bugs within the Larin Gamma5 scheme

[ygtw1]

• Your Mathematica version

  13.1

• Your FeynCalc version

  10.1

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

  Yes

• 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.

(input)

<< FeynCalc`; FCSetDiracGammaScheme["Larin"]; f[ex_] := {TR[ex] /. p2 -> p1, DiracSimplify[DiracTrace[ex]] /. p2 -> p1, TR[ex /. p2 -> p1], DiracSimplify[DiracTrace[ex /. p2 -> p1]]} // Contract // FullSimplify;

example1 = GSD[p1] . DiracSigma[GAD[bb], GAD[Lor1]] . GA[5] . GSD[p2] . GA[5] . DiracSigma[GAD[bb], GAD[Lor1]]; f@%

example2 = FCI[GSD[p1] . GA[5] . GAD[bb] . GA[5] . GSD[p2] . GA[5] . GAD[bb1] . GAD[bb2] . GAD[Lor1] LCD[bb, bb1, bb2, Lor1]]; f@%

example3 = FCI[GSD[p1] . GA[5] . GAD[bb] .(GA[5].)GSD[p2] .(GA[5].) GAD[bb1] . GAD[bb2] . GAD[Lor1] LCD[bb, bb1, bb2, Lor1]]; f@%

(output)

During evaluation of In[1]:= FeynCalc 10.1.0 (stable version). For help, use the online documentation, visit the forum and have a look at the supplied examples. The PDF-version of the manual can be downloaded here.

During evaluation of In[1]:= If you use FeynCalc in your research, please evaluate FeynCalcHowToCite[] to learn how to cite this software.

During evaluation of In[1]:= Please keep in mind that the proper academic attribution of our work is crucial to ensure the future development of this package!

Out[5]= {-(4/3) (D-4) (D-3) (D-2) (D-1) p1^2,-(4/3) (D-4) (D-3) (D-2) (D-1) p1^2,2/3 (D-5) (D-4) (D-3) (D-2) (D-1) p1^2,2/3 (D-5) (D-4) (D-3) (D-2) (D-1) p1^2}

Out[7]= {-(2/3) I (D-3)^2 (D-2) (D-1) (D ((D-23) D+80)-4) p1^2,-(2/3) I (D-3)^2 (D-2)^2 (D-1) ((D-21) D+74) p1^2,-(2/3) I (D-3)^2 (D-2)^2 (D-1) ((D-21) D+74) p1^2,-(2/3) I (D-3)^2 (D-2)^2 (D-1) ((D-21) D+74) p1^2}

Out[9]= {8 I (D-3) (D-2) (D-1) p1^2,8 I (D-3) (D-2) (D-1) p1^2,-4 I (D-6) (D-3) (D-2) (D-1) p1^2,-4 I (D-6) (D-3) (D-2) (D-1) p1^2}

[vsht]

Many thanks for the bug report. Indeed, the implemented formula was not correct, which resulted in the described issues.

Please check if it now works for you.

Concerning the example with an external Levi-Civita, this case is kind of tricky, so I added an extra note to the manual about it: https://github.com/FeynCalc/feyncalc/blob/master/FeynCalc/Documentation/Markdown/Extra/Gamma5.md

[ygtw1]

With the updated version of FeynCalc, example1 and example3 are working correctly, but it seems example2 still has issues. Here is my test.

(input)

FCSetDiracGammaScheme["Larin"]; f[ex_] := Contract[TR[ex] /. p2 -> p1] - Contract[TR[ex /. p2 -> p1]] // FullSimplify;

GSD[p1] . GA[5] . GAD[bb] . GA[5] . GSD[p2] . GA[5] . GAD[bb1] . GAD[bb2] . GAD[Lor1] // f; % // FCE // StandardForm %% LCD[bb, bb1, bb2, Lor1] // Contract // FullSimplify

(output)

-4 I (-3 + D) (5 LCD[bb, bb1, bb2, Lor1] SPD[p1, p1] + (-9 + D) (FVD[p1, Lor1] LCD[bb, bb1, bb2][p1] - FVD[p1, bb2] LCD[bb, bb1, Lor1][p1] + FVD[p1, bb1] LCD[bb, bb2, Lor1][p1]) + 5 FVD[p1, bb] LCD[bb1, bb2, Lor1][p1])

32 I (D-4) (D-3)^2 (D-2) (D-1) p1^2