{
 "metadata": {
  "name": "Matching with Sympy"
 },
 "nbformat": 3,
 "nbformat_minor": 0,
 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Manipulating The Results"
     ]
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "1.1 Simplifications (1)"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "#The first thing to do is to make the Toolbox available\n",
      "import sys\n",
      "#pretty printing\n",
      "%load_ext sympy.interactive.ipythonprinting\n",
      "sys.path.append('Source/Output')\n",
      "from Toolbox import * "
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 1
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "#Now we can load a result which will serve as an example. For that purpose use the load function of the Toolbox\n",
      "RGEs,info = loadmodel('SM-BL.pickle')\n",
      "info"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 2,
       "text": [
        "Generated by PyR@TE for SM."
       ]
      }
     ],
     "prompt_number": 2
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "#Print all the results available in the dictionary\n",
      "RGEs.keys()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 3,
       "text": [
        "[\\lambda_1, SU3c, \\mu_1, SU2L, U1, \\lambda_3, U1BL, m_H, Y_l, Y_H, \\lambda_2, \n",
        "Y_e, Y_d, Y_u]"
       ]
      }
     ],
     "prompt_number": 3
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "#Let's have a look at the lambda_1 coupling and we use the getoneloop function to get rid of the 4pi factors\n",
      "Lambda1 = getoneloop(RGEs['\\\\lambda_1'])\n",
      "Lambda1"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{3}{8} g_{1}^{4} + \\frac{3}{4} g_{1}^{2} g_{SU2L}^{2} - 3 g_{1}^{2} \\lambda_{1} + \\frac{9}{8} g_{SU2L}^{4} - 9 g_{SU2L}^{2} \\lambda_{1} + 24 \\lambda_{1}^{2} + 12 \\lambda_{1} \\operatorname{trace}{\\left (Y_{d}^{\\dagger},Y_{d} \\right )} + 4 \\lambda_{1} \\operatorname{trace}{\\left (Y_{e}^{\\dagger},Y_{e} \\right )} + 4 \\lambda_{1} \\operatorname{trace}{\\left (Y_{l}^{\\dagger},Y_{l} \\right )} + 12 \\lambda_{1} \\operatorname{trace}{\\left (Y_{u}^{\\dagger},Y_{u} \\right )} + \\lambda_{3}^{2} - 6 \\operatorname{trace}{\\left (Y_{d},Y_{d}^{\\dagger},Y_{d},Y_{d}^{\\dagger} \\right )} - 2 \\operatorname{trace}{\\left (Y_{e},Y_{e}^{\\dagger},Y_{e},Y_{e}^{\\dagger} \\right )} - 2 \\operatorname{trace}{\\left (Y_{l},Y_{l}^{\\dagger},Y_{l},Y_{l}^{\\dagger} \\right )} - 6 \\operatorname{trace}{\\left (Y_{u},Y_{u}^{\\dagger},Y_{u},Y_{u}^{\\dagger} \\right )}$$"
       ],
       "output_type": "pyout",
       "png": 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STB29QL80E3q9dO2VbTKeyppHH+vgWsxGfFd9OJTmRZitgXMa8JXE53OAdwPP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\noiiKoiiKoiiKoiiKoiiKoihKj/h/nA/OB1KutgsAAAAASUVORK5CYII=\n",
       "prompt_number": 4,
       "text": [
        "    4       2       2                      4                                  \n",
        "3\u22c5g\u2081    3\u22c5g\u2081 \u22c5g_SU2L        2      9\u22c5g_SU2L            2           2          \n",
        "\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 - 3\u22c5g\u2081 \u22c5\u03bb\u2081 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 - 9\u22c5g_SU2L \u22c5\u03bb\u2081 + 24\u22c5\u03bb\u2081  + 12\u22c5\u03bb\u2081\u22c5t\n",
        "  8           4                        8                                      \n",
        "\n",
        "                                                                              \n",
        "    \u239b   \u2020     \u239e             \u239b  \u2020    \u239e             \u239b   \u2020     \u239e              \u239b  \n",
        "race\u239dY_d , Y_d\u23a0 + 4\u22c5\u03bb\u2081\u22c5trace\u239dY\u2091 , Y\u2091\u23a0 + 4\u22c5\u03bb\u2081\u22c5trace\u239dY_l , Y_l\u23a0 + 12\u22c5\u03bb\u2081\u22c5trace\u239dY\u1d64\n",
        "                                                                              \n",
        "\n",
        "                                                                              \n",
        "\u2020    \u239e     2          \u239b        \u2020          \u2020\u239e          \u239b      \u2020        \u2020\u239e      \n",
        " , Y\u1d64\u23a0 + \u03bb\u2083  - 6\u22c5trace\u239dY_d, Y_d , Y_d, Y_d \u23a0 - 2\u22c5trace\u239dY\u2091, Y\u2091 , Y\u2091, Y\u2091 \u23a0 - 2\u22c5t\n",
        "                                                                              \n",
        "\n",
        "                                                      \n",
        "    \u239b        \u2020          \u2020\u239e          \u239b      \u2020        \u2020\u239e\n",
        "race\u239dY_l, Y_l , Y_l, Y_l \u23a0 - 6\u22c5trace\u239dY\u1d64, Y\u1d64 , Y\u1d64, Y\u1d64 \u23a0\n",
        "                                                      "
       ]
      }
     ],
     "prompt_number": 4
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Let's now imagine that we want to neglect all yukawas but the top quark. Then Yukawa matrix Yu is then a scalar.\n",
      "traces and MatMul are our own declared classes and that therefore they need to be loaded \n",
      "before being able to do any matching with it."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "#Let's declare symbols for each yukawas\n",
      "yl = Symbol('Y_l',commutative=False)\n",
      "ye= Symbol('Y_e',commutative=False)\n",
      "yu= Symbol('Y_u',commutative=False)\n",
      "yd= Symbol('Y_d',commutative=False)\n",
      "yt=Symbol('yt',real=True)\n",
      "#List of substitutions\n",
      "Subs = ((yl,0),(yd,0),(yl,0),(ye,0),(yu,yt))\n",
      "Lambda1 = Lambda1.subs(Subs)\n",
      "Lambda1"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{3}{8} g_{1}^{4} + \\frac{3}{4} g_{1}^{2} g_{SU2L}^{2} - 3 g_{1}^{2} \\lambda_{1} + \\frac{9}{8} g_{SU2L}^{4} - 9 g_{SU2L}^{2} \\lambda_{1} + 24 \\lambda_{1}^{2} + 20 \\lambda_{1} \\operatorname{trace}{\\left (0,0 \\right )} + 12 \\lambda_{1} \\operatorname{trace}{\\left (yt,yt \\right )} + \\lambda_{3}^{2} - 10 \\operatorname{trace}{\\left (0,0,0,0 \\right )} - 6 \\operatorname{trace}{\\left (yt,yt,yt,yt \\right )}$$"
       ],
       "output_type": "pyout",
       "png": 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       "prompt_number": 5,
       "text": [
        "    4       2       2                      4                                  \n",
        "3\u22c5g\u2081    3\u22c5g\u2081 \u22c5g_SU2L        2      9\u22c5g_SU2L            2           2          \n",
        "\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 - 3\u22c5g\u2081 \u22c5\u03bb\u2081 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 - 9\u22c5g_SU2L \u22c5\u03bb\u2081 + 24\u22c5\u03bb\u2081  + 20\u22c5\u03bb\u2081\u22c5t\n",
        "  8           4                        8                                      \n",
        "\n",
        "                                                                              \n",
        "                                     2                                        \n",
        "race(0, 0) + 12\u22c5\u03bb\u2081\u22c5trace(yt, yt) + \u03bb\u2083  - 10\u22c5trace(0, 0, 0, 0) - 6\u22c5trace(yt, yt\n",
        "                                                                              \n",
        "\n",
        "         \n",
        "         \n",
        ", yt, yt)\n",
        "         "
       ]
      }
     ],
     "prompt_number": 5
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "#load the trace and MatMul classes\n",
      "from RGEsmathModule import trace,MatM\n",
      "#declare some wild cards for the mapping\n",
      "p,q,r,s,t,u,v,w = map(Wild,['p','q','r','s','t','u','v','w'])\n",
      "#Remove the 0 traces\n",
      "Lambda1 =Lambda1.replace(trace(p,q),p*q).replace(trace(p,q,r,s),p*q*r*s)\n",
      "Lambda1"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{3}{8} g_{1}^{4} + \\frac{3}{4} g_{1}^{2} g_{SU2L}^{2} - 3 g_{1}^{2} \\lambda_{1} + \\frac{9}{8} g_{SU2L}^{4} - 9 g_{SU2L}^{2} \\lambda_{1} + 24 \\lambda_{1}^{2} + 12 \\lambda_{1} yt^{2} + \\lambda_{3}^{2} - 6 yt^{4}$$"
       ],
       "output_type": "pyout",
       "png": 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atvVTpT5N5q0smulNL9B5mmlnu9RdVlbJhhTXJJwNxEzUwHkb2eMq4N5a36hCgorXNFRw\nhLfluAgVQtH1LdqBY1BNNShUT0Ozou5ELVmHoyo6LW85HA6HwwC70HP9m4XItd8JrCLbrEaHwzSd\nnrccDofDkYNg08oX0CD0Y9HaNVOQC/SRel7LGNughRr3RW7NV9EMM4ejKjo1bzkcDofD4XA4HA6H\nw+FwOBwOh8PhcDgcDofD4XA4HA6Hw+FwOBwdxf8B1qy4q3N4IhAAAAAASUVORK5CYII=\n",
       "prompt_number": 43,
       "text": [
        "    4       2       2                      4                                  \n",
        "3\u22c5g\u2081    3\u22c5g\u2081 \u22c5g_SU2L        2      9\u22c5g_SU2L            2           2          \n",
        "\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 - 3\u22c5g\u2081 \u22c5\u03bb\u2081 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 - 9\u22c5g_SU2L \u22c5\u03bb\u2081 + 24\u22c5\u03bb\u2081  + 12\u22c5\u03bb\u2081\u22c5y\n",
        "  8           4                        8                                      \n",
        "\n",
        "                \n",
        " 2     2       4\n",
        "t  + \u03bb\u2083  - 6\u22c5yt \n",
        "                "
       ]
      }
     ],
     "prompt_number": 43
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "#Now Let's try to obtain the beta function for the parameter lambda_1 - lambda_3 in the same approximation\n",
      "#We first filter the rges for the lambda3 result\n",
      "yh = Symbol('Y_H',real=True)\n",
      "YH = Symbol('Y_H',commutative=False)\n",
      "Lambda3=getoneloop(RGEs['\\\\lambda_3']).subs(Subs).replace(trace(p,q),p*q).replace(trace(p,q,r,s),p*q*r*s).subs(YH,yh)\n",
      "Lambda3"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$4 Y_{H}^{2} \\lambda_{3} - \\frac{3}{2} g_{1}^{2} \\lambda_{3} - \\frac{9}{2} g_{SU2L}^{2} \\lambda_{3} - 24 g_{U1BL}^{2} \\lambda_{3} + 12 \\lambda_{1} \\lambda_{3} + 8 \\lambda_{2} \\lambda_{3} + 4 \\lambda_{3}^{2} + 6 \\lambda_{3} yt^{2}$$"
       ],
       "output_type": "pyout",
       "png": 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       "prompt_number": 44,
       "text": [
        "                2              2                                              \n",
        "     2      3\u22c5g\u2081 \u22c5\u03bb\u2083   9\u22c5g_SU2L \u22c5\u03bb\u2083            2                              \n",
        "4\u22c5Y_H \u22c5\u03bb\u2083 - \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 - \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 - 24\u22c5g_U1BL \u22c5\u03bb\u2083 + 12\u22c5\u03bb\u2081\u22c5\u03bb\u2083 + 8\u22c5\u03bb\u2082\u22c5\u03bb\u2083 + 4\u22c5\u03bb\n",
        "               2            2                                                 \n",
        "\n",
        "             \n",
        " 2          2\n",
        "\u2083  + 6\u22c5\u03bb\u2083\u22c5yt \n",
        "             "
       ]
      }
     ],
     "prompt_number": 44
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "#We can now do the difference\n",
      "Diff = (Lambda3 - Lambda1).expand()\n",
      "Diff"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$4 Y_{H}^{2} \\lambda_{3} - \\frac{3}{8} g_{1}^{4} - \\frac{3}{4} g_{1}^{2} g_{SU2L}^{2} + 3 g_{1}^{2} \\lambda_{1} - \\frac{3}{2} g_{1}^{2} \\lambda_{3} - \\frac{9}{8} g_{SU2L}^{4} + 9 g_{SU2L}^{2} \\lambda_{1} - \\frac{9}{2} g_{SU2L}^{2} \\lambda_{3} - 24 g_{U1BL}^{2} \\lambda_{3} - 24 \\lambda_{1}^{2} + 12 \\lambda_{1} \\lambda_{3} - 12 \\lambda_{1} yt^{2} + 8 \\lambda_{2} \\lambda_{3} + 3 \\lambda_{3}^{2} + 6 \\lambda_{3} yt^{2} + 6 yt^{4}$$"
       ],
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAABIQAAAAfCAYAAACWPBalAAAABHNCSVQICAgIfAhkiAAAFopJREFU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\nKIqiKIrScf4fAvoZ1NKoLw0AAAAASUVORK5CYII=\n",
       "prompt_number": 45,
       "text": [
        "                4       2       2                  2              4           \n",
        "     2      3\u22c5g\u2081    3\u22c5g\u2081 \u22c5g_SU2L        2      3\u22c5g\u2081 \u22c5\u03bb\u2083   9\u22c5g_SU2L            \n",
        "4\u22c5Y_H \u22c5\u03bb\u2083 - \u2500\u2500\u2500\u2500\u2500 - \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + 3\u22c5g\u2081 \u22c5\u03bb\u2081 - \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 - \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + 9\u22c5g_SU2L\n",
        "              8           4                       2           8               \n",
        "\n",
        "               2                                                              \n",
        "2      9\u22c5g_SU2L \u22c5\u03bb\u2083            2           2                      2           \n",
        " \u22c5\u03bb\u2081 - \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 - 24\u22c5g_U1BL \u22c5\u03bb\u2083 - 24\u22c5\u03bb\u2081  + 12\u22c5\u03bb\u2081\u22c5\u03bb\u2083 - 12\u22c5\u03bb\u2081\u22c5yt  + 8\u22c5\u03bb\u2082\u22c5\u03bb\u2083 \n",
        "            2                                                                 \n",
        "\n",
        "                          \n",
        "      2          2       4\n",
        "+ 3\u22c5\u03bb\u2083  + 6\u22c5\u03bb\u2083\u22c5yt  + 6\u22c5yt \n",
        "                          "
       ]
      }
     ],
     "prompt_number": 45
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Note that if one wants to export those results into latex it suffices to right click on the result and then onto Show Math As--> TeX Command"
     ]
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "1.2 Simplifications (2)"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Let's now have a look at the Yukawa rge and try to do some simplification. The goal is to see how to match the Yukawa Matrices that are not inside the traces"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "bYl = getoneloop(RGEs['Y_l'])\n",
      "bYl"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$- \\frac{3}{4} g_{1}^{2} \\operatorname{Y\\_l}{\\left (Lbar_{f},nuR_{f} \\right )} - \\frac{9}{4} g_{SU2L}^{2} \\operatorname{Y\\_l}{\\left (Lbar_{f},nuR_{f} \\right )} - 6 g_{U1BL}^{2} \\operatorname{Y\\_l}{\\left (Lbar_{f},nuR_{f} \\right )} + 3 \\operatorname{Y\\_l}{\\left (Lbar_{f},nuR_{f} \\right )} \\operatorname{trace}{\\left (Y_{d}^{\\dagger},Y_{d} \\right )} + \\operatorname{Y\\_l}{\\left (Lbar_{f},nuR_{f} \\right )} \\operatorname{trace}{\\left (Y_{e}^{\\dagger},Y_{e} \\right )} + \\operatorname{Y\\_l}{\\left (Lbar_{f},nuR_{f} \\right )} \\operatorname{trace}{\\left (Y_{l}^{\\dagger},Y_{l} \\right )} + 3 \\operatorname{Y\\_l}{\\left (Lbar_{f},nuR_{f} \\right )} \\operatorname{trace}{\\left (Y_{u}^{\\dagger},Y_{u} \\right )} - \\frac{3}{2} \\operatorname{MatM}{\\left (\\begin{pmatrix}Y_{e}, & Y_{e}^{\\dagger}, & Y_{l}\\end{pmatrix},Lbar_{f},nuR_{f} \\right )} + 2 \\operatorname{MatM}{\\left (\\begin{pmatrix}Y_{l}, & Y_{H}^{\\dagger}, & Y_{H}\\end{pmatrix},Lbar_{f},nuR_{f} \\right )} + \\frac{3}{2} \\operatorname{MatM}{\\left (\\begin{pmatrix}Y_{l}, & Y_{l}^{\\dagger}, & Y_{l}\\end{pmatrix},Lbar_{f},nuR_{f} \\right )}$$"
       ],
       "output_type": "pyout",
       "prompt_number": 14,
       "text": [
        "      2                              2                                        \n",
        "  3\u22c5g\u2081 \u22c5Y_l(Lbar_f, nuR_f)   9\u22c5g_SU2L \u22c5Y_l(Lbar_f, nuR_f)           2         \n",
        "- \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 - \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 - 6\u22c5g_U1BL \u22c5Y_l(Lbar\n",
        "             4                            4                                   \n",
        "\n",
        "                                                                              \n",
        "                                       \u239b   \u2020     \u239e                           \u239b\n",
        "_f, nuR_f) + 3\u22c5Y_l(Lbar_f, nuR_f)\u22c5trace\u239dY_d , Y_d\u23a0 + Y_l(Lbar_f, nuR_f)\u22c5trace\u239d\n",
        "                                                                              \n",
        "\n",
        "                                                                              \n",
        "  \u2020    \u239e                           \u239b   \u2020     \u239e                             \u239b  \n",
        "Y\u2091 , Y\u2091\u23a0 + Y_l(Lbar_f, nuR_f)\u22c5trace\u239dY_l , Y_l\u23a0 + 3\u22c5Y_l(Lbar_f, nuR_f)\u22c5trace\u239dY\u1d64\n",
        "                                                                              \n",
        "\n",
        "               \u239b\u239b      \u2020     \u239e               \u239e                                \n",
        "\u2020    \u239e   3\u22c5MatM\u239d\u239dY\u2091, Y\u2091 , Y_l\u23a0, Lbar_f, nuR_f\u23a0         \u239b\u239b        \u2020     \u239e      \n",
        " , Y\u1d64\u23a0 - \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + 2\u22c5MatM\u239d\u239dY_l, Y_H , Y_H\u23a0, Lbar\n",
        "                           2                                                  \n",
        "\n",
        "                   \u239b\u239b        \u2020     \u239e               \u239e\n",
        "         \u239e   3\u22c5MatM\u239d\u239dY_l, Y_l , Y_l\u23a0, Lbar_f, nuR_f\u23a0\n",
        "_f, nuR_f\u23a0 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "                                2                   "
       ]
      }
     ],
     "prompt_number": 14
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "#If we want to match the object representing the Yukawa with indices i.e. outside of the traces we need to use functions\n",
      "FYl = Function('Y_l')\n",
      "#Let's get rid of the indices and translate Y_l into the Symbol Yl that we have introduced above\n",
      "bYl = bYl.replace(FYl(p,q),yl)\n",
      "bYl"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$- \\frac{3}{4} g_{1}^{2} Y_{l} - \\frac{9}{4} g_{SU2L}^{2} Y_{l} - 6 g_{U1BL}^{2} Y_{l} + 3 \\operatorname{trace}{\\left (Y_{d}^{\\dagger},Y_{d} \\right )} Y_{l} + \\operatorname{trace}{\\left (Y_{e}^{\\dagger},Y_{e} \\right )} Y_{l} + \\operatorname{trace}{\\left (Y_{l}^{\\dagger},Y_{l} \\right )} Y_{l} + 3 \\operatorname{trace}{\\left (Y_{u}^{\\dagger},Y_{u} \\right )} Y_{l} - \\frac{3}{2} \\operatorname{MatM}{\\left (\\begin{pmatrix}Y_{e}, & Y_{e}^{\\dagger}, & Y_{l}\\end{pmatrix},Lbar_{f},nuR_{f} \\right )} + 2 \\operatorname{MatM}{\\left (\\begin{pmatrix}Y_{l}, & Y_{H}^{\\dagger}, & Y_{H}\\end{pmatrix},Lbar_{f},nuR_{f} \\right )} + \\frac{3}{2} \\operatorname{MatM}{\\left (\\begin{pmatrix}Y_{l}, & Y_{l}^{\\dagger}, & Y_{l}\\end{pmatrix},Lbar_{f},nuR_{f} \\right )}$$"
       ],
       "output_type": "pyout",
       "prompt_number": 23,
       "text": [
        "      2               2                                                       \n",
        "  3\u22c5g\u2081 \u22c5Y_l   9\u22c5g_SU2L \u22c5Y_l           2              \u239b   \u2020     \u239e            \u239b \n",
        "- \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 - \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 - 6\u22c5g_U1BL \u22c5Y_l + 3\u22c5trace\u239dY_d , Y_d\u23a0\u22c5Y_l + trace\u239dY\n",
        "      4             4                                                         \n",
        "\n",
        "                                                                  \u239b\u239b      \u2020   \n",
        " \u2020    \u239e            \u239b   \u2020     \u239e              \u239b  \u2020    \u239e       3\u22c5MatM\u239d\u239dY\u2091, Y\u2091 , Y\n",
        "\u2091 , Y\u2091\u23a0\u22c5Y_l + trace\u239dY_l , Y_l\u23a0\u22c5Y_l + 3\u22c5trace\u239dY\u1d64 , Y\u1d64\u23a0\u22c5Y_l - \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "                                                                              \n",
        "\n",
        "  \u239e               \u239e                                                   \u239b\u239b      \n",
        "_l\u23a0, Lbar_f, nuR_f\u23a0         \u239b\u239b        \u2020     \u239e               \u239e   3\u22c5MatM\u239d\u239dY_l, Y\n",
        "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + 2\u22c5MatM\u239d\u239dY_l, Y_H , Y_H\u23a0, Lbar_f, nuR_f\u23a0 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "2                                                                             \n",
        "\n",
        "  \u2020     \u239e               \u239e\n",
        "_l , Y_l\u23a0, Lbar_f, nuR_f\u23a0\n",
        "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "     2                   "
       ]
      }
     ],
     "prompt_number": 23
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "#Now to be consistent we should remove the indices in the MatM function and also transforms all the Yukawa as scalars\n",
      "bYl.replace(MatM((p,q,r),u,v),MatM((p,q,r)))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$- \\frac{3}{4} g_{1}^{2} Y_{l} - \\frac{9}{4} g_{SU2L}^{2} Y_{l} - 6 g_{U1BL}^{2} Y_{l} + 3 \\operatorname{trace}{\\left (Y_{d}^{\\dagger},Y_{d} \\right )} Y_{l} + \\operatorname{trace}{\\left (Y_{e}^{\\dagger},Y_{e} \\right )} Y_{l} + \\operatorname{trace}{\\left (Y_{l}^{\\dagger},Y_{l} \\right )} Y_{l} + 3 \\operatorname{trace}{\\left (Y_{u}^{\\dagger},Y_{u} \\right )} Y_{l} - \\frac{3}{2} \\operatorname{MatM}{\\left (\\begin{pmatrix}Y_{e}, & Y_{e}^{\\dagger}, & Y_{l}\\end{pmatrix} \\right )} + 2 \\operatorname{MatM}{\\left (\\begin{pmatrix}Y_{l}, & Y_{H}^{\\dagger}, & Y_{H}\\end{pmatrix} \\right )} + \\frac{3}{2} \\operatorname{MatM}{\\left (\\begin{pmatrix}Y_{l}, & Y_{l}^{\\dagger}, & Y_{l}\\end{pmatrix} \\right )}$$"
       ],
       "output_type": "pyout",
       "prompt_number": 30,
       "text": [
        "      2               2                                                       \n",
        "  3\u22c5g\u2081 \u22c5Y_l   9\u22c5g_SU2L \u22c5Y_l           2              \u239b   \u2020     \u239e            \u239b \n",
        "- \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 - \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 - 6\u22c5g_U1BL \u22c5Y_l + 3\u22c5trace\u239dY_d , Y_d\u23a0\u22c5Y_l + trace\u239dY\n",
        "      4             4                                                         \n",
        "\n",
        "                                                                  \u239b\u239b      \u2020   \n",
        " \u2020    \u239e            \u239b   \u2020     \u239e              \u239b  \u2020    \u239e       3\u22c5MatM\u239d\u239dY\u2091, Y\u2091 , Y\n",
        "\u2091 , Y\u2091\u23a0\u22c5Y_l + trace\u239dY_l , Y_l\u23a0\u22c5Y_l + 3\u22c5trace\u239dY\u1d64 , Y\u1d64\u23a0\u22c5Y_l - \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "                                                                      2       \n",
        "\n",
        "  \u239e\u239e                                    \u239b\u239b        \u2020     \u239e\u239e\n",
        "_l\u23a0\u23a0         \u239b\u239b        \u2020     \u239e\u239e   3\u22c5MatM\u239d\u239dY_l, Y_l , Y_l\u23a0\u23a0\n",
        "\u2500\u2500\u2500\u2500 + 2\u22c5MatM\u239d\u239dY_l, Y_H , Y_H\u23a0\u23a0 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "                                             2            "
       ]
      }
     ],
     "prompt_number": 30
    }
   ],
   "metadata": {}
  }
 ]
}