FeynCalc manual (development version)

 
 

DiracTrace

DiracTrace[exp] is the head of Dirac traces. By default the trace is not evaluated. The evaluation occurs only when the option DiracTraceEvaluate is set to True. It is recommended to use DiracSimplify, which will automatically evaluate all Dirac traces in the input expression.

See also

Overview, Contract, DiracEquation, DiracGamma, DiracGammaExpand, DiracTrick, FCGetDiracGammaScheme, FCSetDiracGammaScheme.

Examples

There is no automatic evaluation of Dirac traces

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

tr(γˉμ.γˉν)\text{tr}\left(\bar{\gamma }^{\mu }.\bar{\gamma }^{\nu }\right)

DiracTrace[GA[\[Mu], \[Nu], \[Rho], \[Sigma]]]

tr(γˉμ.γˉν.γˉρ.γˉσ)\text{tr}\left(\bar{\gamma }^{\mu }.\bar{\gamma }^{\nu }.\bar{\gamma }^{\rho }.\bar{\gamma }^{\sigma }\right)

You can either set the option DiracTraceEvaluate to True or use DiracSimplify.

DiracTrace[GA[\[Mu], \[Nu], \[Rho], \[Sigma]], DiracTraceEvaluate -> True]

4(gˉμσgˉνρgˉμρgˉνσ+gˉμνgˉρσ)4 \left(\bar{g}^{\mu \sigma } \bar{g}^{\nu \rho }-\bar{g}^{\mu \rho } \bar{g}^{\nu \sigma }+\bar{g}^{\mu \nu } \bar{g}^{\rho \sigma }\right)

DiracSimplify[DiracTrace[GA[\[Mu], \[Nu], \[Rho], \[Sigma]]]]

4gˉμσgˉνρ4gˉμρgˉνσ+4gˉμνgˉρσ4 \bar{g}^{\mu \sigma } \bar{g}^{\nu \rho }-4 \bar{g}^{\mu \rho } \bar{g}^{\nu \sigma }+4 \bar{g}^{\mu \nu } \bar{g}^{\rho \sigma }

DiracTrace[GS[p, q, r, s]] 
 
DiracSimplify[%]

tr((γˉp).(γˉq).(γˉr).(γˉs))\text{tr}\left(\left(\bar{\gamma }\cdot \overline{p}\right).\left(\bar{\gamma }\cdot \overline{q}\right).\left(\bar{\gamma }\cdot \overline{r}\right).\left(\bar{\gamma }\cdot \overline{s}\right)\right)

4(ps)(qr)4(pr)(qs)+4(pq)(rs)4 \left(\overline{p}\cdot \overline{s}\right) \left(\overline{q}\cdot \overline{r}\right)-4 \left(\overline{p}\cdot \overline{r}\right) \left(\overline{q}\cdot \overline{s}\right)+4 \left(\overline{p}\cdot \overline{q}\right) \left(\overline{r}\cdot \overline{s}\right)

The old methods of evaluating traces by replacing DiracTrace with TR are deprecated and should not be used anymore. In particular, they are slower are less efficient than using DiracSimplify.

Traces involving γ5\gamma^5 or chirality projectors in 44 dimensions are also possible

DiracTrace[GA[\[Mu], \[Nu], \[Rho], \[Sigma], 5]] 
 
DiracSimplify[%]

tr(γˉμ.γˉν.γˉρ.γˉσ.γˉ5)\text{tr}\left(\bar{\gamma }^{\mu }.\bar{\gamma }^{\nu }.\bar{\gamma }^{\rho }.\bar{\gamma }^{\sigma }.\bar{\gamma }^5\right)

4iϵˉμνρσ-4 i \bar{\epsilon }^{\mu \nu \rho \sigma }

DiracTrace[GA[\[Mu], \[Nu], \[Rho], \[Sigma], \[Delta], \[Tau], 5]] 
 
DiracSimplify[%]

tr(γˉμ.γˉν.γˉρ.γˉσ.γˉδ.γˉτ.γˉ5)\text{tr}\left(\bar{\gamma }^{\mu }.\bar{\gamma }^{\nu }.\bar{\gamma }^{\rho }.\bar{\gamma }^{\sigma }.\bar{\gamma }^{\delta }.\bar{\gamma }^{\tau }.\bar{\gamma }^5\right)

4igˉδμϵˉνρστ4igˉδτϵˉμνρσ4igˉμτϵˉδνρσ4igˉνρϵˉδμστ+4igˉνσϵˉδμρτ4igˉρσϵˉδμντ-4 i \bar{g}^{\delta \mu } \bar{\epsilon }^{\nu \rho \sigma \tau }-4 i \bar{g}^{\delta \tau } \bar{\epsilon }^{\mu \nu \rho \sigma }-4 i \bar{g}^{\mu \tau } \bar{\epsilon }^{\delta \nu \rho \sigma }-4 i \bar{g}^{\nu \rho } \bar{\epsilon }^{\delta \mu \sigma \tau }+4 i \bar{g}^{\nu \sigma } \bar{\epsilon }^{\delta \mu \rho \tau }-4 i \bar{g}^{\rho \sigma } \bar{\epsilon }^{\delta \mu \nu \tau }

DiracTrace[GA[\[Mu], \[Nu], \[Rho], \[Sigma], \[Delta], \[Tau], 6]] 
 
DiracSimplify[%]

tr(γˉμ.γˉν.γˉρ.γˉσ.γˉδ.γˉτ.γˉ6)\text{tr}\left(\bar{\gamma }^{\mu }.\bar{\gamma }^{\nu }.\bar{\gamma }^{\rho }.\bar{\gamma }^{\sigma }.\bar{\gamma }^{\delta }.\bar{\gamma }^{\tau }.\bar{\gamma }^6\right)

2gˉδμgˉντgˉρσ+2gˉδμgˉνσgˉρτ2gˉδμgˉνρgˉστ+2gˉδτgˉμσgˉνρ+2gˉδσgˉμτgˉνρ2gˉδτgˉμρgˉνσ2gˉδρgˉμτgˉνσ2gˉδσgˉμρgˉντ+2gˉδρgˉμσgˉντ+2gˉδτgˉμνgˉρσ+2gˉδνgˉμτgˉρσ+2gˉδσgˉμνgˉρτ2gˉδνgˉμσgˉρτ2gˉδρgˉμνgˉστ+2gˉδνgˉμρgˉστ2igˉδμϵˉνρστ2igˉδτϵˉμνρσ2igˉμτϵˉδνρσ2igˉνρϵˉδμστ+2igˉνσϵˉδμρτ2igˉρσϵˉδμντ-2 \bar{g}^{\delta \mu } \bar{g}^{\nu \tau } \bar{g}^{\rho \sigma }+2 \bar{g}^{\delta \mu } \bar{g}^{\nu \sigma } \bar{g}^{\rho \tau }-2 \bar{g}^{\delta \mu } \bar{g}^{\nu \rho } \bar{g}^{\sigma \tau }+2 \bar{g}^{\delta \tau } \bar{g}^{\mu \sigma } \bar{g}^{\nu \rho }+2 \bar{g}^{\delta \sigma } \bar{g}^{\mu \tau } \bar{g}^{\nu \rho }-2 \bar{g}^{\delta \tau } \bar{g}^{\mu \rho } \bar{g}^{\nu \sigma }-2 \bar{g}^{\delta \rho } \bar{g}^{\mu \tau } \bar{g}^{\nu \sigma }-2 \bar{g}^{\delta \sigma } \bar{g}^{\mu \rho } \bar{g}^{\nu \tau }+2 \bar{g}^{\delta \rho } \bar{g}^{\mu \sigma } \bar{g}^{\nu \tau }+2 \bar{g}^{\delta \tau } \bar{g}^{\mu \nu } \bar{g}^{\rho \sigma }+2 \bar{g}^{\delta \nu } \bar{g}^{\mu \tau } \bar{g}^{\rho \sigma }+2 \bar{g}^{\delta \sigma } \bar{g}^{\mu \nu } \bar{g}^{\rho \tau }-2 \bar{g}^{\delta \nu } \bar{g}^{\mu \sigma } \bar{g}^{\rho \tau }-2 \bar{g}^{\delta \rho } \bar{g}^{\mu \nu } \bar{g}^{\sigma \tau }+2 \bar{g}^{\delta \nu } \bar{g}^{\mu \rho } \bar{g}^{\sigma \tau }-2 i \bar{g}^{\delta \mu } \bar{\epsilon }^{\nu \rho \sigma \tau }-2 i \bar{g}^{\delta \tau } \bar{\epsilon }^{\mu \nu \rho \sigma }-2 i \bar{g}^{\mu \tau } \bar{\epsilon }^{\delta \nu \rho \sigma }-2 i \bar{g}^{\nu \rho } \bar{\epsilon }^{\delta \mu \sigma \tau }+2 i \bar{g}^{\nu \sigma } \bar{\epsilon }^{\delta \mu \rho \tau }-2 i \bar{g}^{\rho \sigma } \bar{\epsilon }^{\delta \mu \nu \tau }

DD-dimensional traces that do not involve γ5\gamma^5 are unambiguous.

DiracTrace[(-GSD[q] + SMP["m_e"]) . GAD[\[Nu]] . (GSD[p - q] + SMP["m_e"]) . GAD[\[Mu]]] 
 
DiracSimplify[%]

tr((meγq).γν.(me+γ(pq)).γμ)\text{tr}\left(\left(m_e-\gamma \cdot q\right).\gamma ^{\nu }.\left(m_e+\gamma \cdot (p-q)\right).\gamma ^{\mu }\right)

4me2gμν+4gμν(pq)4q2gμν4pνqμ4pμqν+8qμqν4 m_e^2 g^{\mu \nu }+4 g^{\mu \nu } (p\cdot q)-4 q^2 g^{\mu \nu }-4 p^{\nu } q^{\mu }-4 p^{\mu } q^{\nu }+8 q^{\mu } q^{\nu }

Traces that contain γ5\gamma^5 in DD dimensions are scheme-dependent. The default scheme used in FeynCalc is the naive dimension regularization (NDR), where γ5\gamma^5 is assumed to anticommute with all other Dirac matrices. However, chiral traces are ambiguous in NDR, unless the trace contains an even number of γ5\gamma^5. This is why FeynCalc will leave such traces unevaluated.

DiracTrace[GAD[\[Mu], \[Nu], \[Rho]] . GA[5] . GAD[\[Sigma], \[Delta], \[Tau]] . GA[5]] 
 
DiracSimplify[%]

tr(γμ.γν.γρ.γˉ5.γσ.γδ.γτ.γˉ5)\text{tr}\left(\gamma ^{\mu }.\gamma ^{\nu }.\gamma ^{\rho }.\bar{\gamma }^5.\gamma ^{\sigma }.\gamma ^{\delta }.\gamma ^{\tau }.\bar{\gamma }^5\right)

4gδτgμσgνρ4gδσgμτgνρ+4gδμgνρgστ+4gδτgμρgνσ+4gδρgμτgνσ+4gδσgμρgντ4gδρgμσgντ4gδτgμνgρσ4gδνgμτgρσ+4gδμgντgρσ4gδσgμνgρτ+4gδνgμσgρτ4gδμgνσgρτ+4gδρgμνgστ4gδνgμρgστ-4 g^{\delta \tau } g^{\mu \sigma } g^{\nu \rho }-4 g^{\delta \sigma } g^{\mu \tau } g^{\nu \rho }+4 g^{\delta \mu } g^{\nu \rho } g^{\sigma \tau }+4 g^{\delta \tau } g^{\mu \rho } g^{\nu \sigma }+4 g^{\delta \rho } g^{\mu \tau } g^{\nu \sigma }+4 g^{\delta \sigma } g^{\mu \rho } g^{\nu \tau }-4 g^{\delta \rho } g^{\mu \sigma } g^{\nu \tau }-4 g^{\delta \tau } g^{\mu \nu } g^{\rho \sigma }-4 g^{\delta \nu } g^{\mu \tau } g^{\rho \sigma }+4 g^{\delta \mu } g^{\nu \tau } g^{\rho \sigma }-4 g^{\delta \sigma } g^{\mu \nu } g^{\rho \tau }+4 g^{\delta \nu } g^{\mu \sigma } g^{\rho \tau }-4 g^{\delta \mu } g^{\nu \sigma } g^{\rho \tau }+4 g^{\delta \rho } g^{\mu \nu } g^{\sigma \tau }-4 g^{\delta \nu } g^{\mu \rho } g^{\sigma \tau }

DiracTrace[GAD[\[Mu], \[Nu], \[Rho]] . GA[5] . GAD[\[Sigma], \[Delta], \[Tau]] . GA[7]] 
 
DiracSimplify[%]

tr(γμ.γν.γρ.γˉ5.γσ.γδ.γτ.γˉ7)\text{tr}\left(\gamma ^{\mu }.\gamma ^{\nu }.\gamma ^{\rho }.\bar{\gamma }^5.\gamma ^{\sigma }.\gamma ^{\delta }.\gamma ^{\tau }.\bar{\gamma }^7\right)

12  tr(γμ.γν.γρ.γσ.γδ.γτ.γˉ5)+2gδτgμσgνρ+2gδσgμτgνρ2gδτgμρgνσ2gδρgμτgνσ2gδσgμρgντ+2gδρgμσgντ+2gδτgμνgρσ+2gδνgμτgρσ2gδμgντgρσ+2gδσgμνgρτ2gδνgμσgρτ+2gδμgνσgρτ2gδρgμνgστ+2gδνgμρgστ2gδμgνρgστ-\frac{1}{2} \;\text{tr}\left(\gamma ^{\mu }.\gamma ^{\nu }.\gamma ^{\rho }.\gamma ^{\sigma }.\gamma ^{\delta }.\gamma ^{\tau }.\bar{\gamma }^5\right)+2 g^{\delta \tau } g^{\mu \sigma } g^{\nu \rho }+2 g^{\delta \sigma } g^{\mu \tau } g^{\nu \rho }-2 g^{\delta \tau } g^{\mu \rho } g^{\nu \sigma }-2 g^{\delta \rho } g^{\mu \tau } g^{\nu \sigma }-2 g^{\delta \sigma } g^{\mu \rho } g^{\nu \tau }+2 g^{\delta \rho } g^{\mu \sigma } g^{\nu \tau }+2 g^{\delta \tau } g^{\mu \nu } g^{\rho \sigma }+2 g^{\delta \nu } g^{\mu \tau } g^{\rho \sigma }-2 g^{\delta \mu } g^{\nu \tau } g^{\rho \sigma }+2 g^{\delta \sigma } g^{\mu \nu } g^{\rho \tau }-2 g^{\delta \nu } g^{\mu \sigma } g^{\rho \tau }+2 g^{\delta \mu } g^{\nu \sigma } g^{\rho \tau }-2 g^{\delta \rho } g^{\mu \nu } g^{\sigma \tau }+2 g^{\delta \nu } g^{\mu \rho } g^{\sigma \tau }-2 g^{\delta \mu } g^{\nu \rho } g^{\sigma \tau }

Over the years people invented many different schemes to deal with γ5\gamma^5 in dimensional regularization. Currently, only the t’Hooft-Veltman-Breitenlohner-Maison (BMHV) prescription is fully supported in FeynCalc.

FCSetDiracGammaScheme["BMHV"]; 
 
DiracSimplify[DiracTrace[GAD[\[Mu], \[Nu], \[Rho]] . GA[5] . GAD[\[Sigma], \[Delta], \[Tau]] . GA[7]]]

4iϵˉνρστg^δμ+16g^ντg^ρσg^δμ8gντg^ρσg^δμ16g^νσg^ρτg^δμ+8gνσg^ρτg^δμ8g^ντgρσg^δμ+4gντgρσg^δμ+8g^νσgρτg^δμ4gνσgρτg^δμ+4gνρgστg^δμ4iϵˉμρστg^δν+4iϵˉμνστg^δρ2iϵˉνρστgδμ+2iϵˉμρστgδν2iϵˉμνστgδρ+2iϵˉμνρτgδσ+2iϵˉμνρσgδτ4iϵˉδνρτg^μσ+4iϵˉδνρσg^μτ+2iϵˉδρστgμν2iϵˉδνστgμρ+2iϵˉδνρτgμσ2iϵˉδνρσgμτ+4iϵˉδμρτg^νσ+16g^δρg^μτg^νσ8gδρg^μτg^νσ+4gδτgμρg^νσ8g^δρgμτg^νσ+4gδρgμτg^νσ4iϵˉδμρσg^ντ16g^δρg^μσg^ντ+8gδρg^μσg^ντ+4gδσgμρg^ντ+8g^δρgμσg^ντ4gδρgμσg^ντ+2iϵˉδμστgνρ4gδτg^μσgνρ4gδσg^μτgνρ+2gδτgμσgνρ+2gδσgμτgνρ2iϵˉδμρτgνσ8g^δρg^μτgνσ+4gδρg^μτgνσ2gδτgμρgνσ+4g^δρgμτgνσ2gδρgμτgνσ+2iϵˉδμρσgντ+8g^δρg^μσgντ4gδρg^μσgντ2gδσgμρgντ4g^δρgμσgντ+2gδρgμσgντ4iϵˉδμντg^ρσ16g^δνg^μτg^ρσ+8gδνg^μτg^ρσ4gδτgμνg^ρσ+8g^δνgμτg^ρσ4gδνgμτg^ρσ8gδμg^ντg^ρσ+4gδμgντg^ρσ+4iϵˉδμνσg^ρτ+16g^δνg^μσg^ρτ8gδνg^μσg^ρτ4gδσgμνg^ρτ8g^δνgμσg^ρτ+4gδνgμσg^ρτ+8gδμg^νσg^ρτ4gδμgνσg^ρτ+2iϵˉδμντgρσ+8g^δνg^μτgρσ4gδνg^μτgρσ+2gδτgμνgρσ4g^δνgμτgρσ+2gδνgμτgρσ+4gδμg^ντgρσ2gδμgντgρσ2iϵˉδμνσgρτ8g^δνg^μσgρτ+4gδνg^μσgρτ+2gδσgμνgρτ+4g^δνgμσgρτ2gδνgμσgρτ4gδμg^νσgρτ+2gδμgνσgρτ+2iϵˉδμνρgστ+4g^δρgμνgστ2gδρgμνgστ4g^δνgμρgστ+2gδνgμρgστ2gδμgνρgστ4 i \bar{\epsilon }^{\nu \rho \sigma \tau } \hat{g}^{\delta \mu }+16 \hat{g}^{\nu \tau } \hat{g}^{\rho \sigma } \hat{g}^{\delta \mu }-8 g^{\nu \tau } \hat{g}^{\rho \sigma } \hat{g}^{\delta \mu }-16 \hat{g}^{\nu \sigma } \hat{g}^{\rho \tau } \hat{g}^{\delta \mu }+8 g^{\nu \sigma } \hat{g}^{\rho \tau } \hat{g}^{\delta \mu }-8 \hat{g}^{\nu \tau } g^{\rho \sigma } \hat{g}^{\delta \mu }+4 g^{\nu \tau } g^{\rho \sigma } \hat{g}^{\delta \mu }+8 \hat{g}^{\nu \sigma } g^{\rho \tau } \hat{g}^{\delta \mu }-4 g^{\nu \sigma } g^{\rho \tau } \hat{g}^{\delta \mu }+4 g^{\nu \rho } g^{\sigma \tau } \hat{g}^{\delta \mu }-4 i \bar{\epsilon }^{\mu \rho \sigma \tau } \hat{g}^{\delta \nu }+4 i \bar{\epsilon }^{\mu \nu \sigma \tau } \hat{g}^{\delta \rho }-2 i \bar{\epsilon }^{\nu \rho \sigma \tau } g^{\delta \mu }+2 i \bar{\epsilon }^{\mu \rho \sigma \tau } g^{\delta \nu }-2 i \bar{\epsilon }^{\mu \nu \sigma \tau } g^{\delta \rho }+2 i \bar{\epsilon }^{\mu \nu \rho \tau } g^{\delta \sigma }+2 i \bar{\epsilon }^{\mu \nu \rho \sigma } g^{\delta \tau }-4 i \bar{\epsilon }^{\delta \nu \rho \tau } \hat{g}^{\mu \sigma }+4 i \bar{\epsilon }^{\delta \nu \rho \sigma } \hat{g}^{\mu \tau }+2 i \bar{\epsilon }^{\delta \rho \sigma \tau } g^{\mu \nu }-2 i \bar{\epsilon }^{\delta \nu \sigma \tau } g^{\mu \rho }+2 i \bar{\epsilon }^{\delta \nu \rho \tau } g^{\mu \sigma }-2 i \bar{\epsilon }^{\delta \nu \rho \sigma } g^{\mu \tau }+4 i \bar{\epsilon }^{\delta \mu \rho \tau } \hat{g}^{\nu \sigma }+16 \hat{g}^{\delta \rho } \hat{g}^{\mu \tau } \hat{g}^{\nu \sigma }-8 g^{\delta \rho } \hat{g}^{\mu \tau } \hat{g}^{\nu \sigma }+4 g^{\delta \tau } g^{\mu \rho } \hat{g}^{\nu \sigma }-8 \hat{g}^{\delta \rho } g^{\mu \tau } \hat{g}^{\nu \sigma }+4 g^{\delta \rho } g^{\mu \tau } \hat{g}^{\nu \sigma }-4 i \bar{\epsilon }^{\delta \mu \rho \sigma } \hat{g}^{\nu \tau }-16 \hat{g}^{\delta \rho } \hat{g}^{\mu \sigma } \hat{g}^{\nu \tau }+8 g^{\delta \rho } \hat{g}^{\mu \sigma } \hat{g}^{\nu \tau }+4 g^{\delta \sigma } g^{\mu \rho } \hat{g}^{\nu \tau }+8 \hat{g}^{\delta \rho } g^{\mu \sigma } \hat{g}^{\nu \tau }-4 g^{\delta \rho } g^{\mu \sigma } \hat{g}^{\nu \tau }+2 i \bar{\epsilon }^{\delta \mu \sigma \tau } g^{\nu \rho }-4 g^{\delta \tau } \hat{g}^{\mu \sigma } g^{\nu \rho }-4 g^{\delta \sigma } \hat{g}^{\mu \tau } g^{\nu \rho }+2 g^{\delta \tau } g^{\mu \sigma } g^{\nu \rho }+2 g^{\delta \sigma } g^{\mu \tau } g^{\nu \rho }-2 i \bar{\epsilon }^{\delta \mu \rho \tau } g^{\nu \sigma }-8 \hat{g}^{\delta \rho } \hat{g}^{\mu \tau } g^{\nu \sigma }+4 g^{\delta \rho } \hat{g}^{\mu \tau } g^{\nu \sigma }-2 g^{\delta \tau } g^{\mu \rho } g^{\nu \sigma }+4 \hat{g}^{\delta \rho } g^{\mu \tau } g^{\nu \sigma }-2 g^{\delta \rho } g^{\mu \tau } g^{\nu \sigma }+2 i \bar{\epsilon }^{\delta \mu \rho \sigma } g^{\nu \tau }+8 \hat{g}^{\delta \rho } \hat{g}^{\mu \sigma } g^{\nu \tau }-4 g^{\delta \rho } \hat{g}^{\mu \sigma } g^{\nu \tau }-2 g^{\delta \sigma } g^{\mu \rho } g^{\nu \tau }-4 \hat{g}^{\delta \rho } g^{\mu \sigma } g^{\nu \tau }+2 g^{\delta \rho } g^{\mu \sigma } g^{\nu \tau }-4 i \bar{\epsilon }^{\delta \mu \nu \tau } \hat{g}^{\rho \sigma }-16 \hat{g}^{\delta \nu } \hat{g}^{\mu \tau } \hat{g}^{\rho \sigma }+8 g^{\delta \nu } \hat{g}^{\mu \tau } \hat{g}^{\rho \sigma }-4 g^{\delta \tau } g^{\mu \nu } \hat{g}^{\rho \sigma }+8 \hat{g}^{\delta \nu } g^{\mu \tau } \hat{g}^{\rho \sigma }-4 g^{\delta \nu } g^{\mu \tau } \hat{g}^{\rho \sigma }-8 g^{\delta \mu } \hat{g}^{\nu \tau } \hat{g}^{\rho \sigma }+4 g^{\delta \mu } g^{\nu \tau } \hat{g}^{\rho \sigma }+4 i \bar{\epsilon }^{\delta \mu \nu \sigma } \hat{g}^{\rho \tau }+16 \hat{g}^{\delta \nu } \hat{g}^{\mu \sigma } \hat{g}^{\rho \tau }-8 g^{\delta \nu } \hat{g}^{\mu \sigma } \hat{g}^{\rho \tau }-4 g^{\delta \sigma } g^{\mu \nu } \hat{g}^{\rho \tau }-8 \hat{g}^{\delta \nu } g^{\mu \sigma } \hat{g}^{\rho \tau }+4 g^{\delta \nu } g^{\mu \sigma } \hat{g}^{\rho \tau }+8 g^{\delta \mu } \hat{g}^{\nu \sigma } \hat{g}^{\rho \tau }-4 g^{\delta \mu } g^{\nu \sigma } \hat{g}^{\rho \tau }+2 i \bar{\epsilon }^{\delta \mu \nu \tau } g^{\rho \sigma }+8 \hat{g}^{\delta \nu } \hat{g}^{\mu \tau } g^{\rho \sigma }-4 g^{\delta \nu } \hat{g}^{\mu \tau } g^{\rho \sigma }+2 g^{\delta \tau } g^{\mu \nu } g^{\rho \sigma }-4 \hat{g}^{\delta \nu } g^{\mu \tau } g^{\rho \sigma }+2 g^{\delta \nu } g^{\mu \tau } g^{\rho \sigma }+4 g^{\delta \mu } \hat{g}^{\nu \tau } g^{\rho \sigma }-2 g^{\delta \mu } g^{\nu \tau } g^{\rho \sigma }-2 i \bar{\epsilon }^{\delta \mu \nu \sigma } g^{\rho \tau }-8 \hat{g}^{\delta \nu } \hat{g}^{\mu \sigma } g^{\rho \tau }+4 g^{\delta \nu } \hat{g}^{\mu \sigma } g^{\rho \tau }+2 g^{\delta \sigma } g^{\mu \nu } g^{\rho \tau }+4 \hat{g}^{\delta \nu } g^{\mu \sigma } g^{\rho \tau }-2 g^{\delta \nu } g^{\mu \sigma } g^{\rho \tau }-4 g^{\delta \mu } \hat{g}^{\nu \sigma } g^{\rho \tau }+2 g^{\delta \mu } g^{\nu \sigma } g^{\rho \tau }+2 i \bar{\epsilon }^{\delta \mu \nu \rho } g^{\sigma \tau }+4 \hat{g}^{\delta \rho } g^{\mu \nu } g^{\sigma \tau }-2 g^{\delta \rho } g^{\mu \nu } g^{\sigma \tau }-4 \hat{g}^{\delta \nu } g^{\mu \rho } g^{\sigma \tau }+2 g^{\delta \nu } g^{\mu \rho } g^{\sigma \tau }-2 g^{\delta \mu } g^{\nu \rho } g^{\sigma \tau }

Keep in mind that the BMHV scheme violates axial Ward identities and requires special model-dependent counter-terms to restore those. Therefore, just setting FCSetDiracGammaScheme[“BMHV”] does not automatically resolve all your troubles with γ5\gamma^5 in DD-dimensions. The proper treatment of γ5\gamma^5 in dimensional regularization is an intricate issue that cannot be boiled down to a simple and universal recipe. FeynCalc merely carries out the algebraic operations that you request, but it is still your task to ensure that what you do makes sense.

Traces that are free of γ5\gamma^5 but contain both 44- and DD-dimensional Dirac matrices may appear in calculations that use the BMHV prescription, but they do not make sense in NDR. Therefore, their evaluation will be successful only if the correct scheme is used.

FCSetDiracGammaScheme["NDR"];
DiracTrace[(-GSD[q] + SMP["m_e"]) . GA[\[Nu]] . (GS[p] - GSD[q] + SMP["m_e"]) . GA[\[Mu]]] 
 
DiracSimplify[%]

tr((meγq).γˉν.(γˉp+meγq).γˉμ)\text{tr}\left(\left(m_e-\gamma \cdot q\right).\bar{\gamma }^{\nu }.\left(\bar{\gamma }\cdot \overline{p}+m_e-\gamma \cdot q\right).\bar{\gamma }^{\mu }\right)

1nywe2zsmni95

$Aborted\text{\$Aborted}

FCSetDiracGammaScheme["BMHV"];
ex = DiracSimplify[DiracTrace[(-GSD[q] + SMP["m_e"]) . GA[\[Nu]] . (GS[p] - GSD[q] + SMP["m_e"]) . GA[\[Mu]]] ]

4me2gˉμν+4gˉμν(pq)4q2gˉμν4pνqμ4pμqν+8qμqν4 m_e^2 \bar{g}^{\mu \nu }+4 \bar{g}^{\mu \nu } \left(\overline{p}\cdot \overline{q}\right)-4 q^2 \bar{g}^{\mu \nu }-4 \overline{p}^{\nu } \overline{q}^{\mu }-4 \overline{p}^{\mu } \overline{q}^{\nu }+8 \overline{q}^{\mu } \overline{q}^{\nu }

ex // FCE // StandardForm

(*-4 FV[p, \[Nu]] FV[q, \[Mu]] - 4 FV[p, \[Mu]] FV[q, \[Nu]] + 8 FV[q, \[Mu]] FV[q, \[Nu]] + 4 MT[\[Mu], \[Nu]] SMP["m_e"]^2 + 4 MT[\[Mu], \[Nu]] SP[p, q] - 4 MT[\[Mu], \[Nu]] SPD[q, q]*)
FCSetDiracGammaScheme["NDR"];

Notice that in this case the result contains 44- and DD-dimensional tensors.

Traces involving γ5\gamma^5 in the BMHV scheme are evaluated using West’s formula. It is possible to turn it off by setting the option West to False, but then the evaluation will require much more time.

FCSetDiracGammaScheme["BMHV"]; 
 
AbsoluteTiming[r1 = DiracSimplify[DiracTrace[GAD[\[Mu], \[Nu], \[Rho]] . GA[5] . GAD[\[Sigma], \[Delta], \[Tau]] . GA[7]]];]

{0.252561,Null}\{0.252561,\text{Null}\}

AbsoluteTiming[r2 = DiracSimplify[DiracTrace[GAD[\[Mu], \[Nu], \[Rho]] . GA[5] . GAD[\[Sigma], \[Delta], \[Tau]] . GA[7], 
      West -> False]];]

{2.20889,Null}\{2.20889,\text{Null}\}

r1 === r2

True\text{True}

FCSetDiracGammaScheme["NDR"]; 
 
ClearAll[r1, r2]

If you know that traces with one γ5\gamma^5 do not contribute to your final result, use the new NDR-Discard scheme to put them to zero

FCSetDiracGammaScheme["NDR-Discard"]; 
 
DiracSimplify[DiracTrace[GAD[\[Mu], \[Nu], \[Rho]] . GA[5] . GAD[\[Sigma], \[Delta], \[Tau]] . GA[7]]]

2gδτgμσgνρ+2gδσgμτgνρ2gδμgνρgστ2gδτgμρgνσ2gδρgμτgνσ2gδσgμρgντ+2gδρgμσgντ+2gδτgμνgρσ+2gδνgμτgρσ2gδμgντgρσ+2gδσgμνgρτ2gδνgμσgρτ+2gδμgνσgρτ2gδρgμνgστ+2gδνgμρgστ2 g^{\delta \tau } g^{\mu \sigma } g^{\nu \rho }+2 g^{\delta \sigma } g^{\mu \tau } g^{\nu \rho }-2 g^{\delta \mu } g^{\nu \rho } g^{\sigma \tau }-2 g^{\delta \tau } g^{\mu \rho } g^{\nu \sigma }-2 g^{\delta \rho } g^{\mu \tau } g^{\nu \sigma }-2 g^{\delta \sigma } g^{\mu \rho } g^{\nu \tau }+2 g^{\delta \rho } g^{\mu \sigma } g^{\nu \tau }+2 g^{\delta \tau } g^{\mu \nu } g^{\rho \sigma }+2 g^{\delta \nu } g^{\mu \tau } g^{\rho \sigma }-2 g^{\delta \mu } g^{\nu \tau } g^{\rho \sigma }+2 g^{\delta \sigma } g^{\mu \nu } g^{\rho \tau }-2 g^{\delta \nu } g^{\mu \sigma } g^{\rho \tau }+2 g^{\delta \mu } g^{\nu \sigma } g^{\rho \tau }-2 g^{\delta \rho } g^{\mu \nu } g^{\sigma \tau }+2 g^{\delta \nu } g^{\mu \rho } g^{\sigma \tau }

FCSetDiracGammaScheme["NDR"];

Sorting of the matrices inside 44-dimensional traces helps to avoid some spurious terms.

DiracTrace[GA[\[Mu], \[Nu], 5, \[Rho], \[Sigma], \[Tau], \[Kappa]], DiracTraceEvaluate -> True] - 
   DiracTrace[GA[\[Mu], \[Nu], \[Rho], \[Sigma], \[Tau], \[Kappa], 5],DiracTraceEvaluate -> True] // Expand

00

When the sorting is turned off via Sort to True, one may obtain some spurious terms that vanish by Schouten’s identity.

DiracTrace[GA[\[Mu], \[Nu], 5, \[Rho], \[Sigma], \[Tau], \[Kappa]], DiracTraceEvaluate -> True, Sort -> False] - 
   DiracTrace[GA[\[Mu], \[Nu], \[Rho], \[Sigma], \[Tau], \[Kappa], 5],DiracTraceEvaluate -> True, Sort -> False] // Expand

4igˉκμϵˉνρστ4igˉκνϵˉμρστ4igˉκσϵˉμνρτ+4igˉκτϵˉμνρσ+4igˉμρϵˉκνστ4igˉνρϵˉκμστ+4igˉρσϵˉκμντ4igˉρτϵˉκμνσ4 i \bar{g}^{\kappa \mu } \bar{\epsilon }^{\nu \rho \sigma \tau }-4 i \bar{g}^{\kappa \nu } \bar{\epsilon }^{\mu \rho \sigma \tau }-4 i \bar{g}^{\kappa \sigma } \bar{\epsilon }^{\mu \nu \rho \tau }+4 i \bar{g}^{\kappa \tau } \bar{\epsilon }^{\mu \nu \rho \sigma }+4 i \bar{g}^{\mu \rho } \bar{\epsilon }^{\kappa \nu \sigma \tau }-4 i \bar{g}^{\nu \rho } \bar{\epsilon }^{\kappa \mu \sigma \tau }+4 i \bar{g}^{\rho \sigma } \bar{\epsilon }^{\kappa \mu \nu \tau }-4 i \bar{g}^{\rho \tau } \bar{\epsilon }^{\kappa \mu \nu \sigma }

The trace of the unit matrix in the Dirac space is fixed to 4, which is the standard choice in dimensional regularization.

DiracTrace[1] 
 
DiracSimplify[%]

tr(1)\text{tr}(1)

44

If, for some reason, this value must be modified, one can do so using the option TraceOfOne.

DiracTrace[1, TraceOfOne -> D, DiracTraceEvaluate -> True]

DD

DiracSimplify[DiracTrace[GAD[\[Mu], \[Nu]], TraceOfOne -> D]]

DgμνD g^{\mu \nu }

Since FeynCalc 9.3 it is also possible to compute traces of Dirac matrices with Cartesian or temporal indices. However, the support of nonrelativistic calculations is a very new feature, so that things may not work as smooth as they do for manifestly Lorentz covariant expressions.

DiracTrace[CGAD[i, j, k, l]] 
 
DiracSimplify[%]

tr(γi.γj.γk.γl)\text{tr}\left(\gamma ^i.\gamma ^j.\gamma ^k.\gamma ^l\right)

4δilδjk4δikδjl+4δijδkl4 \delta ^{il} \delta ^{jk}-4 \delta ^{ik} \delta ^{jl}+4 \delta ^{ij} \delta ^{kl}

DiracTrace[CGA[i, j, k, l] . GA[6] . CGA[m, n]] 
 
DiracSimplify[%]

tr(γi.γj.γk.γl.γˉ6.γm.γn)\text{tr}\left(\overline{\gamma }^i.\overline{\gamma }^j.\overline{\gamma }^k.\overline{\gamma }^l.\bar{\gamma }^6.\overline{\gamma }^m.\overline{\gamma }^n\right)

2δˉinδˉjmδˉkl+2δˉimδˉjnδˉkl2δˉijδˉklδˉmn+2δˉinδˉjlδˉkm2δˉilδˉjnδˉkm2δˉimδˉjlδˉkn+2δˉilδˉjmδˉkn2δˉinδˉjkδˉlm+2δˉikδˉjnδˉlm2δˉijδˉknδˉlm+2δˉimδˉjkδˉln2δˉikδˉjmδˉln+2δˉijδˉkmδˉln2δˉilδˉjkδˉmn+2δˉikδˉjlδˉmn-2 \bar{\delta }^{in} \bar{\delta }^{jm} \bar{\delta }^{kl}+2 \bar{\delta }^{im} \bar{\delta }^{jn} \bar{\delta }^{kl}-2 \bar{\delta }^{ij} \bar{\delta }^{kl} \bar{\delta }^{mn}+2 \bar{\delta }^{in} \bar{\delta }^{jl} \bar{\delta }^{km}-2 \bar{\delta }^{il} \bar{\delta }^{jn} \bar{\delta }^{km}-2 \bar{\delta }^{im} \bar{\delta }^{jl} \bar{\delta }^{kn}+2 \bar{\delta }^{il} \bar{\delta }^{jm} \bar{\delta }^{kn}-2 \bar{\delta }^{in} \bar{\delta }^{jk} \bar{\delta }^{lm}+2 \bar{\delta }^{ik} \bar{\delta }^{jn} \bar{\delta }^{lm}-2 \bar{\delta }^{ij} \bar{\delta }^{kn} \bar{\delta }^{lm}+2 \bar{\delta }^{im} \bar{\delta }^{jk} \bar{\delta }^{ln}-2 \bar{\delta }^{ik} \bar{\delta }^{jm} \bar{\delta }^{ln}+2 \bar{\delta }^{ij} \bar{\delta }^{km} \bar{\delta }^{ln}-2 \bar{\delta }^{il} \bar{\delta }^{jk} \bar{\delta }^{mn}+2 \bar{\delta }^{ik} \bar{\delta }^{jl} \bar{\delta }^{mn}