Merge branch 'ws2425-kick-off' into 'main'

Add Exercise 4

See merge request Lehre/tp1_forstudents!11

Exercise04/Exercise04.ipynb

+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "a1c50278",
+   "metadata": {},
+   "source": [
+    "# Übungen zu Teilchenphysik I\n",
+    "## Exercise 03 - QED and EW theory"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "b82039ad",
+   "metadata": {},
+   "source": [
+    "## Preamble\n",
+    "\n",
+    "Since we're asking you to do some calculations, it's up to you whether you write the \"heavy\" calculations in the notebook or not: it's fine if you don't write them down in the notebook, but show us on paper."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "ad70f3aa-e25b-4929-b97b-b33325e649d8",
+   "metadata": {},
+   "source": [
+    "<a name='section_1_0'></a>\n",
+    "<hr style=\"height: 1px;\">\n",
+    "\n",
+    "\n",
+    "## <h1 style=\"border:1px; border-style:solid; padding: 0.25em; color: #FFFFFF; background-color: #FFA500\">Section 1: A massive scalar... photon?</h1>\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "14c1176f",
+   "metadata": {},
+   "source": [
+    "Let's imagine a parallel universe in which the photon, instead of being a massless vector particle (spin-1), is a massive scalar particle (spin-0). The QED vertex in the Feynman rules in this theory would be $-i g_{e} I$, where $I$ is the $4 \\times 4$ unit matrix (to compare with $-i g_{e} \\gamma^{\\mu}$ for the massless vector photon). There would also be no factors for the outer photon lines, since there is no photon polarisation."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "8d20c0c4",
+   "metadata": {},
+   "source": [
+    "<div class=\"alert alert-info\">\n",
+    "<strong>Exercise:</strong> \n",
+    "Assuming that this \"photon\" is heavy enough to decay into a pair of Standard Model particles, calculate the decay rate $\\Gamma$ for $\\gamma \\rightarrow e^{+}e^{-}$ using the helicity spinors (in the centre-of-mass frame). Neglect the electron/positron mass in this calculation.</span>\n",
+    "</div>"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "2c6b3759",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  },
+  {
+   "cell_type": "markdown",
+   "id": "809505cb",
+   "metadata": {},
+   "source": [
+    "<div class=\"alert alert-info\">\n",
+    "<strong>Exercise:</strong> \n",
+    "Is helicity \"conserved\" in high-energy interactions in this theory? Explain how this differs from QED.</span>\n",
+    "</div>"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "5ab2e03e",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  },
+  {
+   "cell_type": "markdown",
+   "id": "9941f5ee",
+   "metadata": {},
+   "source": [
+    "<div class=\"alert alert-info\">\n",
+    "<strong>Exercise:</strong> \n",
+    "Do the calculation again, but this time include the electron/positron mass.</span>\n",
+    "</div>"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "0fb4e658",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  },
+  {
+   "cell_type": "markdown",
+   "id": "49e604a5",
+   "metadata": {},
+   "source": [
+    "<div class=\"alert alert-info\">\n",
+    "<strong>Exercise:</strong> \n",
+    "If $m_{\\gamma} = 3$~GeV, calculate the lifetime of this \"photon\" in seconds.</span>\n",
+    "</div>"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "04764778",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  },
+  {
+   "cell_type": "markdown",
+   "id": "1cab1d98",
+   "metadata": {},
+   "source": [
+    "<a name='section_1_2'></a>\n",
+    "<hr style=\"height: 1px;\">\n",
+    "\n",
+    "\n",
+    "## <h1 style=\"border:1px; border-style:solid; padding: 0.25em; color: #FFFFFF; background-color: #FFA500\">Section 2: Muon decay</h1>"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "ce5dd64c",
+   "metadata": {},
+   "source": [
+    "Let's consider the muon decay $\\mu^- \\to \\nu_\\mu \\bar{\\nu_e} e^-$, which is described by the S=standard model's electroweak interaction lagrangian."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "d68a4fdf",
+   "metadata": {},
+   "source": [
+    "<div class=\"alert alert-info\">\n",
+    "<strong>Exercise:</strong> \n",
+    "Draw the first-order Feynman diagram describing muon decay; consider $m_W \\gg m_\\mu$ and derive the effective lagrangian (also called the Fermi lagrangian), showing the relationship between the electroweak coupling constant $g_W$ and the Fermi constant $G_F$.</span>\n",
+    "</div>"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "88173131",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  },
+  {
+   "cell_type": "markdown",
+   "id": "fcc1c2ae",
+   "metadata": {},
+   "source": [
+    "<div class=\"alert alert-info\">\n",
+    "<strong>Exercise:</strong> \n",
+    "Using the Fermi lagrangian, calculate the decay rate of the muon and its lifetime.</span>\n",
+    "</div>"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "6c67f8f0",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  },
+  {
+   "cell_type": "markdown",
+   "id": "cbc7ad7c",
+   "metadata": {},
+   "source": [
+    "<div class=\"alert alert-info\">\n",
+    "<strong>Exercise:</strong> \n",
+    "Using the derived formula for muon decay, calculate the lifetime of the $\\tau$ lepton and compare it with the measured value (check the most recent value on PDG). Are there any differences between the calculated and the measured value? Why are there differences?</span>\n",
+    "</div>"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "6dd70496",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
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+   "name": "python3"
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+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.8.8"
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