Name: Vladyslav Shtabovenko Date: 07/28/15-05:00:25 PM Z

Hi Nikita,

I would be willing to check everything myself, if this would be a purely
FeynCalc-related issue, but with FeynCalcFormLink you are actually using
FeynCalc as a frontend for FORM. What I mean is that the trace
calculation is done completely in FORM, while FeynCalcFormLink is just
converting the input/output between FeynCalc and FORM notations.

Since FORM is known to be very reliable on the Dirac algebra, I would be
very surprised if there is indeed a bug there.

The attached file is the code for your calculation in FORM. You can run
it via

form trace.frm

and then there is no FeynCalc involved here anywhere. In the result
there is no explicit I, because in FORM

Tr[GA[i,j,k,l].GA[5]] = 4 Eps[i,j,k,l]

So every term with an explicit e_ is imaginary.

You can also modify this code to check other traces that you computed by
hand. As I said, finding a Dirac trace related bug in FORM would be
highly improbable.

On the other hand, there are some parts of your expressions that make we
wonder. For example, some fermionic propagators come with a plus sign,
like (GS[p1]+m). Is this intended? Also (1+GA[5].GS[s]) looks a bit
weird to me, since normally I would expect to see rather
(1+GA[5]).GS[s]. If this is all correct this way, then nevermind.

Cheers,
Vladyslav

Am 28.07.2015 um 15:49 schrieb Nikita Belyaev:
> Hi Vladyslav ,
>
>> 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]
>>
>> Schouten[%]
>>
>> then it is clear that they do not vanish by the Schouten identity.
>
> And this is exactly the thing we are worrying about. We’ve checked by hand that expressions with u^3 are cancelled out and there is totally no imaginary part (you also can check it by hand if you want, it isn’t long calculation).
>
> Moreover we can provide you additional example:
>
> Line29:= (GS[p]-m).GA[\[Beta]1].(GS[p]+GS[p1]+GS[p2]-m).GA[\[Alpha]1].GS[k2].GA[\[Alpha]].(1-GA[5]);
> Line30:= GA[\[Beta]].(GS[p2]-m).GA[\[Beta]1].(GS[p1]+m);
> Line31:= GS[k1].GA[\[Alpha]1].(GS[q]-u).(1+GA[5].GS[s]).GA[\[Beta]].(GS[q]-GS[p1]-GS[p2]-u).GA[\[Alpha]].(1-GA[5]);
> Line32:= (GS[p]+GS[p1]+GS[p2]-m).GA[\[Beta]].(GS[p]-m).GA[\[Alpha]1].GS[k2].GA[\[Alpha]].(1-GA[5]);
> Line33:= GA[\[Beta]].(GS[p2]-m).GA[\[Beta]1].(GS[p1]+m);
> Line34:= GS[k1].GA[\[Alpha]1].(GS[q]-GS[p1]-GS[p2]-u).GA[\[Beta]1].(GS[q]-u).(1+GA[5].GS[s]).GA[\[Alpha]].(1-GA[5]);
>
> We’ve calculated this term by hand and it also contains no imaginary part while FeynCalc give us the same set of it in the output as in the previous case.
>
> Best Regards,
> Nikita Belyaev
>