(* 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[ 191395, 4500] NotebookOptionsPosition[ 177195, 4038] NotebookOutlinePosition[ 178102, 4069] CellTagsIndexPosition[ 178015, 4064] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["A quick introduction to the ColorMath package", "Title", CellChangeTimes->{{3.552303955831121*^9, 3.552303977621805*^9}, 3.5523725985175943`*^9, {3.55237298835699*^9, 3.552372996095862*^9}, { 3.552381558954961*^9, 3.5523815716570387`*^9}, {3.5524889370819693`*^9, 3.552488939341982*^9}}], Cell["\<\ ColorMath is a package for symbolically performing color summed calculations \ in QCD (SU(Nc)). \ \>", "Text", CellChangeTimes->{{3.552490061294855*^9, 3.5524901341531773`*^9}, { 3.5524901668169737`*^9, 3.55249017188241*^9}, 3.5525738947885733`*^9, 3.559549265784073*^9}], Cell[TextData[{ StyleBox["Warning: To input color structure use copy and paste, type in the \ FullForm, or use the short command functions. 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FullForm[", Cell[BoxData[ SubscriptBox[ RowBox[{"(", TemplateBox[{"\[Delta]","Q1"}, "Superscript"], ")"}], "Q2"]], "Output", GeneratedCell->False, CellAutoOverwrite->False, CellChangeTimes->{ 3.5523044731136427`*^9, {3.5523045916794767`*^9, 3.552304612318742*^9}, { 3.552309027541548*^9, 3.552309052165141*^9}, 3.552365838358552*^9, 3.552373022922453*^9, 3.5523837240625*^9, 3.552561174462961*^9, { 3.559299791198843*^9, 3.5592998049372787`*^9}}], "] gives the same result." }], "Text", CellChangeTimes->{{3.5523043343619328`*^9, 3.5523044353670807`*^9}, { 3.552372609940667*^9, 3.552372614244483*^9}, {3.552381601148233*^9, 3.552381621915721*^9}, {3.5525733259739647`*^9, 3.552573374301073*^9}, { 3.559299846607397*^9, 3.559299914525445*^9}, {3.5592999514675426`*^9, 3.559299984454183*^9}, {3.55930004800646*^9, 3.559300050912615*^9}, 3.5593840927124987`*^9, 3.559473718367815*^9, 3.559477275669293*^9, { 3.5602396135843153`*^9, 3.56023962690467*^9}, {3.561285695771389*^9, 3.5612857616778307`*^9}}] 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CellChangeTimes->{{3.552304573891387*^9, 3.552304606307301*^9}, { 3.5593001730332193`*^9, 3.5593001787172117`*^9}}], " for each quark-gluon vertex, and ", Cell[BoxData[ RowBox[{"I", TemplateBox[{"f",RowBox[{"{", RowBox[{"g3", ",", "g2", ",", "g1"}], "}"}]}, "Superscript"]}]], "Input", CellChangeTimes->{{3.552370734811524*^9, 3.552370749053899*^9}, { 3.559543351973974*^9, 3.559543354605855*^9}}], " for each triple gluon vertex, and by making sure that internal indices are \ contracted by appearing twice.\n\n", StyleBox["If you try some basic calculations and fail, see \ \[OpenCurlyDoubleQuote]Common mistakes\[CloseCurlyDoubleQuote] and \ \[OpenCurlyDoubleQuote]Input and FullForm\[CloseCurlyDoubleQuote] below, and \ try out the WhatIsWrong function.", FontColor->RGBColor[1, 0, 0]] }], "Text", CellChangeTimes->{{3.552305171953828*^9, 3.552305199374908*^9}, { 3.552381970026721*^9, 3.5523819784194*^9}, {3.552489099939045*^9, 3.5524891010344267`*^9}, {3.560496014883943*^9, 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The amplitude, having an s-channel contribution, and a t-channel \ contribution, can be written in the form\ \>", "Text", CellChangeTimes->{{3.559303074217882*^9, 3.559303104491644*^9}, { 3.559303181929307*^9, 3.559303255280924*^9}, {3.559303368312055*^9, 3.559303471812831*^9}, {3.5593035036422987`*^9, 3.5593036400906076`*^9}, { 3.559381553130932*^9, 3.559381556078372*^9}, {3.559543499683856*^9, 3.55954353495191*^9}, {3.5602404922784758`*^9, 3.560240494553008*^9}, 3.560513282964179*^9, 3.5605137631296577`*^9, 3.5612998585202303`*^9}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Amp", "=", RowBox[{ RowBox[{"S", SubscriptBox[ TemplateBox[{TemplateBox[{"t", RowBox[{"{", "g", "}"}]}, "Superscript"],"q1"}, "Superscript"], "q2"], SubscriptBox[ TemplateBox[{TemplateBox[{"t", RowBox[{"{", "g", "}"}]}, "Superscript"],"q4"}, "Superscript"], "q3"]}], "+", RowBox[{"T", SubscriptBox[ TemplateBox[{TemplateBox[{"t", RowBox[{"{", "g", "}"}]}, "Superscript"],"q1"}, "Superscript"], "q3"], SubscriptBox[ TemplateBox[{TemplateBox[{"t", RowBox[{"{", "g", "}"}]}, "Superscript"],"q4"}, "Superscript"], "q2"]}]}]}]], "Input", CellChangeTimes->{{3.559021847101935*^9, 3.5590218949998693`*^9}, { 3.5590222373670177`*^9, 3.5590222412930193`*^9}, {3.559022297652206*^9, 3.5590223267286787`*^9}, {3.559549118801992*^9, 3.559549148909272*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"T", " ", SubscriptBox[ TemplateBox[{TemplateBox[{"t", RowBox[{"{", "g", "}"}]}, "Superscript"],"q1"}, "Superscript"], "q3"], " ", SubscriptBox[ TemplateBox[{TemplateBox[{"t", RowBox[{"{", "g", "}"}]}, "Superscript"],"q4"}, "Superscript"], "q2"]}], "+", RowBox[{"S", " ", SubscriptBox[ TemplateBox[{TemplateBox[{"t", RowBox[{"{", "g", "}"}]}, "Superscript"],"q1"}, "Superscript"], "q2"], " ", SubscriptBox[ TemplateBox[{TemplateBox[{"t", RowBox[{"{", "g", "}"}]}, "Superscript"],"q4"}, "Superscript"], "q3"]}]}]], "Output", CellChangeTimes->CompressedData[" 1:eJxTTMoPSmVkYGDQAuLDn418rra8cuxK+cr+FUi3HzyS9w1Ir7vW7fkbSD83 Z3in0PrKUfCxMZcykFbgzFMA0fqxLs0qQJrB5fYMEP3i3O9MDSD9oPzbK00g fSjqsp4WkH50TFPOFUjr6axL8QLST3t3S/sC6bsXv9+pA9IH5n8q+gWka5U5 mv4B6S0bHe4ytb1ydBcINmcG0vt5LMpZgPQmzaqgQCA9+/iuSBDd9WHXXjDd /EW8Ckhfb9+QD6K/zFhkXQukU0w4Ns4D0gujAt4uAtIHbk/TXQKk/cz2LQbR E9LYM5cCad2OY5+k2l85qlxw+pvd8cpRrfF40RQgve/3GZ7pQHrzeaedIHr/ R9H1s4F088aWz4uBtFe+/La9QNqtTvrSz8mvHIVY7n2bN+OVIwAP0bP9 "]] }, Open ]], Cell["where S and T denote momentum space factors.", "Text", CellChangeTimes->{{3.55930364471271*^9, 3.559303654219604*^9}}], Cell["\<\ To square the amplitude we need to multiply it with its conjugate and \ contract the indices. 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This is the case both for gluon resummation calculations, and \ for next to leading order calculations with a loop from gluon exchange \ between external legs. It is calculated using the function CGamma, which \ takes as arguments the basis (i.e. a List of the names of the basis vectors), \ the parton numbers (i.e. the place of the index in the tensor definition), \ and, optionally, the scalar product matrix and its inverse. The result is \ contained in a matrix, where the effect of gluon exchange between parton1 and \ parton2 on vector j is contained in column j. Thus element ij is the i \ component of the resulting vector. By default, consistency checks are made. It is checked that the new vector \ squared and its basis decomposed version squared agree, and that orthonormal \ bases results in symmetric matrices. 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" }], "Text", CellChangeTimes->{{3.5593060444814653`*^9, 3.55930608425007*^9}, { 3.559377165944496*^9, 3.559377198525305*^9}, {3.559377294368121*^9, 3.559377431027995*^9}, {3.559377461547963*^9, 3.559377579822907*^9}, 3.559377721809457*^9, {3.5593819811339903`*^9, 3.5593819849940453`*^9}, { 3.559382017891111*^9, 3.55938217327037*^9}, {3.5593842561027946`*^9, 3.559384259229765*^9}, 3.559474826879715*^9, {3.559478011595634*^9, 3.5594780362226667`*^9}, {3.559925064211372*^9, 3.559925069362376*^9}, { 3.560511859089365*^9, 3.560511862177187*^9}}], Cell["\<\ As partons 1 and 2 are in a manifest multiplet in this basis, the result of \ the exchange is just a constant for each basis vector. Exchanging a gluon \ between parton 1 and 3 we get a less simple result.\ \>", "Text", CellChangeTimes->{{3.5593821596839733`*^9, 3.559382185685863*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"CGamma", "[", RowBox[{"OurBasis", ",", "1", ",", "3", ",", "SPM", ",", RowBox[{"Inverse", "[", "SPM", "]"}], ",", " ", RowBox[{"BasisType", "\[Rule]", " ", "OrthogonalBasis"}], ",", " ", RowBox[{"Verbose", "\[Rule]", " ", "False"}]}], "]"}]], "Input", CellChangeTimes->{{3.559377762449863*^9, 3.5593777733318653`*^9}, { 3.559378130512492*^9, 3.559378135030023*^9}, {3.559478237030208*^9, 3.559478239761443*^9}, {3.5599250876705847`*^9, 3.5599250887562943`*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"0", ",", FractionBox["TR", "Nc"], ",", RowBox[{ RowBox[{"(", RowBox[{"1", "-", FractionBox["4", SuperscriptBox["Nc", "2"]]}], ")"}], " ", SuperscriptBox["TR", "2"]}], ",", SuperscriptBox["TR", "2"]}], "}"}], ",", RowBox[{"{", RowBox[{ FractionBox["TR", "Nc"], ",", "0", ",", RowBox[{ RowBox[{"(", RowBox[{"1", "-", FractionBox["4", SuperscriptBox["Nc", "2"]]}], ")"}], " ", SuperscriptBox["TR", "2"]}], ",", SuperscriptBox["TR", "2"]}], "}"}], ",", RowBox[{"{", RowBox[{ FractionBox["1", "2"], ",", FractionBox["1", "2"], ",", FractionBox[ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "6"}], "+", SuperscriptBox["Nc", "2"]}], ")"}], " ", "TR"}], RowBox[{"2", " ", "Nc"}]], ",", FractionBox[ RowBox[{"Nc", " ", "TR"}], "2"]}], "}"}], ",", RowBox[{"{", RowBox[{ FractionBox["1", "2"], ",", FractionBox["1", "2"], ",", FractionBox[ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "4"}], "+", SuperscriptBox["Nc", "2"]}], ")"}], " ", "TR"}], RowBox[{"2", " ", "Nc"}]], ",", FractionBox[ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "2"}], "+", SuperscriptBox["Nc", "2"]}], ")"}], " ", "TR"}], RowBox[{"2", " ", "Nc"}]]}], "}"}]}], "}"}]], "Output", CellChangeTimes->{ 3.5593777640986547`*^9, {3.559377947212129*^9, 3.55937798169985*^9}, { 3.559378123875436*^9, 3.559378135591547*^9}, 3.559458673013049*^9, 3.5594629947599792`*^9, 3.5594631729504538`*^9, 3.559466011129098*^9, 3.559466184020694*^9, 3.559474854594083*^9, 3.559477080890843*^9, 3.559478067924794*^9, 3.559532786657222*^9, 3.559651102505622*^9, 3.559904170508438*^9, 3.5599214304310503`*^9, 3.559922225386895*^9, 3.5599250945771313`*^9, 3.560082385431693*^9, 3.560082437053277*^9, 3.560082927165169*^9, 3.560166295621731*^9, 3.5601720470051126`*^9, 3.560249229376598*^9, 3.560251768238645*^9, 3.560252703305935*^9, 3.560495058385661*^9, 3.5605032809741163`*^9, 3.5611862432847433`*^9, 3.561273594813683*^9, 3.5612748780463753`*^9, 3.561275141219751*^9, 3.5612831270852222`*^9, 3.561300059818994*^9, 3.561352800755899*^9, 3.567242976567719*^9, 3.569678320552928*^9}] }, Open ]], Cell["\<\ More information about the functions and rules can be obtained using ? Name, \ for example for CGamma we can read about sign conventions:\ \>", "Text", CellChangeTimes->{{3.559478072540719*^9, 3.559478084707551*^9}, { 3.559478217812745*^9, 3.559478230282866*^9}, {3.559478342891217*^9, 3.559478374652446*^9}, {3.559544176658553*^9, 3.559544189163748*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"?", "CGamma"}]], "Input", CellChangeTimes->{{3.5594782319905863`*^9, 3.559478242302483*^9}}], Cell[BoxData[ StyleBox["\<\"CGamma[CB, p1, p2] calculates the effect of exchanging a gluon \ between parton p1 and parton p2 in the color basis CB={C1,..Cn}, where each \ basis vector, \!\(\*SubscriptBox[\(C1\), \({I1, \(\(...\) \(Ik\)\)}\)]\) \ etc., has been defined. The result is contained in a matrix, where column j \ contains the basis decomposed version of the vector resulting after a gluon \ exchange between p1 and p2 in the jth vector. The sign convention of the \ triple gluon vertex is given by ordering the partons as external index, \ internal dummy index, and index of gluon to be exchaged. The quark-gluon \ vertex comes without additional signs. Two more arguments, containing the \ scalar product matrix and its inverse, can be supplied, \ CGamma[CB,p1,p2,spm,Inverse[spm]]. As a self-consistency check, it is checked \ that the vector after gluon exchange has the same norm before and after being \ basis decomposed, unless the option MakeChecks\[Rule]False is used. For \ orthonormal bases it is also by default checked that the resulting matrix is \ symmetric. For more efficient calculations the option BasisType (with default \ value GeneralBasis) may be set as: BasisType \[Rule] OrthonormalBasis (for an \ orthonormal basis), BasisType \[Rule] OrthogonalBasis (for an orthogonal \ basis) or BasisType \[Rule] TraceBasis if the basis a is trace basis, i.e., a \ basis where each basis vector is proportional to one product of open and \ closed quark-lines (as opposed to a sum). For simplifying roots the option \ NcMin can be used, and for turning off progress messages Verbose\[Rule] False \ may be used.\"\>", "MSG"]], "Print", "PrintUsage", CellChangeTimes->{3.569678323591687*^9}, CellTags->"Info3569681923-6013797"] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Input and FullForm", "Section", CellChangeTimes->{{3.559300472875924*^9, 3.559300477593445*^9}, { 3.559543176300959*^9, 3.559543179126445*^9}}], Cell["\<\ To successfully use ColorMath it is essential to use color tensors which have \ the correct FullForm. \ \>", "Text", CellChangeTimes->{{3.5593004855088778`*^9, 3.559300547587034*^9}, { 3.5593006598911657`*^9, 3.5593006643658543`*^9}, {3.559301539499776*^9, 3.5593015700420523`*^9}, {3.5593020453329573`*^9, 3.559302074987124*^9}, 3.559302290675188*^9, 3.5593843000863943`*^9, {3.560082610226427*^9, 3.560082625137759*^9}, {3.560086524819072*^9, 3.560086589463503*^9}, { 3.56127540763622*^9, 3.561275422094206*^9}}], Cell[CellGroupData[{ 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To get started, it is advised to copy \ the color structures, or to write in the FullForm:\ \>", "Text", CellChangeTimes->{{3.559301769345881*^9, 3.5593019030216007`*^9}, { 3.559302110264751*^9, 3.559302181003277*^9}, 3.55930246880497*^9, { 3.559382322668338*^9, 3.5593823794505253`*^9}, 3.5593844539326677`*^9, { 3.560243235314094*^9, 3.56024323714879*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Superscript", "[", RowBox[{"f", ",", RowBox[{"{", RowBox[{"I1", ",", "I2", ",", "I3"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.559301913143116*^9, 3.5593019593534517`*^9}, 3.559302008799097*^9}], Cell[BoxData[ TemplateBox[{"f",RowBox[{"{", RowBox[{"I1", ",", "I2", ",", "I3"}], "}"}]}, "Superscript"]], "Output", CellChangeTimes->{3.559302010744671*^9, 3.559302183141547*^9, 3.559371611122487*^9, 3.559382380717148*^9, 3.559458673381435*^9, 3.5594629954118643`*^9, 3.559463173083334*^9, 3.55946601140483*^9, 3.559466184199383*^9, 3.559477081711596*^9, 3.559532787853045*^9, 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