# How far do Madgraph integrate offshell for virtual particles

Asked by Siyuan Sun on 2018-03-13

This is probably a really simple question but here goes:

I'm attempting to generate:

generate p p > W- W- W+ [QCD]

I can see the pp > W H > WWW diagram in addition to many other diagrams in the html output of madgraph.

However, when I plot the WW mass, I do not see a peak at mH = 125.

Initially, I thought this was because my BW_cut is set to the default 15 so we are only allowing W* to go offshell to ~50 GeV. So I set BW_cut to 40 to allow the full W mass window.

I thought this would allow the pp > WH > WWW* process and therefore the mH = 125 GeV peak. However, this is not what I see.

On top of this, the calculated cross-section of the pp > WWW process did not change when I changed BW_cut from 15 to 40, even if doing so should have allowed another resonant production pp > WH > WWW* to be included.

I do see events with low W boson mass when I increase the BW_cut.

So my question is what is madgraph generating and how is it calculating the cross-section? It's obviously not integrating over the full W mass line shape for all diagrams. Doing so would include the contribution from pp > WH > WWW* where the H is on-shell.

If it's not including the offshell components what exactly is meant by having the pp > WH > WWW* diagram? Is it somehow only calculating pp > WH* > WWW where all 3 W's are on-shell and the tail of the W BW is added later just by tweaking the WWW phase space somehow without actually calculating the exact W line shape and interference from pp> WH > WWW*?

-Sun

## Question information

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Solved by:
Olivier Mattelaer
Solved:
2018-03-14
Last query:
2018-03-14
2018-03-13
 Olivier Mattelaer (olivier-mattelaer) said on 2018-03-13: #1

Hi,

If you run:
> generate p p > W- W- W+ [QCD]

You are asking for three onshell W.
So you will not include any
> pp > WH > WWW*

Cheers,

Olivier

> On 13 Mar 2018, at 18:22, Siyuan Sun <email address hidden> wrote:
>
> New question #665550 on MadGraph5_aMC@NLO:
>
>
> This is probably a really simple question but here goes:
>
> I'm attempting to generate:
>
> generate p p > W- W- W+ [QCD]
>
> I can see the pp > W H > WWW diagram in addition to many other diagrams in the html output of madgraph.
>
> However, when I plot the WW mass, I do not see a peak at mH = 125.
>
> Initially, I thought this was because my BW_cut is set to the default 15 so we are only allowing W* to go offshell to ~50 GeV. So I set BW_cut to 40 to allow the full W mass window.
>
> I thought this would allow the pp > WH > WWW* process and therefore the mH = 125 GeV peak. However, this is not what I see.
>
> On top of this, the calculated cross-section of the pp > WWW process did not change when I changed BW_cut from 15 to 40, even if doing so should have allowed another resonant production pp > WH > WWW* to be included.
>
> I do see events with low W boson mass when I increase the BW_cut.
>
> So my question is what is madgraph generating and how is it calculating the cross-section? It's obviously not integrating over the full W mass line shape for all diagrams. Doing so would include the contribution from pp > WH > WWW* where the H is on-shell.
>
> If it's not including the offshell components what exactly is meant by having the pp > WH > WWW* diagram? Is it somehow only calculating pp > WH* > WWW where all 3 W's are on-shell and the tail of the W BW is added later just by tweaking the WWW phase space somehow without actually calculating the exact W line shape and interference from pp> WH > WWW*?
>
> Thank you for your help,
>
> -Sun
>
>
> --

 Siyuan Sun (ssun) said on 2018-03-13: #2

Hi Olivier,

I just wanted to ask a follow-up. Can you please clarify for me what is meant by on-shell and offshell?

On-shell doesn't mean only W = W pole mass. You can still get a 2 GeV W.

If it doesn't include any pp > WH > WWW* then what does it mean to have the diagram included in the calculation? Why do I see it in the list of diagrams that its computing over?

What is madgraph doing exactly in terms of the actual integral its computing. It's definitely not the path integral integrated from pole - BW_cut*width to pole + BW_cut*width that would expect from computing the feynmann diagrams by hand because that would include pp > WH >WWW* contribution.

Best,

-Sun

 Olivier Mattelaer (olivier-mattelaer) said on 2018-03-13: #3

Hi,

All diffusion theory are based on the S-matrix formalism. It assumes that initial state/final state should be in a asymptotic state.
Therefore both initial state and final state are exactly onshell meaning W = W pole mass.

Additionally any tool able to decay afterwards the W, would need to use the Narrow Width Approximation.
Which is not valid in this context.

Note that the W width actually breaks gauge invariance and to restore gauge invariance you
have to use the complex mass scheme which is only valid if all final state particles have zero width.

> If it doesn't include any pp > WH > WWW* then what does it mean to have
> the diagram included in the calculation? Why do I see it in the list of
> diagrams that its computing over?

This diagram has to be present even if the Higgs is off-shell. It is actually crucial to include since it has
negative interference with the other diagram and reduce the cross-section and fordid the amplitude to cross the unitary threshold.
In this context that diagram is only realised when the Higgs is off-shell (> 2*MW) due to the onshell requirement of the final state.

> What is madgraph doing exactly in terms of the actual integral its
> computing.

In this case they are NO integral on each of the W mass.
The rest of the phase-space integration is a complicated business.
But assuming that the phase-space integration follow the pp > WH > WWW
in that case we will integrate explicitly over the Higgs mass between: 2*MW to \sqrt{s}-MW
Those value are just due to the mass of the final state (such value actually also depends of the cuts on the final state)

Now if you ask the 2 W to decay in 4 lepton, then higgs mass will be integrate between 0 and \sqrt{s}-MW
(assuming no cut on the lepton and the lepton to be massless)

> It's definitely not the path integral integrated from pole -
> BW_cut*width to pole + BW_cut*width that would expect from computing the
> feynmann diagrams by hand because that would include pp > WH >WWW*
> contribution.

Here you seem to confuse stuff:
1) bwcutoff is a parameter when internal propagator is onshell/offshell
The impact of that parameter is not on the cross-section but on how the LHE event is written.
The parton-shower will re-act differently if we write internal propagator inside the LHE file.
and this parameter allows us to choose that.

At NLO, i do not think that the "bwcutoff" parameter is use for anything else (expect maybe in some development branch where
we have to remove some resonant contribution appearing at NLO).

FOR LO diagram generation, you have additional syntax allowing to force a particle to be onshell (and then decay).
and some syntax to force the particle to be offshell
In those case, we also use the bwcutoff for those syntax which are not available at NLO

Cheers,

Olivier

> On 13 Mar 2018, at 18:52, Siyuan Sun <email address hidden> wrote:
>
> Question #665550 on MadGraph5_aMC@NLO changed:
>
>
> Siyuan Sun is still having a problem:
> Hi Olivier,
>
>
> I just wanted to ask a follow-up. Can you please clarify for me what is
> meant by on-shell and offshell?
>
> On-shell doesn't mean only W = W pole mass. You can still get a 2 GeV
> W.
>
> If it doesn't include any pp > WH > WWW* then what does it mean to have
> the diagram included in the calculation? Why do I see it in the list of
> diagrams that its computing over?
>
> What is madgraph doing exactly in terms of the actual integral its
> computing. It's definitely not the path integral integrated from pole -
> BW_cut*width to pole + BW_cut*width that would expect from computing the
> feynmann diagrams by hand because that would include pp > WH >WWW*
> contribution.
>
> Best,
>
> -Sun
>
> --

 Siyuan Sun (ssun) said on 2018-03-14: #4

Hi Oliver,

Thank you very much for your clear and detail response. Everything is clear now.

Best,

-Sun

 Siyuan Sun (ssun) said on 2018-03-14: #5

Thanks Olivier Mattelaer, that solved my question.