7.4. Alignment consistency

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from __future__ import annotations

import logging
import os

import jax.numpy as jnp
import matplotlib.pyplot as plt
import sympy as sp
from ampform_dpd import AmplitudeModel
from ampform_dpd.io import cached, simplify_latex_rendering
from numpy.testing import assert_almost_equal
from tensorwaves.data import SympyDataTransformer
from tqdm.auto import tqdm

from polarimetry.data import create_data_transformer, generate_meshgrid_sample
from polarimetry.io import display_latex, mute_jax_warnings
from polarimetry.lhcb import (
    flip_production_coupling_signs,
    load_model_builder,
    load_model_parameters,
)
from polarimetry.lhcb.particle import load_particles
from polarimetry.plot import (
    add_watermark,
    convert_svg_to_png,
    reduce_svg_size,
    use_mpl_latex_fonts,
)

mute_jax_warnings()
simplify_latex_rendering()

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

model_choice = 0
model_file = "../../data/model-definitions.yaml"
particles = load_particles("../../data/particle-definitions.yaml")
amplitude_builder = load_model_builder(model_file, particles, model_choice)
imported_parameter_values = load_model_parameters(
    model_file, amplitude_builder.decay, model_choice, particles
)
models = {}
for reference_subsystem in [1, 2, 3]:
    models[reference_subsystem] = amplitude_builder.formulate(
        reference_subsystem, cleanup_summations=True
    )
    models[reference_subsystem].parameter_defaults.update(imported_parameter_values)
models[2] = flip_production_coupling_signs(models[2], subsystem_names=["K", "L"])
models[3] = flip_production_coupling_signs(models[3], subsystem_names=["K", "D"])

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display_latex(m.intensity.cleanup() for m in models.values())

\[\begin{split}\displaystyle \begin{array}{c} \sum_{\lambda_{0}=-1/2}^{1/2} \sum_{\lambda_{1}=-1/2}^{1/2}{\left|{\sum_{\lambda_0^{\prime}=-1/2}^{1/2} \sum_{\lambda_1^{\prime}=-1/2}^{1/2}{A^{1}_{\lambda_0^{\prime}, \lambda_1^{\prime}, 0, 0} d^{\frac{1}{2}}_{\lambda_1^{\prime},\lambda_{1}}\left(\zeta^1_{1(1)}\right) d^{\frac{1}{2}}_{\lambda_{0},\lambda_0^{\prime}}\left(\zeta^0_{1(1)}\right) + A^{2}_{\lambda_0^{\prime}, \lambda_1^{\prime}, 0, 0} d^{\frac{1}{2}}_{\lambda_1^{\prime},\lambda_{1}}\left(\zeta^1_{2(1)}\right) d^{\frac{1}{2}}_{\lambda_{0},\lambda_0^{\prime}}\left(\zeta^0_{2(1)}\right) + A^{3}_{\lambda_0^{\prime}, \lambda_1^{\prime}, 0, 0} d^{\frac{1}{2}}_{\lambda_1^{\prime},\lambda_{1}}\left(\zeta^1_{3(1)}\right) d^{\frac{1}{2}}_{\lambda_{0},\lambda_0^{\prime}}\left(\zeta^0_{3(1)}\right)}}\right|^{2}} \\ \sum_{\lambda_{0}=-1/2}^{1/2} \sum_{\lambda_{1}=-1/2}^{1/2}{\left|{\sum_{\lambda_0^{\prime}=-1/2}^{1/2} \sum_{\lambda_1^{\prime}=-1/2}^{1/2}{A^{1}_{\lambda_0^{\prime}, \lambda_1^{\prime}, 0, 0} d^{\frac{1}{2}}_{\lambda_1^{\prime},\lambda_{1}}\left(\zeta^1_{1(2)}\right) d^{\frac{1}{2}}_{\lambda_{0},\lambda_0^{\prime}}\left(\zeta^0_{1(2)}\right) + A^{2}_{\lambda_0^{\prime}, \lambda_1^{\prime}, 0, 0} d^{\frac{1}{2}}_{\lambda_1^{\prime},\lambda_{1}}\left(\zeta^1_{2(2)}\right) d^{\frac{1}{2}}_{\lambda_{0},\lambda_0^{\prime}}\left(\zeta^0_{2(2)}\right) + A^{3}_{\lambda_0^{\prime}, \lambda_1^{\prime}, 0, 0} d^{\frac{1}{2}}_{\lambda_1^{\prime},\lambda_{1}}\left(\zeta^1_{3(2)}\right) d^{\frac{1}{2}}_{\lambda_{0},\lambda_0^{\prime}}\left(\zeta^0_{3(2)}\right)}}\right|^{2}} \\ \sum_{\lambda_{0}=-1/2}^{1/2} \sum_{\lambda_{1}=-1/2}^{1/2}{\left|{\sum_{\lambda_0^{\prime}=-1/2}^{1/2} \sum_{\lambda_1^{\prime}=-1/2}^{1/2}{A^{1}_{\lambda_0^{\prime}, \lambda_1^{\prime}, 0, 0} d^{\frac{1}{2}}_{\lambda_1^{\prime},\lambda_{1}}\left(\zeta^1_{1(3)}\right) d^{\frac{1}{2}}_{\lambda_{0},\lambda_0^{\prime}}\left(\zeta^0_{1(3)}\right) + A^{2}_{\lambda_0^{\prime}, \lambda_1^{\prime}, 0, 0} d^{\frac{1}{2}}_{\lambda_1^{\prime},\lambda_{1}}\left(\zeta^1_{2(3)}\right) d^{\frac{1}{2}}_{\lambda_{0},\lambda_0^{\prime}}\left(\zeta^0_{2(3)}\right) + A^{3}_{\lambda_0^{\prime}, \lambda_1^{\prime}, 0, 0} d^{\frac{1}{2}}_{\lambda_1^{\prime},\lambda_{1}}\left(\zeta^1_{3(3)}\right) d^{\frac{1}{2}}_{\lambda_{0},\lambda_0^{\prime}}\left(\zeta^0_{3(3)}\right)}}\right|^{2}} \\ \end{array}\end{split}\]

See DPD angles for the definition of each \(\zeta^i_{j(k)}\).

Note that a change in reference sub-system requires the production couplings for certain sub-systems to flip sign:

• Sub-system 2 as reference system: flip signs of \(\mathcal{H}^\mathrm{production}_{K^{**}}\) and \(\mathcal{H}^\mathrm{production}_{L^{**}}\)

• Sub-system 3 as reference system: flip signs of \(\mathcal{H}^\mathrm{production}_{K^{**}}\) and \(\mathcal{H}^\mathrm{production}_{D^{**}}\)

unfolded_intensity_exprs = {
    reference_subsystem: cached.unfold(model)
    for reference_subsystem, model in tqdm(models.items(), disable=NO_LOG)
}

subs_intensity_exprs = {
    reference_subsystem: cached.xreplace(
        expr, models[reference_subsystem].parameter_defaults
    )
    for reference_subsystem, expr in unfolded_intensity_exprs.items()
}

intensity_funcs = {
    reference_subsystem: cached.lambdify(expr)
    for reference_subsystem, expr in tqdm(subs_intensity_exprs.items(), disable=NO_LOG)
}

transformer = {}
for reference_subsystem in tqdm([1, 2, 3], disable=NO_LOG):
    model = models[reference_subsystem]
    transformer.update(create_data_transformer(model).functions)
transformer = SympyDataTransformer(transformer)
grid_sample = generate_meshgrid_sample(model.decay, resolution=1000)
grid_sample = transformer(grid_sample)
intensity_grids = {i: func(grid_sample) for i, func in intensity_funcs.items()}

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{i: jnp.nansum(grid) for i, grid in intensity_grids.items()}

{1: Array(2.45658113e+09, dtype=float64),
 2: Array(2.45658113e+09, dtype=float64),
 3: Array(2.45658113e+09, dtype=float64)}

assert_almost_equal(jnp.nansum(intensity_grids[2] - intensity_grids[1]), 0, decimal=6)
assert_almost_equal(jnp.nansum(intensity_grids[2] - intensity_grids[1]), 0, decimal=6)

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def overlay_inset(
    png_file: str, ax, position: tuple[float, float], width: float
) -> None:
    image = plt.imread(png_file)
    res_x, res_y, _ = image.shape
    x_min, x_max = ax.get_xlim()
    y_min, y_max = ax.get_ylim()
    aspect_ratio = res_x / res_y
    aspect_ratio /= (x_max - x_min) / (y_max - y_min)
    extent = [
        position[0],
        position[0] + width,
        position[1],
        position[1] + width / aspect_ratio,
    ]
    ax.imshow(image, aspect="auto", extent=extent, zorder=2)
    ax.set_xlim(x_min, x_max)
    ax.set_ylim(y_min, y_max)


for subsystem in ["K", "D", "L"]:
    convert_svg_to_png(f"../_images/orientation-{subsystem}.svg", dpi=1000)
    del subsystem


def plot_comparison(colorbar: bool, watermark: bool, show: bool = False) -> None:
    plt.ioff()
    x_label = R"$m^2\left(K^-\pi^+\right)$ [GeV$^2$]"
    y_label = R"$m^2\left(pK^-\right)$ [GeV$^2$]"

    plt.rcdefaults()
    plt.rc("font", size=18)
    use_mpl_latex_fonts()
    fig, axes = plt.subplots(
        dpi=200,
        figsize=(20, 6) if colorbar else (18.5, 6),
        ncols=3,
        sharey=True,
        gridspec_kw={"width_ratios": [1, 1, 1.21 if colorbar else 1]},
    )
    fig.patch.set_facecolor("none")
    for ax in axes.flatten():
        ax.patch.set_color("none")
    normalized_intensities = {i: I / jnp.nansum(I) for i, I in intensity_grids.items()}
    global_max = max(map(jnp.nanmax, normalized_intensities.values()))
    axes[0].set_ylabel(y_label)
    subsystem_names = ["K", "L", "D"]
    for i, (ax, name) in enumerate(zip(axes, subsystem_names), 1):
        ax.set_xlabel(x_label)
        ax.set_box_aspect(1)
        mesh = ax.pcolormesh(
            grid_sample["sigma1"],
            grid_sample["sigma2"],
            normalized_intensities[i],
            rasterized=True,
        )
        mesh.set_clim(vmax=global_max)
        if colorbar and ax is axes[-1]:
            c_bar = fig.colorbar(mesh, ax=ax)
            c_bar.ax.set_ylabel("Normalized intensity")
        if watermark:
            add_watermark(ax)
        overlay_inset(
            f"../_images/orientation-{name}.png",
            ax=ax,
            position=(1.05, 3.85),
            width=0.75,
        )
    fig.subplots_adjust(wspace=0)
    output_filename = "intensity-alignment-consistency"
    if watermark:
        output_filename += "-watermark"
    if colorbar:
        output_filename += "-colorbar"
    output_filename = f"../_static/images/{output_filename}.svg"
    fig.savefig(output_filename, bbox_inches="tight")
    reduce_svg_size(output_filename)
    if show:
        plt.show(fig)
    plt.close(fig)
    if show:
        plt.ion()


plt.rc("font", size=18)
plot_comparison(colorbar=True, watermark=False, show=True)
plot_comparison(colorbar=True, watermark=True)
plot_comparison(colorbar=False, watermark=False)
plot_comparison(colorbar=False, watermark=True)