The two most relevant functions for the manipulations of Dirac
matrices are DiracSimplify and DiracTrace.
The goal of DiracSimplify is to eliminate all pairs of
Dirac matrices with the equal indices or contracted with the same 4-vectors
GA[\[Mu]] . GS[p + m] . GA[\[Mu]]
DiracSimplify[%]\bar{\gamma }^{\mu }.\left(\bar{\gamma }\cdot \left(\overline{m}+\overline{p}\right)\right).\bar{\gamma }^{\mu }
-2 \bar{\gamma }\cdot \overline{m}-2 \bar{\gamma }\cdot \overline{p}
GA[\[Mu]] . GS[p + m1] . GA[\[Nu]] . GS[p + m2]
DiracSimplify[%]\bar{\gamma }^{\mu }.\left(\bar{\gamma }\cdot \left(\overline{\text{m1}}+\overline{p}\right)\right).\bar{\gamma }^{\nu }.\left(\bar{\gamma }\cdot \left(\overline{\text{m2}}+\overline{p}\right)\right)
\bar{\gamma }^{\mu }.\left(\bar{\gamma }\cdot \overline{\text{m1}}\right).\bar{\gamma }^{\nu }.\left(\bar{\gamma }\cdot \overline{\text{m2}}\right)+\bar{\gamma }^{\mu }.\left(\bar{\gamma }\cdot \overline{\text{m1}}\right).\bar{\gamma }^{\nu }.\left(\bar{\gamma }\cdot \overline{p}\right)+\bar{\gamma }^{\mu }.\left(\bar{\gamma }\cdot \overline{p}\right).\bar{\gamma }^{\nu }.\left(\bar{\gamma }\cdot \overline{\text{m2}}\right)-\overline{p}^2 \bar{\gamma }^{\mu }.\bar{\gamma }^{\nu }+2 \overline{p}^{\nu } \bar{\gamma }^{\mu }.\left(\bar{\gamma }\cdot \overline{p}\right)
DiracTrace is used for the evaluation of Dirac traces.
The trace is not evaluated by default
DiracTrace[GA[\[Mu], \[Nu]]]\text{tr}\left(\bar{\gamma }^{\mu }.\bar{\gamma }^{\nu }\right)
To obtain the result we can either use the option
DiracTraceEvaluate
DiracTrace[GA[\[Mu], \[Nu]], DiracTraceEvaluate -> True]4 \bar{g}^{\mu \nu }
or use DiracSimplify instead.
By default FeynCalc refuses to compute a D-dimensional trace that contains \gamma^5
DiracTrace[GAD[\[Alpha], \[Beta], \[Mu], \[Nu], \[Rho], \[Sigma], 5]] // DiracSimplify\text{tr}\left(\gamma ^{\alpha }.\gamma ^{\beta }.\gamma ^{\mu }.\gamma ^{\nu }.\gamma ^{\rho }.\gamma ^{\sigma }.\bar{\gamma }^5\right)
This is because by default FeynCalc is using anticommuting \gamma^5 in D-dimensions, a scheme known as Naive Dimensional Regularization (NDR)
DiracSimplify[GAD[\[Mu]] . GA[5] . GAD[\[Nu]]]-\gamma ^{\mu }.\gamma ^{\nu }.\bar{\gamma }^5
In general, a chiral trace is a very ambiguous object in NDR. The results depends on the position of \gamma^5 inside the trace, so that we chose not to produce results that might be potentially inconsistent. However, FeynCalc can also be told to use the Breitenlohner-Maison-t’Hooft-Veltman scheme (BMHV), which is an algebraically consistent scheme (but has other issues, e.g. it breaks Ward identities)
FCSetDiracGammaScheme["BMHV"];Notice that now FeynCalc anticommutes \gamma^5 according to the BMHV algebra, which leads to the appearance of D-4-dimensional Dirac matrices
DiracSimplify[GAD[\[Mu]] . GA[5] . GAD[\[Nu]]]2 \gamma ^{\mu }.\hat{\gamma }^{\nu }.\bar{\gamma }^5-\gamma ^{\mu }.\gamma ^{\nu }.\bar{\gamma }^5
Also Dirac traces are not an issue now
DiracTrace[GAD[\[Alpha], \[Beta], \[Mu], \[Nu], \[Rho], \[Sigma]] . GA[5]] // DiracSimplify-4 i g^{\alpha \beta } \bar{\epsilon }^{\mu \nu \rho \sigma }+4 i g^{\alpha \mu } \bar{\epsilon }^{\beta \nu \rho \sigma }-4 i g^{\alpha \nu } \bar{\epsilon }^{\beta \mu \rho \sigma }+4 i g^{\alpha \rho } \bar{\epsilon }^{\beta \mu \nu \sigma }-4 i g^{\alpha \sigma } \bar{\epsilon }^{\beta \mu \nu \rho }-4 i g^{\beta \mu } \bar{\epsilon }^{\alpha \nu \rho \sigma }+4 i g^{\beta \nu } \bar{\epsilon }^{\alpha \mu \rho \sigma }-4 i g^{\beta \rho } \bar{\epsilon }^{\alpha \mu \nu \sigma }+4 i g^{\beta \sigma } \bar{\epsilon }^{\alpha \mu \nu \rho }-4 i g^{\mu \nu } \bar{\epsilon }^{\alpha \beta \rho \sigma }+4 i g^{\mu \rho } \bar{\epsilon }^{\alpha \beta \nu \sigma }-4 i g^{\mu \sigma } \bar{\epsilon }^{\alpha \beta \nu \rho }-4 i g^{\nu \rho } \bar{\epsilon }^{\alpha \beta \mu \sigma }+4 i g^{\nu \sigma } \bar{\epsilon }^{\alpha \beta \mu \rho }-4 i g^{\rho \sigma } \bar{\epsilon }^{\alpha \beta \mu \nu }
To compute chiral traces in the BMHV scheme, FeynCalc uses West’s formula. Still, NDR is the default scheme in FeynCalc.
In tree-level calculation a useful operation is the so-called
SPVAT-decomposition of Dirac chains. This is done using
DiracReduce
GA[\[Mu], \[Nu], \[Rho]] . GS[p] . GA[\[Alpha]]
DiracReduce[%]\bar{\gamma }^{\mu }.\bar{\gamma }^{\nu }.\bar{\gamma }^{\rho }.\left(\bar{\gamma }\cdot \overline{p}\right).\bar{\gamma }^{\alpha }
-i \bar{g}^{\mu \nu } \bar{\gamma }^{\text{\$MU}(\text{\$68})}.\bar{\gamma }^5 \bar{\epsilon }^{\alpha \rho \;\text{\$MU}(\text{\$68})\overline{p}}+i \bar{g}^{\alpha \rho } \bar{\gamma }^{\text{\$MU}(\text{\$70})}.\bar{\gamma }^5 \bar{\epsilon }^{\mu \nu \;\text{\$MU}(\text{\$70})\overline{p}}+i \overline{p}^{\alpha } \bar{\gamma }^{\text{\$MU}(\text{\$71})}.\bar{\gamma }^5 \bar{\epsilon }^{\mu \nu \rho \;\text{\$MU}(\text{\$71})}+i \overline{p}^{\rho } \bar{\gamma }^{\text{\$MU}(\text{\$72})}.\bar{\gamma }^5 \bar{\epsilon }^{\alpha \mu \nu \;\text{\$MU}(\text{\$72})}+\bar{\gamma }^{\rho } \overline{p}^{\alpha } \bar{g}^{\mu \nu }-\bar{\gamma }^{\nu } \overline{p}^{\alpha } \bar{g}^{\mu \rho }-\bar{\gamma }^{\rho } \overline{p}^{\mu } \bar{g}^{\alpha \nu }+\bar{\gamma }^{\nu } \overline{p}^{\mu } \bar{g}^{\alpha \rho }+\bar{\gamma }^{\mu } \overline{p}^{\alpha } \bar{g}^{\nu \rho }+\bar{\gamma }^{\alpha } \overline{p}^{\mu } \bar{g}^{\nu \rho }+\bar{\gamma }^{\rho } \overline{p}^{\nu } \bar{g}^{\alpha \mu }-\bar{\gamma }^{\mu } \overline{p}^{\nu } \bar{g}^{\alpha \rho }-\bar{\gamma }^{\alpha } \overline{p}^{\nu } \bar{g}^{\mu \rho }-\bar{\gamma }^{\nu } \overline{p}^{\rho } \bar{g}^{\alpha \mu }+\bar{\gamma }^{\mu } \overline{p}^{\rho } \bar{g}^{\alpha \nu }+\bar{\gamma }^{\alpha } \overline{p}^{\rho } \bar{g}^{\mu \nu }-\bar{g}^{\alpha \rho } \bar{g}^{\mu \nu } \bar{\gamma }\cdot \overline{p}+\bar{g}^{\alpha \nu } \bar{g}^{\mu \rho } \bar{\gamma }\cdot \overline{p}-\bar{g}^{\alpha \mu } \bar{g}^{\nu \rho } \bar{\gamma }\cdot \overline{p}-i \bar{\gamma }^{\nu }.\bar{\gamma }^5 \bar{\epsilon }^{\alpha \mu \rho \overline{p}}+i \bar{\gamma }^{\mu }.\bar{\gamma }^5 \bar{\epsilon }^{\alpha \nu \rho \overline{p}}
Gordon’s identities are implemented via
GordonSimplify
SpinorUBar[p1, m1] . GA[\[Mu]] . SpinorU[p2, m2]
GordonSimplify[%]\bar{u}(\text{p1},\text{m1}).\bar{\gamma }^{\mu }.u(\text{p2},\text{m2})
\frac{\left(\overline{\text{p1}}+\overline{\text{p2}}\right)^{\mu } \left(\varphi (\overline{\text{p1}},\text{m1})\right).\left(\varphi (\overline{\text{p2}},\text{m2})\right)}{\text{m1}+\text{m2}}+\frac{i \left(\varphi (\overline{\text{p1}},\text{m1})\right).\sigma ^{\mu \overline{\text{p1}}-\overline{\text{p2}}}.\left(\varphi (\overline{\text{p2}},\text{m2})\right)}{\text{m1}+\text{m2}}