diff --git a/BasicExamples.nb b/BasicExamples.nb new file mode 100644 index 0000000..b392070 --- /dev/null +++ b/BasicExamples.nb @@ -0,0 +1,13610 @@ +(* Content-type: application/vnd.wolfram.mathematica *) + +(*** Wolfram Notebook File ***) +(* http://www.wolfram.com/nb *) + +(* CreatedBy='Mathematica 8.0' *) + +(*CacheID: 234*) +(* Internal cache information: +NotebookFileLineBreakTest +NotebookFileLineBreakTest +NotebookDataPosition[ 157, 7] +NotebookDataLength[ 511037, 13602] +NotebookOptionsPosition[ 473112, 13026] +NotebookOutlinePosition[ 473574, 13044] +CellTagsIndexPosition[ 473531, 13041] +WindowFrame->Normal*) + +(* Beginning of Notebook Content *) +Notebook[{ +Cell[BoxData[ + RowBox[{ + RowBox[{"SetDirectory", "[", + RowBox[{"NotebookDirectory", "[", "]"}], "]"}], ";"}]], "Input", + CellChangeTimes->{{3.781786761328034*^9, 3.781786769223647*^9}, { + 3.7971915793849573`*^9, + 3.7971915838999634`*^9}},ExpressionUUID->"dc453752-83d2-4d0d-9f4c-\ +1a179aba4247"], + 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TraditionalForm]],ExpressionUUID-> + "a7894135-ce55-4c56-baaf-ded5ee5570c0"], + " denotes the modified Bessel function of the third kind." +}], "Text", + CellChangeTimes->{{3.781787514944394*^9, 3.7817875307839003`*^9}, { + 3.7971916628260746`*^9, 3.7971919464224763`*^9}, {3.7979269490491457`*^9, + 3.797926950000276*^9}, {3.797927005647272*^9, + 3.7979270255278463`*^9}},ExpressionUUID->"d93abd0d-e2db-410d-8942-\ +00f2873a61dd"], + +Cell[TextData[{ + "Analogously the double-", + StyleBox["K", + FontSlant->"Italic"], + " integral is\n", + Cell[BoxData[ + FormBox[ + RowBox[{ + RowBox[{ + SubscriptBox["I", + RowBox[{"\[Alpha]", + RowBox[{"{", + RowBox[{ + SubscriptBox["\[Beta]", "1"], + SubscriptBox["\[Beta]", "2"]}], "}"}]}]], "(", "p", ")"}], "=", + RowBox[{ + SuperscriptBox["p", + RowBox[{ + SubscriptBox["\[Beta]", "1"], "+", + SubscriptBox["\[Beta]", "2"]}]], + RowBox[{ + SuperscriptBox[ + SubscriptBox["\[Integral]", "0"], "\[Infinity]"], + RowBox[{"dx", " ", + SuperscriptBox["x", 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The standard momenta magnitudes ", + Cell[BoxData[ + FormBox[ + RowBox[{ + SubscriptBox["p", "1"], ",", " ", + SubscriptBox["p", "2"], ",", " ", + SubscriptBox["p", "3"]}], TraditionalForm]],ExpressionUUID-> + "a1c884d5-eb96-4aff-b6b2-01976499230e"], + " are then assumed as arguments. Parameters can also be inputted as \ +subscripts:" +}], "Text", + CellChangeTimes->{{3.78178760129624*^9, 3.7817876411768093`*^9}, { + 3.794887453569615*^9, 3.794887474569602*^9}, {3.797192060129637*^9, + 3.7971920910846806`*^9}, {3.797927076304063*^9, + 3.7979270957364197`*^9}},ExpressionUUID->"247985cb-d475-4788-b74c-\ +5191783784d8"], + +Cell[CellGroupData[{ + +Cell[BoxData[ + RowBox[{ + RowBox[{ + RowBox[{"i", "[", + RowBox[{"\[Alpha]", ",", + RowBox[{"{", + RowBox[{"\[Beta]1", ",", "\[Beta]2", ",", "\[Beta]3"}], "}"}]}], "]"}], + "-", + RowBox[{ + SubscriptBox["i", + RowBox[{"\[Alpha]", ",", + RowBox[{"{", + RowBox[{"\[Beta]1", ",", "\[Beta]2", ",", "\[Beta]3"}], "}"}]}]], "[", + RowBox[{ + SubscriptBox["p", "1"], ",", + SubscriptBox["p", "2"], ",", + SubscriptBox["p", "3"]}], "]"}]}], "//", "KExpand"}]], "Input", + CellChangeTimes->{{3.7979271023678493`*^9, 3.7979271249277687`*^9}, { + 3.797927579757101*^9, 3.797927596496648*^9}, {3.797935862292218*^9, + 3.79793586293546*^9}},ExpressionUUID->"67200ffc-9223-4cef-8986-\ +5d12adfe9904"], + +Cell[BoxData["0"], "Output", + CellChangeTimes->{3.797927125644553*^9, 3.7979275967663507`*^9, + 3.79793586330826*^9, 3.798248275814546*^9, 3.7986857520311418`*^9, + 3.798685802167979*^9, + 3.799037617375333*^9},ExpressionUUID->"1f84b12d-e519-48ea-8afb-\ +055e3f041eff"] +}, Open ]], + +Cell[TextData[{ + "To evaluate multiple-", + StyleBox["K", + FontSlant->"Italic"], + " integrals explicitly, use ", + StyleBox["KEvaluate", + FontSlant->"Italic"], + ". ", + StyleBox["KEvaluate", + FontSlant->"Italic"], + " can be applied to any expression and replaces all double- and triple-", + StyleBox["K", + FontSlant->"Italic"], + " integrals with explicit expressions, if these are known. The package \ +implements the reduction scheme introduced in [", + ButtonBox["1511.02357", + BaseStyle->"Hyperlink", + ButtonData->{ + URL["https://arxiv.org/pdf/1511.02357.pdf"], None}, + ButtonNote->"https://arxiv.org/pdf/1511.02357.pdf"], + "]. It provides evaluation of the multiple-", + StyleBox["K", + FontSlant->"Italic"], + " integrals in the following cases:" +}], "Text", + CellChangeTimes->{{3.797927151535733*^9, 3.797927172088361*^9}, { + 3.79792721870348*^9, 3.797927272792275*^9}, {3.7979273181041403`*^9, + 3.797927386737391*^9}, {3.798248305988546*^9, 3.798248329732109*^9}, { + 3.798248368291429*^9, + 3.798248383812748*^9}},ExpressionUUID->"c50a115c-ec58-4fca-9718-\ +4fa58b9f53ef"], + +Cell[CellGroupData[{ + +Cell[TextData[{ + "All double-", + StyleBox["K", + FontSlant->"Italic"], + " integrals have analytic expressions." +}], "Item", + CellChangeTimes->{{3.797927395703156*^9, 3.797927409319684*^9}, { + 3.7979276294877443`*^9, + 3.797927639999776*^9}},ExpressionUUID->"9a21bc91-1ec3-4b84-b6fa-\ +219db6be256f"], + +Cell[TextData[{ + "Triple-", + StyleBox["K", + FontSlant->"Italic"], + " integrals with half-integral \[Beta]-indices are expressible in terms of \ +elementary functions and Gamma function." +}], "Item", + CellChangeTimes->{{3.797927395703156*^9, + 3.7979274643038673`*^9}},ExpressionUUID->"274637e8-5dd3-41ff-bfc4-\ +9240e48c6577"], + +Cell[TextData[{ + "Triple-", + StyleBox["K", + FontSlant->"Italic"], + " integrals with two half-integral \[Beta]-indices are expressible in terms \ +of Hypergeometric2F1." +}], "Item", + CellChangeTimes->{{3.797927395703156*^9, + 3.797927490704995*^9}},ExpressionUUID->"adbb3bff-24da-42bc-b179-\ +28c7107848f9"], + +Cell[TextData[{ + "Certain class of triple-", + StyleBox["K", + FontSlant->"Italic"], + " integrals with integral \[Alpha]- and \[Beta]-indices defined as follows. \ +Let ", + Cell[BoxData[ + FormBox[ + RowBox[{ + SubscriptBox["n", + RowBox[{"(", + RowBox[{ + SubscriptBox["\[Sigma]", "1"], + SubscriptBox["\[Sigma]", "2"], + SubscriptBox["\[Sigma]", "3"]}], ")"}]], "=", + RowBox[{ + RowBox[{"-", + FractionBox["1", "2"]}], + RowBox[{"(", + RowBox[{ + SubscriptBox["\[Sigma]", "1"], "|", + SubscriptBox["\[Beta]", "1"], "|", + RowBox[{"+", + SubscriptBox["\[Sigma]", "2"]}], "|", + SubscriptBox["\[Beta]", "2"], "|", + RowBox[{"+", + SubscriptBox["\[Sigma]", "3"]}], "|", + SubscriptBox["\[Beta]", "3"], "|", + RowBox[{ + RowBox[{"+", "\[Alpha]"}], "+", "1"}]}], ")"}]}]}], TraditionalForm]], + ExpressionUUID->"938d1976-ebcf-47cc-8069-b70d6cc9e5b0"], + ", where ", + Cell[BoxData[ + FormBox[ + RowBox[{ + SubscriptBox["\[Sigma]", "j"], "=", + RowBox[{"\[PlusMinus]", "1"}]}], TraditionalForm]],ExpressionUUID-> + "98fdc8d2-4ca9-456d-991a-d951ec8df8ec"], + ". If ", + Cell[BoxData[ + FormBox[ + RowBox[{ + SubscriptBox["n", + RowBox[{"(", + RowBox[{"-", "++"}], ")"}]], ",", + SubscriptBox["n", + RowBox[{"(", + RowBox[{"+", + RowBox[{"-", "+"}]}], ")"}]], ",", + RowBox[{ + SubscriptBox["n", + RowBox[{"(", + RowBox[{"++", "-"}], ")"}]], "<", "0"}]}], TraditionalForm]], + ExpressionUUID->"4c3b48fc-c568-4aec-8316-4cca68af261e"], + ", then the (regulated) triple-", + StyleBox["K", + FontSlant->"Italic"], + " integral is expressible in terms of dilogarithm ", + Cell[BoxData[ + FormBox[ + SubscriptBox["Li", "2"], TraditionalForm]],ExpressionUUID-> + "3892693f-0167-4763-a900-da824dd0081a"], + " and elementary functions." +}], "Item", + CellChangeTimes->{{3.797927395703156*^9, 3.797927526528173*^9}, { + 3.797927657031619*^9, 3.797927657416875*^9}, {3.797927818296541*^9, + 3.797927823088812*^9}, {3.797928735585561*^9, 3.797928854105352*^9}, { + 3.7979290032182827`*^9, 3.797929005641367*^9}, {3.797933080238048*^9, + 3.797933224493533*^9}, 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not have a decent representation, so nothing happens:" +}], "Text", + CellChangeTimes->{{3.797193344162457*^9, 3.797193362874483*^9}, { + 3.79719340347054*^9, 3.797193406686545*^9}, {3.7979361137504673`*^9, + 3.797936114463464*^9}},ExpressionUUID->"8651df40-a0de-4128-9bc5-\ +5a88c10ab305"], + +Cell[CellGroupData[{ + +Cell[BoxData[{ + RowBox[{"IsSolvable", "[", + RowBox[{"i", "[", + RowBox[{"1", ",", + RowBox[{"{", + RowBox[{"1", ",", "1", ",", "1"}], "}"}]}], "]"}], + "]"}], "\[IndentingNewLine]", + RowBox[{"KEvaluate", "[", + RowBox[{"i", "[", + RowBox[{"1", ",", + RowBox[{"{", + RowBox[{"1", ",", "1", ",", "1"}], "}"}]}], "]"}], "]"}]}], "Input", + CellChangeTimes->{{3.797193368915492*^9, 3.7971933727354975`*^9}, { + 3.7979360756913643`*^9, 3.797936081585958*^9}, + 3.798685790337308*^9},ExpressionUUID->"e8ff13db-0330-4019-bbc6-\ +96af4b63dfb6"], + +Cell[BoxData["False"], "Output", + CellChangeTimes->{{3.7971933749555*^9, 3.797193387369518*^9}, + 3.797442320475046*^9, 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If, for a given set of parameters the integral is divergent, a regulator, \ +\[Epsilon], can be used, and the result Laurent-expanded in \[Epsilon]. \ +\[Epsilon] is a special symbol used by the package:" +}], "Text", + CellChangeTimes->{{3.797192044757615*^9, 3.7971921870958157`*^9}, { + 3.7971929451218834`*^9, 3.7971929468468857`*^9}, {3.7971929831779375`*^9, + 3.797192986592942*^9}, {3.7979362121835127`*^9, 3.797936212613793*^9}, + 3.798248459204307*^9},ExpressionUUID->"b8622019-d078-4df3-bb3a-\ +57c7e730de1a"], + +Cell[CellGroupData[{ + +Cell[BoxData["\[Epsilon]"], "Input", + CellChangeTimes->{{3.797192188040817*^9, + 3.7971921883858175`*^9}},ExpressionUUID->"4c6d8e0d-a74e-47e1-806f-\ +8f7b3a9f8e2c"], + +Cell[BoxData["\[Epsilon]"], "Output", + CellChangeTimes->{3.7971921888108177`*^9, 3.7971929885279446`*^9, + 3.797442336518069*^9, 3.797936210289178*^9, 3.798248473772663*^9, + 3.798685754217265*^9, 3.798685804033084*^9, + 3.799037620529797*^9},ExpressionUUID->"8d2d2c10-8f56-4bf5-b36e-\ +41b0777ad0f1"] +}, Open ]], + +Cell["\<\ +If the regulator \[Epsilon] is used in the expression, the integral evaluates \ +to a power series:\ +\>", "Text", + CellChangeTimes->{ + 3.797192565613349*^9, {3.797936216479632*^9, + 3.797936217007654*^9}},ExpressionUUID->"9c8d247a-1517-49fd-bb7f-\ +a0fe574daf5c"], + +Cell[CellGroupData[{ + +Cell[BoxData[ + RowBox[{"KEvaluate", "[", + RowBox[{ + RowBox[{"i", "[", + RowBox[{ + RowBox[{"1", "+", "\[Epsilon]"}], ",", + RowBox[{"{", + RowBox[{"1", ",", "1"}], "}"}]}], "]"}], "[", "p", "]"}], + "]"}]], "Input",ExpressionUUID->"014eed7f-76a5-4f39-b968-ee56fe5d09d8"], + +Cell[BoxData[ + InterpretationBox[ + RowBox[{ + FractionBox["1", "\[Epsilon]"], "+", + RowBox[{"(", + RowBox[{ + RowBox[{"-", + FractionBox["1", "2"]}], "-", "EulerGamma", "+", + RowBox[{"Log", "[", "2", "]"}], "-", + RowBox[{"Log", "[", "p", "]"}]}], ")"}], "+", + InterpretationBox[ + SuperscriptBox[ + RowBox[{"O", "[", "\[Epsilon]", "]"}], "1"], + SeriesData[TripleK`\[Epsilon], 0, {}, -1, 1, 1], + Editable->False]}], + SeriesData[ + TripleK`\[Epsilon], 0, { + 1, Rational[-1, 2] - 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Note, however, that some combinations of the regulating parameters may \ +still yield a divergent integral. If the regulator \[Epsilon] is already \ +present in the expression, the default parameters are ignored:" +}], "Text", + CellChangeTimes->{{3.7971926589624805`*^9, 3.797192710607553*^9}, { + 3.7971928200187073`*^9, 3.7971928218147097`*^9}, {3.797192858250761*^9, + 3.797192895690814*^9}, {3.797936407064107*^9, + 3.797936407743917*^9}},ExpressionUUID->"20882784-813f-4493-b7b7-\ +780e283703af"], + +Cell[CellGroupData[{ + +Cell[BoxData[{ + RowBox[{ + RowBox[{ + RowBox[{"KEvaluate", "[", + RowBox[{ + RowBox[{ + RowBox[{"i", "[", + RowBox[{"1", ",", + RowBox[{"{", + RowBox[{"1", ",", "1"}], "}"}]}], "]"}], "[", "p", "]"}], ",", " ", + RowBox[{"uParameter", "\[Rule]", "1"}], ",", + RowBox[{"vParameters", "\[Rule]", + RowBox[{"{", + RowBox[{"0", ",", "0", ",", "0"}], "}"}]}]}], "]"}], "//", + "FullSimplify"}], "\[IndentingNewLine]", + RowBox[{"(*", " ", + RowBox[{"This", " ", + RowBox[{"diverges", ":"}]}], " ", "*)"}]}], "\[IndentingNewLine]", + RowBox[{ + RowBox[{"KEvaluate", "[", + RowBox[{ + RowBox[{ + RowBox[{"i", "[", + RowBox[{"1", ",", + RowBox[{"{", + RowBox[{"1", ",", "1"}], "}"}]}], "]"}], "[", "p", "]"}], ",", " ", + RowBox[{"uParameter", "\[Rule]", + RowBox[{"2", "v"}]}], ",", + RowBox[{"vParameters", "\[Rule]", + RowBox[{"{", + RowBox[{"v", ",", "v", ",", "v"}], "}"}]}]}], "]"}], "//", + "FullSimplify"}], "\[IndentingNewLine]", + RowBox[{ + RowBox[{"KEvaluate", "[", + RowBox[{ + RowBox[{ + RowBox[{"i", "[", + RowBox[{ + RowBox[{"1", "+", "\[Epsilon]"}], ",", + RowBox[{"{", + RowBox[{"1", ",", "1"}], "}"}]}], "]"}], "[", "p", "]"}], ",", " ", + RowBox[{"uParameter", "\[Rule]", + RowBox[{"2", "v"}]}], ",", + RowBox[{"vParameters", "\[Rule]", + RowBox[{"{", + RowBox[{"v", ",", "v", ",", "v"}], "}"}]}]}], "]"}], "//", + "FullSimplify"}]}], "Input", + CellChangeTimes->{{3.7971927178735633`*^9, 3.79719273003658*^9}, { + 3.797192780907652*^9, 3.7971928551007566`*^9}, {3.7971928912358074`*^9, + 3.797192892395809*^9}, {3.797936420906179*^9, 3.797936431704372*^9}, { + 3.7982488011889477`*^9, + 3.79824883808436*^9}},ExpressionUUID->"dd471db2-0423-41c7-aa78-\ +1288ad5ecb43"], + +Cell[BoxData[ + InterpretationBox[ + RowBox[{ + FractionBox["1", "\[Epsilon]"], "+", + RowBox[{"(", + RowBox[{ + RowBox[{"-", + FractionBox["1", "2"]}], "-", "EulerGamma", "+", + RowBox[{"Log", "[", "2", "]"}], "-", + RowBox[{"Log", "[", "p", "]"}]}], ")"}], "+", + InterpretationBox[ + SuperscriptBox[ + RowBox[{"O", "[", "\[Epsilon]", "]"}], "1"], + SeriesData[TripleK`\[Epsilon], 0, {}, -1, 1, 1], + Editable->False]}], + SeriesData[ + TripleK`\[Epsilon], 0, { + 1, Rational[-1, 2] - 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Option ", + StyleBox["Assumptions", + FontSlant->"Italic"], + " can be used to add additional conditions:" +}], "Text", + CellChangeTimes->{{3.797193096093096*^9, 3.797193138373167*^9}, { + 3.797193178783224*^9, 3.7971932469893203`*^9}, {3.797936452175804*^9, + 3.797936452603304*^9}},ExpressionUUID->"30d753eb-c45a-4ae6-98a2-\ +d0a1a2b61b30"], + +Cell[CellGroupData[{ + +Cell[BoxData[{ + RowBox[{"IsDivergent", "[", + RowBox[{"i", "[", + RowBox[{"n", ",", + RowBox[{"{", + RowBox[{ + RowBox[{"n", "+", "1"}], ",", + RowBox[{"n", "+", "1"}], ",", + RowBox[{"n", "+", "1"}]}], "}"}]}], "]"}], "]"}], "\[IndentingNewLine]", + RowBox[{"IsDivergent", "[", + RowBox[{"i", "[", + RowBox[{ + RowBox[{"n", "+", "1"}], ",", + RowBox[{"{", + RowBox[{"n", ",", "n", ",", "n"}], "}"}]}], "]"}], + "]"}], "\[IndentingNewLine]", + RowBox[{"IsDivergent", "[", + RowBox[{ + RowBox[{"i", "[", + RowBox[{ + RowBox[{"n", "+", "1"}], ",", + RowBox[{"{", + RowBox[{"n", ",", "n", ",", "n"}], "}"}]}], "]"}], ",", " ", + RowBox[{"Assumptions", "\[Rule]", + RowBox[{"{", + RowBox[{ + RowBox[{"n", "\[Element]", "Integers"}], ",", + RowBox[{"n", "\[GreaterEqual]", "1"}]}], "}"}]}]}], "]"}]}], "Input", + CellChangeTimes->{{3.7971931513051853`*^9, 3.7971931668972073`*^9}, { + 3.7971932118632708`*^9, 3.7971932214392843`*^9}, {3.7971932548363314`*^9, + 3.797193285694375*^9}},ExpressionUUID->"638d66c2-1493-480b-9722-\ +441850c76f6a"], + +Cell[BoxData["True"], "Output", + CellChangeTimes->{{3.797193157305194*^9, 3.7971931672072077`*^9}, + 3.797193215543276*^9, 3.7971932862643757`*^9, 3.7971943452218685`*^9, + 3.797442476658267*^9, 3.797936453762128*^9, 3.798248876156975*^9, + 3.7986857546762896`*^9, 3.798685804454108*^9, + 3.799037620964365*^9},ExpressionUUID->"5c72eaee-cf49-4e60-9fe7-\ +d40c188fa53c"], + +Cell[BoxData[ + RowBox[{ + RowBox[{"(", + RowBox[{ + RowBox[{"n", "\[Element]", + TemplateBox[{}, + "Integers"]}], "&&", + RowBox[{"(", + RowBox[{ + RowBox[{"n", "\[LessEqual]", + RowBox[{"-", "1"}]}], "||", + RowBox[{"n", "\[GreaterEqual]", "1"}]}], ")"}]}], ")"}], "||", + RowBox[{"(", + RowBox[{ + RowBox[{ + RowBox[{"2", " ", "n"}], "\[Element]", + TemplateBox[{}, + "Integers"]}], "&&", + RowBox[{ + RowBox[{"1", "+", + RowBox[{"2", " ", "n"}]}], "\[LessEqual]", "0"}]}], ")"}]}]], "Output", + CellChangeTimes->{{3.797193157305194*^9, 3.7971931672072077`*^9}, + 3.797193215543276*^9, 3.7971932862643757`*^9, 3.7971943452218685`*^9, + 3.797442476658267*^9, 3.797936453762128*^9, 3.798248876156975*^9, + 3.7986857546762896`*^9, 3.798685804454108*^9, + 3.7990376211309023`*^9},ExpressionUUID->"6ee4200d-5961-41da-8852-\ +e75c0f72fe7a"], + +Cell[BoxData["True"], "Output", + CellChangeTimes->{{3.797193157305194*^9, 3.7971931672072077`*^9}, + 3.797193215543276*^9, 3.7971932862643757`*^9, 3.7971943452218685`*^9, + 3.797442476658267*^9, 3.797936453762128*^9, 3.798248876156975*^9, + 3.7986857546762896`*^9, 3.798685804454108*^9, + 3.799037621132152*^9},ExpressionUUID->"b2b3c2a7-eab6-4b71-bf51-\ +2fd305f3311e"] +}, Open ]], + +Cell[TextData[{ + "Divergences of double- and triple-", + StyleBox["K", + FontSlant->"Italic"], + " integrals can be evaluated directly without the evaluation of the entire \ +integral. See section 2.5.1 of [", + ButtonBox["1511.02357", + BaseStyle->"Hyperlink", + ButtonData->{ + URL["https://arxiv.org/pdf/1511.02357.pdf"], None}, + ButtonNote->"https://arxiv.org/pdf/1511.02357.pdf"], + "] for details." +}], "Text", + CellChangeTimes->{{3.79719343125058*^9, 3.7971935586127596`*^9}, { + 3.7971937752450647`*^9, 3.7971937804510717`*^9}, 3.797936458504402*^9, { + 3.798248888732459*^9, + 3.798248889004491*^9}},ExpressionUUID->"5afdcbd9-626d-48dd-83b5-\ +3632a7a4b713"], + +Cell[TextData[{ + "To evaluate divergences of a given expression up to and including given \ +order in \[Epsilon] use ", + StyleBox["KDivergence", + FontSlant->"Italic"], + ":" +}], "Text", + CellChangeTimes->{ + 3.7982488919114647`*^9},ExpressionUUID->"b001977a-38d3-4492-8c29-\ +99712bdb041b"], + +Cell[CellGroupData[{ + +Cell[BoxData[{ + RowBox[{"KDivergence", "[", + RowBox[{ + RowBox[{ + RowBox[{"i", "[", + RowBox[{ + RowBox[{"1", "+", "\[Epsilon]"}], ",", + RowBox[{"{", + RowBox[{"1", ",", "1"}], 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For details, consult appendix A.3 of [", + ButtonBox["1304.7760", + BaseStyle->"Hyperlink", + ButtonData->{ + URL["https://arxiv.org/pdf/1304.7760.pdf"], None}, + ButtonNote->"https://arxiv.org/pdf/1304.7760.pdf"], + "] and appendix C of [", + ButtonBox["1902.01251", + BaseStyle->"Hyperlink", + ButtonData->{ + URL["https://arxiv.org/pdf/1902.01251.pdf"], None}, + ButtonNote->"https://arxiv.org/pdf/1902.01251.pdf"], + "]." +}], "Text", + CellChangeTimes->{{3.797195852043998*^9, 3.7971961233043804`*^9}, { + 3.797196238331543*^9, 3.7971963113346457`*^9}, {3.797938170588299*^9, + 3.797938205043384*^9}},ExpressionUUID->"0e569706-9359-4e43-b111-\ +a0a977aca93d"], + +Cell[TextData[{ + "The momentum space integrals above are represented by ", + StyleBox["LoopIntegral", + FontSlant->"Italic"], + ". 2-point function-like integrals are represented as:" +}], "Text", + CellChangeTimes->{{3.781788441696711*^9, 3.781788463337407*^9}, { + 3.797196323265662*^9, 3.797196333672677*^9}, + 3.797938214225676*^9},ExpressionUUID->"cccffcc3-9cd2-450e-b901-\ +0397d673f7a1"], + +Cell[CellGroupData[{ + +Cell[BoxData[ + RowBox[{ + RowBox[{"LoopIntegral", "[", + RowBox[{"d", ",", + RowBox[{"{", + RowBox[{"\[Delta]1", ",", "\[Delta]2"}], "}"}]}], "]"}], "[", + RowBox[{"Numerator", ",", "k", ",", "p"}], "]"}]], "Input", + CellChangeTimes->{{3.7817884845219727`*^9, 3.78178851516964*^9}, { + 3.79719661905608*^9, + 3.797196629192094*^9}},ExpressionUUID->"b919e6fb-849c-4190-860d-\ +f73b53728e1c"], + +Cell[BoxData[ + TagBox[ + RowBox[{"\<\"\\!\\(\\*StyleBox[\\\"\[Integral]\\\",FontSize->18]\\)\"\>", + " ", + TagBox[ + FractionBox[ + RowBox[{ + SuperscriptBox["\<\"\[DifferentialD]\"\>", "d"], " ", "k"}], + SuperscriptBox[ + RowBox[{"(", + RowBox[{"2", " ", "\[Pi]"}], ")"}], "d"]], + HoldForm], " ", + TagBox[ + FractionBox["Numerator", + RowBox[{ + SuperscriptBox["k", + RowBox[{"2", " ", "\[Delta]1"}]], " ", + SuperscriptBox[ + RowBox[{"(", + RowBox[{"k", "-", "p"}], ")"}], + RowBox[{"2", " ", "\[Delta]2"}]]}]], + HoldForm]}], + HoldForm]], "Output", + CellChangeTimes->{ + 3.781788516037409*^9, 3.794555563713339*^9, 3.797196338123684*^9, { + 3.797196622842085*^9, 3.797196630727096*^9}, 3.797444674697467*^9, { + 3.797444854407029*^9, 3.7974448997200933`*^9}, 3.797938215310196*^9, + 3.7982509285336027`*^9, 3.7986857591275425`*^9, 3.7986858079043036`*^9, + 3.799037626075899*^9},ExpressionUUID->"364be254-2195-453b-8aba-\ +7a2c4007c8e4"] +}, Open ]], + +Cell[TextData[{ + "3-point function-like integrals assume standard momentum magnitudes ", + Cell[BoxData[ + FormBox[ + RowBox[{ + SubscriptBox["p", "1"], ",", " ", + SubscriptBox["p", "2"], ",", " ", + SubscriptBox["p", "3"]}], TraditionalForm]],ExpressionUUID-> + "8a4b802c-0778-4d7e-8479-e7341395c995"], + ":" +}], "Text", + CellChangeTimes->{{3.781788469849334*^9, 3.781788481681872*^9}, { + 3.781788526321504*^9, 3.781788541842341*^9}, {3.7971963439136915`*^9, + 3.797196349054699*^9}, + 3.797938221193215*^9},ExpressionUUID->"6964230c-52ed-4857-aee1-\ +94872bc6df48"], + +Cell[CellGroupData[{ + +Cell[BoxData[ + RowBox[{ + RowBox[{"LoopIntegral", "[", + RowBox[{"d", ",", + RowBox[{"{", + RowBox[{"\[Delta]1", ",", "\[Delta]2", ",", "\[Delta]3"}], "}"}]}], + "]"}], "[", + RowBox[{"Numerator", ",", "k"}], "]"}]], "Input", + CellChangeTimes->{{3.781788548961565*^9, 3.7817885531856833`*^9}, + 3.7971966344731016`*^9},ExpressionUUID->"01aa5b81-766f-43ed-90ef-\ +ccb10a0669d1"], + +Cell[BoxData[ + TagBox[ + RowBox[{"\<\"\\!\\(\\*StyleBox[\\\"\[Integral]\\\",FontSize->18]\\)\"\>", + " ", + TagBox[ + FractionBox[ + RowBox[{ + SuperscriptBox["\<\"\[DifferentialD]\"\>", "d"], " ", "k"}], + SuperscriptBox[ + RowBox[{"(", + RowBox[{"2", " ", "\[Pi]"}], ")"}], "d"]], + HoldForm], " ", + TagBox[ + FractionBox["Numerator", + RowBox[{ + SuperscriptBox["k", + RowBox[{"2", " ", "\[Delta]3"}]], " ", + SuperscriptBox[ + RowBox[{"(", + RowBox[{"k", "-", + SubscriptBox["p", "1"]}], ")"}], + RowBox[{"2", " ", "\[Delta]2"}]], " ", + SuperscriptBox[ + RowBox[{"(", + RowBox[{"k", "+", + SubscriptBox["p", "2"]}], ")"}], + RowBox[{"2", " ", "\[Delta]1"}]]}]], + HoldForm]}], + HoldForm]], "Output", + CellChangeTimes->{3.78178855392774*^9, 3.794555565031801*^9, + 3.7971963499997005`*^9, 3.7971966351081023`*^9, 3.7974447908689404`*^9, + 3.797444901086095*^9, 3.797938222248913*^9, 3.798250930928075*^9, + 3.798685759150544*^9, 3.798685807914304*^9, + 3.799037626088697*^9},ExpressionUUID->"af7f5926-b7f6-410a-ac1e-\ +74540e19678d"] +}, Open ]], + +Cell[CellGroupData[{ + +Cell["Momentum manipulations", "Subsection", + CellChangeTimes->{{3.7974426213474703`*^9, 3.797442625932477*^9}, { + 3.797444000441516*^9, + 3.797444054969592*^9}},ExpressionUUID->"e0926b4c-9868-42bb-9d36-\ +4a1d8a6a35e1"], + +Cell[TextData[{ + "When indices are put on momenta, they are treated as vectors. Such \ +expressions can be manipulated by two functions: ", + StyleBox["Contract", + FontSlant->"Italic"], + ", which contracts free or specified indices and ", + StyleBox["Diff", + FontSlant->"Italic"], + ", which takes derivatives with respect to a given momentum." +}], "Text", + CellChangeTimes->{{3.7974433454075937`*^9, + 3.7974434427187304`*^9}},ExpressionUUID->"8e1ae90e-40f3-4b5b-9963-\ +673bbe1aef23"], + +Cell[TextData[{ + "When default momenta ", + Cell[BoxData[ + FormBox[ + SubsuperscriptBox["p", "j", "\[Mu]"], TraditionalForm]],ExpressionUUID-> + "b744f970-0b5d-47f7-ad8c-ca95be0332a7"], + " are used, it is always assumed that ", + Cell[BoxData[ + FormBox[ + RowBox[{ + SubsuperscriptBox["p", "3", "\[Mu]"], "=", + RowBox[{ + RowBox[{"-", + SubsuperscriptBox["p", "1", "\[Mu]"]}], "-", + SubsuperscriptBox["p", "2", "\[Mu]"]}]}], TraditionalForm]], + ExpressionUUID->"d7fd89cb-11e7-4f0a-a4dc-8f0e20a1bfc1"], + ". For this reason all scalar products ", + Cell[BoxData[ + FormBox[ + RowBox[{ + SubscriptBox["p", "i"], "\[CenterDot]", + SubscriptBox["p", "j"]}], TraditionalForm]],ExpressionUUID-> + "4addf5ed-710c-46eb-81ad-112c67c713b3"], + " are expressed in terms of magnitudes:" +}], "Text", + CellChangeTimes->{{3.7974442824749155`*^9, 3.797444381361055*^9}, + 3.797444451317153*^9, {3.797937781969186*^9, + 3.7979377821205673`*^9}},ExpressionUUID->"deae2631-dd54-4073-a2ea-\ +52100380e07a"], + +Cell[CellGroupData[{ + +Cell[BoxData[ + RowBox[{ + RowBox[{ + RowBox[{"p", "[", "1", "]"}], "\[CenterDot]", + RowBox[{"p", "[", "2", "]"}]}], "//", "KSimplify"}]], "Input", + CellChangeTimes->{ + 3.7982481066118193`*^9},ExpressionUUID->"d4519333-3e1e-438b-b90f-\ +2f21f5c9ee26"], + +Cell[BoxData[ + RowBox[{ + FractionBox["1", "2"], " ", + RowBox[{"(", + RowBox[{ + RowBox[{"-", + SubsuperscriptBox["p", "1", "2"]}], "-", + SubsuperscriptBox["p", "2", "2"], "+", + SubsuperscriptBox["p", "3", "2"]}], ")"}]}]], "Output", + CellChangeTimes->{ + 3.797937295441021*^9, {3.7982481068597813`*^9, 3.7982481097862062`*^9}, + 3.7982509407007103`*^9, 3.7986857591755457`*^9, 3.798685807938306*^9, + 3.799037626154673*^9},ExpressionUUID->"ebd36192-abe1-493a-a714-\ +bb0bdc4870ef"] +}, Open ]], + +Cell[TextData[{ + "We used ", + StyleBox["KSimplify", + FontSlant->"Italic"], + " for basic simplifications of expressions containing multiple-", + StyleBox["K", + FontSlant->"Italic"], + " integrals (and also momentum loop integrals):" +}], "Text", + CellChangeTimes->{{3.797937243367976*^9, 3.797937297745038*^9}, { + 3.797937748744872*^9, + 3.7979377526889467`*^9}},ExpressionUUID->"3174dc97-845b-4790-846d-\ +862c31086ff5"], + +Cell[CellGroupData[{ + +Cell[BoxData[{ + RowBox[{ + RowBox[{"i", "[", + RowBox[{"2", ",", + RowBox[{"{", + RowBox[{"3", ",", "1", ",", "1"}], "}"}]}], "]"}], "+", + RowBox[{ + RowBox[{"p", "[", "1", "]"}], + RowBox[{"D", "[", + RowBox[{ + RowBox[{ + RowBox[{"i", "[", + RowBox[{"1", ",", + RowBox[{"{", + RowBox[{"2", ",", "1", ",", "1"}], "}"}]}], "]"}], "[", + RowBox[{ + RowBox[{"p", "[", "1", "]"}], ",", + RowBox[{"p", "[", "2", "]"}], ",", + RowBox[{"p", "[", "3", "]"}]}], "]"}], ",", + RowBox[{"p", "[", "1", "]"}]}], "]"}]}]}], "\[IndentingNewLine]", + RowBox[{"%", "//", "KSimplify"}]}], "Input", + CellChangeTimes->{{3.797937524786771*^9, 3.797937561096759*^9}, { + 3.7979375958967657`*^9, 3.7979376411786737`*^9}, {3.797937681163629*^9, + 3.797937695057639*^9}},ExpressionUUID->"1bb0f6be-f389-4b1c-8a34-\ +6ab8b6314a41"], + +Cell[BoxData[ + RowBox[{ + SubscriptBox["i", + RowBox[{"2", ",", + RowBox[{"{", + RowBox[{"3", ",", "1", ",", "1"}], "}"}]}]], "+", + RowBox[{ + SubscriptBox["p", "1"], " ", + RowBox[{ + SuperscriptBox[ + SubscriptBox["i", + RowBox[{"1", ",", + RowBox[{"{", + RowBox[{"2", ",", "1", ",", "1"}], "}"}]}]], + TagBox[ + RowBox[{"(", + RowBox[{"1", ",", "0", ",", "0"}], ")"}], + Derivative], + MultilineFunction->None], "[", + RowBox[{ + SubscriptBox["p", "1"], ",", + SubscriptBox["p", "2"], ",", + SubscriptBox["p", "3"]}], "]"}]}]}]], "Output", + CellChangeTimes->{{3.797937638430833*^9, 3.797937641454742*^9}, { + 3.797937690614043*^9, 3.797937695477777*^9}, 3.79825094353625*^9, + 3.7986857592005467`*^9, 3.798685807950306*^9, + 3.799037626222373*^9},ExpressionUUID->"9d8d5427-9ff2-4c30-b5e6-\ +2b4c53a81467"], + +Cell[BoxData[ + RowBox[{"4", " ", + SubscriptBox["i", + RowBox[{"1", ",", + RowBox[{"{", + RowBox[{"2", ",", "1", ",", "1"}], "}"}]}]]}]], "Output", + CellChangeTimes->{{3.797937638430833*^9, 3.797937641454742*^9}, { + 3.797937690614043*^9, 3.797937695477777*^9}, 3.79825094353625*^9, + 3.7986857592005467`*^9, 3.798685807950306*^9, + 3.799037626223967*^9},ExpressionUUID->"bff125d8-2c7e-4b1b-9426-\ +82720565902c"] +}, Open ]], + +Cell[TextData[{ + "To contract indices use ", + StyleBox["Contract", + FontSlant->"Italic"], + ". Without any arguments ", + StyleBox["Contract", + FontSlant->"Italic"], + " contracts all free indices appearing on known vectors. By default these \ +are only ", + Cell[BoxData[ + FormBox[ + SubscriptBox["p", "j"], TraditionalForm]],ExpressionUUID-> + "10c1e9c3-5b05-4658-b501-71a6ad3f18ee"], + StyleBox[" ", + FontSlant->"Italic"], + "and integration momenta ", + StyleBox["LoopIntegral", + FontSlant->"Italic"], + "s:" +}], "Text", + CellChangeTimes->{{3.797443486850793*^9, 3.797443589732937*^9}, { + 3.7974436771400604`*^9, 3.7974436782200623`*^9}, {3.7974444132230997`*^9, + 3.7974444343201294`*^9}, {3.79793777770545*^9, + 3.797937778113186*^9}},ExpressionUUID->"9e9a11f0-3e16-4d4d-a3be-\ +f143f1c6dd3d"], + +Cell[CellGroupData[{ + +Cell[BoxData[{ + RowBox[{"Contract", "[", + RowBox[{ + RowBox[{ + RowBox[{"p", "[", "1", "]"}], "[", "\[Mu]", "]"}], + RowBox[{ + RowBox[{"p", "[", "2", "]"}], "[", "\[Mu]", "]"}]}], + "]"}], "\[IndentingNewLine]", + RowBox[{"Contract", "[", + RowBox[{ + RowBox[{ + RowBox[{"p", "[", "1", "]"}], "[", "\[Mu]", "]"}], + RowBox[{"k", "[", "\[Mu]", "]"}]}], "]"}]}], "Input", + CellChangeTimes->{{3.797443466658764*^9, 3.7974434759187775`*^9}, { + 3.7974435952939453`*^9, 3.7974436375700045`*^9}, {3.797444622472394*^9, + 3.7974446226923943`*^9}},ExpressionUUID->"eecc1fa3-d0d5-4dfc-b25e-\ +b42616e3a1ed"], + +Cell[BoxData[ + RowBox[{ + FractionBox["1", "2"], " ", + RowBox[{"(", + RowBox[{ + RowBox[{"-", + SubsuperscriptBox["p", "1", "2"]}], "-", + SubsuperscriptBox["p", "2", "2"], "+", + SubsuperscriptBox["p", "3", "2"]}], ")"}]}]], "Output", + CellChangeTimes->{3.797443476638778*^9, 3.7974436379050055`*^9, + 3.7974446239073963`*^9, 3.797937723281486*^9, 3.798250945831275*^9, + 3.7986857592135477`*^9, 3.7986858079683075`*^9, + 3.799037626238389*^9},ExpressionUUID->"dd7866a4-7211-46b6-9e40-\ +ae780138f13f"], + +Cell[BoxData[ + RowBox[{ + RowBox[{"k", "[", "\[Mu]", "]"}], " ", + TemplateBox[{"p","1","\[Mu]"}, + "Subsuperscript"]}]], "Output", + CellChangeTimes->{3.797443476638778*^9, 3.7974436379050055`*^9, + 3.7974446239073963`*^9, 3.797937723281486*^9, 3.798250945831275*^9, + 3.7986857592135477`*^9, 3.7986858079683075`*^9, + 3.799037626240377*^9},ExpressionUUID->"acc122c1-2fe5-4f0f-81b2-\ +ae54fc24ae32"] +}, Open ]], + +Cell[TextData[{ + "Additional vectors can be specified by option ", + StyleBox["Vectors", + FontSlant->"Italic"], + ". Its argument can be a single vector or a list:" +}], "Text", + CellChangeTimes->{{3.7974436472410183`*^9, 3.797443721021122*^9}, { + 3.797937786392928*^9, + 3.797937786584846*^9}},ExpressionUUID->"be9d5955-2230-4998-9fc5-\ +a28cdf532368"], + +Cell[CellGroupData[{ + +Cell[BoxData[ + RowBox[{"Contract", "[", + RowBox[{ + RowBox[{ + RowBox[{ + RowBox[{"p", "[", "1", "]"}], "[", "\[Mu]", "]"}], + RowBox[{"k", "[", "\[Mu]", "]"}]}], ",", " ", + RowBox[{"Vectors", "\[Rule]", "k"}]}], "]"}]], "Input", + CellChangeTimes->{{3.797443686600074*^9, 3.7974437064331017`*^9}, { + 3.7974438012142353`*^9, + 3.7974438022942367`*^9}},ExpressionUUID->"6547ad10-5b9f-4424-a93b-\ +db357fc90e05"], + +Cell[BoxData[ + RowBox[{"k", "\[CenterDot]", + SubscriptBox["p", "1"]}]], "Output", + CellChangeTimes->{{3.7974436998120923`*^9, 3.797443706688102*^9}, + 3.7979377331707687`*^9, 3.797937770444351*^9, 3.798250947502111*^9, + 3.798685759228548*^9, 3.798685807980308*^9, + 3.799037626277972*^9},ExpressionUUID->"d0b0e33a-ff89-4b7c-8384-\ +abbccd09bc56"] +}, Open ]], + +Cell[TextData[{ + "If option ", + StyleBox["Vectors", + FontSlant->"Italic"], + " is set to ", + StyleBox["All", + FontSlant->"Italic"], + ", all repeated indices are contracted. Note that this can lead to undesired \ +results:" +}], "Text", + CellChangeTimes->{{3.7974437554761705`*^9, 3.7974437993382325`*^9}, { + 3.797937788968882*^9, + 3.797937789144876*^9}},ExpressionUUID->"3e73fdae-5149-4396-a18e-\ +45ed3cad79a7"], + +Cell[CellGroupData[{ + +Cell[BoxData[ + RowBox[{"Contract", "[", + RowBox[{ + RowBox[{ + RowBox[{ + RowBox[{"Sin", "[", "x", "]"}], + RowBox[{"Cos", "[", "x", "]"}]}], "+", + RowBox[{ + RowBox[{ + RowBox[{"p", "[", "1", "]"}], "[", "\[Mu]", "]"}], + RowBox[{"k", "[", "\[Mu]", "]"}]}]}], ",", " ", + RowBox[{"Vectors", "\[Rule]", "All"}]}], "]"}]], "Input", + CellChangeTimes->{{3.797443806875243*^9, 3.797443817427258*^9}, { + 3.7974438745823383`*^9, + 3.797443879508345*^9}},ExpressionUUID->"16d6244e-c4d1-4115-b4a7-\ +f7c9524ce5e4"], + +Cell[BoxData[ + RowBox[{ + RowBox[{"Cos", "\[CenterDot]", "Sin"}], "+", + RowBox[{"k", "\[CenterDot]", + SubscriptBox["p", "1"]}]}]], "Output", + CellChangeTimes->{{3.797443808931246*^9, 3.7974438177122583`*^9}, + 3.7974438798383455`*^9, 3.797937771873611*^9, 3.7982509492799397`*^9, + 3.798685759238549*^9, 3.7986858079893084`*^9, + 3.799037626313621*^9},ExpressionUUID->"1aa3e654-1fd9-490e-a33b-\ +d1608c24a255"] +}, Open ]], + +Cell["\<\ +To contract over a specified index or a list of such indices, the extra \ +argument can be supplied:\ +\>", "Text", + CellChangeTimes->{{3.7974438940003653`*^9, 3.797443938976429*^9}, { + 3.797937792440797*^9, + 3.797937792625169*^9}},ExpressionUUID->"99b07dc7-84d9-43b8-8c93-\ +1246d70a182e"], + +Cell[CellGroupData[{ + +Cell[BoxData[ + RowBox[{"Contract", "[", + RowBox[{ + RowBox[{ + RowBox[{ + RowBox[{"Sin", "[", "x", "]"}], + RowBox[{"Cos", "[", "x", "]"}]}], "+", + RowBox[{ + RowBox[{ + RowBox[{"p", "[", "1", "]"}], "[", "\[Mu]", "]"}], + RowBox[{"k", "[", "\[Mu]", "]"}]}]}], ",", " ", "\[Mu]"}], + "]"}]], "Input", + CellChangeTimes->{{3.797443924129408*^9, + 3.79744392580941*^9}},ExpressionUUID->"6f8a9853-56d6-4e47-958e-\ +4404d92ae176"], + +Cell[BoxData[ + RowBox[{ + RowBox[{"k", "\[CenterDot]", + SubscriptBox["p", "1"]}], "+", + RowBox[{ + RowBox[{"Cos", "[", "x", "]"}], " ", + RowBox[{"Sin", "[", "x", "]"}]}]}]], "Output", + CellChangeTimes->{3.797443926194411*^9, 3.798250950559822*^9, + 3.7986857592475495`*^9, 3.798685808000309*^9, + 3.799037626348784*^9},ExpressionUUID->"a225eca5-c18e-46bf-b617-\ +211c9c8b7588"] +}, Open ]], + +Cell[TextData[{ + "To contract a pair of two distinct indices one can call the following \ +version of ", + StyleBox["Contract:", + FontSlant->"Italic"] +}], "Text", + CellChangeTimes->{{3.7974439492484436`*^9, 3.797443976763482*^9}, { + 3.797937795432889*^9, + 3.797937795609158*^9}},ExpressionUUID->"053230b6-7339-4b10-8ecd-\ +575c18a94cef"], + +Cell[CellGroupData[{ + +Cell[BoxData[ + RowBox[{"Contract", "[", + RowBox[{ + RowBox[{ + RowBox[{ + RowBox[{"p", "[", "1", "]"}], "[", "\[Mu]", "]"}], + RowBox[{"k", "[", "\[Nu]", "]"}]}], ",", " ", "\[Mu]", ",", "\[Nu]"}], + "]"}]], "Input", + CellChangeTimes->{{3.797443984264493*^9, + 3.797443989850501*^9}},ExpressionUUID->"2f013a70-a3c6-4c85-96e9-\ +59ec7b8f87be"], + +Cell[BoxData[ + RowBox[{"k", "\[CenterDot]", + SubscriptBox["p", "1"]}]], "Output", + CellChangeTimes->{3.7974439902455015`*^9, 3.797937798341468*^9, + 3.79825095194415*^9, 3.79868575925655*^9, 3.79868580800931*^9, + 3.799037626384014*^9},ExpressionUUID->"e69da52f-7bb5-432f-a987-\ +9914415f4b92"] +}, Open ]], + +Cell[TextData[{ + "Finally, option ", + StyleBox["Dimension", + FontSlant->"Italic"], + " specifies spacetime dimension:" +}], "Text", + CellChangeTimes->{{3.7982509743335123`*^9, + 3.798250994478236*^9}},ExpressionUUID->"431e9a79-c7a0-4b66-be7d-\ +2bb8f8ed3262"], + +Cell[CellGroupData[{ + +Cell[BoxData[{ + RowBox[{"Contract", "[", + RowBox[{ + RowBox[{"\[Delta]", "[", + RowBox[{"\[Mu]", ",", "\[Nu]"}], "]"}], ",", "\[Mu]", ",", "\[Nu]"}], + "]"}], "\[IndentingNewLine]", + RowBox[{"Contract", "[", + RowBox[{ + RowBox[{"\[Delta]", "[", + RowBox[{"\[Mu]", ",", "\[Nu]"}], "]"}], ",", "\[Mu]", ",", "\[Nu]", ",", + RowBox[{"Dimension", "\[Rule]", "4"}]}], "]"}]}], "Input", + CellChangeTimes->{{3.7982510028313427`*^9, 3.7982510167259293`*^9}, { + 3.7985245803920794`*^9, 3.7985245882488947`*^9}, + 3.7985306569271297`*^9},ExpressionUUID->"4c886d6f-0cb3-4a80-a4f6-\ +1126972a3f9b"], + +Cell[BoxData["d"], "Output", + CellChangeTimes->{3.798251017071315*^9, 3.798524589269126*^9, + 3.7985306574362*^9, 3.798685759268551*^9, 3.79868580801931*^9, + 3.799037626421976*^9},ExpressionUUID->"63d78e31-95be-4182-b106-\ +a27ed7229b60"], + +Cell[BoxData["4"], "Output", + CellChangeTimes->{3.798251017071315*^9, 3.798524589269126*^9, + 3.7985306574362*^9, 3.798685759268551*^9, 3.79868580801931*^9, + 3.7990376264239473`*^9},ExpressionUUID->"6ecc6fe6-7722-46bb-b872-\ +5220a4296c79"] +}, Open ]], + +Cell[TextData[{ + "Any expression can be differentiated with respect to a given vector by \ +using function ", + StyleBox["Diff", + FontSlant->"Italic"], + ". 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+ Rational[1, 2] ( + Rational[1, 32] (2 - 2 EulerGamma) Pi^(-2) + TripleK`\[Delta][$CellContext`\[Mu], $CellContext`\[Nu]] + + Rational[1, 32] Pi^(-2) Log[Pi] + TripleK`\[Delta][$CellContext`\[Mu], $CellContext`\[Nu]] + + Pi^(-2) (Rational[3, 32] Log[2] + TripleK`\[Delta][$CellContext`\[Mu], $CellContext`\[Nu]] + + Rational[1, 16] ( + TripleK`\[Delta][$CellContext`\[Mu], $CellContext`\[Nu]] (-1 + + EulerGamma + Rational[1, 2] (EulerGamma - Log[2]) + 2 TripleK`NL[ + TripleK`p[1], + TripleK`p[2], + TripleK`p[3]] TripleK`p[1]^2 TripleK`p[2]^2 TripleK`p[3]^2 + TripleK`\[Lambda][ + TripleK`p[1], + TripleK`p[2], + TripleK`p[3]]^Rational[-3, 2] + + Rational[1, 2] ( + Log[TripleK`p[3]^2] + TripleK`p[3]^2 (TripleK`p[1]^2 + TripleK`p[2]^2 - + TripleK`p[3]^2) + + Log[TripleK`p[2]^2] + TripleK`p[2]^2 (TripleK`p[1]^2 - TripleK`p[2]^2 + + TripleK`p[3]^2) + + Log[TripleK`p[1]^2] + TripleK`p[1]^2 (-TripleK`p[1]^2 + TripleK`p[2]^2 + + TripleK`p[3]^2))/TripleK`\[Lambda][ + TripleK`p[1], + TripleK`p[2], 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TripleK`p[2], + TripleK`p[3]]^Rational[-1, 2]) + Rational[1, 4] TripleK`NL[ + TripleK`p[1], + TripleK`p[2], + TripleK`p[3]] ((-4) TripleK`p[1]^2 TripleK`p[2] + + 4 TripleK`p[2]^3 - 4 TripleK`p[2] TripleK`p[3]^2) + TripleK`\[Lambda][ + TripleK`p[1], + TripleK`p[2], + TripleK`p[3]]^Rational[-3, 2] - + TripleK`p[2]^(-1) (Log[TripleK`p[2]^2] TripleK`p[2]^2 + + Rational[1, 2] + Log[TripleK`p[3]^2] (TripleK`p[1]^2 - TripleK`p[2]^2 - + TripleK`p[3]^2) + + Rational[1, 2] + Log[TripleK`p[1]^2] (-TripleK`p[1]^2 - TripleK`p[2]^2 + + TripleK`p[3]^2))/TripleK`\[Lambda][ + TripleK`p[1], + TripleK`p[2], + TripleK`p[3]]) - 2 TripleK`p[2] (Rational[-1, 4] TripleK`NL[ + TripleK`p[1], + TripleK`p[2], + TripleK`p[3]] ((-4) TripleK`p[1]^2 TripleK`p[2] + + 4 TripleK`p[2]^3 - 4 TripleK`p[2] TripleK`p[3]^2) + TripleK`\[Lambda][ + TripleK`p[1], + TripleK`p[2], + TripleK`p[3]]^Rational[-3, 2] + + TripleK`p[2]^(-1) (Log[TripleK`p[2]^2] TripleK`p[2]^2 + + Rational[1, 2] + Log[TripleK`p[3]^2] (TripleK`p[1]^2 - TripleK`p[2]^2 - + TripleK`p[3]^2) + + Rational[1, 2] + Log[TripleK`p[1]^2] (-TripleK`p[1]^2 - TripleK`p[2]^2 + + TripleK`p[3]^2))/TripleK`\[Lambda][ + TripleK`p[1], + TripleK`p[2], + TripleK`p[3]])) TripleK`p[1][$CellContext`\[Mu]] + TripleK`p[1][$CellContext`\[Nu]] + + 4 TripleK`p[2] (Rational[-1, 4] TripleK`NL[ + TripleK`p[1], + TripleK`p[2], + TripleK`p[3]] ((-4) TripleK`p[1]^2 TripleK`p[2] + + 4 TripleK`p[2]^3 - 4 TripleK`p[2] TripleK`p[3]^2) + TripleK`\[Lambda][ + TripleK`p[1], + TripleK`p[2], + TripleK`p[3]]^Rational[-3, 2] + + TripleK`p[2]^(-1) (Log[TripleK`p[2]^2] TripleK`p[2]^2 + + Rational[1, 2] + Log[TripleK`p[3]^2] (TripleK`p[1]^2 - TripleK`p[2]^2 - + TripleK`p[3]^2) + + Rational[1, 2] + Log[TripleK`p[1]^2] (-TripleK`p[1]^2 - TripleK`p[2]^2 + + TripleK`p[3]^2))/TripleK`\[Lambda][ + TripleK`p[1], + TripleK`p[2], + TripleK`p[3]]) TripleK`p[1][$CellContext`\[Mu]] + TripleK`p[1][$CellContext`\[Nu]] - 4 + TripleK`p[1] (Rational[-1, 4] TripleK`NL[ + TripleK`p[1], + TripleK`p[2], 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TripleK`p[1] + TripleK`p[2]^2 - 4 TripleK`p[1] TripleK`p[3]^2) ( + Log[TripleK`p[2]^2] TripleK`p[2]^2 + + Rational[1, 2] + Log[TripleK`p[3]^2] (TripleK`p[1]^2 - TripleK`p[2]^2 - + TripleK`p[3]^2) + + Rational[1, 2] + Log[TripleK`p[1]^2] (-TripleK`p[1]^2 - TripleK`p[2]^2 + + TripleK`p[3]^2)) TripleK`\[Lambda][ + TripleK`p[1], + TripleK`p[2], + TripleK`p[3]]^(-2) + + Rational[-1, 2] + TripleK`p[1]^(-1) ((-4) TripleK`p[1]^2 TripleK`p[2] + + 4 TripleK`p[2]^3 - 4 TripleK`p[2] TripleK`p[3]^2) ( + Log[TripleK`p[1]^2] TripleK`p[1]^2 + + Rational[1, 2] + Log[TripleK`p[3]^2] (-TripleK`p[1]^2 + TripleK`p[2]^2 - + TripleK`p[3]^2) + + Rational[1, 2] + Log[TripleK`p[2]^2] (-TripleK`p[1]^2 - TripleK`p[2]^2 + + TripleK`p[3]^2)) TripleK`\[Lambda][ + TripleK`p[1], + TripleK`p[2], + TripleK`p[3]]^(-2) + 2 TripleK`NL[ + TripleK`p[1], + TripleK`p[2], + TripleK`p[3]] TripleK`p[1] TripleK`p[2] TripleK`\[Lambda][ + TripleK`p[1], + TripleK`p[2], + TripleK`p[3]]^Rational[-3, 2] + + Rational[ + 1, 2] (-TripleK`p[1]^(-1) ((-4) TripleK`p[1]^2 TripleK`p[2] + + 4 TripleK`p[2]^3 - 4 TripleK`p[2] TripleK`p[3]^2) ( + Log[TripleK`p[1]^2] TripleK`p[1]^2 + + Rational[1, 2] + Log[TripleK`p[3]^2] (-TripleK`p[1]^2 + TripleK`p[2]^2 - + TripleK`p[3]^2) + + Rational[1, 2] + Log[TripleK`p[2]^2] (-TripleK`p[1]^2 - TripleK`p[2]^2 + + TripleK`p[3]^2)) TripleK`\[Lambda][ + TripleK`p[1], + TripleK`p[2], + TripleK`p[3]]^Rational[-3, 2] + + 2 TripleK`p[1]^(-1) (-Log[TripleK`p[2]^2] TripleK`p[2] + + Log[TripleK`p[3]^2] TripleK`p[2] + + TripleK`p[2]^(-1) (-TripleK`p[1]^2 - TripleK`p[2]^2 + + TripleK`p[3]^2)) TripleK`\[Lambda][ + TripleK`p[1], + TripleK`p[2], + TripleK`p[3]]^Rational[-1, 2]) TripleK`\[Lambda][ + TripleK`p[1], + TripleK`p[2], + TripleK`p[3]]^Rational[-1, 2]) + TripleK`p[1][$CellContext`\[Nu]] TripleK`p[2][$CellContext`\[Mu]] - + TripleK`p[1] TripleK`p[2] (Rational[3, 8] TripleK`NL[ + TripleK`p[1], + TripleK`p[2], + TripleK`p[3]] (4 TripleK`p[1]^3 - 4 TripleK`p[1] TripleK`p[2]^2 - + 4 TripleK`p[1] TripleK`p[3]^2) ((-4) TripleK`p[1]^2 TripleK`p[2] + + 4 TripleK`p[2]^3 - 4 TripleK`p[2] TripleK`p[3]^2) + TripleK`\[Lambda][ + TripleK`p[1], + TripleK`p[2], + TripleK`p[3]]^Rational[-5, 2] + + Rational[-1, 2] + TripleK`p[2]^(-1) (4 TripleK`p[1]^3 - 4 TripleK`p[1] + TripleK`p[2]^2 - 4 TripleK`p[1] TripleK`p[3]^2) ( + Log[TripleK`p[2]^2] TripleK`p[2]^2 + + Rational[1, 2] + Log[TripleK`p[3]^2] (TripleK`p[1]^2 - TripleK`p[2]^2 - + TripleK`p[3]^2) + + Rational[1, 2] + Log[TripleK`p[1]^2] (-TripleK`p[1]^2 - TripleK`p[2]^2 + + TripleK`p[3]^2)) TripleK`\[Lambda][ + TripleK`p[1], + TripleK`p[2], + TripleK`p[3]]^(-2) + + Rational[-1, 2] + TripleK`p[1]^(-1) ((-4) TripleK`p[1]^2 TripleK`p[2] + + 4 TripleK`p[2]^3 - 4 TripleK`p[2] TripleK`p[3]^2) ( + Log[TripleK`p[1]^2] TripleK`p[1]^2 + + Rational[1, 2] + Log[TripleK`p[3]^2] (-TripleK`p[1]^2 + TripleK`p[2]^2 - + TripleK`p[3]^2) + + Rational[1, 2] + Log[TripleK`p[2]^2] (-TripleK`p[1]^2 - TripleK`p[2]^2 + + TripleK`p[3]^2)) TripleK`\[Lambda][ + TripleK`p[1], + TripleK`p[2], + TripleK`p[3]]^(-2) + 2 TripleK`NL[ + TripleK`p[1], + TripleK`p[2], + TripleK`p[3]] TripleK`p[1] TripleK`p[2] TripleK`\[Lambda][ + TripleK`p[1], + TripleK`p[2], + TripleK`p[3]]^Rational[-3, 2] + + Rational[ + 1, 2] (-TripleK`p[1]^(-1) ((-4) TripleK`p[1]^2 TripleK`p[2] + + 4 TripleK`p[2]^3 - 4 TripleK`p[2] TripleK`p[3]^2) ( + Log[TripleK`p[1]^2] TripleK`p[1]^2 + + Rational[1, 2] + Log[TripleK`p[3]^2] (-TripleK`p[1]^2 + TripleK`p[2]^2 - + TripleK`p[3]^2) + + Rational[1, 2] + Log[TripleK`p[2]^2] (-TripleK`p[1]^2 - TripleK`p[2]^2 + + TripleK`p[3]^2)) TripleK`\[Lambda][ + TripleK`p[1], + TripleK`p[2], + TripleK`p[3]]^Rational[-3, 2] + + 2 TripleK`p[1]^(-1) (-Log[TripleK`p[2]^2] TripleK`p[2] + + Log[TripleK`p[3]^2] TripleK`p[2] + + TripleK`p[2]^(-1) (-TripleK`p[1]^2 - TripleK`p[2]^2 + + TripleK`p[3]^2)) TripleK`\[Lambda][ + TripleK`p[1], + TripleK`p[2], + TripleK`p[3]]^Rational[-1, 2]) TripleK`\[Lambda][ + TripleK`p[1], + TripleK`p[2], + 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Therefore we can check the ABJ \ +calculations explicitly, without using any `tricks\[CloseCurlyQuote].\ +\>", "Text", + CellChangeTimes->{{3.79488686636104*^9, 3.794886931369855*^9}, { + 3.7972003781794367`*^9, + 3.797200391654456*^9}},ExpressionUUID->"605dd233-f16a-47ac-8c52-\ +0688c8a3c088"], + +Cell[TextData[{ + "We will work with a single free Weyl fermion \[Psi] and the axial current\n", + Cell[BoxData[ + FormBox[ + RowBox[{ + SuperscriptBox["j", "\[Mu]"], "=", + RowBox[{ + SubscriptBox[ + OverscriptBox["\[Psi]", "_"], + OverscriptBox["\[Alpha]", "."]], + SuperscriptBox[ + OverscriptBox["\[Sigma]", "_"], + RowBox[{"\[Mu]", + OverscriptBox["\[Alpha]", "."], "\[Alpha]"}]], + SubscriptBox["\[Psi]", "\[Alpha]"]}]}], TraditionalForm]], + FormatType->"TraditionalForm",ExpressionUUID-> + "bf7ac359-3d50-43e4-abdc-958327ecd047"], + ".\nUnlike in previous sections, we work in ", + StyleBox["Lorentzian", + FontSlant->"Italic"], + " signature here." +}], "Text", + 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parameters \[Alpha], \\!\\(\\*SubscriptBox[\\(\[Beta]\\), \ +\\(1\\)]\\), \\!\\(\\*SubscriptBox[\\(\[Beta]\\), \\(2\\)]\\), \ +\\!\\(\\*SubscriptBox[\\(\[Beta]\\), \\(3\\)]\\) and momentum magnitudes \ +\\!\\(\\*SubscriptBox[\\nStyleBox[\\\"p\\\",\\nFontSlant->\\\"Italic\\\"], \ +\\(1\\)]\\), \ +\\!\\(\\*SubscriptBox[\\nStyleBox[\\\"p\\\",\\nFontSlant->\\\"Italic\\\"], \ +\\(2\\)]\\), \ +\\!\\(\\*SubscriptBox[\\nStyleBox[\\\"p\\\",\\nFontSlant->\\\"Italic\\\"], \ +\\(3\\)]\\).\"\>"], "Print", "PrintUsage", + CellChangeTimes->{3.7990403554073143`*^9}, + CellTags-> + "Info3813799047555-5679326",ExpressionUUID->"e204aba9-04f8-4e19-8a4f-\ +544945947bdc"] +}, Open ]] +}, Open ]], + +Cell[CellGroupData[{ + +Cell[BoxData[{ + RowBox[{"?", "NL"}], "\[IndentingNewLine]", + RowBox[{"?", "\[Lambda]"}]}], "Input", + CellChangeTimes->{{3.798614330309518*^9, + 3.798614333216633*^9}},ExpressionUUID->"48e774db-2390-46b7-98a5-\ +f264f3e3d286"], + +Cell[CellGroupData[{ + 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+multiple-\\!\\(\\*\\nStyleBox[\\\"K\\\",\\nFontSlant->\\\"Italic\\\"]\\) \ +integrals in \ +\\!\\(\\*\\nStyleBox[\\\"expr\\\",\\nFontSlant->\\\"Italic\\\"]\\).\"\>"], \ +"Print", "PrintUsage", + CellChangeTimes->{3.799040357503878*^9}, + CellTags-> + "Info3953799047557-5679326",ExpressionUUID->"808d6163-c989-410d-be23-\ +9fdb621bff4a"], + +Cell[BoxData["\<\"LoopEvaluate[\\!\\(\\*\\nStyleBox[\\\"expr\\\",\\nFontSlant-\ +>\\\"Italic\\\"]\\)] evaluates all momentum space loop integrals in \\!\\(\\*\ +\\nStyleBox[\\\"expr\\\",\\nFontSlant->\\\"Italic\\\"]\\).\"\>"], "Print", \ +"PrintUsage", + CellChangeTimes->{3.799040357617899*^9}, + CellTags-> + "Info3963799047557-5679326",ExpressionUUID->"883971a9-1f60-4e51-8efc-\ +b2ce995de302"], + +Cell[BoxData["\<\"IsSolvable[\\!\\(\\*\\nStyleBox[\\\"i\\\",\\nFontSlant->\\\"\ +Italic\\\"]\\)[\[Alpha],{\\!\\(\\*SubscriptBox[\\(\[Beta]\\), \ +\\(1\\)]\\),...}]] returns True if the \ +multiple-\\!\\(\\*\\nStyleBox[\\\"K\\\",\\nFontSlant->\\\"Italic\\\"]\\) \ +integral \\!\\(\\*SubscriptBox[\\nStyleBox[\\\"i\\\",\\nFontSlant->\\\"Italic\ +\\\"], \\(\[Alpha], {\\!\\(\\*SubscriptBox[\\(\[Beta]\\), \ +\\(1\\)]\\),...}\\)]\\) is computable by \ +KEvaluate.\\nIsSolvable[\\!\\(\\*\\nStyleBox[\\\"i\\\",\\nFontSlant->\\\"\ +Italic\\\"]\\)[\[Alpha],{\\!\\(\\*SubscriptBox[\\(\[Beta]\\), \ +\\(1\\)]\\),...}][\\!\\(\\*SubscriptBox[\\(p\\), \ +\\(1\\)]\\),\\!\\(\\*SubscriptBox[\\(p\\), \\(2\\)]\\),\\!\\(\\*SubscriptBox[\ +\\(p\\), \\(3\\)]\\)]] returns True if the \ +multiple-\\!\\(\\*\\nStyleBox[\\\"K\\\",\\nFontSlant->\\\"Italic\\\"]\\) \ +integral \\!\\(\\*SubscriptBox[\\nStyleBox[\\\"i\\\",\\nFontSlant->\\\"Italic\ +\\\"], \\(\[Alpha], {\\!\\(\\*SubscriptBox[\\(\[Beta]\\), \ +\\(1\\)]\\),...}\\)]\\)[\\!\\(\\*SubscriptBox[\\(p\\), \ +\\(1\\)]\\),\\!\\(\\*SubscriptBox[\\(p\\), \\(2\\)]\\),\\!\\(\\*SubscriptBox[\ +\\(p\\), \\(3\\)]\\)] is computable by KEvaluate.\"\>"], "Print", "PrintUsage", + CellChangeTimes->{3.7990403577337523`*^9}, + CellTags-> + "Info3973799047557-5679326",ExpressionUUID->"7544fa6f-406c-47d7-a0a7-\ +2f29fadf33e0"] +}, Open ]] +}, Open ]], + +Cell[CellGroupData[{ + +Cell[BoxData[{ + RowBox[{"?", "ScalarKOp"}], "\[IndentingNewLine]", + RowBox[{"?", "ScalarKKOp"}]}], "Input", + CellChangeTimes->{{3.7986148399582376`*^9, + 3.798614848422861*^9}},ExpressionUUID->"9394e450-a885-43f9-a7db-\ +790cce449650"], + +Cell[CellGroupData[{ + +Cell[BoxData["\<\"KOp[\\!\\(\\*\\nStyleBox[\\\"expr\\\",\\nFontSlant->\\\"\ +Italic\\\"]\\), \ +\\!\\(\\*\\nStyleBox[\\\"p\\\",\\nFontSlant->\\\"Italic\\\"]\\), \[Beta]] \ +applies single K\\!\\(\\*\\nStyleBox[\\\"(\\\",\\nFontSize->12,\\nFontSlant->\ +\\\"Italic\\\"]\\)\\!\\(\\*\\nStyleBox[\\\"\[Beta]\\\",\\nFontSize->12,\\\ +nFontSlant->\\\"Italic\\\"]\\)\\!\\(\\*\\nStyleBox[\\\")\\\",\\nFontSize->12,\ +\\nFontSlant->\\\"Italic\\\"]\\)\\!\\(\\*\\nStyleBox[\\\" \ +\\\",\\nFontSize->12]\\) operator to \ +\\!\\(\\*\\nStyleBox[\\\"expr\\\",\\nFontSlant->\\\"Italic\\\"]\\) with \ +respect to momentum magnitude \\!\\(\\*\\nStyleBox[\\\"p\\\",\\nFontSlant->\\\ +\"Italic\\\"]\\) and with parameter \[Beta].\"\>"], "Print", "PrintUsage", + CellChangeTimes->{3.799040357915125*^9}, + CellTags-> + "Info3983799047557-5679326",ExpressionUUID->"e179ae8b-5df2-427b-9f83-\ +62a43f7752e3"], + +Cell[BoxData["\<\"KKOp[\\!\\(\\*\\nStyleBox[\\\"expr\\\",\\nFontSlant->\\\"\ +Italic\\\"]\\), \\!\\(\\*\\nStyleBox[SubscriptBox[\\\"p\\\", \ +\\\"i\\\"],\\nFontSlant->\\\"Italic\\\"]\\), \ +\\!\\(\\*\\nStyleBox[SubscriptBox[\\\"p\\\", \ +\\\"j\\\"],\\nFontSlant->\\\"Italic\\\"]\\), \\!\\(\\*SubscriptBox[\\(\[Beta]\ +\\), \\nStyleBox[\\\"i\\\",\\nFontSlant->\\\"Italic\\\"]]\\), \ +\\!\\(\\*SubscriptBox[\\(\[Beta]\\), \ +\\nStyleBox[\\\"j\\\",\\nFontSlant->\\\"Italic\\\"]]\\)] = \ +KOp[\\!\\(\\*\\nStyleBox[\\\"expr\\\",\\nFontSlant->\\\"Italic\\\"]\\), \ +\\!\\(\\*\\nStyleBox[SubscriptBox[\\\"p\\\", \ +\\\"i\\\"],\\nFontSlant->\\\"Italic\\\"]\\), \\!\\(\\*SubscriptBox[\\(\[Beta]\ +\\), \\nStyleBox[\\\"i\\\",\\nFontSlant->\\\"Italic\\\"]]\\)] - KOp[\\!\\(\\*\ +\\nStyleBox[\\\"expr\\\",\\nFontSlant->\\\"Italic\\\"]\\), \ +\\!\\(\\*\\nStyleBox[SubscriptBox[\\\"p\\\", \ +\\\"j\\\"],\\nFontSlant->\\\"Italic\\\"]\\), \\!\\(\\*SubscriptBox[\\(\[Beta]\ +\\), \\nStyleBox[\\\"j\\\",\\nFontSlant->\\\"Italic\\\"]]\\)] applies the \ +conformal Ward identity operator in its scalar form.\"\>"], "Print", \ +"PrintUsage", + CellChangeTimes->{3.799040358059245*^9}, + CellTags-> + "Info3993799047557-5679326",ExpressionUUID->"ed04491f-2b0b-45a0-8b55-\ +85d9b4bf7762"] +}, Open ]] +}, Open ]] +}, Open ]], + +Cell[CellGroupData[{ + +Cell["Konformal.wl", "Section", + CellChangeTimes->{{3.798856590861178*^9, + 3.798856592885206*^9}},ExpressionUUID->"b9e06ae5-4f21-4f00-82f1-\ +217420e2c630"], + +Cell[BoxData[ + RowBox[{"<<", "Konformal`"}]], "Input", + CellChangeTimes->{{3.798614922748843*^9, + 3.7986149252607*^9}},ExpressionUUID->"23569f8b-e22c-47c0-b32f-6dc5b7e9c936"], + +Cell[CellGroupData[{ + +Cell[BoxData[ + RowBox[{"?", "\[CapitalDelta]"}]], "Input", + CellChangeTimes->{{3.798614936638476*^9, + 3.7986149373696127`*^9}},ExpressionUUID->"2e8567d1-4c01-4d57-99bd-\ +8be8b27e3062"], + +Cell[BoxData["\<\"\[CapitalDelta][\\!\\(\\*\\nStyleBox[\\\"j\\\",\\nFontSlant-\ +>\\\"Italic\\\"]\\)] denotes conformal dimension of the \\!\\(\\*\\nStyleBox[\ +\\\"j\\\",\\nFontSlant->\\\"Italic\\\"]\\)-th operator involved.\"\>"], \ +"Print", "PrintUsage", + CellChangeTimes->{3.7990403583598537`*^9}, + CellTags-> + "Info4013799047558-5679326",ExpressionUUID->"c1c71373-08c9-4956-ab4f-\ +edaba7334c27"] +}, Open ]], + +Cell[CellGroupData[{ + +Cell[BoxData[{ + RowBox[{"?", "pi"}], "\[IndentingNewLine]", + RowBox[{"?", "PI"}]}], "Input", + CellChangeTimes->{{3.79861494266226*^9, + 3.7986149447969503`*^9}},ExpressionUUID->"9c3f99a8-32dd-4d97-b930-\ +733a94dda104"], + +Cell[CellGroupData[{ + +Cell[BoxData["\<\"pi[\\!\\(\\*\\nStyleBox[\\\"n\\\",\\nFontSlant->\\\"Italic\\\ +\"]\\)][\[Mu],\[Nu]] returns the transverse projector with respect to \ +momentum \\!\\(\\*SubscriptBox[\\nStyleBox[\\\"p\\\",\\nFontSlant->\\\"Italic\ +\\\"], \\nStyleBox[\\\"n\\\",\\nFontSlant->\\\"Italic\\\"]]\\).\"\>"], "Print",\ + "PrintUsage", + CellChangeTimes->{3.7990403585140963`*^9}, + CellTags-> + "Info4023799047558-5679326",ExpressionUUID->"f935334f-eee0-4f94-95ee-\ +6beabb84dc94"], + +Cell[BoxData["\<\"PI[\\!\\(\\*\\nStyleBox[\\\"d\\\",\\nFontSlant->\\\"Italic\\\ +\"]\\)][\\!\\(\\*\\nStyleBox[\\\"n\\\",\\nFontSlant->\\\"Italic\\\"]\\)][\[Mu]\ +,\[Nu],\[Rho],\[Sigma]] returns the transverse-traceless projector in \ +\\!\\(\\*\\nStyleBox[\\\"d\\\",\\nFontSlant->\\\"Italic\\\"]\\) dimensions \ +with respect to momentum \\!\\(\\*\\nStyleBox[SubscriptBox[\\\"p\\\", \\\"n\\\ +\"],\\nFontSlant->\\\"Italic\\\"]\\).\"\>"], "Print", "PrintUsage", + CellChangeTimes->{3.7990403586281033`*^9}, + CellTags-> + "Info4033799047558-5679326",ExpressionUUID->"a70999d9-a9cb-4cfb-8443-\ +67cd91ba9043"] +}, Open ]] +}, Open ]], + +Cell[CellGroupData[{ + +Cell[BoxData[{ + RowBox[{"?", "DilOp"}], "\[IndentingNewLine]", + RowBox[{"?", "SingleScalarOp"}], "\[IndentingNewLine]", + RowBox[{"?", "SingleTensorOp"}], "\[IndentingNewLine]", + RowBox[{"?", "CWIOp"}]}], "Input", + CellChangeTimes->{{3.798614959990364*^9, + 3.798614978797133*^9}},ExpressionUUID->"3a4a3dea-b4da-4b8b-bd26-\ +45655bd89af4"], + +Cell[CellGroupData[{ + +Cell[BoxData["\<\"DilOp[\\!\\(\\*\\nStyleBox[\\\"d\\\",\\nFontSlant->\\\"\ +Italic\\\"]\\), \\*\\nStyleBox[\\(\[CapitalDelta]\\!\\(\\*\\nStyleBox[\\\"t\\\ +\",\\nFontSlant->\\\"Italic\\\"]\\)\\)], \ +\\!\\(\\*\\nStyleBox[\\\"n\\\",\\nFontSlant->\\\"Italic\\\"]\\)][\\!\\(\\*\\\ +nStyleBox[\\\"expr\\\",\\nFontSlant->\\\"Italic\\\"]\\)] applies the \ +dilatation operator to \ +\\!\\(\\*\\nStyleBox[\\\"expr\\\",\\nFontSlant->\\\"Italic\\\"]\\) w.r.t. \ +momenta \\!\\(\\*\\nStyleBox[SubscriptBox[\\\"p\\\", \ +\\\"1\\\"],\\nFontSlant->\\\"Italic\\\"]\\), ..., \ +\\!\\(\\*\\nStyleBox[SubscriptBox[\\\"p\\\", \ +\\\"n\\\"],\\nFontSlant->\\\"Italic\\\"]\\) in \ +\\!\\(\\*\\nStyleBox[\\\"d\\\",\\nFontSlant->\\\"Italic\\\"]\\) dimensions \ +and with the sum of conformal dimensions equal to \\*\\nStyleBox[\\(\ +\[CapitalDelta]\\!\\(\\*\\nStyleBox[\\\"t\\\",\\nFontSlant->\\\"Italic\\\"]\\)\ +\\)].\"\>"], "Print", "PrintUsage", + CellChangeTimes->{3.7990403587873*^9}, + CellTags-> + "Info4043799047558-5679326",ExpressionUUID->"ecb60b09-31ff-4e8f-9159-\ +cc26d7c3e67c"], + +Cell[BoxData["\<\"SingleScalarOp[\\!\\(\\*\\nStyleBox[\\\"d\\\",\\nFontSlant->\ +\\\"Italic\\\"]\\), \ +\[CapitalDelta]][\\!\\(\\*\\nStyleBox[\\\"expr\\\",\\nFontSlant->\\\"Italic\\\ +\"]\\), \\!\\(\\*\\nStyleBox[\\\"n\\\",\\nFontSlant->\\\"Italic\\\"]\\), \ +\[Kappa]] applies the single CWI operator \\!\\(\\*SubsuperscriptBox[\\(K\\), \ +\\nStyleBox[\\\"n\\\",\\nFontSlant->\\\"Italic\\\"], \\(\[Kappa]\\)]\\) to \ +\\!\\(\\*\\nStyleBox[\\\"expr\\\",\\nFontSlant->\\\"Italic\\\"]\\) w.r.t. \ +momentum \\!\\(\\*\\nStyleBox[SubscriptBox[\\\"p\\\", \ +\\\"n\\\"],\\nFontSlant->\\\"Italic\\\"]\\).\"\>"], "Print", "PrintUsage", + CellChangeTimes->{3.799040358903698*^9}, + CellTags-> + "Info4053799047558-5679326",ExpressionUUID->"618c049d-027b-4b7b-8ecb-\ +b0f8bbda3949"], + +Cell[BoxData["\<\"SingleTensorOp[\\!\\(\\*\\nStyleBox[\\\"d\\\",\\nFontSlant->\ +\\\"Italic\\\"]\\)][\\!\\(\\*\\nStyleBox[\\\"expr\\\",\\nFontSlant->\\\"\ +Italic\\\"]\\), \ +\\!\\(\\*\\nStyleBox[\\\"n\\\",\\nFontSlant->\\\"Italic\\\"]\\), \[Kappa], \ +\\!\\(\\*\\nStyleBox[\\\"i\\\",\\nFontSlant->\\\"Italic\\\"]\\)] applies the \ +spinorial part of the CWI operator to \ +\\!\\(\\*\\nStyleBox[\\\"expr\\\",\\nFontSlant->\\\"Italic\\\"]\\) w.r.t. \ +momentum \\!\\(\\*\\nStyleBox[SubscriptBox[\\\"p\\\", \ +\\\"n\\\"],\\nFontSlant->\\\"Italic\\\"]\\).\"\>"], "Print", "PrintUsage", + CellChangeTimes->{3.799040359018732*^9}, + CellTags-> + "Info4063799047558-5679326",ExpressionUUID->"22381cd4-4fdf-47bb-8f45-\ +a00abb519176"], + +Cell[BoxData["\<\"CWIOp[\\!\\(\\*\\nStyleBox[\\\"d\\\",\\nFontSlant->\\\"\ +Italic\\\"]\\), {\\!\\(\\*SubscriptBox[\\(\[CapitalDelta]\\), \\(1\\)]\\), \ +...}][\\!\\(\\*\\nStyleBox[\\\"expr\\\",\\nFontSlant->\\\"Italic\\\"]\\), \ +\[Kappa]] applies the full CWI operator \\!\\(\\*SuperscriptBox[\\(K\\), \\(\ +\[Kappa]\\)]\\) to \\!\\(\\*\\nStyleBox[\\\"expr\\\",\\nFontSlant->\\\"Italic\ +\\\"]\\) in \\!\\(\\*\\nStyleBox[\\\"d\\\",\\nFontSlant->\\\"Italic\\\"]\\) \ +dimensions. \ +\\!\\(\\*\\nStyleBox[\\\"expr\\\",\\nFontSlant->\\\"Italic\\\"]\\) can be \ +thought of as an \ +(\\!\\(\\*\\nStyleBox[\\\"n\\\",\\nFontSlant->\\\"Italic\\\"]\\)+1)-point \ +function of scalar operators of conformal dimensions \ +\\!\\(\\*SubscriptBox[\\(\[CapitalDelta]\\), \\(1\\)]\\), ..., \ +\\!\\(\\*SubscriptBox[\\(\[CapitalDelta]\\), \ +\\nStyleBox[\\\"n\\\",\\nFontSlant->\\\"Italic\\\"]]\\).\\nCWIOp[\\!\\(\\*\\\ +nStyleBox[\\\"d\\\",\\nFontSlant->\\\"Italic\\\"]\\), {\\!\\(\\*SubscriptBox[\ +\\(\[CapitalDelta]\\), \\(1\\)]\\), \ +...}][\\!\\(\\*\\nStyleBox[\\\"expr\\\",\\nFontSlant->\\\"Italic\\\"]\\), \ +\[Kappa], {{\\!\\(\\*SubscriptBox[\\(\[Mu]\\), \\(11\\)]\\), ..., \ +\\!\\(\\*SubscriptBox[\\(\[Mu]\\), \\(1 \\*SubscriptBox[\\(m\\), \\(1\\)]\\)]\ +\\)}, ...}] takes into consideration tensor structure determined by \ +{{\\!\\(\\*SubscriptBox[\\(\[Mu]\\), \\(11\\)]\\), ..., \ +\\!\\(\\*SubscriptBox[\\(\[Mu]\\), \\(1 \\*SubscriptBox[\\(m\\), \\(1\\)]\\)]\ +\\)}, ...}. \ +\\!\\(\\*\\nStyleBox[\\\"expr\\\",\\nFontSlant->\\\"Italic\\\"]\\) can be \ +thought of as an \ +(\\!\\(\\*\\nStyleBox[\\\"n\\\",\\nFontSlant->\\\"Italic\\\"]\\)+1)-point \ +function of tensorial operators, each with indices \\!\\(\\*SubscriptBox[\\(\ +\[Mu]\\), \\(j1\\)]\\),...,\\!\\(\\*SubscriptBox[\\(\[Mu]\\), \ +SubscriptBox[\\(jm\\), \\(j\\)]]\\).\"\>"], "Print", "PrintUsage", + CellChangeTimes->{3.799040359132105*^9}, + CellTags-> + "Info4073799047559-5679326",ExpressionUUID->"54a637d6-8db5-4a32-bf4b-\ +9ff033026c14"] +}, Open ]] +}, Open ]], + +Cell[CellGroupData[{ + +Cell[BoxData[{ + RowBox[{"?", "PrimaryCWIs"}], "\[IndentingNewLine]", + RowBox[{"?", "SecondaryCWIsLhs"}], "\[IndentingNewLine]", + RowBox[{"?", "SecondaryCWIsRhs"}], "\[IndentingNewLine]", + RowBox[{"?", "TransverseWIs"}], "\[IndentingNewLine]", + RowBox[{"?", "PrimarySolutions"}]}], "Input", + CellChangeTimes->{{3.798615093150078*^9, + 3.7986151183652487`*^9}},ExpressionUUID->"2fe44eb3-fde1-4c4e-bdcd-\ +f4fbaff0ccab"], + +Cell[CellGroupData[{ + +Cell[BoxData["\<\"PrimaryCWIs[\\!\\(\\*\\nStyleBox[\\\"X\\\",\\nFontSlant->\\\ +\"Italic\\\"]\\)] lists primary CWIs for the correlation function denoted by \ +\\!\\(\\*\\nStyleBox[\\\"X\\\",\\nFontSlant->\\\"Italic\\\"]\\).\"\>"], \ +"Print", "PrintUsage", + CellChangeTimes->{3.799040359338141*^9}, + CellTags-> + "Info4083799047559-5679326",ExpressionUUID->"210faabf-35e9-40ba-bd8c-\ +dcd988ac268a"], + +Cell[BoxData["\<\"SecondaryCWIsLhs[\\!\\(\\*\\nStyleBox[\\\"X\\\",\\\ +nFontSlant->\\\"Italic\\\"]\\)] lists left hand sides of secondary CWIs for \ +the correlation function denoted by \ +\\!\\(\\*\\nStyleBox[\\\"X\\\",\\nFontSlant->\\\"Italic\\\"]\\).\"\>"], \ +"Print", "PrintUsage", + CellChangeTimes->{3.799040359453313*^9}, + CellTags-> + "Info4093799047559-5679326",ExpressionUUID->"b3c11391-8063-4d22-bd1d-\ +bf2a24ed5170"], + +Cell[BoxData["\<\"SecondaryCWIsRhs[\\!\\(\\*\\nStyleBox[\\\"X\\\",\\\ +nFontSlant->\\\"Italic\\\"]\\)] lists right hand sides of secondary CWIs for \ +the correlation function denoted by \ +\\!\\(\\*\\nStyleBox[\\\"X\\\",\\nFontSlant->\\\"Italic\\\"]\\).\"\>"], \ +"Print", "PrintUsage", + CellChangeTimes->{3.7990403595967627`*^9}, + CellTags-> + "Info4103799047559-5679326",ExpressionUUID->"0c3e70bc-991d-40a0-aaec-\ +7a2faa468921"], + +Cell[BoxData["\<\"TransverseWIs[\\!\\(\\*\\nStyleBox[\\\"X\\\",\\nFontSlant->\ +\\\"Italic\\\"]\\)] lists transverse WIs for the correlation function denoted \ +by \\!\\(\\*\\nStyleBox[\\\"X\\\",\\nFontSlant->\\\"Italic\\\"]\\) in a \ +generic case.\"\>"], "Print", "PrintUsage", + CellChangeTimes->{3.79904035971213*^9}, + CellTags-> + "Info4113799047559-5679326",ExpressionUUID->"4d900e49-c25b-4bc2-b5c5-\ +f0600651cfdb"], + +Cell[BoxData["\<\"PrimarySolutions[\\!\\(\\*\\nStyleBox[\\\"X\\\",\\\ +nFontSlant->\\\"Italic\\\"]\\)] lists form factors solving primary WIs for \ +the correlation function denoted by \ +\\!\\(\\*\\nStyleBox[\\\"X\\\",\\nFontSlant->\\\"Italic\\\"]\\).\"\>"], \ +"Print", "PrintUsage", + CellChangeTimes->{3.799040359854439*^9}, + CellTags-> + "Info4123799047559-5679326",ExpressionUUID->"df5d03c0-0b9b-4e82-8819-\ +c488f22d9fbc"] +}, Open ]] +}, Open ]], + +Cell[CellGroupData[{ + +Cell[BoxData[{ + RowBox[{"?", "OOO"}], "\[IndentingNewLine]", + RowBox[{"?", "JOO"}], "\[IndentingNewLine]", + RowBox[{"?", "TOO"}], "\[IndentingNewLine]", + RowBox[{"?", "JJO"}], "\[IndentingNewLine]", + RowBox[{"?", "TJO"}], "\[IndentingNewLine]", + RowBox[{"?", "TTO"}], "\[IndentingNewLine]", + RowBox[{"?", "JJJ"}], "\[IndentingNewLine]", + RowBox[{"?", "TJJ"}], "\[IndentingNewLine]", + RowBox[{"?", "TTJ"}], "\[IndentingNewLine]", + RowBox[{"?", "TTT"}]}], "Input", + CellChangeTimes->{{3.7986151579504547`*^9, + 3.798615182804932*^9}},ExpressionUUID->"30f9b4a7-39e6-42d2-a9dc-\ +5c712cbcea5e"], + +Cell[CellGroupData[{ + +Cell[BoxData["\<\"Index for .\"\>"], "Print", "PrintUsage", + CellChangeTimes->{3.799040360012062*^9}, + CellTags-> + "Info4133799047559-5679326",ExpressionUUID->"9023d9b7-d459-47a7-93b3-\ +2d25fef4f6cb"], + +Cell[BoxData["\<\"Index for .\"\>"], "Print", "PrintUsage", + CellChangeTimes->{3.799040360129002*^9}, + CellTags-> + "Info4143799047560-5679326",ExpressionUUID->"671e2703-7045-4e73-ae7b-\ +055a813a0b9d"], + +Cell[BoxData["\<\"Index for .\"\>"], "Print", "PrintUsage", + CellChangeTimes->{3.799040360269367*^9}, + CellTags-> + "Info4153799047560-5679326",ExpressionUUID->"cc9156bd-c88a-446a-a6d7-\ +8e85cc3698f4"], + +Cell[BoxData["\<\"Index for .\"\>"], "Print", "PrintUsage", + CellChangeTimes->{3.799040360412277*^9}, + CellTags-> + "Info4163799047560-5679326",ExpressionUUID->"347dd1f7-530d-4b13-9289-\ +cb50010c0fad"], + +Cell[BoxData["\<\"Index for .\"\>"], "Print", "PrintUsage", + CellChangeTimes->{3.799040360521016*^9}, + CellTags-> + "Info4173799047560-5679326",ExpressionUUID->"125fd7a3-541c-4df4-a1c1-\ +3aefe00ab4bb"], + +Cell[BoxData["\<\"Index for .\"\>"], "Print", "PrintUsage", + CellChangeTimes->{3.799040360629983*^9}, + CellTags-> + "Info4183799047560-5679326",ExpressionUUID->"1010901d-121d-45b4-bd89-\ +d9c237b1b6f3"], + +Cell[BoxData["\<\"Index for .\"\>"], "Print", "PrintUsage", + CellChangeTimes->{3.799040360738996*^9}, + CellTags-> + "Info4193799047560-5679326",ExpressionUUID->"53bdf005-7af6-47b8-be5e-\ +dcb054b5912f"], + +Cell[BoxData["\<\"Index for .\"\>"], "Print", "PrintUsage", + CellChangeTimes->{3.799040360849009*^9}, + CellTags-> + "Info4203799047560-5679326",ExpressionUUID->"a8de0d74-ae39-443c-b002-\ +4951302c831a"], + +Cell[BoxData["\<\"Index for .\"\>"], "Print", "PrintUsage", + CellChangeTimes->{3.7990403609593678`*^9}, + CellTags-> + "Info4213799047560-5679326",ExpressionUUID->"7655c1b9-93ee-455e-aaf3-\ +41df79b9a57f"], + +Cell[BoxData["\<\"Index for .\"\>"], "Print", "PrintUsage", + CellChangeTimes->{3.799040361070424*^9}, + CellTags-> + "Info4223799047560-5679326",ExpressionUUID->"71675991-1343-4cbc-ad0d-\ +15c612acbd10"] +}, Open ]] +}, Open ]], + +Cell[CellGroupData[{ + +Cell[BoxData[{ + RowBox[{"?", "J"}], "\[IndentingNewLine]", + RowBox[{"?", "suJToI"}], "\[IndentingNewLine]", + RowBox[{"?", "JToI"}]}], "Input", + CellChangeTimes->{{3.798615190886602*^9, 3.798615194244933*^9}, { + 3.798856526717352*^9, + 3.798856527956995*^9}},ExpressionUUID->"f71120d5-1336-47fb-b834-\ +189fa26fe3a2"], + +Cell[CellGroupData[{ + +Cell[BoxData["\<\"J[\\!\\(\\*\\nStyleBox[\\\"N\\\",\\nFontSlant->\\\"Italic\\\ +\"]\\), {\\!\\(\\*SubscriptBox[\\nStyleBox[\\\"k\\\",\\nFontSlant->\\\"Italic\ +\\\"], \\(1\\)]\\), \ +\\!\\(\\*SubscriptBox[\\nStyleBox[\\\"k\\\",\\nFontSlant->\\\"Italic\\\"], \ +\\(2\\)]\\), \ +\\!\\(\\*SubscriptBox[\\nStyleBox[\\\"k\\\",\\nFontSlant->\\\"Italic\\\"], \ +\\(3\\)]\\)}] denotes the \ +triple-\\!\\(\\*\\nStyleBox[\\\"K\\\",\\nFontSlant->\\\"Italic\\\"]\\) \ +integral in the reduced form.\"\>"], "Print", "PrintUsage", + CellChangeTimes->{3.799040361252961*^9}, + CellTags-> + "Info4233799047561-5679326",ExpressionUUID->"671128bd-a1fd-47da-9674-\ +0712d3bafd95"], + +Cell[BoxData["\<\"suJToI[\\!\\(\\*\\nStyleBox[\\\"d\\\",\\nFontSlant->\\\"\ +Italic\\\"]\\), {\\!\\(\\*SubscriptBox[\\(\[CapitalDelta]\\), \\(1\\)]\\), \ +\\!\\(\\*SubscriptBox[\\(\[CapitalDelta]\\), \\(2\\)]\\), \ +\\!\\(\\*SubscriptBox[\\(\[CapitalDelta]\\), \\(3\\)]\\)}] is a substitution \ +list for expressing \\!\\(\\*\\nStyleBox[\\\"J\\\",\\nFontSlant->\\\"Italic\\\ +\"]\\)-integrals as \ +triple-\\!\\(\\*\\nStyleBox[\\\"K\\\",\\nFontSlant->\\\"Italic\\\"]\\) \ +integrals.\"\>"], "Print", "PrintUsage", + CellChangeTimes->{3.799040361369622*^9}, + CellTags-> + "Info4243799047561-5679326",ExpressionUUID->"8f505254-846f-4cf0-8f99-\ +ac3e63bb6f3f"], + +Cell[BoxData["\<\"JToI[d,{\\!\\(\\*SubscriptBox[\\(\[CapitalDelta]\\), \ +\\(1\\)]\\),\\!\\(\\*SubscriptBox[\\(\[CapitalDelta]\\), \ +\\(2\\)]\\),\\!\\(\\*SubscriptBox[\\(\[CapitalDelta]\\), \\(3\\)]\\)}][\\!\\(\ +\\*\\nStyleBox[\\\"expr\\\",\\nFontSlant->\\\"Italic\\\"]\\)] replaces all \ +\\!\\(\\*\\nStyleBox[\\\"J\\\",\\nFontSlant->\\\"Italic\\\"]\\)-integrals in \ +\\!\\(\\*\\nStyleBox[\\\"expr\\\",\\nFontSlant->\\\"Italic\\\"]\\) with \ +triple-\\!\\(\\*\\nStyleBox[\\\"K\\\",\\nFontSlant->\\\"Italic\\\"]\\) \ +intgerals.\"\>"], "Print", "PrintUsage", + CellChangeTimes->{3.7990403615169287`*^9}, + CellTags-> + "Info4253799047561-5679326",ExpressionUUID->"a1926c93-8c2e-4417-9da2-\ +4b455758a99f"] +}, Open ]] +}, Open ]], + +Cell[CellGroupData[{ + +Cell[BoxData[{ + RowBox[{"?", "KOp"}], "\[IndentingNewLine]", + RowBox[{"?", "KKOp"}]}], "Input", + CellChangeTimes->{{3.7986152107106657`*^9, + 3.798615217060775*^9}},ExpressionUUID->"18b337ef-900a-4b43-bafc-\ +66e11f94edb1"], + +Cell[CellGroupData[{ + +Cell[BoxData["\<\"KOp[\\!\\(\\*\\nStyleBox[\\\"expr\\\",\\nFontSlant->\\\"\ +Italic\\\"]\\), \ +\\!\\(\\*\\nStyleBox[\\\"p\\\",\\nFontSlant->\\\"Italic\\\"]\\), \[Beta]] \ +applies single K operator to \\!\\(\\*\\nStyleBox[\\\"expr\\\",\\nFontSlant->\ +\\\"Italic\\\"]\\) with respect to momentum magnitude \\!\\(\\*\\nStyleBox[\\\ +\"p\\\",\\nFontSlant->\\\"Italic\\\"]\\) and with parameter \ +\[Beta].\\nKOp[\\!\\(\\*\\nStyleBox[\\\"expr\\\",\\nFontSlant->\\\"Italic\\\"]\ +\\), \\!\\(\\*\\nStyleBox[\\\"n\\\",\\nFontSlant->\\\"Italic\\\"]\\)] applies \ +KOp to \\!\\(\\*\\nStyleBox[\\\"expr\\\",\\nFontSlant->\\\"Italic\\\"]\\) \ +with respect to momentum \\!\\(\\*\\nStyleBox[SubscriptBox[\\\"p\\\", \\\"n\\\ +\"],\\nFontSlant->\\\"Italic\\\"]\\) and with symbolic parameter \ +\[Beta]=\\!\\(\\*SubscriptBox[\\(\[CapitalDelta]\\), \ +\\nStyleBox[\\\"n\\\",\\nFontSlant->\\\"Italic\\\"]]\\)-\\!\\(\\*FractionBox[\ +\\nStyleBox[\\\"d\\\",\\nFontSlant->\\\"Italic\\\"], \\(2\\)]\\).\"\>"], \ +"Print", "PrintUsage", + CellChangeTimes->{3.799040361672214*^9}, + CellTags-> + "Info4263799047561-5679326",ExpressionUUID->"73194cd3-0741-4f30-b4e8-\ +12e57fce6895"], + +Cell[BoxData["\<\"KKOp[\\!\\(\\*\\nStyleBox[\\\"expr\\\",\\nFontSlant->\\\"\ +Italic\\\"]\\), \\!\\(\\*\\nStyleBox[SubscriptBox[\\\"p\\\", \ +\\\"i\\\"],\\nFontSlant->\\\"Italic\\\"]\\), \ +\\!\\(\\*\\nStyleBox[SubscriptBox[\\\"p\\\", \ +\\\"j\\\"],\\nFontSlant->\\\"Italic\\\"]\\), \\!\\(\\*SubscriptBox[\\(\[Beta]\ +\\), \\nStyleBox[\\\"i\\\",\\nFontSlant->\\\"Italic\\\"]]\\), \ +\\!\\(\\*SubscriptBox[\\(\[Beta]\\), \ +\\nStyleBox[\\\"j\\\",\\nFontSlant->\\\"Italic\\\"]]\\)] = \ +KOp[\\!\\(\\*\\nStyleBox[\\\"expr\\\",\\nFontSlant->\\\"Italic\\\"]\\), \ +\\!\\(\\*\\nStyleBox[SubscriptBox[\\\"p\\\", \ +\\\"i\\\"],\\nFontSlant->\\\"Italic\\\"]\\), \\!\\(\\*SubscriptBox[\\(\[Beta]\ +\\), \\nStyleBox[\\\"i\\\",\\nFontSlant->\\\"Italic\\\"]]\\)] - KOp[\\!\\(\\*\ +\\nStyleBox[\\\"expr\\\",\\nFontSlant->\\\"Italic\\\"]\\), \ +\\!\\(\\*\\nStyleBox[SubscriptBox[\\\"p\\\", \ +\\\"j\\\"],\\nFontSlant->\\\"Italic\\\"]\\), \\!\\(\\*SubscriptBox[\\(\[Beta]\ +\\), \\nStyleBox[\\\"j\\\",\\nFontSlant->\\\"Italic\\\"]]\\)] applies the \ +conformal Ward identity operator in its scalar \ +form.\\nKKOp[\\!\\(\\*\\nStyleBox[\\\"expr\\\",\\nFontSlant->\\\"Italic\\\"]\\\ +), \\!\\(\\*\\nStyleBox[\\\"m\\\",\\nFontSlant->\\\"Italic\\\"]\\), \\!\\(\\*\ +\\nStyleBox[\\\"n\\\",\\nFontSlant->\\\"Italic\\\"]\\)] applies KKOp to \ +\\!\\(\\*\\nStyleBox[\\\"expr\\\",\\nFontSlant->\\\"Italic\\\"]\\) with \ +respect to momenta \\!\\(\\*\\nStyleBox[SubscriptBox[\\\"p\\\", \ +\\\"m\\\"],\\nFontSlant->\\\"Italic\\\"]\\)\\!\\(\\*\\nStyleBox[\\\",\\\",\\\ +nFontSlant->\\\"Italic\\\"]\\)\\!\\(\\*\\nStyleBox[\\\" \\\",\\nFontSlant->\\\ +\"Italic\\\"]\\)\\!\\(\\*\\nStyleBox[SubscriptBox[\\nStyleBox[\\\"p\\\",\\\ +nFontSlant->\\\"Italic\\\"], \\\"n\\\"],\\nFontSlant->\\\"Italic\\\"]\\) and \ +with symbolic parameters \\!\\(\\*SubscriptBox[\\(\[Beta]\\), \\(m\\)]\\) = \ +\\!\\(\\*SubscriptBox[\\(\[CapitalDelta]\\), \ +\\nStyleBox[\\\"m\\\",\\nFontSlant->\\\"Italic\\\"]]\\)-\\!\\(\\*FractionBox[\ +\\nStyleBox[\\\"d\\\",\\nFontSlant->\\\"Italic\\\"], \\(2\\)]\\), \ +\\!\\(\\*SubscriptBox[\\(\[Beta]\\), \\(n\\)]\\) = \\!\\(\\*SubscriptBox[\\(\ +\[CapitalDelta]\\), \ +\\nStyleBox[\\\"n\\\",\\nFontSlant->\\\"Italic\\\"]]\\)-\\!\\(\\*FractionBox[\ +\\nStyleBox[\\\"d\\\",\\nFontSlant->\\\"Italic\\\"], \\(2\\)]\\).\"\>"], \ +"Print", "PrintUsage", + CellChangeTimes->{3.7990403617870817`*^9}, + CellTags-> + "Info4273799047561-5679326",ExpressionUUID->"4be61b5f-d5d0-48ca-b983-\ +a7bdde42aad6"] +}, Open ]] +}, Open ]], + +Cell[CellGroupData[{ + +Cell[BoxData[{ + RowBox[{"?", "LOp"}], "\[IndentingNewLine]", + RowBox[{"?", "LprimeOp"}]}], "Input", + CellChangeTimes->{{3.798615399887067*^9, + 3.798615406317271*^9}},ExpressionUUID->"72b516d0-5f15-48fa-b64c-\ +575d3becb0e1"], + +Cell[CellGroupData[{ + +Cell[BoxData["\<\"LOp[\\!\\(\\*\\nStyleBox[\\\"d\\\",\\nFontSlant->\\\"Italic\ +\\\"]\\), {\\!\\(\\*SubscriptBox[\\(\[CapitalDelta]\\), \ +\\(1\\)]\\),\\!\\(\\*SubscriptBox[\\(\[CapitalDelta]\\), \\(2\\)]\\)}][\\!\\(\ +\\*\\nStyleBox[\\\"expr\\\",\\nFontSlant->\\\"Italic\\\"]\\), \ +\\!\\(\\*\\nStyleBox[\\\"s\\\",\\nFontSlant->\\\"Italic\\\"]\\), \ +\\!\\(\\*\\nStyleBox[\\\"N\\\",\\nFontSlant->\\\"Italic\\\"]\\)] applies the \ +secondary \\!\\(\\*\\nStyleBox[SubscriptBox[\\\"L\\\", \\nRowBox[{\\\"s\\\", \ +\\\",\\\", \ +\\nStyleBox[\\\"N\\\",\\nFontSlant->\\\"Italic\\\"]}]],\\nFontSlant->\\\"\ +Italic\\\"]\\) operator to \\!\\(\\*\\nStyleBox[\\\"expr\\\",\\nFontSlant->\\\ +\"Italic\\\"]\\). \ +\\nLOp[\\!\\(\\*\\nStyleBox[\\\"expr\\\",\\nFontSlant->\\\"Italic\\\"]\\), \ +\\!\\(\\*\\nStyleBox[\\\"s\\\",\\nFontSlant->\\\"Italic\\\"]\\), \ +\\!\\(\\*\\nStyleBox[\\\"N\\\",\\nFontSlant->\\\"Italic\\\"]\\)] uses \ +symbolic parameters \\!\\(\\*\\nStyleBox[\\\"d\\\",\\nFontSlant->\\\"Italic\\\ +\"]\\), \\!\\(\\*SubscriptBox[\\(\[CapitalDelta]\\), \\(1\\)]\\), \ +\\!\\(\\*SubscriptBox[\\(\[CapitalDelta]\\), \\(2\\)]\\).\"\>"], "Print", \ +"PrintUsage", + CellChangeTimes->{3.7990403619529333`*^9}, + CellTags-> + "Info4283799047561-5679326",ExpressionUUID->"a0970563-18fa-4226-a72c-\ +32b98040d3f0"], + +Cell[BoxData["\<\"LprimeOp[\\!\\(\\*\\nStyleBox[\\\"d\\\",\\nFontSlant->\\\"\ +Italic\\\"]\\), {\\!\\(\\*SubscriptBox[\\(\[CapitalDelta]\\), \\(1\\)]\\),\\!\ +\\(\\*SubscriptBox[\\(\[CapitalDelta]\\), \\(2\\)]\\)}][\\!\\(\\*\\nStyleBox[\ +\\\"expr\\\",\\nFontSlant->\\\"Italic\\\"]\\), \ +\\!\\(\\*\\nStyleBox[\\\"s\\\",\\nFontSlant->\\\"Italic\\\"]\\), \ +\\!\\(\\*\\nStyleBox[\\\"N\\\",\\nFontSlant->\\\"Italic\\\"]\\)] applies the \ +secondary \ +\\!\\(\\*\\nStyleBox[\\\"L\\\",\\nFontSlant->\\\"Italic\\\"]\\)\\!\\(\\*\\\ +nStyleBox[SubscriptBox[\\\"'\\\", \\nRowBox[{\\\"s\\\", \\\",\\\", \ +\\\"N\\\"}]],\\nFontSlant->\\\"Italic\\\"]\\) operator to \ +\\!\\(\\*\\nStyleBox[\\\"expr\\\",\\nFontSlant->\\\"Italic\\\"]\\).\\\ +nLprimeOp[\\!\\(\\*\\nStyleBox[\\\"expr\\\",\\nFontSlant->\\\"Italic\\\"]\\), \ +\\!\\(\\*\\nStyleBox[\\\"s\\\",\\nFontSlant->\\\"Italic\\\"]\\), \ +\\!\\(\\*\\nStyleBox[\\\"N\\\",\\nFontSlant->\\\"Italic\\\"]\\)] uses \ +symbolic parameters \\!\\(\\*\\nStyleBox[\\\"d\\\",\\nFontSlant->\\\"Italic\\\ +\"]\\), \\!\\(\\*SubscriptBox[\\(\[CapitalDelta]\\), \\(1\\)]\\), \ +\\!\\(\\*SubscriptBox[\\(\[CapitalDelta]\\), \\(2\\)]\\).\"\>"], "Print", \ +"PrintUsage", + CellChangeTimes->{3.799040362066496*^9}, + CellTags-> + "Info4293799047561-5679326",ExpressionUUID->"5cd2f9a7-0597-47dd-ac8b-\ +55869c214e00"] +}, Open ]] +}, Open ]], + +Cell[CellGroupData[{ + +Cell[BoxData[{ + RowBox[{"?", "ROp"}], "\[IndentingNewLine]", + RowBox[{"?", "RprimeOp"}]}], "Input", + CellChangeTimes->{{3.798615421853623*^9, + 3.798615423092942*^9}},ExpressionUUID->"80797909-3e53-4bb5-ac39-\ +11127d36ba96"], + +Cell[CellGroupData[{ + +Cell[BoxData["\<\"ROp[\\!\\(\\*SubscriptBox[\\(\[CapitalDelta]\\), \ +\\(1\\)]\\)][\\!\\(\\*\\nStyleBox[\\\"expr\\\",\\nFontSlant->\\\"Italic\\\"]\\\ +), \\!\\(\\*\\nStyleBox[\\\"s\\\",\\nFontSlant->\\\"Italic\\\"]\\), \\!\\(\\*\ +\\nStyleBox[\\\"N\\\",\\nFontSlant->\\\"Italic\\\"]\\)] applies the secondary \ +\\!\\(\\*\\nStyleBox[SubscriptBox[\\\"R\\\", \ +\\\"s\\\"],\\nFontSlant->\\\"Italic\\\"]\\) operator to \\!\\(\\*\\nStyleBox[\ +\\\"expr\\\",\\nFontSlant->\\\"Italic\\\"]\\). \\nROp[\\!\\(\\*\\nStyleBox[\\\ +\"expr\\\",\\nFontSlant->\\\"Italic\\\"]\\), \ +\\!\\(\\*\\nStyleBox[\\\"s\\\",\\nFontSlant->\\\"Italic\\\"]\\)] uses \ +symbolic parameter \\!\\(\\*SubscriptBox[\\(\[CapitalDelta]\\), \ +\\(1\\)]\\).\"\>"], "Print", "PrintUsage", + CellChangeTimes->{3.799040362222597*^9}, + CellTags-> + "Info4303799047562-5679326",ExpressionUUID->"4db55b29-e906-4cd9-ac5f-\ +929efd5cb441"], + +Cell[BoxData["\<\"RprimeOp[\\!\\(\\*SubscriptBox[\\(\[CapitalDelta]\\), \ +\\(2\\)]\\)][\\!\\(\\*\\nStyleBox[\\\"expr\\\",\\nFontSlant->\\\"Italic\\\"]\\\ +), \\!\\(\\*\\nStyleBox[\\\"s\\\",\\nFontSlant->\\\"Italic\\\"]\\), \\!\\(\\*\ +\\nStyleBox[\\\"N\\\",\\nFontSlant->\\\"Italic\\\"]\\)] applies the secondary \ +\\!\\(\\*\\nStyleBox[\\\"R\\\",\\nFontSlant->\\\"Italic\\\"]\\)\\!\\(\\*\\\ +nStyleBox[SubscriptBox[\\\"'\\\", \\\"s\\\"],\\nFontSlant->\\\"Italic\\\"]\\) \ +operator to \ +\\!\\(\\*\\nStyleBox[\\\"expr\\\",\\nFontSlant->\\\"Italic\\\"]\\).\\\ +nRprimeOp[\\!\\(\\*\\nStyleBox[\\\"expr\\\",\\nFontSlant->\\\"Italic\\\"]\\), \ +\\!\\(\\*\\nStyleBox[\\\"s\\\",\\nFontSlant->\\\"Italic\\\"]\\)] uses \ +symbolic parameter \\!\\(\\*SubscriptBox[\\(\[CapitalDelta]\\), \ +\\(2\\)]\\).\"\>"], "Print", "PrintUsage", + CellChangeTimes->{3.799040362339521*^9}, + CellTags-> + "Info4313799047562-5679326",ExpressionUUID->"6d10926e-04e7-46c8-8c08-\ +50d8f94b69ff"] +}, Open ]] +}, Open ]], + +Cell[CellGroupData[{ + +Cell[BoxData[ + RowBox[{"?", "suExpandAs"}]], "Input", + CellChangeTimes->{{3.7986154340698013`*^9, + 3.798615437293498*^9}},ExpressionUUID->"dde0f175-f750-4499-adcf-\ +8324a19c4f42"], + +Cell[BoxData["\<\"suExpandAs[\\!\\(\\*\\nStyleBox[\\\"A\\\",\\nFontSlant->\\\"\ +Italic\\\"]\\)] is a substitution list for reapplying standard momenta \\!\\(\ +\\*SubscriptBox[\\nStyleBox[\\\"p\\\",\\nFontSlant->\\\"Italic\\\"], \\(1\\)]\ +\\),\\!\\(\\*SubscriptBox[\\nStyleBox[\\\"p\\\",\\nFontSlant->\\\"Italic\\\"],\ + \\(2\\)]\\),\\!\\(\\*SubscriptBox[\\nStyleBox[\\\"p\\\",\\nFontSlant->\\\"\ +Italic\\\"], \\(3\\)]\\) to form factors \\!\\(\\*\\nStyleBox[SubscriptBox[\\\ +\"A\\\", \\\"j\\\"],\\nFontSlant->\\\"Italic\\\"]\\)\\!\\(\\*\\nStyleBox[\\\" \ +\\\",\\nFontSlant->\\\"Italic\\\"]\\)under CWI operators.\"\>"], "Print", \ +"PrintUsage", + CellChangeTimes->{3.799040362524623*^9}, + CellTags-> + "Info4323799047562-5679326",ExpressionUUID->"4c4b535b-ab93-47e1-93d2-\ +39e471e86ef7"] +}, Open ]] +}, Open ]] +}, +WindowSize->{984, 945}, +WindowMargins->{{Automatic, 75}, {45, Automatic}}, +Magnification:>1.25 Inherited, +FrontEndVersion->"11.2 for Linux x86 (64-bit) (September 10, 2017)", +StyleDefinitions->"Default.nb" +] +(* End of Notebook Content *) + +(* Internal cache information *) +(*CellTagsOutline +CellTagsIndex->{ + "Info3763799047554-5679326"->{ + Cell[1847, 58, 1198, 19, 95, 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:Name: Konformal` *) + + +(* :Title: Repository of results on conformal 3-point functions. *) + + +(* :Author: Adam Bzowski *) + + +(* :Summary: + This package provides basic tools for manipulation of conformal correlation functions. + The package serves as a repository of results on conformal 3-point functions and conformal Ward identities. +*) + + +(* :Context: Konformal` *) + + +(* :Package Version: 1.0 *) + + +(* :History: + Version 1.0 by Adam Bzowski, May 2020. +*) + + +(* :Copyright: GNU General Public License v3.0, Adam Bzowski, 2020 *) + + +(* :Keywords: + Conformal field theory + Conformal Ward identities + Dimensional regularization +*) + + +(* :Mathematica Version: 11.2 *) + + +BeginPackage["Konformal`", "Global`"]; +Unprotect @@ Names["Konformal`*"]; +ClearAll @@ Names["Konformal`*"]; +Get[FileNameJoin[{NotebookDirectory[], "TripleK.wl"}]]; + + +(* ::Section:: *) +(*Interface*) + + +\[CapitalDelta]::usage = "\[CapitalDelta][\!\(\* +StyleBox[\"j\",\nFontSlant->\"Italic\"]\)] denotes conformal dimension of the \!\(\* +StyleBox[\"j\",\nFontSlant->\"Italic\"]\)-th operator involved."; + + +pi::usage = "pi[\!\(\* +StyleBox[\"n\",\nFontSlant->\"Italic\"]\)][\[Mu],\[Nu]] returns the transverse projector with respect to momentum \!\(\*SubscriptBox[ +StyleBox[\"p\",\nFontSlant->\"Italic\"], +StyleBox[\"n\",\nFontSlant->\"Italic\"]]\)."; +PI::usage = "PI[\!\(\* +StyleBox[\"d\",\nFontSlant->\"Italic\"]\)][\!\(\* +StyleBox[\"n\",\nFontSlant->\"Italic\"]\)][\[Mu],\[Nu],\[Rho],\[Sigma]] returns the transverse-traceless projector in \!\(\* +StyleBox[\"d\",\nFontSlant->\"Italic\"]\) dimensions with respect to momentum \!\(\* +StyleBox[SubscriptBox[\"p\", \"n\"],\nFontSlant->\"Italic\"]\)."; + + +DilOp::usage = "DilOp[\!\(\* +StyleBox[\"d\",\nFontSlant->\"Italic\"]\), \* +StyleBox[\(\[CapitalDelta]\!\(\* +StyleBox[\"t\",\nFontSlant->\"Italic\"]\)\)], \!\(\* +StyleBox[\"n\",\nFontSlant->\"Italic\"]\)][\!\(\* +StyleBox[\"expr\",\nFontSlant->\"Italic\"]\)] applies the dilatation operator to \!\(\* +StyleBox[\"expr\",\nFontSlant->\"Italic\"]\) w.r.t. momenta \!\(\* +StyleBox[SubscriptBox[\"p\", \"1\"],\nFontSlant->\"Italic\"]\), ..., \!\(\* +StyleBox[SubscriptBox[\"p\", \"n\"],\nFontSlant->\"Italic\"]\) in \!\(\* +StyleBox[\"d\",\nFontSlant->\"Italic\"]\) dimensions and with the sum of conformal dimensions equal to \* +StyleBox[\(\[CapitalDelta]\!\(\* +StyleBox[\"t\",\nFontSlant->\"Italic\"]\)\)]."; +SingleScalarOp::usage = "SingleScalarOp[\!\(\* +StyleBox[\"d\",\nFontSlant->\"Italic\"]\), \[CapitalDelta]][\!\(\* +StyleBox[\"expr\",\nFontSlant->\"Italic\"]\), \!\(\* +StyleBox[\"n\",\nFontSlant->\"Italic\"]\), \[Kappa]] applies the single CWI operator \!\(\*SubsuperscriptBox[\(K\), +StyleBox[\"n\",\nFontSlant->\"Italic\"], \(\[Kappa]\)]\) to \!\(\* +StyleBox[\"expr\",\nFontSlant->\"Italic\"]\) w.r.t. momentum \!\(\* +StyleBox[SubscriptBox[\"p\", \"n\"],\nFontSlant->\"Italic\"]\)."; +SingleTensorOp::usage = "SingleTensorOp[\!\(\* +StyleBox[\"d\",\nFontSlant->\"Italic\"]\)][\!\(\* +StyleBox[\"expr\",\nFontSlant->\"Italic\"]\), \!\(\* +StyleBox[\"n\",\nFontSlant->\"Italic\"]\), \[Kappa], \!\(\* +StyleBox[\"i\",\nFontSlant->\"Italic\"]\)] applies the spinorial part of the CWI operator to \!\(\* +StyleBox[\"expr\",\nFontSlant->\"Italic\"]\) w.r.t. momentum \!\(\* +StyleBox[SubscriptBox[\"p\", \"n\"],\nFontSlant->\"Italic\"]\)."; +CWIOp::usage = "CWIOp[\!\(\* +StyleBox[\"d\",\nFontSlant->\"Italic\"]\), {\!\(\*SubscriptBox[\(\[CapitalDelta]\), \(1\)]\), ...}][\!\(\* +StyleBox[\"expr\",\nFontSlant->\"Italic\"]\), \[Kappa]] applies the full CWI operator \!\(\*SuperscriptBox[\(K\), \(\[Kappa]\)]\) to \!\(\* +StyleBox[\"expr\",\nFontSlant->\"Italic\"]\) in \!\(\* +StyleBox[\"d\",\nFontSlant->\"Italic\"]\) dimensions. \!\(\* +StyleBox[\"expr\",\nFontSlant->\"Italic\"]\) can be thought of as an (\!\(\* +StyleBox[\"n\",\nFontSlant->\"Italic\"]\)+1)-point function of scalar operators of conformal dimensions \!\(\*SubscriptBox[\(\[CapitalDelta]\), \(1\)]\), ..., \!\(\*SubscriptBox[\(\[CapitalDelta]\), +StyleBox[\"n\",\nFontSlant->\"Italic\"]]\). +CWIOp[\!\(\* +StyleBox[\"d\",\nFontSlant->\"Italic\"]\), {\!\(\*SubscriptBox[\(\[CapitalDelta]\), \(1\)]\), ...}][\!\(\* +StyleBox[\"expr\",\nFontSlant->\"Italic\"]\), \[Kappa], {{\!\(\*SubscriptBox[\(\[Mu]\), \(11\)]\), ..., \!\(\*SubscriptBox[\(\[Mu]\), \(1 \*SubscriptBox[\(m\), \(1\)]\)]\)}, ...}] takes into consideration tensor structure determined by {{\!\(\*SubscriptBox[\(\[Mu]\), \(11\)]\), ..., \!\(\*SubscriptBox[\(\[Mu]\), \(1 \*SubscriptBox[\(m\), \(1\)]\)]\)}, ...}. \!\(\* +StyleBox[\"expr\",\nFontSlant->\"Italic\"]\) can be thought of as an (\!\(\* +StyleBox[\"n\",\nFontSlant->\"Italic\"]\)+1)-point function of tensorial operators, each with indices \!\(\*SubscriptBox[\(\[Mu]\), \(j1\)]\),...,\!\(\*SubscriptBox[\(\[Mu]\), SubscriptBox[\(jm\), \(j\)]]\)."; + + +PrimaryCWIs::usage = "PrimaryCWIs[\!\(\* +StyleBox[\"X\",\nFontSlant->\"Italic\"]\)] lists primary CWIs for the correlation function denoted by \!\(\* +StyleBox[\"X\",\nFontSlant->\"Italic\"]\)."; +SecondaryCWIsLhs::usage = "SecondaryCWIsLhs[\!\(\* +StyleBox[\"X\",\nFontSlant->\"Italic\"]\)] lists left hand sides of secondary CWIs for the correlation function denoted by \!\(\* +StyleBox[\"X\",\nFontSlant->\"Italic\"]\)."; +SecondaryCWIsRhs::usage = "SecondaryCWIsRhs[\!\(\* +StyleBox[\"X\",\nFontSlant->\"Italic\"]\)] lists right hand sides of secondary CWIs for the correlation function denoted by \!\(\* +StyleBox[\"X\",\nFontSlant->\"Italic\"]\)."; +TransverseWIs::usage = "TransverseWIs[\!\(\* +StyleBox[\"X\",\nFontSlant->\"Italic\"]\)] lists transverse WIs for the correlation function denoted by \!\(\* +StyleBox[\"X\",\nFontSlant->\"Italic\"]\) in a generic case."; +PrimarySolutions::usage = "PrimarySolutions[\!\(\* +StyleBox[\"X\",\nFontSlant->\"Italic\"]\)] lists form factors solving primary WIs for the correlation function denoted by \!\(\* +StyleBox[\"X\",\nFontSlant->\"Italic\"]\)."; + + +OOO::usage = "Index for ."; +JOO::usage = "Index for ."; +TOO::usage = "Index for ."; +JJO::usage = "Index for ."; +TJO::usage = "Index for ."; +TTO::usage = "Index for ."; +JJJ::usage = "Index for ."; +TJJ::usage = "Index for ."; +TTJ::usage = "Index for ."; +TTT::usage = "Index for ."; + + +J::usage = "J[\!\(\* +StyleBox[\"N\",\nFontSlant->\"Italic\"]\), {\!\(\*SubscriptBox[ +StyleBox[\"k\",\nFontSlant->\"Italic\"], \(1\)]\), \!\(\*SubscriptBox[ +StyleBox[\"k\",\nFontSlant->\"Italic\"], \(2\)]\), \!\(\*SubscriptBox[ +StyleBox[\"k\",\nFontSlant->\"Italic\"], \(3\)]\)}] denotes the triple-\!\(\* +StyleBox[\"K\",\nFontSlant->\"Italic\"]\) integral in the reduced form."; +suJToI::usage = "suJToI[\!\(\* +StyleBox[\"d\",\nFontSlant->\"Italic\"]\), {\!\(\*SubscriptBox[\(\[CapitalDelta]\), \(1\)]\), \!\(\*SubscriptBox[\(\[CapitalDelta]\), \(2\)]\), \!\(\*SubscriptBox[\(\[CapitalDelta]\), \(3\)]\)}] is a substitution list for expressing \!\(\* +StyleBox[\"J\",\nFontSlant->\"Italic\"]\)-integrals as triple-\!\(\* +StyleBox[\"K\",\nFontSlant->\"Italic\"]\) integrals."; +JToI::usage = "JToI[d,{\!\(\*SubscriptBox[\(\[CapitalDelta]\), \(1\)]\),\!\(\*SubscriptBox[\(\[CapitalDelta]\), \(2\)]\),\!\(\*SubscriptBox[\(\[CapitalDelta]\), \(3\)]\)}][\!\(\* +StyleBox[\"expr\",\nFontSlant->\"Italic\"]\)] replaces all \!\(\* +StyleBox[\"J\",\nFontSlant->\"Italic\"]\)-integrals in \!\(\* +StyleBox[\"expr\",\nFontSlant->\"Italic\"]\) with triple-\!\(\* +StyleBox[\"K\",\nFontSlant->\"Italic\"]\) intgerals."; + + +KOp::usage = "KOp[\!\(\* +StyleBox[\"expr\",\nFontSlant->\"Italic\"]\), \!\(\* +StyleBox[\"p\",\nFontSlant->\"Italic\"]\), \[Beta]] applies single K operator to \!\(\* +StyleBox[\"expr\",\nFontSlant->\"Italic\"]\) with respect to momentum magnitude \!\(\* +StyleBox[\"p\",\nFontSlant->\"Italic\"]\) and with parameter \[Beta]. +KOp[\!\(\* +StyleBox[\"expr\",\nFontSlant->\"Italic\"]\), \!\(\* +StyleBox[\"n\",\nFontSlant->\"Italic\"]\)] applies KOp to \!\(\* +StyleBox[\"expr\",\nFontSlant->\"Italic\"]\) with respect to momentum \!\(\* +StyleBox[SubscriptBox[\"p\", \"n\"],\nFontSlant->\"Italic\"]\) and with symbolic parameter \[Beta]=\!\(\*SubscriptBox[\(\[CapitalDelta]\), +StyleBox[\"n\",\nFontSlant->\"Italic\"]]\)-\!\(\*FractionBox[ +StyleBox[\"d\",\nFontSlant->\"Italic\"], \(2\)]\)."; +KKOp::usage = "KKOp[\!\(\* +StyleBox[\"expr\",\nFontSlant->\"Italic\"]\), \!\(\* +StyleBox[SubscriptBox[\"p\", \"i\"],\nFontSlant->\"Italic\"]\), \!\(\* +StyleBox[SubscriptBox[\"p\", \"j\"],\nFontSlant->\"Italic\"]\), \!\(\*SubscriptBox[\(\[Beta]\), +StyleBox[\"i\",\nFontSlant->\"Italic\"]]\), \!\(\*SubscriptBox[\(\[Beta]\), +StyleBox[\"j\",\nFontSlant->\"Italic\"]]\)] = KOp[\!\(\* +StyleBox[\"expr\",\nFontSlant->\"Italic\"]\), \!\(\* +StyleBox[SubscriptBox[\"p\", \"i\"],\nFontSlant->\"Italic\"]\), \!\(\*SubscriptBox[\(\[Beta]\), +StyleBox[\"i\",\nFontSlant->\"Italic\"]]\)] - KOp[\!\(\* +StyleBox[\"expr\",\nFontSlant->\"Italic\"]\), \!\(\* +StyleBox[SubscriptBox[\"p\", \"j\"],\nFontSlant->\"Italic\"]\), \!\(\*SubscriptBox[\(\[Beta]\), +StyleBox[\"j\",\nFontSlant->\"Italic\"]]\)] applies the conformal Ward identity operator in its scalar form. +KKOp[\!\(\* +StyleBox[\"expr\",\nFontSlant->\"Italic\"]\), \!\(\* +StyleBox[\"m\",\nFontSlant->\"Italic\"]\), \!\(\* +StyleBox[\"n\",\nFontSlant->\"Italic\"]\)] applies KKOp to \!\(\* +StyleBox[\"expr\",\nFontSlant->\"Italic\"]\) with respect to momenta \!\(\* +StyleBox[SubscriptBox[\"p\", \"m\"],\nFontSlant->\"Italic\"]\)\!\(\* +StyleBox[\",\",\nFontSlant->\"Italic\"]\)\!\(\* +StyleBox[\" \",\nFontSlant->\"Italic\"]\)\!\(\* +StyleBox[SubscriptBox[ +StyleBox[\"p\",\nFontSlant->\"Italic\"], \"n\"],\nFontSlant->\"Italic\"]\) and with symbolic parameters \!\(\*SubscriptBox[\(\[Beta]\), \(m\)]\) = \!\(\*SubscriptBox[\(\[CapitalDelta]\), +StyleBox[\"m\",\nFontSlant->\"Italic\"]]\)-\!\(\*FractionBox[ +StyleBox[\"d\",\nFontSlant->\"Italic\"], \(2\)]\), \!\(\*SubscriptBox[\(\[Beta]\), \(n\)]\) = \!\(\*SubscriptBox[\(\[CapitalDelta]\), +StyleBox[\"n\",\nFontSlant->\"Italic\"]]\)-\!\(\*FractionBox[ +StyleBox[\"d\",\nFontSlant->\"Italic\"], \(2\)]\)."; + + +LOp::usage = "LOp[\!\(\* +StyleBox[\"d\",\nFontSlant->\"Italic\"]\), {\!\(\*SubscriptBox[\(\[CapitalDelta]\), \(1\)]\),\!\(\*SubscriptBox[\(\[CapitalDelta]\), \(2\)]\)}][\!\(\* +StyleBox[\"expr\",\nFontSlant->\"Italic\"]\), \!\(\* +StyleBox[\"s\",\nFontSlant->\"Italic\"]\), \!\(\* +StyleBox[\"N\",\nFontSlant->\"Italic\"]\)] applies the secondary \!\(\* +StyleBox[SubscriptBox[\"L\", +RowBox[{\"s\", \",\", +StyleBox[\"N\",\nFontSlant->\"Italic\"]}]],\nFontSlant->\"Italic\"]\) operator to \!\(\* +StyleBox[\"expr\",\nFontSlant->\"Italic\"]\). +LOp[\!\(\* +StyleBox[\"expr\",\nFontSlant->\"Italic\"]\), \!\(\* +StyleBox[\"s\",\nFontSlant->\"Italic\"]\), \!\(\* +StyleBox[\"N\",\nFontSlant->\"Italic\"]\)] uses symbolic parameters \!\(\* +StyleBox[\"d\",\nFontSlant->\"Italic\"]\), \!\(\*SubscriptBox[\(\[CapitalDelta]\), \(1\)]\), \!\(\*SubscriptBox[\(\[CapitalDelta]\), \(2\)]\)."; +LprimeOp::usage = "LprimeOp[\!\(\* +StyleBox[\"d\",\nFontSlant->\"Italic\"]\), {\!\(\*SubscriptBox[\(\[CapitalDelta]\), \(1\)]\),\!\(\*SubscriptBox[\(\[CapitalDelta]\), \(2\)]\)}][\!\(\* +StyleBox[\"expr\",\nFontSlant->\"Italic\"]\), \!\(\* +StyleBox[\"s\",\nFontSlant->\"Italic\"]\), \!\(\* +StyleBox[\"N\",\nFontSlant->\"Italic\"]\)] applies the secondary \!\(\* +StyleBox[\"L\",\nFontSlant->\"Italic\"]\)\!\(\* +StyleBox[SubscriptBox[\"'\", +RowBox[{\"s\", \",\", \"N\"}]],\nFontSlant->\"Italic\"]\) operator to \!\(\* +StyleBox[\"expr\",\nFontSlant->\"Italic\"]\). +LprimeOp[\!\(\* +StyleBox[\"expr\",\nFontSlant->\"Italic\"]\), \!\(\* +StyleBox[\"s\",\nFontSlant->\"Italic\"]\), \!\(\* +StyleBox[\"N\",\nFontSlant->\"Italic\"]\)] uses symbolic parameters \!\(\* +StyleBox[\"d\",\nFontSlant->\"Italic\"]\), \!\(\*SubscriptBox[\(\[CapitalDelta]\), \(1\)]\), \!\(\*SubscriptBox[\(\[CapitalDelta]\), \(2\)]\)."; + + +ROp::usage = "ROp[\!\(\*SubscriptBox[\(\[CapitalDelta]\), \(1\)]\)][\!\(\* +StyleBox[\"expr\",\nFontSlant->\"Italic\"]\), \!\(\* +StyleBox[\"s\",\nFontSlant->\"Italic\"]\), \!\(\* +StyleBox[\"N\",\nFontSlant->\"Italic\"]\)] applies the secondary \!\(\* +StyleBox[SubscriptBox[\"R\", \"s\"],\nFontSlant->\"Italic\"]\) operator to \!\(\* +StyleBox[\"expr\",\nFontSlant->\"Italic\"]\). +ROp[\!\(\* +StyleBox[\"expr\",\nFontSlant->\"Italic\"]\), \!\(\* +StyleBox[\"s\",\nFontSlant->\"Italic\"]\)] uses symbolic parameter \!\(\*SubscriptBox[\(\[CapitalDelta]\), \(1\)]\)."; +RprimeOp::usage = "RprimeOp[\!\(\*SubscriptBox[\(\[CapitalDelta]\), \(2\)]\)][\!\(\* +StyleBox[\"expr\",\nFontSlant->\"Italic\"]\), \!\(\* +StyleBox[\"s\",\nFontSlant->\"Italic\"]\), \!\(\* +StyleBox[\"N\",\nFontSlant->\"Italic\"]\)] applies the secondary \!\(\* +StyleBox[\"R\",\nFontSlant->\"Italic\"]\)\!\(\* +StyleBox[SubscriptBox[\"'\", \"s\"],\nFontSlant->\"Italic\"]\) operator to \!\(\* +StyleBox[\"expr\",\nFontSlant->\"Italic\"]\). +RprimeOp[\!\(\* +StyleBox[\"expr\",\nFontSlant->\"Italic\"]\), \!\(\* +StyleBox[\"s\",\nFontSlant->\"Italic\"]\)] uses symbolic parameter \!\(\*SubscriptBox[\(\[CapitalDelta]\), \(2\)]\)."; + + +suExpandAs::usage = "suExpandAs[\!\(\* +StyleBox[\"A\",\nFontSlant->\"Italic\"]\)] is a substitution list for reapplying standard momenta \!\(\*SubscriptBox[ +StyleBox[\"p\",\nFontSlant->\"Italic\"], \(1\)]\),\!\(\*SubscriptBox[ +StyleBox[\"p\",\nFontSlant->\"Italic\"], \(2\)]\),\!\(\*SubscriptBox[ +StyleBox[\"p\",\nFontSlant->\"Italic\"], \(3\)]\) to form factors \!\(\* +StyleBox[SubscriptBox[\"A\", \"j\"],\nFontSlant->\"Italic\"]\)\!\(\* +StyleBox[\" \",\nFontSlant->\"Italic\"]\)under CWI operators."; + + +Options[PrimaryCWIs] = { FormFactor -> A }; +Options[SecondaryCWIsLhs] = { FormFactor -> A }; +Options[SecondaryCWIsRhs] = { Index -> \[Mu] }; +Options[TransverseWIs] = { RenormalizationScale -> \[Mu], + CentralCharge -> cT, + JJNormalization -> cJ, + OONormalization -> cO, + GaugeCoupling -> g, + dTdgDerivative -> cg, + dTdADerivative -> c3 }; +Options[PrimarySolutions] = { FormFactor -> A, PrimaryConstant -> \[Alpha] }; + + +Begin["Konformal`Private`"]; + + +(* ::Section:: *) +(*Implementation*) + + +DilOp[d_, \[CapitalDelta]t_, n_][F_] := Module[{idx, j}, + (\[CapitalDelta]t-(n-1) d) F - Sum[Contract[p[j][idx] Diff[F,p[j],idx], Dimension->d], {j,n}] +]; +DilOp[F_,n_] := Module[{j}, DilOp[d, Sum[\[CapitalDelta][j],{j,n}], n][F]]; + + +SingleScalarOp[d_, \[CapitalDelta]_][F_, n_, \[Kappa]_] := Module[{idx1, idx2, d1,d2}, + d1 = Diff[F, p[n], idx1]; + d2 = Diff[d1, p[n], idx2]; + (( 2(\[CapitalDelta]-d) d1 - 2 Contract[p[n][idx2] d2, Dimension->d]) /. idx1->\[Kappa]) + p[n][\[Kappa]]Contract[d2,idx1,idx2, Dimension->d] +]; +SingleScalarOp[F_,n_,\[Kappa]_] := SingleScalarOp[d, \[CapitalDelta][n]][F,n,\[Kappa]]; + + +SingleTensorOp[d_][F_, n_, \[Kappa]_, ii_] := 2 \[Delta][ii, \[Kappa]] Contract[Diff[F,p[n],ii], Dimension->d] - 2 Diff[F /. {ii->\[Kappa]}, p[n], ii]; +SingleTensorOp[F_, n_, \[Kappa]_, ii_] := SingleTensorOp[d][F,n,\[Kappa],ii]; + + +CWIOp[d_, \[CapitalDelta]s_List][F_, \[Kappa]_] := Module[{j}, + Sum[SingleScalarOp[d, \[CapitalDelta]s[[j]]][F, j, \[Kappa]], {j, Length[\[CapitalDelta]s]}] +]; +CWIOp[F_,n_,\[Kappa]_] := Module[{j}, CWIOp[d, Table[\[CapitalDelta][j],{j,n-1}]][F,\[Kappa]]]; + + +CWIOp[d_, \[CapitalDelta]s_List][F_, \[Kappa]_, in\[Mu]s_List] := Module[{dds, j, jj, idx, \[Mu]s}, + \[Mu]s = Which[ + Length[in\[Mu]s] === Length[\[CapitalDelta]s], in\[Mu]s, + Length[in\[Mu]s] < Length[\[CapitalDelta]s], Join[in\[Mu]s, Table[{}, Length[\[CapitalDelta]s]-Length[in\[Mu]s]]], + Length[\[CapitalDelta]s] < Length[in\[Mu]s], Drop[in\[Mu]s, -(Length[in\[Mu]s]-Length[\[CapitalDelta]s])]]; + dds = ParallelTable[Diff[F,p[j],idx], {j,Length[\[CapitalDelta]s]}]; + ScalarOpWithD[d,\[CapitalDelta]s][F,dds,idx,\[Kappa]] + Sum[SingleTensorOpWithD[d][F,dds[[j]],idx,\[Kappa],\[Mu]s[[j]][[jj]]], + {j,Length[\[Mu]s]},{jj,Length[\[Mu]s[[j]]]}] +]; +CWIOp[F_,n_,\[Kappa]_,\[Mu]s_List] := Module[{j}, CWIOp[d, Table[\[CapitalDelta][j],{j,n-1}]][F,\[Kappa],\[Mu]s]]; + + +SingleScalarOpWithD[d_,\[CapitalDelta]_][F_,d1_,idx1_,n_,\[Kappa]_] := Module[{idx2,d2}, + d2 = Diff[d1,p[n],idx2]; + ((2(\[CapitalDelta]-d) d1-2 Contract[p[n][idx2] d2, Dimension->d])/.idx1->\[Kappa]) + p[n][\[Kappa]]Contract[d2,idx1,idx2, Dimension->d] + ]; + + +ScalarOpWithD[d_,\[CapitalDelta]s_List][F_,dds_List,idx_,\[Kappa]_] := Module[{j}, + Sum[SingleScalarOpWithD[d,\[CapitalDelta]s[[j]]][F,dds[[j]],idx,j,\[Kappa]], {j,Length[\[CapitalDelta]s]}] +]; + + +SingleTensorOpWithD[d_][F_,d1_,idx_,\[Kappa]_,ii_] := 2 \[Delta][ii,\[Kappa]] Contract[d1 /. {idx->ii}, Dimension->d] - 2 (d1 /. {ii->\[Kappa]} /. {idx->ii}); + + +pi[n_][i_,j_] := \[Delta][i,j] - p[n][i] p[n][j] / p[n]^2; +PI[d_][n_][i_,j_,k_,l_]:=1/2( pi[n][i,k]pi[n][j,l]+pi[n][i,l]pi[n][j,k]) - 1/(d-1) pi[n][i,j] pi[n][k,l]; + + +KOp[F_, p_, \[Beta]_] := D[F,p,p] - (2\[Beta]-1)/p * D[F,p]; +KOp[F_, n_] := KOp[F, p[n], \[CapitalDelta][n]-d/2]; +KKOp[F_,p1_,p2_,\[Beta]1_,\[Beta]2_] := KOp[F, p1, \[Beta]1] - KOp[F, p2, \[Beta]2]; +KKOp[F_,m_,n_] := KOp[F,m] - KOp[F,n]; + + +LOp[d_, {\[CapitalDelta]1_,\[CapitalDelta]2_}][F_,s_,N_] := p[1](p[1]^2+p[2]^2-p[3]^2) D[F,p[1]] + 2 p[1]^2 p[2] D[F,p[2]] + ((2d-\[CapitalDelta]1-2\[CapitalDelta]2+s+N)p[1]^2+(\[CapitalDelta]1-2+s)(p[3]^2-p[2]^2)) F; +LOp[F_,s_,N_] := LOp[d,{\[CapitalDelta][1],\[CapitalDelta][2]}][F,s,N]; +LprimeOp[d_, {\[CapitalDelta]1_,\[CapitalDelta]2_}][F_,s_,N_] := p[2](p[1]^2+p[2]^2-p[3]^2) D[F,p[2]] + 2 p[2]^2 p[1] D[F,p[1]] + ((2d-\[CapitalDelta]2-2\[CapitalDelta]1+s+N)p[2]^2+(\[CapitalDelta]2-2+s)(p[3]^2-p[1]^2)) F; +LprimeOp[F_,s_,N_] := LprimeOp[d,{\[CapitalDelta][1],\[CapitalDelta][2]}][F,s,N]; + + +ROp[\[CapitalDelta]1_][F_,s_] := p[1] D[F,p[1]] - (\[CapitalDelta]1-2+s)F; +ROp[F_,s_] := ROp[\[CapitalDelta][1]][F,s]; +RprimeOp[\[CapitalDelta]2_][F_,s_] := p[2] D[F,p[2]] - (\[CapitalDelta]2-2+s)F; +RprimeOp[F_,s_] := RprimeOp[\[CapitalDelta][2]][F,s]; + + +suJToI[d_,\[CapitalDelta]s_]:={ J[n_,ks_] :> i[d/2-1+n,\[CapitalDelta]s-d/2+ks] }; +suJToI[d_,\[CapitalDelta]s_,u_,vs_]:={ J[n_,ks_] :> i[d/2-1+n,\[CapitalDelta]s-d/2+ks, u,vs] }; + + +suExpandAs[A_] := { Hold[X_][A[n_],x___] :> Hold[X][A[n][p[1],p[2],p[3]],x], + HoldForm[X_][A[n_],x___] :> HoldForm[X][A[n][p[1],p[2],p[3]],x] }; + + +JToI[d_,\[CapitalDelta]s_][exp_] := Module[{ii}, + exp /. suJToI[d, \[CapitalDelta]s] + /. { i[a_,b_][x___] :> ii[a,b][x], + Derivative[k_,m_,n_][i[a_,b_]][x___] :> Derivative[k,m,n][ii[a,b]][x] } + /. { i[a_,b_] :> i[a,b][p[1],p[2],p[3]] } + /. { ii -> i } +]; + + +SubleadingC[c_] := c[0]; +TwoPointFunction[\[CapitalDelta]_, d_][c_, n_, \[Mu]_][] /; (EvenQ[2\[CapitalDelta]-d] && 2\[CapitalDelta]-d >= 0) := p[n]^(2\[CapitalDelta]-d) *( c Log[p[n]^2/\[Mu]^2] + SubleadingC[c] ); +TwoPointFunction[\[CapitalDelta]_, d_][c_, n_, \[Mu]_][] := c p[n]^(2\[CapitalDelta]-d); +TwoPointFunction[\[CapitalDelta]_, d_][c_, n_, \[Mu]_][i_,j_] /; (EvenQ[2\[CapitalDelta]-d] && 2\[CapitalDelta]-d >= 2) := pi[n][i,j] p[n]^(2\[CapitalDelta]-d)*( c Log[p[n]^2/\[Mu]^2] + SubleadingC[c] ); +TwoPointFunction[\[CapitalDelta]_, d_][c_, n_, \[Mu]_][i_,j_] := c pi[n][i,j] p[n]^(2\[CapitalDelta]-d); +TwoPointFunction[\[CapitalDelta]_, d_][c_, n_, \[Mu]_][i_,j_,k_,l_] /; (EvenQ[2\[CapitalDelta]-d] && 2\[CapitalDelta]-d >= 4) := PI[d][n][i,j,k,l] p[n]^(2\[CapitalDelta]-d)*( c Log[p[n]^2/\[Mu]^2] + SubleadingC[c] ); +TwoPointFunction[\[CapitalDelta]_, d_][c_, n_, \[Mu]_][i_,j_,k_,l_] := c PI[d][n][i,j,k,l] p[n]^(2\[CapitalDelta]-d); + + +PrePrimaryCWIs[OOO][A_]:= { + {HoldForm[KKOp][A[1], 1, 3]}, + {HoldForm[KKOp][A[1], 2, 3]}, + {HoldForm[KKOp][A[1], 1, 2]} +}; +PrePrimaryCWIs[JOO][A_]:= { + {HoldForm[KKOp][A[1], 1, 3]}, + {HoldForm[KKOp][A[1], 2, 3]}, + {HoldForm[KKOp][A[1], 1, 2]} +}; +PrePrimaryCWIs[TOO][A_]:= { + {HoldForm[KKOp][A[1], 1, 3]}, + {HoldForm[KKOp][A[1], 2, 3]}, + {HoldForm[KKOp][A[1], 1, 2]} +}; +PrePrimaryCWIs[JJO][A_]:={ + {HoldForm[KKOp][A[1], 1, 3], + HoldForm[KKOp][A[2], 1,3] - 2*A[1][p[1], p[2], p[3]] }, + {HoldForm[KKOp][A[1], 2,3], + HoldForm[KKOp][A[2], 2, 3]- 2*A[1][p[1], p[2], p[3]] }, + {HoldForm[KKOp][A[1], 1,2], + HoldForm[KKOp][A[2], 1,2]} +}; +PrePrimaryCWIs[TJO][A_]:={ + {HoldForm[KKOp][A[1], 1,3], + HoldForm[KKOp][A[2], 1,3]-4*A[1][p[1], p[2], p[3]] }, + {HoldForm[KKOp][A[1], 2,3], + HoldForm[KKOp][A[2], 2,3]-4*A[1][p[1], p[2], p[3]] }, + {HoldForm[KKOp][A[1], 1,2], + HoldForm[KKOp][A[2], 1,2] } +}; +PrePrimaryCWIs[TTO][A_]:={ + {HoldForm[KKOp][A[1], 1,3], + HoldForm[KKOp][A[2], 1,3]-8*A[1][p[1], p[2], p[3]], + HoldForm[KKOp][A[3], 1,3]-2 A[2][p[1], p[2], p[3]] }, + {HoldForm[KKOp][A[1], 2,3], + HoldForm[KKOp][A[2], 2,3]-8*A[1][p[1], p[2], p[3]], + HoldForm[KKOp][A[3], 2,3]-2 A[2][p[1], p[2], p[3]] }, + {HoldForm[KKOp][A[1], 1,2], + HoldForm[KKOp][A[2], 1,2], + HoldForm[KKOp][A[3], 1,2] } +}; +PrePrimaryCWIs[JJJ][A_]:={ + {HoldForm[KKOp][A[1], 1,3], + HoldForm[KKOp][A[2], 1,3]-2*A[1][p[1], p[2], p[3]]}, + {HoldForm[KKOp][A[1], 2,3], + HoldForm[KKOp][A[2], 2,3]-2*A[1][p[1], p[2], p[3]]}, + {HoldForm[KKOp][A[1], 1,2], + HoldForm[KKOp][A[2], 1,2] } +}; +PrePrimaryCWIs[TJJ][A_]:={ + {HoldForm[KKOp][A[1], 1,3], + HoldForm[KKOp][A[2], 1,3]+2*A[1][p[1], p[2], p[3]], + HoldForm[KKOp][A[3], 1,3]-4*A[1][p[1], p[2], p[3]], + HoldForm[KKOp][A[4], 1,3]-2A[3][p[1], p[3], p[2]] }, + {HoldForm[KKOp][A[1], 2,3], + HoldForm[KKOp][A[2], 2,3], + HoldForm[KKOp][A[3], 2,3]-4*A[1][p[1], p[2], p[3]], + HoldForm[KKOp][A[4], 2,3]+2 A[3][p[1], p[2], p[3]] -2A[3][p[1], p[3], p[2]] }, + {HoldForm[KKOp][A[1], 1,2], + HoldForm[KKOp][A[2], 1,2]+ 2*A[1][p[1], p[2], p[3]], + HoldForm[KKOp][A[3], 1,2], + HoldForm[KKOp][A[4], 1,2]-2*A[3][p[1], p[2], p[3]] } +}; +PrePrimaryCWIs[TTJ][A_]:={ + {HoldForm[KKOp][A[1], 1,3], + HoldForm[KKOp][A[2], 1,3]-8*A[1][p[1], p[2], p[3]], + HoldForm[KKOp][A[3], 1,3], + HoldForm[KKOp][A[4], 1,3]-2 A[2][p[1], p[2], p[3]], + HoldForm[KKOp][A[5], 1,3]+2A[2][p[1], p[2], p[3]]+4 A[3][p[2], p[1], p[3]]}, + {HoldForm[KKOp][A[1], 2,3], + HoldForm[KKOp][A[2], 2,3]-8*A[1][p[1], p[2], p[3]], + HoldForm[KKOp][A[3], 2,3]+4*A[1][p[1], p[2], p[3]], + HoldForm[KKOp][A[4], 2,3]-2A[2][p[1], p[2], p[3]], + HoldForm[KKOp][A[5], 2,3]+4 A[3][p[2], p[1], p[3]] }, + {HoldForm[KKOp][A[1], 1,2], + HoldForm[KKOp][A[2], 1,2], + HoldForm[KKOp][A[3], 1,2]-4*A[1][p[1], p[2], p[3]], + HoldForm[KKOp][A[4], 1,2], + HoldForm[KKOp][A[5], 1,2]+2*A[2][p[1], p[2], p[3]]} +}; +PrePrimaryCWIs[TTT][A_]:={ + {HoldForm[KKOp][A[1], 1,3], + HoldForm[KKOp][A[2], 1,3] - 8*A[1][p[1], p[2], p[3]], + HoldForm[KKOp][A[3], 1,3]- 2*A[2][p[1], p[2], p[3]], + HoldForm[KKOp][A[4], 1,3]+ 4*A[2][p[1], p[3], p[2]], + HoldForm[KKOp][A[5], 1,3] - 2*A[4][p[1], p[2], p[3]] + 2*A[4][p[3], p[2], p[1]] }, + {HoldForm[KKOp][A[1], 2,3], + HoldForm[KKOp][A[2], 2,3] - 8*A[1][p[1], p[2], p[3]], + HoldForm[KKOp][A[3], 2, 3] - 2*A[2][p[1], p[2], p[3]], + HoldForm[KKOp][A[4], 2,3] + 4*A[2][p[3], p[2], p[1]], + HoldForm[KKOp][A[5], 2, 3]-2*A[4][p[1], p[2], p[3]] + 2*A[4][p[1], p[3], p[2]] }, + {HoldForm[KKOp][A[1], 1,2], + HoldForm[KKOp][A[2], 1,2], + HoldForm[KKOp][A[3], 1,2], + HoldForm[KKOp][A[4], 1,2] + 4*A[2][p[1], p[3], p[2]] - 4*A[2][p[3], p[2], p[1]], + HoldForm[KKOp][A[5], 1,2] - 2*A[4][p[1], p[3], p[2]] + 2*A[4][p[3], p[2], p[1]] } +}; + + +fcall:PrimaryCWIs[X_, opts___?OptionQ] := Module[{ValidOpts}, + ValidOpts = First /@ Options[PrimaryCWIs]; + Scan[If[!MemberQ[ValidOpts, First[#]], + Message[PrimaryCWIs::optx, ToString[First[#]], ToString[Unevaluated[fcall]]]]&, Flatten[{opts}]]; + PrePrimaryCWIs[X][FormFactor /. Flatten[{opts}] /. Options[PrimaryCWIs]] +]; +PrimaryCWIs[X_, x__] := Null /; Message[PrimaryCWIs::argrx, "PrimaryCWIs", Length[{x}] + 1, 1]; +PrimaryCWIs[] := Null /; Message[PrimaryCWIs::argrx, "PrimaryCWIs", Length[{X}], 1]; + + +PreSecondaryCWIsLhs[OOO][A_] := {}; +PreSecondaryCWIsLhs[JOO][A_] := { HoldForm[LOp][A[1],1,0] }; +PreSecondaryCWIsLhs[TOO][A_] := { HoldForm[LOp][A[1],2,0] }; +PreSecondaryCWIsLhs[JJO][A_] := { HoldForm[LOp][A[1],1,0]+2 HoldForm[ROp][A[2],1] }; +PreSecondaryCWIsLhs[TJO][A_] := { HoldForm[LOp][A[1],2,0]+HoldForm[ROp][A[2],2], + HoldForm[LprimeOp][A[1],1,0]+2 HoldForm[RprimeOp][A[2],1], + HoldForm[LOp][A[2],2,0] }; +PreSecondaryCWIsLhs[TTO][A_] := { HoldForm[LOp][A[1],2,0]+HoldForm[ROp][A[2],2], + HoldForm[LOp][A[2],2,0]+4 HoldForm[ROp][A[3],2] }; +PreSecondaryCWIsLhs[JJJ][A_] := { + HoldForm[LOp][A[1],1,2]+2 HoldForm[ROp][A[2],1]-2 HoldForm[ROp][A[2][p[3],p[1],p[2]],1], + HoldForm[LOp][A[2][p[2],p[3],p[1]],1,0]+2p[1]^2A[2][p[3],p[1],p[2]]-2 p[1]^2 A[2][p[1],p[2],p[3]] } +PreSecondaryCWIsLhs[TJJ][A_] := { + HoldForm[LOp][A[1],2,2]+HoldForm[ROp][A[3],2]-HoldForm[ROp][A[3][p[1],p[3],p[2]],2], + HoldForm[LprimeOp][A[1],1,2]-2 HoldForm[RprimeOp][A[2],1]+2 HoldForm[RprimeOp][A[3],1], + HoldForm[LOp][A[2],2,0]-p[1]^2(A[3][p[1],p[2],p[3]]-A[3][p[1],p[3],p[2]]), + HoldForm[LOp][A[3],2,2]-2 HoldForm[ROp][A[4],2] }; +PreSecondaryCWIsLhs[TTJ][A_] := { + HoldForm[LOp][A[1],2,2]+HoldForm[ROp][A[2],2]-HoldForm[ROp][A[3],2], + HoldForm[LOp][A[2],2,2]+4 HoldForm[ROp][A[4],2]+2 HoldForm[ROp][A[5][p[2],p[1],p[3]],2], + HoldForm[LOp][A[3],2,-2]-2 HoldForm[ROp][A[5][p[2],p[1],p[3]],2], + HoldForm[LOp][A[5],2,0]-2 p[1]^2(2 A[4][p[1],p[2],p[3]]+A[5][p[2],p[1],p[3]]) }; +PreSecondaryCWIsLhs[TTT][A_] := { + HoldForm[LOp][A[1],2,4]+HoldForm[ROp][A[2],2]-HoldForm[ROp][A[2][p[1],p[3],p[2]],2], + HoldForm[LOp][A[2],2,4]+4 HoldForm[ROp][A[3],2]-2 HoldForm[ROp][A[4][p[3],p[2],p[1]],2], + HoldForm[LOp][A[2][p[3],p[2],p[1]],2,2]-HoldForm[ROp][A[4],2]+HoldForm[ROp][A[4][p[1],p[3],p[2]],2]+2 p[1]^2 (-A[2][p[1],p[2],p[3]]+A[2][p[1],p[3],p[2]]), + HoldForm[LOp][A[4][p[1],p[3],p[2]],2,2]-2 HoldForm[ROp][A[5],2]+2 p[1]^2 (-4 A[3][p[1],p[2],p[3]]+A[4][p[3],p[2],p[1]]), + HoldForm[LOp][A[3][p[3],p[2],p[1]],2,0]+p[1]^2 (A[4][p[1],p[2],p[3]]-A[4][p[1],p[3],p[2]]) }; + + +fcall:SecondaryCWIsLhs[X_, opts___?OptionQ] := Module[{ValidOpts}, + ValidOpts = First /@ Options[SecondaryCWIsLhs]; + Scan[If[!MemberQ[ValidOpts, First[#]], + Message[SecondaryCWIsLhs::optx, ToString[First[#]], ToString[Unevaluated[fcall]]]]&, Flatten[{opts}]]; + PreSecondaryCWIsLhs[X][FormFactor /. Flatten[{opts}] /. Options[SecondaryCWIsLhs]] +]; +SecondaryCWIsLhs[X_, x__] := Null /; Message[SecondaryCWIsLhs::argrx, "SecondaryCWIsLhs", Length[{x}] + 1, 1]; +SecondaryCWIsLhs[] := Null /; Message[SecondaryCWIsLhs::argrx, "SecondaryCWIsLhs", Length[{X}], 1]; + + +PreSecondaryCWIsRhs[OOO][\[Mu]_] := {}; +PreSecondaryCWIsRhs[JOO][\[Mu]_] := Module[{j}, + { 2(\[CapitalDelta][1]-1) p[1][j] JOO[j] } /. { j -> Unique[\[Mu]] }]; +PreSecondaryCWIsRhs[TOO][\[Mu]_] := Module[{i,j}, + { 2 \[CapitalDelta][1] HoldForm[Coefficient][p[1][j] TOO[i,j], p[2][i]] } + /. {i -> Unique[\[Mu]], j -> Unique[\[Mu]]}]; +PreSecondaryCWIsRhs[JJO][\[Mu]_] := Module[{i,j}, + { 2(\[CapitalDelta][1]-1) HoldForm[Coefficient][p[1][i] JJO[i][j], p[3][j]] } + /. {i -> Unique[\[Mu]], j -> Unique[\[Mu]]}]; +PreSecondaryCWIsRhs[TJO][\[Mu]_] := Module[{i,j,k}, + { 2 \[CapitalDelta][1] HoldForm[Coefficient][p[1][j] TJO[i,j][k], p[2][i]p[3][k]], + -2 (\[CapitalDelta][2]-1) HoldForm[Coefficient][p[2][k] TJO[i,j][k], p[2][i]p[2][j]], + 4 \[CapitalDelta][1] HoldForm[Coefficient][p[1][j] TJO[i,j][k], \[Delta][i,k]] } + /. {i -> Unique[\[Mu]], j -> Unique[\[Mu]], k -> Unique[\[Mu]]}]; +PreSecondaryCWIsRhs[TTO][\[Mu]_] := Module[{i,j,k,l}, + { 2 \[CapitalDelta][1] HoldForm[Coefficient][p[1][j] TTO[i,j][k,l], p[2][i]p[3][k]p[3][l]], + 8 \[CapitalDelta][1] HoldForm[Coefficient][p[1][j] TTO[i,j][k,l], \[Delta][i,k]p[3][l]] } + /. {i -> Unique[\[Mu]], j -> Unique[\[Mu]], k -> Unique[\[Mu]], l -> Unique[\[Mu]]}]; +PreSecondaryCWIsRhs[JJJ][\[Mu]_] := Module[{i,j,k}, + { 2(\[CapitalDelta][1]-1) HoldForm[Coefficient][p[1][i] JJJ[i][j][k], p[3][j]p[1][k]], + 2(\[CapitalDelta][1]-1) HoldForm[Coefficient][p[1][i] JJJ[i][j][k], \[Delta][j,k]] } + /. {i -> Unique[\[Mu]], j -> Unique[\[Mu]], k -> Unique[\[Mu]]}]; +PreSecondaryCWIsRhs[TJJ][\[Mu]_] := Module[{i,j,k,l}, + { 2 \[CapitalDelta][1] HoldForm[Coefficient][p[1][j] TJJ[i,j][k][l], p[2][i]p[3][k]p[1][l]], + 2 \[CapitalDelta][1] HoldForm[Coefficient][p[2][k] TJJ[i,j][k][l], p[2][i]p[2][j]p[1][l]], + 2 \[CapitalDelta][1] HoldForm[Coefficient][p[1][j] TJJ[i,j][k][l], \[Delta][k,l]p[2][i]], + 4 \[CapitalDelta][1] HoldForm[Coefficient][p[1][j] TJJ[i,j][k][l], \[Delta][i,k]p[1][l]] } + /. {i -> Unique[\[Mu]], j -> Unique[\[Mu]], k -> Unique[\[Mu]], l -> Unique[\[Mu]]}]; +PreSecondaryCWIsRhs[TTJ][\[Mu]_] := Module[{i,j,k,l,m}, + { 2 \[CapitalDelta][1] HoldForm[Coefficient][p[1][j] TTJ[i,j][k,l][m], p[2][i]p[3][k]p[3][l]p[1][m]], + 4 \[CapitalDelta][1] HoldForm[Coefficient][p[1][j] TTJ[i,j][k,l][m], \[Delta][i,m]p[3][k]p[3][l]], + 8 \[CapitalDelta][1] HoldForm[Coefficient][p[1][j] TTJ[i,j][k,l][m], \[Delta][i,k]p[3][l]p[1][m]], + 8 \[CapitalDelta][1] HoldForm[Coefficient][p[1][j] TTJ[i,j][k,l][m], \[Delta][i,l] \[Delta][k,m]] } + /. {i -> Unique[\[Mu]], j -> Unique[\[Mu]], k -> Unique[\[Mu]], l -> Unique[\[Mu]], m -> Unique[\[Mu]]}]; +PreSecondaryCWIsRhs[TTT][\[Mu]_] := Module[{i,j,k,l,m,ni}, + { 2 \[CapitalDelta][1] HoldForm[Coefficient][p[1][j] TTT[i,j][k,l][m,ni], p[2][i]p[3][k]p[3][l]p[1][m]p[1][ni]], + 8 \[CapitalDelta][1] HoldForm[Coefficient][p[1][j] TTT[i,j][k,l][m,ni], \[Delta][i,k]p[3][l]p[1][m]p[1][ni]], + 8 \[CapitalDelta][1] HoldForm[Coefficient][p[1][j] TTT[i,j][k,l][m,ni], \[Delta][k,m]p[2][i]p[3][l]p[1][ni]], + 16 \[CapitalDelta][1] HoldForm[Coefficient][p[1][j] TTT[i,j][k,l][m,ni], \[Delta][i,k]\[Delta][l,m]p[1][ni]], + 4 \[CapitalDelta][1] HoldForm[Coefficient][p[1][j] TTT[i,j][k,l][m,ni], \[Delta][k,m]\[Delta][l,ni]p[2][i]] } + /. {i -> Unique[\[Mu]], j -> Unique[\[Mu]], k -> Unique[\[Mu]], l -> Unique[\[Mu]], m -> Unique[\[Mu]], ni -> Unique[\[Mu]] }]; + + +fcall:SecondaryCWIsRhs[X_, opts___?OptionQ] := Module[{ValidOpts, idx}, + ValidOpts = First /@ Options[SecondaryCWIsRhs]; + Scan[If[!MemberQ[ValidOpts, First[#]], + Message[SecondaryCWIsRhs::optx, ToString[First[#]], ToString[Unevaluated[fcall]]]]&, Flatten[{opts}]]; + PreSecondaryCWIsRhs[X][Index /. Flatten[{opts}] /. Options[SecondaryCWIsRhs]] +]; +SecondaryCWIsRhs[X_, x__] := Null /; Message[SecondaryCWIsRhs::argrx, "SecondaryCWIsRhs", Length[{x}] + 1, 1]; +SecondaryCWIsRhs[] := Null /; Message[SecondaryCWIsRhs::argrx, "SecondaryCWIsRhs", Length[{X}], 1]; + + +PreTransverseWIs[JOO][c_,\[Mu]_] := { p[1][i_]JOO[i_] :> -TwoPointFunction[\[CapitalDelta][2],d][c[2],2,\[Mu]][] + + TwoPointFunction[\[CapitalDelta][3],d][c[3],3,\[Mu]][] }; +PreTransverseWIs[TOO][c_,\[Mu]_] := { p[1][j_]TOO[i_,j_] :> (-p[1][i]-p[2][i]) TwoPointFunction[\[CapitalDelta][3],d][c[3],3,\[Mu]][] + + p[2][i]TwoPointFunction[\[CapitalDelta][2],d][c[2],2,\[Mu]][] }; +PreTransverseWIs[JJO] = { p[1][i_] JJO[i_][j_] :> 0 }; +PreTransverseWIs[TJO] = { p[1][i_] TJO[i_,j_][k_] :> 0, p[2][k_] TJO[i_,j_][k_] :> 0 }; +PreTransverseWIs[TTO] = { p[1][i_] TTO[i_,j_][k_,l_] :> 0 }; +PreTransverseWIs[JJJ][cJ_,g_,\[Mu]_] := { p[1][i_] JJJ[i_][j_][k_] :> ( I g TwoPointFunction[\[CapitalDelta][2],d][cJ,2,\[Mu]][k,j] + -I g TwoPointFunction[\[CapitalDelta][3],d][cJ,3,\[Mu]][k,j] + /. { p[1][j]:>-p[2][j]-p[3][j], p[2][k]:>-p[1][k]-p[3][k] } ) }; +PreTransverseWIs[TJJ][cJ_,c3_,\[Mu]_] := Module[{aa,jj}, + { p[1][j_] TJJ[i_,j_][k_][l_] :> ( \[Delta][l,i] Contract[p[3][aa]TwoPointFunction[\[CapitalDelta][2],d][cJ,2,\[Mu]][k,aa]] + - p[3][i] TwoPointFunction[\[CapitalDelta][2],d][cJ,2,\[Mu]][k,l] + + \[Delta][i,k] Contract[p[2][aa]TwoPointFunction[\[CapitalDelta][3],d][cJ,3,\[Mu]][aa,l]] + - p[2][i] TwoPointFunction[\[CapitalDelta][3],d][cJ,3,\[Mu]][k,l] + + Contract[p[1][jj]*(c3 \[Delta][i,k]TwoPointFunction[\[CapitalDelta][3],d][cJ,3,\[Mu]][jj,l] + + c3 \[Delta][jj,k] TwoPointFunction[\[CapitalDelta][3],d][cJ,3,\[Mu]][i,l] + + c3 \[Delta][i,l] TwoPointFunction[\[CapitalDelta][2],d][cJ,2,\[Mu]][jj,k] + + c3 \[Delta][jj,l] TwoPointFunction[\[CapitalDelta][2],d][cJ,2,\[Mu]][i,k])] + /. { p[3][i]:>-p[1][i]-p[2][i], p[1][k]:>-p[2][k]-p[3][k], p[2][l]:>-p[1][l]-p[3][l] } ), + p[2][k_] TJJ[i_,j_][k_][l_] :> ( \[Delta][i,j] Contract[p[1][aa] TwoPointFunction[\[CapitalDelta][3],d][cJ,3,\[Mu]][aa,l]] + /. { p[3][i]:>-p[1][i]-p[2][i], p[1][k]:>-p[2][k]-p[3][k], p[2][l]:>-p[1][l]-p[3][l] } ) + }]; +PreTransverseWIs[TTJ] = { p[1][i_] TTJ[i_,j_][k_,l_][m_] :> 0 }; +PreTransverseWIs[TTT][cT_,cg_,\[Mu]_] := Module[{aa,jj}, + { p[1][j_] TTT[i_,j_][k_,l_][m_,n_] :> ( p[1][m] TwoPointFunction[\[CapitalDelta][2],d][cT,2,\[Mu]][n,i,k,l] + + p[1][n] TwoPointFunction[\[CapitalDelta][2],d][cT,2,\[Mu]][m,i,k,l] + + p[1][k] TwoPointFunction[\[CapitalDelta][3],d][cT,3,\[Mu]][l,i,m,n] + + p[1][l] TwoPointFunction[\[CapitalDelta][3],d][cT,3,\[Mu]][k,i,m,n] + + \[Delta][m,n] Contract[p[3][aa]TwoPointFunction[\[CapitalDelta][2],d][cT,2,\[Mu]][aa,i,k,l]] + + \[Delta][k,l] Contract[p[2][aa]TwoPointFunction[\[CapitalDelta][3],d][cT,3,\[Mu]][aa,i,m,n]] + - p[3][i] TwoPointFunction[\[CapitalDelta][2],d][cT,2,\[Mu]][k,l,m,n] + - p[2][i]TwoPointFunction[\[CapitalDelta][3],d][cT,3,\[Mu]][k,l,m,n] + + 2 Contract[p[1][jj](cg \[Delta][i,k]TwoPointFunction[\[CapitalDelta][3],d,\[Mu]][cT,3][jj,l,m,n] + + cg \[Delta][i,l] TwoPointFunction[\[CapitalDelta][3],d][cT,3,\[Mu]][jj,k,m,n] + + cg \[Delta][jj,k] TwoPointFunction[\[CapitalDelta][3],d][cT,3,\[Mu]][i,l,m,n] + + cg \[Delta][jj,l] TwoPointFunction[\[CapitalDelta][3],d][cT,3,\[Mu]][i,k,m,n])] + + 2 Contract[p[1][jj](cg \[Delta][i,m]TwoPointFunction[\[CapitalDelta][2],d,\[Mu]][cT,2][jj,n,k,l] + + cg \[Delta][i,n] TwoPointFunction[\[CapitalDelta][2],d][cT,2,\[Mu]][jj,m,k,l] + + cg \[Delta][jj,m] TwoPointFunction[\[CapitalDelta][2],d][cT,2,\[Mu]][i,n,k,l] + + cg \[Delta][jj,n] TwoPointFunction[\[CapitalDelta][2],d][cT,2,\[Mu]][i,m,k,l])] + /. { p[3][i]:>-p[1][i]-p[2][i], + p[1][k]:>-p[2][k]-p[3][k], p[1][l]:>-p[2][l]-p[3][l], + p[2][m]:>-p[1][m]-p[3][m], p[2][n]:>-p[1][n]-p[3][n] } ) }]; + + +PreTransverseWIs[JOO, params_] := PreTransverseWIs[JOO]@@({OONormalization, RenormalizationScale}/.params); +PreTransverseWIs[TOO, params_] := PreTransverseWIs[TOO]@@({OONormalization, RenormalizationScale}/.params); +PreTransverseWIs[JJO, params_] := PreTransverseWIs[JJO]; +PreTransverseWIs[TJO, params_] := PreTransverseWIs[TJO]; +PreTransverseWIs[TTO, params_] := PreTransverseWIs[TTO]; +PreTransverseWIs[JJJ, params_] := PreTransverseWIs[JJJ]@@({JJNormalization, GaugeCoupling, RenormalizationScale}/.params); +PreTransverseWIs[TJJ, params_] := PreTransverseWIs[TJJ]@@({JJNormalization, dTdADerivative, RenormalizationScale}/.params); +PreTransverseWIs[TTJ, params_] := PreTransverseWIs[TTJ]; +PreTransverseWIs[TTT, params_] := PreTransverseWIs[TTT]@@({CentralCharge, dTdgDerivative, RenormalizationScale}/.params); + + +fcall:TransverseWIs[X_, opts___?OptionQ] := Module[{ValidOpts, params, j}, + ValidOpts = First /@ Options[TransverseWIs]; + Scan[If[!MemberQ[ValidOpts, First[#]], + Message[TransverseWIs::optx, ToString[First[#]], ToString[Unevaluated[fcall]]]]&, Flatten[{opts}]]; + params = Table[First[Options[TransverseWIs][[j]]] -> + (First[Options[TransverseWIs][[j]]] /. Flatten[{opts}] /. Options[TransverseWIs]), + {j,Length[Options[TransverseWIs]]}]; + PreTransverseWIs[X, params] +]; +TransverseWIs[X_, x__] := Null /; Message[TransverseWIs::argrx, "TransverseWIs", Length[{x}] + 1, 1]; +TransverseWIs[] := Null /; Message[TransverseWIs::argrx, "TransverseWIs", Length[{X}], 1]; + + +PrePrimarySolutions[OOO][A_, \[Alpha]_] := { A[1] -> J[0,{0,0,0}] \[Alpha][1] }; +PrePrimarySolutions[JOO][A_, \[Alpha]_] := { A[1] -> J[1,{0,0,0}] \[Alpha][1] }; +PrePrimarySolutions[TOO][A_, \[Alpha]_] := { A[1] -> J[2,{0,0,0}] \[Alpha][1] }; +PrePrimarySolutions[JJO][A_, \[Alpha]_] := { A[1] -> \[Alpha][1]J[2,{0,0,0}], + A[2] -> \[Alpha][1] J[1,{0,0,1}] + \[Alpha][2] J[0,{0,0,0}] }; +PrePrimarySolutions[TJO][A_, \[Alpha]_] := { A[1] -> \[Alpha][1] J[3,{0,0,0}], + A[2] -> 2 \[Alpha][1] J[2,{0,0,1}] + \[Alpha][2] J[1,{0,0,0}] }; +PrePrimarySolutions[TTO][A_, \[Alpha]_] := { A[1] -> \[Alpha][1] J[4,{0,0,0}], + A[2] -> 4 \[Alpha][1] J[3,{0,0,1}] + \[Alpha][2] J[2,{0,0,0}], + A[3] -> 2 \[Alpha][1] J[2,{0,0,2}] + \[Alpha][2] J[1,{0,0,1}] + \[Alpha][3]J[0,{0,0,0}] }; +PrePrimarySolutions[JJJ][A_, \[Alpha]_] := { A[1] -> \[Alpha][1] J[3,{0,0,0}], + A[2] -> \[Alpha][1] J[2,{0,0,1}] + \[Alpha][2] J[1,{0,0,0}] }; +PrePrimarySolutions[TJJ][A_, \[Alpha]_] := { A[1] -> \[Alpha][1] J[4,{0,0,0}], + A[2] -> \[Alpha][1] J[3,{1,0,0}] + \[Alpha][2] J[2,{0,0,0}], + A[3] -> 2 \[Alpha][1] J[3,{0,0,1}] + \[Alpha][3] J[2,{0,0,0}], + A[4] -> 2 \[Alpha][1] J[2,{0,1,1}] + \[Alpha][3](J[1,{0,1,0}]+J[1,{0,0,1}]) + \[Alpha][4] J[0,{0,0,0}] } ; +PrePrimarySolutions[TTJ][A_, \[Alpha]_] := { A[1] -> 0, + A[2] -> \[Alpha][2] J[3,{0,0,0}], + A[3] -> \[Alpha][3] J[3,{0,0,0}], + A[4] -> - \[Alpha][2] (J[2,{0,1,0}] + J[2,{1,0,0}]) + \[Alpha][4] J[1,{0,0,0}], + A[5] -> 2 \[Alpha][3] J[2,{0,1,0}] + (\[Alpha][2]+2 \[Alpha][3]) J[2,{1,0,0}] + \[Alpha][5] J[1,{0,0,0}] }; +PrePrimarySolutions[TTT][A_, \[Alpha]_] := { A[1] -> \[Alpha][1] J[6,{0,0,0}], + A[2] -> 4 \[Alpha][1] J[5,{0,0,1}] + \[Alpha][2] J[4,{0,0,0}], + A[3] -> 2 \[Alpha][1] J[4,{0,0,2}] + \[Alpha][2] J[3,{0,0,1}] + \[Alpha][3] J[2,{0,0,0}], + A[4] -> 8 \[Alpha][1] J[4,{1,1,0}] - 2 \[Alpha][2] J[3,{0,0,1}] + \[Alpha][4] J[2,{0,0,0}], + A[5] -> 8 \[Alpha][1] J[3,{1,1,1}] + 2 \[Alpha][2](J[2,{1,1,0}]+J[2,{1,0,1}]+J[2,{0,1,1}]) + \[Alpha][5] J[0,{0,0,0}] }; + + +fcall:PrimarySolutions[X_, opts___?OptionQ] := Module[{ValidOpts, A, alpha}, + ValidOpts = First /@ Options[PrimarySolutions]; + Scan[If[!MemberQ[ValidOpts, First[#]], + Message[PrimarySolutions::optx, ToString[First[#]], ToString[Unevaluated[fcall]]]]&, Flatten[{opts}]]; + {A, alpha} = {FormFactor, PrimaryConstant} /. 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(-1)^(-$CellContext`n) + 2^(3 + $CellContext`n) Gamma[$CellContext`n] TripleK`p[2]^(-2) + TripleK`p[3]^(-2) (TripleK`p[2]^(2 $CellContext`n) TripleK`p[3]^2 + + TripleK`p[2]^2 TripleK`p[3]^(2 $CellContext`n)) + Konformal`\[Alpha][32] + + Rational[-1, 2]^(-$CellContext`n) TripleK`v^2 + Gamma[-1 + $CellContext`n] ((2 EulerGamma^2 + Pi^2 + + 2 HarmonicNumber[-2 + $CellContext`n]^2 - 2 + HarmonicNumber[-2 + $CellContext`n, 2] - 4 EulerGamma Log[2] + + 2 Log[2]^2 - 8 EulerGamma Log[ + TripleK`p[2]] + 8 Log[2] Log[ + TripleK`p[2]] + 8 Log[ + TripleK`p[2]]^2 + + HarmonicNumber[-2 + $CellContext`n] ((-4) EulerGamma + Log[16] + + 8 Log[ + TripleK`p[2]])) + TripleK`p[2]^(-2 + 2 $CellContext`n) + (2 EulerGamma^2 + Pi^2 + + 2 HarmonicNumber[-2 + $CellContext`n]^2 - 2 + HarmonicNumber[-2 + $CellContext`n, 2] - 4 EulerGamma Log[2] + + 2 Log[2]^2 - 8 EulerGamma Log[ + TripleK`p[3]] + 8 Log[2] Log[ + TripleK`p[3]] + 8 Log[ + TripleK`p[3]]^2 + + HarmonicNumber[-2 + $CellContext`n] ((-4) EulerGamma + Log[16] + + 8 Log[ + TripleK`p[3]])) TripleK`p[3]^(-2 + 2 $CellContext`n)) + Konformal`\[Alpha][40] + (-1)^(-$CellContext`n) + 2^(2 + $CellContext`n) + Gamma[-1 + $CellContext`n] ( + TripleK`v (-EulerGamma + HarmonicNumber[-2 + $CellContext`n] + + Log[2] + 2 Log[ + TripleK`p[2]]) TripleK`p[2]^(-2 + 2 $CellContext`n) + + TripleK`v (-EulerGamma + HarmonicNumber[-2 + $CellContext`n] + + Log[2] + 2 Log[ + TripleK`p[3]]) TripleK`p[3]^(-2 + 2 $CellContext`n)) + Konformal`\[Alpha][41] + (-1)^(-$CellContext`n) + 2^(2 + $CellContext`n) Gamma[-1 + $CellContext`n] TripleK`p[2]^(-2) + TripleK`p[3]^(-2) (TripleK`p[2]^(2 $CellContext`n) TripleK`p[3]^2 + + TripleK`p[2]^2 TripleK`p[3]^(2 $CellContext`n)) + Konformal`\[Alpha][42])}, -2, 1, 1], + Editable->False]}]}], "}"}]], "Output", + CellChangeTimes->{ + 3.796581557181799*^9, {3.796581596757062*^9, 3.7965816224135294`*^9}, + 3.7965823150811434`*^9, 3.796582358057601*^9, 3.7966695203662076`*^9, + 3.796670088996612*^9, {3.7966703778750176`*^9, 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"Output",ExpressionUUID->"5473170f-511e-45e5-bef7-543b5faedf61"] +}, Open ]] +}, Closed]] +} +] +*) + diff --git a/TripleK.wl b/TripleK.wl new file mode 100644 index 0000000..e5b4837 --- /dev/null +++ b/TripleK.wl @@ -0,0 +1,1779 @@ +(* ::Package:: *) + +(* :Name: TripleK` *) + + +(* :Title: Tools for manipulation and evaluation of triple-K integrals and conformal correlation functions. *) + + +(* :Author: Adam Bzowski *) + + +(* :Summary: + This package provides basic tools for manipulation and evaluation of triple-K integrals and conformal correlation functions. + The package delivers tools for evaluation of triple-K integrals with integral and half-integral indices as well as divergences of arbitrary triple-K integrals. +*) + + +(* :Context: TripleK` *) + + +(* :Package Version: 1.0 *) + + +(* :History: + Version 1.0 by Adam Bzowski, May 2020. +*) + + +(* :Copyright: GNU General Public License v3.0, Adam Bzowski, 2020 *) + + +(* :Keywords: + Triple-K, + Conformal field theory, + Feynman diagrams, + Loop integrals, + Dimensional regularization +*) + + +(* :Mathematica Version: 11.2 *) + + +BeginPackage["TripleK`", "Global`"]; +Unprotect @@ Names["TripleK`*"]; +ClearAll @@ Names["TripleK`*"]; + + +(* ::Section:: *) +(*Interface*) + + +(* ::Text:: *) +(*Definitions.*) + + +p::usage = "\!\(\*SubsuperscriptBox[ +StyleBox[\"p\",\nFontSlant->\"Italic\"], +StyleBox[\"j\",\nFontSlant->\"Italic\"], \(\[Mu]\)]\) for \!\(\* +StyleBox[\"j\",\nFontSlant->\"Italic\"]\)=1,2 represent external momenta. +\!\(\* +StyleBox[SubscriptBox[\"p\", \"j\"],\nFontSlant->\"Italic\"]\) for \!\(\* +StyleBox[\"j\",\nFontSlant->\"Italic\"]\)=1,2,3 represent magnitudes of vectors \!\(\*SubsuperscriptBox[ +StyleBox[\"p\",\nFontSlant->\"Italic\"], \(1\), \(\[Mu]\)]\), \!\(\*SubsuperscriptBox[ +StyleBox[\"p\",\nFontSlant->\"Italic\"], \(2\), \(\[Mu]\)]\), and \!\(\*SubsuperscriptBox[ +StyleBox[\"p\",\nFontSlant->\"Italic\"], \(3\), \(\[Mu]\)]\)=-\!\(\*SubsuperscriptBox[ +StyleBox[\"p\",\nFontSlant->\"Italic\"], \(1\), \(\[Mu]\)]\)-\!\(\*SubsuperscriptBox[ +StyleBox[\"p\",\nFontSlant->\"Italic\"], \(2\), \(\[Mu]\)]\)."; +\[Delta]::usage = "\!\(\*SubscriptBox[\(\[Delta]\), \(\[Mu]\[Nu]\)]\) represents the Euclidean metric."; +d::usage = "\!\(\* +StyleBox[\"d\",\nFontSlant->\"Italic\"]\) represents the number of spacetime dimensions."; +\[Epsilon]::usage = "\[Epsilon] represents the regulator."; + + +LoopIntegral::usage = "LoopIntegral[\!\(\* +StyleBox[\"d\",\nFontSlant->\"Italic\"]\), {\!\(\*SubscriptBox[\(\[Delta]\), \(1\)]\),\!\(\*SubscriptBox[\(\[Delta]\), \(2\)]\),\!\(\*SubscriptBox[\(\[Delta]\), \(3\)]\)}][\!\(\* +StyleBox[\"numerator\",\nFontSlant->\"Italic\"]\), \!\(\* +StyleBox[\"k\",\nFontSlant->\"Italic\"]\)] represents the 3-point 1-loop integral over momentum \!\(\* +StyleBox[\"k\",\nFontSlant->\"Italic\"]\) in \!\(\* +StyleBox[\"d\",\nFontSlant->\"Italic\"]\) dimensions with denominator \!\(\*SuperscriptBox[ +StyleBox[\"k\",\nFontSlant->\"Italic\"], \(2 \*SubscriptBox[\(\[Delta]\), \(3\)]\)]\)(\!\(\* +StyleBox[\"k\",\nFontSlant->\"Italic\"]\)-\!\(\*SubscriptBox[ +StyleBox[\"p\",\nFontSlant->\"Italic\"], \(1\)]\)\!\(\*SuperscriptBox[\()\), \(2 \*SubscriptBox[\(\[Delta]\), \(2\)]\)]\)(\!\(\* +StyleBox[\"k\",\nFontSlant->\"Italic\"]\)+\!\(\*SubscriptBox[ +StyleBox[\"p\",\nFontSlant->\"Italic\"], \(2\)]\)\!\(\*SuperscriptBox[\()\), \(2 \*SubscriptBox[\(\[Delta]\), \(1\)]\)]\) and the given \!\(\* +StyleBox[\"numerator\",\nFontSlant->\"Italic\"]\). External momenta are \!\(\*SubsuperscriptBox[ +StyleBox[\"p\",\nFontSlant->\"Italic\"], \(1\), \(\[Mu]\)]\), \!\(\*SubsuperscriptBox[ +StyleBox[\"p\",\nFontSlant->\"Italic\"], \(2\), \(\[Mu]\)]\), and \!\(\*SubsuperscriptBox[ +StyleBox[\"p\",\nFontSlant->\"Italic\"], \(3\), \(\[Mu]\)]\)=-\!\(\*SubsuperscriptBox[ +StyleBox[\"p\",\nFontSlant->\"Italic\"], \(1\), \(\[Mu]\)]\)-\!\(\*SubsuperscriptBox[ +StyleBox[\"p\",\nFontSlant->\"Italic\"], \(2\), \(\[Mu]\)]\). +LoopIntegral[\!\(\* +StyleBox[\"d\",\nFontSlant->\"Italic\"]\), {\!\(\*SubscriptBox[\(\[Delta]\), \(1\)]\),\!\(\*SubscriptBox[\(\[Delta]\), \(2\)]\)}][\!\(\* +StyleBox[\"numerator\",\nFontSlant->\"Italic\"]\), \!\(\* +StyleBox[\"k\",\nFontSlant->\"Italic\"]\), \!\(\* +StyleBox[\"p\",\nFontSlant->\"Italic\"]\)] represents the 2-point 1-loop integral over momentum \!\(\* +StyleBox[\"k\",\nFontSlant->\"Italic\"]\) in \!\(\* +StyleBox[\"d\",\nFontSlant->\"Italic\"]\) dimensions with denominator \!\(\*SuperscriptBox[ +StyleBox[\"k\",\nFontSlant->\"Italic\"], \(2 \*SubscriptBox[\(\[Delta]\), \(1\)]\)]\)(\!\(\* +StyleBox[\"k\",\nFontSlant->\"Italic\"]\)-\!\(\* +StyleBox[\"p\",\nFontSlant->\"Italic\"]\)\!\(\*SuperscriptBox[\()\), \(2 \*SubscriptBox[\(\[Delta]\), \(2\)]\)]\) and the given \!\(\* +StyleBox[\"numerator\",\nFontSlant->\"Italic\"]\). External momenta are \!\(\*SuperscriptBox[ +StyleBox[\"p\",\nFontSlant->\"Italic\"], \(\[Mu]\)]\) and -\!\(\*SuperscriptBox[ +StyleBox[\"p\",\nFontSlant->\"Italic\"], \(\[Mu]\)]\)."; +i::usage = "\!\(\* +StyleBox[\"i\",\nFontSlant->\"Italic\"]\)[\!\(\* +StyleBox[\"\[Alpha]\",\nFontSlant->\"Italic\"]\),{\!\(\*SubscriptBox[ +StyleBox[\"\[Beta]\",\nFontSlant->\"Italic\"], \(1\)]\), \!\(\*SubscriptBox[ +StyleBox[\"\[Beta]\",\nFontSlant->\"Italic\"], \(2\)]\)}][\!\(\* +StyleBox[\"p\",\nFontSlant->\"Italic\"]\)] represents the double-\!\(\* +StyleBox[\"K\",\nFontSlant->\"Italic\"]\) integral with parameters \[Alpha], \!\(\*SubscriptBox[\(\[Beta]\), \(1\)]\), \!\(\*SubscriptBox[\(\[Beta]\), \(2\)]\) and momentum \!\(\*SuperscriptBox[ +StyleBox[\"p\",\nFontSlant->\"Italic\"], \(\[Mu]\)]\). +\!\(\* +StyleBox[\"i\",\nFontSlant->\"Italic\"]\)[\!\(\* +StyleBox[\"\[Alpha]\",\nFontSlant->\"Italic\"]\),{\!\(\*SubscriptBox[ +StyleBox[\"\[Beta]\",\nFontSlant->\"Italic\"], \(1\)]\), \!\(\*SubscriptBox[ +StyleBox[\"\[Beta]\",\nFontSlant->\"Italic\"], \(2\)]\), \!\(\*SubscriptBox[ +StyleBox[\"\[Beta]\",\nFontSlant->\"Italic\"], \(3\)]\)}][\!\(\*SubscriptBox[ +StyleBox[\"p\",\nFontSlant->\"Italic\"], \(1\)]\),\!\(\*SubscriptBox[ +StyleBox[\"p\",\nFontSlant->\"Italic\"], \(2\)]\),\!\(\*SubscriptBox[ +StyleBox[\"p\",\nFontSlant->\"Italic\"], \(3\)]\)] represents the triple-\!\(\* +StyleBox[\"K\",\nFontSlant->\"Italic\"]\) integral with parameters \[Alpha], \!\(\*SubscriptBox[\(\[Beta]\), \(1\)]\), \!\(\*SubscriptBox[\(\[Beta]\), \(2\)]\), \!\(\*SubscriptBox[\(\[Beta]\), \(3\)]\) and momentum magnitudes \!\(\*SubscriptBox[ +StyleBox[\"p\",\nFontSlant->\"Italic\"], \(1\)]\), \!\(\*SubscriptBox[ +StyleBox[\"p\",\nFontSlant->\"Italic\"], \(2\)]\), \!\(\*SubscriptBox[ +StyleBox[\"p\",\nFontSlant->\"Italic\"], \(3\)]\)."; +NL::usage = "\!\(\* +StyleBox[\"NL\",\nFontSlant->\"Italic\"]\) represents the non-local part of the 3-point function."; +\[Lambda]::usage = "\[Lambda] represents the K\[ADoubleDot]llen lambda function."; + + +(* ::Text:: *) +(*Simplifications.*) + + +Swap::usage = "Swap[\!\(\* +StyleBox[\"expr\",\nFontSlant->\"Italic\"]\), \!\(\* +StyleBox[\"x\",\nFontSlant->\"Italic\"]\), \!\(\* +StyleBox[\"y\",\nFontSlant->\"Italic\"]\)] swaps \!\(\* +StyleBox[\"x\",\nFontSlant->\"Italic\"]\) and \!\(\* +StyleBox[\"y\",\nFontSlant->\"Italic\"]\) in \!\(\* +StyleBox[\"expr\",\nFontSlant->\"Italic\"]\)."; + + +KToIntegrand::usage = "KToIntegrand[\!\(\* +StyleBox[\"expr\",\nFontSlant->\"Italic\"]\), \!\(\* +StyleBox[\"x\",\nFontSlant->\"Italic\"]\)] replaces each multiple-\!\(\* +StyleBox[\"K\",\nFontSlant->\"Italic\"]\) integral in \!\(\* +StyleBox[\"expr\",\nFontSlant->\"Italic\"]\) by its integrand with the integration variable \!\(\* +StyleBox[\"x\",\nFontSlant->\"Italic\"]\)."; + + +KSimplify::usage = "KSimplify[\!\(\* +StyleBox[\"expr\",\nFontSlant->\"Italic\"]\)] simplifies multiple-\!\(\* +StyleBox[\"K\",\nFontSlant->\"Italic\"]\) integrals and loop integerals in \!\(\* +StyleBox[\"expr\",\nFontSlant->\"Italic\"]\)."; + + +KExpand::usage = "KExpand[\!\(\* +StyleBox[\"expr\",\nFontSlant->\"Italic\"]\)] reduces derivatives of multiple-\!\(\* +StyleBox[\"K\",\nFontSlant->\"Italic\"]\) integrals to a combination of more elementary functions."; + + +KExpand::lev = "Level `1` is an invalid expansion level. Level 1 assumed."; + + +KFullExpand::usage = "KFullExpand[\!\(\* +StyleBox[\"expr\",\nFontSlant->\"Italic\"]\)] expands \!\(\* +StyleBox[\"NL\",\nFontSlant->\"Italic\"]\), its derivatives, and derivatives of multiple-\!\(\* +StyleBox[\"K\",\nFontSlant->\"Italic\"]\) integrals."; + + +PolyGammaToHarmonic::usage = "PolyGammaToHarmonic[\!\(\* +StyleBox[\"expr\",\nFontSlant->\"Italic\"]\)] replaces all PolyGamma functions of integral and half-integral arguments in \!\(\* +StyleBox[\"expr\",\nFontSlant->\"Italic\"]\) by harmonic numbers."; + + +TripleK::nonlin = "Warning: `1` parameter in `2` is non-linear in the regulator, \[Epsilon]. Expansion may be invalid."; + + +(* ::Text:: *) +(*Momentum manipulations.*) + + +Contract::usage = "Contract[\!\(\* +StyleBox[\"expr\",\nFontSlant->\"Italic\"]\)] contracts all repeated indices of recognized vectors in \!\(\* +StyleBox[\"expr\",\nFontSlant->\"Italic\"]\)\!\(\* +StyleBox[\".\",\nFontSlant->\"Italic\"]\) +Contract[\!\(\* +StyleBox[\"expr\",\nFontSlant->\"Italic\"]\), \[Mu]] contracts all repeated \[Mu]'s in \!\(\* +StyleBox[\"expr\",\nFontSlant->\"Italic\"]\). +Contract[\!\(\* +StyleBox[\"expr\",\nFontSlant->\"Italic\"]\), {\[Mu], ...}] contracts all repreated \[Mu], ... in \!\(\* +StyleBox[\"expr\",\nFontSlant->\"Italic\"]\). +Contract[\!\(\* +StyleBox[\"expr\",\nFontSlant->\"Italic\"]\), \[Mu], \[Nu]] contracts indices \[Mu] and \[Nu] in \!\(\* +StyleBox[\"expr\",\nFontSlant->\"Italic\"]\)."; + + +Contract::argbt = "`1` called with `2` arguments; between `3` and `4` arguments are expected."; + + +Diff::usage = "Diff[\!\(\* +StyleBox[\"f\",\nFontSlant->\"Italic\"]\), \!\(\* +StyleBox[\"p\",\nFontSlant->\"Italic\"]\), \[Mu]] calculates the derivative of \!\(\* +StyleBox[\"f\",\nFontSlant->\"Italic\"]\) with respect to \!\(\* +StyleBox[\"p\",\nFontSlant->\"Italic\"]\)[\[Mu]]."; + + +Diff::darg = "Cannot differentiate over the variable `1`, which appears as the integration variable in `2`."; + + +(* ::Text:: *) +(*Momentum integrals to multiple-K.*) + + +LoopToK::usage = "LoopToK[\!\(\* +StyleBox[\"expr\",\nFontSlant->\"Italic\"]\)] replaces all loop integrals by multiple-\!\(\* +StyleBox[\"K\",\nFontSlant->\"Italic\"]\) integrals."; + + +LoopToK::barg = "Option `1` `2` should be a Boolean variable."; +LoopToK::loopfail = "Warning: some loop integrals in `1` have not been converted to triple-K integrals. The resulting expression is likely to be incorrect."; + + +(* ::Text:: *) +(*Divergences.*) + + +IsDivergent::usage = "IsDivergent[i[\[Alpha],{\!\(\*SubscriptBox[\(\[Beta]\), \(1\)]\),...}]] tests if the multiple-\!\(\* +StyleBox[\"K\",\nFontSlant->\"Italic\"]\)\!\(\* +StyleBox[\" \",\nFontSlant->\"Italic\"]\)integral i[\[Alpha],{\!\(\*SubscriptBox[\(\[Beta]\), \(1\)]\),...}] is divergent."; + + +KDivergence::usage = "KDivergence[\!\(\* +StyleBox[\"expr\",\nFontSlant->\"Italic\"]\)] calculates divergent and scheme dependent terms in the \[Epsilon]-expansion of \!\(\* +StyleBox[\"expr\",\nFontSlant->\"Italic\"]\)."; + + +KDivergence::iarg = "Option ExpansionOrder `1` should be an Integer."; +KDivergence::targ = "Option Type `1` should be All, a list of three signs, {\[PlusMinus]1, \[PlusMinus]1, \[PlusMinus]1}, or a list of such lists."; +KDivergence::ksing = "Warning: expression may be indeterminate since it exhibits the `1` singularity."; + + +(* ::Text:: *) +(*Reduction scheme.*) + + +KEvaluate::usage = "KEvaluate[\!\(\* +StyleBox[\"expr\",\nFontSlant->\"Italic\"]\)] evaluates all multiple-\!\(\* +StyleBox[\"K\",\nFontSlant->\"Italic\"]\) integrals in \!\(\* +StyleBox[\"expr\",\nFontSlant->\"Italic\"]\)."; + + +LoopEvaluate::usage = "LoopEvaluate[\!\(\* +StyleBox[\"expr\",\nFontSlant->\"Italic\"]\)] evaluates all momentum space loop integrals in \!\(\* +StyleBox[\"expr\",\nFontSlant->\"Italic\"]\)."; + + +IsSolvable::usage = "IsSolvable[\!\(\* +StyleBox[\"i\",\nFontSlant->\"Italic\"]\)[\[Alpha],{\!\(\*SubscriptBox[\(\[Beta]\), \(1\)]\),...}]] returns True if the multiple-\!\(\* +StyleBox[\"K\",\nFontSlant->\"Italic\"]\) integral \!\(\*SubscriptBox[ +StyleBox[\"i\",\nFontSlant->\"Italic\"], \(\[Alpha], {\!\(\*SubscriptBox[\(\[Beta]\), \(1\)]\),...}\)]\) is computable by KEvaluate. +IsSolvable[\!\(\* +StyleBox[\"i\",\nFontSlant->\"Italic\"]\)[\[Alpha],{\!\(\*SubscriptBox[\(\[Beta]\), \(1\)]\),...}][\!\(\*SubscriptBox[\(p\), \(1\)]\),\!\(\*SubscriptBox[\(p\), \(2\)]\),\!\(\*SubscriptBox[\(p\), \(3\)]\)]] returns True if the multiple-\!\(\* +StyleBox[\"K\",\nFontSlant->\"Italic\"]\) integral \!\(\*SubscriptBox[ +StyleBox[\"i\",\nFontSlant->\"Italic\"], \(\[Alpha], {\!\(\*SubscriptBox[\(\[Beta]\), \(1\)]\),...}\)]\)[\!\(\*SubscriptBox[\(p\), \(1\)]\),\!\(\*SubscriptBox[\(p\), \(2\)]\),\!\(\*SubscriptBox[\(p\), \(3\)]\)] is computable by KEvaluate."; + + +(* ::Text:: *) +(*Conformal operators.*) + + +ScalarKOp::usage = "KOp[\!\(\* +StyleBox[\"expr\",\nFontSlant->\"Italic\"]\), \!\(\* +StyleBox[\"p\",\nFontSlant->\"Italic\"]\), \[Beta]] applies single K\!\(\* +StyleBox[\"(\",\nFontSize->12,\nFontSlant->\"Italic\"]\)\!\(\* +StyleBox[\"\[Beta]\",\nFontSize->12,\nFontSlant->\"Italic\"]\)\!\(\* +StyleBox[\")\",\nFontSize->12,\nFontSlant->\"Italic\"]\)\!\(\* +StyleBox[\" \",\nFontSize->12]\) operator to \!\(\* +StyleBox[\"expr\",\nFontSlant->\"Italic\"]\) with respect to momentum magnitude \!\(\* +StyleBox[\"p\",\nFontSlant->\"Italic\"]\) and with parameter \[Beta]."; +ScalarKKOp::usage = "KKOp[\!\(\* +StyleBox[\"expr\",\nFontSlant->\"Italic\"]\), \!\(\* +StyleBox[SubscriptBox[\"p\", \"i\"],\nFontSlant->\"Italic\"]\), \!\(\* +StyleBox[SubscriptBox[\"p\", \"j\"],\nFontSlant->\"Italic\"]\), \!\(\*SubscriptBox[\(\[Beta]\), +StyleBox[\"i\",\nFontSlant->\"Italic\"]]\), \!\(\*SubscriptBox[\(\[Beta]\), +StyleBox[\"j\",\nFontSlant->\"Italic\"]]\)] = KOp[\!\(\* +StyleBox[\"expr\",\nFontSlant->\"Italic\"]\), \!\(\* +StyleBox[SubscriptBox[\"p\", \"i\"],\nFontSlant->\"Italic\"]\), \!\(\*SubscriptBox[\(\[Beta]\), +StyleBox[\"i\",\nFontSlant->\"Italic\"]]\)] - KOp[\!\(\* +StyleBox[\"expr\",\nFontSlant->\"Italic\"]\), \!\(\* +StyleBox[SubscriptBox[\"p\", \"j\"],\nFontSlant->\"Italic\"]\), \!\(\*SubscriptBox[\(\[Beta]\), +StyleBox[\"j\",\nFontSlant->\"Italic\"]]\)] applies the conformal Ward identity operator in its scalar form."; + + +(* ::Text:: *) +(*Options.*) + + +Options[KExpand] = { Level -> 1 }; +Options[KSimplify] = { Assumptions :> $Assumptions }; +Options[PolyGammaToHarmonic] = { Assumptions :> $Assumptions }; +Options[Contract] = { Dimension -> d, Vectors -> Automatic, Indices -> All }; +Options[LoopToK] = { Recursive -> True }; +Options[IsDivergent] = { Assumptions :> $Assumptions }; +Options[KDivergence] = { Type -> All, ExpansionOrder -> 0, uParameter -> u, vParameters -> {v,v,v}, Assumptions :> $Assumptions }; +Options[KEvaluate] = { uParameter -> u, vParameters -> {v,v,v}, ExpansionLevel -> 1 }; +Options[LoopEvaluate] = { uParameter -> u, vParameters -> {v,v,v}, ExpansionLevel -> 1, Recursive -> True }; + + +Begin["TripleK`Private`"]; + + +(* ::Section:: *) +(*Definitions*) + + +Attributes[p] = { NHoldAll }; +Attributes[\[Delta]] = { Orderless }; +Attributes[NL] = { Orderless, NumericFunction }; +Attributes[\[Lambda]] = { Orderless, NumericFunction }; +Attributes[i] = { NumericFunction }; +Attributes[CenterDot] = { Orderless, OneIdentity }; + + +Attributes[NLfun] = { Orderless, NumericFunction }; +Attributes[\[Lambda]fun] = { Orderless, NumericFunction }; +Attributes[Xfun] = { NumericFunction }; +Attributes[Yfun] = { NumericFunction }; + + +NL /: N[NL[a_,b_,c_], opts___] := N[NLfun[a,b,c], opts]; +\[Lambda] /: N[\[Lambda][a_,b_,c_], opts___] := N[\[Lambda]fun[a,b,c], opts]; +i /: N[i[a_,{b1_,b2_}][p_], opts___] := N[DoubleKValue[a,{b1,b2}][p], opts]; +i /: N[i[a_,{b1_,b2_,b3_}][p1_,p2_,p3_], opts___] /; 00, p[2]>0, p[3]>0 }, + Union[$Assumptions, { p[1]>0, p[2]>0, p[3]>0 }]]; + + +Format[p[j_][\[Mu]_]] := Subsuperscript[p,j,\[Mu]]; +Format[p[j_]] := Subscript[p,j]; +Format[\[Delta][i_,j_]] := Subscript[\[Delta],i,j]; +Format[i[a_,b_]] := Subscript[i,a,b]; +Format[i[a_,b_][p[1],p[2],p[3]]] := Subscript[i,a,b]; +Format[i[a_,b_][Subscript[p,1],Subscript[p,2],Subscript[p,3]]] := Subscript[i,a,b]; +Format[NL[p[1],p[2],p[3]]] = NL; +Format[\[Lambda][p[1],p[2],p[3]]] = \[Lambda]; + + +Format[LoopIntegral[d_,{\[Delta]1_,\[Delta]2_,\[Delta]3_}][num_,k_]] := With[{numref=num /. { k[\[Mu]_]->Superscript[k,\[Mu]] }, pow1=2\[Delta]1, pow2=2\[Delta]2, pow3=2\[Delta]3}, + HoldForm["\!\(\*StyleBox[\"\[Integral]\",FontSize->18]\)"*HoldForm[("\[DifferentialD]"^d*k)/(2*Pi)^d]*HoldForm[numref/(k^(pow3) (k-Subscript[p, 1])^(pow2) (k+Subscript[p, 2])^(pow1))]]]; +Format[LoopIntegral[d_,{\[Delta]1_,\[Delta]2_}][num_,k_,-p_]] := With[{numref=num /. { k[\[Mu]_]->Superscript[k,\[Mu]] }, pow1=2\[Delta]1, pow2=2\[Delta]2}, + HoldForm["\!\(\*StyleBox[\"\[Integral]\",FontSize->18]\)"*HoldForm[("\[DifferentialD]"^d*k)/(2*Pi)^d]*HoldForm[numref/(k^(pow1) (k+p)^(pow2))]]]; +Format[LoopIntegral[d_,{\[Delta]1_,\[Delta]2_}][num_,k_,p_]] := With[{numref=num /. { k[\[Mu]_]->Superscript[k,\[Mu]] }, pow1=2\[Delta]1, pow2=2\[Delta]2}, + HoldForm["\!\(\*StyleBox[\"\[Integral]\",FontSize->18]\)"*HoldForm[("\[DifferentialD]"^d*k)/(2*Pi)^d]*HoldForm[numref/(k^(pow1) (k-p)^(pow2))]]]; + + +(* ::Section:: *) +(*Simplifications*) + + +Swap[exp_,toswap1stterm_,toswap2ndterm_] := Module[{toswap3rdterm},((exp/.toswap1stterm->toswap3rdterm)/.toswap2ndterm->toswap1stterm)/.toswap3rdterm->toswap2ndterm]; +Swap[x___] := Null /; Message[Swap::argrx, "Swap", Length[{x}], 3]; + + +KToIntegrand[exp_, x_] := exp /. { + i[a_, {b1_,b2_}][p_] :> x^a p^(b1+b2) BesselK[b1,p x] BesselK[b2,p x], + i[a_, {b1_,b2_,b3_}][p1_,p2_,p3_] :> + x^a p1^b1 p2^b2 p3^b3 BesselK[b1, p1 x] BesselK[b2, p2 x] BesselK[b3, p3 x] + } /. { + i[a_, {b1_,b2_,b3_}] :> + x^a p[1]^b1 p[2]^b2 p[3]^b3 BesselK[b1, p[1] x] BesselK[b2, p[2] x] BesselK[b3, p[3] x] + }; + + +KToIntegrand[x___] := Null /; Message[KToIntegrand::argrx, "KToIntegrand", Length[{x}], 2]; + + +suMomentumUnformat = { + Subsuperscript[p, j_, \[Mu]_] :> p[j][\[Mu]], + Superscript[Subscript[p, j_], \[Mu]_] :> p[j][\[Mu]], + Subscript[p, j_] :> p[j], + Superscript[p[j_], \[Mu]_] :> p[j][\[Mu]] }; + + +Unformat[exp_] := Module[{tmpNL,tmp\[Lambda],ii}, + exp /. { Subscript[i, a_, b_] :> i[a,b], + Subsuperscript[p, j_, \[Mu]_] :> p[j][\[Mu]], + Superscript[Subscript[p, j_], \[Mu]_] :> p[j][\[Mu]], + Subscript[p, j_] :> p[j], + Superscript[p[j_], \[Mu]_] :> p[j][\[Mu]], + Subscript[\[Delta], i_, j_] :> \[Delta][i,j], + NL[x__] :> tmpNL[x], + Derivative[i__][NL][x__] :> Derivative[i][tmpNL][x], + \[Lambda][x__] :> tmp\[Lambda][x], + Derivative[i__][\[Lambda]][x__] :> Derivative[i][tmp\[Lambda]][x] } + /. { NL -> NL[p[1],p[2],p[3]], \[Lambda] -> \[Lambda][p[1],p[2],p[3]] } + /. { tmpNL -> NL, tmp\[Lambda] -> \[Lambda] } + /. { i[a_,b_][x___] :> ii[a,b][x], + Derivative[k_,m_,n_][i[a_,b_]][x___] :> Derivative[k,m,n][ii[a,b]][x] } + /. { i[a_,b_] :> i[a,b][p[1],p[2],p[3]] } + /. { ii -> i } +]; + + +Unformat[x___] := Null /; Message[Unformat::argrx, "Unformat", Length[{x}], 1]; + + +suKDiff = { + Derivative[k_,0,0][i[a_,{b1_,b2_,b3_}]][p1_,p2_,p3_] :> + -D[p1 * i[a+1, {b1-1,b2,b3}][p1,p2,p3],{p1,k-1}], + Derivative[k_,m_,0][i[a_,{b1_,b2_,b3_}]][p1_,p2_,p3_] :> + -D[p2 * Derivative[k,0,0][i[a+1, {b1,b2-1,b3}]][p1,p2,p3],{p2,m-1}], + Derivative[k_,m_,n_][i[a_,{b1_,b2_,b3_}]][p1_,p2_,p3_] :> + -D[p3 * Derivative[k,m,0][i[a+1, {b1,b2,b3-1}]][p1,p2,p3],{p3,n-1}] }; + + +suKDiffReg = { + Derivative[k_,0,0][i[a_,{b1_,b2_,b3_}, u_,v_]][p1_,p2_,p3_] :> + -D[p1 * i[a+1, {b1-1,b2,b3}, u,v][p1,p2,p3],{p1,k-1}], + Derivative[k_,m_,0][i[a_,{b1_,b2_,b3_}], u_,v_][p1_,p2_,p3_] :> + -D[p2 * Derivative[k,0,0][i[a+1, {b1,b2-1,b3}, u,v]][p1,p2,p3],{p2,m-1}], + Derivative[k_,m_,n_][i[a_,{b1_,b2_,b3_}], u_,v_][p1_,p2_,p3_] :> + -D[p3 * Derivative[k,m,0][i[a+1, {b1,b2,b3-1}, u,v]][p1,p2,p3],{p3,n-1}] }; + + +suKNegative[assume___] := { + i[a_,{b1_,b2_,b3_}][p1_,p2_,p3_] :> + p1^(2b1) i[a,{-b1,b2,b3}][p1,p2,p3] /; Simplify[b1<0 /. \[Epsilon]->0,assume, TimeConstraint->0.1], + i[a_,{b1_,b2_,b3_}][p1_,p2_,p3_] :> + p2^(2b2) i[a,{b1,-b2,b3}][p1,p2,p3] /; Simplify[b2<0 /. \[Epsilon]->0,assume, TimeConstraint->0.1], + i[a_,{b1_,b2_,b3_}][p1_,p2_,p3_] :> + p3^(2b3) i[a,{b1,b2,-b3}][p1,p2,p3] /; Simplify[b3<0 /. \[Epsilon]->0,assume, TimeConstraint->0.1] }; + + +suKNegativeReg = { + i[a_,{b1_,b2_,b3_}, u_,{v1_,v2_,v3_}][p1_,p2_,p3_] :> + p1^(2b1) i[a,{-b1,b2,b3}, u,{-v1,v2,v3}][p1,p2,p3] /; b1<0, + i[a_,{b1_,b2_,b3_}, u_,{v1_,v2_,v3_}][p1_,p2_,p3_] :> + p2^(2b2) i[a,{b1,-b2,b3}, u,{v1,-v2,v3}][p1,p2,p3] /; b2<0, + i[a_,{b1_,b2_,b3_}, u_,{v1_,v2_,v3_}][p1_,p2_,p3_] :> + p3^(2b3) i[a,{b1,b2,-b3}, u,{v1,v2,-v3}][p1,p2,p3] /; b3<0 }; + + +suKOrder[assume___] := { + i[a_,{b1_,b2_,b3_}][p1_,p2_,p3_] :> i[a,{b1,b3,b2}][p1,p3,p2] /; Simplify[b20,assume, TimeConstraint->0.1], + i[a_,{b1_,b2_,b3_}][p1_,p2_,p3_] :> i[a,{b2,b1,b3}][p2,p1,p3] /; Simplify[b10,assume, TimeConstraint->0.1] }; + + +suKInverseOrder = { + i[a_,{b1_,b3_,b2_}][p[1],p[3],p[2]] :> i[a,{b1,b2,b3}][p[1],p[2],p[3]], + i[a_,{b3_,b2_,b1_}][p[3],p[2],p[1]] :> i[a,{b1,b2,b3}][p[1],p[2],p[3]], + i[a_,{b2_,b1_,b3_}][p[2],p[1],p[3]] :> i[a,{b1,b2,b3}][p[1],p[2],p[3]], + i[a_,{b2_,b3_,b1_}][p[2],p[3],p[1]] :> i[a,{b1,b2,b3}][p[1],p[2],p[3]], + i[a_,{b3_,b1_,b2_}][p[3],p[1],p[2]] :> i[a,{b1,b2,b3}][p[1],p[2],p[3]] }; + + +suKOrderReg = { + i[a_,{b1_,b2_,b3_}, u_,{v1_,v2_,v3_}][p1_,p2_,p3_] :> i[a,{b1,b3,b2}, u,{v1,v3,v2}][p1,p3,p2] /; b2 i[a,{b2,b1,b3}, u,{v2,v1,v3}][p2,p1,p3] /; b1 + c p1^2 i[a,{b1p1-1,b2,b3}][p1,p2,p3] /; Simplify[b1p2-b1p1==1 && a-am1==1 && cmtwob1p1==-2b1p1*c, assume, TimeConstraint->0.1], + c_. i[a_,{b1_,b2p2_,b3_}][p1_,p2_,p3_] + cmtwob2p1_. i[am1_,{b1_,b2p1_,b3_}][p1_,p2_,p3_] :> + c p2^2 i[a,{b1,b2p1-1,b3}][p1,p2,p3] /; Simplify[b2p2-b2p1==1 && a-am1==1 && cmtwob2p1==-2b2p1*c, assume, TimeConstraint->0.1], + c_. i[a_,{b1_,b2_,b3p2_}][p1_,p2_,p3_] + cmtwob3p1_. i[am1_,{b1_,b2_,b3p1_}][p1_,p2_,p3_] :> + c p3^2 i[a,{b1,b2,b3p1-1}][p1,p2,p3] /; Simplify[b3p2-b3p1==1 && a-am1==1 && cmtwob3p1==-2b3p1*c, assume, TimeConstraint->0.1], + cp_. i[a_,{b1m1_,b2_,b3_}][p1_,p2_,p3_] + c_. i[am1_,{b1_,b2_,b3_}][p1_,p2_,p3_] :> + c/(2b1) i[a,{b1+1,b2,b3}][p1,p2,p3] /; Simplify[b1-b1m1==1 && a-am1==1 && 2b1 cp==c*p1^2, assume, TimeConstraint->0.1], + cp_. i[a_,{b1_,b2m1_,b3_}][p1_,p2_,p3_] + c_. i[am1_,{b1_,b2_,b3_}][p1_,p2_,p3_] :> + c/(2b2) i[a,{b1,b2+1,b3}][p1,p2,p3] /; Simplify[b2-b2m1==1 && a-am1==1 && 2b2 cp==c*p2^2, assume, TimeConstraint->0.1], + cp_. i[a_,{b1_,b2_,b3m1_}][p1_,p2_,p3_] + c_. i[am1_,{b1_,b2_,b3_}][p1_,p2_,p3_] :> + c/(2b3) i[a,{b1,b2,b3+1}][p1,p2,p3] /; Simplify[b3-b3m1==1 && a-am1==1 && 2b3 cp==c*p3^2, assume, TimeConstraint->0.1], + c_. i[a_,{b1p1_,b2_,b3_}][p1_,p2_,p3_] + cmp1_. i[a_,{b1m1_,b2_,b3_}][p1_,p2_,p3_] :> + 2 c (b1p1-1) i[a-1,{b1p1-1,b2,b3}][p1,p2,p3] /; Simplify[b1p1-b1m1==2 && cmp1==-c p1^2, assume, TimeConstraint->0.1], + c_. i[a_,{b1_,b2p1_,b3_}][p1_,p2_,p3_] + cmp2_. i[a_,{b1_,b2m1_,b3_}][p1_,p2_,p3_] :> + 2 c (b2p1-1) i[a-1,{b1,b2p1-1,b3}][p1,p2,p3] /; Simplify[b2p1-b2m1==2 && cmp2==-c p2^2, assume, TimeConstraint->0.1], + c_. i[a_,{b1_,b2_,b3p1_}][p1_,p2_,p3_] + cmp3_. i[a_,{b1_,b2_,b3m1_}][p1_,p2_,p3_] :> + 2 c (b3p1-1) i[a-1,{b1,b2,b3p1-1}][p1,p2,p3] /; Simplify[b3p1-b3m1==2 && cmp3==-c p3^2, assume, TimeConstraint->0.1], + c1_. i[ap1_,{b1m1_,b2_,b3_}][p1_,p2_,p3_] + c2_. i[ap1_,{b1_,b2m1_,b3_}][p1_,p2_,p3_] + c3_. i[ap1_,{b1_,b2_,b3m1_}][p1_,p2_,p3_] :> + c1/p[1]^2 * (ap1-b1-b2-b3) * i[ap1-1, {b1,b2,b3}][p1,p2,p3] + /; Simplify[b1-b1m1==1 && b2-b2m1==1 && b3-b3m1==1 && c1 p[2]^2==c2 p[1]^2 && c1 p[3]^2==c3 p[1]^2 && !TrueQ[IsDivergent[i[ap1-1,{b1,b2,b3}], assume]], assume, TimeConstraint->0.1] +}; + + +suMomentum3Identities[assume___] := { + c3_. p[3][\[Mu]_] + c1_. p[1][\[Mu]_] :> -c1 p[2][\[Mu]]/; Simplify[c1==c3,assume,TimeConstraint->0.1], + c3_. p[3][\[Mu]_] + c2_. p[2][\[Mu]_] :> -c2 p[1][\[Mu]]/; Simplify[c2==c3,assume,TimeConstraint->0.1], + c2_. p[2][\[Mu]_] + c1_. p[1][\[Mu]_] :> -c1 p[3][\[Mu]]/; Simplify[c1==c2,assume,TimeConstraint->0.1] + }; + + +SimplifyFunctionKIdentities[assume_] := # /. Union[ + suKIdentities[assume], + suKNegative[assume], + suKOrder[assume], + suMomentum3Identities[assume]]&; + + +suMomentum = { + (x_+y_)[j_] :> x[j]+y[j], + (a_*x_)[j_] :> a*x[j] /; NumericQ[a], + (-x_)[j_] :> -x[j], + 0[j_] :> 0, + x_\[CenterDot](y_+z_) :> x\[CenterDot]y + x\[CenterDot]z, + x_\[CenterDot](a_*y_) :> a(x\[CenterDot]y) /; NumericQ[a], + x_\[CenterDot](-y_) :> -(x\[CenterDot]y), + x_\[CenterDot]0 :> 0 }; + + +suMomentumEx[vector_] := { + (x_+y_)[j_] :> x[j]+y[j], + (a_*x_)[j_] :> a*x[j] /; FreeQ[a,p] && FreeQ[a,vector], + (-x_)[j_] :> -x[j], + 0[j_] :> 0, + x_\[CenterDot](y_+z_) :> x\[CenterDot]y + x\[CenterDot]z, + x_\[CenterDot](a_*y_) :> a(x\[CenterDot]y) /; FreeQ[a,p] && FreeQ[a,vector], + x_\[CenterDot](-y_) :> -(x\[CenterDot]y), + x_\[CenterDot]0 :> 0 }; + + +suMomentumEx[vectors_List] := { + (x_+y_)[j_] :> x[j]+y[j], + (a_*x_)[j_] :> a*x[j] /; FreeQ[a,p] && AllTrue[vectors, FreeQ[a,#]&], + (-x_)[j_] :> -x[j], + 0[j_] :> 0, + x_\[CenterDot](y_+z_) :> x\[CenterDot]y + x\[CenterDot]z, + x_\[CenterDot](a_*y_) :> a(x\[CenterDot]y) /; FreeQ[a,p] && AllTrue[vectors, FreeQ[a,#]&], + x_\[CenterDot](-y_) :> -(x\[CenterDot]y), + x_\[CenterDot]0 :> 0 }; + + +suProductsToMagnitudes = { + p[2]\[CenterDot]p[3] -> 1/2 (p[1]^2 - p[2]^2 - p[3]^2), + p[1]\[CenterDot]p[3] -> 1/2 (p[2]^2 - p[1]^2 - p[3]^2), + p[1]\[CenterDot]p[2] -> 1/2 (p[3]^2 - p[2]^2 - p[1]^2), + Sqrt[p[n_]\[CenterDot]p[n_]] :> p[n], + Sqrt[p\[CenterDot]p] -> p, + Sqrt[p[n_]^2] :> p[n], + Sqrt[p^2] -> p, + p[n_]\[CenterDot]p[n_] :> p[n]^2, + p\[CenterDot]p -> p^2 }; + + +suMomentum3 = { + p[3][\[Mu]_] :> -p[1][\[Mu]]-p[2][\[Mu]], + p[3]\[CenterDot]x_ :> -(p[1]\[CenterDot]x)-(p[2]\[CenterDot]x) }; + + +suLoopLinearity = { + LoopIntegral[x__][X_ + Y_, y__] :> LoopIntegral[x][X,y] + LoopIntegral[x][Y,y], + LoopIntegral[x__][-X_, y__] :> -LoopIntegral[x][X,y], + LoopIntegral[x__][0, y__] :> 0, + LoopIntegral[x__][a_*X_, k_, y___] :> a*LoopIntegral[x][X,k,y] /; FreeQ[a,k] && a =!= 1, + LoopIntegral[x__][a_, k_, y___] :> a*LoopIntegral[x][1,k,y] /; FreeQ[a,k] && a =!= 1 +}; + + +suLoopReduction = { + LoopIntegral[d_, {\[Delta]1_,\[Delta]2_}][x_.*(k_\[CenterDot]k_)^n_.,k_,p_] :> + LoopIntegral[d, {\[Delta]1-n//Simplify,\[Delta]2}][x, k, p], + LoopIntegral[d_, {\[Delta]1_,\[Delta]2_}][x_.*(k_\[CenterDot]p_)^n_.,k_,p_] :> + (p\[CenterDot]p /. suProductsToMagnitudes)/2*LoopIntegral[d, {\[Delta]1,\[Delta]2}][x*(k\[CenterDot]p)^(n-1), k,p] + -1/2*LoopIntegral[d, {\[Delta]1,\[Delta]2-1}][x*(k\[CenterDot]p)^(n-1), k,p] + +1/2*LoopIntegral[d, {\[Delta]1-1,\[Delta]2}][x*(k\[CenterDot]p)^(n-1), k,p], + LoopIntegral[d_, {\[Delta]1_,\[Delta]2_,\[Delta]3_}][x_.*(k_\[CenterDot]k_)^n_.,k_] :> + LoopIntegral[d, {\[Delta]1,\[Delta]2,\[Delta]3-n//Simplify}][x, k], + LoopIntegral[d_, {\[Delta]1_,\[Delta]2_,\[Delta]3_}][x_.*(k_\[CenterDot]p[1])^n_.,k_] :> + p[1]^2/2*LoopIntegral[d, {\[Delta]1,\[Delta]2,\[Delta]3}][x*(k\[CenterDot]p[1])^(n-1), k] + -1/2*LoopIntegral[d, {\[Delta]1,\[Delta]2-1,\[Delta]3}][x*(k\[CenterDot]p[1])^(n-1), k] + +1/2*LoopIntegral[d, {\[Delta]1,\[Delta]2,\[Delta]3-1}][x*(k\[CenterDot]p[1])^(n-1),k], + LoopIntegral[d_, {\[Delta]1_,\[Delta]2_,\[Delta]3_}][x_.*(k_\[CenterDot]p[2])^n_., k_] :> + -p[2]^2/2*LoopIntegral[d, {\[Delta]1,\[Delta]2,\[Delta]3}][x*(k\[CenterDot]p[2])^(n-1), k] + +1/2*LoopIntegral[d,{\[Delta]1-1,\[Delta]2,\[Delta]3}][x*(k\[CenterDot]p[2])^(n-1),k] + -1/2*LoopIntegral[d,{\[Delta]1,\[Delta]2,\[Delta]3-1}][x*(k\[CenterDot]p[2])^(n-1),k], + LoopIntegral[d_, {\[Delta]1_,\[Delta]2_,\[Delta]3_}][x_.*(k_\[CenterDot]k_+2 k_\[CenterDot]p[2]+p[2]^2)^n_.,k_] :> + LoopIntegral[d, {\[Delta]1-n//Simplify,\[Delta]2,\[Delta]3}][x, k], + LoopIntegral[d_, {\[Delta]1_,\[Delta]2_,\[Delta]3_}][x_.*(k_\[CenterDot]k_-2 k_\[CenterDot]p[1]+p[1]^2)^n_.,k_] :> + LoopIntegral[d, {\[Delta]1,\[Delta]2-n//Simplify,\[Delta]3}][x, k], + LoopIntegral[d_, {\[Delta]1_,\[Delta]2_}][x_.*(k_\[CenterDot]k_-2 k_\[CenterDot]p_+p_^2)^n_.,k_,p_] :> + LoopIntegral[d, {\[Delta]1,\[Delta]2-n//Simplify}][x, k, p], + LoopIntegral[d_, {\[Delta]1_,\[Delta]2_}][x_.*(k_\[CenterDot]k_+2 k_\[CenterDot]p_+p_^2)^n_.,k_, -p_] :> + LoopIntegral[d, {\[Delta]1,\[Delta]2-n//Simplify}][x, k, -p] +}; + + +suLoopZero = { + LoopIntegral[d_, {n_Integer,\[Delta]2_}][x_,k_,p_] /; n<=0 && FreeQ[x, k\[CenterDot]_]:> 0, + LoopIntegral[d_, {\[Delta]1_,n_Integer}][x_,k_,p_] /; n<=0 && FreeQ[x, k\[CenterDot]_]:> 0, + LoopIntegral[d_, {\[Delta]1_,\[Delta]2_,n_Integer}][x_,k_] /; n<=0 && FreeQ[x, k\[CenterDot]_]:> + LoopIntegral[d, {\[Delta]2,\[Delta]1}][x*(k\[CenterDot]k)^(-n) /. {k -> k+p[1]} //. suMomentumEx[k] /. suProductsToMagnitudes, k, p[3]], + LoopIntegral[d_, {\[Delta]1_,n_Integer,\[Delta]3_}][x_,k_] /; n<=0 && FreeQ[x, k\[CenterDot]_]:> + LoopIntegral[d, {\[Delta]3,\[Delta]1}][x*(k\[CenterDot]k - 2 k\[CenterDot]p[1]+p[1]^2)^(-n), k, -p[2]], + LoopIntegral[d_, {n_Integer,\[Delta]2_,\[Delta]3_}][x_,k_] /; n<=0 && FreeQ[x, k\[CenterDot]_] :> + LoopIntegral[d, {\[Delta]3,\[Delta]2}][x*(k\[CenterDot]k + 2 k\[CenterDot]p[2]+p[2]^2)^(-n), k, p[1]] +}; + + +suLoopZeroRecursive = { + LoopIntegral[d_, {n_Integer,\[Delta]2_}][x_,k_,p_] /; n<=0 && FreeQ[x, LoopIntegral] && FreeQ[x, k\[CenterDot]_] :> 0, + LoopIntegral[d_, {\[Delta]1_,n_Integer}][x_,k_,p_] /; n<=0 && FreeQ[x, LoopIntegral] && FreeQ[x, k\[CenterDot]_] :> 0, + LoopIntegral[d_, {\[Delta]1_,\[Delta]2_,n_Integer}][x_,k_] /; n<=0 && FreeQ[x, LoopIntegral] && FreeQ[x, k\[CenterDot]_] :> + LoopIntegral[d, {\[Delta]2,\[Delta]1}][x*(k\[CenterDot]k)^(-n) /. {k -> k+p[1]} //. suMomentumEx[k] /. suProductsToMagnitudes, k, p[3]], + LoopIntegral[d_, {\[Delta]1_,n_Integer,\[Delta]3_}][x_,k_] /; n<=0 && FreeQ[x, LoopIntegral] && FreeQ[x, k\[CenterDot]_] :> + LoopIntegral[d, {\[Delta]3,\[Delta]1}][x*(k\[CenterDot]k - 2 k\[CenterDot]p[1]+p[1]^2)^(-n), k, -p[2]], + LoopIntegral[d_, {n_Integer,\[Delta]2_,\[Delta]3_}][x_,k_] /; n<=0 && FreeQ[x, LoopIntegral] && FreeQ[x, k\[CenterDot]_] :> + LoopIntegral[d, {\[Delta]3,\[Delta]2}][x*(k\[CenterDot]k + 2 k\[CenterDot]p[2]+p[2]^2)^(-n), k, p[1]] +}; + + +suLoopReductionZero = Union[suLoopReduction, suLoopZero, suLoopLinearity]; +suLoopReductionZeroRecursive = Union[suLoopReduction, suLoopZeroRecursive, suLoopLinearity]; + + +suNLDiff = { + Derivative[i_,j_,k_][NL][a_,b_,c_] :> D[NL[1][a,b,c],{c,k},{b,j},{a,i-1}] /; i>=1, + Derivative[0, j_,k_][NL][a_,b_,c_] :> D[NL[2][a,b,c],{c,k},{b,j-1}] /; j>=1, + Derivative[0, 0, k_][NL][a_,b_,c_] :> D[NL[3][a,b,c],{c,k-1}] /; k>=1, + Derivative[i_,j_,k_][NL[1]][a_,b_,c_] :> D[2/(a Sqrt[\[Lambda][a,b,c]])*(a^2 Log[a^2]+1/2(c^2-a^2-b^2) Log[b^2]+1/2(b^2-a^2-c^2) Log[c^2]),{a,i},{b,j},{c,k}], + Derivative[i_,j_,k_][NL[2]][a_,b_,c_] :> D[2/(b Sqrt[\[Lambda][a,b,c]])*(b^2 Log[b^2]+1/2(c^2-a^2-b^2) Log[a^2]+1/2(a^2-c^2-b^2) Log[c^2]),{a,i},{b,j},{c,k}], + Derivative[i_,j_,k_][NL[3]][a_,b_,c_] :> D[2/(c Sqrt[\[Lambda][a,b,c]])*(c^2 Log[c^2]+1/2(b^2-a^2-c^2) Log[a^2]+1/2(a^2-c^2-b^2)Log[b^2]),{a,i},{b,j},{c,k}], + NL[2][a_,b_,c_]:>2/(b Sqrt[\[Lambda][a,b,c]])*(b^2 Log[b^2]+1/2(c^2-a^2-b^2) Log[a^2]+1/2(a^2-c^2-b^2) Log[c^2]), + NL[3][a_,b_,c_]:>2/(c Sqrt[\[Lambda][a,b,c]])*(c^2 Log[c^2]+1/2(b^2-a^2-c^2) Log[a^2]+1/2(a^2-c^2-b^2) Log[b^2]), + NL[1][a_,b_,c_]:>2/(a Sqrt[\[Lambda][a,b,c]])*(a^2 Log[a^2]+1/2(c^2-a^2-b^2) Log[b^2]+1/2(b^2-a^2-c^2) Log[c^2]), + Derivative[i_,j_,k_][\[Lambda]][a_,b_,c_] :> D[\[Lambda]fun[a,b,c],{a,i},{b,j},{c,k}] +}; + + +suPolyGammaToHarmonic[assume___] := { PolyGamma[0, z_] :> HarmonicNumber[z-1] - EulerGamma + /; Simplify[z>=1 && Element[z,Integers], assume, TimeConstraint->0.1], + PolyGamma[n_Integer, z_] :> (-1)^n *n! * (HarmonicNumber[z-1,n+1] - Zeta[n+1]) + /; Simplify[n > 0 && z>=1 && Element[z,Integers], assume, TimeConstraint->0.1], + PolyGamma[0, z_] :> -EulerGamma - 2Log[2] + 2 HarmonicNumber[2z-1] - HarmonicNumber[z-1/2] + /; Simplify[z>=1/2 && Divisible[2z+1,2], assume, TimeConstraint->0.1], + PolyGamma[n_Integer, z_] :> (-1)^n *n! * ( 2^(n+1)HarmonicNumber[2z-1, n+1] + -HarmonicNumber[z-1/2, n+1] - (2^(n+1)-1) Zeta[n+1] ) + /; Simplify[n>0 && z>=1/2 && Divisible[2z+1,2], assume, TimeConstraint->0.1], + PolyGamma[n_Integer?EvenQ, z_] :> PolyGamma[n, 1-z] + /; Simplify[n>=0 && z<=-1/2 && Divisible[2z+1,2], assume, TimeConstraint->0.1], + PolyGamma[n_Integer?OddQ, z_] :> -PolyGamma[n, 1-z] + (2 Pi)^(n+1) (-1)^((n-1)/2) * (2^(n+1)-1) * BernoulliB[n]/n + /; Simplify[n>=0 && z<=-1/2 && Divisible[2z+1,2], assume, TimeConstraint->0.1] +}; + + +fcall:PolyGammaToHarmonic[exp_, opts___?OptionQ] := Module[{ValidOpts, assume}, + ValidOpts = First /@ Options[PolyGammaToHarmonic]; + Scan[If[!MemberQ[ValidOpts, First[#]], + Message[PolyGammaToHarmonic::optx, ToString[First[#]], ToString[Unevaluated[fcall]]]]&, Flatten[{opts}]]; + assume = {Assumptions} /. Flatten[{opts}] /. Options[PolyGammaToHarmonic] // Flatten; + assume = Assumptions -> If[$Assumptions === True, assume, Union[$Assumptions, assume]]; + + exp /. suPolyGammaToHarmonic[assume] +]; + + +fcall:PolyGammaToHarmonic[exp_, k___] := Null /; Message[PolyGammaToHarmonic::nonopt, Last[{k}], 1, ToString[Unevaluated[fcall]]]; +PolyGammaToHarmonic[] := Null /; Message[PolyGammaToHarmonic::argrx, "PolyGammaToHarmonic", 0, 1]; + + +KExpandOnce[exp_, ks_] := + Expand[exp] /. If[!FreeQ[exp, LoopIntegral], { + LoopIntegral[d_,\[Delta]s_][X_,k_,p_] :> LoopIntegral[d,\[Delta]s][KExpandOnce[X, Union[ks, {k}]], k, p], + LoopIntegral[d_,\[Delta]s_][X_,k_] :> LoopIntegral[d,\[Delta]s][KExpandOnce[X, Union[ks, {k}]], k] + }, {}] /. + { i[a_,b_] :> i[Expand[a], Expand[b]] } //. + suMomentumEx[ks] //. + suLoopLinearity; + + +KExpandFunctions[exp_, inlevel_] := Module[{level, res, Pow}, + level = If[MemberQ[{0,1,2,3,4, D,Diff,\[Lambda],NL,Integer}, inlevel], + inlevel, + Message[KExpand::lev, inlevel]; 1]; + level = Switch[level, D, 1, Diff, 1, Integer, 2, \[Lambda], 3, NL, 4, _, level]; + + res = exp + /. suPolyGammaToHarmonic[] + //. If[1 <= level, suNLDiff, {}] + /. If[3 <= level, { \[Lambda]->\[Lambda]fun }, {}] + /. If[4 <= level, { NL->NLfun }, {}]; + If[level === 2, + res /. { Power[\[Lambda][p__],n_] :> Pow[{p},n] /; !IntegerQ[n] } + /. { \[Lambda][p__]^(n_.) :> \[Lambda]fun[p]^n /; IntegerQ[n] } + /. { Pow[p_,n_] :> (\[Lambda]@@p)^n }, + res] +]; + + +fcall:KExpand[exp_, opts___?OptionQ] := Module[{ValidOpts, level}, + ValidOpts = First /@ Options[KExpand]; + Scan[If[!MemberQ[ValidOpts, First[#]], + Message[KExpand::optx, ToString[First[#]], ToString[Unevaluated[fcall]]]]&, Flatten[{opts}]]; + level = Level /. Flatten[{opts}] /. Options[KExpand]; + + KExpandFunctions[KExpandOnce[Unformat[exp], {}] //. suKDiff, level] +]; +KExpand[exp_List, opts___?OptionQ] := KExpand[#,opts]& /@ exp; +KExpand[exp_SeriesData, opts___?OptionQ] := + SeriesData[exp[[1]],exp[[2]], KExpand[#,opts]& /@ (exp[[3]]), exp[[4]],exp[[5]],exp[[6]]]; + + +fcall:KExpand[exp_, k___] := Null /; Message[KExpand::nonopt, Last[{k}], 1, ToString[Unevaluated[fcall]]]; +KExpand[] := Null /; Message[KExpand::argrx, "KExpand", 0, 1]; + + +KFullExpand[exp_] := KExpand[exp, Level -> NL]; +KFullExpand[x___] := Null /; Message[KFullExpand::argrx, "KFullExpand", Length[{x}], 1]; +KFullExpand[exp_List] := KFullExpand /@ exp; +KFullExpand[exp_SeriesData] := + SeriesData[exp[[1]],exp[[2]], KFullExpand /@ (exp[[3]]), exp[[4]],exp[[5]],exp[[6]]]; + + +KSimplifyOnce[exp_, ks_, assume_] := + Contract[ + exp /. If[!FreeQ[exp, LoopIntegral], { + LoopIntegral[d_,\[Delta]s_][X_,k_,p_] :> LoopIntegral[d,\[Delta]s][KSimplifyOnce[X, Union[ks, {k}], assume], k, p], + LoopIntegral[d_,\[Delta]s_][X_,k_] :> LoopIntegral[d,\[Delta]s][KSimplifyOnce[X, Union[ks, {k}], assume], k] + }, {}] + /. { i[a_,b_] :> i[Simplify[a,assume, TimeConstraint->0.1], Simplify[b,assume, TimeConstraint->0.1]] } + //.suMomentumEx[ks] + //.{ a_*LoopIntegral[x__][X_, k_, y___] :> LoopIntegral[x][a*X,k,y] /; FreeQ[a,k] && a =!= 1 } + ] //. suLoopReductionZeroRecursive; + + +fcall:KSimplify[exp_, opts___?OptionQ] := Module[{ii,LI, ValidOpts, defu,defv,assume}, + ValidOpts = First /@ Options[KSimplify]; + Scan[If[!MemberQ[ValidOpts, First[#]], + Message[KSimplify::optx, ToString[First[#]], ToString[Unevaluated[fcall]]]]&, Flatten[{opts}]]; + assume = {Assumptions} /. Flatten[{opts}] /. Options[KSimplify] // Flatten; + assume = Assumptions -> If[$Assumptions === True, assume, Union[$Assumptions, assume]]; + + Simplify[KSimplifyOnce[Unformat[exp], {}, assume] + //. suKDiff + //. suKNegative[assume] + /. suPolyGammaToHarmonic[assume], + assume, + TransformationFunctions -> {Automatic, SimplifyFunctionKIdentities[assume]} ] + /. suKInverseOrder + //. suNLDiff +]; + + +fcall:KSimplify[exp_, k___] := Null /; Message[KSimplify::nonopt, Last[{k}], 1, ToString[Unevaluated[fcall]]]; +KSimplify[] := Null /; Message[KSimplify::argrx, "KSimplify", 0, 1]; + + +KSimplify[exp_List, opts___?OptionQ] := KSimplify[#,opts]& /@ exp; + + +KSimplify[exp_SeriesData, opts___?OptionQ] := + SeriesData[exp[[1]],exp[[2]], KSimplify[#,opts]& /@ (exp[[3]]), exp[[4]],exp[[5]],exp[[6]]]; + + +PrepareIntegral[a_,{b1_,b2_,b3_}, defu_,defv_, assume___] := Module[{acf0,acf1,bcf0,bcf1}, + If[FreeQ[a,\[Epsilon]]&&FreeQ[b1,\[Epsilon]]&&FreeQ[b2,\[Epsilon]]&&FreeQ[b3,\[Epsilon]], + If[TrueQ[IsDivergent[i[a,{b1,b2,b3}], assume]], + i[a,{b1,b2,b3}, defu,defv], + i[a,{b1,b2,b3}, 0,{0,0,0}] + ], + Quiet[ + acf0 = a /. \[Epsilon]->0; + bcf0 = {b1,b2,b3} /. \[Epsilon]->0; + acf1 = D[a,\[Epsilon]] /. \[Epsilon]->0; + bcf1 = D[{b1,b2,b3},\[Epsilon]] /. \[Epsilon]->0; + ]; + If[!TrueQ[Simplify[a-acf0-acf1 \[Epsilon] == 0, assume, TimeConstraint->0.1]], + Message[TripleK::nonlin, "\[Alpha]", i[a,{b1,b2,b3}]]]; + If[!TrueQ[Simplify[{b1,b2,b3}-bcf0-bcf1 \[Epsilon] == 0, assume, TimeConstraint->0.1]], + Message[TripleK::nonlin, "\[Beta]", i[a,{b1,b2,b3}]]]; + i[acf0,bcf0, acf1,bcf1] + ] +]; + + +PrepareIntegral[a_,{b1_,b2_}, defu_,defv_, assume___] := Module[{acf0,acf1,bcf0,bcf1}, + If[FreeQ[a,\[Epsilon]]&&FreeQ[b1,\[Epsilon]]&&FreeQ[b2,\[Epsilon]], + If[TrueQ[IsDivergent[i[a,{b1,b2}], assume]], + i[a,{b1,b2}, defu, {defv[[1]],defv[[2]]}], + i[a,{b1,b2}, 0,{0,0}] + ], + Quiet[ + acf0 = a /. \[Epsilon]->0; + bcf0 = {b1,b2} /. \[Epsilon]->0; + acf1 = D[a,\[Epsilon]] /. \[Epsilon]->0; + bcf1 = D[{b1,b2},\[Epsilon]] /. \[Epsilon]->0; + ]; + If[TrueQ[Simplify[a-acf0-acf1 \[Epsilon] != 0, assume, TimeConstraint->0.1]], + Message[TripleK::nonlin, "\[Alpha]", i[a,{b1,b2}]]]; + If[TrueQ[Simplify[{b1,b2}-bcf0-bcf1 \[Epsilon] != 0, assume, TimeConstraint->0.1]], + Message[TripleK::nonlin, "\[Beta]", i[a,{b1,b2}]]]; + i[acf0,bcf0, acf1,bcf1] + ] +]; + + +PrepareExpression[exp_, defu_,defv_, assume___] := + Unformat[exp] //. suKDiff /. + { i[a_,{b1_,b2_,b3_}] :> PrepareIntegral[a,{b1,b2,b3}, defu,defv,assume], + i[a_,{b1_,b2_}] :> PrepareIntegral[a,{b1,b2}, defu,defv,assume] } //. + suKNegativeReg //. + suKOrderReg; + + +suPostpareExpression = { i[a_,b_,u_,v_] :> i[a + u \[Epsilon],b + v \[Epsilon]] }; + + +(* ::Section:: *) +(*Momentum manipulations*) + + +suContract[d_] := { + \[Delta][i_,j_]\[Delta][j_,k_] :> \[Delta][i,k] /; !NumericQ[j], + \[Delta][i_,j_]\[Delta][i_,j_] :> d /; !NumericQ[i]&&!NumericQ[j], + \[Delta][i_,i_] :> d /; !NumericQ[i], + x_[i_]y_[i_] :> x\[CenterDot]y /; !NumericQ[i], + \[Delta][i_,j_]x_[j_] :> x[i] /; !NumericQ[j], + x_[i_]^2 :> x\[CenterDot]x /; !NumericQ[i] + }; + + +suContractVectors[d_, vects_] := { + \[Delta][i_,j_]\[Delta][j_,k_] :> \[Delta][i,k] /; !NumericQ[j], + \[Delta][i_,j_]\[Delta][i_,j_] :> d /; !NumericQ[i]&&!NumericQ[j], + \[Delta][i_,i_] :> d /; !NumericQ[i], + x_[i_]y_[i_] :> x\[CenterDot]y /; !NumericQ[i] && MemberQ[vects, x] && MemberQ[vects, y], + \[Delta][i_,j_]x_[j_] :> x[i] /; !NumericQ[j] && MemberQ[vects, x], + x_[i_]^2 :> x\[CenterDot]x /; !NumericQ[i] && MemberQ[vects, x] + }; + + +suContractIndices[d_, js_List] := { + \[Delta][i_,j_]\[Delta][j_,k_] :> \[Delta][i,k] /; MemberQ[js,j], + \[Delta][i_,j_]^2 :> \[Delta][i,i] /; MemberQ[js,j], + \[Delta][j_,j_] :> d /; MemberQ[js,j], + x_[j_]y_[j_] :> x\[CenterDot]y /; MemberQ[js,j], + \[Delta][i_,j_]x_[j_] :> x[i] /; MemberQ[js,j], + x_[j_]^2 :> x\[CenterDot]x /; MemberQ[js,j] + }; + + +suContractIndices[d_, j_] := { + \[Delta][i_,j]\[Delta][j,k_] :> \[Delta][i,k], + \[Delta][i_,j]^2 :> \[Delta][i,i], + \[Delta][j,j] :> d, + x_[j]y_[j] :> x\[CenterDot]y, + \[Delta][i_,j]x_[j] :> x[i], + x_[j]^2 :> x\[CenterDot]x + }; + + +suContractVectorsIndices[d_, vects_, js_List] := { + \[Delta][i_,j_]\[Delta][j_,k_] :> \[Delta][i,k] /; MemberQ[js,j], + \[Delta][i_,j_]^2 :> \[Delta][i,i] /; MemberQ[js,j], + \[Delta][j_,j_] :> d /; MemberQ[js,j], + x_[j_]y_[j_] :> x\[CenterDot]y /; MemberQ[js,j] && MemberQ[vects, x] && MemberQ[vects, y], + \[Delta][i_,j_]x_[j_] :> x[i] /; MemberQ[js,j] && MemberQ[vects, x], + x_[j_]^2 :> x\[CenterDot]x /; MemberQ[js,j] && MemberQ[vects, x] + }; + + +suContractVectorsIndices[d_, vects_, j_] := { + \[Delta][i_,j]\[Delta][j,k_] :> \[Delta][i,k], + \[Delta][i_,j]^2 :> \[Delta][i,i], + \[Delta][j,j] :> d, + x_[j]y_[j] :> x\[CenterDot]y /; MemberQ[vects, x] && MemberQ[vects, y], + \[Delta][i_,j]x_[j] :> x[i] /; MemberQ[vects, x], + x_[j]^2 :> x\[CenterDot]x /; MemberQ[vects, x] + }; + + +ContractOnce[exp_, vects_, dim_, idx_] := + Expand[exp] /. + If[!FreeQ[exp, LoopIntegral], { + LoopIntegral[d_,\[Delta]s_][num_,k_] :> LoopIntegral[d,\[Delta]s][ + ContractOnce[num, Union[vects,{k}], dim, idx], k], + LoopIntegral[d_,\[Delta]s_][num_,k_,p_] :> LoopIntegral[d,\[Delta]s][ + ContractOnce[num, Union[vects,{k}], dim, idx], k, p] + }, {}] //. + If[idx === All, + suContractVectors[dim, vects], + suContractVectorsIndices[dim, vects, idx] + ]; + + +DoContract[exp_, dim_, vects_, idx_] := + If[vects === All, + If[idx === All, + Expand[Unformat[exp]] //. suContract[dim], + Expand[Unformat[exp]] //. suContractIndices[dim, idx] + ], + ContractOnce[ + Unformat[exp], + If[ListQ[vects], Union[vects, {p, p[1],p[2],p[3]}], {vects,p, p[1],p[2],p[3]}], + dim, + idx + ] + ] /. suProductsToMagnitudes; + + +fcall:Contract[exp_, opts___?OptionQ] := Module[{ValidOpts, dim, vects, idx}, + ValidOpts = First /@ Options[Contract]; + Scan[If[!MemberQ[ValidOpts, First[#]], + Message[Contract::optx, ToString[First[#]], ToString[Unevaluated[fcall]]]]&, Flatten[{opts}]]; + {dim, vects, idx} = {Dimension, Vectors, Indices} /. Flatten[{opts}] /. Options[Contract]; + DoContract[exp, dim, If[vects === Automatic, If[idx === All, {}, All], vects], idx] +]; + + +fcall:Contract[exp_, \[Mu]_, opts___?OptionQ] := Module[{ValidOpts, dim, vects}, + ValidOpts = First /@ Options[Contract]; + Scan[If[!MemberQ[ValidOpts, First[#]], + Message[Contract::optx, ToString[First[#]], ToString[Unevaluated[fcall]]]]&, Flatten[{opts}]]; + {dim, vects} = {Dimension, Vectors} /. Flatten[{opts}] /. Options[Contract]; + DoContract[exp, dim, If[vects === Automatic, All, vects], \[Mu]] +]; + + +fcall:Contract[exp_, \[Mu]_, \[Nu]_, opts___?OptionQ] := Module[{ValidOpts, dim, vects}, + ValidOpts = First /@ Options[Contract]; + Scan[If[!MemberQ[ValidOpts, First[#]], + Message[Contract::optx, ToString[First[#]], ToString[Unevaluated[fcall]]]]&, Flatten[{opts}]]; + {dim, vects} = {Dimension, Vectors} /. Flatten[{opts}] /. Options[Contract]; + DoContract[exp /. \[Nu]->\[Mu], dim, If[vects === Automatic, All, vects], \[Mu]] +]; + + +Contract[exp_List, opts___?OptionQ] := Contract[#,opts]& /@ exp; +Contract[exp_List, \[Mu]_, opts___?OptionQ] := Contract[#,\[Mu],opts]& /@ exp; +Contract[exp_List, \[Mu]_, \[Nu]_, opts___?OptionQ] := Contract[#,\[Mu],\[Nu],opts]& /@ exp; + + +Contract[exp_SeriesData, opts___?OptionQ] := + SeriesData[exp[[1]],exp[[2]], Contract[#,opts]& /@ (exp[[3]]), exp[[4]],exp[[5]],exp[[6]]]; +Contract[exp_SeriesData, \[Mu]_, opts___?OptionQ] := + SeriesData[exp[[1]],exp[[2]], Contract[#,\[Mu],opts]& /@ (exp[[3]]), exp[[4]],exp[[5]],exp[[6]]]; +Contract[exp_SeriesData, \[Mu]_, \[Nu]_, opts___?OptionQ] := + SeriesData[exp[[1]],exp[[2]], Contract[#,\[Mu],\[Nu],opts]& /@ (exp[[3]]), exp[[4]],exp[[5]],exp[[6]]]; + + +fcall:Contract[exp_, k___] := Null /; Message[Contract::nonopt, Last[{k}], 1, ToString[Unevaluated[fcall]]]; +Contract[] := Null /; Message[Contract::argbt, "Contract", 0, 1, 3]; + + +OldDiff[exp_, p[mom_], idx_] /; mom===1 || mom===2 := +Module[{Vtemp, Ltemp, L3temp}, + D[Unformat[exp] + //.suMomentum3 + //.suMomentum + /. { p[mom][i_] :> Vtemp[i] } + /. { p[mom] -> Ltemp, p[3] -> L3temp } + //.{ Ltemp\[CenterDot]x_ :> p[mom]\[CenterDot]x } + /. { Ltemp -> Ltemp[p[mom]], + Vtemp[i_] :> Vtemp[p[mom],i], + L3temp -> L3temp[p[mom]] }, + p[mom]] + /. { + Derivative[1][Ltemp][p[mom]] -> p[mom][idx] / p[mom], + Derivative[1][L3temp][p[mom]] -> (p[1][idx] + p[2][idx]) / p[3], + Derivative[1,0][Vtemp][p[mom],i_] :> \[Delta][idx,i], + Derivative[0,1][CenterDot][x_, y_] :> x[idx], + Derivative[1,0][CenterDot][x_, y_] :> y[idx], + Ltemp[p[mom]] -> p[mom], + L3temp[p[mom]] -> p[3], + Vtemp[p[mom],i_] :> p[mom][i] + } +]; + + +OldDiff[exp_, mom_, idx_] := Module[{Vtemp, Ltemp}, + D[exp + //.suMomentumEx[mom] + /. { mom[i_] :> Vtemp[mom,i] } + /. { mom :> Ltemp[mom] } + //.{ Vtemp[Ltemp[mom],i_] :> Vtemp[mom,i], Ltemp[mom]\[CenterDot]x_ :> mom\[CenterDot]x }, + mom] + /. { + Derivative[1][Ltemp][mom] :> mom[idx] / mom, + Derivative[1,0][Vtemp][mom,i_] :> \[Delta][idx,i], + Derivative[0,1][CenterDot][x_, y_] :> x[idx], + Derivative[1,0][CenterDot][x_, y_] :> y[idx], + Ltemp[mom] :> mom, + Vtemp[mom,i_] :> mom[i] + } +]; + + +DiffOnce[exp_, p[mom_], idx_, ks_] /; mom===1 || mom===2 := +Module[{loopint, Vtemp, Ltemp, L3temp, ptemp, itemp}, + loopint = !FreeQ[exp, LoopIntegral]; + D[exp + /. If[loopint, { + LoopIntegral[d_, {\[Delta]1_,\[Delta]2_,\[Delta]3_}][num_,k_] :> + LoopIntegral[d, {\[Delta]1,\[Delta]2,\[Delta]3}][num /. p->ptemp, k][Vtemp[itemp]], + LoopIntegral[d_, {\[Delta]1_,\[Delta]2_}][num_,k_,pp_] :> + LoopIntegral[d, {\[Delta]1,\[Delta]2}][num /. p->ptemp, k, pp /. p->ptemp][Vtemp[itemp]] + }, {}] + //.suMomentum3 + //.suMomentumEx[ks] + /. { p[mom][i_] :> Vtemp[i] } + /. { p[mom] -> Ltemp, p[3] -> L3temp } + //.{ Ltemp\[CenterDot]x_ :> p[mom]\[CenterDot]x } + /. { Ltemp -> Ltemp[p[mom]], + Vtemp[i_] :> Vtemp[p[mom],i], + L3temp -> L3temp[p[mom]] }, + p[mom]] + /. { + Derivative[1][Ltemp][p[mom]] -> p[mom][idx] / p[mom], + Derivative[1][L3temp][p[mom]] -> (p[1][idx] + p[2][idx]) / p[3], + Derivative[1,0][Vtemp][p[mom],i_] :> \[Delta][idx,i], + Derivative[0,1][CenterDot][x_, y_] :> x[idx], + Derivative[1,0][CenterDot][x_, y_] :> y[idx], + Ltemp[p[mom]] -> p[mom], + L3temp[p[mom]] -> p[3], + Vtemp[p[mom],i_] :> p[mom][i] + } + /. If[loopint, { + \[Delta][idx, itemp] -> 1, + Derivative[1][LoopIntegral[d_, {\[Delta]1_,\[Delta]2_,\[Delta]3_}][num_,k_]][p[mom][itemp]] :> + If[mom===1, +2\[Delta]2 LoopIntegral[d, {\[Delta]1,\[Delta]2+1,\[Delta]3}][num*(k[idx]-p[1][idx]) /. ptemp->p, k], 0] + + If[mom===2, -2\[Delta]1 LoopIntegral[d, {\[Delta]1+1,\[Delta]2,\[Delta]3}][num*(p[2][idx]+k[idx]) /. ptemp->p, k], 0] + + LoopIntegral[d, {\[Delta]1,\[Delta]2,\[Delta]3}][DiffOnce[num /. ptemp->p, p[mom], idx, Union[ks, {k}]], k], + Derivative[1][LoopIntegral[d_, {\[Delta]1_,\[Delta]2_}][num_,k_,pp_]][p[mom][itemp]] :> + + 2\[Delta]2 LoopIntegral[d, {\[Delta]1,\[Delta]2+1}][num*(k[idx]-pp[idx]) + /. ptemp->p + /. suMomentum3 + //.suMomentumEx[Union[ks, {k}]], + k, pp /. ptemp->p] * D[pp /. ptemp->p /. p[3]->-p[1]-p[2], p[mom]] + + LoopIntegral[d, {\[Delta]1,\[Delta]2}][DiffOnce[num /. ptemp->p, p[mom], idx, Union[ks, {k}]], k, pp /. ptemp->p], + LoopIntegral[d_,{\[Delta]1_,\[Delta]2_,\[Delta]3_}][num_, k_][p[mom][itemp]] :> + LoopIntegral[d,{\[Delta]1,\[Delta]2,\[Delta]3}][num /. ptemp->p, k], + LoopIntegral[d_,{\[Delta]1_,\[Delta]2_}][num_, k_, pp_][p[mom][itemp]] :> + LoopIntegral[d,{\[Delta]1,\[Delta]2}][num /. ptemp->p, k, pp /. ptemp->p] + }, {}] + /. { LoopIntegral[x__][0, y__] :> 0 } +]; + + +DiffOnce[exp_, mom_, idx_, ks_] := Module[{loopint, Vtemp, Ltemp, ptemp, itemp}, + loopint = !FreeQ[exp, LoopIntegral]; + D[exp + //.suMomentumEx[Flatten[{mom,ks}]] + /. If[loopint, { + LoopIntegral[d_, {\[Delta]1_,\[Delta]2_,\[Delta]3_}][num_,k_] :> + LoopIntegral[d, {\[Delta]1,\[Delta]2,\[Delta]3}][num /. mom->ptemp, k][Vtemp[mom, itemp]], + LoopIntegral[d_, {\[Delta]1_,\[Delta]2_}][num_,k_,pp_] :> + LoopIntegral[d, {\[Delta]1,\[Delta]2}][num /. mom->ptemp, k, pp /. mom->ptemp][Vtemp[mom, itemp]] + }, {}] + /. { mom[i_] :> Vtemp[mom,i] } + /. { mom :> Ltemp[mom] } + //.{ Vtemp[Ltemp[mom],i_] :> Vtemp[mom,i], Ltemp[mom]\[CenterDot]x_ :> mom\[CenterDot]x }, + mom] + /. { + Derivative[1][Ltemp][mom] :> mom[idx] / mom, + Derivative[1,0][Vtemp][mom,i_] :> \[Delta][idx,i], + Derivative[0,1][CenterDot][x_, y_] :> x[idx], + Derivative[1,0][CenterDot][x_, y_] :> y[idx], + Ltemp[mom] :> mom, + Vtemp[mom,i_] :> mom[i] + } + /. If[loopint, { + \[Delta][idx, itemp] -> 1, + Derivative[1][LoopIntegral[d_, {\[Delta]1_,\[Delta]2_,\[Delta]3_}][num_,k_]][mom[itemp]] :> + LoopIntegral[d, {\[Delta]1,\[Delta]2,\[Delta]3}][DiffOnce[num /. ptemp->mom, mom, idx, Union[ks, {k}]], k], + Derivative[1][LoopIntegral[d_, {\[Delta]1_,\[Delta]2_}][num_,k_,pp_]][mom[itemp]] :> + + 2\[Delta]2 LoopIntegral[d, {\[Delta]1,\[Delta]2+1}][num*(k[idx]-pp[idx]) + /. ptemp->mom + //.suMomentumEx[Union[ks, {k}]], + k, pp /. ptemp->mom] * D[pp /. ptemp->mom, mom] + + LoopIntegral[d, {\[Delta]1,\[Delta]2}][DiffOnce[num /. ptemp->mom, mom, idx, Union[ks, {k}]], k, pp /. ptemp->mom], + LoopIntegral[d_,{\[Delta]1_,\[Delta]2_,\[Delta]3_}][num_, k_][mom[itemp]] :> + LoopIntegral[d,{\[Delta]1,\[Delta]2,\[Delta]3}][num /. ptemp->mom, k], + LoopIntegral[d_,{\[Delta]1_,\[Delta]2_}][num_, k_, pp_][mom[itemp]] :> + LoopIntegral[d,{\[Delta]1,\[Delta]2}][num /. ptemp->mom, k, pp /. ptemp->mom] + }, {}] + /. { LoopIntegral[x__][0, y__] :> 0 } +]; + + +Diff[exp_, mom_, idx_] := Module[{pos}, + If[FreeQ[exp, LoopIntegral], + OldDiff[exp, mom /. suMomentumUnformat, idx], + pos = FirstPosition[{exp}, LoopIntegral[__][_,mom,___]]; + If[Head[pos]===Missing, + DiffOnce[Unformat[exp], mom /. suMomentumUnformat, idx, {}], + Message[Diff::darg, mom, Extract[{exp}, pos]]; Null + ] + ] + /. suProductsToMagnitudes +]; + + +Diff[exp_List, mom_, idx_] := Diff[#, mom, idx]& /@ exp; +Diff[exp_SeriesData, mom_, idx_] := + SeriesData[exp[[1]],exp[[2]], Diff[#, mom, idx]& /@ (exp[[3]]), exp[[4]],exp[[5]],exp[[6]]]; + + +Diff[x___] := Null /; Message[Diff::argrx, "Diff", Length[{x}], 3]; + + +(* ::Section:: *) +(*Momentum integrals to multiple-K*) + + +MomentumIntConstant[d_, m_, a_] := 1/( (4 Pi)^(d/2) 2^m ) * a^(-d/2-m); +SchwingerIntegralValue[at_, {\[Delta]1_,\[Delta]2_}][p_] := + 2^(\[Delta]1+\[Delta]2+2at+3)/( Gamma[-3at - 2\[Delta]1 - 2\[Delta]2] ) * + i[-2at - \[Delta]1 - \[Delta]2 - 1, {-2at - \[Delta]1 - 2\[Delta]2, -2at - 2\[Delta]1 - \[Delta]2}][ + Simplify[Sqrt[p\[CenterDot]p] //. suMomentum /. suProductsToMagnitudes, TimeConstraint->0.1]]; +SchwingerIntegralValue[at_, {\[Delta]1_,\[Delta]2_,\[Delta]3_}] := + 2^(at+4)/( Gamma[-2at - \[Delta]1 - \[Delta]2 - \[Delta]3] ) * + i[-at - 1, {-at - \[Delta]2 - \[Delta]3, -at - \[Delta]1 - \[Delta]3, -at - \[Delta]1 - \[Delta]2}][ + p[1],p[2],p[3]]; + + +LoopToSchwinger[exp_] := Module[{l, s,st, b1,b2,b3,bt, counter=1,idx,j}, + Expand[Unformat[Expand[exp]] + /. suProductsToMagnitudes + /. suMomentum3 + /. { LoopIntegral[x__][X_,y__] :> LoopIntegral[x][Expand[X],y] } + //.suMomentum + //.suLoopReductionZero + /. { + LoopIntegral[d_, {\[Delta]1_,\[Delta]2_}][X_,k_,p_] :> + LoopIntegral[d, {\[Delta]1,\[Delta]2}][ + Expand[X /. { k -> l + s[2]p/st } //. suMomentumEx[l]], + l, p], + LoopIntegral[d_, {\[Delta]1_,\[Delta]2_,\[Delta]3_}][X_,k_] :> + LoopIntegral[d, {\[Delta]1,\[Delta]2,\[Delta]3}][ + Expand[X /. { k -> l + (s[2]*p[1])/st - (s[1]*p[2])/st } + //. suMomentumEx[l]], + l] + } + //. suLoopLinearity + /. { l\[CenterDot]x_ :> l[idx[counter]]x[idx[counter++]] } + //. suLoopLinearity + //.{ l[\[Mu]_]^n_ :> l@@ConstantArray[\[Mu],n], l[\[Mu]___]l[\[Nu]___] :> l[\[Mu],\[Nu]] } + /. { + LoopIntegral[d_, {\[Delta]1_,\[Delta]2_}][l[x___],l,p_] :> 0 /; OddQ[Length[{x}]], + LoopIntegral[d_, {\[Delta]1_,\[Delta]2_}][l[x___],l,p_] :> + MomentumIntConstant[d, Length[{x}]/2, st] / (Gamma[\[Delta]1] Gamma[\[Delta]2]) * + l[x] s[1]^b1 * s[2]^b2 * st^bt * + SchwingerIntegral[0, {\[Delta]1, \[Delta]2}][p], + LoopIntegral[d_, {\[Delta]1_,\[Delta]2_}][1,l,p_] :> + MomentumIntConstant[d, 0, st] / (Gamma[\[Delta]1] Gamma[\[Delta]2]) * + s[1]^b1 * s[2]^b2 * st^bt * + SchwingerIntegral[0, {\[Delta]1, \[Delta]2}][p], + LoopIntegral[d_, {\[Delta]1_,\[Delta]2_,\[Delta]3_}][l[x___],l] :> 0 /; OddQ[Length[{x}]], + LoopIntegral[d_, {\[Delta]1_,\[Delta]2_,\[Delta]3_}][l[x___],l] :> + MomentumIntConstant[d, Length[{x}]/2, st] / (Gamma[\[Delta]1] Gamma[\[Delta]2] Gamma[\[Delta]3]) * + l[x] s[1]^b1 * s[2]^b2 * s[3]^b3 * st^bt * + SchwingerIntegral[0, {\[Delta]1, \[Delta]2, \[Delta]3}], + LoopIntegral[d_, {\[Delta]1_,\[Delta]2_,\[Delta]3_}][1,l] :> + MomentumIntConstant[d, 0, st] / (Gamma[\[Delta]1] Gamma[\[Delta]2] Gamma[\[Delta]3]) * + s[1]^b1 * s[2]^b2 * s[3]^b3 * st^bt * + SchwingerIntegral[0, {\[Delta]1, \[Delta]2, \[Delta]3}] + } + //.{ l[x__] :> Sum[\[Delta][First[{x}],{x}[[j]]]l@@Delete[{x},{{1},{j}}], {j,2,Length[{x}]}] } + /. { l[] -> 1 } + ] + /. { s[1]^(a1_) s[2]^(a2_)s[3]^(a3_) st^(at_) SchwingerIntegral[At_, {A1_,A2_,A3_}] :> + SchwingerIntegral[At+at, {A1+a1, A2+a2, A3+a3}], + s[1]^(a1_) s[2]^(a2_) st^(at_) SchwingerIntegral[At_, {A1_,A2_}][p_] :> + SchwingerIntegral[At+at, {A1+a1, A2+a2}][p] } + /. { b1->0, b2->0, b3->0, bt->0 } + //. suContractIndices[d, idx/@Range[counter-1]] +]; + + +LoopToSchwinger[exp_, ks_] := Module[{l, s,st, b1,b2,b3,bt, counter=1,idx,j, SI}, + Expand[Expand[exp] + /. { LoopIntegral[d_,{\[Delta]1_,\[Delta]2_}][X_,k_,p_] :> LoopIntegral[d,{\[Delta]1,\[Delta]2}][ + Expand[X] //. suMomentumEx[Union[ks,{k}]] + /. suProductsToMagnitudes, k, p], + LoopIntegral[d_,{\[Delta]1_,\[Delta]2_,\[Delta]3_}][X_,k_] :> + LoopIntegral[d,{\[Delta]1,\[Delta]2,\[Delta]3}][X /. suMomentum3 + //. suMomentumEx[Union[ks,{k}]] + /. suProductsToMagnitudes, k] } + //. suMomentumEx[ks] + //. suLoopReductionZeroRecursive + /. { + LoopIntegral[d_, {\[Delta]1_,\[Delta]2_}][X_,k_,p_] :> + SI[LoopIntegral[d, {\[Delta]1,\[Delta]2}][ + Expand[X /. { k -> l + s[2]p/st } //. suMomentumEx[Union[ks, {l}]]], + l, p]], + LoopIntegral[d_, {\[Delta]1_,\[Delta]2_,\[Delta]3_}][X_,k_] :> + SI[LoopIntegral[d, {\[Delta]1,\[Delta]2,\[Delta]3}][ + Expand[X /. { k -> l + (s[2]*p[1])/st - (s[1]*p[2])/st } + //. suMomentumEx[Union[ks, {l}]]], + l]] + } + //. suLoopLinearity + /. { l\[CenterDot]x_ :> l[idx[counter]]x[idx[counter++]] } + //. suLoopLinearity + //.{ l[\[Mu]_]^n_ :> l@@ConstantArray[\[Mu],n], l[\[Mu]___]l[\[Nu]___] :> l[\[Mu],\[Nu]] } + /. { + LoopIntegral[d_, {\[Delta]1_,\[Delta]2_}][l[x___],l,p_] :> 0 /; OddQ[Length[{x}]], + LoopIntegral[d_, {\[Delta]1_,\[Delta]2_}][l[x___],l,p_] :> + MomentumIntConstant[d, Length[{x}]/2, st] / (Gamma[\[Delta]1] Gamma[\[Delta]2]) * + l[x] s[1]^b1 * s[2]^b2 * st^bt * + SchwingerIntegral[0, {\[Delta]1, \[Delta]2}][p], + LoopIntegral[d_, {\[Delta]1_,\[Delta]2_}][1,l,p_] :> + MomentumIntConstant[d, 0, st] / (Gamma[\[Delta]1] Gamma[\[Delta]2]) * + s[1]^b1 * s[2]^b2 * st^bt * + SchwingerIntegral[0, {\[Delta]1, \[Delta]2}][p], + LoopIntegral[d_, {\[Delta]1_,\[Delta]2_,\[Delta]3_}][l[x___],l] :> 0 /; OddQ[Length[{x}]], + LoopIntegral[d_, {\[Delta]1_,\[Delta]2_,\[Delta]3_}][l[x___],l] :> + MomentumIntConstant[d, Length[{x}]/2, st] / (Gamma[\[Delta]1] Gamma[\[Delta]2] Gamma[\[Delta]3]) * + l[x] s[1]^b1 * s[2]^b2 * s[3]^b3 * st^bt * + SchwingerIntegral[0, {\[Delta]1, \[Delta]2, \[Delta]3}], + LoopIntegral[d_, {\[Delta]1_,\[Delta]2_,\[Delta]3_}][1,l] :> + MomentumIntConstant[d, 0, st] / (Gamma[\[Delta]1] Gamma[\[Delta]2] Gamma[\[Delta]3]) * + s[1]^b1 * s[2]^b2 * s[3]^b3 * st^bt * + SchwingerIntegral[0, {\[Delta]1, \[Delta]2, \[Delta]3}] + } + /. { s[1]^(a1_) s[2]^(a2_)s[3]^(a3_) st^(at_) SchwingerIntegral[At_, {A1_,A2_,A3_}] :> + SchwingerIntegral[At+at, {A1+a1, A2+a2, A3+a3}], + s[1]^(a1_) s[2]^(a2_) st^(at_) SchwingerIntegral[At_, {A1_,A2_}][p_] :> + SchwingerIntegral[At+at, {A1+a1, A2+a2}][p] } + /. { b1->0, b2->0, b3->0, bt->0 } + /. { SI[x_] :> x } + //.{ l[x__] :> Sum[\[Delta][First[{x}],{x}[[j]]]l@@Delete[{x},{{1},{j}}], {j,2,Length[{x}]}] } + /. { l[] -> 1 } + ] + //. suContractIndices[d, idx/@Range[counter-1]] +]; + + +SchwingerToK[exp_] := exp /. { SchwingerIntegral -> SchwingerIntegralValue } //. + suMomentum /. suProductsToMagnitudes; + + +SchwingerToK[exp_, ks_] := Simplify[exp /. + { SchwingerIntegral -> SchwingerIntegralValue } /. + { i[a_,{b1_,b2_}] :> DoubleKValue[a,{b1,b2}] }//. + suMomentumEx[ks] /. + suProductsToMagnitudes]; + + +LoopToKOnce[exp_, ks_] := Module[{res}, + res = LoopToSchwinger[ + exp /. { LoopIntegral[d_,{\[Delta]1_,\[Delta]2_}][X_,k_,p_] :> LoopIntegral[d,{\[Delta]1,\[Delta]2}][ + LoopToKOnce[X, Union[ks, {k}]], k,p] /; !FreeQ[X, LoopIntegral], + LoopIntegral[d_,{\[Delta]1_,\[Delta]2_,\[Delta]3_}][X_,k_] :> LoopIntegral[d,{\[Delta]1,\[Delta]2,\[Delta]3}][ + LoopToKOnce[X, Union[ks, {k}]], k] /; !FreeQ[X, LoopIntegral] }, + ks]; + If[!FreeQ[res, LoopIntegral], + Message[LoopToK::loopfail, exp]]; + SchwingerToK[res, ks] +]; + + +fcall:LoopToK[exp_, opts___?OptionQ] := Module[{ValidOpts, recursive}, + ValidOpts = First /@ Options[LoopToK]; + Scan[If[!MemberQ[ValidOpts, First[#]], + Message[LoopToK::optx, ToString[First[#]], ToString[Unevaluated[fcall]]]]&, Flatten[{opts}]]; + recursive = Recursive /. Flatten[{opts}] /. Options[LoopToK]; + If [!BooleanQ[recursive], Message[LoopToK::barg, "ExpandDoubleK", recursive]; recursive = ExpandDoubleK /. Options[LoopToK]]; + + If[recursive, + LoopToKOnce[Unformat[exp] /. suProductsToMagnitudes /. suMomentum3, {}], + SchwingerToK[LoopToSchwinger[exp]] + ] +]; + + +fcall:LoopToK[exp_, k___] := Null /; Message[LoopToK::nonopt, Last[{k}], 1, ToString[Unevaluated[fcall]]]; +LoopToK[] := Null /; Message[LoopToK::argrx, "LoopToK", 0, 1]; + + +(* ::Section:: *) +(*Divergences*) + + +IsDivergent[i[a_?NumericQ, {b1_?NumericQ, b2_?NumericQ}]] := ( + (IntegerQ[(1+a-b1-b2)/2] && 1+a-b1-b2 <= 0) || + (IntegerQ[(1+a+b1-b2)/2] && 1+a+b1-b2 <= 0) || + (IntegerQ[(1+a-b1+b2)/2] && 1+a-b1+b2 <= 0) || + (IntegerQ[(1+a+b1+b2)/2] && 1+a+b1+b2 <= 0) ); +IsDivergent[i[a_, {b1_, b2_}], assume___] := Simplify[ + (Element[(1+a-b1-b2)/2, Integers] && 1+a-b1-b2 <= 0) || + (Element[(1+a+b1-b2)/2, Integers] && 1+a+b1-b2 <= 0) || + (Element[(1+a-b1+b2)/2, Integers] && 1+a-b1+b2 <= 0) || + (Element[(1+a+b1+b2)/2, Integers] && 1+a+b1+b2 <= 0), assume, TimeConstraint->0.1 ]; +IsDivergent[i[a_, {b1_, b2_}][p_], assume___] := IsDivergent[i[a, {b1,b2}], assume]; +IsDivergent[i[a_, {b1_, b2_}, u_,v_], assume___] := IsDivergent[i[a, {b1,b2}], assume]; +IsDivergent[i[a_, {b1_, b2_}, u_,v_][p_], assume___] := IsDivergent[i[a, {b1,b2}], assume]; + + +IsDivergent[i[a_?NumericQ, {b1_?NumericQ, b2_?NumericQ, b3_?NumericQ}]] := ( + (IntegerQ[(1+a-b1-b2-b3)/2] && 1+a-b1-b2-b3 <= 0) || + (IntegerQ[(1+a+b1-b2-b3)/2] && 1+a+b1-b2-b3 <= 0) || + (IntegerQ[(1+a-b1+b2-b3)/2] && 1+a-b1+b2-b3 <= 0) || + (IntegerQ[(1+a+b1+b2-b3)/2] && 1+a+b1+b2-b3 <= 0) || + (IntegerQ[(1+a-b1-b2+b3)/2] && 1+a-b1-b2+b3 <= 0) || + (IntegerQ[(1+a+b1-b2+b3)/2] && 1+a+b1-b2+b3 <= 0) || + (IntegerQ[(1+a-b1+b2+b3)/2] && 1+a-b1+b2+b3 <= 0) || + (IntegerQ[(1+a+b1+b2+b3)/2] && 1+a+b1+b2+b3 <= 0) ); +IsDivergent[i[a_, {b1_, b2_, b3_}], assume___] := Simplify[ + (Element[(1+a-b1-b2-b3)/2, Integers] && 1+a-b1-b2-b3 <= 0) || + (Element[(1+a+b1-b2-b3)/2, Integers] && 1+a+b1-b2-b3 <= 0) || + (Element[(1+a-b1+b2-b3)/2, Integers] && 1+a-b1+b2-b3 <= 0) || + (Element[(1+a+b1+b2-b3)/2, Integers] && 1+a+b1+b2-b3 <= 0) || + (Element[(1+a-b1-b2+b3)/2, Integers] && 1+a-b1-b2+b3 <= 0) || + (Element[(1+a+b1-b2+b3)/2, Integers] && 1+a+b1-b2+b3 <= 0) || + (Element[(1+a-b1+b2+b3)/2, Integers] && 1+a-b1+b2+b3 <= 0) || + (Element[(1+a+b1+b2+b3)/2, Integers] && 1+a+b1+b2+b3 <= 0), assume, TimeConstraint->0.1 ]; +IsDivergent[i[a_, {b1_, b2_, b3_}][p1_,p2_,p3_], assume___] := IsDivergent[i[a, {b1,b2,b3}], assume]; +IsDivergent[i[a_, {b1_, b2_, b3_}, u_, v_], assume___] := IsDivergent[i[a, {b1,b2,b3}], assume]; +IsDivergent[i[a_, {b1_, b2_, b3_}, u_, v_][p1_,p2_,p3_], assume___] := IsDivergent[i[a, {b1,b2,b3}], assume]; + + +BesselKCoeff[\[Sigma]_,\[Nu]_][j_]:=(-1)^(j) Gamma[-\[Sigma] \[Nu]-j]/(2^(\[Sigma] \[Nu]+2j+1) j!); +BesselKCoeff[1,n_,v_][j_] := (-1/2)^(1 + 2*j + n)/(v*\[Epsilon]*j!*(j + n)!) + + ((-1)^(2*j + n)*2^(-1 - 2*j - n)* + (Log[2] + PolyGamma[0, 1 + j + n]))/(j!*(j + n)!) - + ((-1)^(2*j + n)*2^(-2 - 2*j - n)*v*\[Epsilon]*(Pi^2 + 3*Log[2]^2 + + 6*Log[2]*PolyGamma[0, 1 + j + n] + + 3*PolyGamma[0, 1 + j + n]^2 - 3*PolyGamma[1, 1 + j + n]))/ + (3*j!*(j + n)!) + ((-1)^(2*j + n)*2^(-2 - 2*j - n)*v^2*\[Epsilon]^2* + (Pi^2*Log[2] + Log[2]^3 + Pi^2*PolyGamma[0, 1 + j + n] + + 3*Log[2]^2*PolyGamma[0, 1 + j + n] + + 3*Log[2]*PolyGamma[0, 1 + j + n]^2 + PolyGamma[0, 1 + j + n]^ + 3 - 3*Log[2]*PolyGamma[1, 1 + j + n] - + 3*PolyGamma[0, 1 + j + n]*PolyGamma[1, 1 + j + n] + + PolyGamma[2, 1 + j + n]))/(3*j!*(j + n)!) - + ((-1)^(2*j + n)*2^(-4 - 2*j - n)*v^3*\[Epsilon]^3* + (7*Pi^4 + 30*Pi^2*Log[2]^2 + 15*Log[2]^4 + + 60*Pi^2*Log[2]*PolyGamma[0, 1 + j + n] + + 60*Log[2]^3*PolyGamma[0, 1 + j + n] + + 30*Pi^2*PolyGamma[0, 1 + j + n]^2 + + 90*Log[2]^2*PolyGamma[0, 1 + j + n]^2 + + 60*Log[2]*PolyGamma[0, 1 + j + n]^3 + + 15*PolyGamma[0, 1 + j + n]^4 - + 30*Pi^2*PolyGamma[1, 1 + j + n] - + 90*Log[2]^2*PolyGamma[1, 1 + j + n] - + 180*Log[2]*PolyGamma[0, 1 + j + n]*PolyGamma[1, 1 + j + n] - + 90*PolyGamma[0, 1 + j + n]^2*PolyGamma[1, 1 + j + n] + + 45*PolyGamma[1, 1 + j + n]^2 + + 60*Log[2]*PolyGamma[2, 1 + j + n] + + 60*PolyGamma[0, 1 + j + n]*PolyGamma[2, 1 + j + n] - + 15*PolyGamma[3, 1 + j + n]))/(45*j!*(j + n)!); +BesselKCoeff[-1,n_,v_][j_] := ((-1)^(2*j + n)*2^(-1 - 2*j + n))/(v*\[Epsilon]*j!*(j - n)!) + + ((-1)^(2*j + n)*2^(-1 - 2*j + n)* + (Log[2] + PolyGamma[0, 1 + j - n]))/(j!*(j - n)!) + + ((-1)^(2*j + n)*2^(-2 - 2*j + n)*v*\[Epsilon]*(Pi^2 + 3*Log[2]^2 + + 6*Log[2]*PolyGamma[0, 1 + j - n] + + 3*PolyGamma[0, 1 + j - n]^2 - 3*PolyGamma[1, 1 + j - n]))/ + (3*j!*(j - n)!) + ((-1)^(2*j + n)*2^(-2 - 2*j + n)*v^2*\[Epsilon]^2* + (Pi^2*Log[2] + Log[2]^3 + Pi^2*PolyGamma[0, 1 + j - n] + + 3*Log[2]^2*PolyGamma[0, 1 + j - n] + + 3*Log[2]*PolyGamma[0, 1 + j - n]^2 + PolyGamma[0, 1 + j - n]^ + 3 - 3*Log[2]*PolyGamma[1, 1 + j - n] - + 3*PolyGamma[0, 1 + j - n]*PolyGamma[1, 1 + j - n] + + PolyGamma[2, 1 + j - n]))/(3*j!*(j - n)!) + + ((-1)^(2*j + n)*2^(-4 - 2*j + n)*v^3*\[Epsilon]^3* + (7*Pi^4 + 30*Pi^2*Log[2]^2 + 15*Log[2]^4 + + 60*Pi^2*Log[2]*PolyGamma[0, 1 + j - n] + + 60*Log[2]^3*PolyGamma[0, 1 + j - n] + + 30*Pi^2*PolyGamma[0, 1 + j - n]^2 + + 90*Log[2]^2*PolyGamma[0, 1 + j - n]^2 + + 60*Log[2]*PolyGamma[0, 1 + j - n]^3 + + 15*PolyGamma[0, 1 + j - n]^4 - + 30*Pi^2*PolyGamma[1, 1 + j - n] - + 90*Log[2]^2*PolyGamma[1, 1 + j - n] - + 180*Log[2]*PolyGamma[0, 1 + j - n]*PolyGamma[1, 1 + j - n] - + 90*PolyGamma[0, 1 + j - n]^2*PolyGamma[1, 1 + j - n] + + 45*PolyGamma[1, 1 + j - n]^2 + + 60*Log[2]*PolyGamma[2, 1 + j - n] + + 60*PolyGamma[0, 1 + j - n]*PolyGamma[2, 1 + j - n] - + 15*PolyGamma[3, 1 + j - n]))/(45*j!*(j - n)!); + + +KDivergenceValue[a_, {b1_,b2_}, u_, {v1_,v2_}, pp_, order_, assume___] /; order <= 0 := + Series[DoubleKValue[a+u \[Epsilon],{b1+v1 \[Epsilon],b2+v2 \[Epsilon]}][pp], {\[Epsilon],0,order}, assume]; + + +KDivergenceValue[type_, a_?NumericQ, {b1_?NumericQ,b2_?NumericQ,b3_?NumericQ}, u_, {v1_,v2_,v3_}, order_] /; order <= 0 := + KDivergenceValue[type, a, {b1,b2,b3}, u, {v1,v2,v3}, order] = + Module[{cutoff, nn, i1,i2,s}, + Sum[ + nn = (-s.{b1,b2,b3} - a - 1)/2; + If[TrueQ[0 <= nn && IntegerQ[nn]], + Sum[Series[cutoff^(-\[Epsilon](u+s.{v1,v2,v3}))/(\[Epsilon](u+s.{v1,v2,v3})) + * p[1]^(2i1 + If[s[[1]]<0,0,2(b1+v1 \[Epsilon])]) + * p[2]^(2i2 + If[s[[2]]<0,0,2(b2+v2 \[Epsilon])]) + * p[3]^(2(nn-i1-i2) + If[s[[3]]<0,0,2(b3+v3 \[Epsilon])]) + * BesselKCoeff[s[[1]],b1+v1 \[Epsilon]][i1] + * BesselKCoeff[s[[2]],b2+v2 \[Epsilon]][i2] + * BesselKCoeff[s[[3]],b3+v3 \[Epsilon]][nn-i1-i2], {\[Epsilon],0,order}], + {i1,0,nn}, {i2,0,nn-i1}], + 0], + {s, type} + ] /. { Log[cutoff]->0 } +]; + + +SingularQ[n_, assume___] := TrueQ[Simplify[Element[n, Integers] && 0<=n, assume, TimeConstraint->0.1]]; + + +KDivergenceValue[type_, a_, {b1_,b2_,b3_}, u_, {v1_,v2_,v3_}, order_, assume___] /; order <= 0 := + KDivergenceValue[type, a, {b1,b2,b3}, u, {v1,v2,v3}, order, assume] = + Module[{cutoff, nn, s, s1,s2,s3, sing0,sing1,sing2,sing3, + j,k, j0,j1,k0,k1, ass, res, factor }, + + ass = Assumptions -> Union[Assumptions /. assume, + {k>=0, k<=k1, j>=0, j<=j1, Element[k,Integers], Element[j,Integers]}]; + + sing0 = SingularQ[(b1+b2+b3-a-1)/2, assume]; + sing1 = SingularQ[#, assume]& /@ { (-b1+b2+b3-a-1)/2, (b1-b2+b3-a-1)/2, (b1+b2-b3-a-1)/2 }; + sing2 = SingularQ[#, assume]& /@ { (-b1-b2+b3-a-1)/2, (-b1+b2-b3-a-1)/2, (b1-b2-b3-a-1)/2 }; + sing3 = SingularQ[(-b1-b2-b3-a-1)/2, assume]; + + factor[special_, ss_] := cutoff^(-\[Epsilon](u+ss.{v1,v2,v3}))/(\[Epsilon](u+ss.{v1,v2,v3})) + * p[1]^(2Subscript[K, 2] + If[ss[[1]]<0,0,2(b1+v1 \[Epsilon])]) + * p[2]^(2(nn-Subscript[K, 1]-Subscript[K, 2]) + If[ss[[2]]<0,0,2(b2+v2 \[Epsilon])]) + * p[3]^(2Subscript[K, 1] + If[ss[[3]]<0,0,2(b3+v3 \[Epsilon])]) + * If[special == 1, BesselKCoeff[ss[[1]],b1,v1][Subscript[K, 2]], BesselKCoeff[ss[[1]],b1+v1 \[Epsilon]][Subscript[K, 2]]] + * If[special == 2, BesselKCoeff[ss[[2]],b2,v2][nn-Subscript[K, 1]-Subscript[K, 2]], BesselKCoeff[ss[[2]],b2+v2 \[Epsilon]][nn-Subscript[K, 1]-Subscript[K, 2]]] + * If[special == 3, BesselKCoeff[ss[[3]],b3,v3][Subscript[K, 1]], BesselKCoeff[ss[[3]],b3+v3 \[Epsilon]][Subscript[K, 1]]]; + + If[sing3, Message[KDivergence::ksing, "(+++)"]]; + If[sing2[[1]], Message[KDivergence::ksing, "(++-)"]]; + If[sing2[[2]], Message[KDivergence::ksing, "(+-+)"]]; + If[sing2[[3]], Message[KDivergence::ksing, "(-++)"]]; + + Sum[ + nn = Simplify[(-s.{b1,b2,b3} - a - 1)/2, assume, TimeConstraint->0.1]; + If[SingularQ[nn], Sum[Series[factor[0, s], {\[Epsilon],0,order}, assume], + {Subscript[K, 1],0,nn}, {Subscript[K, 2],0,nn-Subscript[K, 1]}], 0], + {s, Complement[type, {{-1,-1,-1}}]} + ] + + If[MemberQ[type, {-1,-1,-1}], + s = {-1,-1,-1}; + nn = Simplify[(-s.{b1,b2,b3} - a - 1)/2, assume, TimeConstraint->0.1]; + If[sing0 && sing1[[3]], HoldForm[Sum][Series[factor[3, s], {\[Epsilon],0,order}, assume], + {Subscript[K, 1],b3,nn}, {Subscript[K, 2],0,nn-Subscript[K, 1]}], 0] + + If[sing0 && sing1[[1]], HoldForm[Sum][Series[factor[1, s], {\[Epsilon],0,order}, assume], + {Subscript[K, 1], 0, Simplify[If[sing0 && sing1[[3]], Min[nn, b3-1], nn], assume, TimeConstraint->0.1]}, + {Subscript[K, 2],b1,nn-Subscript[K, 1]}], 0] + + If[sing0 && sing1[[2]], HoldForm[Sum][Series[factor[2, s], {\[Epsilon],0,order}, assume], + {Subscript[K, 1], 0, Simplify[If[sing0 && sing1[[3]], Min[nn-b2, b3-1], nn-b2], assume, TimeConstraint->0.1]}, + {Subscript[K, 2], 0, Simplify[If[sing0 && sing1[[1]], Min[nn-Subscript[K, 1]-b2,b1-1], nn-Subscript[K, 1]-b2], assume, TimeConstraint->0.1]}], 0] + + If[sing0, HoldForm[Sum][Series[factor[0, s], {\[Epsilon],0,order}, assume], + {Subscript[K, 1], 0, Simplify[If[sing0 && sing1[[3]], Min[nn, b3-1], nn], assume, TimeConstraint->0.1]}, + {Subscript[K, 2], Simplify[If[sing0 && sing1[[2]], Max[0, nn-b2+1-Subscript[K, 1]], 0], assume, TimeConstraint->0.1], + Simplify[If[sing0 && sing1[[1]], Min[nn-Subscript[K, 1],b1-1], nn-Subscript[K, 1]], assume, TimeConstraint->0.1]}], 0] + /. { HoldForm[Sum][x_, {j_,j0_,j1_}, {k_,k0_,k1_}] :> + HoldForm[Sum][x /. {j->j+j0, k->k+k0},{ j,0, Simplify[j1-j0/.{k->k+k0},ass,TimeConstraint->0.1]}, + {k,0,Simplify[k1-k0 /. {j->j+j0},ass,TimeConstraint->0.1]}] } + /. { HoldForm[Sum][x_, {j_,0,j1_}, {k_,0,k1_}] :> 0 /; Simplify[j1<0 || k1<0, ass,TimeConstraint->0.1] } + //.{ HoldForm[Sum][x_, {j_,0,j1_}, {k_,0,k1_}] :> + HoldForm[Sum][x /. {k->0}, {j,0,j1/.{k->0}}] /; Simplify[k1==0, ass,TimeConstraint->0.1], + HoldForm[Sum][x_, {j_,0,j1_}, {k_,0,k1_}] :> + HoldForm[Sum][x /. {j->0}, {k,0,k1/.{j->0}}] /; Simplify[j1==0,ass,TimeConstraint->0.1], + HoldForm[Sum][x_, {j_,0,j1_}] :> (x /. {j->0}) /; Simplify[j1==0,ass,TimeConstraint->0.1] }, 0] + /. { Log[cutoff]->0 } +]; + + +AllDivs = Flatten[Table[{s1,s2,s3}, {s1,{-1,1}}, {s2,{-1,1}}, {s3,{-1,1}}],2]; + + +KAllDivergenceValue[a_, {b1_,b2_,b3_}, u_, {v1_,v2_,v3_}, order_, assume___] := + KDivergenceValue[AllDivs, a,{b1,b2,b3}, u,{v1,v2,v3}, order, assume]; + + +fcall:KDivergence[exp_, opts___?OptionQ] := Module[{ValidOpts,type,order,defu,defv,assume}, + ValidOpts = First /@ Options[KDivergence]; + Scan[If[!MemberQ[ValidOpts, First[#]], + Message[KDivergence::optx, ToString[First[#]], ToString[Unevaluated[fcall]]]]&, Flatten[{opts}]]; + {type,order,defu,defv,assume} = {Type, ExpansionOrder, uParameter, vParameters, {Assumptions}} + /. Flatten[{opts}] /. Options[KDivergence]; + If [!IntegerQ[order], Message[KDivergence::iarg, order]]; + assume = Assumptions -> If[$Assumptions === True, Flatten[assume], Union[$Assumptions,Flatten[assume]]]; + type = DeleteDuplicates @ Cases[#, {s1_,s2_,s3_} /; + (s1===1||s1===-1) && + (s2===1||s2===-1) && + (s3===1||s3===-1)]& @ + If[type === All, + AllDivs, + If[!ListQ[type], + Message[KDivergence::targ, type]; {}, + If[Length[type] > 0, + If[ListQ[First[type]], + type, + {type}]]]]; + + PrepareExpression[exp, defu,defv, assume] + /. { i[x__][p__] :> 0 /; !IsDivergent[i@@{x}] } + /. { i[a_?NumericQ, {b1_?NumericQ,b2_?NumericQ,b3_?NumericQ}, u_, {v1_,v2_,v3_}][p1_,p2_,p3_] :> + ( KDivergenceValue[type, a,{b1,b2,b3},u,{v1,v2,v3},order] /. {p[1]->p1, p[2]->p2, p[3]->p3} ) + } + /. { i[a_, {b1_,b2_}, u_, {v1_,v2_}][pp_] :> + KDivergenceValue[a,{b1,b2},u,{v1,v2},pp,order, assume], + i[a_, {b1_,b2_,b3_}, u_, {v1_,v2_,v3_}][p1_,p2_,p3_] :> + ( KDivergenceValue[type, a,{b1,b2,b3},u,{v1,v2,v3},order, assume] /. {p[1]->p1, p[2]->p2, p[3]->p3} ) + } + /. suPolyGammaToHarmonic[assume] + /. suPostpareExpression +]; + + +KDivergence[exp_SeriesData, opts___?OptionQ] := + If[exp[[1]] === \[Epsilon], + KDivergence[Normal[exp], opts], + SeriesData[exp[[1]],exp[[2]], KDivergence[#, opts]& /@ (exp[[3]]), exp[[4]],exp[[5]],exp[[6]]]]; + + +fcall:KDivergence[exp_, k___] := Null /; Message[KDivergence::nonopt, Last[{k}], 1, ToString[Unevaluated[fcall]]]; +KDivergence[] := Null /; Message[KDivergence::argrx, "KDivergence", 0, 1]; + + +(* ::Section:: *) +(*Reduction scheme*) + + +\[Lambda]fun[a_,b_,c_]:=a^4+b^4+c^4 - 2a^2 b^2 - 2 a^2 c^2 - 2 b^2 c^2; +Xfun[a_,b_,c_]:=(-a^2+b^2+c^2-Sqrt[\[Lambda]fun[a,b,c]])/(2c^2); +Yfun[a_,b_,c_]:=(-b^2+a^2+c^2-Sqrt[\[Lambda]fun[a,b,c]])/(2c^2); +NLfun[a_,b_,c_] := Pi^2/6-2Log[a/c]Log[b/c]+Log[Xfun[a,b,c]]Log[Yfun[a,b,c]]-PolyLog[2,Xfun[a,b,c]]-PolyLog[2,Yfun[a,b,c]]; + + +\[Lambda]val = p[1]^4+p[2]^4+p[3]^4 - 2 p[1]^2 p[2]^2-2 p[1]^2 p[3]^2-2 p[2]^2 p[3]^2; +Xval = (-p[1]^2+p[2]^2+p[3]^2-Sqrt[\[Lambda]val])/(2p[3]^2); +Yval = (-p[2]^2+p[1]^2+p[3]^2-Sqrt[\[Lambda]val])/(2p[3]^2); +NLval = Pi^2/6 - 2 Log[p[1]/p[3]]Log[p[2]/p[3]]+Log[Xval]Log[Yval]-PolyLog[2,Xval]-PolyLog[2,Yval]; + + +vt[v_] := v[[1]]+v[[2]]+v[[3]]; +vtt[v_] := v[[1]]^2+v[[2]]^2+v[[3]]^2; + + +i1000 = 1/(2 Sqrt[\[Lambda][p[1],p[2],p[3]]])*NL[p[1], p[2], p[3]] + SeriesData[\[Epsilon],0,{0},0,1,1]; + + +i2111fin = 2 p[1]^2 p[2]^2 p[3]^2 / (\[Lambda][p[1],p[2],p[3]])^(3/2) NL[p[1], p[2], p[3]] + 1/(2 \[Lambda][p[1],p[2],p[3]])( p[1]^2(p[2]^2+p[3]^2-p[1]^2) Log[p[1]^2]+p[2]^2(p[1]^2+p[3]^2-p[2]^2) Log[p[2]^2]+p[3]^2(p[1]^2+p[2]^2-p[3]^2) Log[p[3]^2] ); +i2111div[u_,v_] := 1/((u - vt[v]) \[Epsilon])+u / (u - vt[v])(Log[2]-EulerGamma); +i2111[u_,v_] := i2111fin + i2111div[u,v] + SeriesData[\[Epsilon],0,{0},0,1,1]; + + +i0111m2[u_,v_] := 1/(2(vt[v] - u))Sum[p[j]^2/(u - vt[v] + 2 v[[j]]), {j,1,3}]; +i0111m1[u_,v_] := 1/4 Sum[p[j]^2 Log[p[j]^2]/(u - vt[v] + 2 v[[j]]), {j,1,3}] + (u (1-2EulerGamma+2Log[2])-vt[v]) / (4(vt[v]-u))Sum[p[j]^2/(u-vt[v]+2v[[j]]), {j,1,3}]; +i0111sch[u_,v_] := 1/8 Sum[v[[j]]/(u-vt[v]+2v[[j]])p[j]^2Log[p[j]^2]^2,{j,1,3}]+1/8(u(1-2EulerGamma+2Log[2])-vt[v])Sum[p[j]^2Log[p[j]^2]/(u-vt[v]+2v[[j]]),{j,1,3}]+((vt[v]-u(1-EulerGamma+Log[2]))^2+u^2(EulerGamma-Log[2])^2+1/3 Pi^2 vtt[v])/(8(vt[v]-u))Sum[p[j]^2/(u-vt[v]+2v[[j]]),{j,1,3}]; +i0111fin = -1/8 Sqrt[\[Lambda][p[1],p[2],p[3]]]*NL[p[1],p[2],p[3]]; +i0111scviol = 1/16( (p[3]^2-p[1]^2-p[2]^2)Log[p[1]^2]Log[p[2]^2]+(p[2]^2-p[1]^2-p[3]^2)Log[p[1]^2]Log[p[3]^2]+(p[1]^2-p[2]^2-p[3]^2)Log[p[2]^2]Log[p[3]^2] ); + + +i0111div[u_,v_] := i0111m2[u,v] / \[Epsilon]^2 + i0111m1[u,v] / \[Epsilon] + i0111sch[u,v]; +i0111[u_,v_] := i0111div[u,v] + i0111fin + i0111scviol + SeriesData[\[Epsilon],0,{0},0,1,1]; + + +iahalfhalfhalf[a_] := (Pi/2)^(3/2) * Gamma[a-1/2] (p[1] + p[2] + p[3])^(1/2 - a) + SeriesData[\[Epsilon],0,{0},0,1,1]; +iahalfhalfhalfreg[a_,u_,v_] := Module[{div, div0}, + div = KAllDivergenceValue[a, {1/2,1/2,1/2}, u,v, 0]; + div0 = KAllDivergenceValue[a, {1/2,1/2,1/2}, 1,{0,0,0}, 0]; + iahalfhalfhalf[a + \[Epsilon]] + div - div0 + SeriesData[\[Epsilon],0,{0},0,1,1] + ]; + + +halfBessel[j_,\[Beta]_] := (Abs[\[Beta]]-1/2+j)!/(2^j * j! * (Abs[\[Beta]]-1/2-j)!); + + +iahalfhalfhalf[a_,{b1_,b2_,b3_}] := Module[{k1,k2,k3}, + (Pi/2)^(3/2) * Sum[Gamma[a-1/2-k1-k2-k3] * (p[1] + p[2] + p[3])^(1/2-a+k1+k2+k3) + * p[1]^(b1-k1-1/2) p[2]^(b2-k2-1/2) p[3]^(b3-k3-1/2) + * halfBessel[k1,b1] halfBessel[k2,b2] halfBessel[k3,b3], + {k1,0,Abs[b1]-1/2}, {k2,0,Abs[b2]-1/2}, {k3,0,Abs[b3]-1/2}] + SeriesData[\[Epsilon],0,{0},0,1,1] +]; +iahalfhalfhalfreg[a_,{b1_,b2_,b3_}, u_,v_] := Module[{div, div0}, + div = KAllDivergenceValue[a, {b1,b2,b3}, u,v, 0]; + div0 = KAllDivergenceValue[a, {b1,b2,b3}, 1,{0,0,0}, 0]; + iahalfhalfhalf[a + \[Epsilon], {b1,b2,b3}] + div - div0 + SeriesData[\[Epsilon],0,{0},0,1,1] +]; + + +IntExpK[\[Mu]_,\[Alpha]_,\[Nu]_,\[Beta]_] := 1/(\[Alpha]+\[Beta])^(\[Mu]+\[Nu])Gamma[\[Mu]+\[Nu]]Gamma[\[Mu]-\[Nu]]/Gamma[\[Mu]+1/2]Hypergeometric2F1[\[Mu]+\[Nu],\[Nu]+1/2,\[Mu]+1/2,(\[Alpha]-\[Beta])/(\[Alpha]+\[Beta])]; + + +iaBhalfhalf[a_,{B_,b2_,b3_}] := Module[{k2,k3}, + 2^(B-1)*Pi^(3/2)*p[1]^(2B) * Sum[p[2]^(b2-k2-1/2) p[3]^(b3-k3-1/2) + * halfBessel[k2,b2] halfBessel[k3,b3] * IntExpK[a-k2-k3, p[2]+p[3], B, p[1]], + {k2,0,Abs[b2]-1/2}, {k3,0,Abs[b3]-1/2}] + SeriesData[\[Epsilon],0,{0},0,1,1] +]; +iaBhalfhalfreg[a_,{B_,b2_,b3_}, u_,v_] := Module[{div, div0}, + div = KAllDivergenceValue[a, {B,b2,b3}, u,v, 0]; + div0 = KAllDivergenceValue[a, {B,b2,b3}, 1,{0,0,0}, 0]; + iaBhalfhalf[a + \[Epsilon], {B,b2,b3}] + div - div0 + SeriesData[\[Epsilon],0,{0},0,1,1] +]; + + +TripleKHalf[a_,{1/2,1/2,1/2}, u_,v_] /; OddQ[2a] && 2a<=1 := + TripleKHalf[a,{1/2,1/2,1/2}, u,v] = iahalfhalfhalfreg[a, u, v]; +TripleKHalf[a_,{1/2,1/2,1/2}, u_,v_] := TripleKHalf[a,{1/2,1/2,1/2}, u,v] = iahalfhalfhalf[a]; +TripleKHalf[a_,{b1_,b2_,b3_}, u_,v_] /; OddQ[2b1] && OddQ[2b2] && OddQ[2b3] && IsDivergent[i[a,{b1,b2,b3}]] := + TripleKHalf[a,{b1,b2,b3}, u,v] = iahalfhalfhalfreg[a, {b1,b2,b3}, u, v]; +TripleKHalf[a_,{b1_,b2_,b3_}, u_,v_] /; OddQ[2b1] && OddQ[2b2] && OddQ[2b3] := + TripleKHalf[a,{b1,b2,b3}, u,v] = Series[iahalfhalfhalf[a + \[Epsilon], {b1,b2,b3}],{\[Epsilon],0,0}]; + + +TripleKHalf[a_,{b1_,b2_,b3_}, u_,v_] /; !OddQ[2b1] && OddQ[2b2] && OddQ[2b3] && IsDivergent[i[a,{b1,b2,b3}]] := + TripleKHalf[a,{b1,b2,b3}, u,v] = iaBhalfhalfreg[a, {b1,b2,b3}, u, v]; +TripleKHalf[a_,{b1_,b2_,b3_}, u_,v_] /; !OddQ[2b1] &&OddQ[2b2] && OddQ[2b3] := + TripleKHalf[a,{b1,b2,b3}, u,v] = Series[iaBhalfhalf[a + \[Epsilon], {b1,b2,b3}],{\[Epsilon],0,0}]; +TripleKHalf[a_,{b1_,b2_,b3_}, u_,{v1_,v2_,v3_}] /; OddQ[2b1] && !OddQ[2b2] && OddQ[2b3] := + TripleKHalf[a,{b1,b2,b3}, u,{v1,v2,v3}] = Swap[iaBhalfhalfreg[a, {b2,b1,b3}, u, {v2,v1,v3}], p[2],p[1]]; +TripleKHalf[a_,{b1_,b2_,b3_}, u_,{v1_,v2_,v3_}] /; OddQ[2b1] && OddQ[2b2] && !OddQ[2b3] := + TripleKHalf[a,{b1,b2,b3}, u,{v1,v2,v3}] = Swap[iaBhalfhalfreg[a, {b3,b2,b1}, u, {v3,v2,v1}], p[3],p[1]]; + + +DoubleKValue[a_,{b1_,b2_}][p_] := 2^(a-2)*p^(-1-a+b1+b2) / Gamma[1+a] * + Gamma[(1+a-b1-b2)/2]*Gamma[(1+a+b1-b2)/2]*Gamma[(1+a-b1+b2)/2]*Gamma[(1+a+b1+b2)/2]; +DoubleKValue[a_,{b1_,b2_}, u_,{v1_,v2_}][p_] := + Series[DoubleKValue[a+u \[Epsilon],{b1+v1 \[Epsilon],b2+v2 \[Epsilon]}][p], {\[Epsilon],0,0}]; +(* DoubleKValue[a_,{b1_,b2_}, u_,{v1_,v2_}][p_] := + DoubleKValue[a+u \[Epsilon],{b1+v1 \[Epsilon],b2+v2 \[Epsilon]}][p]; *) + + +MinPos[b_] := Ordering[b,1][[1]]; +MaxPos[b_] := Ordering[b,-1][[1]]; + + +n0[a_,b_] := ( Abs[b[[1]]]+Abs[b[[2]]]+Abs[b[[3]]]-a-1 )/2; +n1[a_,b_] := ( Abs[b[[1]]]+Abs[b[[2]]]+Abs[b[[3]]]-2Min[Abs[b]]-a-1 )/2; +n2[a_,b_] := (-Abs[b[[1]]]-Abs[b[[2]]]-Abs[b[[3]]]+2Max[Abs[b]]-a-1 )/2; +n3[a_,b_] := (-Abs[b[[1]]]-Abs[b[[2]]]-Abs[b[[3]]]-a-1 )/2; +GetNs[i[a_,b_]] := { n0[a,b], n1[a,b], n2[a,b], n3[a,b] }; + + +IsSolvable[i[a_,{b1_,b2_}]] = True; +IsSolvable[i[a_,{b1_,b2_}][p_]] = True; +IsSolvable[i[a_Integer,{b1_Integer,b2_Integer,b3_Integer}]] := Module[{ns}, + ns = GetNs[i[a,{b1,b2,b3}]]; + AllTrue[ns, IntegerQ] && ns[[-2]]<0 +]; +IsSolvable[i[a_,{b1_,b2_,b3_}]] := + (OddQ[2 b1] && OddQ[2 b2]) || (OddQ[2 b1] && OddQ[2 b3]) || (OddQ[2 b2] && OddQ[2 b3]); +IsSolvable[i[a_,{b1_,b2_,b3_}][p1_,p2_,p3_]] := IsSolvable[i[a,{b1,b2,b3}]]; + + +IOperator[j_,\[Beta]_][F_] := p[j]^(2 \[Beta]) F; +MOperator[j_,\[Beta]_][F_] := 2\[Beta] F-(p[j] D[F, p[j]]); +LOperator[j_][F_] := -1/p[j] D[F, p[j]]; +KOperator[j_,\[Beta]_][F_] := D[F, p[j],p[j]] - (2\[Beta] - 1)/p[j] D[F, p[j]]; +BOperator[a_,b_][F_] := PrefBOperator[a,b][ + p[1]^2 MOperator[2, b[[2]]-1][MOperator[3, b[[3]]-1][F]] + + p[2]^2 MOperator[3, b[[3]]-1][MOperator[1, b[[1]]-1][F]] + + p[3]^2 MOperator[1, b[[1]]-1][MOperator[2, b[[2]]-1][F]] ]; +PrefBOperator[a_,b_][F_] := 1/(a+1-b[[1]]-b[[2]]-b[[3]]) * F; + + +TripleKValue[1,{0,0,0}, u_,v_] = i1000; +TripleKValue[2,{1,1,1}, u_,v_] := TripleKValue[2,{1,1,1}, u,v] = i2111[u,v]; +TripleKValue[0,{1,1,1}, u_,v_] := TripleKValue[0,{1,1,1}, u,v] = i0111[u,v]; + + +TripleKValue[a_,{b1_,b2_,b3_}, u_,{v1_,v2_,v3_}] /; b2 + DoubleKValue[a,{b1,b2},u,{v1,v2}][p], + i[a_, {b1_,b2_}, u_, {v1_,v2_}][p_] :> + DoubleKValue[a + u \[Epsilon],{b1 + v1 \[Epsilon], b2 + v2 \[Epsilon]}][p], + i[a_Integer, {b1_Integer,b2_Integer,b3_Integer}, u_, {v1_,v2_,v3_}][p1_,p2_,p3_] :> + ( TripleKValue[a,{b1,b2,b3},u,{v1,v2,v3}] /. {p[1]->p1, p[2]->p2, p[3]->p3} ) + /; TrueQ[IsSolvable[i[a,{b1,b2,b3}]]], + i[a_, {b1_,b2_,b3_}, u_, {v1_,v2_,v3_}][p1_,p2_,p3_] :> + ( TripleKHalf[a,{b1,b2,b3},u,{v1,v2,v3}] /. {p[1]->p1, p[2]->p2, p[3]->p3} ) + /; (OddQ[2 b1] && OddQ[2 b2]) || (OddQ[2 b1] && OddQ[2 b3]) || (OddQ[2 b2] && OddQ[2 b3]) + } + /. suPolyGammaToHarmonic[] + /. suPostpareExpression + // KExpandFunctions[#, level]& +]; + + +KEvaluate[exp_SeriesData, opts___?OptionQ] := + If[exp[[1]] === \[Epsilon], + KEvaluate[Normal[exp], opts], + SeriesData[exp[[1]],exp[[2]], KEvaluate[#, opts]& /@ (exp[[3]]), exp[[4]],exp[[5]],exp[[6]]]]; + + +fcall:KEvaluate[exp_, k___] := Null /; Message[KEvaluate::nonopt, Last[{k}], 1, ToString[Unevaluated[fcall]]]; +KEvaluate[] := Null /; Message[KEvaluate::argrx, "KEvaluate", 0, 1]; + + +fcall:LoopEvaluate[exp_, opts___?OptionQ] := Module[{ValidOpts, defu,defv,level,recursive}, + ValidOpts = First /@ Options[LoopEvaluate]; + Scan[If[!MemberQ[ValidOpts, First[#]], + Message[LoopEvaluate::optx, ToString[First[#]], ToString[Unevaluated[fcall]]]]&, Flatten[{opts}]]; + {recursive, defu,defv, level} = {Recursive, uParameter, vParameters, ExpansionLevel} + /. Flatten[{opts}] /. Options[LoopEvaluate]; + + KEvaluate[LoopToK[exp, Recursive -> recursive], + { uParameter -> defu, vParameters -> defv, ExpansionLevel -> level }] +]; + + +fcall:LoopEvaluate[exp_, k___] := Null /; Message[LoopEvaluate::nonopt, Last[{k}], 1, ToString[Unevaluated[fcall]]]; +LoopEvaluate[] := Null /; Message[LoopEvaluate::argrx, "LoopEvaluate", 0, 1]; + + +(* ::Section:: *) +(*Conformal operators*) + + +ScalarKOp[F_, p_, \[Beta]_] := D[F,p,p] - (2\[Beta]-1)/p * D[F,p]; +ScalarKKOp[F_,p1_,p2_,\[Beta]1_,\[Beta]2_] := ScalarKOp[F, p1, \[Beta]1] - ScalarKOp[F, p2, \[Beta]2]; + + +ScalarKOp[x___] := Null /; Message[ScalarKOp::argrx, "KOp", Length[{x}], 3]; +ScalarKKOp[x___] := Null /; Message[ScalarKOp::argrx, "KOp", Length[{x}], 5]; + + +(* ::Section:: *) +(*End*) + + +End[]; +Protect @@ Names["TripleK`*"]; +EndPackage[];