Issue with DiracSimplify when two 4 dimensional epsilon tensors are contracted with a D dimensional trace.

[fabianwunder]

• Your Mathematica version

  13.1.0 for Linux x86 (64-bit) (June 16, 2022)

• Your FeynCalc version

  9.3.1

• Did you try to reinstall FeynCalc (stable version) using the automatic installer to make sure that you have the latest bugfixes?

  No

• Does your Mathematica initialization file contain statements that might influence the behavior of FeynCalc? Sometimes external packages may modify `init.m` in unusual ways, causing troubles for other codes.

  No

• Please provide a minimal working example that illustrates the problem and works on a fresh kernel. The example should be provided either by writing the code (as `InputForm`!) directly in the issue text or by attaching a Mathematica notebook. Please do not post code samples as screenshots, PDF files etc.: Those essentially require us to retype everything by hand, which is annoying and also time consuming. Please explain the difference between the current behavior and the expected behavior.

<< FeynCalc` term = LC[[Mu], [Nu]][p1, p2] LC[[Rho], [Sigma]][p1, p2] DiracTrace[ FVD[p3, [Sigma]]*GAD[[Mu], [Nu], [Rho]] . GSD[p3]]; FCSetDiracGammaScheme["BMHV"]; termdiracsimplify = term // DiracSimplify; termcontractfirst = term // Contract // DiracSimplify; termdiracsimplify - termcontractfirst // Simplify

The last line should evaluate to zero because contracting and taking the trace should be interchangeable. However FeynCalc gives the result 8(SP[p1, p2]^2 - SP[p1, p1]SP[p2, p2])* (SP[p3, p3] - SPD[p3, p3]).

The error appears in termdiracsimplify. Here, the index of FVD[p3,[Sigma]] is NOT put to four dimensions by the other (four dimensional) [Sigma] and incorrectly a SPD[p3] is produced instead of SP[p3]. This behavior only appears if the external tensors are Epsilon-Tensors and if 4 and D dimensional objects are involved. Interestingly, there is no issue for

termdiracsimplify2 = term // DiracSimplify[#,DiracTraceEvaluate->False]&//DiracSimplify;

It seems as if the error only appears because of the specific order of simplifications within DiracSimplify.

Also there is no problem for termdiracsimplify3 = term/. FVD[p3, [Sigma]]-> FV[p3, [Sigma]] +FVE[p3, [Sigma]] // DiracSimplify,

so the problem seems to be specific to the combination of 4 and D dimensions.

A potential way to consistently avoid the problem should be to have a function that recognizes repeated indices with different dimensionality and makes replacements accordingly e.g. something like LorentzIndex[a,D].....LorentzIndex[a]->LorentzIndex[a].....LorentzIndex[a].

[vsht]

Thanks for the bug report. The property that any 4-dim index should make the whole Eps tensor equally 4-dim was already implemented but obviously not for this particular combination of arguments.

I pushed a fix into the master branch (commit 34062c9) that should take care of this. Could you please test the dev version and let me know whether there are still any relevant issues remaining? Then I could also push it to the stable version.

[fabianwunder]

Thank you very much for the fast bug fix. I tested the dev version, now it seems to work properly.

[vsht]

Thanks for the confirmation. Now the fix should be also in the stable version.