# A Way of Grouping Feynman Diagrams

I have a question on how MadGraph utilizes a coupling order information when it classifies Feynman diagrams.

I tried to dig out all the relevant information here, and found this:

- MadGraph decomposes the Feynman diagrams based on independent phase-space integrations to speed up its computations (https://arxiv.org/abs/hep-ph/0208156).

- It rather looks at the structures of matrix elements (or squared amplitudes), and see whether there is any peak structure, for example, in their integrands. Then it classifies diagrams according to the integration structures.

- As a result, even if two Feynman diagrams have a same coupling order, if their structures of amplitudes are different, e.g s-channel and t-channel, then they are classified differently. The method is called Single-Diagram-Enhanced multi-channel integration.

- Another consequence is that if one uses a four-flavor scheme, diagrams with b-quark are classified differently from other light quarks because they contain mb in their amplitudes, hence classified separately.).

Given above, Is there any pre-step (before even classifying the structure of integration) that utilizes the information of coupling-order (e.g. QCD = 2, QED = 0)? I guess this is too much detail, but I'm wondering if there is any role of coupling order when MadGraph classifies the diagrams.

Thanks!

Best regards,

Han

## Question information

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Solved by:
Olivier Mattelaer
Solved:
2020-11-16
Last query:
2020-11-16
2020-11-15

## This question was reopened

 Olivier Mattelaer (olivier-mattelaer) said on 2020-11-14: #1

Hi,

I can be wrong with that but I would say that no, if two fD only differ by their coupling, they will be integrate simultaneously.

Cheers,

Olivier

 Han (jeonghan-kim) said on 2020-11-15: #2

Hi Olivier,

So what you are saying is if two Feynman diagrams have the identical structure of phase-space integral, then they will be grouped together and integrated together, even if they have a different coupling-order.

I would like to understand this further, in a slightly different angle.
Suppose I generate

generate p p > j j QCD==2 QED==0 @1
add process p p > j j QCD==0 QED==2 @2

As far as I know, if I use the syntax @1 and @2, Feynman diagrams are grouped separately into group 1 and group2 respectively. That doesn't necessarily mean that interferences are calculated among the same group, right? Can you please confirm this?

According to your answer, they will be regrouped according to their structures of phase-space integrals.
It sounds like there is a mapping from
1) Group of Feynman diagrams
to
2) Group of phase-space integrals

If I try to find a role of a coupling-order, it restricts a number of diagrams in 1). This sounds trivial as I can clearly count a number of diagrams by typing a command: "display diagrams”.

At NLO UFO models, it becomes a bit tricky at stage 1). If I don't put the specific coupling-orders at stage 1) it crashes. Could it be related to an orthogonal problem such as pole cancellations or gauge-invariance, where if these conditions are not met, then it crashes? If so, can we generally say that one should be careful in specifying the coupling-oder for NLO processes?

Thanks!

All the best,

Han Olivier Mattelaer (olivier-mattelaer) said on 2020-11-15: #3

Hi,

If you use @X syntax this force MG5aMC to split all the computation in two indepent part that are handle separatly. Therefore, they are no interference between two command generate / add process
(EVEN if you use @0 for both)

They are two level of grouping
The first grouping correspond to various initial-final state
two set of feynman diagram are merged iniside the same directory if
1. they have the same @x output
2. the mass of all final state is the same
3. the spin of all final state is the same
4. potentially other constraint (see madgraph5 paper for more details)

Then inside a given directory you will have a series of matrix1.f, matrix2.f ,....
corresponding to the various initial/final state
if two file have all the same matrix-element then the file is not repeated.

Finally the phase-space integration is also merged between the various matrix.f
if we have the same singularity structure for the diagram 3 of matrix1 and for the diagram 6 of matrix4 then only one channel of integration is used for both.

You will have more details on the following FAQ:

> At NLO UFO models, it becomes a bit tricky at stage 1). If I don't put
> the specific coupling-orders at stage 1) it crashes. Could it be related
> to an orthogonal problem such as pole cancellations or gauge-invariance,
> where if these conditions are not met, then it crashes? If so, can we
> generally say that one should be careful in specifying the coupling-oder
> for NLO processes?

You can obviously asked non-coherent set of diagrams depending of your syntax.
The issue is typically related to the fact that you include EW contribution and that you have then issue that are introduced by EW divergence that are present in the loop computation but not include in the real (and therefore your computation is infitine. The version 3.0 (still beta) is able to include EW correction but is not yet able to generate events at NLO (only to compute cross-section and to have plots)

Cheers,

Olivier

> On 15 Nov 2020, at 09:50, Han <email address hidden> wrote:
>
> Question #693979 on MadGraph5_aMC@NLO changed:
>
>
> Han is still having a problem:
> Hi Olivier,
>
> Thanks for the prompt reply!
>
> So what you are saying is if two Feynman diagrams have the identical
> structure of phase-space integral, then they will be grouped together
> and integrated together, even if they have a different coupling-order.
>
> I would like to understand this further, in a slightly different angle.
> Suppose I generate
>
> generate p p > j j QCD==2 QED==0 @1
> add process p p > j j QCD==0 QED==2 @2
>
> As far as I know, if I use the syntax @1 and @2, Feynman diagrams are
> grouped separately into group 1 and group2 respectively. That doesn't
> necessarily mean that interferences are calculated among the same group,
> right? Can you please confirm this?
>
> According to your answer, they will be regrouped according to their structures of phase-space integrals.
> It sounds like there is a mapping from
> 1) Group of Feynman diagrams
> to
> 2) Group of phase-space integrals
>
> If I try to find a role of a coupling-order, it restricts a number of
> diagrams in 1). This sounds trivial as I can clearly count a number of
> diagrams by typing a command: "display diagrams”.
>
> At NLO UFO models, it becomes a bit tricky at stage 1). If I don't put
> the specific coupling-orders at stage 1) it crashes. Could it be related
> to an orthogonal problem such as pole cancellations or gauge-invariance,
> where if these conditions are not met, then it crashes? If so, can we
> generally say that one should be careful in specifying the coupling-oder
> for NLO processes?
>
> Thanks!
>
> All the best,
>
> Han
>
> --