diff --git a/includes.tex b/includes.tex index 5d790b1..0be4e94 100644 --- a/includes.tex +++ b/includes.tex @@ -542,6 +542,8 @@ \chapter{Loop integrals} \subfile{pages/FCLoopGetEtaSigns.tex} \subfile{pages/FCLoopGetKinematicInvariants.tex} \subfile{pages/FCLoopGLIDifferentiate.tex} +\subfile{pages/FCLoopGLILowerDimension.tex} +\subfile{pages/FCLoopGLIRaiseDimension.tex} \subfile{pages/FCLoopAddScalingParameter.tex} \subfile{pages/FCLoopGLIExpand.tex} \subfile{pages/FCLoopIBPReducableQ.tex} diff --git a/pages/FCLoopGLILowerDimension.tex b/pages/FCLoopGLILowerDimension.tex new file mode 100644 index 0000000..fcffdea --- /dev/null +++ b/pages/FCLoopGLILowerDimension.tex @@ -0,0 +1,55 @@ +% !TeX program = pdflatex +% !TeX root = FCLoopGLILowerDimension.tex + +\documentclass[../FeynCalcManual.tex]{subfiles} +\begin{document} +\hypertarget{fcloopglilowerdimension}{ +\section{FCLoopGLILowerDimension}\label{fcloopglilowerdimension}\index{FCLoopGLILowerDimension}} + +\texttt{FCLoopGLILowerDimension[\allowbreak{}gli,\ \allowbreak{}topo]} +lowers the dimension of the given \texttt{GLI} from \texttt{D} to +\texttt{D-2} and expresses it in terms of \texttt{D}-dimensional loop +integrals returned in the output. + +The algorithm is based on the code of the function \texttt{RaisingDRR} +from R. Lee's LiteRed + +\subsection{See also} + +\hyperlink{toc}{Overview}, +\hyperlink{fcloopgliraisedimension}{FCLoopGLIRaiseDimension}. + +\subsection{Examples} + +\begin{Shaded} +\begin{Highlighting}[] +\NormalTok{topo }\ExtensionTok{=}\NormalTok{ FCTopology}\OperatorTok{[} +\NormalTok{ topo1}\OperatorTok{,} \OperatorTok{\{}\NormalTok{SFAD}\OperatorTok{[}\NormalTok{p1}\OperatorTok{],}\NormalTok{ SFAD}\OperatorTok{[}\NormalTok{p2}\OperatorTok{],}\NormalTok{ SFAD}\OperatorTok{[}\FunctionTok{Q} \SpecialCharTok{{-}}\NormalTok{ p1 }\SpecialCharTok{{-}}\NormalTok{ p2}\OperatorTok{],}\NormalTok{ SFAD}\OperatorTok{[}\FunctionTok{Q} \SpecialCharTok{{-}}\NormalTok{ p2}\OperatorTok{],} +\NormalTok{ SFAD}\OperatorTok{[}\FunctionTok{Q} \SpecialCharTok{{-}}\NormalTok{ p1}\OperatorTok{]\},} \OperatorTok{\{}\NormalTok{p1}\OperatorTok{,}\NormalTok{ p2}\OperatorTok{\},} \OperatorTok{\{}\FunctionTok{Q}\OperatorTok{\},} \OperatorTok{\{}\FunctionTok{Hold}\OperatorTok{[}\NormalTok{SPD}\OperatorTok{[}\FunctionTok{Q}\OperatorTok{]]} \OtherTok{{-}\textgreater{}}\NormalTok{ qq}\OperatorTok{\},} \OperatorTok{\{\}]} +\end{Highlighting} +\end{Shaded} + +\begin{dmath*}\breakingcomma +\text{FCTopology}\left(\text{topo1},\left\{\frac{1}{(\text{p1}^2+i \eta )},\frac{1}{(\text{p2}^2+i \eta )},\frac{1}{((-\text{p1}-\text{p2}+Q)^2+i \eta )},\frac{1}{((Q-\text{p2})^2+i \eta )},\frac{1}{((Q-\text{p1})^2+i \eta )}\right\},\{\text{p1},\text{p2}\},\{Q\},\{\text{Hold}[\text{SPD}(Q)]\to \;\text{qq}\},\{\}\right) +\end{dmath*} + +\begin{Shaded} +\begin{Highlighting}[] +\NormalTok{FCLoopGLILowerDimension}\OperatorTok{[}\NormalTok{GLI}\OperatorTok{[}\NormalTok{topo1}\OperatorTok{,} \OperatorTok{\{}\DecValTok{1}\OperatorTok{,} \DecValTok{1}\OperatorTok{,} \DecValTok{1}\OperatorTok{,} \DecValTok{1}\OperatorTok{,} \DecValTok{1}\OperatorTok{\}],}\NormalTok{ topo}\OperatorTok{]} +\end{Highlighting} +\end{Shaded} + +\begin{dmath*}\breakingcomma +G^{\text{topo1}}(1,1,1,2,2)+G^{\text{topo1}}(1,1,2,1,2)+G^{\text{topo1}}(1,1,2,2,1)+G^{\text{topo1}}(1,2,1,1,2)+G^{\text{topo1}}(1,2,2,1,1)+G^{\text{topo1}}(2,1,1,2,1)+G^{\text{topo1}}(2,1,2,1,1)+G^{\text{topo1}}(2,2,1,1,1) +\end{dmath*} + +\begin{Shaded} +\begin{Highlighting}[] +\NormalTok{FCLoopGLILowerDimension}\OperatorTok{[}\NormalTok{GLI}\OperatorTok{[}\NormalTok{topo1}\OperatorTok{,} \OperatorTok{\{}\NormalTok{n1}\OperatorTok{,}\NormalTok{ n2}\OperatorTok{,}\NormalTok{ n3}\OperatorTok{,} \DecValTok{1}\OperatorTok{,} \DecValTok{1}\OperatorTok{\}],}\NormalTok{ topo}\OperatorTok{]} +\end{Highlighting} +\end{Shaded} + +\begin{dmath*}\breakingcomma +G^{\text{topo1}}(\text{n1},\text{n2},\text{n3},2,2)+\text{n3} G^{\text{topo1}}(\text{n1},\text{n2},\text{n3}+1,1,2)+\text{n3} G^{\text{topo1}}(\text{n1},\text{n2},\text{n3}+1,2,1)+\text{n2} G^{\text{topo1}}(\text{n1},\text{n2}+1,\text{n3},1,2)+\text{n2} \;\text{n3} G^{\text{topo1}}(\text{n1},\text{n2}+1,\text{n3}+1,1,1)+\text{n1} G^{\text{topo1}}(\text{n1}+1,\text{n2},\text{n3},2,1)+\text{n1} \;\text{n3} G^{\text{topo1}}(\text{n1}+1,\text{n2},\text{n3}+1,1,1)+\text{n1} \;\text{n2} G^{\text{topo1}}(\text{n1}+1,\text{n2}+1,\text{n3},1,1) +\end{dmath*} +\end{document} diff --git a/pages/FCLoopGLIRaiseDimension.tex b/pages/FCLoopGLIRaiseDimension.tex new file mode 100644 index 0000000..200c36a --- /dev/null +++ b/pages/FCLoopGLIRaiseDimension.tex @@ -0,0 +1,55 @@ +% !TeX program = pdflatex +% !TeX root = FCLoopGLIRaiseDimension.tex + +\documentclass[../FeynCalcManual.tex]{subfiles} +\begin{document} +\hypertarget{fcloopgliraisedimension}{ +\section{FCLoopGLIRaiseDimension}\label{fcloopgliraisedimension}\index{FCLoopGLIRaiseDimension}} + +\texttt{FCLoopGLIRaiseDimension[\allowbreak{}gli,\ \allowbreak{}topo]} +raises the dimension of the given \texttt{GLI} from N to N+2 and +expresses it in terms of \texttt{N}-dimensional loop integrals returned +in the output. + +The algorithm is based on the code of the function \texttt{RaisingDRR} +from R. Lee's LiteRed + +\subsection{See also} + +\hyperlink{toc}{Overview}, +\hyperlink{fcloopglilowerdimension}{FCLoopGLILowerDimension}. + +\subsection{Examples} + +\begin{Shaded} +\begin{Highlighting}[] +\NormalTok{topo }\ExtensionTok{=}\NormalTok{ FCTopology}\OperatorTok{[} +\NormalTok{ topo1}\OperatorTok{,} \OperatorTok{\{}\NormalTok{SFAD}\OperatorTok{[}\NormalTok{p1}\OperatorTok{],}\NormalTok{ SFAD}\OperatorTok{[}\NormalTok{p2}\OperatorTok{],}\NormalTok{ SFAD}\OperatorTok{[}\FunctionTok{Q} \SpecialCharTok{{-}}\NormalTok{ p1 }\SpecialCharTok{{-}}\NormalTok{ p2}\OperatorTok{],}\NormalTok{ SFAD}\OperatorTok{[}\FunctionTok{Q} \SpecialCharTok{{-}}\NormalTok{ p2}\OperatorTok{],} +\NormalTok{ SFAD}\OperatorTok{[}\FunctionTok{Q} \SpecialCharTok{{-}}\NormalTok{ p1}\OperatorTok{]\},} \OperatorTok{\{}\NormalTok{p1}\OperatorTok{,}\NormalTok{ p2}\OperatorTok{\},} \OperatorTok{\{}\FunctionTok{Q}\OperatorTok{\},} \OperatorTok{\{}\FunctionTok{Hold}\OperatorTok{[}\NormalTok{SPD}\OperatorTok{[}\FunctionTok{Q}\OperatorTok{]]} \OtherTok{{-}\textgreater{}}\NormalTok{ qq}\OperatorTok{\},} \OperatorTok{\{\}]} +\end{Highlighting} +\end{Shaded} + +\begin{dmath*}\breakingcomma +\text{FCTopology}\left(\text{topo1},\left\{\frac{1}{(\text{p1}^2+i \eta )},\frac{1}{(\text{p2}^2+i \eta )},\frac{1}{((-\text{p1}-\text{p2}+Q)^2+i \eta )},\frac{1}{((Q-\text{p2})^2+i \eta )},\frac{1}{((Q-\text{p1})^2+i \eta )}\right\},\{\text{p1},\text{p2}\},\{Q\},\{\text{Hold}[\text{SPD}(Q)]\to \;\text{qq}\},\{\}\right) +\end{dmath*} + +\begin{Shaded} +\begin{Highlighting}[] +\NormalTok{FCLoopGLIRaiseDimension}\OperatorTok{[}\NormalTok{GLI}\OperatorTok{[}\NormalTok{topo1}\OperatorTok{,} \OperatorTok{\{}\DecValTok{1}\OperatorTok{,} \DecValTok{1}\OperatorTok{,} \DecValTok{1}\OperatorTok{,} \DecValTok{1}\OperatorTok{,} \DecValTok{1}\OperatorTok{\}],}\NormalTok{ topo}\OperatorTok{]} +\end{Highlighting} +\end{Shaded} + +\begin{dmath*}\breakingcomma +-\frac{G^{\text{topo1}}(-1,0,1,1,1)}{(1-D) (2-D) Q^2}-\frac{Q^2 G^{\text{topo1}}(1,1,0,1,1)}{(1-D) (2-D)}-\frac{G^{\text{topo1}}(0,-1,1,1,1)}{(1-D) (2-D) Q^2}+\frac{G^{\text{topo1}}(0,0,0,1,1)}{(1-D) (2-D) Q^2}+\frac{G^{\text{topo1}}(0,0,1,0,1)}{(1-D) (2-D) Q^2}+\frac{G^{\text{topo1}}(0,0,1,1,0)}{(1-D) (2-D) Q^2}-\frac{G^{\text{topo1}}(0,1,0,0,1)}{(1-D) (2-D) Q^2}+\frac{G^{\text{topo1}}(0,1,1,0,0)}{(1-D) (2-D) Q^2}-\frac{G^{\text{topo1}}(1,0,0,1,0)}{(1-D) (2-D) Q^2}+\frac{G^{\text{topo1}}(1,0,1,0,0)}{(1-D) (2-D) Q^2}+\frac{G^{\text{topo1}}(1,1,0,0,0)}{(1-D) (2-D) Q^2}-\frac{G^{\text{topo1}}(1,1,1,-1,0)}{(1-D) (2-D) Q^2}-\frac{G^{\text{topo1}}(1,1,1,0,-1)}{(1-D) (2-D) Q^2}+\frac{G^{\text{topo1}}(0,0,1,1,1)}{(1-D) (2-D)}+\frac{G^{\text{topo1}}(0,1,0,1,1)}{(1-D) (2-D)}-\frac{G^{\text{topo1}}(0,1,1,1,0)}{(1-D) (2-D)}+\frac{G^{\text{topo1}}(1,0,0,1,1)}{(1-D) (2-D)}-\frac{G^{\text{topo1}}(1,0,1,0,1)}{(1-D) (2-D)}-\frac{G^{\text{topo1}}(1,1,-1,1,1)}{(1-D) (2-D)}+\frac{G^{\text{topo1}}(1,1,0,0,1)}{(1-D) (2-D)}+\frac{G^{\text{topo1}}(1,1,0,1,0)}{(1-D) (2-D)}+\frac{G^{\text{topo1}}(1,1,1,0,0)}{(1-D) (2-D)} +\end{dmath*} + +\begin{Shaded} +\begin{Highlighting}[] +\NormalTok{FCLoopGLIRaiseDimension}\OperatorTok{[}\NormalTok{GLI}\OperatorTok{[}\NormalTok{topo1}\OperatorTok{,} \OperatorTok{\{}\NormalTok{n1}\OperatorTok{,}\NormalTok{ n2}\OperatorTok{,}\NormalTok{ n3}\OperatorTok{,} \DecValTok{1}\OperatorTok{,} \DecValTok{1}\OperatorTok{\}],}\NormalTok{ topo}\OperatorTok{]} +\end{Highlighting} +\end{Shaded} + +\begin{dmath*}\breakingcomma +-\frac{G^{\text{topo1}}(\text{n1}-2,\text{n2}-1,\text{n3},1,1)}{(1-D) (2-D) Q^2}-\frac{Q^2 G^{\text{topo1}}(\text{n1},\text{n2},\text{n3}-1,1,1)}{(1-D) (2-D)}-\frac{G^{\text{topo1}}(\text{n1}-1,\text{n2}-2,\text{n3},1,1)}{(1-D) (2-D) Q^2}+\frac{G^{\text{topo1}}(\text{n1}-1,\text{n2}-1,\text{n3}-1,1,1)}{(1-D) (2-D) Q^2}+\frac{G^{\text{topo1}}(\text{n1}-1,\text{n2}-1,\text{n3},0,1)}{(1-D) (2-D) Q^2}+\frac{G^{\text{topo1}}(\text{n1}-1,\text{n2}-1,\text{n3},1,0)}{(1-D) (2-D) Q^2}-\frac{G^{\text{topo1}}(\text{n1}-1,\text{n2},\text{n3}-1,0,1)}{(1-D) (2-D) Q^2}+\frac{G^{\text{topo1}}(\text{n1}-1,\text{n2},\text{n3},0,0)}{(1-D) (2-D) Q^2}-\frac{G^{\text{topo1}}(\text{n1},\text{n2}-1,\text{n3}-1,1,0)}{(1-D) (2-D) Q^2}+\frac{G^{\text{topo1}}(\text{n1},\text{n2}-1,\text{n3},0,0)}{(1-D) (2-D) Q^2}+\frac{G^{\text{topo1}}(\text{n1},\text{n2},\text{n3}-1,0,0)}{(1-D) (2-D) Q^2}-\frac{G^{\text{topo1}}(\text{n1},\text{n2},\text{n3},-1,0)}{(1-D) (2-D) Q^2}-\frac{G^{\text{topo1}}(\text{n1},\text{n2},\text{n3},0,-1)}{(1-D) (2-D) Q^2}+\frac{G^{\text{topo1}}(\text{n1}-1,\text{n2}-1,\text{n3},1,1)}{(1-D) (2-D)}+\frac{G^{\text{topo1}}(\text{n1}-1,\text{n2},\text{n3}-1,1,1)}{(1-D) (2-D)}-\frac{G^{\text{topo1}}(\text{n1}-1,\text{n2},\text{n3},1,0)}{(1-D) (2-D)}+\frac{G^{\text{topo1}}(\text{n1},\text{n2}-1,\text{n3}-1,1,1)}{(1-D) (2-D)}-\frac{G^{\text{topo1}}(\text{n1},\text{n2}-1,\text{n3},0,1)}{(1-D) (2-D)}-\frac{G^{\text{topo1}}(\text{n1},\text{n2},\text{n3}-2,1,1)}{(1-D) (2-D)}+\frac{G^{\text{topo1}}(\text{n1},\text{n2},\text{n3}-1,0,1)}{(1-D) (2-D)}+\frac{G^{\text{topo1}}(\text{n1},\text{n2},\text{n3}-1,1,0)}{(1-D) (2-D)}+\frac{G^{\text{topo1}}(\text{n1},\text{n2},\text{n3},0,0)}{(1-D) (2-D)} +\end{dmath*} +\end{document} diff --git a/pages/FeynCalc.tex b/pages/FeynCalc.tex index 97c379c..1785eb2 100644 --- a/pages/FeynCalc.tex +++ b/pages/FeynCalc.tex @@ -1107,6 +1107,11 @@ \section{Loop integrals}\label{loop integrals}\index{Loop integrals}} \hyperlink{../fcloopglidifferentiate}{../FCLoopGLIDifferentiate} - differentiates \hyperlink{../gli}{../GLI}s with respect to a scalar variable. +\item + \hyperlink{../fcloopglilowerdimension}{../FCLoopGLILowerDimension}, + \hyperlink{../fcloopgliraisedimension}{../FCLoopGLIRaiseDimension} - + shifts dimensions of \hyperlink{../gli}{../GLI}s to \(D-2\) or + \(D+2\). \item \hyperlink{../fcloopaddscalingparameter}{../FCLoopAddScalingParameter}, \hyperlink{../fcloopgliexpand}{../FCLoopGLIExpand} - series expansion diff --git a/pages/FrequentlyAskedQuestions.tex b/pages/FrequentlyAskedQuestions.tex index b31cec0..ef0318a 100644 --- a/pages/FrequentlyAskedQuestions.tex +++ b/pages/FrequentlyAskedQuestions.tex @@ -444,9 +444,11 @@ \subsection{How can I define a complex four The presence of an explicit \texttt{I} will make this vector change under \texttt{ComplexConjugate}, such that -\begin{verbatim} -ComplexConjugate[FV[{a,I},mu]]//FCE -\end{verbatim} +\begin{Shaded} +\begin{Highlighting}[] +\NormalTok{ComplexConjugate}\OperatorTok{[}\NormalTok{FV}\OperatorTok{[\{}\FunctionTok{a}\OperatorTok{,}\FunctionTok{I}\OperatorTok{\},}\NormalTok{mu}\OperatorTok{]]}\SpecialCharTok{//}\NormalTok{FCE} +\end{Highlighting} +\end{Shaded} will give you \texttt{FV[\allowbreak{}\{\allowbreak{}a,\ \allowbreak{}-I\},\ \allowbreak{}mu]}. diff --git a/pages/Renormalization.tex b/pages/Renormalization.tex index a1d72ab..96edae5 100644 --- a/pages/Renormalization.tex +++ b/pages/Renormalization.tex @@ -193,11 +193,13 @@ \subsection{Feynman rules}\label{feynman-rules}} \texttt{WriteFeynArtsOutput} we need to set the global variable \texttt{FR\$Loop} to \texttt{True}. For example, -\begin{verbatim} -FR$Loop=True; -SetDirectory[FileNameJoin[{$UserBaseDirectory,"Applications","FeynCalc","FeynArts","Models"}]]; -WriteFeynArtsOutput[LPhi4,Output->"Phi4",CouplingRename->False]; -\end{verbatim} +\begin{Shaded} +\begin{Highlighting}[] +\NormalTok{FR$Loop}\ExtensionTok{=}\ConstantTok{True}\NormalTok{;} +\FunctionTok{SetDirectory}\OperatorTok{[}\FunctionTok{FileNameJoin}\OperatorTok{[\{}\VariableTok{$UserBaseDirectory}\OperatorTok{,}\StringTok{"Applications"}\OperatorTok{,}\StringTok{"FeynCalc"}\OperatorTok{,}\StringTok{"FeynArts"}\OperatorTok{,}\StringTok{"Models"}\OperatorTok{\}]]}\NormalTok{;} +\NormalTok{WriteFeynArtsOutput}\OperatorTok{[}\NormalTok{LPhi4}\OperatorTok{,}\NormalTok{Output}\OtherTok{{-}\textgreater{}}\StringTok{"Phi4"}\OperatorTok{,}\NormalTok{CouplingRename}\OtherTok{{-}\textgreater{}}\ConstantTok{False}\OperatorTok{]}\NormalTok{;} +\end{Highlighting} +\end{Shaded} \hypertarget{renormalization-schemes}{% \subsection{Renormalization schemes}\label{renormalization-schemes}} @@ -320,13 +322,13 @@ \subsubsection{Renormalization conditions for the OS and the renormalized one reads \begin{equation} - \Gamma_R^{\mu \nu} (q) = \Gamma^{\mu \nu} (q) + \text{CT}. + \Gamma_R^{\mu \nu} (q) = \Gamma^{\mu \nu} (q) + \;\text{CT}. \end{equation} For convenience we also introduce \begin{equation} - \tilde{\Gamma}^{\mu \nu}_R(q) = - \Pi^{\mu \nu} (q) + \text{CT} + \tilde{\Gamma}^{\mu \nu}_R(q) = - \Pi^{\mu \nu} (q) + \;\text{CT} \end{equation} which corresponds to what one actually calculates when considering the @@ -438,13 +440,13 @@ \subsubsection{Renormalization conditions for the OS and the renormalized one reads \begin{equation} - \Gamma_R^{\mu \nu} (q) = \Gamma^{\mu \nu} (q) + \text{CT} + \Gamma_R^{\mu \nu} (q) = \Gamma^{\mu \nu} (q) + \;\text{CT} \end{equation} For convenience we also introduce \begin{equation} - \tilde{\Gamma}^{\mu \nu}_R(q) = - \Pi^{\mu \nu} (q) + \text{CT} + \tilde{\Gamma}^{\mu \nu}_R(q) = - \Pi^{\mu \nu} (q) + \;\text{CT} \end{equation} which corresponds to what one actually calculates when considering the