Name: Maksym 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?