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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[ 2641086, 83252]*) (*NotebookOutlinePosition[ 2718757, 85960]*) (* CellTagsIndexPosition[ 2711167, 85736]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["The FeynCalc Book", "Title", TextAlignment->Center], Cell[TextData[StyleBox["FeynCalc 3.1.14", FontWeight->"Bold"]], "Subtitle", PageBreakBelow->True, TextAlignment->Center], Cell["Tools for Feynman diagram calculations", "Subtitle", TextAlignment->Center], Cell["Rolf Mertig", "Subsubtitle", TextAlignment->Center], Cell["March 12, 1999 ", "Subsubtitle", TextAlignment->Center], 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Cell[CellGroupData[{ Cell["Introduction", "Section", PageBreakAbove->True, CellTags->"Short Overview"], Cell[TextData[{ "FeynCalc 3.1 is a ", StyleBox["Mathematica", FontSlant->"Italic"], " 3.0 package for algebraic calculations in high energy physics. Tools are \ provided for Lorentz index contraction, Dirac algebra manipulation, color \ factor calculation, automatic Feynman rule derivation, general noncommutative \ algebra as well as various look-up tables for Feynman parameter integrals, \ Mellin transforms (e.g. all integrals - except 57,58 and 59, and correcting a \ minor misprint in 14) of Appendix 7 from ", ButtonBox["hep-ph/9810241", ButtonData:>{ URL[ "http://xxx.lanl.gov/abs/hep-ph/9810241"], None}, ButtonStyle->"Hyperlink"], " are tabulated in ", ButtonBox["Integrate2", ButtonData:>"Integrate2", ButtonStyle->"Hyperlink"], " , convolutions and Feynman rules. Furthermore special translation \ facilities are provided to change the FeynCalc syntax to and from FORM \ syntax. Optimized FORTRAN generation can be done with ", ButtonBox["Isolate", ButtonData:>"Isolate", ButtonStyle->"Hyperlink"], " and ", ButtonBox["Write2", ButtonData:>"Write2", ButtonStyle->"Hyperlink"], "." }], "Text"], Cell[TextData[{ "The more important functions for input of objects like ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["p", "\[Mu]"], ",", SuperscriptBox["\[Gamma]", "\[Nu]"], ",", SuperscriptBox["g", RowBox[{"\[Mu]", " ", "\[Nu]"}]], ",", " ", SuperscriptBox["\[CurlyEpsilon]", RowBox[{"\[Mu]", " ", "\[Nu]", " ", "\[Rho]", " ", "\[Sigma]"}]], ",", " ", RowBox[{"etc", "."}]}], TraditionalForm]]], " (abbreviations in parentheses) are: ", ButtonBox["DiracMatrix", ButtonData:>"DiracMatrix", ButtonStyle->"Hyperlink"], " (", ButtonBox["GA", ButtonData:>"GA", ButtonStyle->"Hyperlink"], "), ", ButtonBox["DiracSlash", ButtonData:>"DiracSlash", ButtonStyle->"Hyperlink"], " (", ButtonBox["GS", ButtonData:>"GS", ButtonStyle->"Hyperlink"], "), ", ButtonBox["FourVector", ButtonData:>"FourVector", ButtonStyle->"Hyperlink", ButtonNote->"FourVector"], " (", ButtonBox["FV", ButtonData:>"FV", ButtonStyle->"Hyperlink"], "), ", ButtonBox["LeviCivita", ButtonData:>"LeviCivita", ButtonStyle->"Hyperlink"], " (", ButtonBox["LC", ButtonData:>"LC", ButtonStyle->"Hyperlink"], "), ", ButtonBox["MetricTensor", ButtonData:>"MetricTensor", ButtonStyle->"Hyperlink"], " (", ButtonBox["MT", ButtonData:>"MT", ButtonStyle->"Hyperlink"], "), ", ButtonBox["Spinor", ButtonData:>"Spinor", ButtonStyle->"Hyperlink"], ", ", ButtonBox["SUNDelta", ButtonData:>"SUNDelta", ButtonStyle->"Hyperlink"], ", ", ButtonBox["SUNF", ButtonData:>"SUNF", ButtonStyle->"Hyperlink"], ", ", ButtonBox["ScalarProduct", ButtonData:>"ScalarProduct", ButtonStyle->"Hyperlink"], " (", ButtonBox["SP", ButtonData:>"SP", ButtonStyle->"Hyperlink"], "). 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The functions \ OneLoop and OneLoopSum for calculation of Standard Model (-like) 1-loop \ diagrams still work in FeynCalc 3.1 but they have not been developed much \ further compared to the FeynCalc 1.0 version. While ", ButtonBox["OneLoop", ButtonData:>"OneLoop", ButtonStyle->"Hyperlink"], " and OneLoopSum still work fine for self-energy, triangle and box diagrams \ not too complicated the reader should be aware of the package FormCalc which \ is more effective for the adventurous researcher calculating thousands of \ diagrams involving four external gauge bosons. Notice however that FormCalc \ does not reduce the Passarino-Veltman integrals to scalar integrals. Also \ FormCalc is limited to the 't Hooft Feynman gauge. 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For a description of TARCER see ", ButtonBox["hep-ph/9801383", ButtonData:>{ URL[ "http://xxx.lanl.gov/abs/hep-ph/9801383"], None}, ButtonStyle->"Hyperlink"], " or the published version in Computer Physics Communications 111 (1998) \ 265-273." }], "Text"], Cell["\<\ For more detailed information concerning FeynCalc functions you can \ investigate most of the source code in the FeynCalc directory in the \ HighEnergyPhysics directory. The location of the HighEnergyPhysics directory \ is stored upon installation in the fc.m file and you can get its value from \ the global variable $FeynCalcDirectory. A complete list of FeynCalc objects \ is stored in the variable $FeynCalcStuff. Not all functions are documented in \ this notebook. Some are either very special ones (e.g. for tools for 2-loop \ QCD diagrams originating in twist-2 OPE) or still experimental.\ \>", "Text"], Cell[TextData[{ "The ", Cell[BoxData[ FormBox[ SuperscriptBox["\[Gamma]", "5"], TraditionalForm]]], "scheme used in OneLoop is the naive one, i.e., ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"{", RowBox[{ SuperscriptBox["\[Gamma]", "5"], ",", SuperscriptBox["\[Gamma]", "\[Mu]"]}], ")"}], "=", "0"}], TraditionalForm]]], " in 4 and D dimensions. See also ", ButtonBox["ToLarin", ButtonData:>"ToLarin", ButtonStyle->"Hyperlink"] }], "Text"], Cell["The metric used is the one from Bjorken and Drell (+---).", "Text"], Cell[TextData[{ "For suggestions and bug reports please email the author: ", ButtonBox["rolf@mertig.com", ButtonData:>{ URL[ "mailto:rolf@mertig.com"], None}, ButtonStyle->"Hyperlink"], "." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Start FeynCalc", "Section", CellTags->"Load FeynCalc"], Cell["\<\ This is only for timing purposes of this notebook and can be \ omitted in general.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"absolutestarttime", "=", RowBox[{"AbsoluteTime", "[", "]"}]}]], "Input"], Cell[BoxData[ FormBox["3.130269317`15.5162*^9", TraditionalForm]], "Output"] }, Open ]], Cell["This defines typesetting rules for q1 and q2:", "Text"], Cell[BoxData[ RowBox[{ RowBox[{"q1", "/:", RowBox[{"MakeBoxes", "[", RowBox[{"q1", ",", "TraditionalForm"}], "]"}], ":=", RowBox[{"InterpretationBox", "[", RowBox[{ RowBox[{"SubscriptBox", "[", RowBox[{"q", ",", "1"}], "]"}], ",", "q1"}], "]"}]}], ";"}]], "Input"], Cell[BoxData[ RowBox[{ RowBox[{"q2", "/:", RowBox[{"MakeBoxes", "[", RowBox[{"q2", ",", "TraditionalForm"}], "]"}], ":=", RowBox[{"InterpretationBox", "[", RowBox[{ RowBox[{"SubscriptBox", "[", RowBox[{"q", ",", "2"}], "]"}], ",", "q2"}], "]"}]}], ";"}]], "Input"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"{", RowBox[{"q1", ",", "q2"}], "}"}]], "Input"], Cell[BoxData[ FormBox[ RowBox[{"{", RowBox[{ InterpretationBox[ SubscriptBox["q", "1"], q1], ",", InterpretationBox[ SubscriptBox["q", "2"], q2]}], "}"}], TraditionalForm]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"%", "//", "StandardForm"}]], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{"q1", ",", "q2"}], "}"}]], "Output"] }, Open ]], Cell[TextData[{ "By loading FeynCalc the setting of the default output format type is \ automatically set to TraditionalForm. 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Please send suggestions \ to me ...\ \>", "Commentary"], Cell[TextData[{ "The following subsections are examples from various areas of research or \ education. See also ", ButtonBox["www.feyncalc.com/examples", ButtonData:>{ URL[ "http://www.feyncalc.com/examples"], None}, ButtonStyle->"Hyperlink", ButtonNote->"http://www.feyncalc.com/examples"], "." }], "Text"], Cell[CellGroupData[{ Cell["Standard Model", "Subsection"], Cell[CellGroupData[{ Cell[TextData[{ "Tree level ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"\[Gamma]", " ", "\[Gamma]"}], " ", "\[Rule]", " ", RowBox[{ SuperscriptBox["e", "+"], SuperscriptBox["e", "-"]}]}], TraditionalForm]]] }], "Subsubsection", CellTags->"Tree level"], Cell[TextData[{ "Tree level pair production ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"\[Gamma]", " ", "\[Gamma]"}], " ", "\[Rule]", " ", RowBox[{ SuperscriptBox["e", "+"], SuperscriptBox["e", "-"]}]}], TraditionalForm]]] }], "Text"], Cell["Define typesetting rules for p1, p2, k1, k2", "Text"], Cell["\<\ p1/: MakeBoxes[p1, fmt_] := InterpretationBox[SubscriptBox[p, 1], \ p1]\ \>", "Input"], Cell["\<\ k1/: MakeBoxes[k1, fmt_] := InterpretationBox[SubscriptBox[k, 1], \ k1]\ \>", "Input"], Cell["\<\ p2/: MakeBoxes[p2, fmt_] := InterpretationBox[SubscriptBox[p, 2], \ p2]\ \>", "Input"], Cell["\<\ k2/: MakeBoxes[k2, fmt_] := InterpretationBox[SubscriptBox[k, 2], \ k2]\ \>", "Input"], Cell["Define slashes of external momenta to save future typing.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"{", RowBox[{"p1sl", ",", "p2sl", ",", "k1sl", ",", "k2sl"}], "}"}], "=", RowBox[{"DiracSlash", "/@", RowBox[{"{", RowBox[{"p1", ",", "p2", ",", "k1", ",", "k2"}], "}"}]}]}]], "Input"], Cell[BoxData[ FormBox[ RowBox[{"{", RowBox[{ RowBox[{ FormBox["\<\"\[Gamma]\"\>", "TraditionalForm"], "\[NoBreak]", "\[CenterDot]", "\[NoBreak]", FormBox[ InterpretationBox[ SubscriptBox["p", "1"], p1], "TraditionalForm"]}], ",", RowBox[{ FormBox["\<\"\[Gamma]\"\>", "TraditionalForm"], "\[NoBreak]", "\[CenterDot]", "\[NoBreak]", FormBox[ InterpretationBox[ SubscriptBox["p", "2"], p2], "TraditionalForm"]}], ",", RowBox[{ FormBox["\<\"\[Gamma]\"\>", "TraditionalForm"], "\[NoBreak]", "\[CenterDot]", "\[NoBreak]", FormBox[ InterpretationBox[ SubscriptBox["k", "1"], k1], "TraditionalForm"]}], ",", RowBox[{ FormBox["\<\"\[Gamma]\"\>", "TraditionalForm"], "\[NoBreak]", "\[CenterDot]", "\[NoBreak]", FormBox[ InterpretationBox[ SubscriptBox["k", "2"], k2], "TraditionalForm"]}]}], "}"}], TraditionalForm]], "Output"] }, Open ]], Cell[TextData[{ "Kinematic definitions so that only ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["p", "1"], ".", SubscriptBox["k", "1"]}], TraditionalForm]]], ", ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["p", "1"], ".", SubscriptBox["k", "2"]}], TraditionalForm]]], " appear in final results. I.e., define ", Cell[BoxData[ FormBox[ RowBox[{ SubsuperscriptBox["p", "1", "2"], "=", RowBox[{ SubsuperscriptBox["p", "2", "2"], "=", " ", SuperscriptBox["m", "2"]}]}], TraditionalForm]]], " ", Cell[BoxData[ FormBox[ RowBox[{ SubsuperscriptBox["k", "1", "2"], "=", RowBox[{ SubsuperscriptBox["k", "2", "2"], "=", " ", "0"}]}], TraditionalForm]]], " ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubscriptBox["k", "1"], "\[CenterDot]", SubscriptBox["k", "2"]}], " ", "=", " ", RowBox[{ RowBox[{ SubscriptBox["p", "1"], "\[CenterDot]", SubscriptBox["p", "2"]}], "+", SuperscriptBox["m", "2"]}]}], TraditionalForm]]], " ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubscriptBox["k", "1"], "\[CenterDot]", SubscriptBox["p", "2"]}], " ", "=", " ", RowBox[{ SubscriptBox["k", "2"], "\[CenterDot]", SubscriptBox["p", "1"]}]}], TraditionalForm]]], " ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubscriptBox["k", "2"], "\[CenterDot]", SubscriptBox["p", "2"]}], " ", "=", " ", RowBox[{ SubscriptBox["k", "1"], 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x)*x - 4*ln(x)*ln(1 - x)*ln(1 - x) + 4*ln(x)*ln(1 - x)*ln(1 - x)*x + 8*ln(x)*ln(1 - x)*ln(1 - x)*[1+x]^-1 - 8*ln(x)*ln(1 - x)*ln(1 + x) + 8*ln(x)*ln(1 - x)*ln(1 + x)*x + 16*ln(x)*ln(1 - x)*ln(1 + x)*[1+x]^-1 + 4/3*ln(x)*ln(1 + x)*x^-1 + 4*ln(x)*ln(1 + x)*x + 16/3*ln(x)*ln(1 + x)*x^2 + 8*ln(x)*ln(1 + x)*ln(1 + x)*x + 8*ln(x)*ln(1 + x)*ln(1 + x)*[1+x]^-1 - 6*ln(x)*ln(x) - 8/3*ln(x)*ln(x)*x^2 + 2*ln(x)*ln(x)*ln(1 - x) - 2*ln(x)*ln(x)*ln(1 - x)*x - 4*ln(x)*ln(x)*ln(1 - x)*[1+x]^-1 - 6*ln(x)*ln(x)*ln(1 + x) + 2*ln(x)*ln(x)*ln(1 + x)*x + 8*ln(x)*ln(x)*ln(1 + x)*[1+x]^-1 + ln(x)*ln(x)*ln(x) - ln(x)*ln(x)*ln(x)*x - 2*ln(x)*ln(x)*ln(x)*[1+x]^-1 + 4*ln(x)*Li2(1 - x) - 4*ln(x)*Li2(1 - x)*x - 8*ln(x)*Li2(1 - x)*[1+x]^-1 - 4*ln(x)*Li2( - x) + 4*ln(x)*Li2( - x)*x + 8*ln(x)*Li2( - x)*[1+x]^-1 - 4*Li2(1 - x) - 4*Li2(1 - x)*x + 4/3*Li2( - x)*x^-1 + 4*Li2( - x)*x + 16/3*Li2( - x)*x^2 + 8*Li3(1 - x) - 8*Li3(1 - x)*x - 16*Li3(1 - x)*[1+x]^-1 - 4*Li3( - x) - 4*Li3( - x)*x + 4*S12(1 - x) - 4*S12(1 - x)*x - 8*S12(1 - x)*[1+x]^-1 - 8*S12( - x) + 24*S12( - x)*x + 32*S12( - x)*[1+x]^-1 + 4*S12(x^2) - 4*S12(x^2)*x - 8*S12(x^2)*[1+x]^-1 ) + [(-1)^m]*O1*Cf^2 * ( + 82/3 + 12*x*Z2 + 4*x*Z3 - 82/3*x - 32/3*x^2*Z2 - 12*Z2 - 8*Z3*[1+x]^-1 + 12*Z3 + 16*ln(1 - x) - 16*ln(1 - x)*x + 16*ln(1 - x)*Li2(1 - x) - 16*ln(1 - x)*Li2(1 - x)*x - 32*ln(1 - x)*Li2(1 - x)*[1+x]^-1 - 16*ln(1 + x)*x*Z2 - 16*ln(1 + x)*Z2*[1+x]^-1 + 16*ln(1 + x)*Li2(1 - x) - 16*ln(1 + x)*Li2(1 - x)*x - 32*ln(1 + x)*Li2(1 - x)*[1+x]^-1 - 32*ln(1 + x)*Li2( - x)*x - 32*ln(1 + x)*Li2( - x)*[1+x]^-1 + 74/3*ln(x) + 50/3*ln(x)*x + 8*ln(x)*ln(1 - x) + 8*ln(x)*ln(1 - x)*x + 8*ln(x)*ln(1 - x)*ln(1 - x) - 8*ln(x)*ln(1 - x)*ln(1 - x)*x - 16*ln(x)*ln(1 - x)*ln(1 - x)*[1+x]^-1 + 16*ln(x)*ln(1 - x)*ln(1 + x) - 16*ln(x)*ln(1 - x)*ln(1 + x)*x - 32*ln(x)*ln(1 - x)*ln(1 + x)*[1+x]^-1 - 8/3*ln(x)*ln(1 + x)*x^-1 - 8*ln(x)*ln(1 + x)*x - 32/3*ln(x)*ln(1 + x)*x^2 - 16*ln(x)*ln(1 + x)*ln(1 + x)*x - 16*ln(x)*ln(1 + x)*ln(1 + x)*[1+x]^-1 + 12*ln(x)*ln(x) + 16/3*ln(x)*ln(x)*x^2 - 4*ln(x)*ln(x)*ln(1 - x) + 4*ln(x)*ln(x)*ln(1 - x)*x + 8*ln(x)*ln(x)*ln(1 - x)*[1+x]^-1 + 12*ln(x)*ln(x)*ln(1 + x) - 4*ln(x)*ln(x)*ln(1 + x)*x - 16*ln(x)*ln(x)*ln(1 + x)*[1+x]^-1 - 2*ln(x)*ln(x)*ln(x) + 2*ln(x)*ln(x)*ln(x)*x + 4*ln(x)*ln(x)*ln(x)*[1+x]^-1 - 8*ln(x)*Li2(1 - x) + 8*ln(x)*Li2(1 - x)*x + 16*ln(x)*Li2(1 - x)*[1+x]^-1 + 8*ln(x)*Li2( - x) - 8*ln(x)*Li2( - x)*x - 16*ln(x)*Li2( - x)*[1+x]^-1 + 8*Li2(1 - x) + 8*Li2(1 - x)*x - 8/3*Li2( - x)*x^-1 - 8*Li2( - x)*x - 32/3*Li2( - x)*x^2 - 16*Li3(1 - x) + 16*Li3(1 - x)*x + 32*Li3(1 - x)*[1+x]^-1 + 8*Li3( - x) + 8*Li3( - x)*x - 8*S12(1 - x) + 8*S12(1 - x)*x + 16*S12(1 - x)*[1+x]^-1 + 16*S12( - x) - 48*S12( - x)*x - 64*S12( - x)*[1+x]^-1 - 8*S12(x^2) + 8*S12(x^2)*x + 16*S12(x^2)*[1+x]^-1 ) + [(-1)^m]*O2*Ca*Cf * ( + 8/3 - 8/3*x - 16/3*x^2*Z2 - 8/3*ln(x) + 16/3*ln(x)*x + 8/3*ln(x)*ln(1 + x)*x^-1 - 8*ln(x)*ln(1 + x)*x - 16/3*ln(x)*ln(1 + x)*x^2 + 8/3*ln(x)*ln(x)*x^2 + 8/3*Li2( - x)*x^-1 - 8*Li2( - x)*x - 16/3*Li2( - x)*x^2 ) + [(-1)^m]*O2*Cf^2 * ( - 16/3 + 16/3*x + 32/3*x^2*Z2 + 16/3*ln(x) - 32/3*ln(x)*x - 16/3*ln(x)*ln(1 + x)*x^-1 + 16*ln(x)*ln(1 + x)*x + 32/3*ln(x)*ln(1 + x)*x^2 - 16/3*ln(x)*ln(x)*x^2 - 16/3*Li2( - x)*x^-1 + 16*Li2( - x)*x + 32/3*Li2( - x)*x^2 ) + e^-2*delta*O1*Ca*Cf * ( - 22 ) + e^-2*delta*O1*Cf*Tf * ( + 8 ) + e^-2*delta*O1*Cf^2 * ( - 18 + 32*Z2 ) + e^-2*O1*Ca*Cf * ( + 44/3 + 44/3*x - 88/3*[(1-x)+]^-1 ) + e^-2*O1*Cf*Tf * ( - 16/3 - 16/3*x + 32/3*[(1-x)+]^-1 ) + e^-2*O1*Cf^2 * ( + 40 + 8*x - 48*[(1-x)+]^-1 + 32*ln(1 - x) + 32*ln(1 - x)*x - 64*ln(1 - x)*[(1-x)+]^-1 - 24*ln(x) - 24*ln(x)*x + 32*ln(x)*[1-x]^-1 ) + e^-1*delta*O1*Ca*Cf * ( + 325/6 - 44/3*Z2 - 12*Z3 ) + e^-1*delta*O1*Cf*Tf * ( - 58/3 + 16/3*Z2 ) + e^-1*delta*O1*Cf^2 * ( + 87/2 - 36*Z2 - 8*Z3 ) + e^-1*O1*Ca*Cf * ( - 158/9 + 4*x*Z2 + 22/9*x - 8*Z2*[(1-x)+]^-1 + 4*Z2 + 238/9*[(1-x)+]^-1 + 44/3*ln(1 - x) + 44/3*ln(1 - x)*x - 88/3*ln(1 - x)*[(1-x)+]^-1 + 34/3*ln(x) + 34/3*ln(x)*x - 44/3*ln(x)*[1-x]^-1 - 2*ln(x)*ln(x) - 2*ln(x)*ln(x)*x + 4*ln(x)*ln(x)*[1-x]^-1 ) + e^-1*O1*Cf*Tf * ( + 88/9 - 56/9*x - 56/9*[(1-x)+]^-1 - 16/3*ln(1 - x) - 16/3*ln(1 - x)*x + 32/3*ln(1 - x)*[(1-x)+]^-1 - 8/3*ln(x) - 8/3*ln(x)*x + 16/3*ln(x)*[1-x]^-1 ) + e^-1*O1*Cf^2 * ( - 40 - 4*x + 56*[(1-x)+]^-1 - 4*ln(1 - x) + 44*ln(1 - x)*x - 24*ln(1 - x)*[(1-x)+]^-1 + 24*ln(1 - x)*ln(1 - x) + 24*ln(1 - x)*ln(1 - x)*x - 48*ln(1 - x)*ln(1 - x)*[(1-x)+]^-1 + 36*ln(x) - 20*ln(x)*x - 36*ln(x)*[1-x]^-1 - 16*ln(x)*ln(1 - x)*[1-x]^-1 - 14*ln(x)*ln(x) - 14*ln(x)*ln(x)*x + 16*ln(x)*ln(x)*[1-x]^-1 - 8*Li2(1 - x) - 8*Li2(1 - x)*x ) + e^-1*O2*Ca*Cf * ( + 20/3 - 88/3*x ) + e^-1*O2*Cf*Tf * ( - 16/3 + 32/3*x ) + e^-1*O2*Cf^2 * ( - 16 - 8*x - 32*ln(1 - x)*x + 16*ln(x)*x ) + delta*O1*Ca*Cf * ( - 7081/72 + 301/18*Z2 + 49/5*Z2^2 + 28*Z3 ) + delta*O1*Cf*Tf * ( + 569/18 - 46/9*Z2 - 8*Z3 ) + delta*O1*Cf^2 * ( - 541/8 + 97/2*Z2 - 74/5*Z2^2 + 54*Z3 ) + O1*Ca*Cf * ( + 941/27 + 17/3*x*Z2 - 8*x*Z3 - 580/27*x - 16/3*x^2*Z2 - 24*x^2*Z3 - 28/3*Z2*[(1-x)+]^-1 - 1/3*Z2 + 12*Z3*[1-x]^-1 + 22*Z3*[(1-x)+]^-1 - 20*Z3 - 670/27*[(1-x)+]^-1 - 122/9*ln(1 - x) + 6*ln(1 - x)*x*Z2 - 32/9*ln(1 - x)*x + 8*ln(1 - x)*x^2*Z2 - 10*ln(1 - x)*Z2*[(1-x)+]^-1 + 2*ln(1 - x)*Z2 + 238/9*ln(1 - x)*[(1-x)+]^-1 + 22/3*ln(1 - x)*ln(1 - x) + 22/3*ln(1 - x)*ln(1 - x)*x - 44/3*ln(1 - x)*ln(1 - x)*[(1-x)+]^-1 + 2*ln(1 - x)*Li2(1 - x) - 2*ln(1 - x)*Li2(1 - x)*x + 8*ln(1 - x)*Li2(1 - x)*x^2 - 2*ln(1 - x)*Li2(1 - x)*[1-x]^-1 + 14/9*ln(x) - 2*ln(x)*x*Z2 + 176/9*ln(x)*x - 8*ln(x)*x^2*Z2 + 10*ln(x)*Z2*[1-x]^-1 - 6*ln(x)*Z2 + 101/9*ln(x)*[1-x]^-1 + 28/3*ln(x)*ln(1 - x) + 10/3*ln(x)*ln(1 - x)*x - 38/3*ln(x)*ln(1 - x)*[1-x]^-1 - 12*ln(x)*ln(1 + x) - 4/3*ln(x)*ln(1 + x)*x^-1 - 16*ln(x)*ln(1 + x)*x - 16/3*ln(x)*ln(1 + x)*x^2 + 47/6*ln(x)*ln(x) + 47/6*ln(x)*ln(x)*x + 8/3*ln(x)*ln(x)*x^2 - 11/3*ln(x)*ln(x)*[1-x]^-1 + ln(x)*ln(x)*ln(1 - x) - ln(x)*ln(x)*ln(1 - x)*x + 4*ln(x)*ln(x)*ln(1 - x)*x^2 - ln(x)*ln(x)*ln(1 - x)*[1-x]^-1 - ln(x)*ln(x)*ln(x) - ln(x)*ln(x)*ln(x)*x + 2*ln(x)*ln(x)*ln(x)*[1-x]^-1 + 8*ln(x)*Li2(1 - x) + 16*ln(x)*Li2(1 - x)*x^2 - 12*ln(x)*Li2(1 - x)*[1-x]^-1 + 8*ln(x)*Li2( - x) - 8*ln(x)*Li2( - x)*[1-x]^-1 - 12*Li2(1 - x)*x + 4*Li2(1 - x)*[1-x]^-1 - 12*Li2( - x) - 4/3*Li2( - x)*x^-1 - 16*Li2( - x)*x - 16/3*Li2( - x)*x^2 - 6*Li3(1 - x) + 6*Li3(1 - x)*x - 24*Li3(1 - x)*x^2 + 6*Li3(1 - x)*[1-x]^-1 - 16*Li3( - x) + 16*Li3( - x)*[1-x]^-1 + 14*S12(1 - x) - 6*S12(1 - x)*x + 24*S12(1 - x)*x^2 - 14*S12(1 - x)*[1-x]^-1 ) + O1*Cf*Tf * ( - 88/27 - 4/3*x*Z2 + 20/27*x + 8/3*Z2*[(1-x)+]^-1 - 4/3*Z2 + 128/27*[(1-x)+]^-1 + 88/9*ln(1 - x) - 56/9*ln(1 - x)*x - 56/9*ln(1 - x)*[(1-x)+]^-1 - 8/3*ln(1 - x)*ln(1 - x) - 8/3*ln(1 - x)*ln(1 - x)*x + 16/3*ln(1 - x)*ln(1 - x)*[(1-x)+]^-1 + 44/9*ln(x) - 28/9*ln(x)*x - 28/9*ln(x)*[1-x]^-1 - 8/3*ln(x)*ln(1 - x) - 8/3*ln(x)*ln(1 - x)*x + 16/3*ln(x)*ln(1 - x)*[1-x]^-1 - 2/3*ln(x)*ln(x) - 2/3*ln(x)*ln(x)*x + 4/3*ln(x)*ln(x)*[1-x]^-1 ) + O1*Cf^2 * ( + 188/3 - 4*x*Z2 - 24*x*Z3 + 4/3*x + 32/3*x^2*Z2 + 48*x^2*Z3 - 4*Z2*[(1-x)+]^-1 + 20*Z2 - 24*Z3*[1-x]^-1 + 24*Z3 - 56*[(1-x)+]^-1 - 38*ln(1 - x) + 8*ln(1 - x)*x*Z2 + 38*ln(1 - x)*x - 16*ln(1 - x)*x^2*Z2 - 8*ln(1 - x)*Z2*[(1-x)+]^-1 + 8*ln(1 - x)*Z2 + 28*ln(1 - x)*[(1-x)+]^-1 - 13*ln(1 - x)*ln(1 - x) + 31*ln(1 - x)*ln(1 - x)*x - 6*ln(1 - x)*ln(1 - x)*[(1-x)+]^-1 + 28/3*ln(1 - x)*ln(1 - x)*ln(1 - x) + 28/3*ln(1 - x)*ln(1 - x)*ln(1 - x)*x - 56/3*ln(1 - x)*ln(1 - x)*ln(1 - x)*[(1-x)+]^-1 - 12*ln(1 - x)*Li2(1 - x) + 4*ln(1 - x)*Li2(1 - x)*x - 16*ln(1 - x)*Li2(1 - x)*x^2 - 68/3*ln(x) - 14*ln(x)*x*Z2 - 68/3*ln(x)*x + 16*ln(x)*x^2*Z2 + 2*ln(x)*Z2 + 44*ln(x)*[1-x]^-1 - 6*ln(x)*ln(1 - x) + 34*ln(x)*ln(1 - x)*x - 20*ln(x)*ln(1 - x)*[1-x]^-1 + 6*ln(x)*ln(1 - x)*ln(1 - x) + 6*ln(x)*ln(1 - x)*ln(1 - x)*x - 20*ln(x)*ln(1 - x)*ln(1 - x)*[1-x]^-1 + 24*ln(x)*ln(1 + x) + 8/3*ln(x)*ln(1 + x)*x^-1 + 32*ln(x)*ln(1 + x)*x + 32/3*ln(x)*ln(1 + x)*x^2 + 11*ln(x)*ln(x) - 23*ln(x)*ln(x)*x - 16/3*ln(x)*ln(x)*x^2 - 15*ln(x)*ln(x)*[1-x]^-1 - 6*ln(x)*ln(x)*ln(1 - x) + 2*ln(x)*ln(x)*ln(1 - x)*x - 8*ln(x)*ln(x)*ln(1 - x)*x^2 - 4*ln(x)*ln(x)*ln(1 - x)*[1-x]^-1 - 5*ln(x)*ln(x)*ln(x) - 5*ln(x)*ln(x)*ln(x)*x + 16/3*ln(x)*ln(x)*ln(x)*[1-x]^-1 - 8*ln(x)*Li2(1 - x) + 24*ln(x)*Li2(1 - x)*x - 32*ln(x)*Li2(1 - x)*x^2 - 16*ln(x)*Li2(1 - x)*[1-x]^-1 - 16*ln(x)*Li2( - x) + 16*ln(x)*Li2( - x)*[1-x]^-1 - 24*Li2(1 - x) + 48*Li2(1 - x)*x - 4*Li2(1 - x)*[1-x]^-1 + 24*Li2( - x) + 8/3*Li2( - x)*x^-1 + 32*Li2( - x)*x + 32/3*Li2( - x)*x^2 + 20*Li3(1 - x) - 28*Li3(1 - x)*x + 48*Li3(1 - x)*x^2 + 32*Li3( - x) - 32*Li3( - x)*[1-x]^-1 - 8*S12(1 - x) + 56*S12(1 - x)*x - 48*S12(1 - x)*x^2 - 32*S12(1 - x)*[1-x]^-1 ) + O2*Ca*Cf * ( - 26/9 - 8*x*Z2 - 24*x*Z3 + 232/9*x + 16/3*x^2*Z2 + 24*x^2*Z3 + 4*Z2 - 16/3*ln(1 - x) + 8*ln(1 - x)*x*Z2 - 64/3*ln(1 - x)*x - 8*ln(1 - x)*x^2*Z2 + 8*ln(1 - x)*Li2(1 - x)*x - 8*ln(1 - x)*Li2(1 - x)*x^2 + 6*ln(x) - 8*ln(x)*x*Z2 - 28*ln(x)*x + 8*ln(x)*x^2*Z2 - 4*ln(x)*ln(1 - x) + 8*ln(x)*ln(1 - x)*x - 8/3*ln(x)*ln(1 + x)*x^-1 + 8*ln(x)*ln(1 + x)*x + 16/3*ln(x)*ln(1 + x)*x^2 - 8/3*ln(x)*ln(x)*x^2 + 4*ln(x)*ln(x)*ln(1 - x)*x - 4*ln(x)*ln(x)*ln(1 - x)*x^2 + 16*ln(x)*Li2(1 - x)*x - 16*ln(x)*Li2(1 - x)*x^2 - 8*Li2(1 - x) + 16*Li2(1 - x)*x - 8/3*Li2( - x)*x^-1 + 8*Li2( - x)*x + 16/3*Li2( - x)*x^2 - 24*Li3(1 - x)*x + 24*Li3(1 - x)*x^2 + 24*S12(1 - x)*x - 24*S12(1 - x)*x^2 ) + O2*Cf*Tf * ( - 8/9 - 32/9*x - 16/3*ln(1 - x) + 32/3*ln(1 - x)*x - 8/3*ln(x) + 16/3*ln(x)*x ) + O2*Cf^2 * ( - 32/3 + 16*x*Z2 + 48*x*Z3 - 16/3*x - 32/3*x^2*Z2 - 48*x^2*Z3 - 8*Z2 - 16*ln(1 - x)*x*Z2 - 44*ln(1 - x)*x + 16*ln(1 - x)*x^2*Z2 - 24*ln(1 - x)*ln(1 - x)*x - 16*ln(1 - x)*Li2(1 - x)*x + 16*ln(1 - x)*Li2(1 - x)*x^2 - 40/3*ln(x) + 16*ln(x)*x*Z2 + 68/3*ln(x)*x - 16*ln(x)*x^2*Z2 + 8*ln(x)*ln(1 - x) - 32*ln(x)*ln(1 - x)*x + 16/3*ln(x)*ln(1 + x)*x^-1 - 16*ln(x)*ln(1 + x)*x - 32/3*ln(x)*ln(1 + x)*x^2 + 12*ln(x)*ln(x)*x + 16/3*ln(x)*ln(x)*x^2 - 8*ln(x)*ln(x)*ln(1 - x)*x + 8*ln(x)*ln(x)*ln(1 - x)*x^2 - 32*ln(x)*Li2(1 - x)*x + 32*ln(x)*Li2(1 - x)*x^2 + 16*Li2(1 - x) - 48*Li2(1 - x)*x + 16/3*Li2( - x)*x^-1 - 16*Li2( - x)*x - 32/3*Li2( - x)*x^2 + 48*Li3(1 - x)*x - 48*Li3(1 - x)*x^2 - 48*S12(1 - x)*x + 48*S12(1 - x)*x^2 ) \"]//ReleaseHold;\ \>", "Input"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Short", "[", "aqq", "]"}]], "Input"], Cell[BoxData[ FormBox[ TagBox[ RowBox[{ FractionBox[ RowBox[{"O1", " ", RowBox[{"(", RowBox[{ RowBox[{"32", " ", RowBox[{"\[Zeta]", "(", "2", ")"}]}], "-", "18"}], ")"}], " ", RowBox[{"\[Delta]", "(", FormBox[ RowBox[{"1", "-", "x"}], "TraditionalForm"], ")"}], " ", SubsuperscriptBox["C", "F", "2"]}], SuperscriptBox[ TagBox["\[CurlyEpsilon]", TraditionalForm], "2"]], "+", RowBox[{"\[LeftSkeleton]", "29", "\[RightSkeleton]"}], "+", RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"-", "1"}], ")"}], "m"], " ", SubscriptBox["C", "A"], " ", "O1", " ", RowBox[{"(", RowBox[{"\[LeftSkeleton]", "1", "\[RightSkeleton]"}], ")"}], " ", SubscriptBox["C", "F"]}]}], Short], TraditionalForm]], "Output"] }, Open ]], Cell["\<\ Comment: By default all bracketed FORM expressions are translated \ to a Hold[] expression.\ \>", "Commentary"], Cell[CellGroupData[{ Cell["FORM2FeynCalc[\"-[(-1)^m]\"]", "Input"], Cell[BoxData[ FormBox[ RowBox[{"-", RowBox[{"Hold", "(", SuperscriptBox[ RowBox[{"(", RowBox[{"-", "1"}], ")"}], "m"], ")"}]}], TraditionalForm]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"%", "//", "ReleaseHold"}]], "Input"], Cell[BoxData[ FormBox[ RowBox[{"-", SuperscriptBox[ RowBox[{"(", RowBox[{"-", "1"}], ")"}], "m"]}], TraditionalForm]], "Output"] }, Open ]], Cell["Calculate moments", "Text"], Cell["\<\ The first moment should be zero, which is verified by Integrate2 in \ a reasonable amount of time. The variable O2 can be set to zero since it is \ the coefficient of unphysical matrix elements.\ \>", "Text"], Cell[TextData[{ "This calculates for m=1 the integral ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubsuperscriptBox["\[Integral]", "0", RowBox[{"1", " "}]], " ", RowBox[{ SubsuperscriptBox[ StyleBox[ OverscriptBox[ StyleBox["A", FontSlant->"Plain"], "^"], FontSlant->"Plain"], "qq", RowBox[{"NS", ",", "PHYS"}]], " ", RowBox[{"\[DifferentialD]", "x"}]}]}], " "}], TraditionalForm]]] }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Timing", "[", RowBox[{"Integrate2", "[", RowBox[{ RowBox[{ RowBox[{"aqq", "/.", RowBox[{"m", "\[Rule]", "1"}]}], "/.", RowBox[{"O2", "\[Rule]", "0"}]}], ",", RowBox[{"{", RowBox[{"x", ",", "0", ",", "1"}], "}"}]}], "]"}], "]"}]], "Input"], Cell[BoxData[ FormBox[ RowBox[{"{", RowBox[{ RowBox[{"18.6500000000000021`", " ", "Second"}], ",", "0"}], "}"}], TraditionalForm]], "Output"] }, Open ]], Cell["\<\ If you know that no partial fraction 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