Name: Maksym (email_not_shown)
Date: 07/31/17-12:24:25 AM Z

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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