(************** Content-type: application/mathematica ************** Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 9225, 289]*) (*NotebookOutlinePosition[ 9955, 314]*) (* CellTagsIndexPosition[ 9911, 310]*) (*WindowFrame->Normal*) Notebook[{ Cell["QED vacuum polarization to two loops", "Title"], Cell[CellGroupData[{ Cell[BoxData[ \(\(Needs["\"];\)\)], "Input"], Cell[TextData[{ StyleBox["FeynCalc", FontWeight->"Bold"], " ", "4.1.1", " For help, type ?FeynCalc, ", ButtonBox["use the built-in help system", ButtonFunction:>(FrontEndExecute[ FrontEnd`HelpBrowserLookup[ "AddOns", #]]&), ButtonData:>{"Short Overview", "intro"}, ButtonStyle->"AddOnsLink", ButtonNote->"Open the help browser"], " or visit ", ButtonBox["www.feyncalc.org", ButtonData:>{ URL[ "http://www.feyncalc.org"], None}, ButtonStyle->"Hyperlink", ButtonNote->"http://www.feyncalc.org"] }], "Text", GeneratedCell->True, CellAutoOverwrite->True] }, Open ]], Cell["\<\ The expression we wish to consider is the product of the two factors below:\ \>", "Text"], Cell[BoxData[ \(tra = DiracTrace[ GAD[\[Nu], \ \[Sigma], \ \[Mu], \ \[Lambda], \ \[Alpha], \ \[Rho], \ \ \[Beta], \ \[Lambda]]] // FCI\)], "Input"], Cell[BoxData[ \(notrace = \(-e^4\)\ FAD[q, p, p + k, p + q + k, p + q] FVD[p, \[Beta]] FVD[p + k, \[Alpha]] FVD[p + q + k, \[Mu]] FVD[p + q, \[Nu]] /. {\[Alpha] \[Rule] \[Nu], \[Beta] \[Rule] \ \[Mu], \[Nu] \[Rule] \[Alpha], \[Mu] \[Rule] \[Beta]} (*\(Seems\ to\ be\ a\ \ bug\ in\ Itzykson &\) Zuber*) // FCI\)], "Input"], Cell["The trace is easily evaluated:", "Text"], Cell[BoxData[ \(trace = Collect[Simplify\ [ Tr[tra]] //. {\((6 a_ - D\ a_)\) \[Rule] \((D - 6)\) min\ a, \((D\ a_ - 6 a_)\) \[Rule] \((D - 6)\) a, \((2 a_ - D\ a_)\) \[Rule] \((D - 2)\) min\ a, \((D\ a_ - 2 a_)\) \[Rule] \((D - 2)\) a}, {\((D - 2)\), \((D - 6)\)}] /. min \[Rule] \(-1\) // Simplify\)], "Input"], Cell["\<\ The other factor will be Feynman parameterized. First, the structure of the \ expression:\ \>", "Text"], Cell[BoxData[ \(\((\(FeynmanParametrize1[notrace, p, Method \[Rule] Exp, Integrate \[Rule] False, CompleteSquare \[Rule] False, FeynmanParameterNames \[Rule] {\[Alpha]1, \[Alpha]2, \ \[Alpha]3, \[Alpha]4}] // ExpandScalarProduct\) // Simplify)\) /. {\[Alpha]1 + \[Alpha]2 + \[Alpha]3 + \[Alpha]4 \ \[Rule] \[CapitalSigma], \(-\[Alpha]1\) - \[Alpha]2 - \[Alpha]3 - \[Alpha]4 \ \[Rule] \(-\[CapitalSigma]\)} // Simplify\)], "Input"], Cell[BoxData[ \(\((\(FeynmanParametrize1[notrace, p, Method \[Rule] Exp, Integrate \[Rule] False, CompleteSquare \[Rule] True, FeynmanParameterNames \[Rule] {\[Alpha]1, \[Alpha]2, \ \[Alpha]3, \[Alpha]4}] // ExpandScalarProduct\) // Simplify)\) /. {\[Alpha]1 + \[Alpha]2 + \[Alpha]3 + \[Alpha]4 \ \[Rule] \[CapitalSigma], \(-\[Alpha]1\) - \[Alpha]2 - \[Alpha]3 - \[Alpha]4 \ \[Rule] \(-\[CapitalSigma]\)} // Simplify\)], "Input"], Cell[TextData[{ "Tell ", StyleBox["FeynmanParametrize1", FontFamily->"Courier"], " to do the ", StyleBox["p", FontSlant->"Italic"], " integral:" }], "Text"], Cell[BoxData[ \(\(notrace1 = \[IndentingNewLine]\((\(FeynmanParametrize1[notrace, p, Method \[Rule] Exp, Integrate \[Rule] True, FeynmanParameterNames \[Rule] {\[Alpha]1, \[Alpha]2, \ \[Alpha]3, \[Alpha]4}] // ExpandScalarProduct\) // Simplify)\);\)\)], "Input"], Cell["Multiply on the trace and contract Lorentz indices:", "Text"], Cell[BoxData[ \(\(qexpr = notrace1 /. \((a : Dot[Integratedx[__], __])\) \[RuleDelayed] ReplacePart[a, Contract[a[\([\(-1\)]\)] trace], \(-1\)];\)\)], "Input"], Cell[CellGroupData[{ Cell["Simplify a bit:", "Text"], Cell[BoxData[ \(\(qexprcc = qexpr /. \((aa : Dot[Integratedx[__], __])\) \[RuleDelayed] MapAt[\((\((\(Collect[# /. \ 2^\((a_ - D)\) \[Rule] \((2^a)\)\ tD, {tD, e, \[ExponentialE]^_, \[Pi]^_, _^\((\(-\(D\/2\)\) \ + _)\), Pair[__]}] //. p : \((Pair[__]*_)\) \[RuleDelayed] Simplify[p]\) /. tD \[Rule] 2^\((\(-D\))\))\) &)\), Expand[ExpandScalarProduct[aa]], \(-1\)];\)\)], "Input"] }, Open ]], Cell[BoxData[ \(qexprcc\)], "Input"], Cell[TextData[{ "Tell ", StyleBox["FeynmanParametrize1", FontFamily->"Courier"], " to do the ", StyleBox["q", FontSlant->"Italic"], " integral:" }], "Text"], Cell[BoxData[ \(\(pexpr = FeynmanParametrize1[qexprcc, q, Method \[Rule] Exp, Flatten \[Rule] True, Integrate \[Rule] True, FeynmanParameterNames \[Rule] {x}];\)\)], "Input"], Cell[CellGroupData[{ Cell["Simplify a bit:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(pexpr // LeafCount\)], "Input"], Cell[BoxData[ \(TraditionalForm\`345553\)], "Output"] }, Open ]], Cell[BoxData[ \(\(tmp = \(pexpr //. {p : \((Pair[a_LorentzIndex, b_] Pair[a_LorentzIndex, c_])\) \[RuleDelayed] Contract[p], p : \((Pair[_LorentzIndex, _]^2)\) \[RuleDelayed] Contract[p]}\) /. E^a_ \[RuleDelayed] E^Simplify[a];\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(tmp // LeafCount\)], "Input"], Cell[BoxData[ \(TraditionalForm\`345186\)], "Output"] }, Open ]], Cell[BoxData[ \(\(pexprcc = tmp /. \((aa : Dot[Integratedx[__], ___, _?\((FreeQ[#, Integratedx] &)\)])\) \[RuleDelayed] MapAt[\((\((\(\(Collect[# /. \ 2^\((a_ - 2 D)\) \[RuleDelayed] \((2^ a)\)\ t2D, {\[ExponentialE]^_, e, t2D, \[Pi]^_, _Pair, _^\((_?\((\(! FreeQ[#, D]\) &)\))\)}] /. p : \((_Pair*_)\) \[RuleDelayed] Simplify[p]\) /. t2D \[Rule] 2^\((\(-2\) D)\)\) /. a : \((\((_?\((\(! FreeQ[#, x]\) &)\))\)^_)\) \[RuleDelayed] Simplify[a])\) &)\), aa, \(-1\)];\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(pexprcc // LeafCount\)], "Input"], Cell[BoxData[ \(TraditionalForm\`1485\)], "Output"] }, Open ]], Cell[BoxData[ \(\(pexps = pexprcc // Simplify;\)\)], "Input"], Cell[BoxData[ \(\(pexpss = pexps /. p : \((a_Pair*b_)\) \[RuleDelayed] a*FullSimplify[b];\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(pexpss // LeafCount\)], "Input"], Cell[BoxData[ \(TraditionalForm\`966\)], "Output"] }, Open ]] }, Open ]], Cell[TextData[{ "Use ", StyleBox["FeynmanReduce", FontFamily->"Courier"], " to get rid of the exponentials and shrink the integration region:" }], "Text"], Cell[BoxData[ \(rexp = FeynmanReduce[ pexpss, {\[Alpha]1, \[Alpha]2, \[Alpha]3, \[Alpha]4}]\)], "Input"], Cell["\<\ Variable substitutions and integration: ***Takes very long***\ \>", "Text"], Cell[BoxData[ \(\(fdi = CheckF[FeynmanIntegrate[ rexp /. {\[Alpha]3 \[Rule] \[Alpha]4, \[Alpha]4 \[Rule] \ \[Alpha]3}, {k}, {\[Alpha]1, \[Alpha]2, \[Alpha]3, \[Alpha]4, x}], "\"];\)\)], "Input"], Cell[BoxData[ \(sdi = \(Series[fdi /. _Dot \[Rule] 0, {Epsilon, 0, 0}] // Normal\) // FullSimplify\)], "Input"] }, FrontEndVersion->"4.1 for Microsoft Windows", ScreenRectangle->{{0, 1024}, {0, 723}}, CellGrouping->Manual, WindowSize->{710, 630}, WindowMargins->{{138, Automatic}, {Automatic, 45}}, PrintingCopies->1, PrintingPageRange->{Automatic, Automatic} ] (******************************************************************* Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. 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