2. Cross-check with LHCb data

Import Python libraries Hide code cell content

from __future__ import annotations

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

import numpy as np
import sympy as sp
from IPython.display import Markdown, Math
from tqdm.auto import tqdm

from polarimetry.amplitude import AmplitudeModel, simplify_latex_rendering
from polarimetry.data import create_data_transformer
from polarimetry.io import (
    as_latex,
    display_latex,
    mute_jax_warnings,
    perform_cached_doit,
    perform_cached_lambdify,
)
from polarimetry.lhcb import (
    get_conversion_factor,
    get_conversion_factor_ls,
    load_model,
    load_model_builder,
    parameter_key_to_symbol,
)
from polarimetry.lhcb.particle import load_particles


@lru_cache(maxsize=None)
def load_model_cached(model_id: int | str) -> AmplitudeModel:
    return load_model(MODEL_FILE, PARTICLES, model_id)


mute_jax_warnings()
simplify_latex_rendering()
NO_TQDM = "EXECUTE_NB" in os.environ
if NO_TQDM:
    logging.getLogger().setLevel(logging.ERROR)

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 Hide code cell source

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 Hide code cell source

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(as_latex([K892_chain, L1405_chain, L1690_chain]))

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

Values for LHCb-PAPER-2022-002 Hide code cell source

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 Hide code cell source

builder = load_model_builder(MODEL_FILE, PARTICLES, model_id=0)
build_dynamics = lambda c: builder.dynamics_choices.get_builder(c)(c)[0].doit()
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])

\[\begin{split}\displaystyle \begin{array}{c} 2.16872014550901 + 23.5822591700909 i \\ -0.563648141017186 + 0.137636377592249 i \\ -1.5078327158518 + 0.977503639506157 i \\ \end{array}\end{split}\]

Assert that these values are equal Hide code cell source

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 Hide code cell source

amplitude_vars = {k: v for k, v in crosscheck_data["chainvars"].items()}
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()})

\[\begin{split}\displaystyle \begin{array}{rcl} \theta_{23} &=& 1.821341166520149 \\ \theta_{31} &=& 1.8038351483715633 \\ \theta_{12} &=& 1.1139045236042229 \\ \zeta^0_{1(1)} &=& 0.0 \\ \zeta^1_{1(1)} &=& 0.0 \\ \zeta^0_{2(1)} &=& -2.0777687076712614 \\ \zeta^1_{2(1)} &=& 0.22583331080386268 \\ \zeta^0_{3(1)} &=& 2.6540796539955838 \\ \zeta^1_{3(1)} &=& -0.5594175047790548 \\ \sigma_{1} &=& 0.7980703453578917 \\ \sigma_{2} &=& 3.6486261122281745 \\ \sigma_{3} &=& 1.9247541217931925 \\ \end{array}\end{split}\]

Code for creating functions for each sub-amplitude Hide code cell source

@lru_cache(maxsize=None)
def create_amplitude_functions(
    model_id: int | str,
) -> 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: perform_cached_lambdify(
            expr.xreplace(fixed_parameters),
            parameters=production_couplings,
            backend="numpy",
        )
        for k, expr in tqdm(exprs.items(), desc="Performing doit", disable=NO_TQDM)
    }


@lru_cache(maxsize=None)
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)[0]
        for λ_Λc in [-half, +half]
        for λ_p in [-half, +half]
    }
    model = load_model(MODEL_FILE, PARTICLES, model_id)
    return {
        k: perform_cached_doit(expr.doit().xreplace(model.amplitudes))
        for k, expr in tqdm(exprs.items(), desc="Lambdifying", disable=NO_TQDM)
    }


@lru_cache(maxsize=None)
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 Hide code cell source

def plusminus_to_helicity(plusminus: str) -> sp.Rational:
    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 i, (amp_identifier, entry) in enumerate(real_amp_crosscheck.items()):
        coupling = parameter_key_to_symbol(
            amp_identifier.replace("Ar", "A"),
            min_ls,
            particle_definitions=PARTICLES,
        )
        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_name = coupling.indices[0]
    resonance = PARTICLES[str(resonance_name)]
    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

Show code cell source Hide code cell source

create_comparison_table(model_id=0, decimals=13)

Show code cell output Hide code cell output

Tip

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

	Computed	Expected	Difference
ArD(1232)1	\(\mathcal{H}^\mathrm{production}_{D(1232), - \frac{1}{2}, 0}\)
A++	-0.488498+0.517710j	-0.488498+0.517710j	3.08e-14
A+-	0.894898-0.948412j	0.894898-0.948412j	7.14e-15
A-+	0.121490-0.128755j	0.121490-0.128755j	1.80e-14
A--	-0.222563+0.235872j	-0.222563+0.235872j	6.36e-15
ArD(1232)2	\(\mathcal{H}^\mathrm{production}_{D(1232), \frac{1}{2}, 0}\)
A++	-0.222563+0.235872j	-0.222563+0.235872j	6.36e-15
A+-	-0.121490+0.128755j	-0.121490+0.128755j	1.80e-14
A-+	-0.894898+0.948412j	-0.894898+0.948412j	7.14e-15
A--	-0.488498+0.517710j	-0.488498+0.517710j	3.08e-14
ArD(1600)1	\(\mathcal{H}^\mathrm{production}_{D(1600), - \frac{1}{2}, 0}\)
A++	0.289160+0.081910j	0.289160+0.081910j	3.07e-14
A+-	-0.529724-0.150054j	-0.529724-0.150054j	6.87e-15
A-+	-0.071915-0.020371j	-0.071915-0.020371j	1.80e-14
A--	0.131743+0.037319j	0.131743+0.037319j	5.91e-15
ArD(1600)2	\(\mathcal{H}^\mathrm{production}_{D(1600), \frac{1}{2}, 0}\)
A++	0.131743+0.037319j	0.131743+0.037319j	5.91e-15
A+-	0.071915+0.020371j	0.071915+0.020371j	1.80e-14
A-+	0.529724+0.150054j	0.529724+0.150054j	6.87e-15
A--	0.289160+0.081910j	0.289160+0.081910j	3.07e-14
ArD(1700)1	\(\mathcal{H}^\mathrm{production}_{D(1700), - \frac{1}{2}, 0}\)
A++	-0.018885-0.001757j	-0.018885-0.001757j	3.20e-13
A+-	0.315695+0.029366j	0.315695+0.029366j	2.00e-14
A-+	0.004697+0.000437j	0.004697+0.000437j	3.34e-13
A--	-0.078514-0.007303j	-0.078514-0.007303j	6.86e-15
ArD(1700)2	\(\mathcal{H}^\mathrm{production}_{D(1700), \frac{1}{2}, 0}\)
A++	0.078514+0.007303j	0.078514+0.007303j	6.86e-15
A+-	0.004697+0.000437j	0.004697+0.000437j	3.34e-13
A-+	0.315695+0.029366j	0.315695+0.029366j	2.00e-14
A--	0.018885+0.001757j	0.018885+0.001757j	3.20e-13
ArK(892)1	\(\mathcal{H}^\mathrm{production}_{K(892), 0, - \frac{1}{2}}\)
A++	-0.537695-5.846793j	-0.537695-5.846793j	4.88e-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	3.42e-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	3.32e-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	4.88e-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	1.37e-16
ArK(1430)2	\(\mathcal{H}^\mathrm{production}_{K(1430), 0, - \frac{1}{2}}\)
A++	0.909456+0.072819j	0.909456+0.072819j	1.37e-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	\(\mathcal{H}^\mathrm{production}_{L(1405), - \frac{1}{2}, 0}\)
A++	-0.412613+0.100755j	-0.412613+0.100755j	1.49e-15
A+-	-0.256372+0.062603j	-0.256372+0.062603j	3.27e-15
A-+	-0.242818+0.059293j	-0.242818+0.059293j	1.30e-15
A--	-0.150872+0.036841j	-0.150872+0.036841j	3.42e-15
ArL(1405)2	\(\mathcal{H}^\mathrm{production}_{L(1405), \frac{1}{2}, 0}\)
A++	-0.150872+0.036841j	-0.150872+0.036841j	3.42e-15
A+-	0.242818-0.059293j	0.242818-0.059293j	1.30e-15
A-+	0.256372-0.062603j	0.256372-0.062603j	3.27e-15
A--	-0.412613+0.100755j	-0.412613+0.100755j	1.49e-15
ArL(1520)1	\(\mathcal{H}^\mathrm{production}_{L(1520), - \frac{1}{2}, 0}\)
A++	0.257632-0.288056j	0.257632-0.288056j	1.56e-14
A+-	0.731594-0.817988j	0.731594-0.817988j	2.29e-14
A-+	0.151613-0.169517j	0.151613-0.169517j	1.55e-14
A--	0.430534-0.481376j	0.430534-0.481376j	2.30e-14
ArL(1520)2	\(\mathcal{H}^\mathrm{production}_{L(1520), \frac{1}{2}, 0}\)
A++	-0.430534+0.481376j	-0.430534+0.481376j	2.29e-14
A+-	0.151613-0.169517j	0.151613-0.169517j	1.55e-14
A-+	0.731594-0.817988j	0.731594-0.817988j	2.28e-14
A--	-0.257632+0.288056j	-0.257632+0.288056j	1.55e-14
ArL(1600)1	\(\mathcal{H}^\mathrm{production}_{L(1600), - \frac{1}{2}, 0}\)
A++	-0.385436+0.424707j	-0.385436+0.424707j	1.35e-15
A+-	0.382669-0.421658j	0.382669-0.421658j	3.75e-15
A-+	-0.226825+0.249935j	-0.226825+0.249935j	1.60e-15
A--	0.225196-0.248141j	0.225196-0.248141j	3.56e-15
ArL(1600)2	\(\mathcal{H}^\mathrm{production}_{L(1600), \frac{1}{2}, 0}\)
A++	-0.225196+0.248141j	-0.225196+0.248141j	3.60e-15
A+-	-0.226825+0.249935j	-0.226825+0.249935j	1.60e-15
A-+	0.382669-0.421658j	0.382669-0.421658j	3.80e-15
A--	0.385436-0.424707j	0.385436-0.424707j	1.44e-15
ArL(1670)1	\(\mathcal{H}^\mathrm{production}_{L(1670), - \frac{1}{2}, 0}\)
A++	-0.846639+0.064025j	-0.846639+0.064025j	1.18e-15
A+-	-0.526049+0.039781j	-0.526049+0.039781j	3.17e-15
A-+	-0.498237+0.037678j	-0.498237+0.037678j	1.11e-15
A--	-0.309574+0.023411j	-0.309574+0.023411j	3.59e-15
ArL(1670)2	\(\mathcal{H}^\mathrm{production}_{L(1670), \frac{1}{2}, 0}\)
A++	-0.309574+0.023411j	-0.309574+0.023411j	3.59e-15
A+-	0.498237-0.037678j	0.498237-0.037678j	1.11e-15
A-+	0.526049-0.039781j	0.526049-0.039781j	3.17e-15
A--	-0.846639+0.064025j	-0.846639+0.064025j	1.18e-15
ArL(1690)1	\(\mathcal{H}^\mathrm{production}_{L(1690), - \frac{1}{2}, 0}\)
A++	0.232446-0.150691j	0.232446-0.150691j	1.66e-14
A+-	0.660073-0.427915j	0.660073-0.427915j	2.37e-14
A-+	0.136791-0.088680j	0.136791-0.088680j	1.65e-14
A--	0.388445-0.251823j	0.388445-0.251823j	2.37e-14
ArL(1690)2	\(\mathcal{H}^\mathrm{production}_{L(1690), \frac{1}{2}, 0}\)
A++	-0.388445+0.251823j	-0.388445+0.251823j	2.36e-14
A+-	0.136791-0.088680j	0.136791-0.088680j	1.65e-14
A-+	0.660073-0.427915j	0.660073-0.427915j	2.37e-14
A--	-0.232446+0.150691j	-0.232446+0.150691j	1.66e-14
ArL(2000)1	\(\mathcal{H}^\mathrm{production}_{L(2000), - \frac{1}{2}, 0}\)
A++	1.072514+1.195841j	1.072514+1.195841j	1.47e-15
A+-	0.666394+0.743022j	0.666394+0.743022j	2.94e-15
A-+	0.631162+0.703738j	0.631162+0.703738j	1.34e-15
A--	0.392165+0.437260j	0.392165+0.437260j	3.29e-15
ArL(2000)2	\(\mathcal{H}^\mathrm{production}_{L(2000), \frac{1}{2}, 0}\)
A++	0.392165+0.437260j	0.392165+0.437260j	3.29e-15
A+-	-0.631162-0.703738j	-0.631162-0.703738j	1.34e-15
A-+	-0.666394-0.743022j	-0.666394-0.743022j	2.94e-15
A--	1.072514+1.195841j	1.072514+1.195841j	1.47e-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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Tip

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

	Computed	Expected	Difference
ArD(1232)1	\(\mathcal{H}^\mathrm{LS,production}_{D(1232), 1, \frac{3}{2}}\)
A++	0.502796-0.532862j	0.502796-0.532862j	1.91e-14
A+-	-0.546882+0.579585j	-0.546882+0.579585j	5.18e-15
A-+	0.546882-0.579585j	0.546882-0.579585j	5.18e-15
A--	0.502796-0.532862j	0.502796-0.532862j	1.91e-14
ArD(1232)2	\(\mathcal{H}^\mathrm{LS,production}_{D(1232), 2, \frac{3}{2}}\)
A++	-0.180489+0.191282j	-0.180489+0.191282j	5.49e-14
A+-	0.689818-0.731068j	0.689818-0.731068j	2.43e-15
A-+	0.689818-0.731068j	0.689818-0.731068j	2.32e-15
A--	0.180489-0.191282j	0.180489-0.191282j	5.49e-14
ArD(1600)1	\(\mathcal{H}^\mathrm{LS,production}_{D(1600), 1, \frac{3}{2}}\)
A++	-0.297624-0.084307j	-0.297624-0.084307j	1.79e-14
A+-	0.323720+0.091699j	0.323720+0.091699j	3.99e-15
A-+	-0.323720-0.091699j	-0.323720-0.091699j	3.99e-15
A--	-0.297624-0.084307j	-0.297624-0.084307j	1.79e-14
ArD(1600)2	\(\mathcal{H}^\mathrm{LS,production}_{D(1600), 2, \frac{3}{2}}\)
A++	0.143541+0.040660j	0.143541+0.040660j	5.47e-14
A+-	-0.548604-0.155402j	-0.548604-0.155402j	1.92e-15
A-+	-0.548604-0.155402j	-0.548604-0.155402j	1.80e-15
A--	-0.143541-0.040660j	-0.143541-0.040660j	5.44e-14
ArD(1700)1	\(\mathcal{H}^\mathrm{LS,production}_{D(1700), 1, \frac{3}{2}}\)
A++	-0.042164-0.003922j	-0.042164-0.003922j	1.10e-13
A+-	-0.226551-0.021074j	-0.226551-0.021074j	1.42e-14
A-+	-0.226551-0.021074j	-0.226551-0.021074j	1.42e-14
A--	0.042164+0.003922j	0.042164+0.003922j	1.11e-13
ArD(1700)2	\(\mathcal{H}^\mathrm{LS,production}_{D(1700), 2, \frac{3}{2}}\)
A++	-0.105349-0.009800j	-0.105349-0.009800j	5.87e-14
A+-	0.336381+0.031290j	0.336381+0.031290j	2.29e-14
A-+	-0.336381-0.031290j	-0.336381-0.031290j	2.29e-14
A--	-0.105349-0.009800j	-0.105349-0.009800j	5.85e-14
ArK(892)1	\(\mathcal{H}^\mathrm{LS,production}_{K(892), 0, \frac{1}{2}}\)
A++	0.219513+2.386943j	0.219513+2.386943j	4.88e-15
A+-	-0.857733-9.326825j	-0.857733-9.326825j	3.64e-15
A-+	-0.857733-9.326825j	-0.857733-9.326825j	3.64e-15
A--	-0.219513-2.386943j	-0.219513-2.386943j	4.88e-15
ArK(892)2	\(\mathcal{H}^\mathrm{LS,production}_{K(892), 1, \frac{1}{2}}\)
A++	0.219549+2.387337j	0.219549+2.387337j	7.18e-15
A+-	-0.857874-9.328364j	-0.857874-9.328364j	2.84e-15
A-+	0.857874+9.328364j	0.857874+9.328364j	2.78e-15
A--	0.219549+2.387337j	0.219549+2.387337j	7.18e-15
ArK(892)3	\(\mathcal{H}^\mathrm{LS,production}_{K(892), 1, \frac{3}{2}}\)
A++	0.310489+3.376204j	0.310489+3.376204j	4.51e-15
A+-	0.606609+6.596150j	0.606609+6.596150j	2.78e-15
A-+	-0.606609-6.596150j	-0.606609-6.596150j	2.76e-15
A--	0.310489+3.376204j	0.310489+3.376204j	4.51e-15
ArK(892)4	\(\mathcal{H}^\mathrm{LS,production}_{K(892), 2, \frac{3}{2}}\)
A++	0.310629+3.377724j	0.310629+3.377724j	1.38e-14
A+-	0.606882+6.599119j	0.606882+6.599119j	7.79e-15
A-+	0.606882+6.599119j	0.606882+6.599119j	7.79e-15
A--	-0.310629-3.377724j	-0.310629-3.377724j	1.38e-14
ArK(1430)1	\(\mathcal{H}^\mathrm{LS,production}_{K(1430), 0, \frac{1}{2}}\)
A++	0.643091+0.051436j	0.643091+0.051436j	1.29e-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	1.29e-16
ArK(1430)2	\(\mathcal{H}^\mathrm{LS,production}_{K(1430), 1, \frac{1}{2}}\)
A++	-0.643091-0.051436j	-0.643091-0.051436j	2.22e-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	2.22e-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	\(\mathcal{H}^\mathrm{LS,production}_{L(1405), 0, \frac{1}{2}}\)
A++	-0.398444+0.097295j	-0.398444+0.097295j	8.23e-16
A+-	-0.009584+0.002340j	-0.009584+0.002340j	7.99e-14
A-+	0.009584-0.002340j	0.009584-0.002340j	8.05e-14
A--	-0.398444+0.097295j	-0.398444+0.097295j	8.56e-16
ArL(1405)2	\(\mathcal{H}^\mathrm{LS,production}_{L(1405), 1, \frac{1}{2}}\)
A++	0.163270-0.039869j	0.163270-0.039869j	2.06e-14
A+-	0.311387-0.076037j	0.311387-0.076037j	2.50e-14
A-+	0.311387-0.076037j	0.311387-0.076037j	2.50e-14
A--	-0.163270+0.039869j	-0.163270+0.039869j	2.04e-14
ArL(1520)1	\(\mathcal{H}^\mathrm{LS,production}_{L(1520), 1, \frac{3}{2}}\)
A++	0.117387-0.135999j	0.117387-0.135999j	3.12e-14
A+-	-0.599627+0.694701j	-0.599627+0.694701j	1.92e-14
A-+	-0.599627+0.694701j	-0.599627+0.694701j	1.92e-14
A--	-0.117387+0.135999j	-0.117387+0.135999j	3.12e-14
ArL(1520)2	\(\mathcal{H}^\mathrm{LS,production}_{L(1520), 2, \frac{3}{2}}\)
A++	0.330006-0.382330j	0.330006-0.382330j	7.45e-14
A+-	0.278127-0.322225j	0.278127-0.322225j	7.90e-14
A-+	-0.278127+0.322225j	-0.278127+0.322225j	7.90e-14
A--	0.330006-0.382330j	0.330006-0.382330j	7.45e-14
ArL(1600)1	\(\mathcal{H}^\mathrm{LS,production}_{L(1600), 0, \frac{1}{2}}\)
A++	-0.431782+0.475775j	-0.431782+0.475775j	1.40e-15
A+-	0.110199-0.121426j	0.110199-0.121426j	9.54e-15
A-+	0.110199-0.121426j	0.110199-0.121426j	9.90e-15
A--	0.431782-0.475775j	0.431782-0.475775j	1.40e-15
ArL(1600)2	\(\mathcal{H}^\mathrm{LS,production}_{L(1600), 1, \frac{1}{2}}\)
A++	0.102310-0.112734j	0.102310-0.112734j	3.05e-14
A+-	-0.389148+0.428797j	-0.389148+0.428797j	2.21e-14
A-+	0.389148-0.428797j	0.389148-0.428797j	2.22e-14
A--	0.102310-0.112734j	0.102310-0.112734j	3.05e-14
ArL(1670)1	\(\mathcal{H}^\mathrm{LS,production}_{L(1670), 0, \frac{1}{2}}\)
A++	-0.817566+0.061827j	-0.817566+0.061827j	1.69e-16
A+-	-0.019666+0.001487j	-0.019666+0.001487j	7.61e-14
A-+	0.019666-0.001487j	0.019666-0.001487j	7.62e-14
A--	-0.817566+0.061827j	-0.817566+0.061827j	1.69e-16
ArL(1670)2	\(\mathcal{H}^\mathrm{LS,production}_{L(1670), 1, \frac{1}{2}}\)
A++	0.345271-0.026110j	0.345271-0.026110j	1.85e-14
A+-	0.658498-0.049798j	0.658498-0.049798j	2.38e-14
A-+	0.658498-0.049798j	0.658498-0.049798j	2.39e-14
A--	-0.345271+0.026110j	-0.345271+0.026110j	1.87e-14
ArL(1690)1	\(\mathcal{H}^\mathrm{LS,production}_{L(1690), 1, \frac{3}{2}}\)
A++	0.110308-0.071511j	0.110308-0.071511j	3.03e-14
A+-	-0.563468+0.365287j	-0.563468+0.365287j	1.82e-14
A-+	-0.563468+0.365287j	-0.563468+0.365287j	1.83e-14
A--	-0.110308+0.071511j	-0.110308+0.071511j	3.03e-14
ArL(1690)2	\(\mathcal{H}^\mathrm{LS,production}_{L(1690), 2, \frac{3}{2}}\)
A++	0.333287-0.216064j	0.333287-0.216064j	7.66e-14
A+-	0.280891-0.182097j	0.280891-0.182097j	8.10e-14
A-+	-0.280891+0.182097j	-0.280891+0.182097j	8.09e-14
A--	0.333287-0.216064j	0.333287-0.216064j	7.66e-14
ArL(2000)1	\(\mathcal{H}^\mathrm{LS,production}_{L(2000), 0, \frac{1}{2}}\)
A++	1.036314+1.105950j	1.036314+1.105950j	1.14e-15
A+-	0.024928+0.026603j	0.024928+0.026603j	7.91e-14
A-+	-0.024928-0.026603j	-0.024928-0.026603j	7.92e-14
A--	1.036314+1.105950j	1.036314+1.105950j	1.14e-15
ArL(2000)2	\(\mathcal{H}^\mathrm{LS,production}_{L(2000), 1, \frac{1}{2}}\)
A++	-0.529297-0.564863j	-0.529297-0.564863j	1.86e-14
A+-	-1.009471-1.077303j	-1.009471-1.077303j	2.35e-14
A-+	-1.009471-1.077303j	-1.009471-1.077303j	2.36e-14
A--	0.529297+0.564863j	0.529297+0.564863j	1.86e-14