Name: Vladyslav Shtabovenko Date: 08/05/14-10:31:20 AM Z

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Hi Nikita,

the thing is that to compute traces of gamma matrices,
the current version of FeynCalc uses West’s formula

<<http://www.sciencedirect.com/science/article/pii/001046559390011Z>>

This formula was originally derived to treat gamma^5 in dimensional
regularization (using the t’Hooft-Veltman scheme) but it also has the
nice property that in 4 dimensions the results are equivalent to what
you get when using the usual trace of gamma matrices.

Unfortunately, it is not always easy to see that equivalence. Schouten’s
identities implemented in FeynCacl are helpful (using Schouten[expr_]),
but not when there are too many Lorentz indices.

In this sense, the results you get are correct, but they are written
in a somewhat different way as compared to the usual trace.

You can use FeynCalcFormLink to outsource the trace computations to FORM
which should give you the result in the form you want.

For example,

(*run this only once!*)
Import[“http://www.feyncalc.org/formlink/install.m”]
(*——————-*)

Needs[“FeynCalcFormLink`”]
$LeviCivitaSign = -1;
Tr1a = DiracTrace[
    GS[P1].GS[P2].GS[P3].GS[P4].GS[P5].GA[i].(1

• GA[5])];
  Tr2a = DiracTrace[
      GS[Q1].GS[Q2].GS[Q3].GS[Q4].GS[Q5].GA[i].(1
• GA[5])];
  Tr3a = DiracTrace[
      GS[P1].GS[P2].GS[P3].GS[P4].GS[P5].GA[i].GS[Q1].GS[Q2].GS[Q3].GS[
        Q4].GS[Q5].GA[i].(1 - GA[5])];
  FeynCalcFormLink[Tr1a.Tr2a + 2 Tr3a]

gives you zero right away.

Cheers,
Vladyslav

On 04/08/14 11:01, Nikita Belyaev wrote:
> Hi Vladislav,
>
> Thanks for the response, in this particular case it works.
> But if I add two gamma matrixes to the formula
>
*> Tr1a = Tr[GS[P1].GS[P2].GS[P3].GS[P4].GS[P5].GA[i].(1

• GA[5])];*
  *> Tr1a = Tr[GS[P1].GS[P2].GS[P3].GS[P4].GS[P5].GA[i].(1
• GA[5])];*
  *> Tr3a = Tr[GS[P1].GS[P2].GS[P3].GS[P4].GS[P5].GA[i].GS[Q1].GS[Q2].GS[Q3].GS[Q4].GS[Q5].GA[i].(1
• GA[5])];*
  >
  > Result = Simplify[Contract[Tr1a.Tr2a + 2 Tr3a]] // Schouten
  >
  > the result is non zero again.
  >
  > The source of my concern is the following matrix element
  >
  > Tr1e = Tr[GA[i].(GS[p2] - m).GA[k].(GS[p1] + m)];
  > Tr2e = Tr[(GS[p] - m).GA[k].(GS[p] + GS[p1] + GS[p2] -
  > m).GA[l].GS[k2].GA[m].(1 - GA[5])];
  > Tr3e = Tr[
  > GS[k1].GA[l].(GS[q] - u).(1 + GA[5].GS[s]).GA[i].(GS[q] -
  > GS[p1] - GS[p2] - u).GA[m].(1 - GA[5])];
  > Tr4e = Tr[(GS[p] + GS[p1] + GS[p2] - m).GA[i].(GS[p] -
  > m).GA[l].GS[k2].GA[m].(1 - GA[5])];
  > Tr5e = Tr[
  > GS[k1].GA[l].(GS[q] - GS[p1] - GS[p2] - u).GA[k].(GS[q] -
  > u).(1 + GA[5].GS[s]).GA[m].(1 - GA[5])];
  > TrA1A2 = Simplify[Contract[Tr1e.(Tr2e.Tr3e + Tr4e.Tr5e)]] // Schouten
  >
  > which again contains imaginary part while it should be real.
  >
  > Is there some rules I have to follow when calculating terms with a lot of gamma matrixes to get a correct result?
  >
  > Best Regards,
  > Nikita Belyaev
  >

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