# GIFSchool: HiggsGG-LO-mtfinite.nb

File HiggsGG-LO-mtfinite.nb, 12.4 KB (added by anonymous, 6 years ago) |
---|

Line | |
---|---|

1 | (************** Content-type: application/mathematica ************** |

2 | |

3 | Mathematica-Compatible Notebook |

4 | |

5 | This notebook can be used with any Mathematica-compatible |

6 | application, such as Mathematica, MathReader or Publicon. The data |

7 | for the notebook starts with the line containing stars above. |

8 | |

9 | To get the notebook into a Mathematica-compatible application, do |

10 | one of the following: |

11 | |

12 | * Save the data starting with the line of stars above into a file |

13 | with a name ending in .nb, then open the file inside the |

14 | application; |

15 | |

16 | * Copy the data starting with the line of stars above to the |

17 | clipboard, then use the Paste menu command inside the application. |

18 | |

19 | Data for notebooks contains only printable 7-bit ASCII and can be |

20 | sent directly in email or through ftp in text mode. Newlines can be |

21 | CR, LF or CRLF (Unix, Macintosh or MS-DOS style). |

22 | |

23 | NOTE: If you modify the data for this notebook not in a Mathematica- |

24 | compatible application, you must delete the line below containing |

25 | the word CacheID, otherwise Mathematica-compatible applications may |

26 | try to use invalid cache data. |

27 | |

28 | For more information on notebooks and Mathematica-compatible |

29 | applications, contact Wolfram Research: |

30 | web: http://www.wolfram.com |

31 | email: info@wolfram.com |

32 | phone: +1-217-398-0700 (U.S.) |

33 | |

34 | Notebook reader applications are available free of charge from |

35 | Wolfram Research. |

36 | *******************************************************************) |

37 | |

38 | (*CacheID: 232*) |

39 | |

40 | |

41 | (*NotebookFileLineBreakTest |

42 | NotebookFileLineBreakTest*) |

43 | (*NotebookOptionsPosition[ 10709, 318]*) |

44 | (*NotebookOutlinePosition[ 11346, 340]*) |

45 | (* CellTagsIndexPosition[ 11302, 336]*) |

46 | (*WindowFrame->Normal*) |

47 | |

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", |

68 | FontWeight->"Bold"], |

69 | " ", |

70 | "4.1.0.3b", |

71 | " ", |

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

73 | ButtonBox["www.feyncalc.org", |

74 | ButtonData:>{ |

75 | URL[ "http://www.feyncalc.org"], None}, |

76 | ButtonStyle->"Hyperlink", |

77 | ButtonNote->"http://www.feyncalc.org"] |

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[{"(", |

132 | RowBox[{ |

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

134 | SuperscriptBox[ |

135 | FormBox["l", |

136 | "TraditionalForm"], |

137 | FormBox[ |

138 | FormBox["mu", |

139 | "TraditionalForm"], |

140 | "TraditionalForm"]], " ", |

141 | SuperscriptBox[ |

142 | FormBox["l", |

143 | "TraditionalForm"], |

144 | FormBox[ |

145 | FormBox["nu", |

146 | "TraditionalForm"], |

147 | "TraditionalForm"]]}], "+", |

148 | RowBox[{"2", " ", |

149 | SuperscriptBox[ |

150 | FormBox["q2", |

151 | "TraditionalForm"], |

152 | FormBox[ |

153 | FormBox["mu", |

154 | "TraditionalForm"], |

155 | "TraditionalForm"]], " ", |

156 | SuperscriptBox[ |

157 | FormBox["q1", |

158 | "TraditionalForm"], |

159 | FormBox[ |

160 | FormBox["nu", |

161 | "TraditionalForm"], |

162 | "TraditionalForm"]]}], "-", |

163 | RowBox[{"2", " ", |

164 | SuperscriptBox[ |

165 | FormBox["q1", |

166 | "TraditionalForm"], |

167 | FormBox[ |

168 | FormBox["mu", |

169 | "TraditionalForm"], |

170 | "TraditionalForm"]], " ", |

171 | SuperscriptBox[ |

172 | FormBox["q2", |

173 | "TraditionalForm"], |

174 | FormBox[ |

175 | FormBox["nu", |

176 | "TraditionalForm"], |

177 | "TraditionalForm"]]}], "-", |

178 | RowBox[{ |

179 | SuperscriptBox["g", |

180 | RowBox[{ |

181 | FormBox[ |

182 | FormBox["mu", |

183 | "TraditionalForm"], |

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

185 | FormBox[ |

186 | FormBox["nu", |

187 | "TraditionalForm"], |

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

189 | RowBox[{"(", |

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

191 | RowBox[{"2", " ", |

192 | SuperscriptBox[ |

193 | FormBox["l", |

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

195 | ")"}]}], TraditionalForm]], "Output"] |

196 | }, Open ]] |

197 | }, Closed]], |

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

222 | RowBox[{ |

223 | FormBox[\("C"\_"0"\), |

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

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

226 | FormBox["0", |

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

228 | FormBox["0", |

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

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

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

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

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

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

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

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

237 | RowBox[{"(", |

238 | RowBox[{\(mh2\ g\^\(mu\[NoBreak]nu\)\), "-", |

239 | RowBox[{"2", " ", |

240 | SuperscriptBox[ |

241 | FormBox["q2", |

242 | "TraditionalForm"], |

243 | FormBox[ |

244 | FormBox["mu", |

245 | "TraditionalForm"], |

246 | "TraditionalForm"]], " ", |

247 | SuperscriptBox[ |

248 | FormBox["q1", |

249 | "TraditionalForm"], |

250 | FormBox[ |

251 | FormBox["nu", |

252 | "TraditionalForm"], |

253 | "TraditionalForm"]]}]}], ")"}]}], ")"}]}], |

254 | TraditionalForm]], "Output"] |

255 | }, Open ]] |

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 | |

290 | Cell[BoxData[ |

291 | FormBox[ |

292 | RowBox[{"-", |

293 | FractionBox[ |

294 | RowBox[{"as", " ", "deltaAB", " ", |

295 | RowBox[{"(", |

296 | RowBox[{\(mh\^2\ g\^\(mu\[NoBreak]nu\)\), "-", |

297 | RowBox[{"2", " ", |

298 | SuperscriptBox[ |

299 | FormBox["q2", |

300 | "TraditionalForm"], |

301 | FormBox[ |

302 | FormBox["mu", |

303 | "TraditionalForm"], |

304 | "TraditionalForm"]], " ", |

305 | SuperscriptBox[ |

306 | FormBox["q1", |

307 | "TraditionalForm"], |

308 | FormBox[ |

309 | FormBox["nu", |

310 | "TraditionalForm"], |

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

312 | TraditionalForm]], "Output"] |

313 | }, Open ]], |

314 | |

315 | Cell[BoxData[""], "Input"] |

316 | }, Closed]] |

317 | }, Open ]] |

318 | }, |

319 | FrontEndVersion->"4.1 for Macintosh", |

320 | ScreenRectangle->{{0, 1280}, {0, 832}}, |

321 | WindowSize->{710, 706}, |

322 | WindowMargins->{{Automatic, 230}, {Automatic, 4}} |

323 | ] |

324 | |

325 | (******************************************************************* |

326 | Cached data follows. If you edit this Notebook file directly, not |

327 | using Mathematica, you must remove the line containing CacheID at |

328 | the top of the file. The cache data will then be recreated when |

329 | you save this file from within Mathematica. |

330 | *******************************************************************) |

331 | |

332 | (*CellTagsOutline |

333 | CellTagsIndex->{} |

334 | *) |

335 | |

336 | (*CellTagsIndex |

337 | CellTagsIndex->{} |

338 | *) |

339 | |

340 | (*NotebookFileOutline |

341 | Notebook[{ |

342 | |

343 | Cell[CellGroupData[{ |

344 | Cell[1727, 52, 134, 4, 156, "Title"], |

345 | |

346 | Cell[CellGroupData[{ |

347 | Cell[1886, 60, 36, 0, 46, "Subsection"], |

348 | Cell[1925, 62, 62, 1, 27, "Input"], |

349 | Cell[1990, 65, 369, 14, 70, "Text"] |

350 | }, Closed]], |

351 | |

352 | Cell[CellGroupData[{ |

353 | Cell[2396, 84, 35, 0, 30, "Subsection"], |

354 | |

355 | Cell[CellGroupData[{ |

356 | Cell[2456, 88, 41, 0, 42, "Subsubsection"], |

357 | Cell[2500, 90, 407, 8, 139, "Input"] |

358 | }, Open ]] |

359 | }, Closed]], |

360 | |

361 | Cell[CellGroupData[{ |

362 | Cell[2956, 104, 44, 0, 30, "Subsection"], |

363 | |

364 | Cell[CellGroupData[{ |

365 | Cell[3025, 108, 753, 13, 219, "Input"], |

366 | Cell[3781, 123, 2747, 71, 62, "Output"] |

367 | }, Open ]] |

368 | }, Closed]], |

369 | |

370 | Cell[CellGroupData[{ |

371 | Cell[6577, 200, 67, 0, 30, "Subsection"], |

372 | |

373 | Cell[CellGroupData[{ |

374 | Cell[6669, 204, 438, 7, 155, "Input"], |

375 | Cell[7110, 213, 1772, 40, 62, "Output"] |

376 | }, Open ]] |

377 | }, Closed]], |

378 | |

379 | Cell[CellGroupData[{ |

380 | Cell[8931, 259, 141, 3, 48, "Subsection"], |

381 | Cell[9075, 264, 167, 4, 57, "Input"] |

382 | }, Closed]], |

383 | |

384 | Cell[CellGroupData[{ |

385 | Cell[9279, 273, 118, 3, 30, "Subsection"], |

386 | |

387 | Cell[CellGroupData[{ |

388 | Cell[9422, 280, 378, 7, 123, "Input"], |

389 | Cell[9803, 289, 837, 22, 46, "Output"] |

390 | }, Open ]], |

391 | Cell[10655, 314, 26, 0, 27, "Input"] |

392 | }, Closed]] |

393 | }, Open ]] |

394 | } |

395 | ] |

396 | *) |

397 | |

398 | |

399 | |

400 | (******************************************************************* |

401 | End of Mathematica Notebook file. |

402 | *******************************************************************) |

403 |