(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 4.0, MathReader 4.0, or any compatible application. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 455005, 17106]*) (*NotebookOutlinePosition[ 455753, 17133]*) (* CellTagsIndexPosition[ 455709, 17129]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[TextData[{ "Normalization ", Cell[BoxData[ \(TraditionalForm\`\((N\_ab)\)\)]], " and ", Cell[BoxData[ \(TraditionalForm\`\[Theta]\)]], " dependence ", Cell[BoxData[ \(TraditionalForm\`\((F\_ab)\)\)]], "\nfor the ", Cell[BoxData[ \(TraditionalForm\`qq \[Rule] \ \[Gamma]\[Gamma]\)]], " and ", Cell[BoxData[ \(TraditionalForm\`gg \[Rule] \[Gamma]\[Gamma]\)]], " processes" }], "Subsubtitle"], 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" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ SubscriptBox["M", RowBox[{\(u d\&_\), "\[Rule]", FormBox[ RowBox[{ FormBox[\(e\^+\), "TraditionalForm"], \(\[Nu]\_e\)}], "TraditionalForm"]}]], "=", \(\(("\Ene\>" // GetAmplitude)\) /. m\_x_ :> 0 /; x =!= W\)}]], "Input"], Cell[BoxData[ FormBox[ RowBox[{"{", FormBox[ RowBox[{ RowBox[{"(", RowBox[{\(e\^2\), " ", RowBox[{"(", RowBox[{ RowBox[{"\[CurlyPhi]", "\[NoBreak]", "(", "\[NoBreak]", FormBox[\(k\_2\), "TraditionalForm"], "\[NoBreak]", ")"}], " ", \(\[Gamma]\^\[Mu]\), " ", \(\[Gamma]\^7\), " ", RowBox[{"\[CurlyPhi]", "\[NoBreak]", "(", "\[NoBreak]", FormBox[\(-k\_1\), "TraditionalForm"], "\[NoBreak]", ")"}]}], ")"}], " ", RowBox[{"(", RowBox[{ RowBox[{"\[CurlyPhi]", "\[NoBreak]", "(", "\[NoBreak]", FormBox[\(-p\_2\), "TraditionalForm"], "\[NoBreak]", ")"}], " ", \(\[Gamma]\^\[Mu]\), " ", \(\[Gamma]\^7\), " ", RowBox[{"\[CurlyPhi]", "\[NoBreak]", "(", "\[NoBreak]", FormBox[\(p\_1\), "TraditionalForm"], "\[NoBreak]", ")"}]}], ")"}], " ", \(V\_ud\)}], ")"}], "/", \((2\ \((\((p\_1 + p\_2)\)\^2 - m\_W\%2)\)\ s\_w\%2)\)}], "TraditionalForm"], "}"}], TraditionalForm]], "Output"] }, Open ]], Cell["\<\ The Mandelstam variables are defined such that the fermion comes from the \ proton:\ \>", "Text"], Cell[BoxData[ \(\(SetMandelstam[s, t, u, p\_1, p\_2, \(-k\_1\), \(-k\_2\), 0, 0, 0, 0]; \)\)], "Input"], Cell["The amplitude squared is:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ SubscriptBox[\(M\^2\), RowBox[{\(u d\&_\), "\[Rule]", FormBox[ RowBox[{ FormBox[\(e\^+\), "TraditionalForm"], \(\[Nu]\_e\)}], "TraditionalForm"]}]], "=", RowBox[{ RowBox[{"(", RowBox[{ SubscriptBox["M", RowBox[{\(u d\&_\), "\[Rule]", FormBox[ RowBox[{ FormBox[\(e\^+\), "TraditionalForm"], \(\[Nu]\_e\)}], "TraditionalForm"]}]], "//", "AmplitudeSquare"}], ")"}], "[", \([1, 1]\), "]"}]}]], "Input"], Cell[BoxData[ \(TraditionalForm \`\(e\^4\ t\^2\ V\_ud\%2\)\/\(\((s - m\_W\%2)\)\^2\ s\_w\%4\)\)], "Output"] }, Open ]], Cell["In the W CM frame:", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Momenta = {\np\_1 -> \(\@s\/2\) {I, 0, 0, 1}, p\_2 -> \(\@s\/2\) {I, 0, 0, \(-1\)}, \n k\_1 -> \(\@s\/2\) {I, 0, Sin[\[Theta]], Cos[\[Theta]]}, \n k\_2 -> \(\@s\/2\) {I, 0, \(-Sin[\[Theta]]\), \(-Cos[\[Theta]]\)}}; \)\), "\n", \(MandelstamVariables = {\ns -> \(-\((p\_1 + p\_2)\) . \((p\_1 + p\_2)\)\) /. Momenta; \n t -> \((\(\(\(-\((p\_1 - k\_1)\) . \((p\_1 - k\_1)\)\) /. Momenta\) /. Sin[\[Theta]]\^2 -> 1 - Cos[\[Theta]]\^2 // Expand\) // Factor)\), \n u -> \((\(\(\(-\((p\_1 - k\_2)\) . \((p\_1 - k\_2)\)\) /. Momenta\) /. Sin[\[Theta]]\^2 -> 1 - Cos[\[Theta]]\^2 // Expand\) // Factor)\)}\)}], "Input"], Cell[BoxData[ \(TraditionalForm \`{t \[Rule] 1\/2\ s\ \((cos(\[Theta]) - 1)\), u \[Rule] \(-\(1\/2\)\)\ s\ \((cos(\[Theta]) + 1)\)}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ SubscriptBox[\(M\^2\), RowBox[{\(u d\&_\), "\[Rule]", FormBox[ RowBox[{ FormBox[\(e\^+\), "TraditionalForm"], \(\[Nu]\_e\)}], "TraditionalForm"]}]], "/.", "MandelstamVariables"}], " "}]], "Input"], Cell[BoxData[ \(TraditionalForm \`\(e\^4\ s\^2\ \((cos(\[Theta]) - 1)\)\^2\ V\_ud\%2\)\/\(4 \((s - \ m\_W\%2)\)\^2\ s\_w\%4\)\)], "Output"] }, Open ]], Cell[BoxData[ RowBox[{ RowBox[{\(Amp2[uD -> \(e\^+\) \[Nu]\_e]\), "=", FormBox[ FormBox[\(\(e\^4\ s\^2\ \((1 - cos(\[Theta]))\)\^2\ V\_ud\%2\)\/\(4\ \ \((s - m\_W\%2)\)\^2\ s\_w\%4\)\), "TraditionalForm"], "TraditionalForm"]}], ";"}]], "Input"], Cell["\<\ This agrees with my hand calculations (notes: Total Cross Section) and \ C.-P.'s notes (Zeroth Order) and with CompHEP.\ \>", "Text"], Cell[TextData[{ "The averaging factors: initial state color ", Cell[BoxData[ \(TraditionalForm\`\((1\/3)\)\)]], " and spin ", Cell[BoxData[ \(TraditionalForm\`\((\(1\/2\) 1\/2)\)\)]], "." }], "Text"], Cell[CellGroupData[{ Cell["CompHEP result", "Subsubsection"], Cell[BoxData[ \(<< "\<~/CompHEP/comphep_32.18/results/uD-e1n1.m\>"\)], "Input"], Cell[TextData[{ "Taking away the initial state color ", Cell[BoxData[ \(TraditionalForm\`\((1\/3)\)\)]], " and spin ", Cell[BoxData[ \(TraditionalForm\`\((\(1\/2\) 1\/2)\)\)]], " averaging factors and changing notation:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(1\/\(\(1\/3\) \(1\/2\) 1\/2\)\) \((Summ /. {EE\ -> \ e, MZ -> m\_W\/\@\(1 - s\_w\^2\), SW -> s\_w, SC[p1, p3] -> \(-t\)\/2, SC[p1, p2] -> s\/2, s12\^2 -> V\_ud + 1, s13\^2 -> V\_ud + 1} // Simplify)\)\)], "Input"], Cell[BoxData[ \(TraditionalForm \`\(e\^4\ t\^2\ V\_ud\%2\)\/\(\((s - m\_W\%2)\)\^2\ s\_w\%4\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(% /. t -> \(1 - Cos[\[Theta]]\)\/2\)], "Input"], Cell[BoxData[ \(TraditionalForm \`\(e\^4\ \((1 - cos(\[Theta]))\)\^2\ V\_ud\%2\)\/\(4 \((s - \ m\_W\%2)\)\^2\ s\_w\%4\)\)], "Output"] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ Cell[BoxData[ \(TraditionalForm\`q q\&_\)]], " \[Rule] ", Cell[BoxData[ \(TraditionalForm\`\(W\^-\)\)]], " \[Rule] ", Cell[BoxData[ FormBox[ RowBox[{ FormBox[\(e\^-\), "TraditionalForm"], \(\[Nu]\&_\_e\)}], TraditionalForm]]] }], "Section"], Cell[TextData[{ "The three amplitude of ", Cell[BoxData[ \(TraditionalForm\`u d\&_\)]], " \[Rule] ", Cell[BoxData[ \(TraditionalForm\`\(W\^-\)\)]], " \[Rule] ", Cell[BoxData[ FormBox[ RowBox[{ FormBox[\(e\^-\), "TraditionalForm"], \(\[Nu]\&_\_e\)}], TraditionalForm]]], " is:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(CreateAmplitude1[{SM, QCD}, {u\&_, d} -> {\(e\^-\), \[Nu]\&_\_e}, 0] \)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: 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Momenta; \n t -> \((\(\(\(-\((p\_1 - k\_1)\) . \((p\_1 - k\_1)\)\) /. Momenta\) /. Sin[\[Theta]]\^2 -> 1 - Cos[\[Theta]]\^2 // Expand\) // Factor)\), \n u -> \((\(\(\(-\((p\_1 - k\_2)\) . \((p\_1 - k\_2)\)\) /. Momenta\) /. 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We made tha quark charge explicit." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(FAL = "\eE\>" // GetAmplitude;\)\), "\n", \(Amp = \(\((\(Q\_f\/\(2\/3\)\) FAL[\([1, 2]\)] // Contract)\) /. Conjugate[QMM11] -> V\_ud\) /. {MU\ -> \ 0, ME -> 0, MLA -> 0}\)} ], "Input"], Cell[BoxData[ FormBox[ RowBox[{\(e\^2\), " ", RowBox[{"(", RowBox[{ RowBox[{"\[CurlyPhi]", "\[NoBreak]", "(", "\[NoBreak]", FormBox["k1", "TraditionalForm"], "\[NoBreak]", ")"}], " ", \(\[Gamma]\^\[Mu]\), " ", RowBox[{"\[CurlyPhi]", "\[NoBreak]", "(", "\[NoBreak]", FormBox[\(-k2\), "TraditionalForm"], "\[NoBreak]", ")"}]}], ")"}], " ", RowBox[{"(", RowBox[{ RowBox[{"\[CurlyPhi]", "\[NoBreak]", "(", "\[NoBreak]", FormBox[\(-p2\), "TraditionalForm"], "\[NoBreak]", ")"}], " ", \(\[Gamma]\^\[Mu]\), " ", RowBox[{"\[CurlyPhi]", "\[NoBreak]", "(", "\[NoBreak]", FormBox["p1", "TraditionalForm"], "\[NoBreak]", ")"}]}], ")"}], " ", FormBox[\(1\/\((p1 + p2)\)\^2\), "TraditionalForm"], " ", \(Q\_f\)}], TraditionalForm]], "Output"] }, Open ]], Cell["The Mandelstam variables are defined in the standard way:", "Text"], Cell[BoxData[ \(\(SetMandelstam[s, t, u, p1, p2, \(-k1\), \(-k2\), 0, 0, 0, 0];\)\)], "Input"], Cell["The amplitude squared is:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Amp2[0]\ = \(\(\(\((\(ComplexConjugate[Amp] Amp // \n\tFermionSpinSum\) // \n\t DiracSimplify)\) /. \n\tDiracTrace -> Tr // \n\t Simplify\) // \n\tContract\) // PropagatorDenominatorExplicit\) // Simplify\)], "Input"], Cell[BoxData[ \(TraditionalForm\`\(8\ e\^4\ \((t\^2 + u\^2)\)\ Q\_f\%2\)\/s\^2\)], "Output"] }, Open ]], Cell["In the W CM frame:", "Text"], Cell[BoxData[{ \(p\_1 = \(\@s\/2\) {I, 0, 0, 1}; p\_2 = \(\@s\/2\) {I, 0, 0, \(-1\)};\n k\_1 = \(\@s\/2\) {I, 0, Sin[\[Theta]], Cos[\[Theta]]}; k\_2 = \(\@s\/2\) {I, 0, \(-Sin[\[Theta]]\), \(-Cos[\[Theta]]\)};\n s -> \(-\((p\_1 + p\_2)\) . \((p\_1 + p\_2)\)\)\), "\n", \(t -> \(-\((p\_1 - k\_1)\) . \((p\_1 - k\_1)\)\) // Simplify\), "\n", \(u -> \(-\((p\_1 - k\_2)\) . \((p\_1 - k\_2)\)\) // Simplify\)}], "Input"], Cell[BoxData[ \(\(Mandelstam = {t -> s \( 1 - Cos[\[Theta]]\)\/2, u -> s \( 1 + Cos[\[Theta]]\)\/2}; \)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(\(Amp2[0] /. Mandelstam // Expand\) // Factor\)], "Input"], Cell[BoxData[ \(TraditionalForm\`4\ e\^4\ \((\(cos\^2\)(\[Theta]) + 1)\)\ Q\_f\%2\)], "Output"] }, Open ]], Cell[BoxData[ RowBox[{ RowBox[{\(Amp2[qQ -> \(e\^+\) \(e\^-\)]\), "=", FormBox[ FormBox[\(4\ e\^4\ \((\(cos\^2\)(\[Theta]) + 1)\)\ Q\_f\%2\), "TraditionalForm"], "TraditionalForm"]}], ";"}]], "Input"], Cell["This agrees with CompHEP.", "Text"], Cell[TextData[{ "The averaging factors: initial state color ", Cell[BoxData[ \(TraditionalForm\`\((\(1\/3\) 1\/3)\)\)]], " and spin ", Cell[BoxData[ \(TraditionalForm\`\((\(1\/2\) 1\/2)\)\)]], ".\nSummation over colors has not been done yet." }], "Text"], Cell[CellGroupData[{ Cell["CompHEP result", "Subsubsection"], Cell[BoxData[ \(<< "\<~/CompHEP/comphep_32.18/results/uU-eE.m\>"\)], "Input"], Cell[TextData[{ "Taking away the initial state color ", Cell[BoxData[ \(TraditionalForm\`\((\(1\/3\) 1\/3)\)\)]], " and spin ", Cell[BoxData[ \(TraditionalForm\`\((\(1\/2\) 1\/2)\)\)]], " averaging factors, making the quark charge and the color sum explicit \ and changing notation:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(1\/\(\(\((1\/3)\)\^2\) \((1\/2)\)\^2\)\) \(Q\_f\^2\/\((2\/3)\)\^2\) \(N\_C\/3\) \((Summ /. {EE\ -> \ e, SC[p1, p3] -> \(-t\)\/2, SC[p1, p2] -> s\/2} // Simplify)\) // TrickMandelstam[#, {s, t, u, 0}] &\)], "Input"], Cell[BoxData[ \(TraditionalForm \`\(8\ e\^4\ \((t\^2 + u\^2)\)\ N\_C\ Q\_f\%2\)\/s\^2\)], "Output"] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ Cell[BoxData[ \(TraditionalForm\`q q\&_\)]], " \[Rule] \[Gamma] \[Gamma]" }], "Section"], Cell[CellGroupData[{ Cell["FeynCalc", "Subsubsection"], Cell[TextData[{ "The three amplitude of ", Cell[BoxData[ \(TraditionalForm\`u u\&_\)]], " \[Rule] \[Gamma] \[Gamma] is:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(CreateAmplitude1[{SM, QCD}, {u, U}, {A, A}, 0]\)], "Input"], Cell[CellGroupData[{ Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica 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o`040000oooooooo00006?ooo`007Oooo`8000000oooo`000000000eoooo00<0003oooooool0:_oo o`030000oooo000001Soool008?oool300006Oooo`00Woooo`00Woooo`00Woooo`00Woooo`00Wooo o`00Woooo`00Woooo`00Woooo`00Woooo`00Woooo`00Woooo`00Woooo`00Woooo`00Woooo`00Wooo o`00Woooo`00Woooo`00Woooo`00Woooo`00Woooo`00Woooo`00Woooo`00\ \>"], ImageRangeCache->{{{0, 157.875}, {157.875, 0}} -> {-1, -0.842644, 0.196696, 0.196696}}] }, Open ]] }, Open ]], Cell[TextData[{ "The momentum assignment is (as usual with FA): ", Cell[BoxData[ \(TraditionalForm\`\(u(p\_1)\) \(\(u\&_\)(p\_2)\)\)]], " \[Rule] ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ FormBox["\[Gamma]", "TraditionalForm"], "(", \(k\_1\), ")"}], \(\[Gamma](k\_2)\)}], TraditionalForm]]], ".\nWe use 't Hooft-Feynman gauge. The u quark charge is included by FA and \ taken made explicit." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(FAL = "\AA\>" // GetAmplitude;\)\n Amp[1] = \(Q\_f\^2\/\((2\/3)\)\^2\) FAL[\([1, 2]\)] /. {MU\ -> \ 0, MD -> 0, ME -> 0}\), "\n", \(Amp[2] = \(Q\_f\^2\/\((2\/3)\)\^2\) FAL[\([2, 2]\)] /. {MU\ -> \ 0, MD -> 0, ME -> 0}\)}], "Input"], Cell[BoxData[ FormBox[ RowBox[{\(e\^2\), " ", RowBox[{"(", RowBox[{ RowBox[{"\[CurlyPhi]", "\[NoBreak]", "(", "\[NoBreak]", FormBox[\(-p2\), "TraditionalForm"], "\[NoBreak]", ")"}], " ", \(\[Gamma]\^\[Lambda]\), " ", RowBox[{ FormBox["\<\"\[Gamma]\"\>", "TraditionalForm"], "\[NoBreak]", "\[CenterDot]", "\[NoBreak]", "(", "\[NoBreak]", FormBox[\(p1 - k1\), "TraditionalForm"], "\[NoBreak]", ")"}], " ", \(\[Gamma]\^\[Nu]\), " ", RowBox[{"\[CurlyPhi]", "\[NoBreak]", "(", "\[NoBreak]", FormBox["p1", "TraditionalForm"], "\[NoBreak]", ")"}]}], ")"}], " ", \(\(\[CurlyEpsilon]\_\[Lambda]\%*\)(k2)\), " ", \(\(\[CurlyEpsilon]\_\[Nu]\%*\)(k1)\), " ", FormBox[\(1\/\((k1 - p1)\)\^2\), "TraditionalForm"], " ", \(Q\_f\%2\)}], TraditionalForm]], "Output"], Cell[BoxData[ FormBox[ RowBox[{\(e\^2\), " ", RowBox[{"(", RowBox[{ RowBox[{"\[CurlyPhi]", "\[NoBreak]", "(", "\[NoBreak]", FormBox[\(-p2\), "TraditionalForm"], "\[NoBreak]", ")"}], " ", \(\[Gamma]\^\[Nu]\), " ", RowBox[{ FormBox["\<\"\[Gamma]\"\>", "TraditionalForm"], "\[NoBreak]", "\[CenterDot]", "\[NoBreak]", "(", "\[NoBreak]", FormBox[\(k1 - p2\), "TraditionalForm"], "\[NoBreak]", ")"}], " ", \(\[Gamma]\^\[Lambda]\), " ", RowBox[{"\[CurlyPhi]", "\[NoBreak]", "(", "\[NoBreak]", FormBox["p1", "TraditionalForm"], "\[NoBreak]", ")"}]}], ")"}], " ", \(\(\[CurlyEpsilon]\_\[Lambda]\%*\)(k2)\), " ", \(\(\[CurlyEpsilon]\_\[Nu]\%*\)(k1)\), " ", FormBox[\(1\/\((k1 - p2)\)\^2\), "TraditionalForm"], " ", \(Q\_f\%2\)}], TraditionalForm]], "Output"] }, Open ]], Cell[BoxData[ \(\(\(\n \)\(SetMandelstam[s, t, u, p1, p2, \(-k1\), \(-k2\), 0, 0, 0, 0];\)\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Do[ Do[\n\tAmp2[i, j] = \(\((\(\(\(\(\(ComplexConjugate[Amp[i]] Amp[j] // Simplify\) // \n\tSumPolarization[#, k1] &\) // SumPolarization[#, k2] &\) // \n\tContract\) // \n\tFermionSpinSum\) // \n\tDiracSimplify)\) /. \n\ DiracTrace -> Tr // Simplify\) // \n\t TrickMandelstam[#, {s, t, u, 0}] &\n, {i, 2}], {j, 2}];\)\), "\n", \(Amp2[3] = \(\[Sum]\+\(i = 1\)\%2\(\[Sum]\+\(j = 1\)\%2 Amp2[i, j]\) // \n\t\tPropagatorDenominatorExplicit\) // Simplify\)}], "Input"], Cell[BoxData[ \(TraditionalForm\`\(8\ e\^4\ \((t\^2 + u\^2)\)\ Q\_f\%4\)\/\(t\ u\)\)], "Output"] }, Open ]], Cell[TextData[{ "This must still be summed over ", Cell[BoxData[ \(TraditionalForm\`N\_C\)]], "." }], "Text"], Cell["In the \[Gamma]\[Gamma] CM frame:", "Text"], Cell[BoxData[{ \(p\_1 = \(\@s\/2\) {I, 0, 0, 1}; p\_2 = \(\@s\/2\) {I, 0, 0, \(-1\)};\n k\_1 = \(\@s\/2\) {I, 0, Sin[\[Theta]], Cos[\[Theta]]}; k\_2 = \(\@s\/2\) {I, 0, \(-Sin[\[Theta]]\), \(-Cos[\[Theta]]\)};\n s -> \ \(-\((p\_1 + p\_2)\) . \((p\_1 + p\_2)\)\)\), "\n", \(t -> \(-\((p\_1 - k\_1)\) . \((p\_1 - k\_1)\)\) // Simplify\), "\n", \(u -> \(-\((p\_1 - k\_2)\) . \((p\_1 - k\_2)\)\) // Simplify\)}], "Input"], Cell[BoxData[ \(\(Mandelstam = {t -> s \( 1 - Cos[\[Theta]]\)\/2, u -> s \( 1 + Cos[\[Theta]]\)\/2}; \)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(Amp2[3] /. Mandelstam // Simplify[#, Trig -> False] &\)], "Input"], Cell[BoxData[ \(TraditionalForm \`\(-\(\(16\ e\^4\ \((\(cos\^2\)(\[Theta]) + 1)\) Q\_f\%4\)\/\(\(cos\^2\)(\[Theta]) - 1\)\)\)\)], "Output"] }, Open ]], Cell[BoxData[ RowBox[{ RowBox[{\(Amp2[qQ -> \[Gamma]\[Gamma]]\), "=", FormBox[ \(-\(\(16\ e\^4\ \((\(cos\^2\)(\[Theta]) + 1)\) Q\_f\%4\)\/\(\(cos\^2\)(\[Theta]) - 1\)\)\), "TraditionalForm"]}], ";"}]], "Input"], Cell["This agrees with both Ohnemus and Owens and CompHEP.", "Text"], Cell[TextData[{ "The averaging factors: initial state color ", Cell[BoxData[ \(TraditionalForm\`\((\(1\/3\) 1\/3)\)\)]], " and spin ", Cell[BoxData[ \(TraditionalForm\`\((\(1\/2\) 1\/2)\)\)]], " and the identical final state particles factor ", Cell[BoxData[ \(TraditionalForm\`\((1\/2)\)\)]], "." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["CompHEP", "Subsubsection"], Cell[BoxData[ \(<< "\<~/CompHEP/comphep_32.18/results/sum1.m\>"\)], "Input"], Cell[TextData[{ "To obtain the uanveraged amplitude square we take away the initial state \ color ", Cell[BoxData[ \(TraditionalForm\`\((\(1\/3\) 1\/3)\)\)]], " and spin ", Cell[BoxData[ \(TraditionalForm\`\((\(1\/2\) 1\/2)\)\)]], " averaging factors and the identical final state particles factor ", Cell[BoxData[ \(TraditionalForm\`\((1\/2)\)\)]], ", make explicit the quark charges ", Cell[BoxData[ \(TraditionalForm\`\((Q\_f = 2\/3)\)\)]], " and the sum over colors ", Cell[BoxData[ \(TraditionalForm\`\((N\_C = 3\)\)]], "), and change notation:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(1\/\(\(\((1\/3)\)\^2\) \(\((1\/2)\)\^2\) 1\/2\)\) \(Q\_f\^4\/\((2\/3)\)\^4\) \(N\_C\/3\) Summ /. {EE\ -> \ e, SW -> s\_w, SC[p1, p3] -> \(-t\)\/2, SC[p1, p2] -> s\/2} // Factor\) // TrickMandelstam[#, {s, t, u, 0}] &\)], "Input"], Cell[BoxData[ \(TraditionalForm \`\(8\ e\^4\ \((t\^2 + u\^2)\)\ N\_C\ Q\_f\%4\)\/\(t\ u\)\)], "Output"] }, Open ]], Cell["This agrees with both Ohnemus and Owens and FC.", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Papageno", "Subsubsection"], Cell["\<\ In Papageno the following is coded: CONST=AEM**2*16.*PISQ CONST1=QU4 CONST2=QD4 .... A4=CONST*2.*(U2+T2)/U/T/3. A5=CONST*AL**2*11.**2/9**2.*2./4./8./8.*XMAT_PHOT(S,T,U) WT=((SF(23)+SF(24))*CONST1+(SF(27)+SF(28))*CONST2)*A4/2. 1 +SF(9)*A5/2./16./PISQ That is\ \>", "Text"], Cell[BoxData[ \(\(\(\ \)\(WT\_\(q q\&_ \[Rule] \[Gamma]\[Gamma]\) = Q\_q\^4\ \[Alpha]\_EM\^2\ 16\ \[Pi]\^2\ 2 \(\( u\^2 + t\^2\)\/\(u\ t\)\) \(1\/3\) 1\/2 /. \[Alpha]\_EM -> e\^2\/\(4 \[Pi]\);\)\)\)], "Input"], Cell[TextData[{ "After taking out the initial state color ", Cell[BoxData[ \(TraditionalForm\`\((\(1\/3\) 1\/3)\)\)]], " and spin ", Cell[BoxData[ \(TraditionalForm\`\((\(1\/2\) 1\/2)\)\)]], " averaging factors and the identical final state particles factor ", Cell[BoxData[ \(TraditionalForm\`\((1\/2)\)\)]], ", and making the color sum explicit, for the amplitude square (before \ averaging) we obtain:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(\(1\/D\_F\^2\) \(1\/\((N - 2)\)\^2\) 1\/N\_identical\)\/\(\(\((1\/3)\)\^2\) \(\((1\/2)\)\^2\) 1\/2\)\) \(Abs[M\_\(q q\&_ \[Rule] \[Gamma]\[Gamma]\)[s, t, u]]\^2\/\(\(8\ e\^4\ \ \((t\^2 + u\^2)\)\ Q\_q\%4\)\/\(t\ u\)\)\) \(N\_C\/3\) WT\_\(q q\&_ \[Rule] \[Gamma]\[Gamma]\)\)], "Input"], Cell[BoxData[ \(TraditionalForm \`\(\[LeftBracketingBar]M\_\(q\ q\&_ \[Rule] \[Gamma]\[Gamma]\)[s, t, u]\ \[RightBracketingBar]\^2\ N\_C\)\/\(\((N - 2)\)\^2\ \(D\_F\%2\) \ N\_identical\)\)], "Output"] }, Open ]], Cell["which agrees with both FC and CompHEP.", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Berger et.al. NPB239 (1984) 52", "Subsubsection"], Cell["On page 56 they have", "Text"], Cell[BoxData[ \(\(d\[Sigma]\_\(q q\&_ \[Rule] \[Gamma]\[Gamma]\) = \[Pi]\ \[Alpha]\_EM\^2\ \(1\/3\) 2\/s\^2\ \(\(u\^2 + t\^2\)\/\(u\ t\)\) dt /. \[Alpha]\_EM -> e\^2\/\(4 \[Pi]\);\)\)], "Input"], Cell[TextData[{ "After taking out the initial state color ", Cell[BoxData[ \(TraditionalForm\`\((\(1\/3\) 1\/3)\)\)]], " and spin ", Cell[BoxData[ \(TraditionalForm\`\((\(1\/2\) 1\/2)\)\)]], " averaging factors and the identical final state particles factor ", Cell[BoxData[ \(TraditionalForm\`\((1\/2)\)\)]], ", and making the color sum explicit, for the amplitude square (before \ averaging) we obtain:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(1\/\(\(\((1\/3)\)\^2\) \(\((1\/2)\)\^2\) 1\/2\)\) \(N\_C\/3\) d\[Sigma]\_\(q q\&_ \[Rule] \[Gamma]\[Gamma]\)\/dt\)], "Input"], Cell[BoxData[ \(TraditionalForm \`\(e\^4\ \((t\^2 + u\^2)\)\ N\_C\)\/\(\[Pi]\ s\^2\ t\ u\)\)], "Output"] }, Open ]], Cell[TextData[{ "There's a ", Cell[BoxData[ \(TraditionalForm\`\(\[Pi]\ s\^2\)\/8\)]], "factor which I don't understand." }], "Text"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["g g \[Rule] \[Gamma] \[Gamma]", "Section"], Cell[CellGroupData[{ Cell["FeynCalc", "Subsubsection"], Cell[TextData[{ "The box contribution to the ", Cell[BoxData[ \(TraditionalForm\`O(\(\[Alpha]\^2\) \[Alpha]\_S\^2)\)]], " amplitude of ", Cell[BoxData[ \(TraditionalForm\`g\ g\)]], " \[Rule] \[Gamma] \[Gamma] is:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(CreateAmplitude1[{SM, QCD}, {G, G}, {A, A}, 1, \ Boxes\ -> \ True]\)], "Input"], Cell[CellGroupData[{ Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: 1 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics /Helvetica findfont 10 scalefont setfont % Scaling calculations 0.04 0.04 0.04 0.04 [ [ 0 0 0 0 ] [ 1 1 0 0 ] ] MathScale % Start of Graphics 1 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the q ", Cell[BoxData[ \(TraditionalForm\`q\&_\)]], " \[Rule] \[Gamma] \[Gamma] process from Papageno.\nIn Papageno the \ following is coded:\n\n CONST=AEM**2*16.*PISQ\n CONST1=QU4\n \ CONST2=QD4\n ....\n A4=CONST*2.*(U2+T2)/U/T/3.\n \ A5=CONST*AL**2*11.**2/9**2.*2./4./8./8.*XMAT_PHOT(S,T,U)\n \ WT=((SF(23)+SF(24))*CONST1+(SF(27)+SF(28))*CONST2)*A4/2.\n 1 \ +SF(9)*A5/2./16./PISQ\n \nThat is" }], "Text"], Cell[BoxData[ \(\(\(\ \)\(WT\_\(gg \[Rule] \[Gamma]\[Gamma]\) = \(\[Alpha]\_EM\^2\) 16 \( \[Pi]\^2\) \(\[Alpha]\_S\^2\) \(\((11\/9)\)\^2\) 2 \( 1\/4\) \(1\/8\) \(1\/8\) \(Abs[M\_\(gg \[Rule] \[Gamma]\[Gamma]\)[s, t, u]]\^2\) \(1\/2\) \(1\/16\) 1\/\[Pi]\^2;\)\)\)], "Input"], Cell[TextData[{ "After taking out the initial state color ", Cell[BoxData[ \(TraditionalForm\`\((\(1\/8\) 1\/8)\)\)]], " and spin ", Cell[BoxData[ \(TraditionalForm\`\((\(1\/2\) 1\/2)\)\)]], " averaging factors and the identical final state particles factor ", Cell[BoxData[ \(TraditionalForm\`\((1\/2)\)\)]], ", the factor ", Cell[BoxData[ \(TraditionalForm\`\((\[Sum]\+\(i = 1\)\%n\_f Q\_q\^2)\)\^2\)]], " (without the top quark) and making the factor from the color sum/traces \ ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{\(F\_color\), "=", " ", RowBox[{"8", " ", FormBox[\(1\/2\), "TraditionalForm"], " ", FormBox[\(1\/2\), "TraditionalForm"]}]}], ")"}], TraditionalForm]]], " explicit we obtain:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ \(\(\(1\/D\_A\^2\) \(1\/\((N - 2)\)\^2\) 1\/N\_identical\)\/\(\(\((1\/8)\)\^2\) \(\((1\/2)\)\^2\) 1\/2\)\), FractionBox[ FormBox[\(\((\[Sum]\+\(i = 1\)\%n\_f Q\_\(q[i]\)\^2)\)\^2\), "TraditionalForm"], \(\((11\/9)\)\^2\)], FractionBox[\(F\_color\), RowBox[{"8", " ", FormBox[\(1\/2\), "TraditionalForm"], " ", FormBox[\(1\/2\), "TraditionalForm"]}]], " ", \(WT\_\(gg \[Rule] \[Gamma]\[Gamma]\)\)}]], "Input"], Cell[BoxData[ \(TraditionalForm \`\(\[LeftBracketingBar]M\_\(gg \[Rule] \[Gamma]\[Gamma]\)[s, t, u]\ \[RightBracketingBar]\^2\ F\_color\ \(\[Alpha]\_EM\%2\) \(\(\[Alpha]\_S\%2\)(\ \[Sum]\+\(i = 1\)\%n\_f Q\_\(q(i)\)\%2)\)\^2\)\/\(\(\((N - 2)\)\^2\) D\_A\%2\ \ N\_identical\)\)], "Output"] }, Open ]], Cell[TextData[{ "The factor ", Cell[BoxData[ \(TraditionalForm\`\[Sum]\+\(i = 1\)\%n\_f Q\_q\^2\)]], " (without and with the top quark) is" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \({\ 2 \((2\/3)\)\^2\ + \ 3 \((\(-\(1\/3\)\))\)\^2\n, 3 \((2\/3)\)\^2\ + \ 3 \((\(-\(1\/3\)\))\)\^2}\)], "Input"], Cell[BoxData[ \(TraditionalForm\`{11\/9, 5\/3}\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["ResBos", "Subsubsection"], Cell[TextData[{ "In RES.FOR:\n- For H production we have: \t", Cell[BoxData[ \(TraditionalForm \`HCOUPL*\((\(ALPI(C2*S_Q)\)*Q_V)\) ** 2 = \(\(\(\@2\) \(G\_\[Mu]\) \[Pi]\)\/576\) \((\(\(\[Alpha]\_S\)(\(C\_2\) Q)\)\/\[Pi])\)\^2\)]] }], "Text"], Cell[BoxData[ \(\(\(\[Sigma]\^H\)\_0 = \(\(\(\@2\) \(G\_\[Mu]\) \[Pi]\)\/576\) \((\[Alpha]\_S[\(C\_2\) Q]\/\[Pi])\)\^2;\)\)], "Input"], Cell[TextData[{ "- For \[Gamma]\[Gamma] production we have:\t", Cell[BoxData[ \(TraditionalForm \`2*\(PI/8.0\)/8.0*PI ** 2*\(ALPI(C2*S_Q)\) ** 2 = \(\(2 \[Pi]\)\/64\) \(\[Pi]\^2\) \(\(\[Alpha]\_S\^2\)(\(C\_2\) Q)\)\)]] }], "Text"], Cell[BoxData[ \(\(\(\[Sigma]\^\[Gamma]\[Gamma]\)\_0 = \(\(2 \[Pi]\)\/64\) \(\[Pi]\^2\) \(\[Alpha]\_S\^2\)[\(C\_2\) Q]; \)\)], "Input"], Cell["\<\ In ResBos we have: C Identical particle final state and spin average factors & 1.d0/2.d0/2.d0/2.d0* C Effective matrix element & 2.d0*(gWeak*sWeak)**4/(16.0*Pi**2)* CsB t and b quark masses are non-zero in loop: & GGAA(sH,tH,uH)*(SigS+SigY(0)) \ \>", "Text"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ Cell[BoxData[ \(TraditionalForm\`q q\&_\)]], " \[Rule] Z Z" }], "Section"], Cell[CellGroupData[{ Cell["FeynCalc", "Subsubsection"], Cell[TextData[{ "The three amplitude of ", Cell[BoxData[ \(TraditionalForm\`u u\&_\)]], " \[Rule] Z Z is:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(CreateAmplitude1[{SM, QCD}, 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Hooft-Feynman gauge. The u quark charge is included by FA and \ taken made explicit." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(FAL = "\ZZ\>" // GetAmplitude;\)\n \(Repl = {MU\ -> \ 0, DiracGamma[7]\/2 - \(2\ s\_w\%2\)\/3 -> \(T\_\(3, q\)\) DiracGamma[7] - Q\_q\ s\_w\%2};\)\n Amp[1] = FAL[\([1, 2]\)] /. Repl\), "\n", \(Amp[2] = FAL[\([2, 2]\)] /. Repl\)}], "Input"], Cell[BoxData[ FormBox[ RowBox[{\(1\/\(s\_w\%2\ \((1 - s\_w\%2)\)\)\), RowBox[{"(", RowBox[{\(e\^2\), " ", RowBox[{"(", RowBox[{ RowBox[{"\[CurlyPhi]", "\[NoBreak]", "(", "\[NoBreak]", FormBox[\(-p\_2\), "TraditionalForm"], "\[NoBreak]", ")"}], " ", \(\[Gamma]\^\[Lambda]\), " ", \((\[Gamma]\^7\ T\_\(3, q\) - Q\_q\ s\_w\%2)\), " ", RowBox[{ FormBox["\<\"\[Gamma]\"\>", "TraditionalForm"], "\[NoBreak]", "\[CenterDot]", "\[NoBreak]", "(", "\[NoBreak]", FormBox[\(p\_1 - k\_1\), "TraditionalForm"], "\[NoBreak]", ")"}], " ", \(\[Gamma]\^\[Nu]\), " ", \((\[Gamma]\^7\ T\_\(3, q\) - Q\_q\ s\_w\%2)\), " ", RowBox[{"\[CurlyPhi]", "\[NoBreak]", "(", "\[NoBreak]", FormBox[\(p\_1\), "TraditionalForm"], "\[NoBreak]", ")"}]}], ")"}], " ", \(\(\[CurlyEpsilon]\_\[Lambda]\%*\)(k\_2)\), " ", \(\(\[CurlyEpsilon]\_\[Nu]\%*\)(k\_1)\), " ", FormBox[\(1\/\((k\_1 - p\_1)\)\^2\), "TraditionalForm"]}], ")"}]}], TraditionalForm]], "Output"], Cell[BoxData[ FormBox[ RowBox[{\(1\/\(s\_w\%2\ \((1 - s\_w\%2)\)\)\), RowBox[{"(", RowBox[{\(e\^2\), " ", RowBox[{"(", RowBox[{ RowBox[{"\[CurlyPhi]", "\[NoBreak]", "(", "\[NoBreak]", FormBox[\(-p\_2\), "TraditionalForm"], "\[NoBreak]", ")"}], " ", \(\[Gamma]\^\[Nu]\), " ", \((\[Gamma]\^7\ T\_\(3, q\) - Q\_q\ s\_w\%2)\), " ", RowBox[{ FormBox["\<\"\[Gamma]\"\>", "TraditionalForm"], "\[NoBreak]", "\[CenterDot]", "\[NoBreak]", "(", "\[NoBreak]", FormBox[\(k\_1 - p\_2\), "TraditionalForm"], "\[NoBreak]", ")"}], " ", \(\[Gamma]\^\[Lambda]\), " ", \((\[Gamma]\^7\ T\_\(3, q\) - Q\_q\ s\_w\%2)\), " ", RowBox[{"\[CurlyPhi]", "\[NoBreak]", "(", "\[NoBreak]", FormBox[\(p\_1\), "TraditionalForm"], "\[NoBreak]", ")"}]}], ")"}], " ", \(\(\[CurlyEpsilon]\_\[Lambda]\%*\)(k\_2)\), " ", \(\(\[CurlyEpsilon]\_\[Nu]\%*\)(k\_1)\), " ", FormBox[\(1\/\((k\_1 - p\_2)\)\^2\), "TraditionalForm"]}], ")"}]}], TraditionalForm]], "Output"] }, Open ]], Cell[BoxData[ \(\(SetMandelstam[s, t, u, p1, p2, \(-k1\), \(-k2\), 0, 0, m\_Z, m\_Z]; \)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Do[ Do[\n\tAmp2[i, j] = \(\((\(\(\(\(\(ComplexConjugate[Amp[i]] Amp[j] // Simplify\) // \n\tSumPolarization[#, k1] &\) // SumPolarization[#, k2] &\) // \n\tContract\) // \n\tFermionSpinSum\) // \n\tDiracSimplify)\) /. \n\ DiracTrace -> Tr // Simplify\) // \n\t TrickMandelstam[#, {s, t, u, 2\ m\_Z\^2}] &\n, {i, 2}], {j, 2}]; \)\), "\n", \(Amp2[3] = \(\[Sum]\+\(i = 1\)\%2\(\[Sum]\+\(j = 1\)\%2 Amp2[i, j]\) // \n\t\tPropagatorDenominatorExplicit\) // Simplify\)}], "Input"], Cell[BoxData[ \(TraditionalForm\`$Aborted\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(\ \)\(----\)\)\)], "Input"], Cell[BoxData[ \(TraditionalForm\`\(8\ e\^4\ \((t\^2 + u\^2)\)\ Q\_f\%4\)\/\(t\ u\)\)], "Output"] }, Open ]], Cell[TextData[{ "This must still be summed over ", Cell[BoxData[ \(TraditionalForm\`N\_C\)]], "." }], "Text"], Cell["In the \[Gamma]\[Gamma] CM frame:", "Text"], Cell[BoxData[{ \(p\_1 = \(\@s\/2\) {I, 0, 0, 1}; p\_2 = \(\@s\/2\) {I, 0, 0, \(-1\)};\n k\_1 = \(\@s\/2\) {I, 0, Sin[\[Theta]], Cos[\[Theta]]}; k\_2 = \(\@s\/2\) {I, 0, \(-Sin[\[Theta]]\), \(-Cos[\[Theta]]\)};\n s -> \ \(-\((p\_1 + p\_2)\) . \((p\_1 + p\_2)\)\)\), "\n", \(t -> \(-\((p\_1 - k\_1)\) . \((p\_1 - k\_1)\)\) // Simplify\), "\n", \(u -> \(-\((p\_1 - k\_2)\) . \((p\_1 - k\_2)\)\) // Simplify\)}], "Input"], Cell[BoxData[ \(\(Mandelstam = {t -> s \( 1 - Cos[\[Theta]]\)\/2, u -> s \( 1 + Cos[\[Theta]]\)\/2}; \)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(Amp2[3] /. Mandelstam // Simplify[#, Trig -> False] &\)], "Input"], Cell[BoxData[ \(TraditionalForm \`\(-\(\(16\ e\^4\ \((\(cos\^2\)(\[Theta]) + 1)\) Q\_f\%4\)\/\(\(cos\^2\)(\[Theta]) - 1\)\)\)\)], "Output"] }, Open ]], Cell[BoxData[ RowBox[{ RowBox[{\(Amp2[qQ -> \[Gamma]\[Gamma]]\), "=", FormBox[ \(-\(\(16\ e\^4\ \((\(cos\^2\)(\[Theta]) + 1)\) Q\_f\%4\)\/\(\(cos\^2\)(\[Theta]) - 1\)\)\), "TraditionalForm"]}], ";"}]], "Input"], Cell["This agrees with both Ohnemus and Owens and CompHEP.", "Text"], Cell[TextData[{ "The averaging factors: initial state color ", Cell[BoxData[ \(TraditionalForm\`\((\(1\/3\) 1\/3)\)\)]], " and spin ", Cell[BoxData[ \(TraditionalForm\`\((\(1\/2\) 1\/2)\)\)]], " and the identical final state particles factor ", Cell[BoxData[ \(TraditionalForm\`\((1\/2)\)\)]], "." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["CompHEP", "Subsubsection"], Cell[BoxData[ \(<< "\<~/CompHEP/comphep_32.18/results/uU-ZZ.m\>"\)], "Input"], Cell[TextData[{ "We make the initial state color ", Cell[BoxData[ \(TraditionalForm\`\((\(1\/3\) 1\/3)\)\)]], " and spin ", Cell[BoxData[ \(TraditionalForm\`\((\(1\/2\) 1\/2)\)\)]], " averaging factors and the identical final state particles factor ", Cell[BoxData[ \(TraditionalForm\`\((1\/2)\)\)]], ", and the sum over colors ", Cell[BoxData[ \(TraditionalForm\`\((N\_C = 3\)\)]], ") explicit ((the quark charges ", Cell[BoxData[ \(TraditionalForm\`\((Q\_f = 2\/3)\)\)]], ")), and change notation:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\((\(\(\(\(\((1\/N\_C)\)\^2\) \(\((1\/N\_qSpin)\)\^2\) 1\/N\_identical\)\/\(\(\((1\/3)\)\^2\) \(\((1\/2)\)\^2\) 1\/2\)\) \(N\_C\/3\) Summ /. \n\t {EE\ -> \ e, \((SW\^2 - 1)\) -> \(-c\_w\^2\), SW -> s\_w, \n\t\t\tSC[p1, p3] -> \(-\(\(t - m\_Z\^2\)\/2\)\), SC[p1, p2] -> s\/2, MZ -> m\_Z} // Simplify\) // \n\t TrickMandelstam[#, {s, \ t, \ u, \ 2\ m\_Z\^2}] &)\) /. 512\ s\_w\%8 - 768\ s\_w\%6 + 864\ s\_w\%4 - 432\ s\_w\%2 + 81 -> \n\t 16*81 \((\((T\_\(3, q\) - \(Q\_q\) s\_w\^2)\)\^4 + \((\(-Q\_q\) s\_w\^2)\)\^4)\) // Simplify\)], "Input"], Cell[BoxData[ \(TraditionalForm \`\((4\ e\^4\ \((\(-\((t\^2 - 8\ u\ t + u\^2)\)\)\ m\_Z\%4 - 4\ t\ u\ \((t + u)\)\ m\_Z\%2 + t\ u\ \((t\^2 + u\^2)\))\) \((Q\_q\%4\ s\_w\%8 + \((T\_\(3, q\) - Q\_q\ s\_w\%2)\)\^4)\))\)/ \((t\^2\ u\^2\ c\_w\%4\ N\_C\ N\_identical\ N\_qSpin\%2\ s\_w\%4)\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(-\((\(-8\)\ u\ t)\)\)\ m\_Z\%4 - 4\ t\ u\ \((t + u)\)\ m\_Z\%2 // TrickMandelstam[#, {\ t, \ u, s, 2\ m\_Z\^2}] &\)], "Input"], Cell[BoxData[ \(TraditionalForm\`4\ s\ t\ u\ m\_Z\%2\)], "Output"] }, Open ]], Cell["This agrees with Ohnemus and Owens.", "Text"], Cell["Notes:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(16*81 \((1\/2 - \(2\/3\) s\_W\^2)\)\^4 + 16*81 \((\(-\(2\/3\)\) s\_W\^2)\)\^4 // Expand\)], "Input"], Cell[BoxData[ \(TraditionalForm \`512\ s\_W\%8 - 768\ s\_W\%6 + 864\ s\_W\%4 - 432\ s\_W\%2 + 81\)], "Output"] }, Open ]] }, Open ]], Cell["Papageno", "Subsubsection"], Cell["Berger et.al. NPB239 (1984) 52", "Subsubsection"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "Ratios for ", Cell[BoxData[ \(TraditionalForm\`q q\&_\)]], " initial state" }], "Section"], Cell[TextData[{ "Compared to the ", Cell[BoxData[ RowBox[{ FormBox[\(q q\&_\), "TraditionalForm"], "\[Rule]", \(\[Gamma] \[Rule] \(e\^+\) \(e\^-\)\)}]]], " process, in the amplitude square of ", Cell[BoxData[ RowBox[{ FormBox[\(q q\&_\), "TraditionalForm"], "\[Rule]", " ", \(\[Gamma]\ \[Gamma]\)}]]], " there's an extra factor:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Amp2[qQ -> \[Gamma]\[Gamma]]/Amp2[qQ -> \(e\^+\) \(e\^-\)]\)], "Input"], Cell[BoxData[ \(TraditionalForm \`\(-\(\(4\ Q\_f\%2\)\/\(\(cos\^2\)(\[Theta]) - 1\)\)\)\)], "Output"] }, Open ]], Cell[TextData[{ "This means: to obtain the resummation formula for ", Cell[BoxData[ RowBox[{ FormBox[\(q q\&_\), "TraditionalForm"], "\[Rule]", " ", \(\[Gamma]\ \[Gamma]\)}]]], " from the resummation formula for ", Cell[BoxData[ RowBox[{ FormBox[\(q q\&_\), "TraditionalForm"], "\[Rule]", \(\(e\^+\) \(e\^-\)\)}]]], " we have to inclue an extra numerical factor, include extra quark charges \ ", Cell[BoxData[ \(TraditionalForm\`Q\_f\%2\)]], ", and include an extra factor of ", Cell[BoxData[ \(TraditionalForm\`1/\((1 - \(cos\^2\)(\[Theta]))\)\)]], ".\nAn extra numerical factor ", Cell[BoxData[ \(TraditionalForm\`\((1\/2)\)\)]], " comes from the identical particle final state, which means that the \ overall extra factor (for ", Cell[BoxData[ RowBox[{ FormBox[\(q q\&_\), "TraditionalForm"], "\[Rule]", " ", \(\[Gamma]\ \[Gamma]\)}]]], " compared to ", Cell[BoxData[ RowBox[{ FormBox[\(q q\&_\), "TraditionalForm"], "\[Rule]", \(\[Gamma] \[Rule] \(e\^+\) \(e\^-\)\)}]]], ") is 2.\n(Note that the initial state is the same, which means that the \ color ", Cell[BoxData[ \(TraditionalForm\`\((1\/3\^2)\)\)]], " and spin ", Cell[BoxData[ \(TraditionalForm\`\((1\/2\^2)\)\)]], " averaging factors are the same for both process.)" }], "Text"], Cell[TextData[{ "In RES.FOR:\n- the quark charges are adjusted.\nIn ResBos:\n- for \[Gamma]\ \[Gamma]: \t", Cell[BoxData[ \(TraditionalForm \`\(\(\(1\/2\) \(1\/2\) \(1\/2\) 32 \((t\/u + u\/t)\)\)\(=\)\)\)]], " ", Cell[BoxData[ \(TraditionalForm\`\(1\/2\) \(1\/2\) \(1\/2\) 32\)]], "*2", Cell[BoxData[ \(TraditionalForm \`\(1 + \(cos\^2\) \[Theta]\)\/\(1 - cos\[Theta]\)\)]], "=8", Cell[BoxData[ \(TraditionalForm\`\(1 + \(cos\^2\) \[Theta]\)\/\(1 - cos\[Theta]\)\)]] , "is programmed,\n- for ", Cell[BoxData[ \(TraditionalForm\`\(\[Gamma]\^*\)\)]], " \[Rule] \[ScriptL]\[ScriptL]': \t", Cell[BoxData[ \(TraditionalForm\`4 \((1 + \(cos\^2\) \[Theta])\)\)]], " \t\t\t\t\t\tis programmed.\nThis means the constant factor of 2 and the \ extra ", Cell[BoxData[ \(TraditionalForm\`1/\((1 - \(cos\^2\)(\[Theta]))\)\)]], " is taken into account." }], "Text"], Cell[CellGroupData[{ Cell["Notes", "Subsubsection"], Cell[TextData[{ "Compared to the symmetric part of the ", Cell[BoxData[ RowBox[{ FormBox[\(q q\&_\), "TraditionalForm"], "\[Rule]", \(\(e\^+\) \[Nu]\_e\)}]]], " process in the amplitude square of ", Cell[BoxData[ RowBox[{ FormBox[\(q q\&_\), "TraditionalForm"], "\[Rule]", " ", \(\[Gamma]\ \[Gamma]\)}]]], " there's an extra factor:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Amp2[qQ -> \[Gamma]\[Gamma]]/ \((Amp2[uD -> \(e\^+\) \[Nu]\_e] /. \((1 - Cos[\[Theta]])\)\^2 -> 1 + Cos[\[Theta]]\^2)\)\)], "Input"], Cell[BoxData[ \(TraditionalForm \`\(-\(\(64\ \((s - m\_W\%2)\)\^2\ \(Q\_f\%4\) s\_w\%4\)\/\(\((\(cos\^2\)(\[Theta]) - 1)\)\ V\_ud\%2\)\)\)\)], "Output"] }, Open ]], Cell[TextData[{ "This means: to obtain the resummation formula for ", Cell[BoxData[ RowBox[{ FormBox[\(q q\&_\), "TraditionalForm"], "\[Rule]", " ", \(\[Gamma]\ \[Gamma]\)}]]], " from the resummation formula for ", Cell[BoxData[ RowBox[{ FormBox[\(q q\&_\), "TraditionalForm"], "\[Rule]", \(\(e\^+\) \[Nu]\_e\)}]]], " we have to set ", Cell[BoxData[ \(TraditionalForm\`V\_ij\)]], " \[Rule] ", Cell[BoxData[ \(TraditionalForm\`\[Delta]\_\(i i\&_\)\)]], ", inclue an extra numerical factor, get rid of the W propagator, include \ the quark charges, replace ", Cell[BoxData[ \(TraditionalForm\`g\_weak\)]], " with ", Cell[BoxData[ \(TraditionalForm\`e\_EM\)]], ", and include an extra factor of ", Cell[BoxData[ \(TraditionalForm\`1/\((1 - \(cos\^2\)(\[Theta]))\)\)]], "." }], "Text"], Cell["\<\ Extra numerical factor coming in from the different averaging:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"(", RowBox[{\((\(1\/3\) 1\/3)\), RowBox[{"(", " ", FormBox[\(\(1\/2\) 1\/2\), "TraditionalForm"], ")"}], " ", FormBox[\(1\/2\), "TraditionalForm"]}], ")"}], "/", RowBox[{"(", RowBox[{ FormBox[\((1\/3)\), "TraditionalForm"], " ", FormBox[\((\(1\/2\) 1\/2)\), "TraditionalForm"]}], ")"}]}], TraditionalForm]], "Input"], Cell[BoxData[ \(TraditionalForm\`1\/6\)], "Output"] }, Open ]], Cell["\<\ The numerical factors combined give an overall extra factor of:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(64 1\/6\)], "Input"], Cell[BoxData[ \(TraditionalForm\`32\/3\)], "Output"] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Ratios for gg initial state", "Section"], Cell[TextData[{ "Based on Papageno the ratio of the weights of the g g \[Rule] \[Gamma] \ \[Gamma] and q ", Cell[BoxData[ \(TraditionalForm\`q\&_\)]], " \[Rule] \[Gamma] \[Gamma] processes is:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"R", "=", RowBox[{ RowBox[{"(", RowBox[{ \(\(\(1\/D\_A\^2\) \(1\/\((N - 2)\)\^2\) 1\/N\_identical\)\/\(\(\((1\/8)\)\^2\) \(\((1\/2)\)\^2\) 1\/2\)\), FractionBox[ FormBox[\(\((\[Sum]\+\(i = 1\)\%n\_f Q\_\(q[i]\)\^2)\)\^2\), "TraditionalForm"], \(\((11\/9)\)\^2\)], FractionBox[\(F\_color\), RowBox[{"8", " ", FormBox[\(1\/2\), "TraditionalForm"], " ", FormBox[\(1\/2\), "TraditionalForm"]}]], " ", \(WT\_\(gg \[Rule] \[Gamma]\[Gamma]\)\)}], ")"}], "/", \((\(\(\(1\/D\_F\^2\) \(1\/\((N - 2)\)\^2\) 1\/N\_identical\)\/\(\(\((1\/3)\)\^2\) \(\((1\/2)\)\^2\) 1\/2\)\) \(Abs[M\_\(q q\&_ \[Rule] \[Gamma]\[Gamma]\)]\^2\/\(\(8 \((t\^2 + u\^2)\)\)\/\(t\ u\)\)\) \(N\_C\/3\) WT\_\(q q\&_ \[Rule] \[Gamma]\[Gamma]\))\)}]}]], "Input"], Cell[BoxData[ \(TraditionalForm \`\((\[LeftBracketingBar]M\_\(gg \[Rule] \[Gamma]\[Gamma]\)[s, t, u]\ \[RightBracketingBar]\^2\ D\_F\%2\ F\_color\ \(\[Alpha]\_EM\%2\) \(\(\[Alpha]\_S\%2\)(\[Sum]\+\(i = 1\)\%n\_f \ Q\_\(q(i)\)\%2)\)\^2)\)/ \((\(e\^4\) \(\[LeftBracketingBar]M\_\(q\ q\&_ \[Rule] \[Gamma]\[Gamma]\)\ \[RightBracketingBar]\^2\) D\_A\%2\ N\_C\ Q\_q\%4)\)\)], "Output"] }, Open ]], Cell["\<\ From here we see the differences: - the (unaveraged, un-color-summed) amplitude square (not including the \ couplings or \tquark charges), - the initial color averaging factors, - the color factors, - the couplings.\ \>", "Text"], Cell[TextData[{ "The ", Cell[BoxData[ \(TraditionalForm\`F\_\(q q\&_\)\)]], " factor is:" }], "Text"], Cell[BoxData[ \(\(F\_\(q q\&_\) = \(\((\((\(e\ Q\_q\)\/2)\)\^2 + \((\(e\ Q\_q\)\/2)\)\^2)\)\^2\) Abs[M\_\(q q\&_ \[Rule] \[Gamma]\[Gamma]\)]\^2\/16;\)\)], "Input"], Cell[TextData[{ "The ", Cell[BoxData[ \(TraditionalForm\`F\_gg\)]], " factor then should be:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(R*F\_\(q q\&_\)\)], "Input"], Cell[BoxData[ \(TraditionalForm \`\(\(1\/\(64\ \(D\_A\%2\) N\_C\)\)\((\[LeftBracketingBar]M\_\(gg \[Rule] \ \[Gamma]\[Gamma]\)[s, t, u]\[RightBracketingBar]\^2\ D\_F\%2\ F\_color\ \(\ \[Alpha]\_EM\%2\) \(\(\[Alpha]\_S\%2\)(\[Sum]\+\(i = 1\)\%n\_f Q\_\(q(i)\)\%2)\)\^2) \)\)\)], "Output"] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Appendix", "Subsubtitle"], Cell[CellGroupData[{ Cell[TextData[{ Cell[BoxData[ \(TraditionalForm\`q q\&_\)]], " \[Rule] Z Z" }], "Section"], Cell[TextData[{ "The three amplitude of ", Cell[BoxData[ \(TraditionalForm\`u u\&_\)]], " \[Rule] Z Z is:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(CreateAmplitude1[{SM, QCD}, {u, U}, {Z, Z}, 0]\)], "Input"], Cell[CellGroupData[{ 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(*********************************************************************** End of Mathematica Notebook file. ***********************************************************************)