Manually modelling helicity amplitudes

The goal in this notebook is to formulate helicity amplitude model manually and reproduce the result in Amplitude model with ampform-dpd notebook.

Import Python libraries

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import logging
import os
import warnings

import graphviz
import numpy as np
import qrules
import sympy as sp
from ampform.io import aslatex, improve_latex_rendering
from ampform_dpd.adapter.qrules import normalize_state_ids, to_three_body_decay
from IPython.display import Math
from qrules.particle import Particle
from sympy.physics.quantum.spin import Rotation as Wigner

from gluex_nstar import create_pgamma, 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()

Reaction

Firstly, we check the decay topology in strong reaction as a reminder.

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)

Define the initial state

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

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,
)
reaction = normalize_state_ids(reaction)

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

Convert transitions to a three-body decay

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decay = to_three_body_decay(reaction.transitions)
Math(aslatex(decay, with_jp=True))

\[\begin{split}\displaystyle ThreeBodyDecay(states={0: State(name='pgamma1', latex='p\\gamma (s1/2)', spin=1/2, parity=-1, mass=4.101931740046389, width=0.0, index=0), 1: State(name='Lambda', latex='\\Lambda', spin=1/2, parity=1, mass=1.115683, width=2.515e-15, index=1), 2: State(name='K+', latex='K^{+}', spin=0, parity=-1, mass=0.49367700000000003, width=5.317e-17, index=2), 3: State(name='pi0', latex='\\pi^{0}', spin=0, parity=-1, mass=0.1349768, width=7.81e-09, index=3)}, chains=(ThreeBodyDecayChain(decay=IsobarNode(parent=State(name='pgamma1', latex='p\\gamma (s1/2)', spin=1/2, parity=-1, mass=4.101931740046389, width=0.0, index=0), child1=IsobarNode(parent=Particle(name='K(0)*(1430)+', latex='K_{0}^{*}(1430)^{+}', spin=0, parity=1, mass=1.43, width=0.27), child1=State(name='K+', latex='K^{+}', spin=0, parity=-1, mass=0.49367700000000003, width=5.317e-17, index=2), child2=State(name='pi0', latex='\\pi^{0}', spin=0, parity=-1, mass=0.1349768, width=7.81e-09, index=3), interaction=None), child2=State(name='Lambda', latex='\\Lambda', spin=1/2, parity=1, mass=1.115683, width=2.515e-15, index=1), interaction=None)), ThreeBodyDecayChain(decay=IsobarNode(parent=State(name='pgamma1', latex='p\\gamma (s1/2)', spin=1/2, parity=-1, mass=4.101931740046389, width=0.0, index=0), child1=IsobarNode(parent=Particle(name='K(0)*(1950)+', latex='K_{0}^{*}(1950)^{+}', spin=0, parity=1, mass=1.957, width=0.17), child1=State(name='K+', latex='K^{+}', spin=0, parity=-1, mass=0.49367700000000003, width=5.317e-17, index=2), child2=State(name='pi0', latex='\\pi^{0}', spin=0, parity=-1, mass=0.1349768, width=7.81e-09, index=3), interaction=None), child2=State(name='Lambda', latex='\\Lambda', spin=1/2, parity=1, mass=1.115683, width=2.515e-15, index=1), interaction=None)), ThreeBodyDecayChain(decay=IsobarNode(parent=State(name='pgamma1', latex='p\\gamma (s1/2)', spin=1/2, parity=-1, mass=4.101931740046389, width=0.0, index=0), child1=IsobarNode(parent=Particle(name='K(0)*(700)+', latex='K_{0}^{*}(700)^{+}', spin=0, parity=1, mass=0.838, width=0.463), child1=State(name='K+', latex='K^{+}', spin=0, parity=-1, mass=0.49367700000000003, width=5.317e-17, index=2), child2=State(name='pi0', latex='\\pi^{0}', spin=0, parity=-1, mass=0.1349768, width=7.81e-09, index=3), interaction=None), child2=State(name='Lambda', latex='\\Lambda', spin=1/2, parity=1, mass=1.115683, width=2.515e-15, index=1), interaction=None)), ThreeBodyDecayChain(decay=IsobarNode(parent=State(name='pgamma1', latex='p\\gamma (s1/2)', spin=1/2, parity=-1, mass=4.101931740046389, width=0.0, index=0), child1=IsobarNode(parent=Particle(name='K(2)*(1430)+', latex='K_{2}^{*}(1430)^{+}', spin=2, parity=1, mass=1.4273, width=0.1), child1=State(name='K+', latex='K^{+}', spin=0, parity=-1, mass=0.49367700000000003, width=5.317e-17, index=2), child2=State(name='pi0', latex='\\pi^{0}', spin=0, parity=-1, mass=0.1349768, width=7.81e-09, index=3), interaction=None), child2=State(name='Lambda', latex='\\Lambda', spin=1/2, parity=1, mass=1.115683, width=2.515e-15, index=1), interaction=None)), ThreeBodyDecayChain(decay=IsobarNode(parent=State(name='pgamma1', latex='p\\gamma (s1/2)', spin=1/2, parity=-1, mass=4.101931740046389, width=0.0, index=0), child1=IsobarNode(parent=Particle(name='K(2)*(1980)+', latex='K_{2}^{*}(1980)^{+}', spin=2, parity=1, mass=1.99, width=0.348), child1=State(name='K+', latex='K^{+}', spin=0, parity=-1, mass=0.49367700000000003, width=5.317e-17, index=2), child2=State(name='pi0', latex='\\pi^{0}', spin=0, parity=-1, mass=0.1349768, width=7.81e-09, index=3), interaction=None), child2=State(name='Lambda', latex='\\Lambda', spin=1/2, parity=1, mass=1.115683, width=2.515e-15, index=1), interaction=None)), ThreeBodyDecayChain(decay=IsobarNode(parent=State(name='pgamma1', latex='p\\gamma (s1/2)', spin=1/2, parity=-1, mass=4.101931740046389, width=0.0, index=0), child1=IsobarNode(parent=Particle(name='K(3)*(1780)+', latex='K_{3}^{*}(1780)^{+}', spin=3, parity=-1, mass=1.779, width=0.161), child1=State(name='K+', latex='K^{+}', spin=0, parity=-1, mass=0.49367700000000003, width=5.317e-17, index=2), child2=State(name='pi0', latex='\\pi^{0}', spin=0, parity=-1, mass=0.1349768, width=7.81e-09, index=3), interaction=None), child2=State(name='Lambda', latex='\\Lambda', spin=1/2, parity=1, mass=1.115683, width=2.515e-15, index=1), interaction=None)), ThreeBodyDecayChain(decay=IsobarNode(parent=State(name='pgamma1', latex='p\\gamma (s1/2)', spin=1/2, parity=-1, mass=4.101931740046389, width=0.0, index=0), child1=IsobarNode(parent=Particle(name='K(4)*(2045)+', latex='K_{4}^{*}(2045)^{+}', spin=4, parity=1, mass=2.048, width=0.199), child1=State(name='K+', latex='K^{+}', spin=0, parity=-1, mass=0.49367700000000003, width=5.317e-17, index=2), child2=State(name='pi0', latex='\\pi^{0}', spin=0, parity=-1, mass=0.1349768, width=7.81e-09, index=3), interaction=None), child2=State(name='Lambda', latex='\\Lambda', spin=1/2, parity=1, mass=1.115683, width=2.515e-15, index=1), interaction=None)), ThreeBodyDecayChain(decay=IsobarNode(parent=State(name='pgamma1', latex='p\\gamma (s1/2)', spin=1/2, parity=-1, mass=4.101931740046389, width=0.0, index=0), child1=IsobarNode(parent=Particle(name='K*(1410)+', latex='K^{*}(1410)^{+}', spin=1, parity=-1, mass=1.414, width=0.232), child1=State(name='K+', latex='K^{+}', spin=0, parity=-1, mass=0.49367700000000003, width=5.317e-17, index=2), child2=State(name='pi0', latex='\\pi^{0}', spin=0, parity=-1, mass=0.1349768, width=7.81e-09, index=3), interaction=None), child2=State(name='Lambda', latex='\\Lambda', spin=1/2, parity=1, mass=1.115683, width=2.515e-15, index=1), interaction=None)), ThreeBodyDecayChain(decay=IsobarNode(parent=State(name='pgamma1', latex='p\\gamma (s1/2)', spin=1/2, parity=-1, mass=4.101931740046389, width=0.0, index=0), child1=IsobarNode(parent=Particle(name='K*(1680)+', latex='K^{*}(1680)^{+}', spin=1, parity=-1, mass=1.718, width=0.32), child1=State(name='K+', latex='K^{+}', spin=0, parity=-1, mass=0.49367700000000003, width=5.317e-17, index=2), child2=State(name='pi0', latex='\\pi^{0}', spin=0, parity=-1, mass=0.1349768, width=7.81e-09, index=3), interaction=None), child2=State(name='Lambda', latex='\\Lambda', spin=1/2, parity=1, mass=1.115683, width=2.515e-15, index=1), interaction=None)), ThreeBodyDecayChain(decay=IsobarNode(parent=State(name='pgamma1', latex='p\\gamma (s1/2)', spin=1/2, parity=-1, mass=4.101931740046389, width=0.0, index=0), child1=IsobarNode(parent=Particle(name='K*(892)+', latex='K^{*}(892)^{+}', spin=1, parity=-1, mass=0.89188, width=0.0485), child1=State(name='K+', latex='K^{+}', spin=0, parity=-1, mass=0.49367700000000003, width=5.317e-17, index=2), child2=State(name='pi0', latex='\\pi^{0}', spin=0, parity=-1, mass=0.1349768, width=7.81e-09, index=3), interaction=None), child2=State(name='Lambda', latex='\\Lambda', spin=1/2, parity=1, mass=1.115683, width=2.515e-15, index=1), interaction=None)), ThreeBodyDecayChain(decay=IsobarNode(parent=State(name='pgamma1', latex='p\\gamma (s1/2)', spin=1/2, parity=-1, mass=4.101931740046389, width=0.0, index=0), child1=IsobarNode(parent=Particle(name='N(1650)+', latex='N(1650)^{+}', spin=1/2, parity=-1, mass=1.65, width=0.125), child1=State(name='Lambda', latex='\\Lambda', spin=1/2, parity=1, mass=1.115683, width=2.515e-15, index=1), child2=State(name='K+', latex='K^{+}', spin=0, parity=-1, mass=0.49367700000000003, width=5.317e-17, index=2), interaction=None), child2=State(name='pi0', latex='\\pi^{0}', spin=0, parity=-1, mass=0.1349768, width=7.81e-09, index=3), interaction=None)), ThreeBodyDecayChain(decay=IsobarNode(parent=State(name='pgamma1', latex='p\\gamma (s1/2)', spin=1/2, parity=-1, mass=4.101931740046389, width=0.0, index=0), child1=IsobarNode(parent=Particle(name='N(1675)+', latex='N(1675)^{+}', spin=5/2, parity=-1, mass=1.675, width=0.145), child1=State(name='Lambda', latex='\\Lambda', spin=1/2, parity=1, mass=1.115683, width=2.515e-15, index=1), child2=State(name='K+', latex='K^{+}', spin=0, parity=-1, mass=0.49367700000000003, width=5.317e-17, index=2), interaction=None), child2=State(name='pi0', latex='\\pi^{0}', spin=0, parity=-1, mass=0.1349768, width=7.81e-09, index=3), interaction=None)), ThreeBodyDecayChain(decay=IsobarNode(parent=State(name='pgamma1', latex='p\\gamma (s1/2)', spin=1/2, parity=-1, mass=4.101931740046389, width=0.0, index=0), child1=IsobarNode(parent=Particle(name='N(1680)+', latex='N(1680)^{+}', spin=5/2, parity=1, mass=1.685, width=0.12), child1=State(name='Lambda', latex='\\Lambda', spin=1/2, parity=1, mass=1.115683, width=2.515e-15, index=1), child2=State(name='K+', latex='K^{+}', spin=0, parity=-1, mass=0.49367700000000003, width=5.317e-17, index=2), interaction=None), child2=State(name='pi0', latex='\\pi^{0}', spin=0, parity=-1, mass=0.1349768, width=7.81e-09, index=3), interaction=None)), ThreeBodyDecayChain(decay=IsobarNode(parent=State(name='pgamma1', latex='p\\gamma (s1/2)', spin=1/2, parity=-1, mass=4.101931740046389, width=0.0, index=0), child1=IsobarNode(parent=Particle(name='N(1700)+', latex='N(1700)^{+}', spin=3/2, parity=-1, mass=1.72, width=0.2), child1=State(name='Lambda', latex='\\Lambda', spin=1/2, parity=1, mass=1.115683, width=2.515e-15, index=1), child2=State(name='K+', latex='K^{+}', spin=0, parity=-1, mass=0.49367700000000003, width=5.317e-17, index=2), interaction=None), child2=State(name='pi0', latex='\\pi^{0}', spin=0, parity=-1, mass=0.1349768, width=7.81e-09, index=3), interaction=None)), ThreeBodyDecayChain(decay=IsobarNode(parent=State(name='pgamma1', latex='p\\gamma (s1/2)', spin=1/2, parity=-1, mass=4.101931740046389, width=0.0, index=0), child1=IsobarNode(parent=Particle(name='N(1710)+', latex='N(1710)^{+}', spin=1/2, parity=1, mass=1.71, width=0.14), child1=State(name='Lambda', latex='\\Lambda', spin=1/2, parity=1, mass=1.115683, width=2.515e-15, index=1), child2=State(name='K+', latex='K^{+}', spin=0, parity=-1, mass=0.49367700000000003, width=5.317e-17, index=2), interaction=None), child2=State(name='pi0', latex='\\pi^{0}', spin=0, parity=-1, mass=0.1349768, width=7.81e-09, index=3), interaction=None)), ThreeBodyDecayChain(decay=IsobarNode(parent=State(name='pgamma1', latex='p\\gamma (s1/2)', spin=1/2, parity=-1, mass=4.101931740046389, width=0.0, index=0), child1=IsobarNode(parent=Particle(name='N(1720)+', latex='N(1720)^{+}', spin=3/2, parity=1, mass=1.72, width=0.25), child1=State(name='Lambda', latex='\\Lambda', spin=1/2, parity=1, mass=1.115683, width=2.515e-15, index=1), child2=State(name='K+', latex='K^{+}', spin=0, parity=-1, mass=0.49367700000000003, width=5.317e-17, index=2), interaction=None), child2=State(name='pi0', latex='\\pi^{0}', spin=0, parity=-1, mass=0.1349768, width=7.81e-09, index=3), interaction=None)), ThreeBodyDecayChain(decay=IsobarNode(parent=State(name='pgamma1', latex='p\\gamma (s1/2)', spin=1/2, parity=-1, mass=4.101931740046389, width=0.0, index=0), child1=IsobarNode(parent=Particle(name='N(1875)+', latex='N(1875)^{+}', spin=3/2, parity=-1, mass=1.875, width=0.2), child1=State(name='Lambda', latex='\\Lambda', spin=1/2, parity=1, mass=1.115683, width=2.515e-15, index=1), child2=State(name='K+', latex='K^{+}', spin=0, parity=-1, mass=0.49367700000000003, width=5.317e-17, index=2), interaction=None), child2=State(name='pi0', latex='\\pi^{0}', spin=0, parity=-1, mass=0.1349768, width=7.81e-09, index=3), interaction=None)), ThreeBodyDecayChain(decay=IsobarNode(parent=State(name='pgamma1', latex='p\\gamma (s1/2)', spin=1/2, parity=-1, mass=4.101931740046389, width=0.0, index=0), child1=IsobarNode(parent=Particle(name='N(1880)+', latex='N(1880)^{+}', spin=1/2, parity=1, mass=1.88, width=0.3), child1=State(name='Lambda', latex='\\Lambda', spin=1/2, parity=1, mass=1.115683, width=2.515e-15, index=1), child2=State(name='K+', latex='K^{+}', spin=0, parity=-1, mass=0.49367700000000003, width=5.317e-17, index=2), interaction=None), child2=State(name='pi0', latex='\\pi^{0}', spin=0, parity=-1, mass=0.1349768, width=7.81e-09, index=3), interaction=None)), ThreeBodyDecayChain(decay=IsobarNode(parent=State(name='pgamma1', latex='p\\gamma (s1/2)', spin=1/2, parity=-1, mass=4.101931740046389, width=0.0, index=0), child1=IsobarNode(parent=Particle(name='N(1895)+', latex='N(1895)^{+}', spin=1/2, parity=-1, mass=1.895, width=0.12), child1=State(name='Lambda', latex='\\Lambda', spin=1/2, parity=1, mass=1.115683, width=2.515e-15, index=1), child2=State(name='K+', latex='K^{+}', spin=0, parity=-1, mass=0.49367700000000003, width=5.317e-17, index=2), interaction=None), child2=State(name='pi0', latex='\\pi^{0}', spin=0, parity=-1, mass=0.1349768, width=7.81e-09, index=3), interaction=None)), ThreeBodyDecayChain(decay=IsobarNode(parent=State(name='pgamma1', latex='p\\gamma (s1/2)', spin=1/2, parity=-1, mass=4.101931740046389, width=0.0, index=0), child1=IsobarNode(parent=Particle(name='N(1900)+', latex='N(1900)^{+}', spin=3/2, parity=1, mass=1.92, width=0.2), child1=State(name='Lambda', latex='\\Lambda', spin=1/2, parity=1, mass=1.115683, width=2.515e-15, index=1), child2=State(name='K+', latex='K^{+}', spin=0, parity=-1, mass=0.49367700000000003, width=5.317e-17, index=2), interaction=None), child2=State(name='pi0', latex='\\pi^{0}', spin=0, parity=-1, mass=0.1349768, width=7.81e-09, index=3), interaction=None)), ThreeBodyDecayChain(decay=IsobarNode(parent=State(name='pgamma1', latex='p\\gamma (s1/2)', spin=1/2, parity=-1, mass=4.101931740046389, width=0.0, index=0), child1=IsobarNode(parent=Particle(name='N(2060)+', latex='N(2060)^{+}', spin=5/2, parity=-1, mass=2.1, width=0.4), child1=State(name='Lambda', latex='\\Lambda', spin=1/2, parity=1, mass=1.115683, width=2.515e-15, index=1), child2=State(name='K+', latex='K^{+}', spin=0, parity=-1, mass=0.49367700000000003, width=5.317e-17, index=2), interaction=None), child2=State(name='pi0', latex='\\pi^{0}', spin=0, parity=-1, mass=0.1349768, width=7.81e-09, index=3), interaction=None)), ThreeBodyDecayChain(decay=IsobarNode(parent=State(name='pgamma1', latex='p\\gamma (s1/2)', spin=1/2, parity=-1, mass=4.101931740046389, width=0.0, index=0), child1=IsobarNode(parent=Particle(name='N(2190)+', latex='N(2190)^{+}', spin=7/2, parity=-1, mass=2.18, width=0.4), child1=State(name='Lambda', latex='\\Lambda', spin=1/2, parity=1, mass=1.115683, width=2.515e-15, index=1), child2=State(name='K+', latex='K^{+}', spin=0, parity=-1, mass=0.49367700000000003, width=5.317e-17, index=2), interaction=None), child2=State(name='pi0', latex='\\pi^{0}', spin=0, parity=-1, mass=0.1349768, width=7.81e-09, index=3), interaction=None)), ThreeBodyDecayChain(decay=IsobarNode(parent=State(name='pgamma1', latex='p\\gamma (s1/2)', spin=1/2, parity=-1, mass=4.101931740046389, width=0.0, index=0), child1=IsobarNode(parent=Particle(name='Sigma(1385)0', latex='\\Sigma(1385)^{0}', spin=3/2, parity=1, mass=1.3838, width=0.044), child1=State(name='Lambda', latex='\\Lambda', spin=1/2, parity=1, mass=1.115683, width=2.515e-15, index=1), child2=State(name='pi0', latex='\\pi^{0}', spin=0, parity=-1, mass=0.1349768, width=7.81e-09, index=3), interaction=None), child2=State(name='K+', latex='K^{+}', spin=0, parity=-1, mass=0.49367700000000003, width=5.317e-17, index=2), interaction=None)), ThreeBodyDecayChain(decay=IsobarNode(parent=State(name='pgamma1', latex='p\\gamma (s1/2)', spin=1/2, parity=-1, mass=4.101931740046389, width=0.0, index=0), child1=IsobarNode(parent=Particle(name='Sigma(1660)0', latex='\\Sigma(1660)^{0}', spin=1/2, parity=1, mass=1.66, width=0.2), child1=State(name='Lambda', latex='\\Lambda', spin=1/2, parity=1, mass=1.115683, width=2.515e-15, index=1), child2=State(name='pi0', latex='\\pi^{0}', spin=0, parity=-1, mass=0.1349768, width=7.81e-09, index=3), interaction=None), child2=State(name='K+', latex='K^{+}', spin=0, parity=-1, mass=0.49367700000000003, width=5.317e-17, index=2), interaction=None)), ThreeBodyDecayChain(decay=IsobarNode(parent=State(name='pgamma1', latex='p\\gamma (s1/2)', spin=1/2, parity=-1, mass=4.101931740046389, width=0.0, index=0), child1=IsobarNode(parent=Particle(name='Sigma(1670)0', latex='\\Sigma(1670)^{0}', spin=3/2, parity=-1, mass=1.675, width=0.07), child1=State(name='Lambda', latex='\\Lambda', spin=1/2, parity=1, mass=1.115683, width=2.515e-15, index=1), child2=State(name='pi0', latex='\\pi^{0}', spin=0, parity=-1, mass=0.1349768, width=7.81e-09, index=3), interaction=None), child2=State(name='K+', latex='K^{+}', spin=0, parity=-1, mass=0.49367700000000003, width=5.317e-17, index=2), interaction=None)), ThreeBodyDecayChain(decay=IsobarNode(parent=State(name='pgamma1', latex='p\\gamma (s1/2)', spin=1/2, parity=-1, mass=4.101931740046389, width=0.0, index=0), child1=IsobarNode(parent=Particle(name='Sigma(1750)0', latex='\\Sigma(1750)^{0}', spin=1/2, parity=-1, mass=1.75, width=0.15), child1=State(name='Lambda', latex='\\Lambda', spin=1/2, parity=1, mass=1.115683, width=2.515e-15, index=1), child2=State(name='pi0', latex='\\pi^{0}', spin=0, parity=-1, mass=0.1349768, width=7.81e-09, index=3), interaction=None), child2=State(name='K+', latex='K^{+}', spin=0, parity=-1, mass=0.49367700000000003, width=5.317e-17, index=2), interaction=None)), ThreeBodyDecayChain(decay=IsobarNode(parent=State(name='pgamma1', latex='p\\gamma (s1/2)', spin=1/2, parity=-1, mass=4.101931740046389, width=0.0, index=0), child1=IsobarNode(parent=Particle(name='Sigma(1775)0', latex='\\Sigma(1775)^{0}', spin=5/2, parity=-1, mass=1.775, width=0.12), child1=State(name='Lambda', latex='\\Lambda', spin=1/2, parity=1, mass=1.115683, width=2.515e-15, index=1), child2=State(name='pi0', latex='\\pi^{0}', spin=0, parity=-1, mass=0.1349768, width=7.81e-09, index=3), interaction=None), child2=State(name='K+', latex='K^{+}', spin=0, parity=-1, mass=0.49367700000000003, width=5.317e-17, index=2), interaction=None)), ThreeBodyDecayChain(decay=IsobarNode(parent=State(name='pgamma1', latex='p\\gamma (s1/2)', spin=1/2, parity=-1, mass=4.101931740046389, width=0.0, index=0), child1=IsobarNode(parent=Particle(name='Sigma(1910)0', latex='\\Sigma(1940)^{0}', spin=3/2, parity=-1, mass=1.91, width=0.22), child1=State(name='Lambda', latex='\\Lambda', spin=1/2, parity=1, mass=1.115683, width=2.515e-15, index=1), child2=State(name='pi0', latex='\\pi^{0}', spin=0, parity=-1, mass=0.1349768, width=7.81e-09, index=3), interaction=None), child2=State(name='K+', latex='K^{+}', spin=0, parity=-1, mass=0.49367700000000003, width=5.317e-17, index=2), interaction=None)), ThreeBodyDecayChain(decay=IsobarNode(parent=State(name='pgamma1', latex='p\\gamma (s1/2)', spin=1/2, parity=-1, mass=4.101931740046389, width=0.0, index=0), child1=IsobarNode(parent=Particle(name='Sigma(1915)0', latex='\\Sigma(1915)^{0}', spin=5/2, parity=1, mass=1.915, width=0.12), child1=State(name='Lambda', latex='\\Lambda', spin=1/2, parity=1, mass=1.115683, width=2.515e-15, index=1), child2=State(name='pi0', latex='\\pi^{0}', spin=0, parity=-1, mass=0.1349768, width=7.81e-09, index=3), interaction=None), child2=State(name='K+', latex='K^{+}', spin=0, parity=-1, mass=0.49367700000000003, width=5.317e-17, index=2), interaction=None)), ThreeBodyDecayChain(decay=IsobarNode(parent=State(name='pgamma1', latex='p\\gamma (s1/2)', spin=1/2, parity=-1, mass=4.101931740046389, width=0.0, index=0), child1=IsobarNode(parent=Particle(name='Sigma(2030)0', latex='\\Sigma(2030)^{0}', spin=7/2, parity=1, mass=2.03, width=0.18), child1=State(name='Lambda', latex='\\Lambda', spin=1/2, parity=1, mass=1.115683, width=2.515e-15, index=1), child2=State(name='pi0', latex='\\pi^{0}', spin=0, parity=-1, mass=0.1349768, width=7.81e-09, index=3), interaction=None), child2=State(name='K+', latex='K^{+}', spin=0, parity=-1, mass=0.49367700000000003, width=5.317e-17, index=2), interaction=None))))\end{split}\]

Model implementation

Initial state:

\(0: p \gamma \)

Final states:

\(1: \Lambda, \)

\(2:K^+, \)

\(3:\pi^0\)

Resonances:

Final states 23 ->

\(1: K^*\)

Final states 31 ->

\(2: \Sigma^*\)

Final states 12 ->

\(3: N^*\)

Intensity

half = sp.Rational(1, 2)

A1 = sp.IndexedBase("A^1")
A2 = sp.IndexedBase("A^2")
A3 = sp.IndexedBase("A^3")

λ0, λ1 = sp.symbols(r"\lambda_0:2", rational=True)
λ0prime, λ1prime = sp.symbols(r"\lambda'_0:2", rational=True)

zeta_1_0 = sp.Symbol(r"\zeta_{1(1)}^0")
zeta_1_1 = sp.Symbol(r"\zeta_{1(1)}^1")
zeta_2_0 = sp.Symbol(r"\zeta_{2(1)}^0")
zeta_2_1 = sp.Symbol(r"\zeta_{2(1)}^1")
zeta_3_0 = sp.Symbol(r"\zeta_{3(1)}^0")
zeta_3_1 = sp.Symbol(r"\zeta_{3(1)}^1")

A1_aligned = (
    A1[λ0prime, λ1prime]
    * Wigner.d(half, λ1prime, λ1, zeta_1_1)
    * Wigner.d(half, λ0prime, λ0, zeta_1_0)
)
A2_aligned = (
    A2[λ0prime, λ1prime]
    * Wigner.d(half, λ1prime, λ1, zeta_2_1)
    * Wigner.d(half, λ0prime, λ0, zeta_2_0)
)
A3_aligned = (
    A3[λ0prime, λ1prime]
    * Wigner.d(half, λ1prime, λ1, zeta_3_1)
    * Wigner.d(half, λ0prime, λ0, zeta_3_0)
)

intensity = sp.Sum(
    sp.Pow(
        sp.Abs(
            sp.Sum(
                A1_aligned + A2_aligned + A3_aligned,
                [λ0prime, -half, half],
                [λ1prime, -half, half],
            )
        ),
        2,
    ),
    [λ0, -half, half],
    [λ1, -half, half],
)
intensity

\[\begin{split}\displaystyle \sum_{\substack{- \frac{1}{2} \leq \lambda_{0} \leq \frac{1}{2}\\- \frac{1}{2} \leq \lambda_{1} \leq \frac{1}{2}}} \left|{\sum_{\substack{- \frac{1}{2} \leq \lambda'_{0} \leq \frac{1}{2}\\- \frac{1}{2} \leq \lambda'_{1} \leq \frac{1}{2}}} \left(A^{1}_{\lambda'_{0}, \lambda'_{1}} d^{\frac{1}{2}}_{\lambda'_{0},\lambda_{0}}\left(\zeta_{1(1)}^0\right) d^{\frac{1}{2}}_{\lambda'_{1},\lambda_{1}}\left(\zeta_{1(1)}^1\right) + A^{2}_{\lambda'_{0}, \lambda'_{1}} d^{\frac{1}{2}}_{\lambda'_{0},\lambda_{0}}\left(\zeta_{2(1)}^0\right) d^{\frac{1}{2}}_{\lambda'_{1},\lambda_{1}}\left(\zeta_{2(1)}^1\right) + A^{3}_{\lambda'_{0}, \lambda'_{1}} d^{\frac{1}{2}}_{\lambda'_{0},\lambda_{0}}\left(\zeta_{3(1)}^0\right) d^{\frac{1}{2}}_{\lambda'_{1},\lambda_{1}}\left(\zeta_{3(1)}^1\right)\right)}\right|^{2}\end{split}\]

\(A^1 \equiv A^{K^*}\)

s23, m_k, gamma_k, c_k, j_k = sp.symbols(r"s_{23} m_{K^*} \Gamma_{K^*} C_{K^*} J_{K^*}")
theta23 = sp.Symbol(r"theta_23")
lambda_R = sp.symbols(r"\lambda_K^*")

A1_expr = sp.Sum(
    (
        gamma_k
        * m_k
        * c_k
        * sp.KroneckerDelta(λ0prime, lambda_R - λ1prime)
        * (-1) ** (half - λ1prime)
        * Wigner.d(j_k, lambda_R, 0, theta23)
        / (-sp.I * gamma_k * m_k + m_k**2 - s23)
    ),
    (lambda_R, -j_k, j_k),
)
A1_expr

\[\displaystyle \sum_{\lambda^{*}_{K}=- J_{K^*}}^{J_{K^*}} \frac{\left(-1\right)^{\frac{1}{2} - \lambda'_{1}} C_{K^*} \Gamma_{K^*} m_{K^*} \delta_{\lambda'_{0}, - \lambda'_{1} + \lambda^{*}_{K}} d^{J_{K^*}}_{\lambda^{*}_{K},0}\left(\theta_{23}\right)}{- i \Gamma_{K^*} m_{K^*} + \left(m_{K^*}\right)^{2} - s_{23}}\]

A1_expr.subs({
    c_k: sp.Symbol(r"C_{K^*(1410)}"),
    m_k: sp.Symbol(r"m_{K^*(1410)}"),
    gamma_k: sp.Symbol(r"\Gamma_{K^*(1410)}"),
})

\[\displaystyle \sum_{\lambda^{*}_{K}=- J_{K^*}}^{J_{K^*}} \frac{\left(-1\right)^{\frac{1}{2} - \lambda'_{1}} C_{K^*(1410)} \Gamma_{K^*(1410)} m_{K^*(1410)} \delta_{\lambda'_{0}, - \lambda'_{1} + \lambda^{*}_{K}} d^{J_{K^*}}_{\lambda^{*}_{K},0}\left(\theta_{23}\right)}{- i \Gamma_{K^*(1410)} m_{K^*(1410)} + \left(m_{K^*(1410)}\right)^{2} - s_{23}}\]

def formulate_chain_amplitude_a1(
    resonance: Particle,
    λ0: sp.Expr,
    λ1: sp.Expr,
) -> sp.Expr:
    c_k = sp.Symbol(rf"C_{{{resonance.latex}}}")
    m_k = sp.Symbol(rf"m_{{{resonance.latex}}}")
    gamma_k = sp.Symbol(rf"\Gamma_{{{resonance.latex}}}")
    theta23 = sp.Symbol("theta_23")
    s23, lambda_R = sp.symbols(r"s_{23} \lambda_{K^*}")
    return sp.Sum(
        (
            gamma_k
            * m_k
            * c_k
            * sp.KroneckerDelta(λ0, lambda_R - λ1)
            * (-1) ** (half - λ1)
            * Wigner.d(resonance.spin, lambda_R, 0, theta23)
            / (-sp.I * gamma_k * m_k + m_k**2 - s23)
        ),
        (lambda_R, -resonance.spin, resonance.spin),
    )

def get_resonances(reaction: qrules.ReactionInfo, *, recoil_id: int) -> set[Particle]:
    resonances = 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()
        transition_recoil_id, *_ = top_decay_products - {resonance_id}
        if transition_recoil_id == recoil_id:
            resonances.add(resonance.particle)
    return resonances


def formulate_a1(
    reaction: qrules.ReactionInfo,
    λ0: sp.Expr,
    λ1: sp.Expr,
) -> sp.Expr:
    resonances = get_resonances(reaction, recoil_id=1)
    amplitudes = (formulate_chain_amplitude_a1(r, λ0, λ1) for r in resonances)
    return sum(amplitudes, start=sp.S.Zero)


A1_expr = formulate_a1(reaction, λ0prime, λ1prime)
A1_expr

\[\displaystyle \sum_{\lambda_{K^*}=-1}^{1} \frac{\left(-1\right)^{\frac{1}{2} - \lambda'_{1}} C_{K^{*}(1410)^{+}} \Gamma_{K^{*}(1410)^{+}} m_{K^{*}(1410)^{+}} \delta_{\lambda'_{0}, - \lambda'_{1} + \lambda_{K^*}} d^{1}_{\lambda_{K^*},0}\left(\theta_{23}\right)}{- i \Gamma_{K^{*}(1410)^{+}} m_{K^{*}(1410)^{+}} + \left(m_{K^{*}(1410)^{+}}\right)^{2} - s_{23}} + \sum_{\lambda_{K^*}=-1}^{1} \frac{\left(-1\right)^{\frac{1}{2} - \lambda'_{1}} C_{K^{*}(1680)^{+}} \Gamma_{K^{*}(1680)^{+}} m_{K^{*}(1680)^{+}} \delta_{\lambda'_{0}, - \lambda'_{1} + \lambda_{K^*}} d^{1}_{\lambda_{K^*},0}\left(\theta_{23}\right)}{- i \Gamma_{K^{*}(1680)^{+}} m_{K^{*}(1680)^{+}} + \left(m_{K^{*}(1680)^{+}}\right)^{2} - s_{23}} + \sum_{\lambda_{K^*}=-1}^{1} \frac{\left(-1\right)^{\frac{1}{2} - \lambda'_{1}} C_{K^{*}(892)^{+}} \Gamma_{K^{*}(892)^{+}} m_{K^{*}(892)^{+}} \delta_{\lambda'_{0}, - \lambda'_{1} + \lambda_{K^*}} d^{1}_{\lambda_{K^*},0}\left(\theta_{23}\right)}{- i \Gamma_{K^{*}(892)^{+}} m_{K^{*}(892)^{+}} + \left(m_{K^{*}(892)^{+}}\right)^{2} - s_{23}} + \sum_{\lambda_{K^*}=0}^{0} \frac{\left(-1\right)^{\frac{1}{2} - \lambda'_{1}} C_{K_{0}^{*}(1430)^{+}} \Gamma_{K_{0}^{*}(1430)^{+}} m_{K_{0}^{*}(1430)^{+}} \delta_{\lambda'_{0}, - \lambda'_{1} + \lambda_{K^*}} d^{0}_{\lambda_{K^*},0}\left(\theta_{23}\right)}{- i \Gamma_{K_{0}^{*}(1430)^{+}} m_{K_{0}^{*}(1430)^{+}} + \left(m_{K_{0}^{*}(1430)^{+}}\right)^{2} - s_{23}} + \sum_{\lambda_{K^*}=0}^{0} \frac{\left(-1\right)^{\frac{1}{2} - \lambda'_{1}} C_{K_{0}^{*}(1950)^{+}} \Gamma_{K_{0}^{*}(1950)^{+}} m_{K_{0}^{*}(1950)^{+}} \delta_{\lambda'_{0}, - \lambda'_{1} + \lambda_{K^*}} d^{0}_{\lambda_{K^*},0}\left(\theta_{23}\right)}{- i \Gamma_{K_{0}^{*}(1950)^{+}} m_{K_{0}^{*}(1950)^{+}} + \left(m_{K_{0}^{*}(1950)^{+}}\right)^{2} - s_{23}} + \sum_{\lambda_{K^*}=0}^{0} \frac{\left(-1\right)^{\frac{1}{2} - \lambda'_{1}} C_{K_{0}^{*}(700)^{+}} \Gamma_{K_{0}^{*}(700)^{+}} m_{K_{0}^{*}(700)^{+}} \delta_{\lambda'_{0}, - \lambda'_{1} + \lambda_{K^*}} d^{0}_{\lambda_{K^*},0}\left(\theta_{23}\right)}{- i \Gamma_{K_{0}^{*}(700)^{+}} m_{K_{0}^{*}(700)^{+}} + \left(m_{K_{0}^{*}(700)^{+}}\right)^{2} - s_{23}} + \sum_{\lambda_{K^*}=-2}^{2} \frac{\left(-1\right)^{\frac{1}{2} - \lambda'_{1}} C_{K_{2}^{*}(1430)^{+}} \Gamma_{K_{2}^{*}(1430)^{+}} m_{K_{2}^{*}(1430)^{+}} \delta_{\lambda'_{0}, - \lambda'_{1} + \lambda_{K^*}} d^{2}_{\lambda_{K^*},0}\left(\theta_{23}\right)}{- i \Gamma_{K_{2}^{*}(1430)^{+}} m_{K_{2}^{*}(1430)^{+}} + \left(m_{K_{2}^{*}(1430)^{+}}\right)^{2} - s_{23}} + \sum_{\lambda_{K^*}=-2}^{2} \frac{\left(-1\right)^{\frac{1}{2} - \lambda'_{1}} C_{K_{2}^{*}(1980)^{+}} \Gamma_{K_{2}^{*}(1980)^{+}} m_{K_{2}^{*}(1980)^{+}} \delta_{\lambda'_{0}, - \lambda'_{1} + \lambda_{K^*}} d^{2}_{\lambda_{K^*},0}\left(\theta_{23}\right)}{- i \Gamma_{K_{2}^{*}(1980)^{+}} m_{K_{2}^{*}(1980)^{+}} + \left(m_{K_{2}^{*}(1980)^{+}}\right)^{2} - s_{23}} + \sum_{\lambda_{K^*}=-3}^{3} \frac{\left(-1\right)^{\frac{1}{2} - \lambda'_{1}} C_{K_{3}^{*}(1780)^{+}} \Gamma_{K_{3}^{*}(1780)^{+}} m_{K_{3}^{*}(1780)^{+}} \delta_{\lambda'_{0}, - \lambda'_{1} + \lambda_{K^*}} d^{3}_{\lambda_{K^*},0}\left(\theta_{23}\right)}{- i \Gamma_{K_{3}^{*}(1780)^{+}} m_{K_{3}^{*}(1780)^{+}} + \left(m_{K_{3}^{*}(1780)^{+}}\right)^{2} - s_{23}} + \sum_{\lambda_{K^*}=-4}^{4} \frac{\left(-1\right)^{\frac{1}{2} - \lambda'_{1}} C_{K_{4}^{*}(2045)^{+}} \Gamma_{K_{4}^{*}(2045)^{+}} m_{K_{4}^{*}(2045)^{+}} \delta_{\lambda'_{0}, - \lambda'_{1} + \lambda_{K^*}} d^{4}_{\lambda_{K^*},0}\left(\theta_{23}\right)}{- i \Gamma_{K_{4}^{*}(2045)^{+}} m_{K_{4}^{*}(2045)^{+}} + \left(m_{K_{4}^{*}(2045)^{+}}\right)^{2} - s_{23}}\]

Math(aslatex(A1_expr, terms_per_line=1))

\[\begin{split}\displaystyle \begin{aligned} & \sum_{\lambda_{K^*}=-1}^{1} \frac{\left(-1\right)^{\frac{1}{2} - \lambda'_{1}} C_{K^{*}(1410)^{+}} \Gamma_{K^{*}(1410)^{+}} m_{K^{*}(1410)^{+}} \delta_{\lambda'_{0}, - \lambda'_{1} + \lambda_{K^*}} d^{1}_{\lambda_{K^*},0}\left(\theta_{23}\right)}{- i \Gamma_{K^{*}(1410)^{+}} m_{K^{*}(1410)^{+}} + \left(m_{K^{*}(1410)^{+}}\right)^{2} - s_{23}} \\ & \;+\; \sum_{\lambda_{K^*}=-1}^{1} \frac{\left(-1\right)^{\frac{1}{2} - \lambda'_{1}} C_{K^{*}(1680)^{+}} \Gamma_{K^{*}(1680)^{+}} m_{K^{*}(1680)^{+}} \delta_{\lambda'_{0}, - \lambda'_{1} + \lambda_{K^*}} d^{1}_{\lambda_{K^*},0}\left(\theta_{23}\right)}{- i \Gamma_{K^{*}(1680)^{+}} m_{K^{*}(1680)^{+}} + \left(m_{K^{*}(1680)^{+}}\right)^{2} - s_{23}} \\ & \;+\; \sum_{\lambda_{K^*}=-1}^{1} \frac{\left(-1\right)^{\frac{1}{2} - \lambda'_{1}} C_{K^{*}(892)^{+}} \Gamma_{K^{*}(892)^{+}} m_{K^{*}(892)^{+}} \delta_{\lambda'_{0}, - \lambda'_{1} + \lambda_{K^*}} d^{1}_{\lambda_{K^*},0}\left(\theta_{23}\right)}{- i \Gamma_{K^{*}(892)^{+}} m_{K^{*}(892)^{+}} + \left(m_{K^{*}(892)^{+}}\right)^{2} - s_{23}} \\ & \;+\; \sum_{\lambda_{K^*}=0}^{0} \frac{\left(-1\right)^{\frac{1}{2} - \lambda'_{1}} C_{K_{0}^{*}(1430)^{+}} \Gamma_{K_{0}^{*}(1430)^{+}} m_{K_{0}^{*}(1430)^{+}} \delta_{\lambda'_{0}, - \lambda'_{1} + \lambda_{K^*}} d^{0}_{\lambda_{K^*},0}\left(\theta_{23}\right)}{- i \Gamma_{K_{0}^{*}(1430)^{+}} m_{K_{0}^{*}(1430)^{+}} + \left(m_{K_{0}^{*}(1430)^{+}}\right)^{2} - s_{23}} \\ & \;+\; \sum_{\lambda_{K^*}=0}^{0} \frac{\left(-1\right)^{\frac{1}{2} - \lambda'_{1}} C_{K_{0}^{*}(1950)^{+}} \Gamma_{K_{0}^{*}(1950)^{+}} m_{K_{0}^{*}(1950)^{+}} \delta_{\lambda'_{0}, - \lambda'_{1} + \lambda_{K^*}} d^{0}_{\lambda_{K^*},0}\left(\theta_{23}\right)}{- i \Gamma_{K_{0}^{*}(1950)^{+}} m_{K_{0}^{*}(1950)^{+}} + \left(m_{K_{0}^{*}(1950)^{+}}\right)^{2} - s_{23}} \\ & \;+\; \sum_{\lambda_{K^*}=0}^{0} \frac{\left(-1\right)^{\frac{1}{2} - \lambda'_{1}} C_{K_{0}^{*}(700)^{+}} \Gamma_{K_{0}^{*}(700)^{+}} m_{K_{0}^{*}(700)^{+}} \delta_{\lambda'_{0}, - \lambda'_{1} + \lambda_{K^*}} d^{0}_{\lambda_{K^*},0}\left(\theta_{23}\right)}{- i \Gamma_{K_{0}^{*}(700)^{+}} m_{K_{0}^{*}(700)^{+}} + \left(m_{K_{0}^{*}(700)^{+}}\right)^{2} - s_{23}} \\ & \;+\; \sum_{\lambda_{K^*}=-2}^{2} \frac{\left(-1\right)^{\frac{1}{2} - \lambda'_{1}} C_{K_{2}^{*}(1430)^{+}} \Gamma_{K_{2}^{*}(1430)^{+}} m_{K_{2}^{*}(1430)^{+}} \delta_{\lambda'_{0}, - \lambda'_{1} + \lambda_{K^*}} d^{2}_{\lambda_{K^*},0}\left(\theta_{23}\right)}{- i \Gamma_{K_{2}^{*}(1430)^{+}} m_{K_{2}^{*}(1430)^{+}} + \left(m_{K_{2}^{*}(1430)^{+}}\right)^{2} - s_{23}} \\ & \;+\; \sum_{\lambda_{K^*}=-2}^{2} \frac{\left(-1\right)^{\frac{1}{2} - \lambda'_{1}} C_{K_{2}^{*}(1980)^{+}} \Gamma_{K_{2}^{*}(1980)^{+}} m_{K_{2}^{*}(1980)^{+}} \delta_{\lambda'_{0}, - \lambda'_{1} + \lambda_{K^*}} d^{2}_{\lambda_{K^*},0}\left(\theta_{23}\right)}{- i \Gamma_{K_{2}^{*}(1980)^{+}} m_{K_{2}^{*}(1980)^{+}} + \left(m_{K_{2}^{*}(1980)^{+}}\right)^{2} - s_{23}} \\ & \;+\; \sum_{\lambda_{K^*}=-3}^{3} \frac{\left(-1\right)^{\frac{1}{2} - \lambda'_{1}} C_{K_{3}^{*}(1780)^{+}} \Gamma_{K_{3}^{*}(1780)^{+}} m_{K_{3}^{*}(1780)^{+}} \delta_{\lambda'_{0}, - \lambda'_{1} + \lambda_{K^*}} d^{3}_{\lambda_{K^*},0}\left(\theta_{23}\right)}{- i \Gamma_{K_{3}^{*}(1780)^{+}} m_{K_{3}^{*}(1780)^{+}} + \left(m_{K_{3}^{*}(1780)^{+}}\right)^{2} - s_{23}} \\ & \;+\; \sum_{\lambda_{K^*}=-4}^{4} \frac{\left(-1\right)^{\frac{1}{2} - \lambda'_{1}} C_{K_{4}^{*}(2045)^{+}} \Gamma_{K_{4}^{*}(2045)^{+}} m_{K_{4}^{*}(2045)^{+}} \delta_{\lambda'_{0}, - \lambda'_{1} + \lambda_{K^*}} d^{4}_{\lambda_{K^*},0}\left(\theta_{23}\right)}{- i \Gamma_{K_{4}^{*}(2045)^{+}} m_{K_{4}^{*}(2045)^{+}} + \left(m_{K_{4}^{*}(2045)^{+}}\right)^{2} - s_{23}} \\ \end{aligned}\end{split}\]

\(A^2 \equiv A^{\Sigma^*}\)

s31, m_sigma, gamma_sigma, c_sigma, j_sigma = sp.symbols(
    r"s_{31} m_{\Sigma^*} \Gamma_{\Sigma^*} C_{\Sigma^*} J_{\Sigma^*}"
)
theta31 = sp.Symbol(r"theta_31")
lambda_R = sp.Symbol(r"\lambda_{\Sigma^*}")

A2_expr = sp.Sum(
    (
        gamma_sigma
        * m_sigma
        * c_sigma
        * sp.KroneckerDelta(λ0prime, lambda_R)
        * Wigner.d(j_sigma, lambda_R, -λ1prime, theta31)
        * (-1) ** (half - λ1prime)
        / (-sp.I * gamma_sigma * m_sigma + m_sigma**2 - s31)
    ),
    (lambda_R, -j_sigma, j_sigma),
)
A2_expr

\[\displaystyle \sum_{\lambda_{\Sigma^*}=- J_{\Sigma^*}}^{J_{\Sigma^*}} \frac{\left(-1\right)^{\frac{1}{2} - \lambda'_{1}} C_{\Sigma^*} \Gamma_{\Sigma^*} m_{\Sigma^*} \delta_{\lambda'_{0} \lambda_{\Sigma^*}} d^{J_{\Sigma^*}}_{\lambda_{\Sigma^*},- \lambda'_{1}}\left(\theta_{31}\right)}{- i \Gamma_{\Sigma^*} m_{\Sigma^*} + \left(m_{\Sigma^*}\right)^{2} - s_{31}}\]

def formulate_chain_amplitude_a2(
    resonance: Particle,
    λ0: sp.Expr,
    λ1: sp.Expr,
) -> sp.Expr:
    c_sigma = sp.Symbol(rf"C_{{{resonance.latex}}}")
    m_sigma = sp.Symbol(rf"m_{{{resonance.latex}}}")
    gamma_sigma = sp.Symbol(rf"\Gamma_{{{resonance.latex}}}")
    theta31 = sp.Symbol("theta_31")
    s31, lambda_R = sp.symbols(r"s_{31} \lambda_{\Sigma^*}")
    return sp.Sum(
        (
            gamma_sigma
            * m_sigma
            * c_sigma
            * sp.KroneckerDelta(λ0, lambda_R)
            * Wigner.d(resonance.spin, lambda_R, -λ1, theta31)
            * (-1) ** (half - λ1)
            / (-sp.I * gamma_sigma * m_sigma + m_sigma**2 - s31)
        ),
        (lambda_R, -resonance.spin, resonance.spin),
    )


formulate_chain_amplitude_a2(
    particle_db["Sigma(1385)0"],
    λ0prime,
    λ1prime,
)

\[\displaystyle \sum_{\lambda_{\Sigma^*}=- \frac{3}{2}}^{\frac{3}{2}} \frac{\left(-1\right)^{\frac{1}{2} - \lambda'_{1}} C_{\Sigma(1385)^{0}} \Gamma_{\Sigma(1385)^{0}} m_{\Sigma(1385)^{0}} \delta_{\lambda'_{0} \lambda_{\Sigma^*}} d^{\frac{3}{2}}_{\lambda_{\Sigma^*},- \lambda'_{1}}\left(\theta_{31}\right)}{- i \Gamma_{\Sigma(1385)^{0}} m_{\Sigma(1385)^{0}} + \left(m_{\Sigma(1385)^{0}}\right)^{2} - s_{31}}\]

def formulate_a2(
    reaction: qrules.ReactionInfo,
    λ0: sp.Expr,
    λ1: sp.Expr,
) -> sp.Expr:
    resonances = get_resonances(reaction, recoil_id=2)
    amplitudes = (formulate_chain_amplitude_a2(r, λ0, λ1) for r in resonances)
    return sum(amplitudes, start=sp.S.Zero)


A2_expr = formulate_a2(reaction, λ0prime, λ1prime)
A2_expr

\[\displaystyle \sum_{\lambda_{\Sigma^*}=- \frac{3}{2}}^{\frac{3}{2}} \frac{\left(-1\right)^{\frac{1}{2} - \lambda'_{1}} C_{\Sigma(1385)^{0}} \Gamma_{\Sigma(1385)^{0}} m_{\Sigma(1385)^{0}} \delta_{\lambda'_{0} \lambda_{\Sigma^*}} d^{\frac{3}{2}}_{\lambda_{\Sigma^*},- \lambda'_{1}}\left(\theta_{31}\right)}{- i \Gamma_{\Sigma(1385)^{0}} m_{\Sigma(1385)^{0}} + \left(m_{\Sigma(1385)^{0}}\right)^{2} - s_{31}} + \sum_{\lambda_{\Sigma^*}=- \frac{1}{2}}^{\frac{1}{2}} \frac{\left(-1\right)^{\frac{1}{2} - \lambda'_{1}} C_{\Sigma(1660)^{0}} \Gamma_{\Sigma(1660)^{0}} m_{\Sigma(1660)^{0}} \delta_{\lambda'_{0} \lambda_{\Sigma^*}} d^{\frac{1}{2}}_{\lambda_{\Sigma^*},- \lambda'_{1}}\left(\theta_{31}\right)}{- i \Gamma_{\Sigma(1660)^{0}} m_{\Sigma(1660)^{0}} + \left(m_{\Sigma(1660)^{0}}\right)^{2} - s_{31}} + \sum_{\lambda_{\Sigma^*}=- \frac{3}{2}}^{\frac{3}{2}} \frac{\left(-1\right)^{\frac{1}{2} - \lambda'_{1}} C_{\Sigma(1670)^{0}} \Gamma_{\Sigma(1670)^{0}} m_{\Sigma(1670)^{0}} \delta_{\lambda'_{0} \lambda_{\Sigma^*}} d^{\frac{3}{2}}_{\lambda_{\Sigma^*},- \lambda'_{1}}\left(\theta_{31}\right)}{- i \Gamma_{\Sigma(1670)^{0}} m_{\Sigma(1670)^{0}} + \left(m_{\Sigma(1670)^{0}}\right)^{2} - s_{31}} + \sum_{\lambda_{\Sigma^*}=- \frac{1}{2}}^{\frac{1}{2}} \frac{\left(-1\right)^{\frac{1}{2} - \lambda'_{1}} C_{\Sigma(1750)^{0}} \Gamma_{\Sigma(1750)^{0}} m_{\Sigma(1750)^{0}} \delta_{\lambda'_{0} \lambda_{\Sigma^*}} d^{\frac{1}{2}}_{\lambda_{\Sigma^*},- \lambda'_{1}}\left(\theta_{31}\right)}{- i \Gamma_{\Sigma(1750)^{0}} m_{\Sigma(1750)^{0}} + \left(m_{\Sigma(1750)^{0}}\right)^{2} - s_{31}} + \sum_{\lambda_{\Sigma^*}=- \frac{5}{2}}^{\frac{5}{2}} \frac{\left(-1\right)^{\frac{1}{2} - \lambda'_{1}} C_{\Sigma(1775)^{0}} \Gamma_{\Sigma(1775)^{0}} m_{\Sigma(1775)^{0}} \delta_{\lambda'_{0} \lambda_{\Sigma^*}} d^{\frac{5}{2}}_{\lambda_{\Sigma^*},- \lambda'_{1}}\left(\theta_{31}\right)}{- i \Gamma_{\Sigma(1775)^{0}} m_{\Sigma(1775)^{0}} + \left(m_{\Sigma(1775)^{0}}\right)^{2} - s_{31}} + \sum_{\lambda_{\Sigma^*}=- \frac{5}{2}}^{\frac{5}{2}} \frac{\left(-1\right)^{\frac{1}{2} - \lambda'_{1}} C_{\Sigma(1915)^{0}} \Gamma_{\Sigma(1915)^{0}} m_{\Sigma(1915)^{0}} \delta_{\lambda'_{0} \lambda_{\Sigma^*}} d^{\frac{5}{2}}_{\lambda_{\Sigma^*},- \lambda'_{1}}\left(\theta_{31}\right)}{- i \Gamma_{\Sigma(1915)^{0}} m_{\Sigma(1915)^{0}} + \left(m_{\Sigma(1915)^{0}}\right)^{2} - s_{31}} + \sum_{\lambda_{\Sigma^*}=- \frac{3}{2}}^{\frac{3}{2}} \frac{\left(-1\right)^{\frac{1}{2} - \lambda'_{1}} C_{\Sigma(1940)^{0}} \Gamma_{\Sigma(1940)^{0}} m_{\Sigma(1940)^{0}} \delta_{\lambda'_{0} \lambda_{\Sigma^*}} d^{\frac{3}{2}}_{\lambda_{\Sigma^*},- \lambda'_{1}}\left(\theta_{31}\right)}{- i \Gamma_{\Sigma(1940)^{0}} m_{\Sigma(1940)^{0}} + \left(m_{\Sigma(1940)^{0}}\right)^{2} - s_{31}} + \sum_{\lambda_{\Sigma^*}=- \frac{7}{2}}^{\frac{7}{2}} \frac{\left(-1\right)^{\frac{1}{2} - \lambda'_{1}} C_{\Sigma(2030)^{0}} \Gamma_{\Sigma(2030)^{0}} m_{\Sigma(2030)^{0}} \delta_{\lambda'_{0} \lambda_{\Sigma^*}} d^{\frac{7}{2}}_{\lambda_{\Sigma^*},- \lambda'_{1}}\left(\theta_{31}\right)}{- i \Gamma_{\Sigma(2030)^{0}} m_{\Sigma(2030)^{0}} + \left(m_{\Sigma(2030)^{0}}\right)^{2} - s_{31}}\]

Math(aslatex(A2_expr, terms_per_line=1))

\[\begin{split}\displaystyle \begin{aligned} & \sum_{\lambda_{\Sigma^*}=- \frac{3}{2}}^{\frac{3}{2}} \frac{\left(-1\right)^{\frac{1}{2} - \lambda'_{1}} C_{\Sigma(1385)^{0}} \Gamma_{\Sigma(1385)^{0}} m_{\Sigma(1385)^{0}} \delta_{\lambda'_{0} \lambda_{\Sigma^*}} d^{\frac{3}{2}}_{\lambda_{\Sigma^*},- \lambda'_{1}}\left(\theta_{31}\right)}{- i \Gamma_{\Sigma(1385)^{0}} m_{\Sigma(1385)^{0}} + \left(m_{\Sigma(1385)^{0}}\right)^{2} - s_{31}} \\ & \;+\; \sum_{\lambda_{\Sigma^*}=- \frac{1}{2}}^{\frac{1}{2}} \frac{\left(-1\right)^{\frac{1}{2} - \lambda'_{1}} C_{\Sigma(1660)^{0}} \Gamma_{\Sigma(1660)^{0}} m_{\Sigma(1660)^{0}} \delta_{\lambda'_{0} \lambda_{\Sigma^*}} d^{\frac{1}{2}}_{\lambda_{\Sigma^*},- \lambda'_{1}}\left(\theta_{31}\right)}{- i \Gamma_{\Sigma(1660)^{0}} m_{\Sigma(1660)^{0}} + \left(m_{\Sigma(1660)^{0}}\right)^{2} - s_{31}} \\ & \;+\; \sum_{\lambda_{\Sigma^*}=- \frac{3}{2}}^{\frac{3}{2}} \frac{\left(-1\right)^{\frac{1}{2} - \lambda'_{1}} C_{\Sigma(1670)^{0}} \Gamma_{\Sigma(1670)^{0}} m_{\Sigma(1670)^{0}} \delta_{\lambda'_{0} \lambda_{\Sigma^*}} d^{\frac{3}{2}}_{\lambda_{\Sigma^*},- \lambda'_{1}}\left(\theta_{31}\right)}{- i \Gamma_{\Sigma(1670)^{0}} m_{\Sigma(1670)^{0}} + \left(m_{\Sigma(1670)^{0}}\right)^{2} - s_{31}} \\ & \;+\; \sum_{\lambda_{\Sigma^*}=- \frac{1}{2}}^{\frac{1}{2}} \frac{\left(-1\right)^{\frac{1}{2} - \lambda'_{1}} C_{\Sigma(1750)^{0}} \Gamma_{\Sigma(1750)^{0}} m_{\Sigma(1750)^{0}} \delta_{\lambda'_{0} \lambda_{\Sigma^*}} d^{\frac{1}{2}}_{\lambda_{\Sigma^*},- \lambda'_{1}}\left(\theta_{31}\right)}{- i \Gamma_{\Sigma(1750)^{0}} m_{\Sigma(1750)^{0}} + \left(m_{\Sigma(1750)^{0}}\right)^{2} - s_{31}} \\ & \;+\; \sum_{\lambda_{\Sigma^*}=- \frac{5}{2}}^{\frac{5}{2}} \frac{\left(-1\right)^{\frac{1}{2} - \lambda'_{1}} C_{\Sigma(1775)^{0}} \Gamma_{\Sigma(1775)^{0}} m_{\Sigma(1775)^{0}} \delta_{\lambda'_{0} \lambda_{\Sigma^*}} d^{\frac{5}{2}}_{\lambda_{\Sigma^*},- \lambda'_{1}}\left(\theta_{31}\right)}{- i \Gamma_{\Sigma(1775)^{0}} m_{\Sigma(1775)^{0}} + \left(m_{\Sigma(1775)^{0}}\right)^{2} - s_{31}} \\ & \;+\; \sum_{\lambda_{\Sigma^*}=- \frac{5}{2}}^{\frac{5}{2}} \frac{\left(-1\right)^{\frac{1}{2} - \lambda'_{1}} C_{\Sigma(1915)^{0}} \Gamma_{\Sigma(1915)^{0}} m_{\Sigma(1915)^{0}} \delta_{\lambda'_{0} \lambda_{\Sigma^*}} d^{\frac{5}{2}}_{\lambda_{\Sigma^*},- \lambda'_{1}}\left(\theta_{31}\right)}{- i \Gamma_{\Sigma(1915)^{0}} m_{\Sigma(1915)^{0}} + \left(m_{\Sigma(1915)^{0}}\right)^{2} - s_{31}} \\ & \;+\; \sum_{\lambda_{\Sigma^*}=- \frac{3}{2}}^{\frac{3}{2}} \frac{\left(-1\right)^{\frac{1}{2} - \lambda'_{1}} C_{\Sigma(1940)^{0}} \Gamma_{\Sigma(1940)^{0}} m_{\Sigma(1940)^{0}} \delta_{\lambda'_{0} \lambda_{\Sigma^*}} d^{\frac{3}{2}}_{\lambda_{\Sigma^*},- \lambda'_{1}}\left(\theta_{31}\right)}{- i \Gamma_{\Sigma(1940)^{0}} m_{\Sigma(1940)^{0}} + \left(m_{\Sigma(1940)^{0}}\right)^{2} - s_{31}} \\ & \;+\; \sum_{\lambda_{\Sigma^*}=- \frac{7}{2}}^{\frac{7}{2}} \frac{\left(-1\right)^{\frac{1}{2} - \lambda'_{1}} C_{\Sigma(2030)^{0}} \Gamma_{\Sigma(2030)^{0}} m_{\Sigma(2030)^{0}} \delta_{\lambda'_{0} \lambda_{\Sigma^*}} d^{\frac{7}{2}}_{\lambda_{\Sigma^*},- \lambda'_{1}}\left(\theta_{31}\right)}{- i \Gamma_{\Sigma(2030)^{0}} m_{\Sigma(2030)^{0}} + \left(m_{\Sigma(2030)^{0}}\right)^{2} - s_{31}} \\ \end{aligned}\end{split}\]

\(A^3 \equiv A^{N^*}\)

s12, m_n, gamma_n, c_n, j_n = sp.symbols(r"s_{12} m_{N^*} \Gamma_{N^*} C_{N^*} J_{N^*}")
theta12 = sp.Symbol(r"theta_12")
lambda_R = sp.Symbol(r"\lambda_{N^*}")

A3_expr = sp.Sum(
    (
        gamma_n
        * m_n
        * c_n
        * sp.KroneckerDelta(λ0prime, lambda_R)
        * Wigner.d(j_n, lambda_R, λ1prime, theta12)
        / (-sp.I * gamma_n * m_n + m_n**2 - s12)
    ),
    (lambda_R, -j_n, j_n),
)
A3_expr

\[\displaystyle \sum_{\lambda_{N^*}=- J_{N^*}}^{J_{N^*}} \frac{C_{N^*} \Gamma_{N^*} m_{N^*} \delta_{\lambda'_{0} \lambda_{N^*}} d^{J_{N^*}}_{\lambda_{N^*},\lambda'_{1}}\left(\theta_{12}\right)}{- i \Gamma_{N^*} m_{N^*} + \left(m_{N^*}\right)^{2} - s_{12}}\]

def formulate_chain_amplitude_a3(
    resonance: Particle,
    λ0: sp.Expr,
    λ1: sp.Expr,
) -> sp.Expr:
    c_n = sp.Symbol(rf"C_{{{resonance.latex}}}")
    m_n = sp.Symbol(rf"m_{{{resonance.latex}}}")
    gamma_n = sp.Symbol(rf"\Gamma_{{{resonance.latex}}}")
    theta12 = sp.Symbol("theta_12")
    s12, lambda_R = sp.symbols(r"s_{12} \lambda_{N^*}")
    return sp.Sum(
        (
            gamma_n
            * m_n
            * c_n
            * sp.KroneckerDelta(λ0, lambda_R)
            * Wigner.d(resonance.spin, lambda_R, λ1, theta12)
            / (-sp.I * gamma_n * m_n + m_n**2 - s12)
        ),
        (lambda_R, -resonance.spin, resonance.spin),
    )


formulate_chain_amplitude_a3(
    particle_db["N(1650)+"],
    λ0prime,
    λ1prime,
)

\[\displaystyle \sum_{\lambda_{N^*}=- \frac{1}{2}}^{\frac{1}{2}} \frac{C_{N(1650)^{+}} \Gamma_{N(1650)^{+}} m_{N(1650)^{+}} \delta_{\lambda'_{0} \lambda_{N^*}} d^{\frac{1}{2}}_{\lambda_{N^*},\lambda'_{1}}\left(\theta_{12}\right)}{- i \Gamma_{N(1650)^{+}} m_{N(1650)^{+}} + \left(m_{N(1650)^{+}}\right)^{2} - s_{12}}\]

def formulate_a3(
    reaction: qrules.ReactionInfo,
    λ0: sp.Expr,
    λ1: sp.Expr,
) -> sp.Expr:
    resonances = get_resonances(reaction, recoil_id=3)
    amplitudes = (formulate_chain_amplitude_a3(r, λ0, λ1) for r in resonances)
    return sum(amplitudes, start=sp.S.Zero)


A3_expr = formulate_a3(reaction, λ0prime, λ1prime)
A3_expr

\[\displaystyle \sum_{\lambda_{N^*}=- \frac{1}{2}}^{\frac{1}{2}} \frac{C_{N(1650)^{+}} \Gamma_{N(1650)^{+}} m_{N(1650)^{+}} \delta_{\lambda'_{0} \lambda_{N^*}} d^{\frac{1}{2}}_{\lambda_{N^*},\lambda'_{1}}\left(\theta_{12}\right)}{- i \Gamma_{N(1650)^{+}} m_{N(1650)^{+}} + \left(m_{N(1650)^{+}}\right)^{2} - s_{12}} + \sum_{\lambda_{N^*}=- \frac{5}{2}}^{\frac{5}{2}} \frac{C_{N(1675)^{+}} \Gamma_{N(1675)^{+}} m_{N(1675)^{+}} \delta_{\lambda'_{0} \lambda_{N^*}} d^{\frac{5}{2}}_{\lambda_{N^*},\lambda'_{1}}\left(\theta_{12}\right)}{- i \Gamma_{N(1675)^{+}} m_{N(1675)^{+}} + \left(m_{N(1675)^{+}}\right)^{2} - s_{12}} + \sum_{\lambda_{N^*}=- \frac{5}{2}}^{\frac{5}{2}} \frac{C_{N(1680)^{+}} \Gamma_{N(1680)^{+}} m_{N(1680)^{+}} \delta_{\lambda'_{0} \lambda_{N^*}} d^{\frac{5}{2}}_{\lambda_{N^*},\lambda'_{1}}\left(\theta_{12}\right)}{- i \Gamma_{N(1680)^{+}} m_{N(1680)^{+}} + \left(m_{N(1680)^{+}}\right)^{2} - s_{12}} + \sum_{\lambda_{N^*}=- \frac{3}{2}}^{\frac{3}{2}} \frac{C_{N(1700)^{+}} \Gamma_{N(1700)^{+}} m_{N(1700)^{+}} \delta_{\lambda'_{0} \lambda_{N^*}} d^{\frac{3}{2}}_{\lambda_{N^*},\lambda'_{1}}\left(\theta_{12}\right)}{- i \Gamma_{N(1700)^{+}} m_{N(1700)^{+}} + \left(m_{N(1700)^{+}}\right)^{2} - s_{12}} + \sum_{\lambda_{N^*}=- \frac{1}{2}}^{\frac{1}{2}} \frac{C_{N(1710)^{+}} \Gamma_{N(1710)^{+}} m_{N(1710)^{+}} \delta_{\lambda'_{0} \lambda_{N^*}} d^{\frac{1}{2}}_{\lambda_{N^*},\lambda'_{1}}\left(\theta_{12}\right)}{- i \Gamma_{N(1710)^{+}} m_{N(1710)^{+}} + \left(m_{N(1710)^{+}}\right)^{2} - s_{12}} + \sum_{\lambda_{N^*}=- \frac{3}{2}}^{\frac{3}{2}} \frac{C_{N(1720)^{+}} \Gamma_{N(1720)^{+}} m_{N(1720)^{+}} \delta_{\lambda'_{0} \lambda_{N^*}} d^{\frac{3}{2}}_{\lambda_{N^*},\lambda'_{1}}\left(\theta_{12}\right)}{- i \Gamma_{N(1720)^{+}} m_{N(1720)^{+}} + \left(m_{N(1720)^{+}}\right)^{2} - s_{12}} + \sum_{\lambda_{N^*}=- \frac{3}{2}}^{\frac{3}{2}} \frac{C_{N(1875)^{+}} \Gamma_{N(1875)^{+}} m_{N(1875)^{+}} \delta_{\lambda'_{0} \lambda_{N^*}} d^{\frac{3}{2}}_{\lambda_{N^*},\lambda'_{1}}\left(\theta_{12}\right)}{- i \Gamma_{N(1875)^{+}} m_{N(1875)^{+}} + \left(m_{N(1875)^{+}}\right)^{2} - s_{12}} + \sum_{\lambda_{N^*}=- \frac{1}{2}}^{\frac{1}{2}} \frac{C_{N(1880)^{+}} \Gamma_{N(1880)^{+}} m_{N(1880)^{+}} \delta_{\lambda'_{0} \lambda_{N^*}} d^{\frac{1}{2}}_{\lambda_{N^*},\lambda'_{1}}\left(\theta_{12}\right)}{- i \Gamma_{N(1880)^{+}} m_{N(1880)^{+}} + \left(m_{N(1880)^{+}}\right)^{2} - s_{12}} + \sum_{\lambda_{N^*}=- \frac{1}{2}}^{\frac{1}{2}} \frac{C_{N(1895)^{+}} \Gamma_{N(1895)^{+}} m_{N(1895)^{+}} \delta_{\lambda'_{0} \lambda_{N^*}} d^{\frac{1}{2}}_{\lambda_{N^*},\lambda'_{1}}\left(\theta_{12}\right)}{- i \Gamma_{N(1895)^{+}} m_{N(1895)^{+}} + \left(m_{N(1895)^{+}}\right)^{2} - s_{12}} + \sum_{\lambda_{N^*}=- \frac{3}{2}}^{\frac{3}{2}} \frac{C_{N(1900)^{+}} \Gamma_{N(1900)^{+}} m_{N(1900)^{+}} \delta_{\lambda'_{0} \lambda_{N^*}} d^{\frac{3}{2}}_{\lambda_{N^*},\lambda'_{1}}\left(\theta_{12}\right)}{- i \Gamma_{N(1900)^{+}} m_{N(1900)^{+}} + \left(m_{N(1900)^{+}}\right)^{2} - s_{12}} + \sum_{\lambda_{N^*}=- \frac{5}{2}}^{\frac{5}{2}} \frac{C_{N(2060)^{+}} \Gamma_{N(2060)^{+}} m_{N(2060)^{+}} \delta_{\lambda'_{0} \lambda_{N^*}} d^{\frac{5}{2}}_{\lambda_{N^*},\lambda'_{1}}\left(\theta_{12}\right)}{- i \Gamma_{N(2060)^{+}} m_{N(2060)^{+}} + \left(m_{N(2060)^{+}}\right)^{2} - s_{12}} + \sum_{\lambda_{N^*}=- \frac{7}{2}}^{\frac{7}{2}} \frac{C_{N(2190)^{+}} \Gamma_{N(2190)^{+}} m_{N(2190)^{+}} \delta_{\lambda'_{0} \lambda_{N^*}} d^{\frac{7}{2}}_{\lambda_{N^*},\lambda'_{1}}\left(\theta_{12}\right)}{- i \Gamma_{N(2190)^{+}} m_{N(2190)^{+}} + \left(m_{N(2190)^{+}}\right)^{2} - s_{12}}\]

Math(aslatex(A3_expr, terms_per_line=1))

\[\begin{split}\displaystyle \begin{aligned} & \sum_{\lambda_{N^*}=- \frac{1}{2}}^{\frac{1}{2}} \frac{C_{N(1650)^{+}} \Gamma_{N(1650)^{+}} m_{N(1650)^{+}} \delta_{\lambda'_{0} \lambda_{N^*}} d^{\frac{1}{2}}_{\lambda_{N^*},\lambda'_{1}}\left(\theta_{12}\right)}{- i \Gamma_{N(1650)^{+}} m_{N(1650)^{+}} + \left(m_{N(1650)^{+}}\right)^{2} - s_{12}} \\ & \;+\; \sum_{\lambda_{N^*}=- \frac{5}{2}}^{\frac{5}{2}} \frac{C_{N(1675)^{+}} \Gamma_{N(1675)^{+}} m_{N(1675)^{+}} \delta_{\lambda'_{0} \lambda_{N^*}} d^{\frac{5}{2}}_{\lambda_{N^*},\lambda'_{1}}\left(\theta_{12}\right)}{- i \Gamma_{N(1675)^{+}} m_{N(1675)^{+}} + \left(m_{N(1675)^{+}}\right)^{2} - s_{12}} \\ & \;+\; \sum_{\lambda_{N^*}=- \frac{5}{2}}^{\frac{5}{2}} \frac{C_{N(1680)^{+}} \Gamma_{N(1680)^{+}} m_{N(1680)^{+}} \delta_{\lambda'_{0} \lambda_{N^*}} d^{\frac{5}{2}}_{\lambda_{N^*},\lambda'_{1}}\left(\theta_{12}\right)}{- i \Gamma_{N(1680)^{+}} m_{N(1680)^{+}} + \left(m_{N(1680)^{+}}\right)^{2} - s_{12}} \\ & \;+\; \sum_{\lambda_{N^*}=- \frac{3}{2}}^{\frac{3}{2}} \frac{C_{N(1700)^{+}} \Gamma_{N(1700)^{+}} m_{N(1700)^{+}} \delta_{\lambda'_{0} \lambda_{N^*}} d^{\frac{3}{2}}_{\lambda_{N^*},\lambda'_{1}}\left(\theta_{12}\right)}{- i \Gamma_{N(1700)^{+}} m_{N(1700)^{+}} + \left(m_{N(1700)^{+}}\right)^{2} - s_{12}} \\ & \;+\; \sum_{\lambda_{N^*}=- \frac{1}{2}}^{\frac{1}{2}} \frac{C_{N(1710)^{+}} \Gamma_{N(1710)^{+}} m_{N(1710)^{+}} \delta_{\lambda'_{0} \lambda_{N^*}} d^{\frac{1}{2}}_{\lambda_{N^*},\lambda'_{1}}\left(\theta_{12}\right)}{- i \Gamma_{N(1710)^{+}} m_{N(1710)^{+}} + \left(m_{N(1710)^{+}}\right)^{2} - s_{12}} \\ & \;+\; \sum_{\lambda_{N^*}=- \frac{3}{2}}^{\frac{3}{2}} \frac{C_{N(1720)^{+}} \Gamma_{N(1720)^{+}} m_{N(1720)^{+}} \delta_{\lambda'_{0} \lambda_{N^*}} d^{\frac{3}{2}}_{\lambda_{N^*},\lambda'_{1}}\left(\theta_{12}\right)}{- i \Gamma_{N(1720)^{+}} m_{N(1720)^{+}} + \left(m_{N(1720)^{+}}\right)^{2} - s_{12}} \\ & \;+\; \sum_{\lambda_{N^*}=- \frac{3}{2}}^{\frac{3}{2}} \frac{C_{N(1875)^{+}} \Gamma_{N(1875)^{+}} m_{N(1875)^{+}} \delta_{\lambda'_{0} \lambda_{N^*}} d^{\frac{3}{2}}_{\lambda_{N^*},\lambda'_{1}}\left(\theta_{12}\right)}{- i \Gamma_{N(1875)^{+}} m_{N(1875)^{+}} + \left(m_{N(1875)^{+}}\right)^{2} - s_{12}} \\ & \;+\; \sum_{\lambda_{N^*}=- \frac{1}{2}}^{\frac{1}{2}} \frac{C_{N(1880)^{+}} \Gamma_{N(1880)^{+}} m_{N(1880)^{+}} \delta_{\lambda'_{0} \lambda_{N^*}} d^{\frac{1}{2}}_{\lambda_{N^*},\lambda'_{1}}\left(\theta_{12}\right)}{- i \Gamma_{N(1880)^{+}} m_{N(1880)^{+}} + \left(m_{N(1880)^{+}}\right)^{2} - s_{12}} \\ & \;+\; \sum_{\lambda_{N^*}=- \frac{1}{2}}^{\frac{1}{2}} \frac{C_{N(1895)^{+}} \Gamma_{N(1895)^{+}} m_{N(1895)^{+}} \delta_{\lambda'_{0} \lambda_{N^*}} d^{\frac{1}{2}}_{\lambda_{N^*},\lambda'_{1}}\left(\theta_{12}\right)}{- i \Gamma_{N(1895)^{+}} m_{N(1895)^{+}} + \left(m_{N(1895)^{+}}\right)^{2} - s_{12}} \\ & \;+\; \sum_{\lambda_{N^*}=- \frac{3}{2}}^{\frac{3}{2}} \frac{C_{N(1900)^{+}} \Gamma_{N(1900)^{+}} m_{N(1900)^{+}} \delta_{\lambda'_{0} \lambda_{N^*}} d^{\frac{3}{2}}_{\lambda_{N^*},\lambda'_{1}}\left(\theta_{12}\right)}{- i \Gamma_{N(1900)^{+}} m_{N(1900)^{+}} + \left(m_{N(1900)^{+}}\right)^{2} - s_{12}} \\ & \;+\; \sum_{\lambda_{N^*}=- \frac{5}{2}}^{\frac{5}{2}} \frac{C_{N(2060)^{+}} \Gamma_{N(2060)^{+}} m_{N(2060)^{+}} \delta_{\lambda'_{0} \lambda_{N^*}} d^{\frac{5}{2}}_{\lambda_{N^*},\lambda'_{1}}\left(\theta_{12}\right)}{- i \Gamma_{N(2060)^{+}} m_{N(2060)^{+}} + \left(m_{N(2060)^{+}}\right)^{2} - s_{12}} \\ & \;+\; \sum_{\lambda_{N^*}=- \frac{7}{2}}^{\frac{7}{2}} \frac{C_{N(2190)^{+}} \Gamma_{N(2190)^{+}} m_{N(2190)^{+}} \delta_{\lambda'_{0} \lambda_{N^*}} d^{\frac{7}{2}}_{\lambda_{N^*},\lambda'_{1}}\left(\theta_{12}\right)}{- i \Gamma_{N(2190)^{+}} m_{N(2190)^{+}} + \left(m_{N(2190)^{+}}\right)^{2} - s_{12}} \\ \end{aligned}\end{split}\]