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address to @redeboer comments until equation 3
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shenvitor committed Jun 28, 2024
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":::{card} PWA101: Amplitude Analysis with Python basics\n",
"TR-999\n",
"^^^\n",
"This document introduces Amplitude Analysis / Partial Wave Analysis (PWA) by demonstrating its application to a specific reaction channel and amplitude model. Basic Python programming and libraries (e.g. `numpy`, `scipy`, etc.) are used to illustrate the general process and full workflow of PWA in hadron physics.\n",
"This document introduces Amplitude Analysis / Partial Wave Analysis (PWA) by demonstrating its application to a specific reaction channel and amplitude model. Basic Python programming and libraries (e.g. `numpy`, `scipy`, etc.) are used to illustrate the more fundamental steps of PWA in hadron physics.\n",
"+++\n",
"✅ [ComPWA/RUB-EP1-AG#93](https://github.com/ComPWA/RUB-EP1-AG/issues/93), [compwa.github.io#217](https://github.com/ComPWA/compwa.github.io/pull/217)\n",
":::\n",
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"tags": []
},
"source": [
"# Amplitude Analysis 101"
"# Amplitude Analysis 101 (PWA 101)"
]
},
{
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"tags": []
},
"source": [
"- This document introduces Amplitude Analysis / Partial Wave Analysis (PWA) by demonstrating its application to a specific reaction channel and amplitude model.\n",
"- Only basic Python programming and libraries (e.g. `numpy`, `scipy`, etc.) are used to illustrate the general process in PWA.\n",
"- Before advancing to the complexities of symbolic expressions (computations) with e.g. `sympy` later, see [here](https://github.com/ComPWA/gluex-nstar/issues/1), and as a comparison .\n",
"- This tutorial aims to equip readers with a basic understanding of PWA methodologies and full workflow in hadron physics through a practical, hands-on example."
":::{note}Summary\n",
"- This document introduces Amplitude Analysis / Partial Wave Analysis (PWA) by demonstrating its application to a specific reaction channel and amplitude model. It aims to equip readers with a basic understanding of the full workflow and methodologies of PWA in hadron physics through a practical, hands-on example.\n",
"- Only basic Python programming and libraries (e.g. `numpy`, `scipy`, etc.) are used to illustrate the more fundamental steps in a PWA.\n",
"- Before advancing to the complexities of symbolic expressions (computations) with tools such as `sympy` as illustrated in PWA101 v2.0 (see [here](https://compwa.github.io/gluex-nstar)), we will focus on Python basics. This will allow us to draw a comparison between basic Python programming and symbolic computations.\n",
":::"
]
},
{
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"<!-- cspell:ignore Mathieu -->\n",
"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.\n",
"\n",
"The photo-production reaction is $ \\gamma p \\to \\eta \\pi^0 p$, which is one of the reaction channels in experiment such as [the GlueX experiment](http://www.gluex.org/). The decays are described by an amplitude model with three possible resonances: $a_2$, $\\Delta^+$, and $N^*$. "
"The photo-production reaction $ \\gamma p \\to \\eta \\pi^0 p$ is one of the reaction channels in experiment such as [the GlueX experiment](http://www.gluex.org/). For simplicity, the decays are described by an amplitude model with three possible resonances: $a_2$, $\\Delta^+$, and $N^*$. "
]
},
{
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"tags": []
},
"source": [
"The Amplitude $A$ has three parts in this case:"
"Given these three subsystems in this particle transition, we can identify three amplitudes $A^{12}$, $A^{23}$, and $A^{31}$:"
]
},
{
Expand All @@ -156,13 +157,13 @@
"source": [
"$$\n",
"\\begin{eqnarray}\n",
"A^{12} &=& \\frac{\\sum a_m Y_2^m (\\Omega_1)}{s-m^2_{a_2}+im_{a_2} \\Gamma_{a_2}} \\times s^{0.5+0.9u_3} \\nonumber \\\\\n",
"A^{23} &=& \\frac{\\sum b_m Y_1^m (\\Omega_2)}{s-m^2_{\\Delta}+im_{\\Delta} \\Gamma_{\\Delta}} \\times s^{0.5+0.9t_1} \\nonumber \\\\\n",
"A^{31} &=& \\frac{c_0}{s-m^2_{N^*}+im_{N^*} \\Gamma_{N^*}} \\times s^{1.08+0.2t_2} \n",
"A^{12} &=& \\frac{\\sum a_m Y_2^m (\\Omega_1)}{s_{12}-m^2_{a_2}+im_{a_2} \\Gamma_{a_2}} \\times s_{12}^{0.5+0.9u_3} \\nonumber \\\\\n",
"A^{23} &=& \\frac{\\sum b_m Y_1^m (\\Omega_2)}{s_{23}-m^2_{\\Delta}+im_{\\Delta} \\Gamma_{\\Delta}} \\times s_{23}^{0.5+0.9t_1} \\nonumber \\\\\n",
"A^{31} &=& \\frac{c_0}{s_{31}-m^2_{N^*}+im_{N^*} \\Gamma_{N^*}} \\times s_{31}^{1.08+0.2t_2} \n",
"\\end{eqnarray}\n",
"$$ (full_model_label)\n",
"\n",
"where $s, t, u$ are the Mandelstam variables $s_{ij}=(p_i+p_j)^2$, $t_i=(p_a-p_i)^2$, and $u_i=(p_b-p_i)^2$, 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"
"where $s, t, u$ are the Mandelstam variables $s_{ij}=(p_i+p_j)^2$, $t_i=(p_a-p_i)^2$, and $u_i=(p_b-p_i)^2$, m is the mass, $\\Gamma$ is the width, $Y^m_l$ is the spherical harmonics, $\\Omega_i$ is the decay angles which is a pair of Euler angles (polar angle $\\theta$ and azimuthal angle $\\phi$), and $a_i$, $b_i$, and $c_i$ are coefficients"
]
},
{
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"tags": []
},
"source": [
"The original full amplitude model from the [Lecture 11 in STRONG2020 HaSP School](https://indico.ific.uv.es/event/6803/contributions/21223/) is shown in equation {eq}`full_model_label`.\n",
"\n",
"*In this report, only the Breit-Wigner and Spherical harmonics terms are kept for doing PWA eventually (The exponential mandelstam variable term is abandoned), as shown in equation {eq}`BW_SH_label`."
"The original full amplitude model from the [Lecture 11 in STRONG2020 HaSP School](https://indico.ific.uv.es/event/6803/contributions/21223/) is shown in Equation {eq}`full_model_label`.\n",
"*In this report, only the Breit-Wigner and Spherical harmonics terms are kept for doing PWA eventually (The exponential mandelstam variable term is abandoned), as shown in Equation {eq}`BW_SH_label`.*"
]
},
{
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"source": [
"$$\n",
"\\begin{eqnarray}\n",
"A^{12} &=& \\frac{\\sum a_m Y_2^m (\\Omega_1)}{s-m^2_{a_2}+im_{a_2} \\Gamma_{a_2}} \\\\\n",
"A^{23} &=& \\frac{\\sum b_m Y_1^m (\\Omega_2)}{s-m^2_{\\Delta}+im_{\\Delta} \\Gamma_{\\Delta}} \\\\\n",
"A^{12} &=& \\frac{\\sum a_m Y_2^m (\\Omega_1)}{s_{12}-m^2_{a_2}+im_{a_2} \\Gamma_{a_2}} \\\\\n",
"A^{23} &=& \\frac{\\sum b_m Y_1^m (\\Omega_2)}{s_{23}-m^2_{\\Delta}+im_{\\Delta} \\Gamma_{\\Delta}} \\\\\n",
"\\\n",
"A^{31} &=& \\frac{c_0}{s-m^2_{N^*}+im_{N^*} \\Gamma_{N^*}} \n",
"A^{31} &=& \\frac{c_0}{s_{31}-m^2_{N^*}+im_{N^*} \\Gamma_{N^*}} \n",
"\\end{eqnarray}\n",
"$$ (BW_SH_label)"
]
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"tags": []
},
"source": [
"with intensity $I$ and amplitude $A$:\n",
"The intensity $I$ that describes our measured distributions is then expressed as a coherent sum of the amplitudes $A^{ij}$:\n",
"\n",
"$$\n",
"\\begin{eqnarray}\n",
Expand All @@ -207,7 +207,7 @@
"\\end{eqnarray}\n",
"$$ (123_label)\n",
"\n",
"where $\\quad 1 \\equiv \\eta ; \\quad 2 \\equiv \\pi^0 ; \\quad 3 \\equiv p$"
"where $\\quad 1 \\equiv \\eta , \\quad 2 \\equiv \\pi^0 , \\quad 3 \\equiv p$."
]
},
{
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},
"source": [
":::{note}\n",
"The ultimate choice of the amplitude model (equations (2) and (3)) for PWA in this tutorial consists of three resonances, and each of them are formed by two terms: Breit-Wigner with Spherical harmonics ($l = 2, 1, 0$).\n",
"The ultimate choice of the amplitude model (Equations {eq}`BW_SH_label` and {eq}`123_label`) for PWA in this tutorial consists of three resonances, and each of them are formed by two terms: Breit-Wigner with spherical harmonics ($l = 2, 1, 0$).\n",
":::"
]
},
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},
"source": [
":::{important}\n",
"The spin of the $\\eta$ meson and the $\\pi^0$ meson are all 0. But the spin of the proton is not 0, it is spin-$\\frac{1}{2}$.\n",
"\n",
"In this amplitude model the **spin** of baryon (proton in this example reaction) is simplified, treated as spin-0 particle, and thus not realistic.\n",
"Additionally, the mesons $\\eta$ and $\\pi^0$ are originally spin-0 particles.\n",
"The spin of the $\\eta$ meson and the $\\pi^0$ meson are all 0, but the spin of the proton is not 0, it is spin-$\\frac{1}{2}$.\n",
"In this amplitude model the **spin** of baryon (proton in this example reaction) is simplified, \n",
"by treating it as a spin-0 particle.\n",
"Overall,\n",
"the $\\eta$, $\\pi^0$ and $p$ are all treated as spin-0 particles.\n",
"\n",
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},
"outputs": [],
"source": [
"phsp_events = 1_000_000\n",
"data_events = 500_000"
"phsp_events = 800_000\n",
"data_events = 400_000"
]
},
{
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