# Problem calculating the cross-section when there is a four particle vertex involved in the process

Asked by Arka Santra on 2018-01-26

Hi,
I am trying to generate spin 0 magnetic monopoles from photon fusion. The process has three diagrams:

For diagram 2 and 3, the coupling of photon-boson-boson is proportional to g. So the matrix element is proportional to g^2 and the cross-section varies as g^4. This part is correctly calculated by the MadGraph.

My problem lies with diagram 1. Here the coupling order of photon-photon-boson-boson should be 2, so the coupling will be proportional to g^2. Then the matrix element should also be proportional to g^2 and eventually, the cross-section will be proportional to g^4. But I see that the MG cross-section for only diagram 1 is proportional to g^2, not g^4.

The couplings are defined here:

Here GC_109 is the coupling of photon-boson-boson. GC_110 is the coupling of photon-photon-boson-boson.

The coupling of photon-photon-boson-boson also involves a user-defined form factor. The form factor is defined here:

and it is used in the Lorentz structure of the VVSS in the lorentz.py:

But since the cross-section from diagram 1 is only proportional to g^2, not g^4 as expected, I wanted to change the coupling by hand:

a. The coupling GC_110 is just the square of the previous value:
b. The form factor is also squared (but not the Lorentz structure) inside the lorentz.py in this line:

Now the cross-section coming from only diagram 1 is proportional to g^4. But unfortunately, the total cross-section of the process is off by a factor of 1.6 to 0.6 (depending on the mass of the monopole) from the theoretical prediction.

My questions are the following:

1. How do I tell MG that the coupling order of diagram 1 is 2, not 1 – so that MG knows that the matrix element will be proportional to g^2 for diagram 1?

2. I was wondering if squaring this value of GC_110 'by hand' disturbs MG’s calculation of interference term when it needs to consider all three diagrams of the process. Is there any way of fixing the coupling order of the four particle vertex in MG (the usual coupling order parameter in MG only filters the diagrams, it does nothing to the coupling value)?

I really appreciate your help. Thank you,
Arka

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2018-01-26
 Olivier Mattelaer (olivier-mattelaer) said on 2018-01-26: #1

Hi Arka,

>1. How do I tell MG that the coupling order of diagram 1 is 2, not 1 – so that MG knows that the matrix element will be proportional to g^2 for diagram 1?

Madgraphs add the coupling order associated to each interaction. So in your case all your diagram are at the second order in QED and do not any power for QCD. This is defined via the couplings:
order = {'QED':2}
saying that such coupling has two power of QED and nothing else.

MadGraph will display this information and you can use such information to select your diagram/interference/...
Now if a coupling has the wrong information compare to the numerical value will not be detected by us.

>2. I was wondering if squaring this value of GC_110 'by hand' disturbs MG’s calculation of interference term when it needs to consider all three diagrams of the process. Is there any way of fixing the coupling order of the four particle vertex in MG (the usual coupling order parameter in MG only filters the diagrams, it does nothing to the coupling value)?

Well, this sounds very weird to square a coupling. but of course, this will change the computation accordingly.
if you do not change the order = {'QED':2} this will not change what is calculate since we use that information to select what to compute.

Cheers,

Olivier

PS: Pointless to say that editing a model like that sounds a very bad idea. If this coupling is wrong, you need to understand why and correct the model which was creating such coupling.

 Arka Santra (santra-arka) said on 2018-01-26: #2

Hi Olivier,
I am a bit confused about how I should incorporate the four particle vertex in my model. The diagram is as follows:

The coupling is g^2*f^2 where g is an integer and f is a form factor. How do you suggest to write this vertex in my model?

I was doing this:
a. The vertex definition:

b. The g^2 (written as GCH in my model, with a prefactor) is written in the couplings.py:

c. The f^2 (written as mymdl_FormFactor(MMM)) is written in the lorentz.py:

Does this recipe look okay to you? If not, please suggest the best way to implement this vertex in my model.

Thank,
Arka

 Olivier Mattelaer (olivier-mattelaer) said on 2018-01-26: #3

Hi Arka,

The typical method for any vertex is to use FeynRules to implement the associated Lagrangian (typically without the form-factor)
such that you do not have to add any vertex defiition/coupling but only have to add the form-factor structure.

Now if you follow carefully all the convention, you can obviously doing this by hand as well, but the previous thread makes me worry.

Cheers,

Olivier
> On 26 Jan 2018, at 17:22, Arka Santra <email address hidden> wrote:
>
> Question #663625 on MadGraph5_aMC@NLO changed:
>
>
> Arka Santra is still having a problem:
> Hi Olivier,
> I am a bit confused about how I should incorporate the four particle vertex in my model. The diagram is as follows:
>
> The coupling is g^2*f^2 where g is an integer and f is a form factor.
> How do you suggest to write this vertex in my model?
>
> I was doing this:
> a. The vertex definition:
>
> b. The g^2 (written as GCH in my model, with a prefactor) is written in the couplings.py:
>
> c. The f^2 (written as mymdl_FormFactor(MMM)) is written in the lorentz.py:
>
>
> Does this recipe look okay to you? If not, please suggest the best way to implement this vertex in my model.
>
>
> Thank,
> Arka
>
> --