triangle_loop.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "96d2a1cf",
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import Symbol, symbols\n",
    "from sympy import I, simplify,pi\n",
    "from sympy import DiracDelta\n",
    "from sympy import integrate,oo,latex,factor"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "f554b7db",
   "metadata": {},
   "outputs": [],
   "source": [
    "mA,mB,mC = symbols('m_A m_B m_C')#mass\n",
    "g = Symbol('g')# coupling\n",
    "q = Symbol('q') # internal momentum\n",
    "k1,k2,k3 = symbols('k_1:4')\n",
    "p1,p2,p3,p4 = symbols('p_1:5') # external momentum"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "37b300a3",
   "metadata": {},
   "outputs": [],
   "source": [
    "# feynman rules\n",
    "class FeynmanRules:\n",
    "    def __init__():\n",
    "        pass\n",
    "    \n",
    "    def propagator(q,m):\n",
    "        prop = I/(q**2 - m**2)\n",
    "        return prop\n",
    "\n",
    "    def vertex():\n",
    "        return -I*g\n",
    "    \n",
    "    def external_line():\n",
    "        return 1\n",
    "    \n",
    "    def delta(k1, k2, k3):\n",
    "        delta = (2*pi)**4*DiracDelta(k1 + k2 + k3)\n",
    "        return delta\n",
    "    \n",
    "    def remove_factor():\n",
    "        return I/((2*pi)**4*DiracDelta(0))\n",
    "    \n",
    "    def integral_factor():\n",
    "        return 1/((2*pi)**4)\n",
    "    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "83de7f01",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{g^{4} \\delta\\left(- k_{1} - k_{2} + q\\right) \\delta\\left(k_{1} - k_{3} - p_{3}\\right) \\delta\\left(k_{2} + k_{3} - p_{4}\\right) \\delta\\left(p_{1} + p_{2} - q\\right)}{\\left(k_{1}^{2} - m_{C}^{2}\\right) \\left(k_{2}^{2} - m_{B}^{2}\\right) \\left(k_{3}^{2} - m_{A}^{2}\\right) \\left(- m_{C}^{2} + q^{2}\\right)}$"
      ],
      "text/plain": [
       "g**4*DiracDelta(-k_1 - k_2 + q)*DiracDelta(k_1 - k_3 - p_3)*DiracDelta(k_2 + k_3 - p_4)*DiracDelta(p_1 + p_2 - q)/((k_1**2 - m_C**2)*(k_2**2 - m_B**2)*(k_3**2 - m_A**2)*(-m_C**2 + q**2))"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fr = FeynmanRules\n",
    "ME = fr.external_line()**4 \n",
    "ME = ME*fr.vertex()**4\n",
    "ME = ME*fr.delta(p1,p2,-q)\n",
    "ME = ME*fr.delta(q,-k1,-k2)\n",
    "ME = ME*fr.delta(k2,k3,-p4)\n",
    "ME = ME*fr.delta(k1,-k3,-p3)\n",
    "ME = ME*fr.propagator(q,mC) \n",
    "ME = ME*fr.propagator(k1,mC) \n",
    "ME = ME*fr.propagator(k2,mB) \n",
    "ME = ME*fr.propagator(k3,mA) \n",
    "ME = ME*fr.integral_factor()**4\n",
    "ME"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "8a29fff2",
   "metadata": {},
   "outputs": [],
   "source": [
    "ME = integrate(ME,(k1,-oo,oo))\n",
    "ME = integrate(ME,(k2,-oo,oo))\n",
    "ME = integrate(ME,(q,-oo,oo))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "0293605f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'- \\\\frac{g^{4} \\\\delta\\\\left(p_{1} + p_{2} - p_{3} - p_{4}\\\\right)}{\\\\left(k_{3} - m_{A}\\\\right) \\\\left(k_{3} + m_{A}\\\\right) \\\\left(k_{3} - m_{B} - p_{4}\\\\right) \\\\left(k_{3} + m_{B} - p_{4}\\\\right) \\\\left(m_{C} - p_{1} - p_{2}\\\\right) \\\\left(m_{C} + p_{1} + p_{2}\\\\right) \\\\left(k_{3} - m_{C} + p_{1} + p_{2} - p_{4}\\\\right) \\\\left(k_{3} + m_{C} + p_{1} + p_{2} - p_{4}\\\\right)}'"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "latex(factor(ME))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "99fe48bc",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'\\\\frac{g^{4} \\\\delta\\\\left(p_{1} + p_{2} - p_{3} - p_{4}\\\\right)}{- k_{3}^{6} m_{C}^{2} + k_{3}^{6} \\\\left(p_{1} + p_{2}\\\\right)^{2} - 2 k_{3}^{5} m_{C}^{2} p_{1} - 2 k_{3}^{5} m_{C}^{2} p_{2} + 4 k_{3}^{5} m_{C}^{2} p_{4} - 4 k_{3}^{5} p_{4} \\\\left(p_{1} + p_{2}\\\\right)^{2} + 2 k_{3}^{5} \\\\left(p_{1} + p_{2}\\\\right)^{3} + k_{3}^{4} m_{A}^{2} m_{C}^{2} - k_{3}^{4} m_{A}^{2} \\\\left(p_{1} + p_{2}\\\\right)^{2} + k_{3}^{4} m_{B}^{2} m_{C}^{2} - k_{3}^{4} m_{B}^{2} \\\\left(p_{1} + p_{2}\\\\right)^{2} + k_{3}^{4} m_{C}^{4} + 6 k_{3}^{4} m_{C}^{2} p_{1} p_{4} + 6 k_{3}^{4} m_{C}^{2} p_{2} p_{4} - 6 k_{3}^{4} m_{C}^{2} p_{4}^{2} - 2 k_{3}^{4} m_{C}^{2} \\\\left(p_{1} + p_{2}\\\\right)^{2} + 6 k_{3}^{4} p_{4}^{2} \\\\left(p_{1} + p_{2}\\\\right)^{2} - 6 k_{3}^{4} p_{4} \\\\left(p_{1} + p_{2}\\\\right)^{3} + k_{3}^{4} \\\\left(p_{1} + p_{2}\\\\right)^{4} + 2 k_{3}^{3} m_{A}^{2} m_{C}^{2} p_{1} + 2 k_{3}^{3} m_{A}^{2} m_{C}^{2} p_{2} - 4 k_{3}^{3} m_{A}^{2} m_{C}^{2} p_{4} + 4 k_{3}^{3} m_{A}^{2} p_{4} \\\\left(p_{1} + p_{2}\\\\right)^{2} - 2 k_{3}^{3} m_{A}^{2} \\\\left(p_{1} + p_{2}\\\\right)^{3} + 2 k_{3}^{3} m_{B}^{2} m_{C}^{2} p_{1} + 2 k_{3}^{3} m_{B}^{2} m_{C}^{2} p_{2} - 2 k_{3}^{3} m_{B}^{2} m_{C}^{2} p_{4} + 2 k_{3}^{3} m_{B}^{2} p_{4} \\\\left(p_{1} + p_{2}\\\\right)^{2} - 2 k_{3}^{3} m_{B}^{2} \\\\left(p_{1} + p_{2}\\\\right)^{3} - 2 k_{3}^{3} m_{C}^{4} p_{4} - 6 k_{3}^{3} m_{C}^{2} p_{1} p_{4}^{2} - 6 k_{3}^{3} m_{C}^{2} p_{2} p_{4}^{2} + 4 k_{3}^{3} m_{C}^{2} p_{4}^{3} + 4 k_{3}^{3} m_{C}^{2} p_{4} \\\\left(p_{1} + p_{2}\\\\right)^{2} - 4 k_{3}^{3} p_{4}^{3} \\\\left(p_{1} + p_{2}\\\\right)^{2} + 6 k_{3}^{3} p_{4}^{2} \\\\left(p_{1} + p_{2}\\\\right)^{3} - 2 k_{3}^{3} p_{4} \\\\left(p_{1} + p_{2}\\\\right)^{4} - k_{3}^{2} m_{A}^{2} m_{B}^{2} m_{C}^{2} + k_{3}^{2} m_{A}^{2} m_{B}^{2} \\\\left(p_{1} + p_{2}\\\\right)^{2} - k_{3}^{2} m_{A}^{2} m_{C}^{4} - 6 k_{3}^{2} m_{A}^{2} m_{C}^{2} p_{1} p_{4} - 6 k_{3}^{2} m_{A}^{2} m_{C}^{2} p_{2} p_{4} + 6 k_{3}^{2} m_{A}^{2} m_{C}^{2} p_{4}^{2} + 2 k_{3}^{2} m_{A}^{2} m_{C}^{2} \\\\left(p_{1} + p_{2}\\\\right)^{2} - 6 k_{3}^{2} m_{A}^{2} p_{4}^{2} \\\\left(p_{1} + p_{2}\\\\right)^{2} + 6 k_{3}^{2} m_{A}^{2} p_{4} \\\\left(p_{1} + p_{2}\\\\right)^{3} - k_{3}^{2} m_{A}^{2} \\\\left(p_{1} + p_{2}\\\\right)^{4} - k_{3}^{2} m_{B}^{2} m_{C}^{4} - 2 k_{3}^{2} m_{B}^{2} m_{C}^{2} p_{1} p_{4} - 2 k_{3}^{2} m_{B}^{2} m_{C}^{2} p_{2} p_{4} + k_{3}^{2} m_{B}^{2} m_{C}^{2} p_{4}^{2} + 2 k_{3}^{2} m_{B}^{2} m_{C}^{2} \\\\left(p_{1} + p_{2}\\\\right)^{2} - k_{3}^{2} m_{B}^{2} p_{4}^{2} \\\\left(p_{1} + p_{2}\\\\right)^{2} + 2 k_{3}^{2} m_{B}^{2} p_{4} \\\\left(p_{1} + p_{2}\\\\right)^{3} - k_{3}^{2} m_{B}^{2} \\\\left(p_{1} + p_{2}\\\\right)^{4} + k_{3}^{2} m_{C}^{4} p_{4}^{2} + 2 k_{3}^{2} m_{C}^{2} p_{1} p_{4}^{3} + 2 k_{3}^{2} m_{C}^{2} p_{2} p_{4}^{3} - k_{3}^{2} m_{C}^{2} p_{4}^{4} - 2 k_{3}^{2} m_{C}^{2} p_{4}^{2} \\\\left(p_{1} + p_{2}\\\\right)^{2} + k_{3}^{2} p_{4}^{4} \\\\left(p_{1} + p_{2}\\\\right)^{2} - 2 k_{3}^{2} p_{4}^{3} \\\\left(p_{1} + p_{2}\\\\right)^{3} + k_{3}^{2} p_{4}^{2} \\\\left(p_{1} + p_{2}\\\\right)^{4} - 2 k_{3} m_{A}^{2} m_{B}^{2} m_{C}^{2} p_{1} - 2 k_{3} m_{A}^{2} m_{B}^{2} m_{C}^{2} p_{2} + 2 k_{3} m_{A}^{2} m_{B}^{2} m_{C}^{2} p_{4} - 2 k_{3} m_{A}^{2} m_{B}^{2} p_{4} \\\\left(p_{1} + p_{2}\\\\right)^{2} + 2 k_{3} m_{A}^{2} m_{B}^{2} \\\\left(p_{1} + p_{2}\\\\right)^{3} + 2 k_{3} m_{A}^{2} m_{C}^{4} p_{4} + 6 k_{3} m_{A}^{2} m_{C}^{2} p_{1} p_{4}^{2} + 6 k_{3} m_{A}^{2} m_{C}^{2} p_{2} p_{4}^{2} - 4 k_{3} m_{A}^{2} m_{C}^{2} p_{4}^{3} - 4 k_{3} m_{A}^{2} m_{C}^{2} p_{4} \\\\left(p_{1} + p_{2}\\\\right)^{2} + 4 k_{3} m_{A}^{2} p_{4}^{3} \\\\left(p_{1} + p_{2}\\\\right)^{2} - 6 k_{3} m_{A}^{2} p_{4}^{2} \\\\left(p_{1} + p_{2}\\\\right)^{3} + 2 k_{3} m_{A}^{2} p_{4} \\\\left(p_{1} + p_{2}\\\\right)^{4} + m_{A}^{2} m_{B}^{2} m_{C}^{4} + 2 m_{A}^{2} m_{B}^{2} m_{C}^{2} p_{1} p_{4} + 2 m_{A}^{2} m_{B}^{2} m_{C}^{2} p_{2} p_{4} - m_{A}^{2} m_{B}^{2} m_{C}^{2} p_{4}^{2} - 2 m_{A}^{2} m_{B}^{2} m_{C}^{2} \\\\left(p_{1} + p_{2}\\\\right)^{2} + m_{A}^{2} m_{B}^{2} p_{4}^{2} \\\\left(p_{1} + p_{2}\\\\right)^{2} - 2 m_{A}^{2} m_{B}^{2} p_{4} \\\\left(p_{1} + p_{2}\\\\right)^{3} + m_{A}^{2} m_{B}^{2} \\\\left(p_{1} + p_{2}\\\\right)^{4} - m_{A}^{2} m_{C}^{4} p_{4}^{2} - 2 m_{A}^{2} m_{C}^{2} p_{1} p_{4}^{3} - 2 m_{A}^{2} m_{C}^{2} p_{2} p_{4}^{3} + m_{A}^{2} m_{C}^{2} p_{4}^{4} + 2 m_{A}^{2} m_{C}^{2} p_{4}^{2} \\\\left(p_{1} + p_{2}\\\\right)^{2} - m_{A}^{2} p_{4}^{4} \\\\left(p_{1} + p_{2}\\\\right)^{2} + 2 m_{A}^{2} p_{4}^{3} \\\\left(p_{1} + p_{2}\\\\right)^{3} - m_{A}^{2} p_{4}^{2} \\\\left(p_{1} + p_{2}\\\\right)^{4}}'"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "latex(ME)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "3e47ac8c",
   "metadata": {},
   "outputs": [],
   "source": []
  }
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