Dirac algebra
See also
Overview.
Simplifications
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[%]
γˉμ.(γˉ⋅(m+p)).γˉμ
−2γˉ⋅m−2γˉ⋅p
GA[\[Mu]] . GS[p + m1] . GA[\[Nu]] . GS[p + m2]
DiracSimplify[%]
γˉμ.(γˉ⋅(m1+p)).γˉν.(γˉ⋅(m2+p))
γˉμ.(γˉ⋅m1).γˉν.(γˉ⋅m2)+γˉμ.(γˉ⋅m1).γˉν.(γˉ⋅p)+γˉμ.(γˉ⋅p).γˉν.(γˉ⋅m2)−p2γˉμ.γˉν+2pνγˉμ.(γˉ⋅p)
DiracTrace
is used for the evaluation of Dirac traces. The trace is not evaluated by default
DiracTrace[GA[\[Mu], \[Nu]]]
tr(γˉμ.γˉν)
To obtain the result we can either use the option DiracTraceEvaluate
DiracTrace[GA[\[Mu], \[Nu]], DiracTraceEvaluate -> True]
4gˉμν
or use DiracSimplify
instead.
By default FeynCalc refuses to compute a D-dimensional trace that contains γ5
DiracTrace[GAD[\[Alpha], \[Beta], \[Mu], \[Nu], \[Rho], \[Sigma], 5]] // DiracSimplify
tr(γα.γβ.γμ.γν.γρ.γσ.γˉ5)
This is because by default FeynCalc is using anticommuting γ5 in D-dimensions, a scheme known as Naive Dimensional Regularization (NDR)
DiracSimplify[GAD[\[Mu]] . GA[5] . GAD[\[Nu]]]
−γμ.γν.γˉ5
In general, a chiral trace is a very ambiguous object in NDR. The results depends on the position of γ5 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 γ5 according to the BMHV algebra, which leads to the appearance of D−4-dimensional Dirac matrices
DiracSimplify[GAD[\[Mu]] . GA[5] . GAD[\[Nu]]]
2γμ.γ^ν.γˉ5−γμ.γν.γˉ5
Also Dirac traces are not an issue now
DiracTrace[GAD[\[Alpha], \[Beta], \[Mu], \[Nu], \[Rho], \[Sigma]] . GA[5]] // DiracSimplify
−4igαβϵˉμνρσ+4igαμϵˉβνρσ−4igανϵˉβμρσ+4igαρϵˉβμνσ−4igασϵˉβμνρ−4igβμϵˉανρσ+4igβνϵˉαμρσ−4igβρϵˉαμνσ+4igβσϵˉαμνρ−4igμνϵˉαβρσ+4igμρϵˉαβνσ−4igμσϵˉαβνρ−4igνρϵˉαβμσ+4igνσϵˉαβμρ−4igρσϵˉαβμν
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[%]
γˉμ.γˉν.γˉρ.(γˉ⋅p).γˉα
−igˉμνγˉ$MU($68).γˉ5ϵˉαρ$MU($68)p+igˉαργˉ$MU($70).γˉ5ϵˉμν$MU($70)p+ipαγˉ$MU($71).γˉ5ϵˉμνρ$MU($71)+ipργˉ$MU($72).γˉ5ϵˉαμν$MU($72)+γˉρpαgˉμν−γˉνpαgˉμρ−γˉρpμgˉαν+γˉνpμgˉαρ+γˉμpαgˉνρ+γˉαpμgˉνρ+γˉρpνgˉαμ−γˉμpνgˉαρ−γˉαpνgˉμρ−γˉνpρgˉαμ+γˉμpρgˉαν+γˉαpρgˉμν−gˉαρgˉμνγˉ⋅p+gˉανgˉμργˉ⋅p−gˉαμgˉνργˉ⋅p−iγˉν.γˉ5ϵˉαμρp+iγˉμ.γˉ5ϵˉανρp
Gordon’s identities are implemented via GordonSimplify
SpinorUBar[p1, m1] . GA[\[Mu]] . SpinorU[p2, m2]
GordonSimplify[%]
uˉ(p1,m1).γˉμ.u(p2,m2)
m1+m2(p1+p2)μ(φ(p1,m1)).(φ(p2,m2))+m1+m2i(φ(p1,m1)).σμp1−p2.(φ(p2,m2))