FeynCalc manual (development version)

 

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 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 }

2γˉm2γˉp-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.(γˉm1).γˉν.(γˉm2)+γˉμ.(γˉm1).γˉν.(γˉp)+γˉμ.(γˉp).γˉν.(γˉm2)p2γˉμ.γˉν+2pνγˉμ.(γˉp)\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.

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 D4D-4-dimensional Dirac matrices

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

2γμ.γ^ν.γˉ5γμ.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+4igαμϵˉβνρσ4igανϵˉβμρσ+4igαρϵˉβμνσ4igασϵˉβμνρ4igβμϵˉανρσ+4igβνϵˉαμρσ4igβρϵˉαμνσ+4igβσϵˉαμνρ4igμνϵˉαβρσ+4igμρϵˉαβνσ4igμσϵˉαβνρ4igνρϵˉαβμσ+4igνσϵˉαβμρ4igρσϵˉαβμν-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.(γˉp).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($70)p+ipαγˉ$MU($71).γˉ5ϵˉμνρ  $MU($71)+ipργˉ$MU($72).γˉ5ϵˉαμν  $MU($72)+γˉρpαgˉμνγˉνpαgˉμργˉρpμgˉαν+γˉνpμgˉαρ+γˉμpαgˉνρ+γˉαpμgˉνρ+γˉρpνgˉαμγˉμpνgˉαργˉαpνgˉμργˉνpρgˉαμ+γˉμpρgˉαν+γˉαpρgˉμνgˉαρgˉμνγˉp+gˉανgˉμργˉpgˉαμgˉνργˉpiγˉν.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)).σμp1p2.(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}}