Axial/Vectorial coupling to the Z

After having done some math expension,
I guess that the first one is the vectorial and the second is axial,
but a confirmation would be appreciated.

Thanks,
Loic

Hi Loic,

I don't agree with you:
If I look at the interactions definition:
mg5> display interactions z ta+ ta-
Interactions 34 : ta+ ta- z has the following property:
{
    'id': 42,
    'particles': [-15,15,23],
    'color': [1 ],
    'lorentz': ['FFV2', 'FFV4'],
    'couplings': {(0, 1): 'GC_59', (0, 0): 'GC_50'},
    'orders': {'QED': 1},
    'loop_particles': None,
    'type': 'base',
    'perturbation_type': None
}

So this means that the interaction is
GC_50 * FFV2 + GC_59 * FFV4

And this is the definition of FFV2/FFV4:

MG5>display lorentz FFV2
structure : Gamma(3,2,-1)*ProjM(-1,1)
name : FFV2
spins : [2, 2, 3]
MG5>display lorentz FFV4
structure : Gamma(3,2,-1)*ProjM(-1,1) + 2*Gamma(3,2,-1)*ProjP(-1,1)
name : FFV4
spins : [2, 2, 3]
MG5>

So little algebra means that your interaction is for the Gamma_mu part:
1/2 (GC_50+ 3* GC_59)
and the Gamma_mu*Gamma_5 part:
1/2 (GC_59 - GC_50)

So in order to remove the axial contribution you have to put GC_59 equal to GC_50.

I would STRONGLY suggest that you do instead the following:

An easier way would be to modify the UFO MODEL:
1) cd models
2) cp -r sm sm_MOD
3) edit the tau+ tau- z interactions (vertices.py) from

V_109 = Vertex(name = 'V_109',
               particles = [ P.ta__plus__, P.ta__minus__, P.Z ],
               color = [ '1' ],
               lorentz = [ L.FFV2, L.FFV4 ],
               couplings = {(0,0):C.GC_50,(0,1):C.GC_59})
to
V_109 = Vertex(name = 'V_109',
               particles = [ P.ta__plus__, P.ta__minus__, P.Z ],
               color = [ '1' ],
               lorentz = [ L.FFV1 ],
               couplings = {(0,0):C.GC_50})

4) change the definition of GC_50 (in couplings.py) if this is important for you.

This is MUCH nicer!!! (and as fast)

Cheers,

Olivier

Salut Olivier,

Can you detail how you get to:
"
So little algebra means that your interaction is for the Gamma_mu part:
 1/2 (GC_50+ 3* GC_59)
 and the Gamma_mu*Gamma_5 part:
 1/2 (GC_59 - GC_50)
"
I am not completely sure of what Gamma(3,2,-1), ProjM(-1,1) and ProjP(-1,1) means.

Thanks for your proposal of implementation,
can you tell me why I should replace " lorentz = [ L.FFV2, L.FFV4 ]," by "lorentz = [ L.FFV1 ]," in the vertex definition?
That's probably related to the first question.
About 4), what should be my coupling ? I want to have the vectorial component of SM. Should it be 1/2(GC50 + 3*GC59) ?
That seems weird...

Thanks for help,
Loic

Hi Loic,

Gamma is the gamma matrices
Gamma(3,2,-1) is the gamma matrices with lorentz index the symbol 3 (which correspond to the third particle of the interactions
and with spin indices 2 and -1 (2 is the index contracted with the second particles of the interactions and -1 a indices contracted with an other part of the expression). In short you don't care that much about those [but if you write your own expression]

ProjM is 1/2*(1-gamma_5)
ProjM is 1/2*(1+gamma_5)

so for tau+ tau- z the expression is
GC_50 Gamma^mu (1- gamma_5)/2 + GC_59 (Gamma^mu (1- gamma_5)/2 + Gamma^mu (1+ gamma_5)
so this is equal to
Gamma^mu (GC_50/2 + GC_59 /2 + GC_59) + Gamma^mu gamma_5 ( -1/2 GC_50 - 1/2 GC_59 +GC_59)

>can you tell me why I should replace " lorentz = [ L.FFV2, L.FFV4 ]," by "lorentz = [ L.FFV1 ]," in the

if you look at the definition of FFV1, this is Gamma(3,2,1) so exactly what you want (if I understand you correctly)

>About 4), what should be my coupling ? I want to have the vectorial component of SM. Should it be 1/2(GC50 + 3*GC59) ?
>That seems weird...

If you want the SM vectorial coupling of the z to tau+ tau-, yes this is correct. Note that I have NO idea how much such value is consistent on the theoretical side. (gauge invariance/ lorentz invariance/unitary bound/re-normalizable/...) Note that if you break gauge invariance, MG will NOT produce lorentz invariance result.
If you modify the UFO model.
You can use the command check (for example check p p > tau+ tau-) in order to see if your model is gauge/lorentz invariant.

Cheers,

Olivier

Looks like that they are one small typo in the above:

the following
ProjM is 1/2*(1-gamma_5)
ProjM is 1/2*(1+gamma_5)
should be read as
ProjM is 1/2*(1-gamma_5)
ProjP is 1/2*(1+gamma_5)

Cheers,

Olivier

Thanks for the clarfications Olivier,

I am able to reproduce the numbers I got from Pythia in the same conditions,
which indicates that the setup is correct!

Thanks again,
Loic