First of all, when you square the amplitude, you do not rename the dummy
indices. Thus, the resulting expressions violate the Einstein convention
and hence make no sense.
If you look at the example shipped with FeynCalc (e.g. QED and QCD tree
level calculations), you will see that all the calculations of the
matrix element squared always contain FCRenameDummyIndices in the right
place.
Second, 3 vs 7 terms looks very much like something related to the
Schouten identity. C.f. the discussions in
http://www.feyncalc.org/forum/0956.html
and
http://www.feyncalc.org/forum/0967.html
Cheers,
Vladyslav
Am 31.07.2017 um 00:24 schrieb Maksym:
> Sorry, this is my inattention, I've just missed the dot between the gamma matrices.
>
> I would be grateful if you'll help me with another problem.
>
> I have two matrix elements f1, f2:
>
> FCClearScalarProducts[];
> {ScalarProduct[p1, p1] =
> ScalarProduct[p2, p2] =
> ScalarProduct[p3, p3] =
> ScalarProduct[k1, k1] = ScalarProduct[k2, k2] = 0};
>
> f1 = PolarizationVector[
> k2, \[Mu]] SpinorUBar[k1, 0].GA[\[Nu]].(1 - GA5).SpinorU[p1,
> 0] SpinorVBar[p2, m].GA[\[Mu]].GS[
> p2 - k2].GA[\[Nu]].(1 - GA5).SpinorU[p3,
> m]/(ScalarProduct[p2 - k2, p2 - k2] - m^2)
>
> f2 = PolarizationVector[
> k2, \[Mu]] SpinorUBar[k1, 0].GA[\[Nu]].(1 - GA5).SpinorU[p1,
> 0] SpinorVBar[p2, m].GA[\[Nu]].(1 - GA5).GS[
> p3 - k2].GA[\[Mu]].SpinorU[p3,
> m]/(ScalarProduct[p3 - k2, p3 - k2] - m^2)
>
> I need to calculate their interference term. If I write
>
> m12 = FermionSpinSum[f1 ComplexConjugate[f2] + f2 ComplexConjugate[f1]] /. DiracTrace -> TR // Contract // Simplify ,
>
> then the output contains only 3 inequivalent summands. This contradicts the known result (there must be 7 inequivalent terms).
>
> If, however, I directly evaluate the trace,
>
> Expand[(Limit[
> TR[ FermionSpinSum[
> SpinorVBar[p2, m].GA[\[Mu]].GS[
> p2 - k2].GA[\[Nu]].(1 - GA5).SpinorU[p3,
> m].ComplexConjugate[
> SpinorVBar[p2, m].GA[\[Nu]C].(1 - GA5).GS[
> p3 - k2].GA[\[Mu]].SpinorU[p3, m]]]] TR[
> FermionSpinSum[
> SpinorUBar[k1, 0].GA[\[Nu]].(1 - GA5).SpinorU[p1,
> 0].ComplexConjugate[
> SpinorUBar[k1, 0].GA[\[Nu]C].(1 - GA5).SpinorU[p1,
> 0]]]] // DiracTrick // Contract // Simplify, m -> 0] +
> ComplexConjugate[
> Limit[TR[
> FermionSpinSum[
> SpinorVBar[p2, m].GA[\[Mu]].GS[
> p2 - k2].GA[\[Nu]].(1 - GA5).SpinorU[p3,
> m].ComplexConjugate[
> SpinorVBar[p2, m].GA[\[Nu]C].(1 - GA5).GS[
> p3 - k2].GA[\[Mu]].SpinorU[p3, m]]]] TR[
> FermionSpinSum[
> SpinorUBar[k1, 0].GA[\[Nu]].(1 - GA5).SpinorU[p1,
> 0].ComplexConjugate[
> SpinorUBar[k1, 0].GA[\[Nu]C].(1 - GA5).SpinorU[p1,
> 0]]]] // DiracTrick // Contract // Simplify,
> m -> 0]]) // Simplify]
>
> then the output contains 7 inequivalent terms with correct factors. What is the reason for this?
>
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