) \to N(1680)^{+}_{+1/2} \pi^{0}_{0}; N(1680)^{+} \to K^{+}_{0} \Lambda_{+1/2}} \;&=\; 1+0i \\ m_{N(1710)^{+}} \;&=\; 1.71 \\ \Gamma_{N(1710)^{+}} \;&=\; 0.14 \\ C_{p\gamma (s1/2) \to N(1710)^{+}_{+1/2} \pi^{0}_{0}; N(1710)^{+} \to K^{+}_{0} \Lambda_{+1/2}} \;&=\; 1+0i \\ m_{N(1720)^{+}} \;&=\; 1.72 \\ \Gamma_{N(1720)^{+}} \;&=\; 0.25 \\ C_{p\gamma (s1/2) \to N(1720)^{+}_{+1/2} \pi^{0}_{0}; N(1720)^{+} \to K^{+}_{0} \Lambda_{+1/2}} \;&=\; 1+0i \\ m_{N(1700)^{+}} \;&=\; 1.72 \\ \Gamma_{N(1700)^{+}} \;&=\; 0.2 \\ C_{p\gamma (s1/2) \to N(1700)^{+}_{+1/2} \pi^{0}_{0}; N(1700)^{+} \to K^{+}_{0} \Lambda_{+1/2}} \;&=\; 1+0i \\ m_{N(1875)^{+}} \;&=\; 1.875 \\ \Gamma_{N(1875)^{+}} \;&=\; 0.2 \\ C_{p\gamma (s1/2) \to N(1875)^{+}_{+1/2} \pi^{0}_{0}; N(1875)^{+} \to K^{+}_{0} \Lambda_{+1/2}} \;&=\; 1+0i \\ m_{N(1880)^{+}} \;&=\; 1.88 \\ \Gamma_{N(1880)^{+}} \;&=\; 0.3 \\ C_{p\gamma (s1/2) \to N(1880)^{+}_{+1/2} \pi^{0}_{0}; N(1880)^{+} \to K^{+}_{0} \Lambda_{+1/2}} \;&=\; 1+0i \\ m_{N(1895)^{+}} \;&=\; 1.895 \\ \Gamma_{N(1895)^{+}} \;&=\; 0.12 \\ C_{p\gamma (s1/2) \to N(1895)^{+}_{+1/2} \pi^{0}_{0}; N(1895)^{+} \to K^{+}_{0} \Lambda_{+1/2}} \;&=\; 1+0i \\ m_{N(1900)^{+}} \;&=\; 1.92 \\ \Gamma_{N(1900)^{+}} \;&=\; 0.2 \\ C_{p\gamma (s1/2) \to N(1900)^{+}_{+1/2} \pi^{0}_{0}; N(1900)^{+} \to K^{+}_{0} \Lambda_{+1/2}} \;&=\; 1+0i \\ m_{N(2060)^{+}} \;&=\; 2.1 \\ \Gamma_{N(2060)^{+}} \;&=\; 0.4 \\ C_{p\gamma (s1/2) \to N(2060)^{+}_{+1/2} \pi^{0}_{0}; N(2060)^{+} \to K^{+}_{0} \Lambda_{+1/2}} \;&=\; 1+0i \\ m_{N(2190)^{+}} \;&=\; 2.18 \\ \Gamma_{N(2190)^{+}} \;&=\; 0.4 \\ C_{p\gamma (s1/2) \to N(2190)^{+}_{+1/2} \pi^{0}_{0}; N(2190)^{+} \to K^{+}_{0} \Lambda_{+1/2}} \;&=\; 1+0i \\ m_{0} \;&=\; 1.115683 \\ m_{1} \;&=\; 0.49367700000000003 \\ m_{2} \;&=\; 0.1349768 \\ m_{012} \;&=\; 4.101931740046389 \\ \end{aligned}\end{split}\]

Kinematic variable definitions

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Math(aslatex(model.kinematic_variables))

\[\begin{split}\displaystyle \begin{aligned} m_{01} \;&=\; m_{{p}_{01}} \\ m_{02} \;&=\; m_{{p}_{02}} \\ m_{12} \;&=\; m_{{p}_{12}} \\ \phi_{0} \;&=\; \phi\left({p}_{12}\right) \\ \phi^{01}_{0} \;&=\; \phi\left(\boldsymbol{B_z}\left(\frac{\left|\vec{{p}_{01}}\right|}{E\left({p}_{01}\right)}\right) \boldsymbol{R_y}\left(- \theta\left({p}_{01}\right)\right) \boldsymbol{R_z}\left(- \phi\left({p}_{01}\right)\right) p_{0}\right) \\ \phi^{02}_{0} \;&=\; \phi\left(\boldsymbol{B_z}\left(\frac{\left|\vec{{p}_{02}}\right|}{E\left({p}_{02}\right)}\right) \boldsymbol{R_y}\left(- \theta\left({p}_{02}\right)\right) \boldsymbol{R_z}\left(- \phi\left({p}_{02}\right)\right) p_{0}\right) \\ \phi_{01} \;&=\; \phi\left({p}_{01}\right) \\ \phi^{12}_{1} \;&=\; \phi\left(\boldsymbol{B_z}\left(\frac{\left|\vec{{p}_{12}}\right|}{E\left({p}_{12}\right)}\right) \boldsymbol{R_y}\left(- \theta\left({p}_{12}\right)\right) \boldsymbol{R_z}\left(- \phi\left({p}_{12}\right)\right) p_{1}\right) \\ \phi_{02} \;&=\; \phi\left({p}_{02}\right) \\ \theta_{0} \;&=\; \theta\left({p}_{12}\right) \\ \theta^{01}_{0} \;&=\; \theta\left(\boldsymbol{B_z}\left(\frac{\left|\vec{{p}_{01}}\right|}{E\left({p}_{01}\right)}\right) \boldsymbol{R_y}\left(- \theta\left({p}_{01}\right)\right) \boldsymbol{R_z}\left(- \phi\left({p}_{01}\right)\right) p_{0}\right) \\ \theta^{02}_{0} \;&=\; \theta\left(\boldsymbol{B_z}\left(\frac{\left|\vec{{p}_{02}}\right|}{E\left({p}_{02}\right)}\right) \boldsymbol{R_y}\left(- \theta\left({p}_{02}\right)\right) \boldsymbol{R_z}\left(- \phi\left({p}_{02}\right)\right) p_{0}\right) \\ \theta_{01} \;&=\; \theta\left({p}_{01}\right) \\ \theta^{12}_{1} \;&=\; \theta\left(\boldsymbol{B_z}\left(\frac{\left|\vec{{p}_{12}}\right|}{E\left({p}_{12}\right)}\right) \boldsymbol{R_y}\left(- \theta\left({p}_{12}\right)\right) \boldsymbol{R_z}\left(- \phi\left({p}_{12}\right)\right) p_{1}\right) \\ \theta_{02} \;&=\; \theta\left({p}_{02}\right) \\ \end{aligned}\end{split}\]

Visualization

unfolded_expression = model.expression.doit()
intensity_func = create_parametrized_function(
    expression=unfolded_expression,
    parameters=model.parameter_defaults,
    backend="jax",
)

phsp_event = 500_000
rng = TFUniformRealNumberGenerator(seed=0)
phsp_generator = TFPhaseSpaceGenerator(
    initial_state_mass=reaction.initial_state[-1].mass,
    final_state_masses={i: p.mass for i, p in reaction.final_state.items()},
)
phsp_momenta = phsp_generator.generate(phsp_event, rng)

helicity_transformer = SympyDataTransformer.from_sympy(
    model.kinematic_variables,
    backend="jax",
)
phsp = helicity_transformer(phsp_momenta)

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resonances = defaultdict(set)
for transition in reaction.transitions:
    topology = transition.topology
    top_decay_products = topology.get_edge_ids_outgoing_from_node(0)
    (resonance_id, resonance), *_ = transition.intermediate_states.items()
    recoil_id, *_ = top_decay_products - {resonance_id}
    resonances[recoil_id].add(resonance.particle)
resonances = {k: sorted(v, key=lambda p: p.mass) for k, v in resonances.items()}
{k: [p.name for p in v] for k, v in resonances.items()}

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{0: ['K(0)*(700)+',
  'K*(892)+',
  'K*(1410)+',
  'K(2)*(1430)+',
  'K(0)*(1430)+',
  'K*(1680)+',
  'K(3)*(1780)+',
  'K(0)*(1950)+',
  'K(2)*(1980)+',
  'K(4)*(2045)+'],
 1: ['Sigma(1385)0',
  'Sigma(1660)0',
  'Sigma(1670)0',
  'Sigma(1750)0',
  'Sigma(1775)0',
  'Sigma(1910)0',
  'Sigma(1915)0',
  'Sigma(2030)0'],
 2: ['N(1650)+',
  'N(1675)+',
  'N(1680)+',
  'N(1710)+',
  'N(1700)+',
  'N(1720)+',
  'N(1875)+',
  'N(1880)+',
  'N(1895)+',
  'N(1900)+',
  'N(2060)+',
  'N(2190)+']}

Design slider UI

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sliders = {}
categorized_sliders_m = defaultdict(list)
categorized_sliders_gamma = defaultdict(list)
categorized_cphi_pair = defaultdict(list)

for symbol, value in model.parameter_defaults.items():
    if symbol.name.startswith(R"\Gamma_{"):
        slider = w.FloatSlider(
            description=Rf"\({sp.latex(symbol)}\)",
            min=0.0,
            max=1.0,
            step=0.01,
            value=value,
            continuous_update=False,
        )
        sliders[symbol.name] = slider
        if symbol.name.startswith(R"\Gamma_{K"):
            categorized_sliders_gamma[0].append(slider)
        elif symbol.name.startswith(R"\Gamma_{\S"):
            categorized_sliders_gamma[1].append(slider)
        elif symbol.name.startswith(R"\Gamma_{N"):
            categorized_sliders_gamma[2].append(slider)

    if symbol.name.startswith("m_{"):
        slider = w.FloatSlider(
            description=Rf"\({sp.latex(symbol)}\)",
            min=0.63,
            max=4,
            step=0.01,
            value=value,
            continuous_update=False,
        )
        sliders[symbol.name] = slider
        if symbol.name.startswith("m_{K"):
            categorized_sliders_m[0].append(slider)
        elif symbol.name.startswith(R"m_{\S"):
            categorized_sliders_m[1].append(slider)
        elif symbol.name.startswith("m_{N"):
            categorized_sliders_m[2].append(slider)

    if symbol.name.startswith("C_{"):
        c_latex = sp.latex(symbol)
        phi_latex = c_latex.replace("C", R"\phi", 1)

        slider_c = w.FloatSlider(
            description=Rf"\({c_latex}\)",
            min=0,
            max=10,
            step=0.01,
            value=abs(value),
            continuous_update=False,
        )
        slider_phi = w.FloatSlider(
            description=Rf"\({phi_latex}\)",
            min=-np.pi,
            max=+np.pi,
            step=0.01,
            value=np.angle(value),
            continuous_update=False,
        )

        sliders[symbol.name] = slider_c
        sliders[symbol.name.replace("C", "phi", 1)] = slider_phi

        cphi_hbox = w.HBox([slider_c, slider_phi])
        if "Sigma" in symbol.name:
            categorized_cphi_pair[1].append(cphi_hbox)
        elif R"\to N" in symbol.name:
            categorized_cphi_pair[2].append(cphi_hbox)
        else:
            categorized_cphi_pair[0].append(cphi_hbox)


assert len(categorized_sliders_gamma) == 3
assert len(categorized_sliders_m) == 3
assert len(categorized_cphi_pair) == 3

subtabs = {}
for category, resonance_list in resonances.items():
    subtabs[category] = []
    for particle in resonance_list:
        m_sliders = [
            slider
            for slider in categorized_sliders_m[category]
            if particle.latex in slider.description
        ]
        gamma_sliders = [
            slider
            for slider in categorized_sliders_gamma[category]
            if particle.latex in slider.description
        ]
        cphi_pairs = [
            hbox
            for hbox in categorized_cphi_pair[category]
            if particle.latex in hbox.children[0].description
        ]
        pole_pair = w.HBox(m_sliders + gamma_sliders)
        resonance_tab = w.VBox([pole_pair, *cphi_pairs])
        subtabs[category].append(resonance_tab)
assert len(subtabs) == 3

main_tabs = []
for category, slider_boxes in subtabs.items():
    sub_tab_widget = w.Tab(children=slider_boxes)
    for i, particle in enumerate(resonances[category]):
        sub_tab_widget.set_title(i, particle.name)

    main_tabs.append(sub_tab_widget)

UI = w.Tab(children=main_tabs, titles=("K*", "Σ*", "N*"))

Define functions for inserting phase

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def insert_phi(parameters: dict) -> dict:
    updated_parameters = {}
    for key, value in parameters.items():
        if key.startswith("phi_"):
            continue
        if key.startswith("C_"):
            phi_key = key.replace("C_", "phi_")
            if phi_key in parameters:
                phi = parameters[phi_key]
                value *= np.exp(1j * phi)  # ruff: ignore[redefined-loop-name]
        updated_parameters[key] = value
    return updated_parameters

%matplotlib widget

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fig_2d, ax_2d = plt.subplots(dpi=200)
ax_2d.set_title("Model-weighted Phase space Dalitz Plot")
ax_2d.set_xlabel(R"$m^2(\Lambda K^+)\;\left[\mathrm{GeV}^2\right]$")
ax_2d.set_ylabel(R"$m^2(K^+ \pi^0)\;\left[\mathrm{GeV}^2\right]$")
fig_2d.canvas.toolbar_visible = False
fig_2d.canvas.header_visible = False
fig_2d.canvas.footer_visible = False

mesh = None


def update_histogram(**parameters):
    global mesh
    parameters = insert_phi(parameters)
    intensity_func.update_parameters(parameters)
    intensity_weights = intensity_func(phsp)
    bin_values, xedges, yedges = jnp.histogram2d(
        phsp["m_01"].real ** 2,
        phsp["m_12"].real ** 2,
        bins=200,
        weights=intensity_weights,
        density=True,
    )
    bin_values = jnp.where(bin_values < 1e-6, jnp.nan, bin_values)
    x, y = jnp.meshgrid(xedges[:-1], yedges[:-1])
    if mesh is None:
        mesh = ax_2d.pcolormesh(x, y, bin_values.T, cmap="jet", vmax=0.15)
    else:
        mesh.set_array(bin_values.T)
    fig_2d.canvas.draw()


w.interactive_output(update_histogram, sliders)
fig_2d.colorbar(mesh, ax=ax_2d)

if STATIC_PAGE:
    filename = "dalitz-plot.png"
    fig_2d.savefig(filename)
    plt.close(fig_2d)
    display(UI, Image(filename))
else:
    display(UI, fig_2d.canvas)

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fig, axes = plt.subplots(figsize=(11, 3.5), ncols=3, sharey=True)
fig.canvas.toolbar_visible = False
fig.canvas.header_visible = False
fig.canvas.footer_visible = False
ax1, ax2, ax3 = axes

for recoil_id, ax in enumerate(axes):
    decay_products = sorted({0, 1, 2} - {recoil_id})
    product_latex = " ".join([reaction.final_state[i].latex for i in decay_products])
    ax.set_xlabel(f"$m({product_latex})$ [GeV]")

LINES = 3 * [None]
RESONANCE_LINE = [None] * len(reaction.get_intermediate_particles())


def update_plot(**parameters):
    parameters = insert_phi(parameters)
    intensity_func.update_parameters(parameters)
    intensities = intensity_func(phsp)
    max_value = 0
    color_id = 0
    for recoil_id, ax in enumerate(axes):
        decay_products = sorted({0, 1, 2} - {recoil_id})
        key = f"m_{''.join(str(i) for i in decay_products)}"
        bin_values, bin_edges = jax.numpy.histogram(
            phsp[key].real,
            bins=120,
            density=True,
            weights=intensities,
        )
        max_value = max(max_value, bin_values.max())

        if LINES[recoil_id] is None:
            LINES[recoil_id] = ax.step(bin_edges[:-1], bin_values, alpha=0.5)[0]
        else:
            LINES[recoil_id].set_ydata(bin_values)

        for resonance in resonances[recoil_id]:
            key = f"m_{{{resonance.latex}}}"
            val = parameters.get(key, resonance.mass)
            if RESONANCE_LINE[color_id] is None:
                RESONANCE_LINE[color_id] = ax.axvline(
                    val,
                    c=f"C{color_id}",
                    ls="dotted",
                    label=resonance.name,
                )
            else:
                RESONANCE_LINE[color_id].set_xdata([val, val])
            color_id += 1

    for ax in axes:
        ax.set_ylim(0, max_value * 1.1)

    fig.canvas.draw()


w.interactive_output(update_plot, sliders)
for ax in axes:
    ax.legend(fontsize="small")

if STATIC_PAGE:
    filename = "histogram.svg"
    fig.savefig(filename)
    plt.close(fig)
    display(UI, SVG(filename))
else:
    display(UI, fig.canvas)