Syntax for off-shell decays/production

Difficult to answer.

If the width of h2 is named G2 and follow the following property
  G2 << (M2-M1) and G2 << (M2-MZ)
then Narrow Width Approximation (NWA) should be valid for he
and given h2 is a scalar (and therefore spin-correlation is not a point).
Then the best way to move forward
is to do
generate p p > h2 j j $$ z w+ w- / a j h1 h2 h3 h+ QCD=0 QED=4
then use madspin in "spinmode=none"
in that mode you should be able to ask
h2 > l+ l- b b~
Note that this syntax does not force "l+l-" to come from the Z (as you did in one of your syntax but none in any of the other). I did not do it since it would likely require to have NWA valid for either h1 or Z which sounds unlikely in your case.

Concerning your syntax question.
All your syntax are doing a selection of diagram in the sense that all your syntax put some restriction on the presence of some particle in S-channel. Any procedure of selection of diagram (with dedicated syntax or via a sorting algorithm) as the potential to break gauge invariance. Often such breaking occurs only "far" from the resonance when NWA is valid. Which is the reason why the syntax with the "," is keeping only the onshell contribution via dedicated cut.

The syntax "> >" is not associated to a phase-space cut and is therefore in general more problematic in term of gauge invariance.

Cheers,

Olivier

Hi Olivier,

Thank you for your reply. I think I need to use an alternative, since I'm interested in studying diagrams where the vertices ZZh2 and h2h1Z are both present. I was thinking about one of these:

1) generate p p > h2 j j $$ z w+ w- / a j h1 h2 h3 h+ QCD=0 QED=4, (h2 > z > h1 l+ l-, h1 > b b~)

2) generate p p > h2 j j $$ z w+ w- / a j h1 h2 h3 h+ QCD=0 QED=4, (h2 > h1 > z b b~, z > l+ l-)

3) generate p p > h2 j j $$ z w+ w- / a j h1 h2 h3 h+ QCD=0 QED=4, (h2 > h1 z > l+ l- b b~)

From what I understand, in all of the above I'm selecting diagrams which have the h2h1Z interaction that I want and nothing else, the difference is which set of particles can be off-shell (z, h1 or both). Technically there is another difference, in 3) the b quarks can come from the Z and the leptons from h1 as well. Other than that, is there more differences between these 3 options? Should I use one in particular or doesn't really matter?

I also saw that there is this bw_cut parameter used to evaluate if a particle is on-shell or not, which takes the value 15 by default. I guess I could increase it to a really large number, but honestly don't know if doing that makes any sense, probably not. In my model, z is the SM Z boson, h2 the SM Higgs and h1 has a mass changing between 120 and 50 GeV (I'm still trying to see if it can be lower), and the width changes a lot depending on the mass and other parameters, but is relatively small (10^-3 at most).

Hi,

Personally I would avoid to use any restriction on the decay and would use
h2 > l+ l- b b~
Using any of your three syntax that you provide sounds dangerous to me and for each of them, they need to be use with justification that such syntax is indeed appropriate and why . (Like you likely have in order to apply the following (and weird)
$$ z w+ w- / a j h1 h2 h3 h+
restriction for the production.)

Certainly your three syntax are quite different in term of prediction and if it is not clear which one you should use, it likely means that you should not use any of the first two.

> I also saw that there is this bw_cut parameter used to evaluate if a particle is on-shell or not, which takes the value 15 by default. I guess I could increase it to a really large number, but honestly don't know if doing that makes any sense, probably not. In my model, z is the SM Z boson, h2 the SM Higgs and h1 has a mass changing between 120 and 50 GeV (I'm still trying to see if it can be lower), and the width changes a lot depending on the mass and other parameters, but is relatively small (10^-3 at most).

That's the opposite actually, this parameter might be decrease but not increase.
Increasing it means that you trust the NWA more and more outside of his region of validity.
But in your case you do have case where mh2 > mh1 + mz, which means that you are not in Narrow-width approximation.
So increasing that parameter will allow madgraph to return a number.
But at first sight, I would say that the question that you ask MG5aMC does not make sense and therefore I would not trust the answer to the question.

Cheers,

Olivier

Thank you again Olivier. Just a final question about a comment you made. Why is this restriction ($$ z w+ w- / a j h1 h2 h3 h+) for the production "weird". It was the only way that I managed to have only VBF Higgs production for the 2HDM that I'm working with. Without this syntax I get s-channels with Zs (this is not VBF) or t-channels where h2 comes a h3Zh2 vertex, for example. Is something wrong with the syntax that I used?

Cheers,
Rodrigo

The syntax "$$" and "/" can , as you say above, break gauge invariance... So they need to be use carefully which means they need to have a justification on the type:
1) this is fine because the gauge invariance is preserved because ...
2) this is fine because the breaking of gauge invariance is suppressed and below ... accuracy (potentially within/thanks to the cuts).

In the Standard Model, one can use what some call the "VBF approximation" which is a justification of the second type to reduce the amount of diagram. But note that such approximation is not valid on the full phase-space and strong cut are needed to make breaking of gauge invariance under control. I personally do not know the case of 2HDM to say if this holds or not (or when).

Your syntax is particularly weird because you have a particle in the final state (of the production part) AND in the forbidden particle list. This is a syntax that has a high probability to break gauge invariance.
Additionaly, such syntax is what is considered of undefined behavior. The way it work is more a side effect of the current implementation and is neither defined anywhere nor tested by our test suite. Which means that it can technically change at any time (and that we are allowed to do it).

Additionally, you also forbid "a" so the photon but allow the Z (in T-channel). I would expect such type of diagram to have sensible interference contribution between them that are "likely" to create issue (likely gauge invariance issue but not only). So again, this is the type of diagram removal that need a theoretical justification.

Let me be clear, I do not say that you can not do it, I say that you can, if you know why you can do it.
If you do know why then do not hesitate to teach me. If you do not, do not hesitate to ask your supervisor/collaborator (or check litterature).

Cheers,

Olivier