Spin-3/2 particle propagator

I'll try to find solution of this puzzle during this week. Now CalCHEP
produces positive width for spin 3/2 particle. If I change sign of
propagator and matrix density, width becomes negative. In general sing
of propagator is defined by sign of Lagrangian. Here there is no
ambiguity because depending on the sign one gets canonical
commutation relations X^2=1 or X^2=-1. The last one can not be
realized in Hilbert space with positive metric.

Best
    Alexander Pukhov

On 11/06/2016 12:57 AM, Nurhan wrote:
> New question #403820 on CalcHEP:
> https://answers.launchpad.net/calchep/+question/403820
>
> Hi Dr. Pukhov,
>
> We use Calchep for a new model including spin-3/2 particle. In the paper https://arxiv.org/pdf/1308.1668v1.pdf, it is written that the propagator used in Calchep is Eq. (98). So, we did our manual calculations by using this propagator. However, in some papers in literature, the sign of the propagator is taken (+) and in some papers it is taken (-) as in Calchep.
>
> I read your answer to Ivan Sobolev (question title: negative cross-section in processes with goldstino). I am confused. In our model, spin-3/2 particle is considered as an exotic spinor. Now, our numerical and manual calculations with the negative sign propagator in Calchep is true? or not?
>
> Thanks
>

I have tested carefully spin 3/2 propagator and have to conclude that in
many (all?) papers
propagator of spin 3/2 particle and free Lagrangian have wrong sign.
For instance 1208.5811 Eq(2.6 2.7). Surely, sing of propagator is
related to sign of Lagrangian. As well as there are some
uncertainties caused by position of "i" in propagator.

But if for Dirac fermion we have
            (\not{p}+M)/(p^2-M^2)
then for spin 3.2 case one has to expect something like
            -(\not{p}+M)(g_{\mu\nu} .......)/(p^2-M^2)

Sign of propagator of fermion particle is defined by condition of
positiveness of canonical(equal-time) anti-commutation relations or
positiveness of Wightman functions.
To test it first of all we have to multiply propagator of \gamma_0 to
replace Dirac conjugation of complex one. Also we can simplify
testing of positiveness performing a sum over fermion states.
  In case of Dirac propagator it gives
   Tr( gamma_0 (\not{p} +M) ) =4 p_0
  In case of spin 3/2 propagator 1208.5811 Eq(2.6) the trace is
     4p_0 (g_{\mu\nu} -p_{\mu}p_{\nu}/M^2}
which is in general negative.

When we calculate squared matrix elements for simplest processes
using amplitude technique, amplitude is squared, summation over
outgoing states is done implicitly and result is correct. When we
calculate processes with photon or gluon radiation we get extra minus
because of propagator 3/2 and extra minus because of wrong sign in
Lagrangian. So, at first sight calculation of gauge boson radiation
should be correct as well. Because in 1208.5811 we have wrong sings
both for propagator and Lagrangian, it works. But in general case, for
instance, calculating interference of diagram with spin 3/2 and without
it we can get wrong result.

  CalcHEP uses squared diagram method. Density matrix for any particle
is calculated by the same formula as propagator. So, changing sing of
propagator
I immediately get negative width of spin 3/2 particle.

I am a little bit surprised why in many (all that I saw) papers spin 3/2
propagator and Lagrangian are presented with wrong sing. But look at my
simple arguments above.
Let me know, please, if you have some objections.

Best
     Alexander Pukhov.

PS: The models 1208.5811 is implemented in CalcHEP (really by KC).
Recently I have implemented and option to calculate processes with
massless vector particles using projection on physical
polarization(without ghosts). It gives a possibility to check gauge
invariance. I'll write you about it later.

Best
    Alexander Pukhov.

On 11/06/2016 12:57 AM, Nurhan wrote:
> New question #403820 on CalcHEP:
> https://answers.launchpad.net/calchep/+question/403820
>
> Hi Dr. Pukhov,
>
> We use Calchep for a new model including spin-3/2 particle. In the paper https://arxiv.org/pdf/1308.1668v1.pdf, it is written that the propagator used in Calchep is Eq. (98). So, we did our manual calculations by using this propagator. However, in some papers in literature, the sign of the propagator is taken (+) and in some papers it is taken (-) as in Calchep.
>
> I read your answer to Ivan Sobolev (question title: negative cross-section in processes with goldstino). I am confused. In our model, spin-3/2 particle is considered as an exotic spinor. Now, our numerical and manual calculations with the negative sign propagator in Calchep is true? or not?
>
> Thanks
>