{ "cells": [ { "cell_type": "code", "execution_count": null, "metadata": { "hideCode": true, "hideOutput": true, "hidePrompt": true, "jupyter": { "source_hidden": true }, "slideshow": { "slide_type": "skip" }, "tags": [ "remove-cell" ] }, "outputs": [], "source": [ "%%capture\n", "%config Completer.use_jedi = False\n", "%config InlineBackend.figure_formats = ['svg']\n", "import os\n", "\n", "STATIC_WEB_PAGE = {\"EXECUTE_NB\", \"READTHEDOCS\"}.intersection(os.environ)\n", "\n", "# Install on Google Colab\n", "import subprocess\n", "import sys\n", "\n", "from IPython import get_ipython\n", "\n", "install_packages = \"google.colab\" in str(get_ipython())\n", "if install_packages:\n", " for package in [\"tensorwaves[doc]\", \"graphviz\"]:\n", " subprocess.check_call(\n", " [sys.executable, \"-m\", \"pip\", \"install\", package]\n", " )" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Analytic continuation" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "jupyter": { "source_hidden": true }, "tags": [ "hide-cell" ] }, "outputs": [], "source": [ "import logging\n", "import warnings\n", "\n", "import ampform\n", "import graphviz\n", "import matplotlib.pyplot as plt\n", "import pandas as pd\n", "import qrules\n", "from IPython.display import Math\n", "\n", "from tensorwaves.data import generate_data, generate_phsp\n", "from tensorwaves.data.transform import HelicityTransformer\n", "from tensorwaves.model import LambdifiedFunction, SympyModel\n", "\n", "logger = logging.getLogger()\n", "logger.setLevel(logging.ERROR)\n", "warnings.filterwarnings(\"ignore\")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Sometimes {mod}`qrules` finds resonances that lie outside phase space, because resonances can 'leak' into phase space. The example below is simplified and the two selected resonances are a bit unusual, but it serves to illustrate how to handle these sub-threshold resonances." ] }, { "cell_type": "code", "execution_count": null, "metadata": { "tags": [] }, "outputs": [], "source": [ "reaction = qrules.generate_transitions(\n", " initial_state=\"D0\",\n", " final_state=[\"K-\", \"K+\", \"K0\"],\n", " allowed_intermediate_particles=[\"a(0)(980)0\", \"a(2)(1320)+\"],\n", " formalism=\"canonical-helicity\",\n", ")\n", "dot = qrules.io.asdot(\n", " reaction,\n", " collapse_graphs=True,\n", " render_node=False,\n", ")\n", "graphviz.Source(dot)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Of the two resonances, $a_0(980)$ lies just below threshold ― it's mass is smaller than the the masses of the two decay products combined:" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "jupyter": { "source_hidden": true }, "tags": [ "hide-input" ] }, "outputs": [], "source": [ "pdg = qrules.load_pdg()\n", "a_meson = pdg[\"a(0)(980)0\"]\n", "k_plus = pdg[\"K+\"]\n", "k_minus = pdg[\"K+\"]\n", "two_k_mass = round(k_minus.mass + k_plus.mass, 3)\n", "display(\n", " Math(\n", " fR\"m_{{{a_meson.latex}}} = {a_meson.mass} \\pm\"\n", " fR\" {a_meson.width}\\;\\mathrm{{GeV}}\"\n", " ),\n", " Math(\n", " fR\"m_{{{k_plus.latex}}} + m_{{{k_minus.latex}}} =\"\n", " fR\" {two_k_mass}\\;\\mathrm{{GeV}}\"\n", " ),\n", ")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "To correctly describe the dynamics for this resonance, we should use make use of {doc}`analytic continuation `. As opposed to {doc}`/usage/step1`, we now use {func}`~ampform.dynamics.builder.create_analytic_breit_wigner` (a relativistic Breit-Wigner with analytic continuation) instead of {func}`~ampform.dynamics.builder.create_relativistic_breit_wigner_with_ff`:" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "from ampform.dynamics.builder import (\n", " create_analytic_breit_wigner,\n", " create_non_dynamic_with_ff,\n", " create_relativistic_breit_wigner_with_ff,\n", ")\n", "\n", "model_builder = ampform.get_builder(reaction)\n", "model_builder.set_dynamics(\n", " \"J/psi(1S)\",\n", " create_non_dynamic_with_ff,\n", ")\n", "model_builder.set_dynamics(\n", " \"a(0)(980)0\",\n", " create_analytic_breit_wigner,\n", ")\n", "model_builder.set_dynamics(\n", " \"a(2)(1320)+\",\n", " create_relativistic_breit_wigner_with_ff,\n", ")\n", "model = model_builder.formulate()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The effect can be seen once we generate data. Despite the fact that the resonance lies outside phase space, there is still a contribution to the intensity:" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "sympy_model = SympyModel(\n", " expression=model.expression.doit(),\n", " parameters=model.parameter_defaults,\n", ")\n", "intensity = LambdifiedFunction(sympy_model, backend=\"jax\")\n", "data_converter = HelicityTransformer(model.adapter)\n", "reaction_info = model.adapter.reaction_info\n", "initial_state_mass = reaction_info.initial_state[-1].mass\n", "final_state_masses = {i: p.mass for i, p in reaction_info.final_state.items()}\n", "phsp_sample = generate_phsp(100_000, initial_state_mass, final_state_masses)\n", "data_sample = generate_data(\n", " 2_000,\n", " initial_state_mass,\n", " final_state_masses,\n", " data_converter,\n", " intensity,\n", ")" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "jupyter": { "source_hidden": true }, "tags": [ "hide-cell" ] }, "outputs": [], "source": [ "import numpy as np\n", "from matplotlib import cm\n", "\n", "reaction_info = model.adapter.reaction_info\n", "intermediate_states = sorted(\n", " (\n", " p\n", " for p in model.particles\n", " if p not in reaction_info.final_state.values()\n", " and p not in reaction_info.initial_state.values()\n", " ),\n", " key=lambda p: p.mass,\n", ")\n", "\n", "evenly_spaced_interval = np.linspace(0, 1, len(intermediate_states))\n", "colors = [cm.rainbow(x) for x in evenly_spaced_interval]\n", "\n", "\n", "def indicate_masses():\n", " plt.xlabel(\"$m_{12}$ [GeV]\")\n", " for i, p in enumerate(intermediate_states):\n", " plt.gca().axvline(\n", " x=p.mass, linestyle=\"dotted\", label=p.name, color=colors[i]\n", " )" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "phsp_set = data_converter.transform(phsp_sample)\n", "data_set = data_converter.transform(data_sample)\n", "data_frame = pd.DataFrame(data_set)\n", "data_frame[\"m_12\"].hist(bins=50, alpha=0.5, density=True)\n", "indicate_masses()\n", "plt.legend();" ] } ], "metadata": { "kernelspec": { "display_name": "Python 3 (ipykernel)", "language": "python", "name": "python3" } }, "nbformat": 4, "nbformat_minor": 4 }