FeynCalc manual (development version)

Dirac algebra

See also

Overview.

Simplifications of Dirac matrix chains

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 44-vectors

GA[\[Mu]] . GS[p + m] . GA[\[Mu]]
DiracSimplify[%]

γˉμ.(γˉ⋅(m‾+p‾)).γˉμ\bar{\gamma }^{\mu }.\left(\bar{\gamma }\cdot \left(\overline{m}+\overline{p}\right)\right).\bar{\gamma }^{\mu }

−2γˉ⋅m‾−2γˉ⋅p‾-2 \bar{\gamma }\cdot \overline{m}-2 \bar{\gamma }\cdot \overline{p}

GA[\[Mu]] . GS[p + m1] . GA[\[Nu]] . GS[p + m2]
DiracSimplify[%]

γˉμ.(γˉ⋅(m1‾+p‾)).γˉν.(γˉ⋅(m2‾+p‾))\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)

γˉμ.(γˉ⋅m1‾).γˉν.(γˉ⋅m2‾)+γˉμ.(γˉ⋅m1‾).γˉν.(γˉ⋅p‾)+γˉμ.(γˉ⋅p‾).γˉν.(γˉ⋅m2‾)−p‾2γˉμ.γˉν+2p‾νγˉμ.(γˉ⋅p‾)\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]]]

tr(γˉμ.γˉν)\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]

4gˉμν4 \bar{g}^{\mu \nu }

or use DiracSimplify instead.

DiracTrace[GA[\[Mu], \[Nu]]] // DiracSimplify

4gˉμν4 \bar{g}^{\mu \nu }

By default FeynCalc refuses to compute a DD-dimensional trace that contains γ5\gamma^5

DiracTrace[GAD[\[Alpha], \[Beta], \[Mu], \[Nu], \[Rho], \[Sigma], 5]] // DiracSimplify

tr(γα.γβ.γμ.γν.γρ.γσ.γˉ5)\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 γ5\gamma^5 in DD-dimensions, a scheme known as Naive Dimensional Regularization (NDR)

DiracSimplify[GAD[\[Mu]] . GA[5] . GAD[\[Nu]]]

−γμ.γν.γˉ5-\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 γ5\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 γ5\gamma^5 according to the BMHV algebra, which leads to the appearance of D−4D-4-dimensional Dirac matrices

DiracSimplify[GAD[\[Mu]] . GA[5] . GAD[\[Nu]]]

2γμ.γ^ν.γˉ5−γμ.γν.γˉ52 \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

−4igαβϵˉμνρσ+4igαμϵˉβνρσ−4igανϵˉβμρσ+4igαρϵˉβμνσ−4igασϵˉβμνρ−4igβμϵˉανρσ+4igβνϵˉαμρσ−4igβρϵˉαμνσ+4igβσϵˉαμνρ−4igμνϵˉαβρσ+4igμρϵˉαβνσ−4igμσϵˉαβνρ−4igνρϵˉαβμσ+4igνσϵˉαβμρ−4igρσϵˉαβμν-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[%]

γˉμ.γˉν.γˉρ.(γˉ⋅p‾).γˉα\bar{\gamma }^{\mu }.\bar{\gamma }^{\nu }.\bar{\gamma }^{\rho }.\left(\bar{\gamma }\cdot \overline{p}\right).\bar{\gamma }^{\alpha }

−igˉμνγˉ$MU($68).γˉ5ϵˉαρ  $MU($68)p‾+igˉαργˉ$MU($70).γˉ5ϵˉμν  $MU($70)p‾+ip‾αγˉ$MU($71).γˉ5ϵˉμνρ  $MU($71)+ip‾ργˉ$MU($72).γˉ5ϵˉαμν  $MU($72)+γˉρp‾αgˉμν−γˉνp‾αgˉμρ−γˉρp‾μgˉαν+γˉνp‾μgˉαρ+γˉμp‾αgˉνρ+γˉαp‾μgˉνρ+γˉρp‾νgˉαμ−γˉμp‾νgˉαρ−γˉαp‾νgˉμρ−γˉνp‾ρgˉαμ+γˉμp‾ρgˉαν+γˉαp‾ρgˉμν−gˉαρgˉμνγˉ⋅p‾+gˉανgˉμργˉ⋅p‾−gˉαμgˉνργˉ⋅p‾−iγˉν.γˉ5ϵˉαμρp‾+iγˉμ.γˉ5ϵˉανρp‾-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[%]

uˉ(p1,m1).γˉμ.u(p2,m2)\bar{u}(\text{p1},\text{m1}).\bar{\gamma }^{\mu }.u(\text{p2},\text{m2})

(p1‾+p2‾)μ(φ(p1‾,m1)).(φ(p2‾,m2))m1+m2+i(φ(p1‾,m1)).σμp1‾−p2‾.(φ(p2‾,m2))m1+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}}

It is possible to reorder the free indices in a chain of Dirac matrices, which can sometimes help to simplify the expressions

DiracOrder[GA[\[Mu], \[Nu], \[Rho]], {\[Nu], \[Mu]}]

2γˉρgˉμν−γˉν.γˉμ.γˉρ2 \bar{\gamma }^{\rho } \bar{g}^{\mu \nu }-\bar{\gamma }^{\nu }.\bar{\gamma }^{\mu }.\bar{\gamma }^{\rho }

However, since this procedure is computationally expensive, DiracSimplify will not apply it by default

DiracSimplify[GAD[\[Mu], \[Nu]] + GAD[\[Nu], \[Mu]]]

γμ.γν+γν.γμ\gamma ^{\mu }.\gamma ^{\nu }+\gamma ^{\nu }.\gamma ^{\mu }

It can be activated via the option DiracOrder

DiracSimplify[GAD[\[Mu], \[Nu]] + GAD[\[Nu], \[Mu]], DiracOrder -> True]

2gμν2 g^{\mu \nu }