# Frascati: HiggsGG-LO-mtfinite.nb

File HiggsGG-LO-mtfinite.nb, 12.4 KB (added by anonymous, 6 years ago) |
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1 | (************** Content-type: application/mathematica ************** |

2 | |

3 | Mathematica-Compatible Notebook |

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48 | |

49 | |

50 | Notebook[{ |

51 | |

52 | Cell[CellGroupData[{ |

53 | Cell[TextData[{ |

54 | "Calculation for ", |

55 | StyleBox["gg > Higgs at LO with full top-mass dependence", |

56 | "DisplayFormula"] |

57 | }], "Title"], |

58 | |

59 | Cell[CellGroupData[{ |

60 | |

61 | Cell["Input FeynCalc", "Subsection"], |

62 | |

63 | Cell[BoxData[ |

64 | \(\(<< HighEnergyPhysics`fc`;\)\)], "Input"], |

65 | |

66 | Cell[TextData[{ |

67 | StyleBox["FeynCalc", |

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69 | " ", |

70 | "4.1.0.3b", |

71 | " ", |

72 | " Evaluate ?FeynCalc for help or visit ", |

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78 | }], "Text", |

79 | GeneratedCell->True, |

80 | CellAutoOverwrite->True] |

81 | }, Closed]], |

82 | |

83 | Cell[CellGroupData[{ |

84 | |

85 | Cell["Preliminaries", "Subsection"], |

86 | |

87 | Cell[CellGroupData[{ |

88 | |

89 | Cell["Kinematics 2->1 ", "Subsubsection"], |

90 | |

91 | Cell[BoxData[ |

92 | \(\(\(\[IndentingNewLine]\)\(\(ScalarProduct[q1, q1] = |

93 | 0;\)\[IndentingNewLine] |

94 | \(ScalarProduct[q2, q2] = 0;\)\[IndentingNewLine] |

95 | \(ScalarProduct[q1, q2] = mh2/2;\)\[IndentingNewLine] |

96 | \(ScalarProduct[q, q1] = mh2/2;\)\[IndentingNewLine] |

97 | \(ScalarProduct[q, q2] = mh2/2;\)\[IndentingNewLine] |

98 | \(ScalarProduct[q, q] = mh2;\)\[IndentingNewLine] |

99 | \)\)\)], "Input"] |

100 | }, Open ]] |

101 | }, Closed]], |

102 | |

103 | Cell[CellGroupData[{ |

104 | |

105 | Cell["Amplitude (2 diagrams)", "Subsection"], |

106 | |

107 | Cell[CellGroupData[{ |

108 | |

109 | Cell[BoxData[ |

110 | \(\(\(\[IndentingNewLine]\)\(\(Amp = \((\(-I\))\) \((\(\((\(-\((\(-I\)\ \ |

111 | gs)\)^2\)\ \ \((\(-\ I\)\ mt\ /v)\)\ *\ I^3*deltaAB/2* |

112 | Tr[\((GSD[l + q1] + mt)\) . |

113 | GAD[mu] . \((GSD[l] + mt)\) . |

114 | GAD[nu] . \((GSD[l - q2] + |

115 | mt)\) + \((GSD[l + q2] + mt)\) . |

116 | GAD[nu] . \((GSD[l] + mt)\) . |

117 | GAD[mu] . \((GSD[l - q1] + mt)\)]\ // |

118 | DiracSimplify)\) /. \ |

119 | Pair[Momentum[q2], LorentzIndex[nu]] \[Rule] 0\) /. \ |

120 | Pair[Momentum[q1], LorentzIndex[mu]] \[Rule] 0)\)\ // |

121 | Simplify\)\(\[IndentingNewLine]\) |

122 | \)\)\)], "Input"], |

123 | |

124 | Cell[BoxData[ |

125 | FormBox[ |

126 | RowBox[{\(1\/v\), |

127 | RowBox[{"(", |

128 | RowBox[{ |

129 | "2", " ", "\[ImaginaryI]", " ", "deltaAB", " ", \(gs\^2\), |

130 | " ", \(mt\^2\), " ", |

131 | RowBox[{"(", |

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133 | RowBox[{"8", " ", |

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139 | "TraditionalForm"], |

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143 | "TraditionalForm"], |

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146 | "TraditionalForm"], |

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151 | "TraditionalForm"], |

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154 | "TraditionalForm"], |

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176 | "TraditionalForm"], |

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183 | "TraditionalForm"], |

184 | "TraditionalForm"], "\[NoBreak]", |

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186 | FormBox["nu", |

187 | "TraditionalForm"], |

188 | "TraditionalForm"]}]], " ", |

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190 | RowBox[{\(\(-2\)\ mt\^2\), "+", "mh2", "+", |

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194 | "TraditionalForm"], "2"]}]}], ")"}]}]}], ")"}]}], |

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198 | |

199 | Cell[CellGroupData[{ |

200 | |

201 | Cell["Let's ask FeynCalc to do the tensor reduction", "Subsection"], |

202 | |

203 | Cell[CellGroupData[{ |

204 | |

205 | Cell[BoxData[ |

206 | \(\(\(\[IndentingNewLine]\)\(\(res = \(\(1/\((2\ Pi)\)^4* |

207 | OneLoop[l, |

208 | FAD[{l, mt}, {l + q1, mt}, {l - q2, mt}]\ Amp // Contract] // |

209 | PaVeReduce\) // Factor\) // Simplify;\)\[IndentingNewLine] |

210 | res = \((\(res /. \ Pair[Momentum[q2], LorentzIndex[nu]] \[Rule] 0\) /. \ |

211 | Pair[Momentum[q1], LorentzIndex[mu]] \[Rule] 0\ )\) // |

212 | Simplify\)\)\)], "Input"], |

213 | |

214 | Cell[BoxData[ |

215 | FormBox[ |

216 | RowBox[{\(1\/\(8\ mh2\ \[Pi]\^2\ v\)\), |

217 | RowBox[{"(", |

218 | RowBox[{"deltaAB", " ", \(gs\^2\), " ", \(mt\^2\), " ", |

219 | RowBox[{"(", |

220 | RowBox[{ |

221 | RowBox[{\((mh2 - 4\ mt\^2)\), " ", |

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224 | "TraditionalForm"], "\[NoBreak]", "(", "\[NoBreak]", |

225 | "mh2", "\[NoBreak]", ",", "\[NoBreak]", |

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227 | "TraditionalForm"], "\[NoBreak]", ",", "\[NoBreak]", |

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229 | "TraditionalForm"], "\[NoBreak]", ",", "\[NoBreak]", |

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231 | "TraditionalForm"], "\[NoBreak]", ",", "\[NoBreak]", |

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233 | "TraditionalForm"], "\[NoBreak]", ",", "\[NoBreak]", |

234 | FormBox[\(mt\^2\), |

235 | "TraditionalForm"], "\[NoBreak]", ")"}]}], "-", "2"}], |

236 | ")"}], " ", |

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256 | }, Closed]], |

257 | |

258 | Cell[CellGroupData[{ |

259 | |

260 | Cell["\<\ |

261 | The scalar integral C0, can be evaluated with the help of the \ |

262 | Feynman parameters (by hand) and the result is:\ |

263 | \>", "Subsection"], |

264 | |

265 | Cell["\<\ |

266 | c0=-I/(16 Pi^2)*1/mt^2*Integrate[1/(1-4 \[Tau] x \ |

267 | y),{x,0,1},{y,0,1-x}, Assumptions \[Rule] {\[Tau]<1}]//Simplify; |

268 | c0FC=(2 Pi)^4/(I Pi^2) c0;\ |

269 | \>", "Input"] |

270 | }, Closed]], |

271 | |

272 | Cell[CellGroupData[{ |

273 | |

274 | Cell["\<\ |

275 | Let's take the mt->Infinity limit and see that the amplitude does \ |

276 | not depend on m_top:\ |

277 | \>", "Subsection"], |

278 | |

279 | Cell[CellGroupData[{ |

280 | |

281 | Cell[BoxData[ |

282 | \(myamp = |

283 | Normal[Series[\(\(\(\((\ |

284 | res\ /. \ C0[x__]\ \[Rule] c0FC // Simplify)\) /. \ |

285 | gs^2\ \[Rule] \ as\ 4\ Pi\) /. \ |

286 | mh2 \[Rule] mh^2\) /. \ \[Tau] \[Rule] \ \(mh^2/4\)/mt^2 // |

287 | PowerExpand\) // FullSimplify, {mt, Infinity, 4}]] // |

288 | Simplify\)], "Input"], |

289 | |

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300 | "TraditionalForm"], |

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307 | "TraditionalForm"], |

308 | FormBox[ |

309 | FormBox["nu", |

310 | "TraditionalForm"], |

311 | "TraditionalForm"]]}]}], ")"}]}], \(6\ \[Pi]\ v\)]}], |

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