| 1 | (************** Content-type: application/mathematica **************
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| 2 |
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| 3 | Mathematica-Compatible Notebook
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| 4 |
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| 5 | This notebook can be used with any Mathematica-compatible
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| 6 | application, such as Mathematica, MathReader or Publicon. The data
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| 7 | for the notebook starts with the line containing stars above.
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| 8 |
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| 9 | To get the notebook into a Mathematica-compatible application, do
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| 10 | one of the following:
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| 11 |
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| 12 | * Save the data starting with the line of stars above into a file
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| 13 | with a name ending in .nb, then open the file inside the
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| 14 | application;
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| 15 |
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| 16 | * Copy the data starting with the line of stars above to the
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| 17 | clipboard, then use the Paste menu command inside the application.
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| 18 |
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| 19 | Data for notebooks contains only printable 7-bit ASCII and can be
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| 20 | sent directly in email or through ftp in text mode. Newlines can be
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| 22 |
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| 23 | NOTE: If you modify the data for this notebook not in a Mathematica-
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| 24 | compatible application, you must delete the line below containing
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| 25 | the word CacheID, otherwise Mathematica-compatible applications may
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| 26 | try to use invalid cache data.
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| 27 |
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| 28 | For more information on notebooks and Mathematica-compatible
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| 29 | applications, contact Wolfram Research:
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| 30 | web: http://www.wolfram.com
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| 31 | email: info@wolfram.com
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| 32 | phone: +1-217-398-0700 (U.S.)
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| 33 |
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| 34 | Notebook reader applications are available free of charge from
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| 35 | Wolfram Research.
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| 36 | *******************************************************************)
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| 37 |
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| 38 | (*CacheID: 232*)
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| 39 |
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| 40 |
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| 41 | (*NotebookFileLineBreakTest
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| 42 | NotebookFileLineBreakTest*)
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| 43 | (*NotebookOptionsPosition[ 32589, 988]*)
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| 44 | (*NotebookOutlinePosition[ 33288, 1012]*)
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| 45 | (* CellTagsIndexPosition[ 33244, 1008]*)
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| 46 | (*WindowFrame->Normal*)
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| 47 |
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| 48 |
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| 49 |
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| 50 | Notebook[{
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| 51 |
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| 52 | Cell[CellGroupData[{
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| 53 | Cell["gg \[Rule] H at NLO in the EFT", "Title",
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| 54 | PageWidth->PaperWidth],
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| 55 |
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| 56 | Cell[CellGroupData[{
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| 57 |
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| 58 | Cell["Input FeynCalc", "Subsection",
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| 59 | PageWidth->PaperWidth],
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| 60 |
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| 61 | Cell[BoxData[
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| 62 | \(<< HighEnergyPhysics`fc`\)], "Input",
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| 63 | PageWidth->PaperWidth],
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| 64 |
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| 65 | Cell[TextData[{
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| 66 | StyleBox["FeynCalc",
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| 67 | FontWeight->"Bold"],
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| 68 | " ",
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| 69 | "4.1.0.3b",
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| 70 | " ",
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| 71 | " Evaluate ?FeynCalc for help or visit ",
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| 72 | ButtonBox["www.feyncalc.org",
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| 73 | ButtonData:>{
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| 74 | URL[ "http://www.feyncalc.org"], None},
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| 75 | ButtonStyle->"Hyperlink",
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| 76 | ButtonNote->"http://www.feyncalc.org"]
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| 77 | }], "Text",
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| 78 | GeneratedCell->True,
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| 79 | CellAutoOverwrite->True],
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| 80 |
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| 81 | Cell[BoxData[
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| 82 | \(\($LimitTo4 = False;\)\)], "Input",
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| 83 | PageWidth->PaperWidth]
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| 84 | }, Closed]],
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| 85 |
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| 86 | Cell[CellGroupData[{
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| 87 |
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| 88 | Cell["Virtual diagrams :preliminaries", "Section",
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| 89 | PageWidth->PaperWidth],
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| 90 |
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| 91 | Cell[CellGroupData[{
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| 92 |
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| 93 | Cell["Kinematics", "Subsection",
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| 94 | PageWidth->PaperWidth],
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| 95 |
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| 96 | Cell["\<\
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| 97 | -----I take all momenta outgoing
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| 98 |
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| 99 | p1 + p2 + p3 = 0
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| 100 |
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| 101 | p1^2=0
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| 102 | p2^2=0
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| 103 | p3^3=Q^2
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| 104 |
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| 105 |
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| 106 | \
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| 107 | \>", "Text",
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| 108 | PageWidth->PaperWidth],
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| 109 |
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| 110 | Cell[BoxData[{
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| 111 | \(\(ScalarProduct[p1, p1] = 0;\)\), "\[IndentingNewLine]",
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| 112 | \(\(ScalarProduct[p2, p2] = 0;\)\), "\[IndentingNewLine]",
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| 113 | \(\(ScalarProduct[p3, p3] = Q2;\)\), "\[IndentingNewLine]",
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| 114 | \(\(ScalarProduct[p1, p3] = \(-Q2\)/2;\)\), "\[IndentingNewLine]",
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| 115 | \(\(ScalarProduct[p1, p2] = Q2/2;\)\), "\[IndentingNewLine]",
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| 116 | \(\(ScalarProduct[p2, p3] = \(-\ Q2\)/2;\)\), "\[IndentingNewLine]",
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| 117 | \(\(ScalarProduct[p1, e1] = 0;\)\), "\[IndentingNewLine]",
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| 118 | \(\(ScalarProduct[p2, e2] = 0;\)\), "\[IndentingNewLine]",
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| 119 | \(\(ScalarProduct[p1, e2] = 0;\)\), "\[IndentingNewLine]",
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| 120 | \(\(\(ScalarProduct[p2, e1] = 0;\)\(\[IndentingNewLine]\)
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| 121 | \)\), "\[IndentingNewLine]",
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| 122 | \(\(ScalarProduct[p1, p1, Dimension \[Rule] D] =
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| 123 | 0;\)\), "\[IndentingNewLine]",
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| 124 | \(\(ScalarProduct[p2, p2, Dimension \[Rule] D] =
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| 125 | 0;\)\), "\[IndentingNewLine]",
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| 126 | \(\(ScalarProduct[p3, p3, Dimension \[Rule] D] =
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| 127 | Q2;\)\), "\[IndentingNewLine]",
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| 128 | \(\(ScalarProduct[p1, p3, Dimension \[Rule] D] = \(-Q2\)/
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| 129 | 2;\)\), "\[IndentingNewLine]",
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| 130 | \(\(ScalarProduct[p1, p2, Dimension \[Rule] D] =
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| 131 | Q2/2;\)\), "\[IndentingNewLine]",
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| 132 | \(\(ScalarProduct[p2, p3, Dimension \[Rule] D] = \(-\ Q2\)/
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| 133 | 2;\)\), "\[IndentingNewLine]",
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| 134 | \(\(ScalarProduct[p1, e1, Dimension \[Rule] D] =
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| 135 | 0;\)\), "\[IndentingNewLine]",
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| 136 | \(\(ScalarProduct[p2, e2, Dimension \[Rule] D] =
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| 137 | 0;\)\), "\[IndentingNewLine]",
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| 138 | \(\(ScalarProduct[p1, e2, Dimension \[Rule] D] =
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| 139 | 0;\)\), "\[IndentingNewLine]",
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| 140 | \(\(ScalarProduct[p2, e1, Dimension \[Rule] D] =
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| 141 | 0;\)\), "\[IndentingNewLine]",
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| 142 | \(\)}], "Input",
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| 143 | PageWidth->PaperWidth]
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| 144 | }, Closed]],
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| 145 |
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| 146 | Cell[CellGroupData[{
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| 147 |
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| 148 | Cell["Verteces and Propagators", "Subsection",
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| 149 | PageWidth->PaperWidth],
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| 150 |
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| 151 | Cell[TextData[{
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| 152 | StyleBox["GGG is the kinematic part of the three-gluon vtx (momenta \
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| 153 | outgoing, clockwise ordering):\nVTX(ggg) = (-\[ImaginaryI] ",
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| 154 | FontSize->14],
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| 155 | Cell[BoxData[
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| 156 | \(TraditionalForm\`g\_s\)],
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| 157 | FontSize->14],
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| 158 | StyleBox[" ) (\[ImaginaryI] ",
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| 159 | FontSize->14],
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| 160 | Cell[BoxData[
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| 161 | \(TraditionalForm\`f\^abc\)],
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| 162 | FontSize->14],
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| 163 | StyleBox[") GGG\nVTX(qqg) = ( -\[ImaginaryI] ",
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| 164 | FontSize->14],
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| 165 | Cell[BoxData[
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| 166 | \(TraditionalForm\`g\_s\)],
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| 167 | FontSize->14],
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| 168 | ")",
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| 169 | StyleBox[" ",
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| 170 | FontSize->14],
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| 171 | Cell[BoxData[
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| 172 | \(TraditionalForm\`\((T\^a)\)\_ij\)],
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| 173 | FontSize->14],
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| 174 | StyleBox[" ",
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| 175 | FontSize->14],
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| 176 | Cell[BoxData[
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| 177 | \(TraditionalForm\`\[Gamma]\^\[Mu]\)],
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| 178 | FontSize->14],
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| 179 | StyleBox["\nGluon Propagator= ",
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| 180 | FontSize->14],
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| 181 | Cell[BoxData[
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| 182 | FormBox[
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| 183 | FractionBox[
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| 184 | StyleBox[\(\(-\[ImaginaryI]\)\ g\^\[Mu]\[Nu]\),
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| 185 | FontSize->16], \(p\^2\)], TraditionalForm]],
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| 186 | FontSize->14],
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| 187 | "\nQuark ",
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| 188 | StyleBox["Propagator= ",
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| 189 | FontSize->14],
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| 190 | Cell[BoxData[
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| 191 | FormBox[
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| 192 | FractionBox[
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| 193 | StyleBox[\(\(\[ImaginaryI]\)\(\ \)\),
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| 194 | FontSize->16], \(p\&^\)], TraditionalForm]],
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| 195 | FontSize->14]
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| 196 | }], "Text",
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| 197 | PageWidth->PaperWidth],
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| 198 |
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| 199 | Cell[BoxData[{
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| 200 | \(\(GGG[p1_, p2_, p3_, m1_, m2_, m3_] :=
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| 201 | FV[p1 - p2, m3]\ MT[m1, m2] + FV[p2 - p3, m1]\ MT[m2, m3] +
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| 202 | FV[p3 - p1, m2]\ MT[m1, m3];\)\), "\[IndentingNewLine]",
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| 203 | \(\(GGGD[p1_, p2_, p3_, m1_, m2_, m3_] :=
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| 204 | FVD[p1 - p2, m3]\ MTD[m1, m2] + FVD[p2 - p3, m1]\ MTD[m2, m3] +
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| 205 | FVD[p3 - p1, m2]\ MTD[m1, m3];\)\), "\[IndentingNewLine]",
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| 206 | \(\(GGGG[m1_, m2_, m3_, m4_] :=
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| 207 | 2\ MT[m1, m2]\ MT[m3, m4] - MT[m1, m3]\ MT[m2, m4] -
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| 208 | MT[m1, m4]\ MT[m2, m3];\)\), "\[IndentingNewLine]",
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| 209 | \(\(GGGGD[m1_, m2_, m3_, m4_] :=
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| 210 | 2\ MTD[m1, m2]\ MTD[m3, m4] - MTD[m1, m3]\ MTD[m2, m4] -
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| 211 | MTD[m1, m4]\ MTD[m2, m3];\)\), "\[IndentingNewLine]",
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| 212 | \(\(PropQuark = I;\)\), "\[IndentingNewLine]",
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| 213 | \(\(PropGluon\ = \(-I\);\)\), "\[IndentingNewLine]",
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| 214 | \(\(vtx = \(-I\)\ gs;\)\)}], "Input",
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| 215 | PageWidth->PaperWidth]
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| 216 | }, Closed]],
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| 217 |
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| 218 | Cell[CellGroupData[{
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| 219 |
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| 220 | Cell["Born Matrix element ", "Subsection",
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| 221 | PageWidth->PaperWidth],
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| 222 |
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| 223 | Cell[CellGroupData[{
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| 224 |
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| 225 | Cell[BoxData[{
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| 226 | \(factborn = gs2\), "\[IndentingNewLine]",
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| 227 | \(\(Born =
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| 228 | factborn \((\
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| 229 | MTD[mu, nu]\ SPD[p1, p2] - FVD[p1, nu]\ FVD[p2, mu])\)\ FVD[e1,
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| 230 | mu]\ FVD[e2, nu] // Contract;\)\), "\[IndentingNewLine]",
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| 231 | \(\(Born4 = Born /. \ D \[Rule] 4;\)\), "\[IndentingNewLine]",
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| 232 | \(Born = Born\)}], "Input",
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| 233 | PageWidth->PaperWidth],
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| 234 |
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| 235 | Cell[BoxData[
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| 236 | \(TraditionalForm\`gs2\)], "Output"],
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| 237 |
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| 238 | Cell[BoxData[
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| 239 | FormBox[
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| 240 | RowBox[{\(1\/2\), " ", "gs2", " ", "Q2", " ",
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| 241 | RowBox[{
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| 242 | FormBox["e1",
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| 243 | "TraditionalForm"], "\[NoBreak]", "\[CenterDot]", "\[NoBreak]",
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| 244 | FormBox["e2",
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| 245 | "TraditionalForm"]}]}], TraditionalForm]], "Output"]
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| 246 | }, Open ]],
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| 247 |
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| 248 | Cell[CellGroupData[{
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| 249 |
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| 250 | Cell[BoxData[{
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| 251 | \(BornSq =
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| 252 | Normal[Series[\((\[IndentingNewLine]gs2^2\ /\((D - 2)\)^2*\((\
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| 253 | MTD[mu, nu]\ SPD[p1, p2] -
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| 254 | FVD[p1, nu]\ FVD[p2, mu])\) \((\
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| 255 | MTD[mu, nu]\ SPD[p1, p2] -
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| 256 | FVD[p1, nu]\ FVD[p2, mu])\)\ // Contract)\)\ /. \
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| 257 | D \[Rule] 4 - 2 e, {e, 0, 2}]] //
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| 258 | Simplify\), "\[IndentingNewLine]",
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| 259 | \(\(BornSq4 = BornSq /. \ e \[Rule] 0;\)\)}], "Input"],
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| 260 |
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| 261 | Cell[BoxData[
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| 262 | \(TraditionalForm\`1\/8\ \((e\^2 + e + 1)\)\ gs2\^2\ Q2\^2\)], "Output"]
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| 263 | }, Open ]]
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| 264 | }, Closed]],
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| 265 |
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| 266 | Cell[CellGroupData[{
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| 267 |
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| 268 | Cell["My Scalar Integrals", "Subsection",
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| 269 | PageWidth->PaperWidth],
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| 270 |
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| 271 | Cell[BoxData[{
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| 272 | \(\(subInt[expr_] :=
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| 273 | expr /. \ {\[IndentingNewLine]B0[x__] \[Rule] \
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| 274 | DUPI/\((I\ Pi^2)\)\ MyB0[x], \[IndentingNewLine]C0[
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| 275 | x__] \[Rule] \
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| 276 | DUPI/\((I\ Pi^2)\)\ MyC0[x], \[IndentingNewLine]D0[
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| 277 | x__] \[Rule] \
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| 278 | DUPI/\((I\ Pi^2)\)\ MyD0[x]};\)\), "\[IndentingNewLine]",
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| 279 | \(\(MyC0[0, 0, Q2, 0, 0,
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| 280 | 0] = \((c\[CapitalGamma]*\((2/e^2 - Pi^2 - 2/e*Log[Q2/mu2] +
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| 281 | Log[Q2/mu2]^2)\))\)/\((2*
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| 282 | Q2)\);\)\), "\[IndentingNewLine]",
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| 283 | \(\(MyB0[0, 0, 0] = 0;\)\), "\[IndentingNewLine]",
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| 284 | \(\(MyB0[Q2, 0, 0] =
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| 285 | c\[CapitalGamma]\ \((1/e + 2 - Log[Q2/mu2])\);\)\), "\n",
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| 286 | \(\)}], "Input",
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| 287 | PageWidth->PaperWidth],
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| 288 |
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| 289 | Cell[BoxData[""], "Input"],
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| 290 |
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| 291 | Cell[BoxData[""], "Input"]
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| 292 | }, Closed]]
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| 293 | }, Closed]],
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| 294 |
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| 295 | Cell[CellGroupData[{
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| 296 |
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| 297 | Cell["CDR", "Section"],
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| 298 |
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| 299 | Cell[CellGroupData[{
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| 300 |
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| 301 | Cell["Vertex Diagram ", "Subsection",
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| 302 | PageWidth->PaperWidth],
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| 303 |
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| 304 | Cell[CellGroupData[{
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| 305 |
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| 306 | Cell[BoxData[{
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| 307 | \(\(\(colorvtx = SUNF[b, x, y] SUNF[y, a, x]\ // SUNSimplify\)\(\n\)
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| 308 | \)\), "\[IndentingNewLine]",
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| 309 | \(\(num =
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| 310 | gs2^2*\((\(-gs2\))\)*
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| 311 | I*\[IndentingNewLine]\((\
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| 312 | MTD[al, be]\ SPD[\(-l\) + p1, p2 + l] - \ \ FVD[\(-l\) + p1,
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| 313 | be]\ FVD[p2 + l, al])\)\ *\
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| 314 | GGGD[\(-l\), p1, l - p1, ro, mu, al]*\[IndentingNewLine]GGGD[p2,
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| 315 | l, \(-l\) - p2, nu, ro, be]*\((\
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| 316 | MTD[mu, nu]\ SPD[p1, p2] - FVD[p1, nu]\ FVD[p2, mu])\)\ //
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| 317 | Contract;\)\), "\n",
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| 318 | \(\(dens = FAD[l, p1 - l, l + p2];\)\), "\[IndentingNewLine]",
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| 319 | \(\(amp = CA\ dens*\ num/\((D - 2)\)^2;\)\), "\n",
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| 320 | \(inte1 = \((\(\((\(OneLoop[l, amp/DUPI] // PaVeReduce\) // Simplify)\) //
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| 321 | Factor\) // Simplify)\) /. \ B0[0, 0, 0] \[Rule] 0\ //
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| 322 | Simplify\)}], "Input",
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| 323 | PageWidth->PaperWidth],
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| 324 |
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| 325 | Cell[BoxData[
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| 326 | FormBox[
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| 327 | RowBox[{\(C\_A\), " ",
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| 328 | SubscriptBox["\[Delta]",
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| 329 | RowBox[{
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| 330 | FormBox[
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| 331 | FormBox["a",
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| 332 | "TraditionalForm"],
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| 333 | "TraditionalForm"], "\[NoBreak]",
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| 334 | FormBox[
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| 335 | FormBox["b",
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| 336 | "TraditionalForm"],
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| 337 | "TraditionalForm"]}]]}], TraditionalForm]], "Output"],
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| 338 |
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| 339 | Cell[BoxData[
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| 340 | FormBox[
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| 341 | RowBox[{"-",
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| 342 | RowBox[{
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| 343 | RowBox[{"(",
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| 344 |
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| 345 | RowBox[{\(C\_A\), " ", \(gs2\^3\), " ", \(\[Pi]\^2\),
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| 346 | " ", \(Q2\^2\), " ",
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| 347 | RowBox[{"(",
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| 348 |
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| 349 | RowBox[{\(\((\(-20\)\ D\^2 + 73\ D - 52)\)\ \(\(B\_0\)(Q2, 0,
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| 350 | 0)\)\), "+",
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| 351 | RowBox[{"8", " ", \((D\^2 - 3\ D + 2)\), " ", "Q2", " ",
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| 352 | RowBox[{
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| 353 | FormBox[\("C"\_"0"\),
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| 354 | "TraditionalForm"], "\[NoBreak]", "(", "\[NoBreak]",
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| 355 | FormBox["0",
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| 356 | "TraditionalForm"], "\[NoBreak]", ",", "\[NoBreak]",
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| 357 | FormBox["0",
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| 358 | "TraditionalForm"], "\[NoBreak]", ",", "\[NoBreak]",
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| 359 | "Q2", "\[NoBreak]", ",", "\[NoBreak]",
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| 360 | FormBox["0",
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| 361 | "TraditionalForm"], "\[NoBreak]", ",", "\[NoBreak]",
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| 362 | FormBox["0",
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| 363 | "TraditionalForm"], "\[NoBreak]", ",", "\[NoBreak]",
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| 364 | FormBox["0",
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| 365 | "TraditionalForm"], "\[NoBreak]", ")"}]}]}], ")"}]}],
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| 366 | ")"}], "/", \((16\ \((D - 2)\)\^2\ \((D - 1)\)\ DUPI)\)}]}],
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| 367 | TraditionalForm]], "Output"]
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| 368 | }, Open ]],
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| 369 |
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| 370 | Cell[BoxData[""], "Input"]
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| 371 | }, Open ]],
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| 372 |
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| 373 | Cell[CellGroupData[{
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| 374 |
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| 375 | Cell["4-gluon Diagram ", "Subsection",
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| 376 | PageWidth->PaperWidth],
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| 377 |
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| 378 | Cell[CellGroupData[{
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| 379 |
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| 380 | Cell[BoxData[{
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| 381 | \(\(color = SUNF[b, c, y] SUNF[y, a, c]\ // SUNSimplify;\)\), "\n",
|
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| 382 | \(\(num =
|
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| 383 | gs2^2*I*\
|
|---|
| 384 | gs2*\((\ MTD[ro, si]\ SPD[l, \(-l\) + p1 + p2] - \ \ FVD[l,
|
|---|
| 385 | si]\ FVD[\(-l\) + p1 + p2, ro])\)\ *\
|
|---|
| 386 | GGGGD[nu, mu, ro, si]\ *\((\
|
|---|
| 387 | MTD[mu, nu]\ SPD[p1, p2] - FVD[p1, nu]\ FVD[p2, mu])\)\ //
|
|---|
| 388 | Contract;\)\), "\n",
|
|---|
| 389 | \(\(dens = FAD[l, l - p1 - p2];\)\), "\n",
|
|---|
| 390 | \(\(amp = CA/2\ \ dens*num/\((D - 2)\)^2;\)\), "\[IndentingNewLine]",
|
|---|
| 391 | \(inte2 = \(\((\(OneLoop[l, amp/DUPI] // PaVeReduce\) // Simplify)\) //
|
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| 392 | Factor\) // Simplify\)}], "Input",
|
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| 393 | PageWidth->PaperWidth],
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| 394 |
|
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| 395 | Cell[BoxData[
|
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| 396 | \(TraditionalForm\`\(-\(\(C\_A\ \((2\ D - 3)\)\ \((2\ D\^2 - 5\ D +
|
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| 397 | 4)\)\ gs2\^3\ \[Pi]\^2\ Q2\^2\ \(\(B\_0\)(Q2, 0,
|
|---|
| 398 | 0)\)\)\/\(16\ \((D - 2)\)\^2\ \((D -
|
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| 399 | 1)\)\ DUPI\)\)\)\)], "Output"]
|
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| 400 | }, Open ]]
|
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| 401 | }, Open ]],
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| 402 |
|
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| 403 | Cell[CellGroupData[{
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| 404 |
|
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| 405 | Cell["Sum the virtuals", "Subsection"],
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| 406 |
|
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| 407 | Cell[CellGroupData[{
|
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| 408 |
|
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| 409 | Cell[BoxData[
|
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| 410 | \(\(\(\[IndentingNewLine]\)\(res = \ \(\(\(-1\)/
|
|---|
| 411 | 2*\[IndentingNewLine]Normal[
|
|---|
| 412 | Series[\(\(\((inte1 + inte2)\)/BornSq\)/
|
|---|
| 413 | I\)/\((mu2/Q2)\)^\((e)\)\ /. \
|
|---|
| 414 | D \[Rule] 4 - 2 e // \ subInt, {e, 0, 0}]] //
|
|---|
| 415 | Simplify\) // PowerExpand\) // Expand\)\)\)], "Input",
|
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| 416 | PageWidth->PaperWidth],
|
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| 417 |
|
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| 418 | Cell[BoxData[
|
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| 419 | \(TraditionalForm\`1\/2\ C\_A\ c\[CapitalGamma]\ gs2\ \[Pi]\^2 - \(C\_A\ \
|
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| 420 | c\[CapitalGamma]\ gs2\)\/e\^2\)], "Output"]
|
|---|
| 421 | }, Open ]],
|
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| 422 |
|
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| 423 | Cell[CellGroupData[{
|
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| 424 |
|
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| 425 | Cell[BoxData[{
|
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| 426 | \(\(add =
|
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| 427 | 1/2 \((Normal[
|
|---|
| 428 | Series[\((\((1 + \ \(19/
|
|---|
| 429 | 4\)/\[Pi]\ gs2/\((4 \[Pi])\)\ )\)/\((1 +
|
|---|
| 430 | 2\ \(gs2/\((4 \[Pi])\)\)/\[Pi])\))\), {gs2, 0,
|
|---|
| 431 | 1}]] - 1)\) c\[CapitalGamma]\ CA/3\ 16
|
|---|
| 432 | Pi^2\ ;\)\), "\[IndentingNewLine]",
|
|---|
| 433 | \(\(virt = add + res;\)\[IndentingNewLine]\), "\[IndentingNewLine]",
|
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| 434 | \(add\)}], "Input",
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| 435 | PageWidth->PaperWidth],
|
|---|
| 436 |
|
|---|
| 437 | Cell[BoxData[
|
|---|
| 438 | \(TraditionalForm\`11\/6\ C\_A\ c\[CapitalGamma]\ gs2\)], "Output"]
|
|---|
| 439 | }, Open ]],
|
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| 440 |
|
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| 441 | Cell[CellGroupData[{
|
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| 442 |
|
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| 443 | Cell[BoxData[
|
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| 444 | \(virt = \(\((\((virt*\(2/gs2\)/c\[CapitalGamma])\) //
|
|---|
| 445 | Expand)\)\(*\)\(as\/\(2 \[Pi]\)\) \(c\[CapitalGamma]\)\(\ \
|
|---|
| 446 | \)\)\)], "Input"],
|
|---|
| 447 |
|
|---|
| 448 | Cell[BoxData[
|
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| 449 | \(TraditionalForm\`\(as\ c\[CapitalGamma]\ \((\(-\(\(2\ C\_A\)\/e\^2\)\) \
|
|---|
| 450 | + \[Pi]\^2\ C\_A + \(11\ C\_A\)\/3)\)\)\/\(2\ \[Pi]\)\)], "Output"]
|
|---|
| 451 | }, Open ]],
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| 452 |
|
|---|
| 453 | Cell[BoxData[
|
|---|
| 454 | \(\[IndentingNewLine]\)], "Input"],
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|---|
| 455 |
|
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| 456 | Cell[CellGroupData[{
|
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| 457 |
|
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| 458 | Cell[BoxData[
|
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| 459 | \(\(\(32/256\)/3\)/\((1 - e)\)^2\)], "Input"],
|
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| 460 |
|
|---|
| 461 | Cell[BoxData[
|
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| 462 | \(TraditionalForm\`1\/\(24\ \((1 - e)\)\^2\)\)], "Output"]
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|---|
| 463 | }, Open ]]
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| 464 | }, Open ]]
|
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| 465 | }, Closed]],
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| 466 |
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| 467 | Cell[CellGroupData[{
|
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| 468 |
|
|---|
| 469 | Cell["The real contributions", "Section",
|
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| 470 | PageWidth->PaperWidth],
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| 471 |
|
|---|
| 472 | Cell[CellGroupData[{
|
|---|
| 473 |
|
|---|
| 474 | Cell["Kinematics", "Subsection",
|
|---|
| 475 | PageWidth->PaperWidth],
|
|---|
| 476 |
|
|---|
| 477 | Cell["\<\
|
|---|
| 478 | -----I take all momenta outgoing
|
|---|
| 479 |
|
|---|
| 480 | p1 + p2 + p3 + p4 = 0
|
|---|
| 481 |
|
|---|
| 482 | p1^2=0
|
|---|
| 483 | p2^2=0
|
|---|
| 484 | p3^3=0
|
|---|
| 485 | p4^2=mh^2
|
|---|
| 486 |
|
|---|
| 487 | -----invariants
|
|---|
| 488 |
|
|---|
| 489 | s = (p1 + p2)^2 = (p3 + p4)^2=2 p1.p2=mh^2+2 p3.p4
|
|---|
| 490 | t = (p1 + p3)^2 = (p2+ p4)^2 =2 p1.p3=mh^2+2 p2.p4
|
|---|
| 491 | u = (p2 + p3)^2 = (p1 + p4)^2=2 p1.p4+mh^2=+2 p2.p3
|
|---|
| 492 |
|
|---|
| 493 | s + t + u = mh^2
|
|---|
| 494 |
|
|---|
| 495 | \[Sigma]+\[Tau]+\[Upsilon]=1
|
|---|
| 496 |
|
|---|
| 497 | -----scalar products
|
|---|
| 498 |
|
|---|
| 499 | p1.p2=s/2
|
|---|
| 500 | p1.p3=(t)/2
|
|---|
| 501 | p1.p4= (u-mh^2)/2
|
|---|
| 502 | p2.p3= (u)/2
|
|---|
| 503 | p2.p4=(t-mh^2)/2
|
|---|
| 504 | p3.p4=(s-mh^2)/2
|
|---|
| 505 |
|
|---|
| 506 | -----physical region for production p1+p2=-p3-p4
|
|---|
| 507 |
|
|---|
| 508 | s>(mh)^2 ;
|
|---|
| 509 | t<0;
|
|---|
| 510 | u<0;
|
|---|
| 511 |
|
|---|
| 512 | \
|
|---|
| 513 | \>", "Text",
|
|---|
| 514 | PageWidth->PaperWidth],
|
|---|
| 515 |
|
|---|
| 516 | Cell[BoxData[{
|
|---|
| 517 | \(\(ScalarProduct[p1, p1] = 0;\)\), "\[IndentingNewLine]",
|
|---|
| 518 | \(\(ScalarProduct[p2, p2] = 0;\)\), "\[IndentingNewLine]",
|
|---|
| 519 | \(\(ScalarProduct[p3, p3] = 0;\)\), "\[IndentingNewLine]",
|
|---|
| 520 | \(\(ScalarProduct[q, p3] = u/2;\)\ (*q =
|
|---|
| 521 | p2 + p3*) \), "\[IndentingNewLine]",
|
|---|
| 522 | \(\(ScalarProduct[q, p2] = u/2;\)\ (*q =
|
|---|
| 523 | p2 + p3*) \), "\[IndentingNewLine]",
|
|---|
| 524 | \(\(ScalarProduct[q, q] = u;\)\ (*q =
|
|---|
| 525 | p2 + p3*) \), "\[IndentingNewLine]",
|
|---|
| 526 | \(\(ScalarProduct[p4, p4] = mh2;\)\), "\[IndentingNewLine]",
|
|---|
| 527 | \(\(ScalarProduct[p1, p2] = s/2;\)\), "\[IndentingNewLine]",
|
|---|
| 528 | \(\(ScalarProduct[p1, p3] = t/2;\)\), "\[IndentingNewLine]",
|
|---|
| 529 | \(\(ScalarProduct[p1, p4] = \((u - mh2)\)/2;\)\), "\[IndentingNewLine]",
|
|---|
| 530 | \(\(ScalarProduct[p2, p3] = u/2;\)\), "\[IndentingNewLine]",
|
|---|
| 531 | \(\(ScalarProduct[p2, p4] = \ \((t - mh2)\)/
|
|---|
| 532 | 2;\)\), "\[IndentingNewLine]",
|
|---|
| 533 | \(\(ScalarProduct[p3, p4] = \((s - mh2)\)/
|
|---|
| 534 | 2\ \ ;\)\), "\[IndentingNewLine]",
|
|---|
| 535 | \(\(ScalarProduct[p3, e3] = 0\ ;\)\), "\[IndentingNewLine]",
|
|---|
| 536 | \(\(ScalarProduct[p1, e1] = 0\ ;\)\), "\[IndentingNewLine]",
|
|---|
| 537 | \(\(ScalarProduct[p2, e2] = 0\ ;\)\), "\[IndentingNewLine]",
|
|---|
| 538 | \(\(s13 = t;\)\), "\[IndentingNewLine]",
|
|---|
| 539 | \(\(s23 = u;\)\)}], "Input",
|
|---|
| 540 | PageWidth->PaperWidth]
|
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| 541 | }, Open ]],
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| 542 |
|
|---|
| 543 | Cell[CellGroupData[{
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| 544 |
|
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| 545 | Cell["Verteces and Propagators", "Subsection",
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| 546 | PageWidth->PaperWidth],
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| 547 |
|
|---|
| 548 | Cell[TextData[{
|
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| 549 | StyleBox["GGG is the kinematic part of the three-gluon vtx (momenta \
|
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| 550 | outgoing, clockwise ordering):\nVTX(ggg) = (-\[ImaginaryI] ",
|
|---|
| 551 | FontSize->14],
|
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| 552 | Cell[BoxData[
|
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| 553 | \(TraditionalForm\`g\_s\)],
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| 554 | FontSize->14],
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| 555 | StyleBox[" ) (\[ImaginaryI] ",
|
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| 556 | FontSize->14],
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| 557 | Cell[BoxData[
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| 558 | \(TraditionalForm\`f\^abc\)],
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| 559 | FontSize->14],
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| 560 | StyleBox[") GGG\nVTX(qqg) = ( -\[ImaginaryI] ",
|
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| 561 | FontSize->14],
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| 562 | Cell[BoxData[
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| 563 | \(TraditionalForm\`g\_s\)],
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| 564 | FontSize->14],
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| 565 | ")",
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| 566 | StyleBox[" ",
|
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| 567 | FontSize->14],
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| 568 | Cell[BoxData[
|
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| 569 | \(TraditionalForm\`\((T\^a)\)\_ij\)],
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| 570 | FontSize->14],
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| 571 | StyleBox[" ",
|
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| 572 | FontSize->14],
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| 573 | Cell[BoxData[
|
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| 574 | \(TraditionalForm\`\[Gamma]\^\[Mu]\)],
|
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| 575 | FontSize->14],
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| 576 | StyleBox["\nGluon Propagator= ",
|
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| 577 | FontSize->14],
|
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| 578 | Cell[BoxData[
|
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| 579 | FormBox[
|
|---|
| 580 | FractionBox[
|
|---|
| 581 | StyleBox[\(\(-\[ImaginaryI]\)\ g\^\[Mu]\[Nu]\),
|
|---|
| 582 | FontSize->16], \(p\^2\)], TraditionalForm]],
|
|---|
| 583 | FontSize->14],
|
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| 584 | "\nQuark ",
|
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| 585 | StyleBox["Propagator= ",
|
|---|
| 586 | FontSize->14],
|
|---|
| 587 | Cell[BoxData[
|
|---|
| 588 | FormBox[
|
|---|
| 589 | FractionBox[
|
|---|
| 590 | StyleBox[\(\(\[ImaginaryI]\)\(\ \)\),
|
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| 591 | FontSize->16], \(p\&^\)], TraditionalForm]],
|
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| 592 | FontSize->14]
|
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| 593 | }], "Text",
|
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| 594 | PageWidth->PaperWidth],
|
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| 595 |
|
|---|
| 596 | Cell[BoxData[{
|
|---|
| 597 | \(\(GGGD[p1_, p2_, p3_, m1_, m2_, m3_] :=
|
|---|
| 598 | FourVector[p1 - p2, m3, \ Dimension\ \[Rule] D]\ MTD[m1, m2] +
|
|---|
| 599 | FourVector[p2 - p3, m1, \ Dimension\ \[Rule] \ D]\ MTD[m2, m3] +
|
|---|
| 600 | FourVector[p3 - p1, m2, \ Dimension\ \[Rule] \ D]\ MTD[m1,
|
|---|
| 601 | m3];\)\), "\[IndentingNewLine]",
|
|---|
| 602 | \(\(H[p1_, p2_, m1_, m2_] :=
|
|---|
| 603 | MTD[m1, m2]\ SPD[p1, p2]\ - \
|
|---|
| 604 | FourVector[p1, m2, \ Dimension \[Rule] D]*\[IndentingNewLine]\
|
|---|
| 605 | FourVector[p2, m1, \
|
|---|
| 606 | Dimension\ \[Rule] D];\)\), "\[IndentingNewLine]",
|
|---|
| 607 | \(\(PropQuark = I;\)\), "\[IndentingNewLine]",
|
|---|
| 608 | \(\(PropGluon\ = \(-I\);\)\), "\[IndentingNewLine]",
|
|---|
| 609 | \(\(vtx = \(-I\)\ gs;\)\)}], "Input",
|
|---|
| 610 | PageWidth->PaperWidth]
|
|---|
| 611 | }, Open ]],
|
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| 612 |
|
|---|
| 613 | Cell[CellGroupData[{
|
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| 614 |
|
|---|
| 615 | Cell["Sum over the four Feynman diagrams", "Subsection"],
|
|---|
| 616 |
|
|---|
| 617 | Cell["\<\
|
|---|
| 618 |
|
|---|
| 619 | uno= - gs (-f123) GGGD[p1,p2,-p1-p2,m1,m2,mu] (-I MTD[mu,nu]/s) (I A) \
|
|---|
| 620 | H[p3,p1+p2,m3,nu]//Contract;
|
|---|
| 621 | tre= - gs (-f123) GGGD[p3,p1,-p3-p1,m3,m1,mu] (-I MTD[mu,nu]/t) (I A) \
|
|---|
| 622 | H[p2,p1+p3,m2,nu]//Contract;
|
|---|
| 623 | qua= - gs (-f123) GGGD[p2,p3,-p3-p2,m2,m3,mu] (-I MTD[mu,nu]/u) (I A) \
|
|---|
| 624 | H[p1,p2+p3,m1,nu]//Contract;
|
|---|
| 625 | due= - A gs (-f123) GGGD[p1,p2,p3,m1,m2,m3];
|
|---|
| 626 | res=uno+due+tre+qua//ExpandScalarProduct;\
|
|---|
| 627 | \>", "Input"]
|
|---|
| 628 | }, Open ]],
|
|---|
| 629 |
|
|---|
| 630 | Cell[CellGroupData[{
|
|---|
| 631 |
|
|---|
| 632 | Cell["\<\
|
|---|
| 633 | Now I have to square the amplitude. The sum is performed over the physical \
|
|---|
| 634 | polarizations of the gluons:\
|
|---|
| 635 | \>", "Subsection"],
|
|---|
| 636 |
|
|---|
| 637 | Cell["\<\
|
|---|
| 638 | res1=res;
|
|---|
| 639 | res2=res /. m1->m1p /. m2->m2p /. m3-> m3p;
|
|---|
| 640 | tot=res1*(-MTD[m1,m1p]+(FVD[p1,m1] FVD[p2,m1p]+FVD[p1,m1p] \
|
|---|
| 641 | FVD[p2,m1])/SPD[p1,p2])//Contract;
|
|---|
| 642 | tot=tot*(-MTD[m2,m2p]+(FVD[p2,m2] FVD[p3,m2p]+FVD[p2,m2p] \
|
|---|
| 643 | FVD[p3,m2])/SPD[p3,p2])//Contract;
|
|---|
| 644 | tot=tot*(-MTD[m3,m3p]+(FVD[p1,m3] FVD[p3,m3p]+FVD[p1,m3p] \
|
|---|
| 645 | FVD[p3,m3])/SPD[p3,p1])//Contract;
|
|---|
| 646 | tot=tot*res2//Contract//Expand;\
|
|---|
| 647 | \>", "Input"],
|
|---|
| 648 |
|
|---|
| 649 | Cell[CellGroupData[{
|
|---|
| 650 |
|
|---|
| 651 | Cell["\<\
|
|---|
| 652 | Amp2=((tot//Contract//Factor)/. A-> as/3/Pi/v /. gs^2-> 4 Pi as/. f123^2-> 24 \
|
|---|
| 653 | //Simplify)//Factor\
|
|---|
| 654 | \>", "Input"],
|
|---|
| 655 |
|
|---|
| 656 | Cell[BoxData[
|
|---|
| 657 | \(TraditionalForm\`\(\(1\/\(3\ \[Pi]\ s\ t\ u\ v\^2\)\)\((32\ as\^3\ \((D\
|
|---|
| 658 | \ s\^4 - 2\ s\^4 + 2\ D\ t\ s\^3 - 4\ t\ s\^3 + 2\ D\ u\ s\^3 - 4\ u\ s\^3 +
|
|---|
| 659 | 3\ D\ t\^2\ s\^2 - 6\ t\^2\ s\^2 + 3\ D\ u\^2\ s\^2 -
|
|---|
| 660 | 6\ u\^2\ s\^2 + 8\ D\ t\ u\ s\^2 - 20\ t\ u\ s\^2 +
|
|---|
| 661 | 2\ D\ t\^3\ s - 4\ t\^3\ s + 2\ D\ u\^3\ s - 4\ u\^3\ s +
|
|---|
| 662 | 8\ D\ t\ u\^2\ s - 20\ t\ u\^2\ s + 8\ D\ t\^2\ u\ s -
|
|---|
| 663 | 20\ t\^2\ u\ s + D\ t\^4 - 2\ t\^4 + D\ u\^4 - 2\ u\^4 +
|
|---|
| 664 | 2\ D\ t\ u\^3 - 4\ t\ u\^3 + 3\ D\ t\^2\ u\^2 - 6\ t\^2\ u\^2 +
|
|---|
| 665 | 2\ D\ t\^3\ u - 4\ t\^3\ u)\))\)\)\)], "Output"]
|
|---|
| 666 | }, Open ]],
|
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| 667 |
|
|---|
| 668 | Cell["\<\
|
|---|
| 669 |
|
|---|
| 670 | Amp2=Amp2 /. D->4-2e//Simplify;\
|
|---|
| 671 | \>", "Input"]
|
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| 672 | }, Open ]],
|
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| 673 |
|
|---|
| 674 | Cell[CellGroupData[{
|
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| 675 |
|
|---|
| 676 | Cell["Real Amplitude Squared in D dimensions", "Subsection",
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| 677 | PageWidth->PaperWidth],
|
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| 678 |
|
|---|
| 679 | Cell[BoxData[
|
|---|
| 680 | \(\(\(\[IndentingNewLine]\)\(\(Emme =
|
|---|
| 681 | 1/s \(\(\((\((mh2^4 + s^4 + t^4 + u^4)\)\ \((1 - 2\ e)\) + \
|
|---|
| 682 | e/2\ \((mh2^2 + s^2 + t^2 + u^2)\)^2)\)/s\)/t\)/
|
|---|
| 683 | u;\)\[IndentingNewLine]
|
|---|
| 684 | \(RealD = Emme/\((1 - e)\)^2;\)\[IndentingNewLine]
|
|---|
| 685 | \(Real4 = RealD /. \ e \[Rule] 0;\)\)\)\)], "Input",
|
|---|
| 686 | PageWidth->PaperWidth]
|
|---|
| 687 | }, Open ]],
|
|---|
| 688 |
|
|---|
| 689 | Cell[CellGroupData[{
|
|---|
| 690 |
|
|---|
| 691 | Cell["Phase space in D dimensions", "Subsection",
|
|---|
| 692 | PageWidth->PaperWidth],
|
|---|
| 693 |
|
|---|
| 694 | Cell[BoxData[{
|
|---|
| 695 | \(\[IndentingNewLine]\(PS = \(1\/\(8 \[Pi]\)\) \(\((\(\(4\)\(\ \)\(\[Pi]\
|
|---|
| 696 | \)\(\ \)\)\/mh2)\)\^e\)
|
|---|
| 697 | 1\/Gamma[1 - e]\ \((mh2\/s)\)\^e\ \ \((1 - mh2\/s)\)\^\(1 - 2 e\)\ \
|
|---|
| 698 | v\^\(-e\)\ \((omv)\)\^\(-e\);\)\), "\[IndentingNewLine]",
|
|---|
| 699 | \(\(substu = {t \[Rule] \ \(-s\)\ \((1 - mh2\/s)\) \((omv)\), \
|
|---|
| 700 | u \[Rule] \ \(-s\)\ \((1 - mh2\/s)\) v\ ,
|
|---|
| 701 | s \[Rule] \ mh2/z};\)\), "\[IndentingNewLine]",
|
|---|
| 702 | \(\(cGamma = \((1/16)\)/Pi^2\ mh2^\((\(-e\))\) \((4\ Pi)\)^e\ Gamma[
|
|---|
| 703 | 1 + e]\ Gamma[1 - e]^2/Gamma[1 - 2\ e];\)\), "\n",
|
|---|
| 704 | \(\(pgg =
|
|---|
| 705 | 2 \((z\ PlusDistribution[1/\((1 - z)\)] + \((1 - z)\)/z +
|
|---|
| 706 | z \((1 - z)\) +
|
|---|
| 707 | 11/12\ DeltaFunction[1 - z])\);\)\), "\[IndentingNewLine]",
|
|---|
| 708 | \(\(s0 = z;\)\)}], "Input",
|
|---|
| 709 | PageWidth->PaperWidth]
|
|---|
| 710 | }, Closed]],
|
|---|
| 711 |
|
|---|
| 712 | Cell[CellGroupData[{
|
|---|
| 713 |
|
|---|
| 714 | Cell["CDR", "Subsection",
|
|---|
| 715 | PageWidth->PaperWidth],
|
|---|
| 716 |
|
|---|
| 717 | Cell[CellGroupData[{
|
|---|
| 718 |
|
|---|
| 719 | Cell[BoxData[{
|
|---|
| 720 | \(intando =
|
|---|
| 721 | FullSimplify[
|
|---|
| 722 | FullSimplify[\(RealD\ PS\ c\[CapitalGamma]\)\/cGamma] \
|
|---|
| 723 | //. \[InvisibleSpace]substu] /. \[InvisibleSpace]omv \[Rule]
|
|---|
| 724 | 1 - v; \), "\n",
|
|---|
| 725 | \(intando = intando\/\((1 - z)\)\^\(\(-2\)\ e - 1\)\)}], "Input"],
|
|---|
| 726 |
|
|---|
| 727 | Cell[BoxData[
|
|---|
| 728 | \(TraditionalForm\`\(-\(\((c\[CapitalGamma]\ \((1\/mh2)\)\^e\ mh2\^e\ \
|
|---|
| 729 | \[Pi]\ \((1 - v)\)\^\(\(-e\) - 1\)\ v\^\(\(-e\) - 1\)\ z\^e\ \((e\ \((3\ \((1 \
|
|---|
| 730 | - v)\)\^4\ \((z - 1)\)\^4 + 3\ v\^4\ \((z - 1)\)\^4 -
|
|---|
| 731 | 2\ v\^2\ \((z\^2 + 1)\)\ \((z - 1)\)\^2 -
|
|---|
| 732 | 2\ \((1 - v)\)\^2\ \((v\^2\ \((z - 1)\)\^2 + z\^2 +
|
|---|
| 733 | 1)\)\ \((z - 1)\)\^2 + 3\ z\^4 - 2\ z\^2 + 3)\) -
|
|---|
| 734 | 2\ \((\((1 - v)\)\^4\ \((z - 1)\)\^4 + v\^4\ \((z - 1)\)\^4 +
|
|---|
| 735 | z\^4 + 1)\))\)\ \(\[CapitalGamma](
|
|---|
| 736 | 1 - 2\ e)\))\)/\((\((e - 1)\)\^2\ \(\[CapitalGamma](1 - e)\)\^3\
|
|---|
| 737 | \ \(\[CapitalGamma](e + 1)\))\)\)\)\)], "Output"]
|
|---|
| 738 | }, Open ]],
|
|---|
| 739 |
|
|---|
| 740 | Cell[BoxData[{
|
|---|
| 741 | \(\(\[Sigma]r =
|
|---|
| 742 | Integrate[intando, {v, 0, 1}, \ GenerateConditions \[Rule] False] //
|
|---|
| 743 | PowerExpand;\)\), "\n",
|
|---|
| 744 | \(\(\[Sigma]r = \(Normal[Series[\[Sigma]r, {e, 0, 1}]] // Factor\) //
|
|---|
| 745 | FullSimplify;\)\), "\n",
|
|---|
| 746 | \(\(\[Sigma]r\ = \[Sigma]r\ \ *\((\(\(-1\)\/\(2 e\)\)
|
|---|
| 747 | DeltaFunction[1 - z] + PlusDistribution[1/\((1 - z)\)] -
|
|---|
| 748 | 2\ e\ PlusDistribution[Log[1 - z]/\((1 - z)\)])\);\)\), "\n",
|
|---|
| 749 | \(\(\[Sigma]r =
|
|---|
| 750 | Normal[Series[\[Sigma]r/\((1 + e + e^2)\), {e, 0, 0}]] //
|
|---|
| 751 | Expand;\)\)}], "Input",
|
|---|
| 752 | PageWidth->PaperWidth]
|
|---|
| 753 | }, Closed]]
|
|---|
| 754 | }, Closed]],
|
|---|
| 755 |
|
|---|
| 756 | Cell[CellGroupData[{
|
|---|
| 757 |
|
|---|
| 758 | Cell["Results", "Section"],
|
|---|
| 759 |
|
|---|
| 760 | Cell[CellGroupData[{
|
|---|
| 761 |
|
|---|
| 762 | Cell["Virtual+Real+Counterterms in CDR", "Subsection"],
|
|---|
| 763 |
|
|---|
| 764 | Cell[BoxData[{
|
|---|
| 765 | \(\(\[Sigma]rn = \(\(\[Sigma]r/c\[CapitalGamma]\)/\[Pi]\)/2\ CA -
|
|---|
| 766 | 1/e\ 2\ Pgg\ z\ CA + 1/e\ 2\ pgg\ z\ CA;\)\), "\n",
|
|---|
| 767 | \(\(\[Sigma]rn = \((\((\((Coefficient[\[Sigma]rn, DeltaFunction[1 - z]] /.
|
|---|
| 768 | z \[Rule] 1 // FullSimplify)\)*
|
|---|
| 769 | DeltaFunction[
|
|---|
| 770 | 1 - z])\) + \((\((\((Coefficient[\[Sigma]rn,
|
|---|
| 771 | PlusDistribution[1/\((1 - z)\)]] //
|
|---|
| 772 | FullSimplify)\))\)
|
|---|
| 773 | PlusDistribution[
|
|---|
| 774 | 1/\((1 - z)\)])\) + \((\((\((Coefficient[\[Sigma]rn,
|
|---|
| 775 | PlusDistribution[Log[1 - z]/\((1 - z)\)]] //
|
|---|
| 776 | FullSimplify)\))\)
|
|---|
| 777 | PlusDistribution[
|
|---|
| 778 | Log[1 - z]/\((1 - z)\)])\) + \((\(\(\[Sigma]rn /.
|
|---|
| 779 | DeltaFunction[x_] \[Rule] 0\) /.
|
|---|
| 780 | PlusDistribution[Log[1 - z]/\((1 - z)\)] \[Rule] 0\) /.
|
|---|
| 781 | PlusDistribution[1/\((1 - z)\)] \[Rule] 0)\))\);\)\), "\n",
|
|---|
| 782 | \(\(\[Sigma]rn = \((\((\((Coefficient[\[Sigma]rn,
|
|---|
| 783 | 1/e\ PlusDistribution[1/\((1 - z)\)]]/\((1 - z)\) //
|
|---|
| 784 | FullSimplify)\))\) 1/e)\) + \((\[Sigma]rn -
|
|---|
| 785 | Coefficient[\[Sigma]rn, 1/e\ PlusDistribution[1/\((1 - z)\)]]
|
|---|
| 786 | 1/e\ PlusDistribution[1/\((1 - z)\)])\) //
|
|---|
| 787 | Expand;\)\)}], "Input",
|
|---|
| 788 | PageWidth->PaperWidth],
|
|---|
| 789 |
|
|---|
| 790 | Cell[CellGroupData[{
|
|---|
| 791 |
|
|---|
| 792 | Cell[BoxData[
|
|---|
| 793 | \(divD =
|
|---|
| 794 | Coefficient[\[Sigma]rn,
|
|---|
| 795 | DeltaFunction[1 - z]]*\(as/2\)/\[Pi]\ c\[CapitalGamma]\)], "Input"],
|
|---|
| 796 |
|
|---|
| 797 | Cell[BoxData[
|
|---|
| 798 | \(TraditionalForm\`\(as\ c\[CapitalGamma]\ \((\(11\ C\_A\)\/\(3\ e\) + \
|
|---|
| 799 | \(2\ C\_A\)\/e\^2 - \(\[Pi]\^2\ C\_A\)\/3)\)\)\/\(2\ \[Pi]\)\)], "Output"]
|
|---|
| 800 | }, Open ]],
|
|---|
| 801 |
|
|---|
| 802 | Cell[CellGroupData[{
|
|---|
| 803 |
|
|---|
| 804 | Cell[BoxData[
|
|---|
| 805 | \(UVC = \ \(\(1/
|
|---|
| 806 | e\)\(\ \)\(4\)\(*\)\(\(-11\)\/6\) \(CA\)\(\ \)\(as\/\(4 \[Pi]\)\) \
|
|---|
| 807 | \(c\[CapitalGamma]\)\(\ \)\)\)], "Input"],
|
|---|
| 808 |
|
|---|
| 809 | Cell[BoxData[
|
|---|
| 810 | \(TraditionalForm\`\(-\(\(11\ as\ C\_A\ c\[CapitalGamma]\)\/\(6\ e\ \[Pi]\
|
|---|
| 811 | \)\)\)\)], "Output"]
|
|---|
| 812 | }, Open ]],
|
|---|
| 813 |
|
|---|
| 814 | Cell[CellGroupData[{
|
|---|
| 815 |
|
|---|
| 816 | Cell[BoxData[
|
|---|
| 817 | \(\((\(\((divD + UVC)\)/as\)/c\[CapitalGamma]\ 2\ \[Pi] //
|
|---|
| 818 | Expand)\)\ as\ \(c\[CapitalGamma]/2\)/\[Pi]\)], "Input"],
|
|---|
| 819 |
|
|---|
| 820 | Cell[BoxData[
|
|---|
| 821 | \(TraditionalForm\`\(as\ c\[CapitalGamma]\ \((\(2\ C\_A\)\/e\^2 - \(\[Pi]\
|
|---|
| 822 | \^2\ C\_A\)\/3 + C\_A\/3)\)\)\/\(2\ \[Pi]\)\)], "Output"]
|
|---|
| 823 | }, Open ]],
|
|---|
| 824 |
|
|---|
| 825 | Cell[BoxData[
|
|---|
| 826 | \(\(sally =
|
|---|
| 827 | as/\[Pi]\ \((\(-11\)/2\ \((1 - z)\)^3 +
|
|---|
| 828 | 6\ \((1 + z^4 + \((1 - z)\)^4)\)\ PlusDistribution[
|
|---|
| 829 | Log[1 - z]/\((1 - z)\)] -
|
|---|
| 830 | 6\ \((z^2\ PlusDistribution[1/\((1 - z)\)] + \((1 - z)\) +
|
|---|
| 831 | z^2 \((1 - z)\))\) Log[z])\);\)\)], "Input"],
|
|---|
| 832 |
|
|---|
| 833 | Cell[BoxData[
|
|---|
| 834 | \(\((\(\(\((\[Sigma]rn*\(as/2\)/\[Pi] /. DeltaFunction[x_] \[Rule] 0)\) -
|
|---|
| 835 | sally /. CA \[Rule] 3\) /. Pgg \[Rule] 0 // Expand\) //
|
|---|
| 836 | Simplify)\) /.
|
|---|
| 837 | PlusDistribution[1/\((1 - z)\)] \[Rule] 1/\((1 - z)\) //
|
|---|
| 838 | Factor\)], "Input"]
|
|---|
| 839 | }, Open ]]
|
|---|
| 840 | }, Closed]],
|
|---|
| 841 |
|
|---|
| 842 | Cell[CellGroupData[{
|
|---|
| 843 |
|
|---|
| 844 | Cell["Gluon initiated process", "Section",
|
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| 845 | PageWidth->PaperWidth],
|
|---|
| 846 |
|
|---|
| 847 | Cell[CellGroupData[{
|
|---|
| 848 |
|
|---|
| 849 | Cell["Kinematics for the Born", "Subsection",
|
|---|
| 850 | PageWidth->PaperWidth],
|
|---|
| 851 |
|
|---|
| 852 | Cell["\<\
|
|---|
| 853 | -----I take all momenta outgoing
|
|---|
| 854 |
|
|---|
| 855 | p1 + p2 + p3 = 0
|
|---|
| 856 |
|
|---|
| 857 | p1^2=0
|
|---|
| 858 | p2^2=0
|
|---|
| 859 | p3^3=Q^2
|
|---|
| 860 |
|
|---|
| 861 |
|
|---|
| 862 | \
|
|---|
| 863 | \>", "Text",
|
|---|
| 864 | PageWidth->PaperWidth],
|
|---|
| 865 |
|
|---|
| 866 | Cell[BoxData[{
|
|---|
| 867 | \(\(ScalarProduct[p1, p1] = 0;\)\), "\[IndentingNewLine]",
|
|---|
| 868 | \(\(ScalarProduct[p2, p2] = 0;\)\), "\[IndentingNewLine]",
|
|---|
| 869 | \(\(ScalarProduct[p3, p3] = Q2;\)\), "\[IndentingNewLine]",
|
|---|
| 870 | \(\(ScalarProduct[p1, p3] = \(-Q2\)/2;\)\), "\[IndentingNewLine]",
|
|---|
| 871 | \(\(ScalarProduct[p1, p2] = Q2/2;\)\), "\[IndentingNewLine]",
|
|---|
| 872 | \(\(ScalarProduct[p2, p3] = \(-\ Q2\)/2;\)\), "\[IndentingNewLine]",
|
|---|
| 873 | \(\(ScalarProduct[p1, e1] = 0;\)\), "\[IndentingNewLine]",
|
|---|
| 874 | \(\(ScalarProduct[p2, e2] = 0;\)\), "\[IndentingNewLine]",
|
|---|
| 875 | \(\(ScalarProduct[p1, e2] = 0;\)\), "\[IndentingNewLine]",
|
|---|
| 876 | \(\(\(ScalarProduct[p2, e1] = 0;\)\(\[IndentingNewLine]\)
|
|---|
| 877 | \)\), "\[IndentingNewLine]",
|
|---|
| 878 | \(\(ScalarProduct[p1, p1, Dimension \[Rule] D] =
|
|---|
| 879 | 0;\)\), "\[IndentingNewLine]",
|
|---|
| 880 | \(\(ScalarProduct[p2, p2, Dimension \[Rule] D] =
|
|---|
| 881 | 0;\)\), "\[IndentingNewLine]",
|
|---|
| 882 | \(\(ScalarProduct[p3, p3, Dimension \[Rule] D] =
|
|---|
| 883 | Q2;\)\), "\[IndentingNewLine]",
|
|---|
| 884 | \(\(ScalarProduct[p1, p3, Dimension \[Rule] D] = \(-Q2\)/
|
|---|
| 885 | 2;\)\), "\[IndentingNewLine]",
|
|---|
| 886 | \(\(ScalarProduct[p1, p2, Dimension \[Rule] D] =
|
|---|
| 887 | Q2/2;\)\), "\[IndentingNewLine]",
|
|---|
| 888 | \(\(ScalarProduct[p2, p3, Dimension \[Rule] D] = \(-\ Q2\)/
|
|---|
| 889 | 2;\)\), "\[IndentingNewLine]",
|
|---|
| 890 | \(\(ScalarProduct[p1, e1, Dimension \[Rule] D] =
|
|---|
| 891 | 0;\)\), "\[IndentingNewLine]",
|
|---|
| 892 | \(\(ScalarProduct[p2, e2, Dimension \[Rule] D] =
|
|---|
| 893 | 0;\)\), "\[IndentingNewLine]",
|
|---|
| 894 | \(\(ScalarProduct[p1, e2, Dimension \[Rule] D] =
|
|---|
| 895 | 0;\)\), "\[IndentingNewLine]",
|
|---|
| 896 | \(\(ScalarProduct[p2, e1, Dimension \[Rule] D] =
|
|---|
| 897 | 0;\)\), "\[IndentingNewLine]",
|
|---|
| 898 | \(\)}], "Input",
|
|---|
| 899 | PageWidth->PaperWidth]
|
|---|
| 900 | }, Open ]],
|
|---|
| 901 |
|
|---|
| 902 | Cell[CellGroupData[{
|
|---|
| 903 |
|
|---|
| 904 | Cell["Real Amplitude", "Subsection",
|
|---|
| 905 | PageWidth->PaperWidth],
|
|---|
| 906 |
|
|---|
| 907 | Cell[BoxData[{
|
|---|
| 908 | \(\(Emme =
|
|---|
| 909 | CF\ 1/s \((\((s^2 + u^2)\)\ - e\ \((s + u)\)^2)\)/
|
|---|
| 910 | t;\)\), "\[IndentingNewLine]",
|
|---|
| 911 | \(\(RealD = Emme/\((1 - e)\);\)\), "\[IndentingNewLine]",
|
|---|
| 912 | \(\(Real4 = RealD /. \ e \[Rule] 0;\)\)}], "Input"]
|
|---|
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|
|---|
| 914 |
|
|---|
| 915 | Cell[CellGroupData[{
|
|---|
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|
|---|
| 917 | Cell["Integration over the phase space in D dimensions", "Subsection",
|
|---|
| 918 | PageWidth->PaperWidth],
|
|---|
| 919 |
|
|---|
| 920 | Cell[BoxData[{
|
|---|
| 921 | \(\(PS = \(1\/\(8 \[Pi]\)\) \(\((\(\(4\)\(\ \)\(\[Pi]\)\(\ \
|
|---|
| 922 | \)\)\/mh2)\)\^e\)
|
|---|
| 923 | 1\/Gamma[1 - e]\ \((mh2\/s)\)\^e\ \ \((1 - mh2\/s)\)\^\(1 - 2 e\)\ \
|
|---|
| 924 | v\^\(-e\)\ \((1 - v)\)\^\(-e\);\)\), "\[IndentingNewLine]",
|
|---|
| 925 | \(\(substu = {t \[Rule] \ \(-s\)\ \((1 - mh2\/s)\) \((1 - v)\), \
|
|---|
| 926 | u \[Rule] \ \(-s\)\ \((1 - mh2\/s)\) v\ ,
|
|---|
| 927 | s \[Rule] \ mh2/z};\)\), "\[IndentingNewLine]",
|
|---|
| 928 | \(\(cGamma = \((1/16)\)/Pi^2\ mh2^\((\(-e\))\) \((4\ Pi)\)^e\ Gamma[
|
|---|
| 929 | 1 + e]\ Gamma[1 - e]^2/Gamma[1 - 2\ e];\)\), "\n",
|
|---|
| 930 | \(\(pgq =
|
|---|
| 931 | CF \((\((1 - z)\)\^2 + 1)\)/z\ //
|
|---|
| 932 | Factor;\)\), "\[IndentingNewLine]",
|
|---|
| 933 | \(\)}], "Input",
|
|---|
| 934 | PageWidth->PaperWidth],
|
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|
|---|
| 936 | Cell[CellGroupData[{
|
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| 937 |
|
|---|
| 938 | Cell["dim-reg", "Subsubsection",
|
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| 939 | PageWidth->PaperWidth],
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| 940 |
|
|---|
| 941 | Cell[CellGroupData[{
|
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| 942 |
|
|---|
| 943 | Cell[BoxData[{
|
|---|
| 944 | \(\(\[Sigma]r =
|
|---|
| 945 | Integrate[\((RealD\ PS\ c\[CapitalGamma]/cGamma\ //
|
|---|
| 946 | Simplify)\) //. \ substu // Expand, {v, 0, 1}, \
|
|---|
| 947 | GenerateConditions \[Rule] False] // Simplify;\)\), "\n",
|
|---|
| 948 | \(\(\[Sigma]r =
|
|---|
| 949 | Normal[Series[\[Sigma]r/\((1 + \ e)\), {e, 0, 0}]] /. \
|
|---|
| 950 | gs^2\ \[Rule] \ as\ 4\ \[Pi]\ // FullSimplify;\)\), "\n",
|
|---|
| 951 | \(\(\[Sigma]r = \(\(\(-\[Sigma]r\)/c\[CapitalGamma]\)/\[Pi]\)/2 -
|
|---|
| 952 | 1/e\ z\ \ \ Pgq\ + 1/e\ \ z\ \ pgq;\)\), "\n",
|
|---|
| 953 | \(\[Sigma]r = \(\[Sigma]r // PowerExpand\) // Expand\)}], "Input",
|
|---|
| 954 | PageWidth->PaperWidth],
|
|---|
| 955 |
|
|---|
| 956 | Cell[BoxData[
|
|---|
| 957 | \(TraditionalForm\`\(-\(1\/2\)\)\ C\_F\ z\^2 +
|
|---|
| 958 | 2\ C\_F\ \(log(1 - z)\)\ z\^2 - C\_F\ \(log(z)\)\ z\^2 +
|
|---|
| 959 | 3\ C\_F\ z - \(Pgq\ z\)\/e - 4\ C\_F\ \(log(1 - z)\)\ z +
|
|---|
| 960 | 2\ C\_F\ \(log(z)\)\ z - \(3\ C\_F\)\/2 + 4\ C\_F\ \(log(1 - z)\) -
|
|---|
| 961 | 2\ C\_F\ \(log(z)\)\)], "Output"]
|
|---|
| 962 | }, Open ]],
|
|---|
| 963 |
|
|---|
| 964 | Cell[CellGroupData[{
|
|---|
| 965 |
|
|---|
| 966 | Cell[BoxData[{
|
|---|
| 967 | \(sally =
|
|---|
| 968 | 2*\((\(-\ \ \((1 - z)\)\)\ \((7 - 3 z)\)/3\ +
|
|---|
| 969 | 1/2\ z\ pgq\ \((1 + 2 Log[1 - z] -
|
|---|
| 970 | Log[z])\))\)\), "\[IndentingNewLine]",
|
|---|
| 971 | \(\((\(\((\[Sigma]r - \ sally)\) /. \ CF -> 4/3\) /. \ Pgq \[Rule] 0)\) //
|
|---|
| 972 | Simplify\)}], "Input"],
|
|---|
| 973 |
|
|---|
| 974 | Cell[BoxData[
|
|---|
| 975 | \(TraditionalForm\`2\ \((1\/3\ \((7 - 3\ z)\)\ \((z - 1)\) +
|
|---|
| 976 | 1\/2\ C\_F\ \((z\^2 - 2\ z + 2)\)\ \((2\ \(log(1 - z)\) - log(z) +
|
|---|
| 977 | 1)\))\)\)], "Output"],
|
|---|
| 978 |
|
|---|
| 979 | Cell[BoxData[
|
|---|
| 980 | \(TraditionalForm\`0\)], "Output"]
|
|---|
| 981 | }, Open ]],
|
|---|
| 982 |
|
|---|
| 983 | Cell[BoxData[""], "Input"]
|
|---|
| 984 | }, Open ]]
|
|---|
| 985 | }, Open ]]
|
|---|
| 986 | }, Closed]]
|
|---|
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|
|---|
| 988 | },
|
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| 989 | FrontEndVersion->"4.1 for Macintosh",
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| 994 | PrintingPageRange->{Automatic, Automatic}
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|---|
| 995 | ]
|
|---|
| 996 |
|
|---|
| 997 | (*******************************************************************
|
|---|
| 998 | Cached data follows. If you edit this Notebook file directly, not
|
|---|
| 999 | using Mathematica, you must remove the line containing CacheID at
|
|---|
| 1000 | the top of the file. The cache data will then be recreated when
|
|---|
| 1001 | you save this file from within Mathematica.
|
|---|
| 1002 | *******************************************************************)
|
|---|
| 1003 |
|
|---|
| 1004 | (*CellTagsOutline
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|---|
| 1005 | CellTagsIndex->{}
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| 1006 | *)
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| 1008 | (*CellTagsIndex
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| 1010 | *)
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| 1017 |
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| 1018 | Cell[CellGroupData[{
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| 1142 |
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|---|
| 1189 |
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|---|
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|---|
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|---|
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|---|
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|---|
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|---|
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|---|
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| 1221 | Cell[32511, 982, 26, 0, 27, "Input"]
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|---|
| 1222 | }, Open ]]
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|---|
| 1223 | }, Open ]]
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|---|
| 1224 | }, Closed]]
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|---|
| 1225 | }, Open ]]
|
|---|
| 1226 | }
|
|---|
| 1227 | ]
|
|---|
| 1228 | *)
|
|---|
| 1229 |
|
|---|
| 1230 |
|
|---|
| 1231 |
|
|---|
| 1232 | (*******************************************************************
|
|---|
| 1233 | End of Mathematica Notebook file.
|
|---|
| 1234 | *******************************************************************)
|
|---|
| 1235 |
|
|---|
