Name: Maksym (email_not_shown)
Date: 08/20/17-09:41:51 PM Z

Hi!

I also have the same problem for the toy case

f = SpinorUBar[p1,
mp].GA[\[Nu]].(1 - GA5).(DiracSlash[k1 + k2] +
mp).GA[\[Mu]].SpinorU[k1, mp] PolarizationVector[
k2, \[Mu]] PolarizationVector[p2, \[Nu]]

corresponding to the simplified process k1 + k2 ->p1+ p2, where k1 and p1 correspond to a particle with mass mp, while k2,p2 correspond to gauge-like fields (k2-particle is coupled through vector-like coupling, while p2-particle is coupled through axial-vector-like couplings). For simplicity, I've neglected the second diagram, which isn't relevant for the present discussion and just complifies it.

With the scalar products evaluated at p1+p2 center of mass frame,

{ScalarProduct[k1, k1] = mp^2, ScalarProduct[k2, k2] = 0,
ScalarProduct[p1, p1] = mp^2, ScalarProduct[p2, p2] = 0,
ScalarProduct[k1, k2] =
ScalarProduct[p1, p2] = (Sqrt[p^2 + mp^2] + p)^2/2 - mp^2/2,
ScalarProduct[k1, p2] =
ScalarProduct[k2, p1] = Sqrt[p^2 + mp^2]*p + p^2*Cos[\[Theta]],
ScalarProduct[k1, p1] = p^2 - mp^2 - p^2*Cos[\[Theta]],
ScalarProduct[k2, p2] = p^2 (1 - Cos[\[Theta]])};

and the code

fstar = ComplexConjugate[f] /. {\[Mu] -> \[Mu]C, \[Nu] -> \[Nu]C}
ms = FermionSpinSum[f fstar] /. DiracTrace -> TR // Contract //
Simplify ;
msquaredneutral =
DoPolarizationSums[DoPolarizationSums[ms, k2, 0], p2, 0] // Simplify
Plot[msquaredneutral /. {mp -> 1, p -> 2}, {\[Theta], 0, Pi}]

I obtain that the plot for the squared matrix element msquaredneutral is negative for values of Theta close to Pi. For other values of p it may even not pass zero, which is also incorrect.

However, if I replace DiracSlash[k1 + k2] + mp by DiracSlash[k1 + k2] - mp, then the plot is positive for all Theta's and always tends to zero at Theta -> Pi. This is very strange for me.

What is the problem with my code?

P.S. I've realized that the problem belonds only to the case of axial-vector coupling, i.e. when the 1-GA5 projector is present. If there is no this projector, then anything is ok.

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