Name: Vladyslav Shtabovenko Date: 08/21/17-04:03:31 AM Z

Hi,

I think that the problem comes from the wrong summation over the
polarizations of the gauge fields.

You do know, how one correctly sums over the polarizations of a massless
vector boson including the choice of an auxiliary vector (c.f. e.g. the
appendix A.1.1.6 of Boehm Denner and Joos, “Gauge Theories of the Strong
and Electroweak Interaction”)

The replacement eps^mu eps^*nu -> - g^{mu nu} which corresponds to
DoPolarizationSums[exp, k, 0] is valid only in QED, where the
Ward-Identities ensure that cancellation of the unphysical degrees of
freedom. In QCD this replacement alone already does not work, unless one
adds extra diagrams with external ghost-fields.

Since you added unphysical longitudinal on-shell degrees of freedom to
you matrix element squared, no wonder that the unitarity is violated.

How about this:

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

Cheers,
Vladyslav

Am 21.08.2017 um 03:41 schrieb Maksym:
> 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.
>