box_diagram.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,factor,latex"
   ]
  },
  {
   "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,k4 = symbols('k_1:5') # external momentum\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": 6,
   "id": "d68689bd",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\\frac{g^{4} \\delta\\left(- k_{1} - k_{2} + p_{1}\\right) \\delta\\left(k_{1} - k_{3} + p_{2}\\right) \\delta\\left(k_{2} - k_{4} - p_{3}\\right) \\delta\\left(k_{3} + k_{4} - p_{4}\\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(k_{4}^{2} - m_{C}^{2}\\right)}\n"
     ]
    }
   ],
   "source": [
    "fr = FeynmanRules\n",
    "ME = fr.external_line()**4\n",
    "ME = ME * fr.vertex()**4\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.propagator(k4,mC)\n",
    "ME = ME * fr.delta(p1,-k1,-k2)\n",
    "ME = ME * fr.delta(k2,-p3,-k4)\n",
    "ME = ME * fr.delta(k4,k3,-p4)\n",
    "ME = ME * fr.delta(p2,k1,-k3)\n",
    "ME = ME *fr.integral_factor()**4\n",
    "ME\n",
    "print(latex(ME))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "a67e857b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{g^{4} \\delta\\left(k_{2} - k_{4} - p_{3}\\right) \\delta\\left(k_{3} + k_{4} - p_{4}\\right) \\delta\\left(- k_{2} - k_{3} + p_{1} + p_{2}\\right)}{- k_{2}^{2} k_{3}^{2} k_{4}^{2} m_{C}^{2} + k_{2}^{2} k_{3}^{2} k_{4}^{2} \\left(- k_{2} + p_{1}\\right)^{2} + k_{2}^{2} k_{3}^{2} m_{C}^{4} - k_{2}^{2} k_{3}^{2} m_{C}^{2} \\left(- k_{2} + p_{1}\\right)^{2} + k_{2}^{2} k_{4}^{2} m_{A}^{2} m_{C}^{2} - k_{2}^{2} k_{4}^{2} m_{A}^{2} \\left(- k_{2} + p_{1}\\right)^{2} - k_{2}^{2} m_{A}^{2} m_{C}^{4} + k_{2}^{2} m_{A}^{2} m_{C}^{2} \\left(- k_{2} + p_{1}\\right)^{2} + k_{3}^{2} k_{4}^{2} m_{B}^{2} m_{C}^{2} - k_{3}^{2} k_{4}^{2} m_{B}^{2} \\left(- k_{2} + p_{1}\\right)^{2} - k_{3}^{2} m_{B}^{2} m_{C}^{4} + k_{3}^{2} m_{B}^{2} m_{C}^{2} \\left(- k_{2} + p_{1}\\right)^{2} - k_{4}^{2} m_{A}^{2} m_{B}^{2} m_{C}^{2} + k_{4}^{2} m_{A}^{2} m_{B}^{2} \\left(- k_{2} + p_{1}\\right)^{2} + m_{A}^{2} m_{B}^{2} m_{C}^{4} - m_{A}^{2} m_{B}^{2} m_{C}^{2} \\left(- k_{2} + p_{1}\\right)^{2}}$"
      ],
      "text/plain": [
       "g**4*DiracDelta(k_2 - k_4 - p_3)*DiracDelta(k_3 + k_4 - p_4)*DiracDelta(-k_2 - k_3 + p_1 + p_2)/(-k_2**2*k_3**2*k_4**2*m_C**2 + k_2**2*k_3**2*k_4**2*(-k_2 + p_1)**2 + k_2**2*k_3**2*m_C**4 - k_2**2*k_3**2*m_C**2*(-k_2 + p_1)**2 + k_2**2*k_4**2*m_A**2*m_C**2 - k_2**2*k_4**2*m_A**2*(-k_2 + p_1)**2 - k_2**2*m_A**2*m_C**4 + k_2**2*m_A**2*m_C**2*(-k_2 + p_1)**2 + k_3**2*k_4**2*m_B**2*m_C**2 - k_3**2*k_4**2*m_B**2*(-k_2 + p_1)**2 - k_3**2*m_B**2*m_C**4 + k_3**2*m_B**2*m_C**2*(-k_2 + p_1)**2 - k_4**2*m_A**2*m_B**2*m_C**2 + k_4**2*m_A**2*m_B**2*(-k_2 + p_1)**2 + m_A**2*m_B**2*m_C**4 - m_A**2*m_B**2*m_C**2*(-k_2 + p_1)**2)"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ME = integrate(ME, (k1,-oo,oo))\n",
    "ME"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "8ade1696",
   "metadata": {},
   "outputs": [],
   "source": [
    "ME = integrate(ME, (k2,-oo,oo))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "2b5903f7",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{g^{4} \\delta\\left(p_{1} + p_{2} - p_{3} - p_{4}\\right)}{2 k_{3}^{3} m_{B}^{2} m_{C}^{2} p_{1} - 2 k_{3}^{3} m_{B}^{2} m_{C}^{2} p_{3} - 2 k_{3}^{3} m_{C}^{4} p_{3} + 2 k_{3}^{3} m_{C}^{2} p_{1}^{2} p_{3} - 6 k_{3}^{3} m_{C}^{2} p_{1} p_{3}^{2} + 4 k_{3}^{3} m_{C}^{2} p_{3}^{3} - k_{3}^{2} m_{B}^{2} m_{C}^{4} + k_{3}^{2} m_{B}^{2} m_{C}^{2} p_{1}^{2} - 2 k_{3}^{2} m_{B}^{2} m_{C}^{2} p_{1} p_{3} - 2 k_{3}^{2} m_{B}^{2} m_{C}^{2} p_{1} p_{4} + k_{3}^{2} m_{B}^{2} m_{C}^{2} p_{3}^{2} + 2 k_{3}^{2} m_{B}^{2} m_{C}^{2} p_{3} p_{4} + 2 k_{3}^{2} m_{B}^{2} m_{C}^{2} \\left(- k_{3} + p_{4}\\right)^{2} - k_{3}^{2} m_{B}^{2} p_{1}^{2} \\left(- k_{3} + p_{4}\\right)^{2} + 2 k_{3}^{2} m_{B}^{2} p_{1} p_{3} \\left(- k_{3} + p_{4}\\right)^{2} + 2 k_{3}^{2} m_{B}^{2} p_{1} \\left(- k_{3} + p_{4}\\right)^{3} - k_{3}^{2} m_{B}^{2} p_{3}^{2} \\left(- k_{3} + p_{4}\\right)^{2} - 2 k_{3}^{2} m_{B}^{2} p_{3} \\left(- k_{3} + p_{4}\\right)^{3} - k_{3}^{2} m_{B}^{2} \\left(- k_{3} + p_{4}\\right)^{4} + k_{3}^{2} m_{C}^{4} p_{3}^{2} + 2 k_{3}^{2} m_{C}^{4} p_{3} p_{4} + k_{3}^{2} m_{C}^{4} \\left(- k_{3} + p_{4}\\right)^{2} - k_{3}^{2} m_{C}^{2} p_{1}^{2} p_{3}^{2} - 2 k_{3}^{2} m_{C}^{2} p_{1}^{2} p_{3} p_{4} - k_{3}^{2} m_{C}^{2} p_{1}^{2} \\left(- k_{3} + p_{4}\\right)^{2} + 2 k_{3}^{2} m_{C}^{2} p_{1} p_{3}^{3} + 6 k_{3}^{2} m_{C}^{2} p_{1} p_{3}^{2} p_{4} + 6 k_{3}^{2} m_{C}^{2} p_{1} p_{3} \\left(- k_{3} + p_{4}\\right)^{2} + 2 k_{3}^{2} m_{C}^{2} p_{1} \\left(- k_{3} + p_{4}\\right)^{3} - k_{3}^{2} m_{C}^{2} p_{3}^{4} - 4 k_{3}^{2} m_{C}^{2} p_{3}^{3} p_{4} - 7 k_{3}^{2} m_{C}^{2} p_{3}^{2} \\left(- k_{3} + p_{4}\\right)^{2} - 6 k_{3}^{2} m_{C}^{2} p_{3} \\left(- k_{3} + p_{4}\\right)^{3} - 2 k_{3}^{2} m_{C}^{2} \\left(- k_{3} + p_{4}\\right)^{4} + k_{3}^{2} p_{1}^{2} p_{3}^{2} \\left(- k_{3} + p_{4}\\right)^{2} + 2 k_{3}^{2} p_{1}^{2} p_{3} \\left(- k_{3} + p_{4}\\right)^{3} + k_{3}^{2} p_{1}^{2} \\left(- k_{3} + p_{4}\\right)^{4} - 2 k_{3}^{2} p_{1} p_{3}^{3} \\left(- k_{3} + p_{4}\\right)^{2} - 6 k_{3}^{2} p_{1} p_{3}^{2} \\left(- k_{3} + p_{4}\\right)^{3} - 6 k_{3}^{2} p_{1} p_{3} \\left(- k_{3} + p_{4}\\right)^{4} - 2 k_{3}^{2} p_{1} \\left(- k_{3} + p_{4}\\right)^{5} + k_{3}^{2} p_{3}^{4} \\left(- k_{3} + p_{4}\\right)^{2} + 4 k_{3}^{2} p_{3}^{3} \\left(- k_{3} + p_{4}\\right)^{3} + 6 k_{3}^{2} p_{3}^{2} \\left(- k_{3} + p_{4}\\right)^{4} + 4 k_{3}^{2} p_{3} \\left(- k_{3} + p_{4}\\right)^{5} + k_{3}^{2} \\left(- k_{3} + p_{4}\\right)^{6} - 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_{3} + 2 k_{3} m_{A}^{2} m_{C}^{4} p_{3} - 2 k_{3} m_{A}^{2} m_{C}^{2} p_{1}^{2} p_{3} + 6 k_{3} m_{A}^{2} m_{C}^{2} p_{1} p_{3}^{2} - 4 k_{3} m_{A}^{2} m_{C}^{2} p_{3}^{3} + m_{A}^{2} m_{B}^{2} m_{C}^{4} - m_{A}^{2} m_{B}^{2} m_{C}^{2} p_{1}^{2} + 2 m_{A}^{2} m_{B}^{2} m_{C}^{2} p_{1} p_{3} + 2 m_{A}^{2} m_{B}^{2} m_{C}^{2} p_{1} p_{4} - m_{A}^{2} m_{B}^{2} m_{C}^{2} p_{3}^{2} - 2 m_{A}^{2} m_{B}^{2} m_{C}^{2} p_{3} p_{4} - 2 m_{A}^{2} m_{B}^{2} m_{C}^{2} \\left(- k_{3} + p_{4}\\right)^{2} + m_{A}^{2} m_{B}^{2} p_{1}^{2} \\left(- k_{3} + p_{4}\\right)^{2} - 2 m_{A}^{2} m_{B}^{2} p_{1} p_{3} \\left(- k_{3} + p_{4}\\right)^{2} - 2 m_{A}^{2} m_{B}^{2} p_{1} \\left(- k_{3} + p_{4}\\right)^{3} + m_{A}^{2} m_{B}^{2} p_{3}^{2} \\left(- k_{3} + p_{4}\\right)^{2} + 2 m_{A}^{2} m_{B}^{2} p_{3} \\left(- k_{3} + p_{4}\\right)^{3} + m_{A}^{2} m_{B}^{2} \\left(- k_{3} + p_{4}\\right)^{4} - m_{A}^{2} m_{C}^{4} p_{3}^{2} - 2 m_{A}^{2} m_{C}^{4} p_{3} p_{4} - m_{A}^{2} m_{C}^{4} \\left(- k_{3} + p_{4}\\right)^{2} + m_{A}^{2} m_{C}^{2} p_{1}^{2} p_{3}^{2} + 2 m_{A}^{2} m_{C}^{2} p_{1}^{2} p_{3} p_{4} + m_{A}^{2} m_{C}^{2} p_{1}^{2} \\left(- k_{3} + p_{4}\\right)^{2} - 2 m_{A}^{2} m_{C}^{2} p_{1} p_{3}^{3} - 6 m_{A}^{2} m_{C}^{2} p_{1} p_{3}^{2} p_{4} - 6 m_{A}^{2} m_{C}^{2} p_{1} p_{3} \\left(- k_{3} + p_{4}\\right)^{2} - 2 m_{A}^{2} m_{C}^{2} p_{1} \\left(- k_{3} + p_{4}\\right)^{3} + m_{A}^{2} m_{C}^{2} p_{3}^{4} + 4 m_{A}^{2} m_{C}^{2} p_{3}^{3} p_{4} + 7 m_{A}^{2} m_{C}^{2} p_{3}^{2} \\left(- k_{3} + p_{4}\\right)^{2} + 6 m_{A}^{2} m_{C}^{2} p_{3} \\left(- k_{3} + p_{4}\\right)^{3} + 2 m_{A}^{2} m_{C}^{2} \\left(- k_{3} + p_{4}\\right)^{4} - m_{A}^{2} p_{1}^{2} p_{3}^{2} \\left(- k_{3} + p_{4}\\right)^{2} - 2 m_{A}^{2} p_{1}^{2} p_{3} \\left(- k_{3} + p_{4}\\right)^{3} - m_{A}^{2} p_{1}^{2} \\left(- k_{3} + p_{4}\\right)^{4} + 2 m_{A}^{2} p_{1} p_{3}^{3} \\left(- k_{3} + p_{4}\\right)^{2} + 6 m_{A}^{2} p_{1} p_{3}^{2} \\left(- k_{3} + p_{4}\\right)^{3} + 6 m_{A}^{2} p_{1} p_{3} \\left(- k_{3} + p_{4}\\right)^{4} + 2 m_{A}^{2} p_{1} \\left(- k_{3} + p_{4}\\right)^{5} - m_{A}^{2} p_{3}^{4} \\left(- k_{3} + p_{4}\\right)^{2} - 4 m_{A}^{2} p_{3}^{3} \\left(- k_{3} + p_{4}\\right)^{3} - 6 m_{A}^{2} p_{3}^{2} \\left(- k_{3} + p_{4}\\right)^{4} - 4 m_{A}^{2} p_{3} \\left(- k_{3} + p_{4}\\right)^{5} - m_{A}^{2} \\left(- k_{3} + p_{4}\\right)^{6}}$"
      ],
      "text/plain": [
       "g**4*DiracDelta(p_1 + p_2 - p_3 - p_4)/(2*k_3**3*m_B**2*m_C**2*p_1 - 2*k_3**3*m_B**2*m_C**2*p_3 - 2*k_3**3*m_C**4*p_3 + 2*k_3**3*m_C**2*p_1**2*p_3 - 6*k_3**3*m_C**2*p_1*p_3**2 + 4*k_3**3*m_C**2*p_3**3 - k_3**2*m_B**2*m_C**4 + k_3**2*m_B**2*m_C**2*p_1**2 - 2*k_3**2*m_B**2*m_C**2*p_1*p_3 - 2*k_3**2*m_B**2*m_C**2*p_1*p_4 + k_3**2*m_B**2*m_C**2*p_3**2 + 2*k_3**2*m_B**2*m_C**2*p_3*p_4 + 2*k_3**2*m_B**2*m_C**2*(-k_3 + p_4)**2 - k_3**2*m_B**2*p_1**2*(-k_3 + p_4)**2 + 2*k_3**2*m_B**2*p_1*p_3*(-k_3 + p_4)**2 + 2*k_3**2*m_B**2*p_1*(-k_3 + p_4)**3 - k_3**2*m_B**2*p_3**2*(-k_3 + p_4)**2 - 2*k_3**2*m_B**2*p_3*(-k_3 + p_4)**3 - k_3**2*m_B**2*(-k_3 + p_4)**4 + k_3**2*m_C**4*p_3**2 + 2*k_3**2*m_C**4*p_3*p_4 + k_3**2*m_C**4*(-k_3 + p_4)**2 - k_3**2*m_C**2*p_1**2*p_3**2 - 2*k_3**2*m_C**2*p_1**2*p_3*p_4 - k_3**2*m_C**2*p_1**2*(-k_3 + p_4)**2 + 2*k_3**2*m_C**2*p_1*p_3**3 + 6*k_3**2*m_C**2*p_1*p_3**2*p_4 + 6*k_3**2*m_C**2*p_1*p_3*(-k_3 + p_4)**2 + 2*k_3**2*m_C**2*p_1*(-k_3 + p_4)**3 - k_3**2*m_C**2*p_3**4 - 4*k_3**2*m_C**2*p_3**3*p_4 - 7*k_3**2*m_C**2*p_3**2*(-k_3 + p_4)**2 - 6*k_3**2*m_C**2*p_3*(-k_3 + p_4)**3 - 2*k_3**2*m_C**2*(-k_3 + p_4)**4 + k_3**2*p_1**2*p_3**2*(-k_3 + p_4)**2 + 2*k_3**2*p_1**2*p_3*(-k_3 + p_4)**3 + k_3**2*p_1**2*(-k_3 + p_4)**4 - 2*k_3**2*p_1*p_3**3*(-k_3 + p_4)**2 - 6*k_3**2*p_1*p_3**2*(-k_3 + p_4)**3 - 6*k_3**2*p_1*p_3*(-k_3 + p_4)**4 - 2*k_3**2*p_1*(-k_3 + p_4)**5 + k_3**2*p_3**4*(-k_3 + p_4)**2 + 4*k_3**2*p_3**3*(-k_3 + p_4)**3 + 6*k_3**2*p_3**2*(-k_3 + p_4)**4 + 4*k_3**2*p_3*(-k_3 + p_4)**5 + k_3**2*(-k_3 + p_4)**6 - 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_3 + 2*k_3*m_A**2*m_C**4*p_3 - 2*k_3*m_A**2*m_C**2*p_1**2*p_3 + 6*k_3*m_A**2*m_C**2*p_1*p_3**2 - 4*k_3*m_A**2*m_C**2*p_3**3 + m_A**2*m_B**2*m_C**4 - m_A**2*m_B**2*m_C**2*p_1**2 + 2*m_A**2*m_B**2*m_C**2*p_1*p_3 + 2*m_A**2*m_B**2*m_C**2*p_1*p_4 - m_A**2*m_B**2*m_C**2*p_3**2 - 2*m_A**2*m_B**2*m_C**2*p_3*p_4 - 2*m_A**2*m_B**2*m_C**2*(-k_3 + p_4)**2 + m_A**2*m_B**2*p_1**2*(-k_3 + p_4)**2 - 2*m_A**2*m_B**2*p_1*p_3*(-k_3 + p_4)**2 - 2*m_A**2*m_B**2*p_1*(-k_3 + p_4)**3 + m_A**2*m_B**2*p_3**2*(-k_3 + p_4)**2 + 2*m_A**2*m_B**2*p_3*(-k_3 + p_4)**3 + m_A**2*m_B**2*(-k_3 + p_4)**4 - m_A**2*m_C**4*p_3**2 - 2*m_A**2*m_C**4*p_3*p_4 - m_A**2*m_C**4*(-k_3 + p_4)**2 + m_A**2*m_C**2*p_1**2*p_3**2 + 2*m_A**2*m_C**2*p_1**2*p_3*p_4 + m_A**2*m_C**2*p_1**2*(-k_3 + p_4)**2 - 2*m_A**2*m_C**2*p_1*p_3**3 - 6*m_A**2*m_C**2*p_1*p_3**2*p_4 - 6*m_A**2*m_C**2*p_1*p_3*(-k_3 + p_4)**2 - 2*m_A**2*m_C**2*p_1*(-k_3 + p_4)**3 + m_A**2*m_C**2*p_3**4 + 4*m_A**2*m_C**2*p_3**3*p_4 + 7*m_A**2*m_C**2*p_3**2*(-k_3 + p_4)**2 + 6*m_A**2*m_C**2*p_3*(-k_3 + p_4)**3 + 2*m_A**2*m_C**2*(-k_3 + p_4)**4 - m_A**2*p_1**2*p_3**2*(-k_3 + p_4)**2 - 2*m_A**2*p_1**2*p_3*(-k_3 + p_4)**3 - m_A**2*p_1**2*(-k_3 + p_4)**4 + 2*m_A**2*p_1*p_3**3*(-k_3 + p_4)**2 + 6*m_A**2*p_1*p_3**2*(-k_3 + p_4)**3 + 6*m_A**2*p_1*p_3*(-k_3 + p_4)**4 + 2*m_A**2*p_1*(-k_3 + p_4)**5 - m_A**2*p_3**4*(-k_3 + p_4)**2 - 4*m_A**2*p_3**3*(-k_3 + p_4)**3 - 6*m_A**2*p_3**2*(-k_3 + p_4)**4 - 4*m_A**2*p_3*(-k_3 + p_4)**5 - m_A**2*(-k_3 + p_4)**6)"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ME = integrate(ME, (k4,-oo,oo))\n",
    "ME"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "04961526",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\\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_{C} - p_{4}\\right) \\left(k_{3} + m_{C} - p_{4}\\right) \\left(k_{3} - m_{B} - p_{3} - p_{4}\\right) \\left(k_{3} + m_{B} - p_{3} - p_{4}\\right) \\left(k_{3} - m_{C} + p_{1} - p_{3} - p_{4}\\right) \\left(k_{3} + m_{C} + p_{1} - p_{3} - p_{4}\\right)}\n"
     ]
    }
   ],
   "source": [
    "#ME = ME.subs({p3: -p4 +p1 + p2})\n",
    "print(latex(factor(ME)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "a6dc9558",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "0c1e6c57",
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