# Frascati: HiggsGG-NLO.nb

File HiggsGG-NLO.nb, 38.8 KB (added by anonymous, 7 years ago) |
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1 | (************** Content-type: application/mathematica ************** |

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38 | (*CacheID: 232*) |

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40 | |

41 | (*NotebookFileLineBreakTest |

42 | NotebookFileLineBreakTest*) |

43 | (*NotebookOptionsPosition[ 32589, 988]*) |

44 | (*NotebookOutlinePosition[ 33288, 1012]*) |

45 | (* CellTagsIndexPosition[ 33244, 1008]*) |

46 | (*WindowFrame->Normal*) |

47 | |

48 | |

49 | |

50 | Notebook[{ |

51 | |

52 | Cell[CellGroupData[{ |

53 | Cell["gg \[Rule] H at NLO in the EFT", "Title", |

54 | PageWidth->PaperWidth], |

55 | |

56 | Cell[CellGroupData[{ |

57 | |

58 | Cell["Input FeynCalc", "Subsection", |

59 | PageWidth->PaperWidth], |

60 | |

61 | Cell[BoxData[ |

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

63 | PageWidth->PaperWidth], |

64 | |

65 | Cell[TextData[{ |

66 | StyleBox["FeynCalc", |

67 | FontWeight->"Bold"], |

68 | " ", |

69 | "4.1.0.3b", |

70 | " ", |

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

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

73 | ButtonData:>{ |

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

75 | ButtonStyle->"Hyperlink", |

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

77 | }], "Text", |

78 | GeneratedCell->True, |

79 | CellAutoOverwrite->True], |

80 | |

81 | Cell[BoxData[ |

82 | \(\($LimitTo4 = False;\)\)], "Input", |

83 | PageWidth->PaperWidth] |

84 | }, Closed]], |

85 | |

86 | Cell[CellGroupData[{ |

87 | |

88 | Cell["Virtual diagrams :preliminaries", "Section", |

89 | PageWidth->PaperWidth], |

90 | |

91 | Cell[CellGroupData[{ |

92 | |

93 | Cell["Kinematics", "Subsection", |

94 | PageWidth->PaperWidth], |

95 | |

96 | Cell["\<\ |

97 | -----I take all momenta outgoing |

98 | |

99 | p1 + p2 + p3 = 0 |

100 | |

101 | p1^2=0 |

102 | p2^2=0 |

103 | p3^3=Q^2 |

104 | |

105 | |

106 | \ |

107 | \>", "Text", |

108 | PageWidth->PaperWidth], |

109 | |

110 | Cell[BoxData[{ |

111 | \(\(ScalarProduct[p1, p1] = 0;\)\), "\[IndentingNewLine]", |

112 | \(\(ScalarProduct[p2, p2] = 0;\)\), "\[IndentingNewLine]", |

113 | \(\(ScalarProduct[p3, p3] = Q2;\)\), "\[IndentingNewLine]", |

114 | \(\(ScalarProduct[p1, p3] = \(-Q2\)/2;\)\), "\[IndentingNewLine]", |

115 | \(\(ScalarProduct[p1, p2] = Q2/2;\)\), "\[IndentingNewLine]", |

116 | \(\(ScalarProduct[p2, p3] = \(-\ Q2\)/2;\)\), "\[IndentingNewLine]", |

117 | \(\(ScalarProduct[p1, e1] = 0;\)\), "\[IndentingNewLine]", |

118 | \(\(ScalarProduct[p2, e2] = 0;\)\), "\[IndentingNewLine]", |

119 | \(\(ScalarProduct[p1, e2] = 0;\)\), "\[IndentingNewLine]", |

120 | \(\(\(ScalarProduct[p2, e1] = 0;\)\(\[IndentingNewLine]\) |

121 | \)\), "\[IndentingNewLine]", |

122 | \(\(ScalarProduct[p1, p1, Dimension \[Rule] D] = |

123 | 0;\)\), "\[IndentingNewLine]", |

124 | \(\(ScalarProduct[p2, p2, Dimension \[Rule] D] = |

125 | 0;\)\), "\[IndentingNewLine]", |

126 | \(\(ScalarProduct[p3, p3, Dimension \[Rule] D] = |

127 | Q2;\)\), "\[IndentingNewLine]", |

128 | \(\(ScalarProduct[p1, p3, Dimension \[Rule] D] = \(-Q2\)/ |

129 | 2;\)\), "\[IndentingNewLine]", |

130 | \(\(ScalarProduct[p1, p2, Dimension \[Rule] D] = |

131 | Q2/2;\)\), "\[IndentingNewLine]", |

132 | \(\(ScalarProduct[p2, p3, Dimension \[Rule] D] = \(-\ Q2\)/ |

133 | 2;\)\), "\[IndentingNewLine]", |

134 | \(\(ScalarProduct[p1, e1, Dimension \[Rule] D] = |

135 | 0;\)\), "\[IndentingNewLine]", |

136 | \(\(ScalarProduct[p2, e2, Dimension \[Rule] D] = |

137 | 0;\)\), "\[IndentingNewLine]", |

138 | \(\(ScalarProduct[p1, e2, Dimension \[Rule] D] = |

139 | 0;\)\), "\[IndentingNewLine]", |

140 | \(\(ScalarProduct[p2, e1, Dimension \[Rule] D] = |

141 | 0;\)\), "\[IndentingNewLine]", |

142 | \(\)}], "Input", |

143 | PageWidth->PaperWidth] |

144 | }, Closed]], |

145 | |

146 | Cell[CellGroupData[{ |

147 | |

148 | Cell["Verteces and Propagators", "Subsection", |

149 | PageWidth->PaperWidth], |

150 | |

151 | Cell[TextData[{ |

152 | StyleBox["GGG is the kinematic part of the three-gluon vtx (momenta \ |

153 | outgoing, clockwise ordering):\nVTX(ggg) = (-\[ImaginaryI] ", |

154 | FontSize->14], |

155 | Cell[BoxData[ |

156 | \(TraditionalForm\`g\_s\)], |

157 | FontSize->14], |

158 | StyleBox[" ) (\[ImaginaryI] ", |

159 | FontSize->14], |

160 | Cell[BoxData[ |

161 | \(TraditionalForm\`f\^abc\)], |

162 | FontSize->14], |

163 | StyleBox[") GGG\nVTX(qqg) = ( -\[ImaginaryI] ", |

164 | FontSize->14], |

165 | Cell[BoxData[ |

166 | \(TraditionalForm\`g\_s\)], |

167 | FontSize->14], |

168 | ")", |

169 | StyleBox[" ", |

170 | FontSize->14], |

171 | Cell[BoxData[ |

172 | \(TraditionalForm\`\((T\^a)\)\_ij\)], |

173 | FontSize->14], |

174 | StyleBox[" ", |

175 | FontSize->14], |

176 | Cell[BoxData[ |

177 | \(TraditionalForm\`\[Gamma]\^\[Mu]\)], |

178 | FontSize->14], |

179 | StyleBox["\nGluon Propagator= ", |

180 | FontSize->14], |

181 | Cell[BoxData[ |

182 | FormBox[ |

183 | FractionBox[ |

184 | StyleBox[\(\(-\[ImaginaryI]\)\ g\^\[Mu]\[Nu]\), |

185 | FontSize->16], \(p\^2\)], TraditionalForm]], |

186 | FontSize->14], |

187 | "\nQuark ", |

188 | StyleBox["Propagator= ", |

189 | FontSize->14], |

190 | Cell[BoxData[ |

191 | FormBox[ |

192 | FractionBox[ |

193 | StyleBox[\(\(\[ImaginaryI]\)\(\ \)\), |

194 | FontSize->16], \(p\&^\)], TraditionalForm]], |

195 | FontSize->14] |

196 | }], "Text", |

197 | PageWidth->PaperWidth], |

198 | |

199 | Cell[BoxData[{ |

200 | \(\(GGG[p1_, p2_, p3_, m1_, m2_, m3_] := |

201 | FV[p1 - p2, m3]\ MT[m1, m2] + FV[p2 - p3, m1]\ MT[m2, m3] + |

202 | FV[p3 - p1, m2]\ MT[m1, m3];\)\), "\[IndentingNewLine]", |

203 | \(\(GGGD[p1_, p2_, p3_, m1_, m2_, m3_] := |

204 | FVD[p1 - p2, m3]\ MTD[m1, m2] + FVD[p2 - p3, m1]\ MTD[m2, m3] + |

205 | FVD[p3 - p1, m2]\ MTD[m1, m3];\)\), "\[IndentingNewLine]", |

206 | \(\(GGGG[m1_, m2_, m3_, m4_] := |

207 | 2\ MT[m1, m2]\ MT[m3, m4] - MT[m1, m3]\ MT[m2, m4] - |

208 | MT[m1, m4]\ MT[m2, m3];\)\), "\[IndentingNewLine]", |

209 | \(\(GGGGD[m1_, m2_, m3_, m4_] := |

210 | 2\ MTD[m1, m2]\ MTD[m3, m4] - MTD[m1, m3]\ MTD[m2, m4] - |

211 | MTD[m1, m4]\ MTD[m2, m3];\)\), "\[IndentingNewLine]", |

212 | \(\(PropQuark = I;\)\), "\[IndentingNewLine]", |

213 | \(\(PropGluon\ = \(-I\);\)\), "\[IndentingNewLine]", |

214 | \(\(vtx = \(-I\)\ gs;\)\)}], "Input", |

215 | PageWidth->PaperWidth] |

216 | }, Closed]], |

217 | |

218 | Cell[CellGroupData[{ |

219 | |

220 | Cell["Born Matrix element ", "Subsection", |

221 | PageWidth->PaperWidth], |

222 | |

223 | Cell[CellGroupData[{ |

224 | |

225 | Cell[BoxData[{ |

226 | \(factborn = gs2\), "\[IndentingNewLine]", |

227 | \(\(Born = |

228 | factborn \((\ |

229 | MTD[mu, nu]\ SPD[p1, p2] - FVD[p1, nu]\ FVD[p2, mu])\)\ FVD[e1, |

230 | mu]\ FVD[e2, nu] // Contract;\)\), "\[IndentingNewLine]", |

231 | \(\(Born4 = Born /. \ D \[Rule] 4;\)\), "\[IndentingNewLine]", |

232 | \(Born = Born\)}], "Input", |

233 | PageWidth->PaperWidth], |

234 | |

235 | Cell[BoxData[ |

236 | \(TraditionalForm\`gs2\)], "Output"], |

237 | |

238 | Cell[BoxData[ |

239 | FormBox[ |

240 | RowBox[{\(1\/2\), " ", "gs2", " ", "Q2", " ", |

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242 | FormBox["e1", |

243 | "TraditionalForm"], "\[NoBreak]", "\[CenterDot]", "\[NoBreak]", |

244 | FormBox["e2", |

245 | "TraditionalForm"]}]}], TraditionalForm]], "Output"] |

246 | }, Open ]], |

247 | |

248 | Cell[CellGroupData[{ |

249 | |

250 | Cell[BoxData[{ |

251 | \(BornSq = |

252 | Normal[Series[\((\[IndentingNewLine]gs2^2\ /\((D - 2)\)^2*\((\ |

253 | MTD[mu, nu]\ SPD[p1, p2] - |

254 | FVD[p1, nu]\ FVD[p2, mu])\) \((\ |

255 | MTD[mu, nu]\ SPD[p1, p2] - |

256 | FVD[p1, nu]\ FVD[p2, mu])\)\ // Contract)\)\ /. \ |

257 | D \[Rule] 4 - 2 e, {e, 0, 2}]] // |

258 | Simplify\), "\[IndentingNewLine]", |

259 | \(\(BornSq4 = BornSq /. \ e \[Rule] 0;\)\)}], "Input"], |

260 | |

261 | Cell[BoxData[ |

262 | \(TraditionalForm\`1\/8\ \((e\^2 + e + 1)\)\ gs2\^2\ Q2\^2\)], "Output"] |

263 | }, Open ]] |

264 | }, Closed]], |

265 | |

266 | Cell[CellGroupData[{ |

267 | |

268 | Cell["My Scalar Integrals", "Subsection", |

269 | PageWidth->PaperWidth], |

270 | |

271 | Cell[BoxData[{ |

272 | \(\(subInt[expr_] := |

273 | expr /. \ {\[IndentingNewLine]B0[x__] \[Rule] \ |

274 | DUPI/\((I\ Pi^2)\)\ MyB0[x], \[IndentingNewLine]C0[ |

275 | x__] \[Rule] \ |

276 | DUPI/\((I\ Pi^2)\)\ MyC0[x], \[IndentingNewLine]D0[ |

277 | x__] \[Rule] \ |

278 | DUPI/\((I\ Pi^2)\)\ MyD0[x]};\)\), "\[IndentingNewLine]", |

279 | \(\(MyC0[0, 0, Q2, 0, 0, |

280 | 0] = \((c\[CapitalGamma]*\((2/e^2 - Pi^2 - 2/e*Log[Q2/mu2] + |

281 | Log[Q2/mu2]^2)\))\)/\((2* |

282 | Q2)\);\)\), "\[IndentingNewLine]", |

283 | \(\(MyB0[0, 0, 0] = 0;\)\), "\[IndentingNewLine]", |

284 | \(\(MyB0[Q2, 0, 0] = |

285 | c\[CapitalGamma]\ \((1/e + 2 - Log[Q2/mu2])\);\)\), "\n", |

286 | \(\)}], "Input", |

287 | PageWidth->PaperWidth], |

288 | |

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

290 | |

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

292 | }, Closed]] |

293 | }, Closed]], |

294 | |

295 | Cell[CellGroupData[{ |

296 | |

297 | Cell["CDR", "Section"], |

298 | |

299 | Cell[CellGroupData[{ |

300 | |

301 | Cell["Vertex Diagram ", "Subsection", |

302 | PageWidth->PaperWidth], |

303 | |

304 | Cell[CellGroupData[{ |

305 | |

306 | Cell[BoxData[{ |

307 | \(\(\(colorvtx = SUNF[b, x, y] SUNF[y, a, x]\ // SUNSimplify\)\(\n\) |

308 | \)\), "\[IndentingNewLine]", |

309 | \(\(num = |

310 | gs2^2*\((\(-gs2\))\)* |

311 | I*\[IndentingNewLine]\((\ |

312 | MTD[al, be]\ SPD[\(-l\) + p1, p2 + l] - \ \ FVD[\(-l\) + p1, |

313 | be]\ FVD[p2 + l, al])\)\ *\ |

314 | GGGD[\(-l\), p1, l - p1, ro, mu, al]*\[IndentingNewLine]GGGD[p2, |

315 | l, \(-l\) - p2, nu, ro, be]*\((\ |

316 | MTD[mu, nu]\ SPD[p1, p2] - FVD[p1, nu]\ FVD[p2, mu])\)\ // |

317 | Contract;\)\), "\n", |

318 | \(\(dens = FAD[l, p1 - l, l + p2];\)\), "\[IndentingNewLine]", |

319 | \(\(amp = CA\ dens*\ num/\((D - 2)\)^2;\)\), "\n", |

320 | \(inte1 = \((\(\((\(OneLoop[l, amp/DUPI] // PaVeReduce\) // Simplify)\) // |

321 | Factor\) // Simplify)\) /. \ B0[0, 0, 0] \[Rule] 0\ // |

322 | Simplify\)}], "Input", |

323 | PageWidth->PaperWidth], |

324 | |

325 | Cell[BoxData[ |

326 | FormBox[ |

327 | RowBox[{\(C\_A\), " ", |

328 | SubscriptBox["\[Delta]", |

329 | RowBox[{ |

330 | FormBox[ |

331 | FormBox["a", |

332 | "TraditionalForm"], |

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

334 | FormBox[ |

335 | FormBox["b", |

336 | "TraditionalForm"], |

337 | "TraditionalForm"]}]]}], TraditionalForm]], "Output"], |

338 | |

339 | Cell[BoxData[ |

340 | FormBox[ |

341 | RowBox[{"-", |

342 | RowBox[{ |

343 | RowBox[{"(", |

344 | |

345 | RowBox[{\(C\_A\), " ", \(gs2\^3\), " ", \(\[Pi]\^2\), |

346 | " ", \(Q2\^2\), " ", |

347 | RowBox[{"(", |

348 | |

349 | RowBox[{\(\((\(-20\)\ D\^2 + 73\ D - 52)\)\ \(\(B\_0\)(Q2, 0, |

350 | 0)\)\), "+", |

351 | RowBox[{"8", " ", \((D\^2 - 3\ D + 2)\), " ", "Q2", " ", |

352 | RowBox[{ |

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

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

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

357 | FormBox["0", |

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

359 | "Q2", "\[NoBreak]", ",", "\[NoBreak]", |

360 | FormBox["0", |

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

362 | FormBox["0", |

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

364 | FormBox["0", |

365 | "TraditionalForm"], "\[NoBreak]", ")"}]}]}], ")"}]}], |

366 | ")"}], "/", \((16\ \((D - 2)\)\^2\ \((D - 1)\)\ DUPI)\)}]}], |

367 | TraditionalForm]], "Output"] |

368 | }, Open ]], |

369 | |

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

371 | }, Open ]], |

372 | |

373 | Cell[CellGroupData[{ |

374 | |

375 | Cell["4-gluon Diagram ", "Subsection", |

376 | PageWidth->PaperWidth], |

377 | |

378 | Cell[CellGroupData[{ |

379 | |

380 | Cell[BoxData[{ |

381 | \(\(color = SUNF[b, c, y] SUNF[y, a, c]\ // SUNSimplify;\)\), "\n", |

382 | \(\(num = |

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)\) // |

392 | Factor\) // Simplify\)}], "Input", |

393 | PageWidth->PaperWidth], |

394 | |

395 | Cell[BoxData[ |

396 | \(TraditionalForm\`\(-\(\(C\_A\ \((2\ D - 3)\)\ \((2\ D\^2 - 5\ D + |

397 | 4)\)\ gs2\^3\ \[Pi]\^2\ Q2\^2\ \(\(B\_0\)(Q2, 0, |

398 | 0)\)\)\/\(16\ \((D - 2)\)\^2\ \((D - |

399 | 1)\)\ DUPI\)\)\)\)], "Output"] |

400 | }, Open ]] |

401 | }, Open ]], |

402 | |

403 | Cell[CellGroupData[{ |

404 | |

405 | Cell["Sum the virtuals", "Subsection"], |

406 | |

407 | Cell[CellGroupData[{ |

408 | |

409 | Cell[BoxData[ |

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", |

416 | PageWidth->PaperWidth], |

417 | |

418 | Cell[BoxData[ |

419 | \(TraditionalForm\`1\/2\ C\_A\ c\[CapitalGamma]\ gs2\ \[Pi]\^2 - \(C\_A\ \ |

420 | c\[CapitalGamma]\ gs2\)\/e\^2\)], "Output"] |

421 | }, Open ]], |

422 | |

423 | Cell[CellGroupData[{ |

424 | |

425 | Cell[BoxData[{ |

426 | \(\(add = |

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]", |

434 | \(add\)}], "Input", |

435 | PageWidth->PaperWidth], |

436 | |

437 | Cell[BoxData[ |

438 | \(TraditionalForm\`11\/6\ C\_A\ c\[CapitalGamma]\ gs2\)], "Output"] |

439 | }, Open ]], |

440 | |

441 | Cell[CellGroupData[{ |

442 | |

443 | Cell[BoxData[ |

444 | \(virt = \(\((\((virt*\(2/gs2\)/c\[CapitalGamma])\) // |

445 | Expand)\)\(*\)\(as\/\(2 \[Pi]\)\) \(c\[CapitalGamma]\)\(\ \ |

446 | \)\)\)], "Input"], |

447 | |

448 | Cell[BoxData[ |

449 | \(TraditionalForm\`\(as\ c\[CapitalGamma]\ \((\(-\(\(2\ C\_A\)\/e\^2\)\) \ |

450 | + \[Pi]\^2\ C\_A + \(11\ C\_A\)\/3)\)\)\/\(2\ \[Pi]\)\)], "Output"] |

451 | }, Open ]], |

452 | |

453 | Cell[BoxData[ |

454 | \(\[IndentingNewLine]\)], "Input"], |

455 | |

456 | Cell[CellGroupData[{ |

457 | |

458 | Cell[BoxData[ |

459 | \(\(\(32/256\)/3\)/\((1 - e)\)^2\)], "Input"], |

460 | |

461 | Cell[BoxData[ |

462 | \(TraditionalForm\`1\/\(24\ \((1 - e)\)\^2\)\)], "Output"] |

463 | }, Open ]] |

464 | }, Open ]] |

465 | }, Closed]], |

466 | |

467 | Cell[CellGroupData[{ |

468 | |

469 | Cell["The real contributions", "Section", |

470 | PageWidth->PaperWidth], |

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] |

541 | }, Open ]], |

542 | |

543 | Cell[CellGroupData[{ |

544 | |

545 | Cell["Verteces and Propagators", "Subsection", |

546 | PageWidth->PaperWidth], |

547 | |

548 | Cell[TextData[{ |

549 | StyleBox["GGG is the kinematic part of the three-gluon vtx (momenta \ |

550 | outgoing, clockwise ordering):\nVTX(ggg) = (-\[ImaginaryI] ", |

551 | FontSize->14], |

552 | Cell[BoxData[ |

553 | \(TraditionalForm\`g\_s\)], |

554 | FontSize->14], |

555 | StyleBox[" ) (\[ImaginaryI] ", |

556 | FontSize->14], |

557 | Cell[BoxData[ |

558 | \(TraditionalForm\`f\^abc\)], |

559 | FontSize->14], |

560 | StyleBox[") GGG\nVTX(qqg) = ( -\[ImaginaryI] ", |

561 | FontSize->14], |

562 | Cell[BoxData[ |

563 | \(TraditionalForm\`g\_s\)], |

564 | FontSize->14], |

565 | ")", |

566 | StyleBox[" ", |

567 | FontSize->14], |

568 | Cell[BoxData[ |

569 | \(TraditionalForm\`\((T\^a)\)\_ij\)], |

570 | FontSize->14], |

571 | StyleBox[" ", |

572 | FontSize->14], |

573 | Cell[BoxData[ |

574 | \(TraditionalForm\`\[Gamma]\^\[Mu]\)], |

575 | FontSize->14], |

576 | StyleBox["\nGluon Propagator= ", |

577 | FontSize->14], |

578 | Cell[BoxData[ |

579 | FormBox[ |

580 | FractionBox[ |

581 | StyleBox[\(\(-\[ImaginaryI]\)\ g\^\[Mu]\[Nu]\), |

582 | FontSize->16], \(p\^2\)], TraditionalForm]], |

583 | FontSize->14], |

584 | "\nQuark ", |

585 | StyleBox["Propagator= ", |

586 | FontSize->14], |

587 | Cell[BoxData[ |

588 | FormBox[ |

589 | FractionBox[ |

590 | StyleBox[\(\(\[ImaginaryI]\)\(\ \)\), |

591 | FontSize->16], \(p\&^\)], TraditionalForm]], |

592 | FontSize->14] |

593 | }], "Text", |

594 | PageWidth->PaperWidth], |

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 ]], |

612 | |

613 | Cell[CellGroupData[{ |

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 ]], |

667 | |

668 | Cell["\<\ |

669 | |

670 | Amp2=Amp2 /. D->4-2e//Simplify;\ |

671 | \>", "Input"] |

672 | }, Open ]], |

673 | |

674 | Cell[CellGroupData[{ |

675 | |

676 | Cell["Real Amplitude Squared in D dimensions", "Subsection", |

677 | PageWidth->PaperWidth], |

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", |

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"] |

913 | }, Open ]], |

914 | |

915 | Cell[CellGroupData[{ |

916 | |

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], |

935 | |

936 | Cell[CellGroupData[{ |

937 | |

938 | Cell["dim-reg", "Subsubsection", |

939 | PageWidth->PaperWidth], |

940 | |

941 | Cell[CellGroupData[{ |

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 = |

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997 | (******************************************************************* |

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

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1232 | (******************************************************************* |

1233 | End of Mathematica Notebook file. |

1234 | *******************************************************************) |

1235 |