Fixed a severe bug in the implementation of the Larin scheme

(Issue #283, thanks to ygtw1)

FeynCalc/Dirac/DiracTrace.m

@@ -703,8 +703,8 @@ we define it is a function (traceNo5fun) so that the result can be retrieved ver
 	]/; EvenQ[Length[{x,y}]];
 
 
-
-spur5LarinMVV[x__DiracGamma, y:DiracGamma[_[_,dim_],dim_], DiracGamma[5]]:=
+(*Implements Eq.10 from 1506.04517 *)
+spur5LarinMVVold[x__DiracGamma, y:DiracGamma[_[_,dim_],dim_], DiracGamma[5]]:=
 	Block[{inds, epsInds, gInds, indPartitions, res, signs},
 		inds = First/@{x};
 		epsInds = Subsets[inds, {3}];
@@ -718,6 +718,22 @@ we define it is a function (traceNo5fun) so that the result can be retrieved ver
 		fastExpand[res]
 	]/; EvenQ[Length[{x,y}]] && Length[{x,y}]>=4;
 
+(*Implements Eq.11 from 1506.04517 *)
+spur5LarinMVV[x__DiracGamma, y : DiracGamma[_[_, dim_], dim_], DiracGamma[5]] :=
+Block[{inds, epsInds, gInds, indPartitions, res, signs},
+	inds = First /@ {x, y};
+	epsInds = Subsets[inds, {4}];
+	gInds = SelectFree[inds, #] & /@ epsInds;
+	indPartitions = MapThread[Join[#1, #2] &, {gInds, epsInds}];
+	signs = Signature /@ (indPartitions /.	Thread[Rule[inds, Range[Length[inds]]]]);
+
+	res =
+	Total[MapThread[$LeviCivitaSign*
+		I*#2*((traceNo5Wrap @@ #1[[;; -5]]))*
+		Apply[Eps, #1[[-4 ;;]]] &, {indPartitions, signs}]];
+	fastExpand[res]
+	] /; EvenQ[Length[{x, y}]] && Length[{x, y}] >= 4;
+
 spur5BMHVWest[x__DiracGamma, DiracGamma[5]]:=
 	Block[{spx = {x,DiracGamma[5]},spt,res},
 		res = 2/(Length[spx]-5) Sum[(-1)^(i+j+1) FCUseCache[ExpandScalarProduct,{Pair[spx[[i]][[1]],spx[[j]][[1]]]},{FCI->False}] *

FeynCalc/Documentation/Markdown/Extra/Gamma5.md

@@ -113,3 +113,23 @@ FeynCalc implement the so-called Moch-Vermaseren-Vogt MVV formula from [Moch:201
 ```mathematica
 FCSetDiracGammaScheme["Larin"]
 ```
+
+The usage of this scheme implies that all axial-vector matrices from the Feynman rules $ \gamma^\mu \gamma^5$ should be entered as
+\begin{equation}
+\gamma^\mu \gamma^5 \to \frac{1}{2} \left ( \gamma^\mu \gamma^5 -  \gamma^5 \gamma^\mu \right )
+\end{equation}
+
+If the trace contains more than one $\gamma^5$, the code will insert
+\begin{equation}
+\gamma^\mu \gamma^5 = \frac{i}{6} \, \texttt{\$LeviCivitaSign} \, \varepsilon^{\mu \nu \rho \sigma} \gamma_\nu \gamma_\rho \gamma_\sigma
+\end{equation}
+for all but the right-most $\gamma^5$. Then the resulting trace will be evaluated according to Eq.(11) from [Moch:2015usa](https://arxiv.org/pdf/1506.04517.pdf)
+ \begin{equation}
+\mathrm{Tr}(\gamma^{\mu_1} \ldots \gamma^{\mu_{2m}} \gamma^5) =  4 i \, \texttt{\$LeviCivitaSign} \, g^{\mu_1 \mu_2} \ldots g^{\mu_{2m-5} \mu_{2m-4}} \varepsilon^{\mu_{2m-3} \mu_{2m-2} \mu_{2m-1} \mu_{2m}} + \textrm{permutations of} \quad \mu_{1} \ldots {\mu_{2m}}
+\end{equation}
+
+Notice that according to [Moch:2015usa](https://arxiv.org/pdf/1506.04517.pdf) one should distinguish between Levi-Civita tensors appearing in the calculating from traces over axial-vector matrices and those introduced e.g. via projectors. The "axial-vector" Levi-Civitas should be contracted first to avoid incorrect results.
+
+Since FeynCalc has no way to know the origin of $\varepsilon$-tensors in the input expression, it is advised to rename the unrelated Levi-Civitas to something else while doing the trace calculations and reintroduce them after the traces have been succesfully evaluated.
+
+