Added a new page on Gamma 5.

includes.tex

@@ -5,13 +5,14 @@ \chapter{Useful information}
 \subfile{pages/Load.tex}
 \subfile{pages/FrequentlyAskedQuestions.tex}
 \subfile{pages/FeynArts.tex}
+\subfile{pages/FeynArtsSigns.tex}
 \subfile{pages/Parallelization.tex}
 \subfile{pages/UpperLowerIndices.tex}
-\subfile{pages/MasterIntegrals.tex}
-\subfile{pages/FeynArtsSigns.tex}
 \subfile{pages/CustomModels.tex}
+\subfile{pages/Gamma5.tex}
 \subfile{pages/DiracAlgebraFormulas.tex}
 \subfile{pages/Renormalization.tex}
+\subfile{pages/MasterIntegrals.tex}
 \subfile{pages/Development.tex}
 
 \chapter{Tutorials}

pages/FeynCalc.tex

@@ -19,20 +19,22 @@ \section{Useful information}\label{useful information}\index{Useful information}
   \hyperlink{frequentlyaskedquestions}{FrequentlyAskedQuestions}
 \item
   \hyperlink{feynarts}{FeynArts}
+\item
+  \hyperlink{feynartssigns}{FeynArtsSigns}
 \item
   \hyperlink{parallelization}{Parallelization}
 \item
   \hyperlink{upperlowerindices}{UpperLowerIndices}
-\item
-  \hyperlink{masterintegrals}{MasterIntegrals}
-\item
-  \hyperlink{feynartssigns}{FeynArtsSigns}
 \item
   \hyperlink{custommodels}{CustomModels}
+\item
+  \hyperlink{gamma5}{Gamma5}
 \item
   \hyperlink{diracalgebraformulas}{DiracAlgebraFormulas}
 \item
   \hyperlink{renormalization}{Renormalization}
+\item
+  \hyperlink{masterintegrals}{MasterIntegrals}
 \item
   \hyperlink{development}{Development}
 \end{itemize}

pages/Gamma5.tex

@@ -0,0 +1,228 @@
+% !TeX program = pdflatex
+% !TeX root = Gamma5.tex
+
+\documentclass[../FeynCalcManual.tex]{subfiles}
+\begin{document}
+\hypertarget{treatment of gamma5 in d dimensions}{
+\section{Treatment of gamma5 in D dimensions}\label{treatment of gamma5 in d dimensions}\index{Treatment of gamma5 in D dimensions}}
+
+\subsection{See also}
+
+\hyperlink{toc}{Overview}.
+
+\hypertarget{nature-of-the-problem}{%
+\subsection{Nature of the problem}\label{nature-of-the-problem}}
+
+It is a well-known fact (cf.~eg.
+\href{https://arxiv.org/pdf/hep-th/0005255}{Jegerlehner:2000dz}) that
+the definition of \(\gamma^5\) in 4 dimensions cannot be consistently
+extended to \(D\) dimensions without giving up either the
+anticommutativity property
+
+\begin{equation}
+\{\gamma^5, \gamma^\mu\} = 0
+\end{equation}
+
+or the cyclicity of the Dirac trace, e.g.~that
+
+\begin{equation}
+\mathrm{Tr}( \gamma^{\mu_1} \ldots \gamma^{\mu_{2n}} \gamma^5 ) = \mathrm{Tr}( \gamma^{\mu_2} \ldots \gamma^{\mu_{2n}} \gamma^5 \gamma^{\mu_1} ) = \mathrm{Tr}( \gamma^{\mu_3} \ldots \gamma^{\mu_{2n}} \gamma^5 \gamma^{\mu_1} \gamma^{\mu_2} ) = \ldots
+\end{equation}
+
+This explains the existence of multiple prescriptions (called
+\(\gamma^5\)-schemes) that aim at avoiding these issues and obtaining
+physical results in the \emph{given calculation}.
+
+Indeed, as of now there is no simple solution or cookbook recipe that
+can be readily applied to any theory at any loop order in a fully
+automatic fashion.
+
+The reason for this is that calculations involving \(\gamma^5\) are not
+limited to the algebraic manipulations of Dirac matrices. In general,
+once \(\gamma^5\) shows up in \(D\)-dimensional amplitudes, there is a
+high chance that the final result will violate some of the essential
+symmetries, such as generalized Ward identities or Bose symmetry.
+
+Once this happens, symmetries violated due to the chosen \(\gamma^5\)
+scheme must be restored by hand, e.g.~by introducing special finite
+counterterms. Unfortunately, an explicit determination of such
+counterterms for a given model is a nontrivial task, especially beyond
+1-loop. This explains why people usually try to avoid this situation and
+would rather opt for figuring out special tricks that work only for this
+particular calculation but manage to preserve the symmetries.
+
+Further discussions on this topic can be found e.g.~in
+
+\begin{itemize}
+\tightlist
+\item
+  chapter D of \href{https://arxiv.org/pdf/1809.01830}{Blondel:2018mad}
+\item
+  \href{https://arxiv.org/pdf/hep-ph/9504315.pdf}{Trueman:1995ca}
+\item
+  \href{https://arxiv.org/pdf/1912.06823.pdf}{Denner:2019vbn}
+\item
+  \href{https://arxiv.org/abs/1705.01827}{Gnendiger:2017pys}
+\item
+  \href{https://arxiv.org/abs/2312.11291}{Stockinger:2023ndm}
+\end{itemize}
+
+\hypertarget{feyncalc-implementation}{%
+\subsection{FeynCalc implementation}\label{feyncalc-implementation}}
+
+FeynCalc has built-in support for several \(\gamma^5\)-schemes in the
+sense that it can manipulate \(D\)-dimensional algebraic expressions
+involving \(\gamma^5\) in accordance with the rules provided by the
+scheme authors.
+
+The nonalgebraic part of a typical \(\gamma^5\)-calculation,
+e.g.~checking for violated symmetries and restoring them is \textbf{not
+handled} by FeynCalc. This is also not something easy to automatize (due
+to the reasons explained above) so that here we expect the user to
+employ their understanding of physics and common sense.
+
+The responsibility of FeynCalc is to ensure that algebraic manipulations
+of Dirac matrices (including \(\gamma^5\)) are consistent within the
+chosen scheme. For the purpose of dealing with \(\gamma^5\) in \(D\)
+dimensions FeynCalc implements three different schemes.
+
+\hypertarget{ndr}{%
+\subsubsection{NDR}\label{ndr}}
+
+The Naive or Conventional Dimensional Regularization (NDR or CDR
+respectively)
+\href{https://doi.org/10.1016/0550-3213(79)90333-X}{Chanowitz:1979zu}
+simply \emph{assumes} that one can define a \(D\)-dimensional
+\(\gamma^5\) that anticommutes with any other Dirac matrix and does not
+break the cyclicity of the trace. For FeynCalc this means that in every
+string of Dirac matrices all \(\gamma^5\) can be safely anticommuted to
+the right end of the string. In the course of this operation FeynCalc
+can always apply \((\gamma^5)^2 = 1\).
+
+Consequently, all Dirac traces with an even number of \(\gamma^5\) can
+be rewritten as traces that involve only the first four
+\(\gamma\)-matrices and evaluated directly, e.g.
+
+\begin{equation}
+\mathrm{Tr}( \gamma^{\mu_1} \gamma^{\mu_2} \gamma^5 \gamma^{\mu_3} \ldots \gamma^{\mu_{2n}} \gamma^5 ) = 
+\mathrm{Tr}( \gamma^{\mu_1} \gamma^{\mu_2} \ldots \gamma^{\mu_{2n}}  )
+\end{equation}
+
+The problematic cases are \(\gamma^5\)-odd traces with an even number of
+other Dirac matrices, where the \(\mathcal{O}(D-4)\) pieces of the
+result depend on the initial position of \(\gamma^5\) in the string.
+Using the anticommutativity property they can be always rewritten as
+traces of a string of other Dirac matrices and one \(\gamma^5\). If the
+number of the other Dirac matrices is odd, such a trace is put to zero
+i.e. \begin{equation}
+\mathrm{Tr}(\gamma^{\mu_1} \ldots \gamma^{\mu_{2n-1}} \gamma^5) = 0, \quad n \in \mathbb{N}
+\end{equation} If the number is even, the trace \begin{equation}
+\mathrm{Tr}(\gamma^{\mu_1} \ldots \gamma^{\mu_{2n}} \gamma^5)
+\end{equation} is returned unevaluated, since FeynCalc does not know how
+to calculate it in a consistent way. A user who knows how these
+ambiguous objects should be treated in the particular calculation can
+still take care of the remaining traces by hand. This ensures that the
+output produced by FeynCalc is algebraically consistent to the maximal
+extent possible in the NDR scheme without extra assumptions.
+
+In FeynCalc, this scheme the default choice. It can also be explicitly
+activated via
+
+\begin{Shaded}
+\begin{Highlighting}[]
+\NormalTok{FCSetDiracGammaScheme}\OperatorTok{[}\StringTok{"NDR"}\OperatorTok{]}
+\end{Highlighting}
+\end{Shaded}
+
+Sometimes \(\gamma^5\) may show up in the calculation as an artifact of
+using a particular set of operators or projectors even though the
+results itself is not supposed to be affected by the
+\(\gamma^5\)-problem. For such cases FeynCalc offers a variety of the
+NDR scheme, where all traces of the form \begin{equation}
+\mathrm{Tr}(\gamma^{\mu_1} \ldots \gamma^{\mu_{2n}} \gamma^5)
+\end{equation} are simply put to zero. It can be used to e.g.~examine
+the effects of the chosen scheme on the final result and can be
+activated via
+
+\begin{Shaded}
+\begin{Highlighting}[]
+\NormalTok{FCSetDiracGammaScheme}\OperatorTok{[}\StringTok{"NDR{-}Discard"}\OperatorTok{]}
+\end{Highlighting}
+\end{Shaded}
+
+\hypertarget{bmhv}{%
+\subsection{BMHV}\label{bmhv}}
+
+FeynCalc also supports the Breitenlohner-Maison implementation
+\href{https://doi.org/10.1007/BF01609069}{Breitenlohner:1977hr} of the
+t'Hooft-Veltman
+\href{https://doi.org/10.1016/0550-3213(72)90279-9}{tHooft:1972tcz}
+prescription, often abbreviated as BMHV, HVBM, HV or BM scheme. In this
+approach \(\gamma^5\) is treated as a purely 4-dimensional object, while
+\(D\)-dimensional Dirac matrices and 4-vectors are decomposed into
+\(4\)- and \(D-4\)-dimensional components. Following
+\href{https://doi.org/10.1016/0550-3213(90)90223-Z}{Buras:1989xd}
+FeynCalc typesets the former with a bar and the latter with a hat e.g.
+
+\begin{equation}
+\gamma^\mu = \bar{\gamma}^\mu + \hat{\gamma}^\mu, \quad p^\mu = \bar{p}^\mu + \hat{p}^\mu
+\end{equation}
+
+The main advantage of the BMHV scheme is that the Dirac algebra
+(including traces) can be evaluated without any algebraic ambiguities.
+However, calculations involving tensors from three different spaces
+(\(D\), \(4\) and \(D-4\)) often turn out to be rather cumbersome, even
+when using computer codes. Moreover, this prescription is known to
+artificially violate Ward identities in chiral theories, which is
+something that can be often avoided when using NDR. Within BMHV FeynCalc
+can simplify arbitrary strings of Dirac matrices and calculate arbitrary
+traces out-of-the-box. The evaluation of \(\gamma^5\)-odd Dirac traces
+is performed using the West-formula from
+\href{https://doi.org/10.1016/0010-4655(93)90011-Z}{West:1991xv}. It is
+worth noting that \(D-4\)-dimensional components of external momenta are
+not set to zero by default, as it is conventionally done in the
+literature. If this is required, the user should evaluate
+\texttt{Momentum[\allowbreak{}pi,\ \allowbreak{}D-4]=0} for each
+relevant momentum \(p_i\). To remove such assignments one should use
+\texttt{FCClearScalarProducts[\allowbreak{}]}.
+
+This scheme is activated by evaluating
+
+\begin{Shaded}
+\begin{Highlighting}[]
+\NormalTok{FCSetDiracGammaScheme}\OperatorTok{[}\StringTok{"BMHV"}\OperatorTok{]}
+\end{Highlighting}
+\end{Shaded}
+
+\hypertarget{larins-scheme}{%
+\subsection{Larin's scheme}\label{larins-scheme}}
+
+Larin's scheme
+\href{https://arxiv.org/pdf/hep-ph/9302240.pdf}{Larin:1993tq} is a
+variety of the BMHV scheme that has been extensively used in QCD
+calculations involving axial vector currents. The main idea is to
+replace the products of \(\gamma^\mu\) and \(\gamma^5\) in a chiral
+trace as in
+
+\begin{equation}
+\gamma^\mu \gamma^5 \to \frac{1}{6} i \varepsilon^{\mu \nu \rho \sigma} \gamma_\nu \gamma_\rho \gamma_\sigma
+\end{equation}
+
+and then calculate the resulting trace. Then, all
+\(\varepsilon^{\mu \nu \rho \sigma}\)-tensors occurring in the amplitude
+should be evaluated in \(D\) dimensions. Together with the correct
+counterterm, this prescription is known to give the same result as when
+using the full BMHV scheme.
+
+FeynCalc implement the so-called Moch-Vermaseren-Vogt MVV formula from
+\href{https://arxiv.org/pdf/1506.04517.pdf}{Moch:2015usa} for
+calculating \(\gamma^5\)-traces in this scheme. The scheme itself is
+activated by setting
+
+\begin{Shaded}
+\begin{Highlighting}[]
+\NormalTok{FCSetDiracGammaScheme}\OperatorTok{[}\StringTok{"Larin"}\OperatorTok{]}
+\end{Highlighting}
+\end{Shaded}
+
+\end{document}