2. Cross-check with LHCb data

Import Python libraries

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import json
import logging
import os
from functools import cache
from textwrap import dedent

import numpy as np
import sympy as sp
from ampform_dpd import AmplitudeModel
from ampform_dpd.io import aslatex, cached, simplify_latex_rendering
from IPython.display import Markdown, Math
from tqdm.auto import tqdm

from polarimetry.data import create_data_transformer
from polarimetry.io import display_latex, mute_jax_warnings
from polarimetry.lhcb import (
    ModelName,
    get_conversion_factor,
    get_conversion_factor_ls,
    load_model,
    load_model_builder,
    parameter_key_to_symbol,
)
from polarimetry.lhcb.particle import load_particles


@cache
def load_model_cached(model_id: int | ModelName) -> AmplitudeModel:
    return load_model(MODEL_FILE, PARTICLES, model_id)


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

MODEL_FILE = "../data/model-definitions.yaml"
PARTICLES = load_particles("../data/particle-definitions.yaml")
DEFAULT_MODEL = load_model_cached(model_id=0)

with open("../data/crosscheck.json") as stream:
    crosscheck_data = json.load(stream)

2.1. Lineshape comparison

We compute a few lineshapes for the following point in phase space and compare it with the values from [1]:

Load phase space point

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σ1, σ2, σ3 = sp.symbols("sigma1:4", nonnegative=True)
lineshape_vars = crosscheck_data["mainvars"]
lineshape_subs = {
    σ1: lineshape_vars["m2kpi"],
    σ2: lineshape_vars["m2pk"],
    **DEFAULT_MODEL.parameter_defaults,
}
lineshape_vars

{'costhetap': -0.9949949110827053,
 'm2kpi': 0.7980703453578917,
 'm2pk': 3.6486261122281745,
 'phikpi': -0.4,
 'phip': -0.3}

The lineshapes are computed for the following decay chains:

Load selected decay chains

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K892_chain = DEFAULT_MODEL.decay.find_chain("K(892)")
L1405_chain = DEFAULT_MODEL.decay.find_chain("L(1405)")
L1690_chain = DEFAULT_MODEL.decay.find_chain("L(1690)")
Math(aslatex([K892_chain, L1405_chain, L1690_chain]))

\[\begin{split}\displaystyle \begin{array}{c} \Lambda_c^+ \xrightarrow[S=1/2]{L=0} \left(K(892) \xrightarrow[S=0]{L=1} \pi^+ K^-\right) p \\ \Lambda_c^+ \xrightarrow[S=1/2]{L=0} \left(\Lambda(1405) \xrightarrow[S=1/2]{L=0} K^- p\right) \pi^+ \\ \Lambda_c^+ \xrightarrow[S=3/2]{L=1} \left(\Lambda(1690) \xrightarrow[S=1/2]{L=2} K^- p\right) \pi^+ \\ \end{array}\end{split}\]

Values for LHCb-PAPER-2022-002

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crosscheck_data["lineshapes"]

{'BW_K(892)_p^1_q^0': '(2.1687201455088894+23.58225917009096j)',
 'BW_L(1405)_p^0_q^0': '(-0.5636481410171861+0.13763637759224928j)',
 'BW_L(1690)_p^2_q^1': '(-1.5078327158518026+0.9775036395061584j)'}

Values as computed by this framework

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def build_dynamics(c) -> sp.Expr:
    return builder.dynamics_choices.get_builder(c)(c).expression.doit()


builder = load_model_builder(MODEL_FILE, PARTICLES, model_id=0)
K892_bw_val = build_dynamics(K892_chain).xreplace(lineshape_subs).n()
L1405_bw_val = build_dynamics(L1405_chain).xreplace(lineshape_subs).n()
L1690_bw_val = build_dynamics(L1690_chain).xreplace(lineshape_subs).n()
display_latex([K892_bw_val, L1405_bw_val, L1690_bw_val])

\[\begin{split}\displaystyle \begin{array}{c} 2.16872014550901 + 23.5822591700909 i \\ -0.563648141017186 + 0.137636377592249 i \\ -1.5078327158518 + 0.977503639506159 i \\ \end{array}\end{split}\]

Assert that these values are equal

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lineshape_decimals = 13
np.testing.assert_array_almost_equal(
    np.array(list(map(complex, crosscheck_data["lineshapes"].values()))),
    np.array(list(map(complex, [K892_bw_val, L1405_bw_val, L1690_bw_val]))),
    decimal=lineshape_decimals,
)
src = f"""
:::{{tip}}
These values are **equal up to {lineshape_decimals} decimals**.
:::
"""
Markdown(src)

Tip

These values are equal up to 13 decimals.

2.2. Amplitude comparison

The amplitude for each decay chain and each outer state helicity combination are evaluated on the following point in phase space:

Load phase space point as in DPD coordinates

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amplitude_vars = dict(crosscheck_data["chainvars"])
transformer = create_data_transformer(DEFAULT_MODEL)
input_data = {
    str(σ1): amplitude_vars["m2kpi"],
    str(σ2): amplitude_vars["m2pk"],
    str(σ3): amplitude_vars["m2ppi"],
}
input_data = {k: float(v) for k, v in transformer(input_data).items()}
display_latex({sp.Symbol(k): v for k, v in input_data.items()})

\[\begin{split}\displaystyle \begin{aligned} \theta_{23} \;&=\; 1.8213411665201502 \\ \theta_{31} \;&=\; 1.803835148371573 \\ \theta_{12} \;&=\; 1.1139045236042195 \\ \zeta^0_{1(1)} \;&=\; 0.0 \\ \zeta^1_{1(1)} \;&=\; 0.0 \\ \zeta^0_{2(1)} \;&=\; -2.077768707671287 \\ \zeta^1_{2(1)} \;&=\; 0.22583331080384336 \\ \zeta^0_{3(1)} \;&=\; 2.6540796539955913 \\ \zeta^1_{3(1)} \;&=\; -0.5594175047790458 \\ \sigma_{1} \;&=\; 0.7980703453578917 \\ \sigma_{2} \;&=\; 3.6486261122281745 \\ \sigma_{3} \;&=\; 1.9247541217931925 \\ \end{aligned}\end{split}\]

Code for creating functions for each sub-amplitude

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@cache
def create_amplitude_functions(
    model_id: int | ModelName,
) -> dict[tuple[sp.Rational, sp.Rational], sp.Expr]:
    model = load_model(MODEL_FILE, PARTICLES, model_id)
    production_couplings = get_production_couplings(model_id)
    fixed_parameters = {
        s: v
        for s, v in model.parameter_defaults.items()
        if s not in production_couplings
    }
    exprs = formulate_amplitude_expressions(model_id)
    return {
        k: cached.lambdify(
            cached.xreplace(expr, fixed_parameters),
            parameters=production_couplings,
            backend="numpy",
        )
        for k, expr in tqdm(exprs.items(), desc="Performing doit", disable=NO_LOG)
    }


@cache
def formulate_amplitude_expressions(
    model_id: int | str,
) -> dict[tuple[sp.Rational, sp.Rational], sp.Expr]:
    builder = load_model_builder(MODEL_FILE, PARTICLES, model_id)
    half = sp.Rational(1, 2)
    exprs = {
        (λ_Λc, λ_p): builder.formulate_aligned_amplitude(
            λ_Λc, λ_p, 0, 0, reference_subsystem=1
        )[0]
        for λ_Λc in [-half, +half]
        for λ_p in [-half, +half]
    }
    model = load_model(MODEL_FILE, PARTICLES, model_id)
    return {
        k: cached.unfold(expr, model.amplitudes)
        for k, expr in tqdm(exprs.items(), desc="Unfolding expressions", disable=NO_LOG)
    }


@cache
def get_production_couplings(model_id: int | str) -> dict[sp.Indexed, complex]:
    model = load_model(MODEL_FILE, PARTICLES, model_id)
    return {
        symbol: value
        for symbol, value in model.parameter_defaults.items()
        if isinstance(symbol, sp.Indexed)
        if "production" in str(symbol)
    }

Code for creating a comparison table

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def plusminus_to_helicity(plusminus: str) -> sp.Expr:
    half = sp.Rational(1, 2)
    if plusminus == "+":
        return +half
    if plusminus == "-":
        return -half
    raise NotImplementedError(plusminus)


def create_comparison_table(
    model_id: int | str, decimals: int | None = None
) -> Markdown:
    min_ls = not is_ls_model(model_id)
    amplitude_funcs = create_amplitude_functions(model_id)
    real_amp_crosscheck = {
        k: v
        for k, v in get_amplitude_crosscheck_data(model_id).items()
        if k.startswith("Ar")
    }
    production_couplings = get_production_couplings(model_id)
    couplings_to_zero = {str(symbol): 0 for symbol in production_couplings}

    src = ""
    if decimals is not None:
        src += dedent(
            f"""
            :::{{tip}}
            Computed amplitudes are equal to LHCb amplitudes up to **{decimals} decimals**.
            :::
            """
        )
    src += dedent(
        """
        |     | Computed | Expected | Difference |
        | ---:| --------:| --------:| ----------:|
        """
    )
    for amp_identifier, entry in real_amp_crosscheck.items():
        coupling = parameter_key_to_symbol(
            amp_identifier.replace("Ar", "A"),
            particle_definitions=PARTICLES,
            min_ls=min_ls,
        )
        src += f"| **`{amp_identifier}`** | ${sp.latex(coupling)}$ |\n"
        for matrix_key, expected in entry.items():
            matrix_suffix = matrix_key[1:]  # ++, +-, -+, --
            λ_Λc, λ_p = map(plusminus_to_helicity, matrix_suffix)
            func = amplitude_funcs[λ_Λc, -λ_p]
            func.update_parameters(couplings_to_zero)
            func.update_parameters({str(coupling): 1})
            computed = complex(func(input_data))
            computed *= determine_conversion_factor(coupling, λ_p, min_ls)
            expected = complex(expected)
            if abs(expected) != 0.0:
                diff = abs(computed - expected) / abs(expected)
                if diff < 1e-6:
                    diff = f"{diff:.2e}"
                else:
                    diff = f'<span style="color:red;">{diff:.2e}</span>'
            else:
                diff = ""
            src += f"| `{matrix_key}` | {computed:>.6f} | {expected:>.6f} | {diff} |\n"
            if decimals is not None:
                np.testing.assert_array_almost_equal(
                    computed,
                    expected,
                    decimal=decimals,
                    err_msg=f"  {amp_identifier} {matrix_key}",
                )
    return Markdown(src)


def determine_conversion_factor(
    coupling: sp.Indexed, λ_p: sp.Rational, min_ls: bool
) -> int:
    resonance_latex = str(coupling.indices[0])
    resonance, *_ = (p for p in PARTICLES.values() if p.latex == resonance_latex)
    if min_ls:
        factor = get_conversion_factor(resonance)
    else:
        _, L, S = coupling.indices
        factor = get_conversion_factor_ls(resonance, L, S)
    half = sp.Rational(1, 2)
    factor *= int((-1) ** (half + λ_p))  # # additional sign flip for amplitude
    return factor


def is_ls_model(model_id: int | str) -> bool:
    if isinstance(model_id, int):
        return model_id == 17
    return "LS couplings" in model_id


def get_amplitude_crosscheck_data(model_id: int | str) -> dict[str, complex]:
    if is_ls_model(model_id):
        return crosscheck_data["chains_LS"]
    return crosscheck_data["chains"]

2.2.1. Default model

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create_comparison_table(model_id=0, decimals=13)

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Tip

Computed amplitudes are equal to LHCb amplitudes up to 13 decimals.

	Computed	Expected	Difference
ArD(1232)1	\(\mathcal{H}^\mathrm{production}_{\Delta(1232), - \frac{1}{2}, 0}\)
A++	-0.488498+0.517710j	-0.488498+0.517710j	2.05e-14
A+-	0.894898-0.948412j	0.894898-0.948412j	4.12e-15
A-+	0.121490-0.128755j	0.121490-0.128755j	2.36e-14
A--	-0.222563+0.235872j	-0.222563+0.235872j	4.63e-15
ArD(1232)2	\(\mathcal{H}^\mathrm{production}_{\Delta(1232), \frac{1}{2}, 0}\)
A++	-0.222563+0.235872j	-0.222563+0.235872j	4.63e-15
A+-	-0.121490+0.128755j	-0.121490+0.128755j	2.37e-14
A-+	-0.894898+0.948412j	-0.894898+0.948412j	4.42e-15
A--	-0.488498+0.517710j	-0.488498+0.517710j	2.05e-14
ArD(1600)1	\(\mathcal{H}^\mathrm{production}_{\Delta(1600), - \frac{1}{2}, 0}\)
A++	0.289160+0.081910j	0.289160+0.081910j	2.55e-14
A+-	-0.529724-0.150054j	-0.529724-0.150054j	4.53e-15
A-+	-0.071915-0.020371j	-0.071915-0.020371j	2.90e-14
A--	0.131743+0.037319j	0.131743+0.037319j	8.03e-15
ArD(1600)2	\(\mathcal{H}^\mathrm{production}_{\Delta(1600), \frac{1}{2}, 0}\)
A++	0.131743+0.037319j	0.131743+0.037319j	8.03e-15
A+-	0.071915+0.020371j	0.071915+0.020371j	2.89e-14
A-+	0.529724+0.150054j	0.529724+0.150054j	4.59e-15
A--	0.289160+0.081910j	0.289160+0.081910j	2.55e-14
ArD(1700)1	\(\mathcal{H}^\mathrm{production}_{\Delta(1700), - \frac{1}{2}, 0}\)
A++	-0.018885-0.001757j	-0.018885-0.001757j	1.57e-13
A+-	0.315695+0.029366j	0.315695+0.029366j	1.17e-14
A-+	0.004697+0.000437j	0.004697+0.000437j	1.54e-13
A--	-0.078514-0.007303j	-0.078514-0.007303j	1.47e-14
ArD(1700)2	\(\mathcal{H}^\mathrm{production}_{\Delta(1700), \frac{1}{2}, 0}\)
A++	0.078514+0.007303j	0.078514+0.007303j	1.47e-14
A+-	0.004697+0.000437j	0.004697+0.000437j	1.54e-13
A-+	0.315695+0.029366j	0.315695+0.029366j	1.17e-14
A--	0.018885+0.001757j	0.018885+0.001757j	1.57e-13
ArK(892)1	\(\mathcal{H}^\mathrm{production}_{K(892), 0, - \frac{1}{2}}\)
A++	-0.537695-5.846793j	-0.537695-5.846793j	3.05e-15
A+-	0.000000+0.000000j	0.000000+0.000000j
A-+	-0.000000+0.000000j	0.000000+0.000000j
A--	0.000000+0.000000j	0.000000+0.000000j
ArK(892)2	\(\mathcal{H}^\mathrm{production}_{K(892), -1, - \frac{1}{2}}\)
A++	-0.000000+0.000000j	0.000000+0.000000j
A+-	0.000000+0.000000j	0.000000+0.000000j
A-+	1.485636+16.154534j	1.485636+16.154534j	2.58e-15
A--	0.000000+0.000000j	0.000000+0.000000j
ArK(892)3	\(\mathcal{H}^\mathrm{production}_{K(892), 1, \frac{1}{2}}\)
A++	-0.000000+0.000000j	0.000000+0.000000j
A+-	-1.485636-16.154534j	-1.485636-16.154534j	2.58e-15
A-+	-0.000000+0.000000j	0.000000+0.000000j
A--	0.000000+0.000000j	0.000000+0.000000j
ArK(892)4	\(\mathcal{H}^\mathrm{production}_{K(892), 0, \frac{1}{2}}\)
A++	-0.000000+0.000000j	0.000000+0.000000j
A+-	0.000000+0.000000j	0.000000+0.000000j
A-+	-0.000000+0.000000j	0.000000+0.000000j
A--	-0.537695-5.846793j	-0.537695-5.846793j	3.05e-15
ArK(1430)1	\(\mathcal{H}^\mathrm{production}_{K(1430), 0, \frac{1}{2}}\)
A++	-0.000000+0.000000j	0.000000+0.000000j
A+-	0.000000+0.000000j	0.000000+0.000000j
A-+	-0.000000+0.000000j	0.000000+0.000000j
A--	0.909456+0.072819j	0.909456+0.072819j	1.22e-16
ArK(1430)2	\(\mathcal{H}^\mathrm{production}_{K(1430), 0, - \frac{1}{2}}\)
A++	0.909456+0.072819j	0.909456+0.072819j	1.22e-16
A+-	0.000000+0.000000j	0.000000+0.000000j
A-+	-0.000000+0.000000j	0.000000+0.000000j
A--	0.000000+0.000000j	0.000000+0.000000j
ArK(700)1	\(\mathcal{H}^\mathrm{production}_{K(700), 0, \frac{1}{2}}\)
A++	-0.000000+0.000000j	0.000000+0.000000j
A+-	0.000000+0.000000j	0.000000+0.000000j
A-+	-0.000000+0.000000j	0.000000+0.000000j
A--	-1.708879+3.380634j	-1.708879+3.380634j	4.97e-16
ArK(700)2	\(\mathcal{H}^\mathrm{production}_{K(700), 0, - \frac{1}{2}}\)
A++	-1.708879+3.380634j	-1.708879+3.380634j	4.97e-16
A+-	0.000000+0.000000j	0.000000+0.000000j
A-+	-0.000000+0.000000j	0.000000+0.000000j
A--	0.000000+0.000000j	0.000000+0.000000j
ArL(1405)1	\(\mathcal{H}^\mathrm{production}_{\Lambda(1405), - \frac{1}{2}, 0}\)
A++	-0.412613+0.100755j	-0.412613+0.100755j	5.67e-15
A+-	-0.256372+0.062603j	-0.256372+0.062603j	1.21e-14
A-+	-0.242818+0.059293j	-0.242818+0.059293j	2.39e-14
A--	-0.150872+0.036841j	-0.150872+0.036841j	1.73e-14
ArL(1405)2	\(\mathcal{H}^\mathrm{production}_{\Lambda(1405), \frac{1}{2}, 0}\)
A++	-0.150872+0.036841j	-0.150872+0.036841j	1.73e-14
A+-	0.242818-0.059293j	0.242818-0.059293j	2.39e-14
A-+	0.256372-0.062603j	0.256372-0.062603j	1.21e-14
A--	-0.412613+0.100755j	-0.412613+0.100755j	5.67e-15
ArL(1520)1	\(\mathcal{H}^\mathrm{production}_{\Lambda(1520), - \frac{1}{2}, 0}\)
A++	0.257632-0.288056j	0.257632-0.288056j	6.95e-14
A+-	0.731594-0.817988j	0.731594-0.817988j	2.37e-14
A-+	0.151613-0.169517j	0.151613-0.169517j	9.87e-14
A--	0.430534-0.481376j	0.430534-0.481376j	6.59e-15
ArL(1520)2	\(\mathcal{H}^\mathrm{production}_{\Lambda(1520), \frac{1}{2}, 0}\)
A++	-0.430534+0.481376j	-0.430534+0.481376j	6.59e-15
A+-	0.151613-0.169517j	0.151613-0.169517j	9.88e-14
A-+	0.731594-0.817988j	0.731594-0.817988j	2.35e-14
A--	-0.257632+0.288056j	-0.257632+0.288056j	6.95e-14
ArL(1600)1	\(\mathcal{H}^\mathrm{production}_{\Lambda(1600), - \frac{1}{2}, 0}\)
A++	-0.385436+0.424707j	-0.385436+0.424707j	2.02e-14
A+-	0.382669-0.421658j	0.382669-0.421658j	4.78e-15
A-+	-0.226825+0.249935j	-0.226825+0.249935j	9.38e-15
A--	0.225196-0.248141j	0.225196-0.248141j	3.43e-14
ArL(1600)2	\(\mathcal{H}^\mathrm{production}_{\Lambda(1600), \frac{1}{2}, 0}\)
A++	-0.225196+0.248141j	-0.225196+0.248141j	3.42e-14
A+-	-0.226825+0.249935j	-0.226825+0.249935j	9.38e-15
A-+	0.382669-0.421658j	0.382669-0.421658j	4.71e-15
A--	0.385436-0.424707j	0.385436-0.424707j	2.02e-14
ArL(1670)1	\(\mathcal{H}^\mathrm{production}_{\Lambda(1670), - \frac{1}{2}, 0}\)
A++	-0.846639+0.064025j	-0.846639+0.064025j	5.89e-15
A+-	-0.526049+0.039781j	-0.526049+0.039781j	1.20e-14
A-+	-0.498237+0.037678j	-0.498237+0.037678j	2.36e-14
A--	-0.309574+0.023411j	-0.309574+0.023411j	1.72e-14
ArL(1670)2	\(\mathcal{H}^\mathrm{production}_{\Lambda(1670), \frac{1}{2}, 0}\)
A++	-0.309574+0.023411j	-0.309574+0.023411j	1.72e-14
A+-	0.498237-0.037678j	0.498237-0.037678j	2.36e-14
A-+	0.526049-0.039781j	0.526049-0.039781j	1.20e-14
A--	-0.846639+0.064025j	-0.846639+0.064025j	5.89e-15
ArL(1690)1	\(\mathcal{H}^\mathrm{production}_{\Lambda(1690), - \frac{1}{2}, 0}\)
A++	0.232446-0.150691j	0.232446-0.150691j	7.09e-14
A+-	0.660073-0.427915j	0.660073-0.427915j	2.23e-14
A-+	0.136791-0.088680j	0.136791-0.088680j	1.00e-13
A--	0.388445-0.251823j	0.388445-0.251823j	7.26e-15
ArL(1690)2	\(\mathcal{H}^\mathrm{production}_{\Lambda(1690), \frac{1}{2}, 0}\)
A++	-0.388445+0.251823j	-0.388445+0.251823j	7.31e-15
A+-	0.136791-0.088680j	0.136791-0.088680j	1.00e-13
A-+	0.660073-0.427915j	0.660073-0.427915j	2.21e-14
A--	-0.232446+0.150691j	-0.232446+0.150691j	7.09e-14
ArL(2000)1	\(\mathcal{H}^\mathrm{production}_{\Lambda(2000), - \frac{1}{2}, 0}\)
A++	1.072514+1.195841j	1.072514+1.195841j	5.50e-15
A+-	0.666394+0.743022j	0.666394+0.743022j	1.19e-14
A-+	0.631162+0.703738j	0.631162+0.703738j	2.41e-14
A--	0.392165+0.437260j	0.392165+0.437260j	1.77e-14
ArL(2000)2	\(\mathcal{H}^\mathrm{production}_{\Lambda(2000), \frac{1}{2}, 0}\)
A++	0.392165+0.437260j	0.392165+0.437260j	1.77e-14
A+-	-0.631162-0.703738j	-0.631162-0.703738j	2.41e-14
A-+	-0.666394-0.743022j	-0.666394-0.743022j	1.19e-14
A--	1.072514+1.195841j	1.072514+1.195841j	5.50e-15

2.2.2. LS-model

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create_comparison_table(
    "Alternative amplitude model obtained using LS couplings",
    decimals=13,
)

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Computed amplitudes are equal to LHCb amplitudes up to 13 decimals.

	Computed	Expected	Difference
ArD(1232)1	\(\mathcal{H}^\mathrm{LS,production}_{\Delta(1232), 1, \frac{3}{2}}\)
A++	0.502796-0.532862j	0.502796-0.532862j	1.13e-14
A+-	-0.546882+0.579585j	-0.546882+0.579585j	9.81e-15
A-+	0.546882-0.579585j	0.546882-0.579585j	9.81e-15
A--	0.502796-0.532862j	0.502796-0.532862j	1.11e-14
ArD(1232)2	\(\mathcal{H}^\mathrm{LS,production}_{\Delta(1232), 2, \frac{3}{2}}\)
A++	-0.180489+0.191282j	-0.180489+0.191282j	3.22e-14
A+-	0.689818-0.731068j	0.689818-0.731068j	4.38e-15
A-+	0.689818-0.731068j	0.689818-0.731068j	4.53e-15
A--	0.180489-0.191282j	0.180489-0.191282j	3.27e-14
ArD(1600)1	\(\mathcal{H}^\mathrm{LS,production}_{\Delta(1600), 1, \frac{3}{2}}\)
A++	-0.297624-0.084307j	-0.297624-0.084307j	1.87e-14
A+-	0.323720+0.091699j	0.323720+0.091699j	1.63e-15
A-+	-0.323720-0.091699j	-0.323720-0.091699j	1.70e-15
A--	-0.297624-0.084307j	-0.297624-0.084307j	1.85e-14
ArD(1600)2	\(\mathcal{H}^\mathrm{LS,production}_{\Delta(1600), 2, \frac{3}{2}}\)
A++	0.143541+0.040660j	0.143541+0.040660j	3.83e-14
A+-	-0.548604-0.155402j	-0.548604-0.155402j	5.53e-15
A-+	-0.548604-0.155402j	-0.548604-0.155402j	5.58e-15
A--	-0.143541-0.040660j	-0.143541-0.040660j	3.85e-14
ArD(1700)1	\(\mathcal{H}^\mathrm{LS,production}_{\Delta(1700), 1, \frac{3}{2}}\)
A++	-0.042164-0.003922j	-0.042164-0.003922j	6.88e-14
A+-	-0.226551-0.021074j	-0.226551-0.021074j	8.83e-15
A-+	-0.226551-0.021074j	-0.226551-0.021074j	8.95e-15
A--	0.042164+0.003922j	0.042164+0.003922j	6.91e-14
ArD(1700)2	\(\mathcal{H}^\mathrm{LS,production}_{\Delta(1700), 2, \frac{3}{2}}\)
A++	-0.105349-0.009800j	-0.105349-0.009800j	2.01e-14
A+-	0.336381+0.031290j	0.336381+0.031290j	1.23e-14
A-+	-0.336381-0.031290j	-0.336381-0.031290j	1.23e-14
A--	-0.105349-0.009800j	-0.105349-0.009800j	2.03e-14
ArK(892)1	\(\mathcal{H}^\mathrm{LS,production}_{K(892), 0, \frac{1}{2}}\)
A++	0.219513+2.386943j	0.219513+2.386943j	3.09e-15
A+-	-0.857733-9.326825j	-0.857733-9.326825j	2.51e-15
A-+	-0.857733-9.326825j	-0.857733-9.326825j	2.51e-15
A--	-0.219513-2.386943j	-0.219513-2.386943j	3.21e-15
ArK(892)2	\(\mathcal{H}^\mathrm{LS,production}_{K(892), 1, \frac{1}{2}}\)
A++	0.219549+2.387337j	0.219549+2.387337j	5.74e-15
A+-	-0.857874-9.328364j	-0.857874-9.328364j	4.06e-15
A-+	0.857874+9.328364j	0.857874+9.328364j	3.91e-15
A--	0.219549+2.387337j	0.219549+2.387337j	5.74e-15
ArK(892)3	\(\mathcal{H}^\mathrm{LS,production}_{K(892), 1, \frac{3}{2}}\)
A++	0.310489+3.376204j	0.310489+3.376204j	4.48e-15
A+-	0.606609+6.596150j	0.606609+6.596150j	3.90e-15
A-+	-0.606609-6.596150j	-0.606609-6.596150j	4.01e-15
A--	0.310489+3.376204j	0.310489+3.376204j	4.48e-15
ArK(892)4	\(\mathcal{H}^\mathrm{LS,production}_{K(892), 2, \frac{3}{2}}\)
A++	0.310629+3.377724j	0.310629+3.377724j	4.48e-15
A+-	0.606882+6.599119j	0.606882+6.599119j	3.06e-15
A-+	0.606882+6.599119j	0.606882+6.599119j	3.06e-15
A--	-0.310629-3.377724j	-0.310629-3.377724j	4.48e-15
ArK(1430)1	\(\mathcal{H}^\mathrm{LS,production}_{K(1430), 0, \frac{1}{2}}\)
A++	0.643091+0.051436j	0.643091+0.051436j	1.08e-16
A+-	0.000000+0.000000j	0.000000+0.000000j
A-+	-0.000000+0.000000j	0.000000+0.000000j
A--	0.643091+0.051436j	0.643091+0.051436j	1.08e-16
ArK(1430)2	\(\mathcal{H}^\mathrm{LS,production}_{K(1430), 1, \frac{1}{2}}\)
A++	-0.643091-0.051436j	-0.643091-0.051436j	2.09e-16
A+-	0.000000+0.000000j	0.000000+0.000000j
A-+	-0.000000+0.000000j	0.000000+0.000000j
A--	0.643091+0.051436j	0.643091+0.051436j	2.09e-16
ArK(700)1	\(\mathcal{H}^\mathrm{LS,production}_{K(700), 0, \frac{1}{2}}\)
A++	-1.070937+2.282902j	-1.070937+2.282902j	3.94e-16
A+-	0.000000+0.000000j	0.000000+0.000000j
A-+	-0.000000+0.000000j	0.000000+0.000000j
A--	-1.070937+2.282902j	-1.070937+2.282902j	3.94e-16
ArK(700)2	\(\mathcal{H}^\mathrm{LS,production}_{K(700), 1, \frac{1}{2}}\)
A++	1.070937-2.282902j	1.070937-2.282902j	4.40e-16
A+-	0.000000+0.000000j	0.000000+0.000000j
A-+	-0.000000+0.000000j	0.000000+0.000000j
A--	-1.070937+2.282902j	-1.070937+2.282902j	4.40e-16
ArL(1405)1	\(\mathcal{H}^\mathrm{LS,production}_{\Lambda(1405), 0, \frac{1}{2}}\)
A++	-0.398444+0.097295j	-0.398444+0.097295j	2.89e-16
A+-	-0.009584+0.002340j	-0.009584+0.002340j	6.53e-13
A-+	0.009584-0.002340j	0.009584-0.002340j	6.53e-13
A--	-0.398444+0.097295j	-0.398444+0.097295j	2.89e-16
ArL(1405)2	\(\mathcal{H}^\mathrm{LS,production}_{\Lambda(1405), 1, \frac{1}{2}}\)
A++	0.163270-0.039869j	0.163270-0.039869j	1.89e-14
A+-	0.311387-0.076037j	0.311387-0.076037j	5.50e-15
A-+	0.311387-0.076037j	0.311387-0.076037j	5.50e-15
A--	-0.163270+0.039869j	-0.163270+0.039869j	1.90e-14
ArL(1520)1	\(\mathcal{H}^\mathrm{LS,production}_{\Lambda(1520), 1, \frac{3}{2}}\)
A++	0.117387-0.135999j	0.117387-0.135999j	8.65e-14
A+-	-0.599627+0.694701j	-0.599627+0.694701j	3.12e-15
A-+	-0.599627+0.694701j	-0.599627+0.694701j	3.12e-15
A--	-0.117387+0.135999j	-0.117387+0.135999j	8.63e-14
ArL(1520)2	\(\mathcal{H}^\mathrm{LS,production}_{\Lambda(1520), 2, \frac{3}{2}}\)
A++	0.330006-0.382330j	0.330006-0.382330j	3.13e-14
A+-	0.278127-0.322225j	0.278127-0.322225j	5.42e-14
A-+	-0.278127+0.322225j	-0.278127+0.322225j	5.42e-14
A--	0.330006-0.382330j	0.330006-0.382330j	3.13e-14
ArL(1600)1	\(\mathcal{H}^\mathrm{LS,production}_{\Lambda(1600), 0, \frac{1}{2}}\)
A++	-0.431782+0.475775j	-0.431782+0.475775j	1.93e-16
A+-	0.110199-0.121426j	0.110199-0.121426j	1.86e-15
A-+	0.110199-0.121426j	0.110199-0.121426j	1.74e-15
A--	0.431782-0.475775j	0.431782-0.475775j	1.93e-16
ArL(1600)2	\(\mathcal{H}^\mathrm{LS,production}_{\Lambda(1600), 1, \frac{1}{2}}\)
A++	0.102310-0.112734j	0.102310-0.112734j	9.63e-14
A+-	-0.389148+0.428797j	-0.389148+0.428797j	6.59e-15
A-+	0.389148-0.428797j	0.389148-0.428797j	6.59e-15
A--	0.102310-0.112734j	0.102310-0.112734j	9.64e-14
ArL(1670)1	\(\mathcal{H}^\mathrm{LS,production}_{\Lambda(1670), 0, \frac{1}{2}}\)
A++	-0.817566+0.061827j	-0.817566+0.061827j	4.28e-16
A+-	-0.019666+0.001487j	-0.019666+0.001487j	6.52e-13
A-+	0.019666-0.001487j	0.019666-0.001487j	6.52e-13
A--	-0.817566+0.061827j	-0.817566+0.061827j	3.03e-16
ArL(1670)2	\(\mathcal{H}^\mathrm{LS,production}_{\Lambda(1670), 1, \frac{1}{2}}\)
A++	0.345271-0.026110j	0.345271-0.026110j	1.86e-14
A+-	0.658498-0.049798j	0.658498-0.049798j	6.08e-15
A-+	0.658498-0.049798j	0.658498-0.049798j	5.91e-15
A--	-0.345271+0.026110j	-0.345271+0.026110j	1.88e-14
ArL(1690)1	\(\mathcal{H}^\mathrm{LS,production}_{\Lambda(1690), 1, \frac{3}{2}}\)
A++	0.110308-0.071511j	0.110308-0.071511j	8.99e-14
A+-	-0.563468+0.365287j	-0.563468+0.365287j	3.67e-15
A-+	-0.563468+0.365287j	-0.563468+0.365287j	3.67e-15
A--	-0.110308+0.071511j	-0.110308+0.071511j	8.99e-14
ArL(1690)2	\(\mathcal{H}^\mathrm{LS,production}_{\Lambda(1690), 2, \frac{3}{2}}\)
A++	0.333287-0.216064j	0.333287-0.216064j	3.08e-14
A+-	0.280891-0.182097j	0.280891-0.182097j	5.44e-14
A-+	-0.280891+0.182097j	-0.280891+0.182097j	5.46e-14
A--	0.333287-0.216064j	0.333287-0.216064j	3.08e-14
ArL(2000)1	\(\mathcal{H}^\mathrm{LS,production}_{\Lambda(2000), 0, \frac{1}{2}}\)
A++	1.036314+1.105950j	1.036314+1.105950j	1.17e-15
A+-	0.024928+0.026603j	0.024928+0.026603j	6.53e-13
A-+	-0.024928-0.026603j	-0.024928-0.026603j	6.54e-13
A--	1.036314+1.105950j	1.036314+1.105950j	1.17e-15
ArL(2000)2	\(\mathcal{H}^\mathrm{LS,production}_{\Lambda(2000), 1, \frac{1}{2}}\)
A++	-0.529297-0.564863j	-0.529297-0.564863j	1.85e-14
A+-	-1.009471-1.077303j	-1.009471-1.077303j	6.35e-15
A-+	-1.009471-1.077303j	-1.009471-1.077303j	6.44e-15
A--	0.529297+0.564863j	0.529297+0.564863j	1.84e-14