From 5c3746c498b1b0946a9e72747824c07f5bc12d84 Mon Sep 17 00:00:00 2001
From: Vitor Shen <17490173+shenvitor@users.noreply.github.com>
Date: Mon, 17 Jun 2024 15:12:04 +0200
Subject: [PATCH] reduce diff with main branch

---
 README.md               |   11 +
 docs/conf.py            |   25 +-
 docs/hello.ipynb        |   43 --
 docs/index.md           |   19 +-
 docs/pgamma-state.ipynb |  780 +++++++++++++++++++++++++++
 docs/reaction.md        |   73 +++
 pixi.lock               | 1124 +++++++++++++++++++++++++++++++++------
 pixi.toml               |   13 +-
 8 files changed, 1848 insertions(+), 240 deletions(-)
 create mode 100644 README.md
 delete mode 100644 docs/hello.ipynb
 create mode 100644 docs/pgamma-state.ipynb
 create mode 100644 docs/reaction.md

diff --git a/README.md b/README.md
new file mode 100644
index 0000000..dd975b7
--- /dev/null
+++ b/README.md
@@ -0,0 +1,11 @@
+# Research Project on Amplitude Models Involving N-Star and K-Star Resonances in the GlueX Experiment
+
+This repository is under active development.
+
+We are doing amplitude analysis / Partial Wave Analysis (PWA) with the use of symbolic expressions, which is included in the ComPWA packages, or CAS(Computer Algebra System)-assisted model building in general (the python package SymPy in this case).
+
+The first demonstration example is: $p \gamma \to$ Resonances (e.g. N\*) $\to \eta \pi^0 p$
+
+The main focus of this research would be: $p \gamma \to$ Resonances (e.g. N\*, K\*) $\to K^+ \pi^0 \Lambda$
+
+Current long term plan: This research project focuses on developing and applying amplitude models to analyze data from the GlueX experiment, specifically targeting N-star (N*) and K-star (K*) resonances.
diff --git a/docs/conf.py b/docs/conf.py
index bed179b..70fccf8 100644
--- a/docs/conf.py
+++ b/docs/conf.py
@@ -1,6 +1,3 @@
-project = "GlueX-Nstar"
-author = "ComPWA"
-
 exclude_patterns = [
     "**.ipynb_checkpoints",
     ".DS_Store",
@@ -8,29 +5,15 @@
     "Thumbs.db",
     "_build",
 ]
-extensions = [
-    "myst_nb",
-    "sphinx.ext.duration",
-    "sphinx.ext.doctest",
-    "sphinx.ext.autodoc",
-    "sphinx.ext.autosummary",
-    "sphinx.ext.intersphinx",
-    "sphinx.ext.napoleon",
-    "sphinx.ext.viewcode",
-]
+extensions = ["myst_nb"]
 html_theme = "sphinx_book_theme"
 master_doc = "index"
 myst_enable_extensions = [
+    "amsmath",
     "colon_fence",
-    "deflist",
     "dollarmath",
-    "html_admonition",
     "html_image",
-    "linkify",
-    "substitution",
-    "tasklist",
+    "smartquotes",
 ]
-
-nb_execution_mode = "auto"
 nitpicky = True
-version = "0.1.0"
+project = "GlueX N-Star"
diff --git a/docs/hello.ipynb b/docs/hello.ipynb
deleted file mode 100644
index 38a1baa..0000000
--- a/docs/hello.ipynb
+++ /dev/null
@@ -1,43 +0,0 @@
-{
- "cells": [
-  {
-   "cell_type": "markdown",
-   "id": "0",
-   "metadata": {},
-   "source": [
-    "# Hello"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": null,
-   "id": "1",
-   "metadata": {},
-   "outputs": [],
-   "source": [
-    "print(\"Hello World!\")"
-   ]
-  }
- ],
- "metadata": {
-  "kernelspec": {
-   "display_name": "Python 3 (ipykernel)",
-   "language": "python",
-   "name": "python3"
-  },
-  "language_info": {
-   "codemirror_mode": {
-    "name": "ipython",
-    "version": 3
-   },
-   "file_extension": ".py",
-   "mimetype": "text/x-python",
-   "name": "python",
-   "nbconvert_exporter": "python",
-   "pygments_lexer": "ipython3",
-   "version": "3.12.3"
-  }
- },
- "nbformat": 4,
- "nbformat_minor": 5
-}
diff --git a/docs/index.md b/docs/index.md
index 38385b6..b08e62e 100644
--- a/docs/index.md
+++ b/docs/index.md
@@ -1,13 +1,20 @@
-# Welcome Page
+# Amplitude Analysis 101 / PWA101 (v2.0)
 
-# Welcome to GlueX-Nstar documentation!
+## CAS-assisted Amplitude Models building for photoproduction reaction
 
-Investigation of the N\* Resonances in the GlueX Experiment
+This document is a follow-up of PWA101 (v1.0) (temporarily see [here](https://compwa--217.org.readthedocs.build/report/999.html)),
+as an introduction to Amplitude Analysis / Partial Wave Analysis (PWA) by demonstrating its application to a specific reaction channel and amplitude model.
 
-> **Note:** This project is under active development.
+Now the focus is the use of symbolic expressions (computations), with the use of [ComPWA packages](https://compwa.github.io/), or CAS(Computer Algebra System)-assisted model building in general (the python package `SymPy` in this case) to illustrate the general process in PWA.
 
-## Contents
+This tutorial aims to equip readers with deeper understanding of PWA methodologies and full workflow in the context of hadron physics through a practical, hands-on example with symbolic expressions via [ComPWA](https://compwa.github.io/).
+
+Furthermore,
+address to the issue ✅&nbsp;[ComPWA/gluex-nstar#1](https://github.com/ComPWA/gluex-nstar/issues/1), this document is PWA101(v2.0), which shows PWA methodologies and full workflow in the context of hadron physics with symbolic expressions via [ComPWA](https://compwa.github.io/).
+
+# Table of contents:
 
 ```{toctree}
-hello
+reaction
+pgamma-state
 ```
diff --git a/docs/pgamma-state.ipynb b/docs/pgamma-state.ipynb
new file mode 100644
index 0000000..24fa056
--- /dev/null
+++ b/docs/pgamma-state.ipynb
@@ -0,0 +1,780 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "0",
+   "metadata": {
+    "editable": true,
+    "slideshow": {
+     "slide_type": ""
+    },
+    "tags": []
+   },
+   "source": [
+    "# Virtual particle that represents the state pγ "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "1",
+   "metadata": {
+    "editable": true,
+    "slideshow": {
+     "slide_type": ""
+    },
+    "tags": []
+   },
+   "source": [
+    "Address to issue ✅&nbsp;[ComPWA/gluex-nstar#5](https://github.com/ComPWA/gluex-nstar/issues/5), this section is an investigation of creating a virtual particle that represents the state $p \\gamma$ in `QRules`"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "2",
+   "metadata": {
+    "editable": true,
+    "slideshow": {
+     "slide_type": ""
+    },
+    "tags": []
+   },
+   "source": [
+    "As an example of side note, create a virtual particle that represents state $e^+ e^-$ can be seen [here](https://qrules.readthedocs.io/0.10.x/usage/particle/#adding-custom-particle-definitions-through-python)."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "3",
+   "metadata": {},
+   "source": [
+    "## Finding particles"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "4",
+   "metadata": {},
+   "source": [
+    "Our reaction is $p + \\gamma \\to \\eta + \\pi^0 + p$, so we can look for each of the reaction and decay particles"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "5",
+   "metadata": {
+    "editable": true,
+    "slideshow": {
+     "slide_type": ""
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "from __future__ import annotations\n",
+    "from IPython.display import Math\n",
+    "from qrules.particle import load_pdg\n",
+    "from ampform.io import aslatex, improve_latex_rendering\n",
+    "\n",
+    "improve_latex_rendering()\n",
+    "particle_db = load_pdg()\n",
+    "print(\"Number of loaded particles:\", len(particle_db))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "6",
+   "metadata": {
+    "editable": true,
+    "slideshow": {
+     "slide_type": ""
+    },
+    "tags": []
+   },
+   "source": [
+    "The [`ParticleCollection`](https://qrules.readthedocs.io/0.10.x/api/qrules.particle/#qrules.particle.ParticleCollection) class offers some methods to search for particles by name or by PID, name, or other properties."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "7",
+   "metadata": {},
+   "source": [
+    "### Proton"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "8",
+   "metadata": {},
+   "source": [
+    "Search by Particle ID (PID) for proton:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "9",
+   "metadata": {
+    "editable": true,
+    "slideshow": {
+     "slide_type": ""
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "particle_db.find(2212)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "10",
+   "metadata": {},
+   "source": [
+    "Search by Particle name for proton:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "11",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "particle_db.find(\"p\")\n",
+    "proton = particle_db[\"p\"]"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "12",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "from IPython.display import Math\n",
+    "\n",
+    "Math(proton.latex)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "13",
+   "metadata": {},
+   "source": [
+    "### $\\gamma$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "14",
+   "metadata": {},
+   "source": [
+    "similarly for $\\gamma$:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "15",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "particle_db.find(\"gamma\")"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "16",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "gamma = particle_db[\"gamma\"]\n",
+    "Math(gamma.latex)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "17",
+   "metadata": {},
+   "source": [
+    "### $\\eta$"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "18",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "particle_db.find(\"eta\")"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "19",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "eta = particle_db[\"eta\"]\n",
+    "Math(gamma.latex)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "20",
+   "metadata": {},
+   "source": [
+    "### $\\pi^0$"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "21",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "particle_db.find(\"pi0\")"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "22",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "pi0 = particle_db[\"pi0\"]\n",
+    "Math(pi0.latex)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "23",
+   "metadata": {},
+   "source": [
+    "# Adding custom particle"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "24",
+   "metadata": {},
+   "source": [
+    "We add the virtual particle $p\\gamma$ to represent the state of $p-\\gamma$ system in the photoproduction reaction.\n",
+    "See PWA101 (temporary [here](https://compwa--217.org.readthedocs.build/report/999.html)), the value of the mass of the system $m_{p\\gamma}$ in CM frame is about 4.102 GeV in this analysis.\n",
+    "\n",
+    "\n",
+    "We use `Particle` from `QRules` for adding custom particle, so that some quantum numbers we need to determine for the $p\\gamma$ virtual state particle:\n",
+    "\n",
+    "* (Total) Spin\n",
+    "    * $J_{p\\gamma} = \\frac{1}{2}$ or $\\frac{3}{2}$\n",
+    "      \n",
+    "      because $J_{p\\gamma}$ = from $|s_p-s_{\\gamma}|$ to $s_p + s_{\\gamma}$ = from $|\\frac{1}{2}-1|$ to $\\frac{1}{2} + 1$\n",
+    "      \n",
+    "* Mass\n",
+    "    * $m_{p\\gamma} = E_{p\\gamma} \\approx 4.102$\n",
+    "    \n",
+    "* Charge\n",
+    "    * $Q_{p\\gamma} = Q_p + Q_{\\gamma} = 1 $\n",
+    "      \n",
+    "* Isospin and $3^{rd}$ component of isospin\n",
+    "    * $I_{p\\gamma} = I_p = \\frac{1}{2}$\n",
+    "    * $I_{3_{p\\gamma}} = I_{3_p} = +\\frac{1}{2}$\n",
+    "  \n",
+    "* Baryon Number\n",
+    "    * $B_{p\\gamma} = B_p  = 1$\n",
+    "\n",
+    "* Parity\n",
+    "    * $P_{p\\gamma} = P_p \\times P_{\\gamma} = +1 \\times (-1) = -1$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "25",
+   "metadata": {},
+   "source": [
+    "These quantum numbers provide a complete and accurate description of the virtual particle representing the p−γ system with a mass of 4.102 GeV.\n",
+    "The pγ virtual particle has two possibilities for the total spin, while the other quantum numbers are fixed."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "26",
+   "metadata": {},
+   "source": [
+    "We add the spin-$\\frac{1}{2}$ $p\\gamma$ virtual particle via `QRules` and named it $p\\gamma$ with a make-up pid number 99990"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "27",
+   "metadata": {
+    "editable": true,
+    "slideshow": {
+     "slide_type": ""
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "from qrules.particle import Particle, Spin\n",
+    "\n",
+    "pgamma = Particle(\n",
+    "    name=\"pgamma\",\n",
+    "    latex=r\"p\\gamma (s1/2)\",\n",
+    "    spin=0.5,\n",
+    "    mass=4.101931071854584,\n",
+    "    charge=1,\n",
+    "    isospin=Spin(1 / 2, +1 / 2),\n",
+    "    baryon_number=1,\n",
+    "    parity=-1,\n",
+    "    pid=99990,\n",
+    ")\n",
+    "pgamma"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "28",
+   "metadata": {},
+   "source": [
+    "We add the spin-$\\frac{3}{2}$ $p\\gamma$ virtual particle via `QRules` and named it $p\\gamma (s3/2)$ with a make-up pid number 99991"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "29",
+   "metadata": {
+    "editable": true,
+    "slideshow": {
+     "slide_type": ""
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "pgamma2 = Particle(\n",
+    "    name=\"pgamma2\",\n",
+    "    latex=R\"p\\gamma (s3/2)\",\n",
+    "    spin=1.5,\n",
+    "    mass=4.101931071854584,\n",
+    "    charge=1,\n",
+    "    isospin=Spin(1 / 2, +1 / 2),\n",
+    "    baryon_number=1,\n",
+    "    parity=-1,\n",
+    "    pid=99991,\n",
+    ")\n",
+    "pgamma2"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "30",
+   "metadata": {
+    "editable": true,
+    "slideshow": {
+     "slide_type": ""
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "Math(pgamma.latex)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "31",
+   "metadata": {
+    "editable": true,
+    "slideshow": {
+     "slide_type": ""
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "Math(pgamma2.latex)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "32",
+   "metadata": {
+    "editable": true,
+    "slideshow": {
+     "slide_type": ""
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "particle_db.add(pgamma)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "33",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "particle_db.add(pgamma2)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "34",
+   "metadata": {
+    "editable": true,
+    "slideshow": {
+     "slide_type": ""
+    },
+    "tags": []
+   },
+   "source": [
+    "The $`p\\gamma`$ virtual states generated via `QRules` will be used in later stages, both possible spin states $(J= \\frac{1}{2}$ and $\\frac{3}{2})$ should be taken into account if possible, otherwise simplified if necessary."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "35",
+   "metadata": {},
+   "source": [
+    ":::{important}\n",
+    "It is possible to add any kind of custom Particle, as long as its quantum numbers comply with the [gellmann_nishijima()](https://qrules.readthedocs.io/0.10.x/api/qrules.conservation_rules/#qrules.conservation_rules.gellmann_nishijima) rule:\n",
+    ":::\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "36",
+   "metadata": {
+    "editable": true,
+    "slideshow": {
+     "slide_type": ""
+    },
+    "tags": []
+   },
+   "source": [
+    ":::{note}\n",
+    "[Gell-Mann-Nishijima formula](https://en.wikipedia.org/wiki/Gell-Mann–Nishijima_formula):\n",
+    "$Q = I_3 + \\frac{1}{2}(B+S+C+B'+T)$\n",
+    "\n",
+    "where \n",
+    "Q is charge, \n",
+    "$I_3$ is Spin projection of isospin, \n",
+    " B is baryon number, \n",
+    " S is strangeness, \n",
+    " C is charmness, \n",
+    " B' is bottomness, and \n",
+    " T is topness.\n",
+    ":::\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "37",
+   "metadata": {
+    "editable": true,
+    "slideshow": {
+     "slide_type": ""
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "import qrules\n",
+    "import graphviz\n",
+    "\n",
+    "reaction1_all = qrules.generate_transitions(\n",
+    "    initial_state=(\"pgamma\"),\n",
+    "    final_state=[\"eta\", \"pi0\", \"p\"],\n",
+    "    # allowed_intermediate_particles=[\"a(2)\", \"Delta\", \"N\"],\n",
+    "    allowed_interaction_types=[\"strong\", \"EM\", \"weak\"],\n",
+    "    formalism=\"canonical-helicity\",\n",
+    "    particle_db=particle_db,\n",
+    ")"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "38",
+   "metadata": {
+    "editable": true,
+    "slideshow": {
+     "slide_type": ""
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "dot_all = qrules.io.asdot(reaction1_all, collapse_graphs=True)\n",
+    "graphviz.Source(dot_all)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "39",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "reaction1_strong_EM = qrules.generate_transitions(\n",
+    "    initial_state=(\"pgamma\"),\n",
+    "    final_state=[\"eta\", \"pi0\", \"p\"],\n",
+    "    # allowed_intermediate_particles=[\"a(2)\", \"Delta\", \"N\"],\n",
+    "    allowed_interaction_types=[\"strong\", \"EM\"],\n",
+    "    formalism=\"canonical-helicity\",\n",
+    "    particle_db=particle_db,\n",
+    ")"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "40",
+   "metadata": {
+    "editable": true,
+    "slideshow": {
+     "slide_type": ""
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "dot_se = qrules.io.asdot(reaction1_strong_EM, collapse_graphs=True)\n",
+    "graphviz.Source(dot_se)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "41",
+   "metadata": {
+    "editable": true,
+    "slideshow": {
+     "slide_type": ""
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "reaction1_strong = qrules.generate_transitions(\n",
+    "    initial_state=(\"pgamma\"),\n",
+    "    final_state=[\"eta\", \"pi0\", \"p\"],\n",
+    "    # allowed_intermediate_particles=[\"a(2)\", \"Delta\", \"N\"],\n",
+    "    allowed_interaction_types=[\"strong\"],\n",
+    "    formalism=\"canonical-helicity\",\n",
+    "    particle_db=particle_db,\n",
+    ")"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "42",
+   "metadata": {
+    "editable": true,
+    "slideshow": {
+     "slide_type": ""
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "dot_s = qrules.io.asdot(reaction1_strong, collapse_graphs=True)\n",
+    "graphviz.Source(dot_s)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "43",
+   "metadata": {
+    "editable": true,
+    "slideshow": {
+     "slide_type": ""
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "from qrules import InteractionType, StateTransitionManager\n",
+    "\n",
+    "stm = StateTransitionManager(\n",
+    "    initial_state=[\"pgamma\"],\n",
+    "    final_state=[\"eta\", \"pi0\", \"p\"],\n",
+    "    formalism=\"canonical-helicity\",\n",
+    "    particle_db=particle_db,\n",
+    ")"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "44",
+   "metadata": {
+    "editable": true,
+    "slideshow": {
+     "slide_type": ""
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "problem_sets = stm.create_problem_sets()\n",
+    "sorted(problem_sets, reverse=True)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "45",
+   "metadata": {
+    "editable": true,
+    "slideshow": {
+     "slide_type": ""
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "from qrules import io\n",
+    "\n",
+    "some_problem_set = problem_sets[60.0][0]\n",
+    "dot = io.asdot(some_problem_set, render_node=True)\n",
+    "graphviz.Source(dot)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "46",
+   "metadata": {
+    "editable": true,
+    "slideshow": {
+     "slide_type": ""
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "some_problem_set = problem_sets[1][0]\n",
+    "dot = io.asdot(some_problem_set, render_node=True)\n",
+    "graphviz.Source(dot)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "47",
+   "metadata": {
+    "editable": true,
+    "slideshow": {
+     "slide_type": ""
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "some_problem_set = problem_sets[0.0001][1]\n",
+    "dot = io.asdot(some_problem_set, render_node=True)\n",
+    "graphviz.Source(dot)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "48",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "reaction = stm.find_solutions(problem_sets)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "49",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "print(\"found\", len(reaction.transitions), \"solutions!\")\n",
+    "reaction.get_intermediate_particles().names"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "50",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "stm.set_allowed_interaction_types([InteractionType.STRONG])\n",
+    "problem_sets = stm.create_problem_sets()\n",
+    "reaction = stm.find_solutions(problem_sets)\n",
+    "\n",
+    "print(\"found\", len(reaction.transitions), \"solutions!\")\n",
+    "reaction.get_intermediate_particles().names"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "51",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "stm.set_allowed_interaction_types([InteractionType.STRONG, InteractionType.EM])\n",
+    "problem_sets = stm.create_problem_sets()\n",
+    "reaction = stm.find_solutions(problem_sets)\n",
+    "\n",
+    "print(\"found\", len(reaction.transitions), \"solutions!\")\n",
+    "reaction.get_intermediate_particles().names"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "52",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3 (ipykernel)",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.12.3"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
diff --git a/docs/reaction.md b/docs/reaction.md
new file mode 100644
index 0000000..17f3519
--- /dev/null
+++ b/docs/reaction.md
@@ -0,0 +1,73 @@
+# Reaction and Models
+
+<!-- cspell:ignore Mathieu -->
+<!-- This amplitude model is adapted from the [Lecture 11 in STRONG2020 HaSP School](https://indico.ific.uv.es/event/6803/contributions/21223/) by Vincent Mathieu. -->
+
+## Reaction with Resonances and Decay
+
+The (photo-production) reaction is $ \gamma p \to \eta \pi^0 p$, its decay is described by an amplitude model with three possible resonances: $a_2$, $\Delta^+$, and $N^*$.
+
+```{image} https://github.com/ComPWA/compwa-org/assets/17490173/ec6bf191-bd5f-43b0-a6cb-da470b071630
+:width: 100%
+```
+
+## Amplitude Models
+
+The amplitude model is adapted from the [Lecture 11 in STRONG2020 HaSP School](https://indico.ific.uv.es/event/6803/contributions/21223/), only the Breit-Wigner and Spherical harmonics terms are kept for doing PWA eventually, as shown in equation {eq}`BW_SH_label`.
+
+$$
+\begin{eqnarray}
+A^{12} &=& \frac{\sum a_m Y_2^m (\Omega_1)}{s-m^2_{a_2}+im_{a_2} \Gamma_{a_2}} \\
+A^{23} &=& \frac{\sum b_m Y_1^m (\Omega_2)}{s-m^2_{\Delta}+im_{\Delta} \Gamma_{\Delta}}  \\
+\
+A^{31} &=& \frac{c_0}{s-m^2_{N^*}+im_{N^*} \Gamma_{N^*}}
+\end{eqnarray}
+$$ (BW_SH_label)
+
+where s is the Mandelstam variable, m is the mass, $\Gamma$ is the width, $Y^m_l$ is the spherical harmonics, $\Omega_i$ is the decay angles (a pair of Euler angles), and $a_i$, $b_i$, and $c_i$ are coefficients
+
+:::{note}
+Mandelstam variables $s_{ij}=(p_i+p_j)^2$, $t_i=(p_a-p_i)^2$, and $u_i=(p_b-p_i)^2$.
+:::
+
+with  intensity $I$ and amplitude $A$:
+
+
+$$
+
+\begin{eqnarray}
+I &=& |A|^2 \nonumber \\
+A &=& A^{12} + A^{23} + A^{31} \\
+\end{eqnarray}
+
+$$
+(123_label)
+
+where $\quad 1 \equiv \eta ; \quad  2 \equiv \pi^0 ; \quad 3 \equiv p$
+
+:::{note}
+The choice of the amplitude model (equations (1) and (2)) for PWA in this tutorial consists of three resonances, and each of them are formed by two terms: Breit-Wigner with Spherical harmonics ($l = 2, 1, 0$).
+:::
+
+:::{important}
+The spin of $\eta$ meson and $\pi^0$ meson are all 0. But the spin of proton is not 0, it is spin-$\frac{1}{2}$.
+
+In this amplitude model **spin** of baryon is simplified (not realistic):
+$\eta$, $\pi^0$ and $p$ are all treated as spin-0 particles.
+
+This means that total intrinsic spin $s$ is ignored in this model,
+the total angular momentum
+$J$ of the system or any subsystems within this model will solely depend on the orbital angular momentum
+$L$, characterized by quantum number $l$.
+And this simplifies the use of spherical harmonics $Y_l^m(\theta,\phi)$,
+since only the orbital angular momentum component is involved, and thus the combination of contribution is not considered (e.g. Clebsch-Gordan Coefficients).
+
+In our case:
+- $A^{12}$ amplitude represents a d-wave interaction, as indicated by $l=2$.
+    - The possible $m$ values are $−2,−1,0,1,2$. Each of these values corresponds to different orientations of the d-wave. The wave type is solely determined by $l$ and all these $m$ values still describe d-wave characteristics.
+- $A^{23}$ amplitude represents a p-wave interaction, as indicated by $l=1$.
+    - The possible $m$ values are $−1,0,1$. Each of these values corresponds to different orientations of the p-wave. Similarly, these values are all p-wave orientations.
+- $A^{31}$ amplitude represents a s-wave interaction, as indicated by $l=0$.
+    - The only possible $m$ value is 0, which is consistent with the spherical symmetry of s-waves.
+:::
+$$
diff --git a/pixi.lock b/pixi.lock
index a1a853c..6e93627 100644
--- a/pixi.lock
+++ b/pixi.lock
@@ -9,6 +9,7 @@ environments:
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       - conda: https://conda.anaconda.org/conda-forge/noarch/alabaster-0.7.16-pyhd8ed1ab_0.conda
+      - conda: https://conda.anaconda.org/conda-forge/noarch/ampform-0.15.4-pyhd8ed1ab_0.conda
       - conda: https://conda.anaconda.org/conda-forge/noarch/anyio-4.3.0-pyhd8ed1ab_0.conda
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       - conda: https://conda.anaconda.org/conda-forge/noarch/defusedxml-0.7.1-pyhd8ed1ab_0.tar.bz2
+      - conda: https://conda.anaconda.org/conda-forge/noarch/deprecated-1.2.14-pyh1a96a4e_0.conda
       - conda: https://conda.anaconda.org/conda-forge/noarch/distlib-0.3.8-pyhd8ed1ab_0.conda
-      - conda: https://conda.anaconda.org/conda-forge/linux-64/docutils-0.20.1-py312h7900ff3_3.conda
+      - conda: https://conda.anaconda.org/conda-forge/noarch/docutils-0.21.2-pyhd8ed1ab_0.conda
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-      - conda: https://conda.anaconda.org/conda-forge/linux-64/gdk-pixbuf-2.42.11-hb9ae30d_0.conda
+      - conda: https://conda.anaconda.org/conda-forge/linux-64/gdk-pixbuf-2.42.12-hb9ae30d_0.conda
       - conda: https://conda.anaconda.org/conda-forge/linux-64/giflib-5.2.2-hd590300_0.conda
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       - conda: https://conda.anaconda.org/conda-forge/noarch/gitpython-3.1.43-pyhd8ed1ab_0.conda
+      - conda: https://conda.anaconda.org/conda-forge/linux-64/gmp-6.3.0-h59595ed_1.conda
+      - conda: https://conda.anaconda.org/conda-forge/linux-64/gmpy2-2.1.5-py312h1d5cde6_1.conda
       - conda: https://conda.anaconda.org/conda-forge/linux-64/graphite2-1.3.13-h59595ed_1003.conda
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@@ -66,7 +70,8 @@ environments:
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       - conda: https://conda.anaconda.org/conda-forge/noarch/h2-4.1.0-pyhd8ed1ab_0.tar.bz2
-      - conda: https://conda.anaconda.org/conda-forge/linux-64/harfbuzz-8.4.0-h3d44ed6_0.conda
+      - conda: https://conda.anaconda.org/conda-forge/linux-64/harfbuzz-8.5.0-hfac3d4d_0.conda
+      - conda: https://conda.anaconda.org/conda-forge/noarch/hepunits-2.3.3-pyhd8ed1ab_0.conda
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+      - conda: https://conda.anaconda.org/conda-forge/noarch/ipython_genutils-0.2.0-py_1.tar.bz2
+      - conda: https://conda.anaconda.org/conda-forge/noarch/ipywidgets-8.1.2-pyhd8ed1ab_1.conda
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-      - conda: https://conda.anaconda.org/conda-forge/noarch/jax-0.4.26-pyhd8ed1ab_0.conda
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+      - conda: https://conda.anaconda.org/conda-forge/noarch/jupyterlab_code_formatter-2.2.1-pyhd8ed1ab_0.conda
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+      - conda: https://conda.anaconda.org/conda-forge/noarch/jupyterlab_widgets-3.0.10-pyhd8ed1ab_0.conda
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-      - conda: https://conda.anaconda.org/conda-forge/noarch/linkify-it-py-2.0.3-pyhd8ed1ab_0.conda
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       - conda: https://conda.anaconda.org/conda-forge/linux-64/ml_dtypes-0.4.0-py312hfb8ada1_0.conda
+      - conda: https://conda.anaconda.org/conda-forge/linux-64/mpc-1.3.1-hfe3b2da_0.conda
+      - conda: https://conda.anaconda.org/conda-forge/linux-64/mpfr-4.2.1-h9458935_1.conda
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+  url: https://conda.anaconda.org/conda-forge/noarch/widgetsnbextension-4.0.10-pyhd8ed1ab_0.conda
+  sha256: 981b06c76a1a86bb84be09522768be0458274926b22f4b0225dfcdd30a6593e0
+  md5: 521f489e3babeddeec638c2add7e9e64
+  depends:
+  - python >=3.7
+  license: BSD-3-Clause
+  license_family: BSD
+  purls:
+  - pkg:pypi/widgetsnbextension
+  size: 886369
+  timestamp: 1707420479741
+- kind: conda
+  name: wrapt
+  version: 1.16.0
+  build: py312h98912ed_0
+  subdir: linux-64
+  url: https://conda.anaconda.org/conda-forge/linux-64/wrapt-1.16.0-py312h98912ed_0.conda
+  sha256: dc8431b343961347ad93b33d2d8270e8c15d8825382f4f2540835c94aba2de05
+  md5: fa957a1c7bee7e47ad44633caf7be8bc
+  depends:
+  - libgcc-ng >=12
+  - python >=3.12,<3.13.0a0
+  - python_abi 3.12.* *_cp312
+  license: BSD-2-Clause
+  license_family: BSD
+  purls:
+  - pkg:pypi/wrapt
+  size: 62482
+  timestamp: 1699532968076
 - kind: conda
   name: xorg-kbproto
   version: 1.0.7
@@ -4818,12 +5610,12 @@ packages:
 - kind: conda
   name: zeromq
   version: 4.3.5
-  build: h75354e8_3
-  build_number: 3
+  build: h75354e8_4
+  build_number: 4
   subdir: linux-64
-  url: https://conda.anaconda.org/conda-forge/linux-64/zeromq-4.3.5-h75354e8_3.conda
-  sha256: c2f2db5d19b603546db025b47fb71765f8dda0a0fe8feb42bd4e6a46194a5590
-  md5: 1b0ea5d6674e4e7dde0537c890813edb
+  url: https://conda.anaconda.org/conda-forge/linux-64/zeromq-4.3.5-h75354e8_4.conda
+  sha256: bc9aaee39e7be107d7daff237435dfd8f791aca460a98583a36a263615205262
+  md5: 03cc8d9838ad9dd0060ab532e81ccb21
   depends:
   - krb5 >=1.21.2,<1.22.0a0
   - libgcc-ng >=12
@@ -4831,8 +5623,8 @@ packages:
   - libstdcxx-ng >=12
   license: MPL-2.0
   license_family: MOZILLA
-  size: 351803
-  timestamp: 1714545110790
+  size: 353229
+  timestamp: 1715607188837
 - kind: conda
   name: zipp
   version: 3.17.0
@@ -4846,6 +5638,8 @@ packages:
   - python >=3.8
   license: MIT
   license_family: MIT
+  purls:
+  - pkg:pypi/zipp
   size: 18954
   timestamp: 1695255262261
 - kind: conda
diff --git a/pixi.toml b/pixi.toml
index fd94169..a6442bd 100644
--- a/pixi.toml
+++ b/pixi.toml
@@ -20,25 +20,28 @@ sphinx-autobuild \
 """
 
 [dependencies]
+ampform = ">=0.15.4,<0.16"
 black = ">=24.4.2,<24.5"
+graphviz = ">=9.0.0,<9.1"
+ipython = ">=8.24.0,<8.25"
+ipywidgets = ">=8.1.2,<8.2"
 isort = ">=5.13.2,<5.14"
 jax = ">=0.4.26,<0.5"
 jaxlib = ">=0.4.23,<0.5"
+jupyter = ">=1.0.0,<1.1"
 jupyterlab = ">=4.2.0,<4.3"
 jupyterlab-git = ">=0.50.0,<0.51"
 jupyterlab-lsp = ">=5.1.0,<5.2"
 jupyterlab-myst = ">=2.4.2,<2.5"
-linkify-it-py = ">=2.0.3,<2.1"
+jupyterlab_code_formatter = ">=2.2.1,<2.3"
 myst-nb = ">=1.1.0,<1.2"
-myst-parser = ">=3.0.1,<3.1"
-nbsphinx = ">=0.9.4,<0.10"
 numpy = ">=1.26.4,<1.27"
 pre-commit = ">=3.7.1,<3.8"
 pydata-sphinx-theme = ">=0.15.2,<0.16"
 python = ">=3.12.3,<3.13"
 python-graphviz = ">=0.20.3,<0.21"
+qrules = ">=0.10.1,<0.11"
 scipy = ">=1.13.0,<1.14"
-sphinx = ">=7.3.7,<7.4"
 sphinx-autobuild = ">=2024.4.16,<2024.5"
-sphinx_rtd_theme = ">=2.0.0,<2.1"
+sphinx-book-theme = ">=1.1.2,<1.2"
 sympy = ">=1.12,<2"