Name: Vladyslav Shtabovenko (email_not_shown)
Date: 07/27/15-12:22:47 AM Z

Hi Nikita,

actually I have some doubts that your expression should have no
imaginary part. At least, if I look at the pieces of the imaginary part
that are proportional to u^3:

SelectNotFree[FCE[TrA2B2], Complex];
u3Piece = SelectNotFree[%, u^3] // EpsEvaluate

-512 I u^3 SP[p1, p2] LC[][k1, k2, p, s] +
 512 I u^3 SP[p, p2] LC[][k1, k2, p1, s] -
 512 I u^3 SP[p, p1] LC[][k1, k2, p2, s] +
 512 I u^3 SP[k1, p2] LC[][k2, p, p1, s] -
 512 I u^3 SP[k1, p] LC[][k2, p1, p2, s]


then it is clear that they do not vanish by the Schouten identity.
However since these are the only terms that contain u^3, I don't think
that they can be cancelled by other terms in the imaginary part.


Am 25.07.2015 um 18:39 schrieb Nikita Belyaev:
> Hello, Vladyslav
> We've faced with a problem that we couldn't solve. It is related to imaginary parts of the traces. We have some of them to calculate and a lot of them were already calculated correctly but for one specific term we cannot get the result without imaginary part.
> We've spent a lot of time using a FeynCalcFormLink procedure and also trying to apply the Scouten identity as you told us some time ago but we did not succeed because as I understand there is no common algorithm to check such kind of equivalence in the general case.
> Could you probably help us to check that the trace included contains no imaginary part?
> Here is the code of our calculation:
> Clear["Global`*"];
> <<HighEnergyPhysics`FeynCalc`
> Needs["FeynCalcFormLink`"]
> $LeviCivitaSign = -1;
> ScalarProduct[p,p]=m^2;
> ScalarProduct[p1,p1]=m^2;
> ScalarProduct[p2,p2]=m^2;
> ScalarProduct[k1,k1]=0;
> ScalarProduct[k2,k2]=0;
> ScalarProduct[q,q] = u^2;
> ScalarProduct[q,s] =0;
> Line13:= (GS[p2]-m).GA[\[Beta]1].(GS[p1]+m).GA[\[Beta]].(GS[p]-m).GA[\[Alpha]1].GS[k2].GA[\[Alpha]].(1-GA[5]);
> Line14:= GS[k1].GA[\[Alpha]1].(GS[q]-GS[p1]-GS[p2]-u).GA[\[Beta]1].(GS[q]-u).(1+GA[5].GS[s]).GA[\[Beta]].(GS[q]-GS[p1]-GS[p]-u).GA[\[Alpha]].(1-GA[5]);
> Line15:= (GS[p]-m).GA[\[Beta]1].(GS[p1]+m).GA[\[Beta]].(GS[p2]-m).GA[\[Alpha]1].GS[k2].GA[\[Alpha]].(1-GA[5]);
> Line16:= GS[k1].GA[\[Alpha]1].(GS[q]-GS[p1]-GS[p]-u).GA[\[Beta]1].(GS[q]-u).(1+GA[5].GS[s]).GA[\[Beta]].(GS[q]-GS[p1]-GS[p2]-u).GA[\[Alpha]].(1-GA[5]);
> Tr13= DiracTrace[Line13];
> Tr14= DiracTrace[Line14];
> Tr15= DiracTrace[Line15];
> Tr16= DiracTrace[Line16];
> TrA2B2=FeynCalcFormLink[Tr13.Tr14+Tr15.Tr16];
> ComplexExpand[FullSimplify[TrA2B2/1024]] //Schouten
> Thanks for your help!
> With the Best Regards,
> Nikita Belyaev

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