r4lv/Sheet 3 - Class Demo.ipynb Secret

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

## Sheet 3 - Class Demo.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": 1,
   "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(np.sqrt(s),\n",
    "                                            [Ap_Mass, photon_Mass]).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": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{'p_0': <tf.Tensor: shape=(1000000, 4), dtype=float64, numpy=\n",
       " array([[-0.37532898,  0.22631087,  0.47645784,  2.10216568],\n",
       "        [-0.30168791,  0.38343113,  0.42551792,  2.10216568],\n",
       "        [ 0.07943709,  0.12522052,  0.63016672,  2.10216568],\n",
       "        ...,\n",
       "        [ 0.09240029,  0.07989894, -0.63575064,  2.10216568],\n",
       "        [-0.35995081, -0.51702018, -0.14908413,  2.10216568],\n",
       "        [-0.41131191,  0.10709213, -0.48831784,  2.10216568]])>,\n",
       " 'p_1': <tf.Tensor: shape=(1000000, 4), dtype=float64, numpy=\n",
       " array([[ 0.37532898, -0.22631087, -0.47645784,  0.64737974],\n",
       "        [ 0.30168791, -0.38343113, -0.42551792,  0.64737974],\n",
       "        [-0.07943709, -0.12522052, -0.63016672,  0.64737974],\n",
       "        ...,\n",
       "        [-0.09240029, -0.07989894,  0.63575064,  0.64737974],\n",
       "        [ 0.35995081,  0.51702018,  0.14908413,  0.64737974],\n",
       "        [ 0.41131191, -0.10709213,  0.48831784,  0.64737974]])>}"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "particles"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "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": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(1000000, 4)"
      ]
     },
     "execution_count": 7,
     "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": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[-0.37532898,  0.22631087,  0.47645784,  2.10216568],\n",
       "       [-0.30168791,  0.38343113,  0.42551792,  2.10216568],\n",
       "       [ 0.07943709,  0.12522052,  0.63016672,  2.10216568],\n",
       "       ...,\n",
       "       [ 0.09240029,  0.07989894, -0.63575064,  2.10216568],\n",
       "       [-0.35995081, -0.51702018, -0.14908413,  2.10216568],\n",
       "       [-0.41131191,  0.10709213, -0.48831784,  2.10216568]])"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p_Ap"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 0.37532898, -0.22631087, -0.47645784,  0.64737974],\n",
       "       [ 0.30168791, -0.38343113, -0.42551792,  0.64737974],\n",
       "       [-0.07943709, -0.12522052, -0.63016672,  0.64737974],\n",
       "       ...,\n",
       "       [-0.09240029, -0.07989894,  0.63575064,  0.64737974],\n",
       "       [ 0.35995081,  0.51702018,  0.14908413,  0.64737974],\n",
       "       [ 0.41131191, -0.10709213,  0.48831784,  0.64737974]])"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p_gamma"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "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": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "# 3 momentum\n",
    "pvec_Ap = p_Ap[:,:3]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(1000000, 3)"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pvec_Ap.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[-0.37532898,  0.22631087,  0.47645784],\n",
       "       [-0.30168791,  0.38343113,  0.42551792],\n",
       "       [ 0.07943709,  0.12522052,  0.63016672],\n",
       "       ...,\n",
       "       [ 0.09240029,  0.07989894, -0.63575064],\n",
       "       [-0.35995081, -0.51702018, -0.14908413],\n",
       "       [-0.41131191,  0.10709213, -0.48831784]])"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pvec_Ap"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "E_Ap = p_Ap[:,3]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([2.10216568, 2.10216568, 2.10216568, ..., 2.10216568, 2.10216568,\n",
       "       2.10216568])"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "E_Ap"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "How would you compute the momentum magnitude? Try in the empty cell below. \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": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Compute your momentum magnitude here\n",
    "P_Ap = np.sqrt(np.sum(pvec_Ap**2,axis=1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.64737974, 0.64737974, 0.64737974, ..., 0.64737974, 0.64737974,\n",
       "       0.64737974])"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P_Ap"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Compute here the invariant mass np.sqrt() using E_Ap and P_Ap\n",
    "# you should get 1.2 as expected\n",
    "M_Ap = np.sqrt(E_Ap**2 - P_Ap**2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([2., 2., 2., ..., 2., 2., 2.])"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M_Ap"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "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": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Compute phi using  np.arctan() of the relevant momenta\n",
    "phi_Ap = np.arctan(py_Ap/px_Ap)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Compute theta using np.arctan\n",
    "theta_Ap = np.arctan(np.sqrt(px_Ap**2 + py_Ap**2) / pz_Ap)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "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": 23,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Here \n",
    "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": 24,
   "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 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": 25,
   "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": 26,
   "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()\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "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": 28,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Now boost into the lab frame\n",
    "pz_Ap_lab = gamma * (pz_Ap - beta *E_Ap)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "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": 30,
   "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": 31,
   "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": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAYkAAAENCAYAAAD6/JlzAAAABHNCSVQICAgIfAhkiAAAAAlwSFlzAAALEgAACxIB0t1+/AAAADh0RVh0U29mdHdhcmUAbWF0cGxvdGxpYiB2ZXJzaW9uMy4xLjIsIGh0dHA6Ly9tYXRwbG90bGliLm9yZy8li6FKAAASgUlEQVR4nO3df6zd9V3H8efLVtwP3SijIdgWi9po2KKOXQGzxSyiUJixmGwEYqRbyGoy0PkjccWY1GzDMKNOMBOto64sc4zglOqYWNmI+geMlpExwMkNP6RNgc524FwU2d7+cT4dx8v9FHrP/dVzno/k5n6/7+/ne87nu+84r34+3+/53lQVkiTN5juWugOSpOXLkJAkdRkSkqQuQ0KS1GVISJK6DAlJUtdLhkSSHUmeTvLlodpJSXYnebj9XtXqSXJdkukkX0py5tA+m1v7h5NsHqq/Kcn9bZ/rkuRo7yFJWjwvZyTxMWDjjNpW4I6q2gDc0dYBLgA2tJ8twPUw+MAHtgFnA2cB24Y+9K8H3j2038aXeA9J0iJ5yZCoqn8CDs0obwJ2tuWdwEVD9Rtr4C7gxCSnAucDu6vqUFUdBnYDG9u211TVXTX4Vt+NM15rtveQJC2SlXPc75SqOtCWnwROactrgCeG2u1rtaPV981SP9p7HNXJJ59c69evf3lHIUkCYO/evV+tqtUz63MNiW+rqkqyoM/2eKn3SLKFwfQWp512Gnv27FnI7kjS2Eny+Gz1ud7d9FSbKqL9frrV9wPrhtqtbbWj1dfOUj/ae7xIVW2vqqmqmlq9+kVBKEmao7mGxC7gyB1Km4Fbh+qXtbuczgGeaVNGtwPnJVnVLlifB9zetj2b5Jx2V9NlM15rtveQJC2Sl5xuSvJJ4K3AyUn2MbhL6Rrg5iSXA48DF7fmtwEXAtPAN4B3AVTVoSQfAO5p7d5fVUcuhr+HwR1UrwQ+2344yntIkhZJxu1R4VNTU+U1CUk6Nkn2VtXUzLrfuJYkdRkSkqQuQ0KS1GVISJK6Rv4ynSRp8azf+plZ649d87YFeT9HEpKkLkNCktRlSEiSugwJSVKXISFJ6jIkJEldhoQkqcuQkCR1GRKSpC5DQpLUZUhIkroMCUlSlyEhSeoyJCRJXYaEJKnLkJAkdRkSkqQuQ0KS1GVISJK6DAlJUpchIUnqMiQkSV2GhCSpy5CQJHUZEpKkLkNCktRlSEiSugwJSVKXISFJ6hopJJL8WpIHknw5ySeTvCLJ6UnuTjKd5FNJTmhtv6utT7ft64de56pW/0qS84fqG1ttOsnWUfoqSTp2cw6JJGuAXwGmquoNwArgEuBDwIer6geBw8DlbZfLgcOt/uHWjiRntP1eD2wE/iTJiiQrgI8AFwBnAJe2tpKkRTLqdNNK4JVJVgKvAg4APwXc0rbvBC5qy5vaOm37uUnS6jdV1f9U1aPANHBW+5muqkeq6jngptZWkrRI5hwSVbUf+H3g3xmEwzPAXuBrVfV8a7YPWNOW1wBPtH2fb+1fN1yfsU+vLklaJKNMN61i8C/704HvBV7NYLpo0SXZkmRPkj0HDx5cii5I0lgaZbrpp4FHq+pgVf0v8GngzcCJbfoJYC2wvy3vB9YBtO2vBf5juD5jn179Rapqe1VNVdXU6tWrRzgkSdKwUULi34FzkryqXVs4F3gQ+Dzw9tZmM3BrW97V1mnbP1dV1eqXtLufTgc2AF8A7gE2tLulTmBwcXvXCP2VJB2jlS/dZHZVdXeSW4B7geeBLwLbgc8ANyX5YKvd0Ha5Afh4kmngEIMPfarqgSQ3MwiY54ErquqbAEmuBG5ncOfUjqp6YK79lSQduzmHBEBVbQO2zSg/wuDOpJlt/xt4R+d1rgaunqV+G3DbKH2UJM2d37iWJHUZEpKkLkNCktRlSEiSugwJSVKXISFJ6jIkJEldhoQkqcuQkCR1GRKSpC5DQpLUZUhIkroMCUlSlyEhSeoyJCRJXYaEJKnLkJAkdRkSkqQuQ0KS1GVISJK6DAlJUpchIUnqMiQkSV2GhCSpy5CQJHUZEpKkLkNCktRlSEiSugwJSVKXISFJ6jIkJEldhoQkqcuQkCR1GRKSpC5DQpLUNVJIJDkxyS1J/jXJQ0l+IslJSXYnebj9XtXaJsl1SaaTfCnJmUOvs7m1fzjJ5qH6m5Lc3/a5LklG6a8k6diMOpK4Fvj7qvph4EeBh4CtwB1VtQG4o60DXABsaD9bgOsBkpwEbAPOBs4Cth0Jltbm3UP7bRyxv5KkYzDnkEjyWuAngRsAquq5qvoasAnY2ZrtBC5qy5uAG2vgLuDEJKcC5wO7q+pQVR0GdgMb27bXVNVdVVXAjUOvJUlaBKOMJE4HDgJ/keSLST6a5NXAKVV1oLV5EjilLa8Bnhjaf1+rHa2+b5b6iyTZkmRPkj0HDx4c4ZAkScNGCYmVwJnA9VX1RuC/eGFqCYA2AqgR3uNlqartVTVVVVOrV69e6LeTpIkxSkjsA/ZV1d1t/RYGofFUmyqi/X66bd8PrBvaf22rHa2+dpa6JGmRzDkkqupJ4IkkP9RK5wIPAruAI3cobQZubcu7gMvaXU7nAM+0aanbgfOSrGoXrM8Dbm/bnk1yTrur6bKh15IkLYKVI+7/y8AnkpwAPAK8i0Hw3JzkcuBx4OLW9jbgQmAa+EZrS1UdSvIB4J7W7v1Vdagtvwf4GPBK4LPtR5K0SEYKiaq6D5iaZdO5s7Qt4IrO6+wAdsxS3wO8YZQ+SpLmzm9cS5K6DAlJUpchIUnqMiQkSV2GhCSpy5CQJHUZEpKkLkNCktRlSEiSugwJSVKXISFJ6jIkJEldhoQkqcuQkCR1GRKSpC5DQpLUZUhIkroMCUlSlyEhSeoyJCRJXYaEJKnLkJAkdRkSkqQuQ0KS1GVISJK6DAlJUpchIUnqMiQkSV2GhCSpy5CQJHUZEpKkLkNCktRlSEiSulYudQckSS+2futnlroLwDyMJJKsSPLFJH/X1k9PcneS6SSfSnJCq39XW59u29cPvcZVrf6VJOcP1Te22nSSraP2VZJ0bOZjuum9wEND6x8CPlxVPwgcBi5v9cuBw63+4daOJGcAlwCvBzYCf9KCZwXwEeAC4Azg0tZWkrRIRgqJJGuBtwEfbesBfgq4pTXZCVzUlje1ddr2c1v7TcBNVfU/VfUoMA2c1X6mq+qRqnoOuKm1lSQtklFHEn8E/Cbwrbb+OuBrVfV8W98HrGnLa4AnANr2Z1r7b9dn7NOrv0iSLUn2JNlz8ODBEQ9JknTEnEMiyc8CT1fV3nnsz5xU1faqmqqqqdWrVy91dyRpbIxyd9ObgZ9LciHwCuA1wLXAiUlWttHCWmB/a78fWAfsS7ISeC3wH0P1I4b36dUlSYtgziOJqrqqqtZW1XoGF54/V1W/AHweeHtrthm4tS3vauu07Z+rqmr1S9rdT6cDG4AvAPcAG9rdUie099g11/5Kko7dQnxP4n3ATUk+CHwRuKHVbwA+nmQaOMTgQ5+qeiDJzcCDwPPAFVX1TYAkVwK3AyuAHVX1wAL0V5LUMS8hUVV3Ane25UcY3Jk0s81/A+/o7H81cPUs9duA2+ajj5KkY+djOSRJXYaEJKnLkJAkdRkSkqQuQ0KS1GVISJK6DAlJUpd/dEiSltBy+eNCPY4kJEldhoQkqcuQkCR1GRKSpC5DQpLUZUhIkroMCUlSlyEhSeryy3SStAiW+5fmehxJSJK6DAlJUpchIUnqMiQkSV1euJakeXS8XqDucSQhSeoyJCRJXU43SdIcjNu0Uo8jCUlSlyEhSepyukmSjmJSppV6DAlJwjDoMSQkTRTD4NgYEpLGkmEwPwwJScueH/hLx5CQNDI/xMeXISFNsN6H+2PXvO2Y2mt8zTkkkqwDbgROAQrYXlXXJjkJ+BSwHngMuLiqDicJcC1wIfAN4J1VdW97rc3Ab7eX/mBV7Wz1NwEfA14J3Aa8t6pqrn2WxsFifFAbBjpilJHE88BvVNW9Sb4H2JtkN/BO4I6quibJVmAr8D7gAmBD+zkbuB44u4XKNmCKQdjsTbKrqg63Nu8G7mYQEhuBz47QZ2nZ8QNZy9mcQ6KqDgAH2vJ/JnkIWANsAt7amu0E7mQQEpuAG9tI4K4kJyY5tbXdXVWHAFrQbExyJ/Caqrqr1W8ELsKQ0DLnh77Gybxck0iyHngjg3/xn9ICBOBJBtNRMAiQJ4Z229dqR6vvm6UuLQuGgSbByCGR5LuBvwJ+taqeHVx6GKiqSrLg1xCSbAG2AJx22mkL/XaaMIaBJtlIIZHkOxkExCeq6tOt/FSSU6vqQJtOerrV9wPrhnZf22r7eWF66kj9zlZfO0v7F6mq7cB2gKmpKS9s66j80JdevlHubgpwA/BQVf3h0KZdwGbgmvb71qH6lUluYnDh+pkWJLcDv5tkVWt3HnBVVR1K8myScxhMY10G/PFc+6vJYxhIoxtlJPFm4BeB+5Pc12q/xSAcbk5yOfA4cHHbdhuD21+nGdwC+y6AFgYfAO5p7d5/5CI28B5euAX2s3jRWpIW1Sh3N/0LkM7mc2dpX8AVndfaAeyYpb4HeMNc+6jJ4IhBWjh+41rHDcNAWnz+ZTpJUpcjCS0rjhak5cWQ0JIwDKTjgyGhBWUYSMc3r0lIkrocSWheOGKQxpMjCUlSlyMJHRNHDNJkcSQhSepyJKFZOWKQBI4kJElH4UhiwjlikHQ0jiQkSV2OJCaEIwZJc+FIQpLU5UhizDhikDSfHElIkroMCUlSl9NNxymnlSQtBkcSkqQuRxLLnCMGSUvJkYQkqcuRxDLgaEHScuVIQpLUZUhIkrqcblpETitJOt44kpAkdRkSkqQup5sWgNNKksaFIwlJUpcjiRE4YpA07hxJSJK6DAlJUpfTTS+D00qSJtWyH0kk2ZjkK0mmk2xd6v5I0iRZ1iGRZAXwEeAC4Azg0iRnLG2vJGlyLPfpprOA6ap6BCDJTcAm4MGFeDOnlSTp/1vuIbEGeGJofR9w9sxGSbYAW9rq15N8ZY7vdzLw1TnuezyatOOFyTvmSTtemNBjzodGPubvm6243EPiZamq7cD2UV8nyZ6qmpqHLh0XJu14YfKOedKOFzzm+basr0kA+4F1Q+trW02StAiWe0jcA2xIcnqSE4BLgF1L3CdJmhjLerqpqp5PciVwO7AC2FFVDyzgW448ZXWcmbTjhck75kk7XvCY51WqaqFeW5J0nFvu002SpCVkSEiSugwJJvPRH0keS3J/kvuS7Fnq/iyEJDuSPJ3ky0O1k5LsTvJw+71qKfs4nzrH+ztJ9rfzfF+SC5eyj/Mpybokn0/yYJIHkry31cf5HPeOecHO88Rfk2iP/vg34GcYfFnvHuDSqlqQb3UvF0keA6aqamy/dJTkJ4GvAzdW1Rta7feAQ1V1TfsHwaqqet9S9nO+dI73d4CvV9XvL2XfFkKSU4FTq+reJN8D7AUuAt7J+J7j3jFfzAKdZ0cSQ4/+qKrngCOP/tBxrqr+CTg0o7wJ2NmWdzL4D2wsdI53bFXVgaq6ty3/J/AQg6c0jPM57h3zgjEkZn/0x4L+j75MFPAPSfa2x5pMilOq6kBbfhI4ZSk7s0iuTPKlNh01NlMvw5KsB94I3M2EnOMZxwwLdJ4Nicn1lqo6k8ETdq9oUxUTpQZzreM+33o98APAjwEHgD9Y2u7MvyTfDfwV8KtV9ezwtnE9x7Mc84KdZ0NiQh/9UVX72++ngb9mMO02CZ5q87pH5nefXuL+LKiqeqqqvllV3wL+nDE7z0m+k8GH5Seq6tOtPNbneLZjXsjzbEhM4KM/kry6XfQiyauB84AvH32vsbEL2NyWNwO3LmFfFtyRD8vm5xmj85wkwA3AQ1X1h0ObxvYc9455Ic/zxN/dBNBuF/sjXnj0x9VL3KUFleT7GYweYPBolr8cx2NO8kngrQweHf0UsA34G+Bm4DTgceDiqhqLi72d430rgymIAh4Dfmlovv64luQtwD8D9wPfauXfYjBHP67nuHfMl7JA59mQkCR1Od0kSeoyJCRJXYaEJKnLkJAkdRkSkqQuQ0KS1GVISPMoyYok17bHON/fvpMiHbcMCWl+XQU8UlWvB64D3rPE/ZFGsnKpOyCNi/aIk5+vqje10qPA25awS9LIDAlp/vw0sC7JfW39JOAfl7A/0sh8LIc0T5JsA56qqj9t6x8FvgQ8C3y1qv5uRvt3zlaXlhOvSUjzZxXwDYAkKxk8Xfdvj2xM8vokv5vkhiQ/0cqXJvmDJGPx5zU1fgwJaf78G3BOW/414DNV9ejQ9ueAVzB4Qusvtto/VNVvAD+exP8etez4f0pp/nwSODPJNPAjwK/P2P4rDB5J/2fAq2Zsc95Xy5IXrqV5UlWHeWEkMZvPA+9jMJI44vwkPwrsaX9VTFpWvHAtSepyukmS1GVISJK6DAlJUpchIUnqMiQkSV2GhCSpy5CQJHUZEpKkLkNCktT1f3gziw8putonAAAAAElFTkSuQmCC\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": [
    "What is the probability that the particle has a $\\theta_{\\rm}$ within the long baseline detectors acceptance ($0 < \\theta_{\\rm lab} < 15^{\\circ}$).\n",
    "\n",
    "Now repeat this for a different mass hypthesis of the $A'$. Try repeat this calculation for $M_{A'} = 1.4$ and $M_{A'} = 1.2$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.418931"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#more  formally using the MC method we can compute this as\n",
    "\n",
    "len(theta[theta*180/np.pi < 15]) / len(theta)"
   ]
  },
  {
   "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": 34,
   "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": 35,
   "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": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.419796"
      ]
     },
     "execution_count": 36,
     "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": 37,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "5.095980556481348\n"
     ]
    },
    {
     "data": {
      "image/png": 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wkCRhGEh9S/KxJC+1Wy4/l+SPu+5JOlb+zkDqU5J/Bp6oqi8kOY/xm+S9u+u+pGPhmYHUv/fy/7e+GAWWdNiLNCOGgdS/s4Gn2z2g/hTwNhg67hgGUh+SrAFOAXYDDwHLgM0967+a5C86ak+aNm9UJ/XnbGBvVb3pdtlJNjB+lvDbc96VdIw8M5D6817g+0cWk7wd+IOq+jfgXXPelXSMDAOpP2cDj01S/0vglCRfAN7T/seCNG85TST1oar+8Mhakl8GhqrqyvZ6K+NnEAvxfzJrgfB3BpIkp4kkSYaBJAnDQJKEYSBJwjCQJGEYSJIwDCRJGAaSJAwDSRLwfx37BkViFIUPAAAAAElFTkSuQmCC\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": 38,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.6579433820236495\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": 39,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.276202\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": 40,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.6756758044383463\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": 41,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.283646\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": 42,
   "metadata": {},
   "outputs": [],
   "source": [
    "taus = np.linspace(0.1, 10, 100)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "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": 44,
   "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-autonumbering": false,
  "toc-showcode": false,
  "toc-showmarkdowntxt": false,
  "toc-showtags": false
 },
 "nbformat": 4,
 "nbformat_minor": 4
}