r4lv/Sheet 3 - Class Demo (empty).ipynb Secret

## requirements.txt
phasespace==1.1.1
numpy==1.18.4
matplotlib==3.2.1
scipy==1.4.1

## Sheet 3 - Class Demo (empty).ipynb
{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Tutorial 3: Toy simulation of $e^{+} e^{-} \\rightarrow \\gamma A'$ using phase space\n",
    "\n",
    "The idea of the notebook will be to study a basic phase space simulaton of the process  $e^{+}e^{-} \\rightarrow \\gamma A'$. Here we will assume that the electron travels along the $z$ axis with $E_{e^{-}} = 6.3\\,\\text{GeV}$ and the positron travels in the exact opposite direction with  $E_{e^{+}} = 0.3$ GeV. You may neglect the mass of electrons and take $\\sqrt{s} = \\sqrt{7.56}$  and $\\beta_{e^{+} e^{-}} = 0.9091$.\n",
    "\n",
    "\n",
    "We will generate the decay in the centre-of-mass (CM) frame and compute the momentums and energies of the photon and the $A'$. We will next study the angular distribution of the $A'$ in the CM frame. Finally we will compute the probability that the particle decays in some hypothetical detector volume. The detector is placed $R=5\\,\\text{m}$ in the the $z$ direction from the collision point such that in spherical coordinates it covers the ranges $5 < r < 45\\,\\text{m}$, $0 < \\theta < 15^{\\circ}$ and $0 < \\phi < 360^{\\circ}$. Below you can see the detector. An inner hermetic detector is tasked at reconstructing the $\\gamma$ and cases in which the $A'$ is short lived. Meanwhile, the long baseline detector gives sensitivity to the scenario in which the $A'$ has a longer lifetime."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<div style=\"text-align: center\">\n",
    "    <img alt=\"\" style=\"width:100%; max-width: 1000px;\" 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/>\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 1. Centre-of-mass frame"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In order to simulate the decay we will use the `phasespace` package in python. Given we assume the CM frame, we set a starting the mass to $\\sqrt{s}$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "import phasespace\n",
    "import numpy as np\n",
    "\n",
    "E1 = 6.3\n",
    "E2 = 0.3\n",
    "beta = (E1-E2)/(E1+E2)\n",
    "gamma = 1/np.sqrt(1-beta**2)\n",
    "s = (E1+E2)**2 - (E1-E2)**2\n",
    "Ap_Mass = 2\n",
    "photon_Mass = 0\n",
    "\n",
    "# here we generate two particles A' with mass Ap_Mass\n",
    "# and a photon with mass = 0 \n",
    "\n",
    "# We generate 1,000,000 events\n",
    "Ngen = 1_000_000\n",
    "weights, particles = phasespace.nbody_decay(\n",
    "    np.sqrt(s),\n",
    "    [Ap_Mass, photon_Mass]\n",
    ").generate(n_events=Ngen)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's inspect what we have made. We are returned a dictionary with keys `p_0` for the $A'$ and `p_1` for the $\\gamma$. In the dictionary these keys map to tensorflow tensors, however we will simply convert these to numpy arrays:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{'p_0': <tf.Tensor: shape=(1000000, 4), dtype=float64, numpy=\n",
       " array([[-0.20166105,  0.44953769, -0.41993954,  2.10216568],\n",
       "        [-0.28557604, -0.04906947, -0.57891195,  2.10216568],\n",
       "        [-0.36529661, -0.50665934,  0.17016235,  2.10216568],\n",
       "        ...,\n",
       "        [-0.20418927, -0.35987075,  0.49789588,  2.10216568],\n",
       "        [-0.61410058, -0.17815279,  0.10120567,  2.10216568],\n",
       "        [-0.23312916, -0.59497121,  0.10373321,  2.10216568]])>,\n",
       " 'p_1': <tf.Tensor: shape=(1000000, 4), dtype=float64, numpy=\n",
       " array([[ 0.20166105, -0.44953769,  0.41993954,  0.64737974],\n",
       "        [ 0.28557604,  0.04906947,  0.57891195,  0.64737974],\n",
       "        [ 0.36529661,  0.50665934, -0.17016235,  0.64737974],\n",
       "        ...,\n",
       "        [ 0.20418927,  0.35987075, -0.49789588,  0.64737974],\n",
       "        [ 0.61410058,  0.17815279, -0.10120567,  0.64737974],\n",
       "        [ 0.23312916,  0.59497121, -0.10373321,  0.64737974]])>}"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "particles"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "# here we get the 4-momentum of the A' and the gamma as numpy arrays\n",
    "p_Ap = particles['p_0'].numpy()\n",
    "p_gamma = particles['p_1'].numpy()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here we have converted the tensors to numpy arrays that contain the 4-momenta of all generate particles lets inspect a bit further. It is very useful to examine the shape of our data first (e.g p_Ap.shape)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(1000000, 4)"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p_Ap.shape"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We see the shape is (100000,4), in fact we have a matrix where the rows represent the particles generated (Ngen= 100000) and the columns are actually the 4-momentum components px, py, pz, E. Hence why the shape is (100000,4). Now lets have a look at the 4-momenta for $A'$ and $\\gamma$ for our generate events"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[-0.20166105,  0.44953769, -0.41993954,  2.10216568],\n",
       "       [-0.28557604, -0.04906947, -0.57891195,  2.10216568],\n",
       "       [-0.36529661, -0.50665934,  0.17016235,  2.10216568],\n",
       "       ...,\n",
       "       [-0.20418927, -0.35987075,  0.49789588,  2.10216568],\n",
       "       [-0.61410058, -0.17815279,  0.10120567,  2.10216568],\n",
       "       [-0.23312916, -0.59497121,  0.10373321,  2.10216568]])"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p_Ap"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 0.20166105, -0.44953769,  0.41993954,  0.64737974],\n",
       "       [ 0.28557604,  0.04906947,  0.57891195,  0.64737974],\n",
       "       [ 0.36529661,  0.50665934, -0.17016235,  0.64737974],\n",
       "       ...,\n",
       "       [ 0.20418927,  0.35987075, -0.49789588,  0.64737974],\n",
       "       [ 0.61410058,  0.17815279, -0.10120567,  0.64737974],\n",
       "       [ 0.23312916,  0.59497121, -0.10373321,  0.64737974]])"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p_gamma"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### &rarr; Why is the 4th component, Energy, always constant for our $\\gamma$ and $A'$?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here we can use numpy slicing/indexing syntax to retrieve all generate events but either the momentum of energy components"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [],
   "source": [
    "# 3 momentum\n",
    "pvec_Ap = p_Ap[: , :3]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(1000000, 3)"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pvec_Ap.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[-0.20166105,  0.44953769, -0.41993954],\n",
       "       [-0.28557604, -0.04906947, -0.57891195],\n",
       "       [-0.36529661, -0.50665934,  0.17016235],\n",
       "       ...,\n",
       "       [-0.20418927, -0.35987075,  0.49789588],\n",
       "       [-0.61410058, -0.17815279,  0.10120567],\n",
       "       [-0.23312916, -0.59497121,  0.10373321]])"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pvec_Ap"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [],
   "source": [
    "E_Ap = p_Ap[: , 2]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([2.10216568, 2.10216568, 2.10216568, ..., 2.10216568, 2.10216568,\n",
       "       2.10216568])"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "E_Ap"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### &rarr; How would you compute the momentum magnitude?\n",
    "\n",
    "*Hint:* you will need to square the momentum components which can be done very simply with one operation.\n",
    "\n",
    "Then you will need to do to use np.sum(v,axis=1) (a sum over the 3 momentum coordinates)\n",
    "\n",
    "Finally, you can apply np.sqrt()\n",
    "\n",
    "Your final result should be constant for each generated particle and the shape is (100000,1). What is your momentum magnitude?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Compute your momentum magnitude here\n",
    "P_Ap = ..."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<br><br><br><br><br><br><br><br><br><br><br><br><br><br>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.64737974, 0.64737974, 0.64737974, ..., 0.64737974, 0.64737974,\n",
       "       0.64737974])"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P_Ap"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### &rarr; Compute the invariant mass using E_Ap and P_Ap"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Compute here the invariant mass using E_Ap and P_Ap\n",
    "M_Ap = ..."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<br><br><br><br><br><br><br><br><br><br><br><br><br><br>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([2., 2., 2., ..., 2., 2., 2.])"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M_Ap"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [],
   "source": [
    "px_Ap = p_Ap[:,0]\n",
    "py_Ap = p_Ap[:,1]\n",
    "pz_Ap = p_Ap[:,2]\n",
    "E_Ap = p_Ap[:,3]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### &rarr; Compute phi using  np.arctan() of the relevant momenta\n",
    "\n",
    "### &rarr; Compute theta using np.arctan"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Compute phi using  np.arctan() of the relevant momenta\n",
    "phi_Ap = ..."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Compute theta using np.arctan\n",
    "theta_Ap = ..."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<br><br><br><br><br><br><br><br><br><br><br><br><br><br>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-0.84278099,  0.453094  , -0.34763959, ..., -0.13313967,\n",
       "        0.44634645,  1.17749926])"
      ]
     },
     "execution_count": 59,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "phi_Ap"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-1.41185037, -1.28066406,  1.46044395, ...,  1.00415133,\n",
       "       -0.5524766 ,  1.55378453])"
      ]
     },
     "execution_count": 60,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "theta_Ap"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [],
   "source": [
    "# here we do a simple transform for theta so that we are in the interval 0 -> pi\n",
    "theta_Ap[theta_Ap < 0 ] =  theta_Ap[theta_Ap < 0 ] + np.pi"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [],
   "source": [
    "vals = np.linspace(0,np.pi,100)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's now plot $\\theta$ and $\\phi$ in the CM frame. Given that the decay happens according to phase space e.g isoptopic in $\\theta$ and $\\phi$  you should see $\\sin \\theta$ dependence and a uniform $\\phi$ dependence"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAYMAAAEKCAYAAADw2zkCAAAABHNCSVQICAgIfAhkiAAAAAlwSFlzAAALEgAACxIB0t1+/AAAADh0RVh0U29mdHdhcmUAbWF0cGxvdGxpYiB2ZXJzaW9uMy4xLjIsIGh0dHA6Ly9tYXRwbG90bGliLm9yZy8li6FKAAAU60lEQVR4nO3de6yc9Z3f8fenBrJRLosJLvLaztpJvK1M1BpyRKg2G6XJFgzbrkmbsmarxU3RelcxUtJN1TWbqtAkSKFViBaVkBJhxaySGJqLsLJkHS+LGkUqlwNxMIYQTrgIWwY7mEuibMma/faP+R1lOJz7ZWYO5/2SRvPMd55n5jvPuXzOc/udVBWSpKXtH/S7AUlS/xkGkiTDQJJkGEiSMAwkSRgGkiTgpH43MFunn356rV27tt9tSNKict999/2kqlaMrS/aMFi7di3Dw8P9bkOSFpUkT45XdzeRJMkwkCQZBpIkDANJEoaBJAnDQJKEYSBJwjCQJLGILzqTBtXaHX85o/mf+MzvLFAn0vQZBtIszfSXvjTIDAOpGbRf7hP145aEFoJhIPXZoIWQliYPIEuS3DLQ0uNf4tKrTbllkORXktyT5AdJDib5b62+LsndSUaS3JLklFZ/XXs80p5f2/VaV7T6I0nO76pvarWRJDvm/2NKkiYznS2Dl4D3V9XPkpwMfC/Jt4E/AT5XVbuTfAG4DLih3T9XVe9IsgW4Bvi9JBuALcCZwK8Bf53kN9p7XA/8C+AQcG+SPVX10Dx+Tuk1Y7ItGw8ua7amDIOqKuBn7eHJ7VbA+4Hfb/VdwFV0wmBzmwb4GvA/k6TVd1fVS8DjSUaAc9p8I1X1GECS3W1ew0Bz4u4gafqmdQA5ybIk+4GjwD7gx8DzVXWizXIIWNWmVwFPAbTnXwDe0l0fs8xE9fH62JZkOMnwsWPHptO6JGkaphUGVfVyVW0EVtP5a/4fL2hXE/dxY1UNVdXQihWv+heekqRZmtGppVX1PHAn8M+AU5OM7mZaDRxu04eBNQDt+V8Fnu2uj1lmorokqUemPGaQZAXwd1X1fJLX0znQew2dUPgQsBvYCtzWFtnTHv/f9vzfVFUl2QN8Jcm1dA4grwfuAQKsT7KOTghs4ZfHIiTNgFcta7amczbRSmBXkmV0tiRurapvJXkI2J3k08D3gZva/DcBf9EOEB+n88udqjqY5FY6B4ZPANur6mWAJJcDe4FlwM6qOjhvn1CveR4oluZuOmcTPQCcNU79MX55NlB3/f8B/3aC17oauHqc+u3A7dPoV5K0AByOQpJkGEiSHJtIi4THBebGA8uailsGkiTDQJJkGEiSMAwkSXgAWQPGA8W95YFljXLLQJJkGEiSDANJEoaBJAkPIEsahweWlx63DCRJbhmoPzyFVBosbhlIkgwDSZJhIEnCMJAkYRhIkvBsIkkz4PUHr11uGUiSDANJkruJtMC8uExaHKbcMkiyJsmdSR5KcjDJR1v9qiSHk+xvtwu7lrkiyUiSR5Kc31Xf1GojSXZ01dclubvVb0lyynx/UEnSxKazm+gE8PGq2gCcC2xPsqE997mq2thutwO057YAZwKbgM8nWZZkGXA9cAGwAbik63Wuaa/1DuA54LJ5+nySpGmYMgyq6khV3d+mfwo8DKyaZJHNwO6qeqmqHgdGgHPabaSqHquqXwC7gc1JArwf+Fpbfhdw0Ww/kCRp5mZ0ADnJWuAs4O5WujzJA0l2JlneaquAp7oWO9RqE9XfAjxfVSfG1Md7/21JhpMMHzt2bCatS5ImMe0wSPJG4OvAx6rqReAG4O3ARuAI8NkF6bBLVd1YVUNVNbRixYqFfjtJWjKmdTZRkpPpBMGXq+obAFX1TNfzXwS+1R4eBtZ0Lb661Zig/ixwapKT2tZB9/ySFgEvRlv8pgyDtk//JuDhqrq2q76yqo60hx8EHmzTe4CvJLkW+DVgPXAPEGB9knV0ftlvAX6/qirJncCH6BxH2ArcNh8fTr3jKaTS4jadLYPfBP4AOJBkf6v9GZ2zgTYCBTwB/BFAVR1McivwEJ0zkbZX1csASS4H9gLLgJ1VdbC93p8Cu5N8Gvg+nfCRJPXIlGFQVd+j81f9WLdPsszVwNXj1G8fb7mqeozO2UaSpD5wOApJkmEgSTIMJEkYBpIkHLVUM+QppJoJrz9YPNwykCQZBpIkw0CShGEgScIwkCRhGEiSMAwkSRgGkiS86EwT8OIyLSQvRhs8bhlIkgwDSZJhIEnCMJAkYRhIkjAMJEkYBpIkDANJEl50tuR5cZkGiRej9Y9bBpKkqcMgyZokdyZ5KMnBJB9t9dOS7EvyaLtf3upJcl2SkSQPJDm767W2tvkfTbK1q/6uJAfaMtclyUJ8WEnS+KazZXAC+HhVbQDOBbYn2QDsAO6oqvXAHe0xwAXA+nbbBtwAnfAArgTeDZwDXDkaIG2eP+xabtPcP5okabqmDIOqOlJV97fpnwIPA6uAzcCuNtsu4KI2vRm4uTruAk5NshI4H9hXVcer6jlgH7CpPffmqrqrqgq4ueu1JEk9MKNjBknWAmcBdwNnVNWR9tTTwBltehXwVNdih1ptsvqhcerjvf+2JMNJho8dOzaT1iVJk5h2GCR5I/B14GNV9WL3c+0v+prn3l6lqm6sqqGqGlqxYsVCv50kLRnTCoMkJ9MJgi9X1Tda+Zm2i4d2f7TVDwNruhZf3WqT1VePU5ck9ch0ziYKcBPwcFVd2/XUHmD0jKCtwG1d9UvbWUXnAi+03Ul7gfOSLG8Hjs8D9rbnXkxybnuvS7teS5LUA9O56Ow3gT8ADiTZ32p/BnwGuDXJZcCTwMXtuduBC4ER4OfAhwGq6niSTwH3tvk+WVXH2/RHgC8Brwe+3W6SBHgxWi9MGQZV9T1govP+PzDO/AVsn+C1dgI7x6kPA++cqhdJ0sJwOIolwmEnJE3G4SgkSYaBJMkwkCRhGEiSMAwkSRgGkiQMA0kShoEkCS86e83x4jJJs2EYSFq0HLNo/ribSJJkGEiSDANJEoaBJAnDQJKEYSBJwjCQJGEYSJIwDCRJeAXyouWwE9LEJvv58Ork8bllIEkyDCRJhoEkiWmEQZKdSY4mebCrdlWSw0n2t9uFXc9dkWQkySNJzu+qb2q1kSQ7uurrktzd6rckOWU+P6AkaWrT2TL4ErBpnPrnqmpju90OkGQDsAU4sy3z+STLkiwDrgcuADYAl7R5Aa5pr/UO4Dngsrl8IEnSzE0ZBlX1XeD4NF9vM7C7ql6qqseBEeCcdhupqseq6hfAbmBzkgDvB77Wlt8FXDTDzyBJmqO5HDO4PMkDbTfS8lZbBTzVNc+hVpuo/hbg+ao6MaY+riTbkgwnGT527NgcWpckdZttGNwAvB3YCBwBPjtvHU2iqm6sqqGqGlqxYkUv3lKSloRZXXRWVc+MTif5IvCt9vAwsKZr1tWtxgT1Z4FTk5zUtg6655ck9cistgySrOx6+EFg9EyjPcCWJK9Lsg5YD9wD3Ausb2cOnULnIPOeqirgTuBDbfmtwG2z6UmSNHtTbhkk+SrwPuD0JIeAK4H3JdkIFPAE8EcAVXUwya3AQ8AJYHtVvdxe53JgL7AM2FlVB9tb/CmwO8mnge8DN83bp3sNcNgJSb2Qzh/ni8/Q0FANDw/3u40FZxhIvbFUxixKcl9VDY2tewWyJMkwkCQZBpIkDANJEoaBJAnDQJKEYSBJwjCQJGEYSJIwDCRJGAaSJGY5hLXml+MPSf030c/hUhmzyC0DSZJhIEkyDCRJGAaSJAwDSRKGgSQJw0CShGEgScIwkCRhGEiScDiKnnLYCUmDyjCQpEkslTGLptxNlGRnkqNJHuyqnZZkX5JH2/3yVk+S65KMJHkgydldy2xt8z+aZGtX/V1JDrRlrkuS+f6QkqTJTeeYwZeATWNqO4A7qmo9cEd7DHABsL7dtgE3QCc8gCuBdwPnAFeOBkib5w+7lhv7XpKkBTZlGFTVd4HjY8qbgV1tehdwUVf95uq4Czg1yUrgfGBfVR2vqueAfcCm9tybq+quqirg5q7XkiT1yGzPJjqjqo606aeBM9r0KuCprvkOtdpk9UPj1MeVZFuS4STDx44dm2XrkqSx5nxqafuLvuahl+m8141VNVRVQytWrOjFW0rSkjDbMHim7eKh3R9t9cPAmq75VrfaZPXV49QlST002zDYA4yeEbQVuK2rfmk7q+hc4IW2O2kvcF6S5e3A8XnA3vbci0nObWcRXdr1WpKkHpnyOoMkXwXeB5ye5BCds4I+A9ya5DLgSeDiNvvtwIXACPBz4MMAVXU8yaeAe9t8n6yq0YPSH6FzxtLrgW+3mySph6YMg6q6ZIKnPjDOvAVsn+B1dgI7x6kPA++cqg9J0sJxbCJJkmEgSXJsogXhgHTSa99rbcwitwwkSYaBJMkwkCRhGEiSMAwkSRgGkiQMA0kShoEkCcNAkoRhIEnCMJAk4dhEkjSvFuuYRYbBHDggnaTXCncTSZIMA0mSYSBJwjCQJGEYSJIwDCRJGAaSJAwDSRJzDIMkTyQ5kGR/kuFWOy3JviSPtvvlrZ4k1yUZSfJAkrO7Xmdrm//RJFvn9pEkSTM1H1sG/7yqNlbVUHu8A7ijqtYDd7THABcA69ttG3ADdMIDuBJ4N3AOcOVogEiSemMhdhNtBna16V3ARV31m6vjLuDUJCuB84F9VXW8qp4D9gGbFqAvSdIE5jo2UQHfSVLA/6qqG4EzqupIe/5p4Iw2vQp4qmvZQ602Uf1Vkmyjs1XBW9/61jm2Pn2OQSTptW6uYfCeqjqc5B8C+5L8sPvJqqoWFPOihc2NAENDQ/P2upK00AZ9NNM57SaqqsPt/ijwTTr7/J9pu39o90fb7IeBNV2Lr261ieqSpB6ZdRgkeUOSN41OA+cBDwJ7gNEzgrYCt7XpPcCl7ayic4EX2u6kvcB5SZa3A8fntZokqUfmspvoDOCbSUZf5ytV9VdJ7gVuTXIZ8CRwcZv/duBCYAT4OfBhgKo6nuRTwL1tvk9W1fE59CVJmqFZh0FVPQb803HqzwIfGKdewPYJXmsnsHO2vUiS5sYrkCVJhoEkyTCQJGEYSJIwDCRJGAaSJOY+HIUkaQ4GZZgKw6CLA9JJWqrcTSRJMgwkSYaBJAnDQJKEYSBJwjCQJGEYSJIwDCRJGAaSJAwDSRKGgSQJw0CSxBIdqM4B6SQNul6PZuqWgSTJMJAkGQaSJAYoDJJsSvJIkpEkO/rdjyQtJQMRBkmWAdcDFwAbgEuSbOhvV5K0dAxEGADnACNV9VhV/QLYDWzuc0+StGQMyqmlq4Cnuh4fAt49dqYk24Bt7eHPkjwyy/c7HfjJLJftJfucf4ulV/ucf4ul10n7zDVzfv1fH684KGEwLVV1I3DjXF8nyXBVDc1DSwvKPuffYunVPuffYum1X30Oym6iw8CarserW02S1AODEgb3AuuTrEtyCrAF2NPnniRpyRiI3URVdSLJ5cBeYBmws6oOLuBbznlXU4/Y5/xbLL3a5/xbLL32pc9UVT/eV5I0QAZlN5EkqY8MA0nS0gqDQR3yIsmaJHcmeSjJwSQfbfWrkhxOsr/dLux3rwBJnkhyoPU03GqnJdmX5NF2v7zPPf6jrvW2P8mLST42KOs0yc4kR5M82FUbdx2m47r2fftAkrP73Of/SPLD1ss3k5za6muT/G3Xuv1Cn/uc8Gud5Iq2Ph9Jcn6v+pyk11u6+nwiyf5W7906raolcaNzYPrHwNuAU4AfABv63VfrbSVwdpt+E/AjOsNyXAX8p373N06/TwCnj6n9d2BHm94BXNPvPsd87Z+mc7HNQKxT4L3A2cCDU61D4ELg20CAc4G7+9znecBJbfqarj7Xds83AOtz3K91+9n6AfA6YF37vbCsn72Oef6zwH/t9TpdSlsGAzvkRVUdqar72/RPgYfpXJW9mGwGdrXpXcBFfexlrA8AP66qJ/vdyKiq+i5wfEx5onW4Gbi5Ou4CTk2ysl99VtV3qupEe3gXneuC+mqC9TmRzcDuqnqpqh4HRuj8fuiJyXpNEuBi4Ku96mfUUgqD8Ya8GLhfuEnWAmcBd7fS5W1zfGe/d710KeA7Se5rQ4QAnFFVR9r008AZ/WltXFt45Q/XIK5TmHgdDvL37n+gs9Uyal2S7yf5P0l+q19NdRnvaz3I6/O3gGeq6tGuWk/W6VIKg4GX5I3A14GPVdWLwA3A24GNwBE6m4+D4D1VdTadUWa3J3lv95PV2b4diHOW20WMvwv871Ya1HX6CoO0DieS5BPACeDLrXQEeGtVnQX8CfCVJG/uV38skq/1GJfwyj9cerZOl1IYDPSQF0lOphMEX66qbwBU1TNV9XJV/T3wRXq4KTuZqjrc7o8C36TT1zOjuy7a/dH+dfgKFwD3V9UzMLjrtJloHQ7c926Sfw/8S+DfteCi7XZ5tk3fR2df/G/0q8dJvtYDtz4BkpwE/GvgltFaL9fpUgqDgR3you0nvAl4uKqu7ap37xf+IPDg2GV7LckbkrxpdJrOwcQH6azLrW22rcBt/enwVV7xl9YgrtMuE63DPcCl7ayic4EXunYn9VySTcB/Bn63qn7eVV+Rzv8mIcnbgPXAY/3pctKv9R5gS5LXJVlHp897et3fOH4b+GFVHRot9HSd9uoI+iDc6JyV8SM66fqJfvfT1dd76OwSeADY324XAn8BHGj1PcDKAej1bXTOxPgBcHB0PQJvAe4AHgX+GjhtAHp9A/As8KtdtYFYp3QC6gjwd3T2WV820TqkcxbR9e379gAw1Oc+R+jscx/9Xv1Cm/fftO+J/cD9wL/qc58Tfq2BT7T1+QhwQb+/9q3+JeCPx8zbs3XqcBSSpCW1m0iSNAHDQJJkGEiSDANJEoaBJAnDQJKEYSDNSpJlSf48nSHHD7QLgqRFyzCQZucK4LGqOhO4DvhIn/uR5uSkfjcgLTZtGI4PVtW7Wulx4Hf62JI0Z4aBNHO/DawZ/W9UwGl0ho+QFi13E0kzt5HOf6LaWFUbge/QGTtGWrTcMpBmbjmdXUOjww6fB1yd5NeBT9MZevo5OiNM/i2dQclOBt4JXFyd/7QnDRS3DKSZ+xGd/0UM8B+Bv6zOv0/cDnyyqj5OZzTKvVX1x8B7q+q/0Bkm+cx+NCxNxTCQZu6rwNlJRoB/Quc/UEFnqOm/75rvxXZ/rN3/gs4/YZcGjruJpBmqquf45ZZBt88DVyU5QicIftLTxqQ58P8ZSJLcTSRJMgwkSRgGkiQMA0kShoEkCcNAkoRhIEnCMJAkYRhIkoD/D1Ya9MQe5C8+AAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# here you can plot your theta\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "# here we convert radians to degrees\n",
    "plt.hist(theta_Ap*180/np.pi, bins=50)\n",
    "plt.xlabel(r'$\\theta_{\\rm cm}$');\n",
    "#plt.savefig('theta.pdf')\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Here you can plot your phi\n",
    "import matplotlib.pyplot as plt\n",
    "plt.hist(phi_Ap*180/np.pi, bins=50)\n",
    "plt.xlabel(r'$\\phi_{\\rm cm}$');\n",
    "#plt.savefig('phi.pdf')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 2. $e^{+} e^{-} \\rightarrow \\gamma A'$  in the Lab frame \n",
    "\n",
    "So far we have been studying our phase space simulation in the CM frame. Let us now boost into the CM frame.\n",
    "\n",
    "Note that we have a boost of $\\beta_{e^{+} e^{-}} = -0.9091$ in the $z$ direction. Also note the equations for how the momentum components change under a boost in the $z$ direction:\n",
    "\\begin{align}\n",
    " p^{\\rm lab}_{x} &= p^{\\rm cm}_{x} \\\\\n",
    " p^{\\rm lab}_{y} &= p^{\\rm cm}_{y} \\\\\n",
    " p^{\\rm lab}_{z} &= \\gamma( p^{\\rm cm}_{z} - \\beta E^{\\rm cm}) \\\\\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [],
   "source": [
    "import phasespace\n",
    "import numpy as np\n",
    "\n",
    "E1 = 6.3\n",
    "E2 = 0.3\n",
    "beta= -(E1-E2)/(E1+E2)\n",
    "gamma = 1/np.sqrt(1-beta**2)\n",
    "s = (E1+E2)**2 - (E1-E2)**2\n",
    "Ap_Mass = 1.2\n",
    "photon_Mass = 0\n",
    "\n",
    "# here we generate two particles A' with mass Ap_Mass\n",
    "# and a photon with mass = 0 \n",
    "\n",
    "# We generate 1,000,000 events\n",
    "Ngen = 1_000_000\n",
    "weights, particles = phasespace.nbody_decay(np.sqrt(s),\n",
    "                                            [Ap_Mass, photon_Mass]).generate(n_events=Ngen)\n",
    "\n",
    "# here we get the 4-momentum of the A' and the gamma as numpy arrays\n",
    "p_Ap = particles['p_0'].numpy()\n",
    "p_gamma = particles['p_1'].numpy()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Given that we have Beta and hence gamma compute p^lab_z\n",
    "px_Ap = p_Ap[:,0]\n",
    "py_Ap = p_Ap[:,1]\n",
    "pz_Ap = p_Ap[:,2]\n",
    "E_Ap = p_Ap[:,3]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Now boost into the lab frame\n",
    "pz_Ap_lab = gamma * (pz_Ap - beta *E_Ap)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Now compute momentum again \n",
    "P_Ap = np.sqrt(px_Ap**2 + py_Ap**2 + pz_Ap_lab**2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Is the momentum still constant?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "plt.hist(P_Ap, bins=50)\n",
    "plt.xlabel(r'$P_{A}$');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's also inspect the angular distribution in $\\theta_{\\rm lab}$. Why would $\\phi$ on the other hand be unchanged?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [],
   "source": [
    "theta = np.arctan(np.sqrt(px_Ap**2+py_Ap**2)/pz_Ap_lab)\n",
    "theta[theta < 0 ] =  theta[theta < 0 ] + np.pi"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "plt.hist(theta*180/np.pi, bins=50)\n",
    "plt.xlabel(r'$\\theta_{\\rm lab}$');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### &rarr; What is the probability that the particle has a $\\theta_{\\rm}$ within the long baseline detectors acceptance ($0 < \\theta_{\\rm lab} < 15^{\\circ}$)?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "probab = ..."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<br><br><br><br><br><br><br><br><br><br><br><br><br><br>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.419725"
      ]
     },
     "execution_count": 62,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "probab"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now repeat this for a different mass hypthesis of the $A'$. Try to repeat this calculation for $M_{A'} = 1.4$ and $M_{A'} = 1.2$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<br><br><br><br><br><br><br><br><br><br><br><br><br><br>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 3. Simulating the decay flight of $A'$\n",
    "\n",
    "In this third part we will look at simulating the decay flight of the $A'$ to understand if the $A'$ would decay in the detector volume. Let's take the case where $M_{A'} = 1.2$ GeV as you will also perform this calculation for this mass analytically in the homework"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {},
   "outputs": [],
   "source": [
    "import phasespace\n",
    "import numpy as np\n",
    "\n",
    "E1 = 6.3\n",
    "E2 = 0.3\n",
    "beta= -(E1-E2)/(E1+E2)\n",
    "gamma = 1/np.sqrt(1-beta**2)\n",
    "s= (E1+E2)**2 - (E1-E2)**2\n",
    "Ap_Mass = 1.2\n",
    "photon_Mass = 0\n",
    "\n",
    "# here we generate two particles A' with mass Ap_Mass\n",
    "# and a photon with mass = 0 \n",
    "\n",
    "# We generate 1,000,000 events\n",
    "Ngen = 1_000_000\n",
    "weights, particles = phasespace.nbody_decay(np.sqrt(s),\n",
    "                                            [Ap_Mass, photon_Mass]).generate(n_events=Ngen)\n",
    "\n",
    "\n",
    "# here we get the 4-momentum of the A' and the gamma as numpy arrays\n",
    "p_Ap = particles['p_0'].numpy()\n",
    "p_gamma = particles['p_1'].numpy()\n",
    "\n",
    "# Given that we have Beta and hence gamma compute p^lab_\n",
    "px_Ap = p_Ap[:,0]\n",
    "py_Ap = p_Ap[:,1]\n",
    "pz_Ap = p_Ap[:,2]\n",
    "E_Ap = p_Ap[:,3]\n",
    "\n",
    "# Now boost into the lab frame\n",
    "pz_Ap_lab = gamma * (pz_Ap - beta*E_Ap)\n",
    "\n",
    "#Now compute momentum again \n",
    "P_Ap_lab = np.sqrt(px_Ap**2 + py_Ap**2 + pz_Ap_lab**2)\n",
    "\n",
    "# energy is \n",
    "E_Ap_lab = np.sqrt(Ap_Mass**2 + P_Ap_lab**2)\n",
    "\n",
    "# theta lab\n",
    "theta_Ap_lab = np.arctan(np.sqrt(px_Ap**2+py_Ap**2)/pz_Ap_lab)\n",
    "theta_Ap_lab[theta_Ap_lab < 0 ] =  theta_Ap_lab[theta_Ap_lab < 0 ] + np.pi"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Given the angular acceptance we will select momentum values where theta  iswithin the detecor acceptance\n",
    "Psel = P_Ap_lab[theta_Ap_lab*180/np.pi < 15]\n",
    "Esel = np.sqrt(Ap_Mass**2 + Psel**2)\n",
    "# we compute the correspoding beta and gamma\n",
    "beta = Psel / Esel\n",
    "gamma = 1/np.sqrt(1-beta**2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.418797"
      ]
     },
     "execution_count": 65,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# We see the probability of this angular acceptance from before\n",
    "len(Psel) / len(P_Ap_lab)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "5.0931424906417035\n"
     ]
    },
    {
     "data": {
      "image/png": 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JEoaBJAnDQBpYkg8nea49cvmpJH/UdU/SifJ3BtKAkvwT8EhVfS7JeUw9JO8tXfclnQjPDKTBvYP/f/TFJLCkw16kWTEMpMG9HXi8PQPqTwAfg6GTjmEgDSDJauA0YBdwH7AM2Nwz/8tJ/ryj9qQZ80F10mDeDuypqlc9LjvJBqbOEn5r5F1JJ8gzA2kw7wC+f3QxyeuB36+qfwXeNPKupBNkGEiDeTvwUJ/6XwCnJfkc8Nb2PxakecthImkAVfUHR9eSnAmMV9UV7fNWps4gFuL/ZNYC4e8MJEkOE0mSDANJEoaBJAnDQJKEYSBJwjCQJGEYSJIwDCRJGAaSJOB/AWkrDRTBnuegAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Lets again plot P_Ap_lab\n",
    "import matplotlib.pyplot as plt\n",
    "plt.hist(Psel, bins=50)\n",
    "plt.xlabel(r'$P_{A}$');\n",
    "\n",
    "# We can also you compute the average momentum\n",
    "print(np.mean(Psel))\n",
    "# In the analytical computation of the homework this value will be used of roughly 5.1 GeV/c"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In the homework you will derive the following formula for the the probability the decay happens between to values of $r$:\n",
    "\\begin{equation}\n",
    " P(r_{1} < r_{\\rm decay} < r_{2}) = e^{-r_{1} / (\\beta \\gamma c\\tau )} - e^{-r_{2} / (\\beta \\gamma c\\tau )} \n",
    "\\end{equation}\n",
    "\n",
    "where here $\\tau$ is the lifetime. \n",
    "\n",
    "With this and the accept reject method we can run a small simulation to find the probability that decays end up within the detector volumen $5<r<45$ (remember we have already selection the $P$ values in the required angular range.\n",
    "\n",
    "We will actually do this for the value of $c \\tau$ that roughly maximises the probability that the decay hapens in the detector volume this can be found by differentiating the expression above and is a question in the homework. The value will be roughly $c\\tau = 4.29\\,\\text{m}$. This is approximate as the homework will assume the average $P^{\\rm lab}_{A'}$ value.\n",
    "\n",
    "To simulate the decays we compute the probability, $ P(5< r_{\\rm decay} < 45)$, for each generated $A'$ decays in the $5<r<45$ based on the $\\beta$ and $\\gamma$ values and the $r$ interval. Now we generate a random uniform number, $u$, where $0 < u < 1$. When $ u <  P(5< r_{\\rm decay} < 45)$ we accept the event."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.6572945842496484\n"
     ]
    }
   ],
   "source": [
    "ctau = 4.283\n",
    "r1 = 5\n",
    "r2 = 45\n",
    "prob_decay = np.exp(-r1/(beta*gamma*ctau))  - np.exp(-r2/(beta*gamma*ctau))\n",
    "u = np.random.uniform(0,1,len(Psel))\n",
    "print(len(Psel[u<prob_decay]) / len(Psel))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.275273\n"
     ]
    }
   ],
   "source": [
    "# The total efficiency is now found to be:\n",
    "print(len(Psel[u<prob_decay]) / len(P_Ap_lab))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is slightly lower than the value in the homework as we don't have a fixed momentum."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.6735124654665625\n"
     ]
    }
   ],
   "source": [
    "# HW calculaton but using toys\n",
    "betahw = 5.1/np.sqrt(5.1**2+1.2**2)\n",
    "gammahw = 1/np.sqrt(1-betahw**2)\n",
    "\n",
    "prob_decay = np.exp(-r1/(betahw*gammahw*ctau))  - np.exp(-r2/(betahw*gammahw*ctau))\n",
    "u = np.random.uniform(0, 1, len(Psel))\n",
    "print(len(Psel[u<prob_decay]) / len(Psel))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.282065\n"
     ]
    }
   ],
   "source": [
    "# The total efficiency is now found to be:\n",
    "print(len(Psel[u<prob_decay]) / len(P_Ap_lab))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can also use our toy decay simulation approach to get the probability $P(5< r_{\\rm decay} < 45)$ as a function of $c\\tau$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {},
   "outputs": [],
   "source": [
    "taus = np.linspace(0.1, 10, 100)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {},
   "outputs": [],
   "source": [
    "r1 = 10\n",
    "r2 = 40\n",
    "\n",
    "P = []\n",
    "for tau in taus:\n",
    "    prob_decay = np.exp(-r1/(beta*gamma*tau))  - np.exp(-r2/(beta*gamma*tau))\n",
    "    k = np.random.uniform(0, 1, len(Psel))\n",
    "    P.append(len(Psel[k<prob_decay]) / len(Psel))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(taus,P)\n",
    "plt.ylabel(r'$P(5< r_{\\rm decay} < 45)$')\n",
    "plt.xlabel(r'$c \\tau$');"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "3.7.6 (Framework)",
   "language": "python",
   "name": "3.7.6-framework"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.7.6"
  },
  "toc": {
   "base_numbering": 1,
   "nav_menu": {},
   "number_sections": false,
   "sideBar": true,
   "skip_h1_title": true,
   "title_cell": "Table of Contents",
   "title_sidebar": "Contents",
   "toc_cell": false,
   "toc_position": {},
   "toc_section_display": true,
   "toc_window_display": true
  },
  "toc-autonumbering": false,
  "toc-showcode": false,
  "toc-showmarkdowntxt": false,
  "toc-showtags": false
 },
 "nbformat": 4,
 "nbformat_minor": 4
}