# cross-section calculation

I attempted to approach a process in two different way defined as follows,

1)
process proc_ee_ZHH = e1, E1 => Z, H, H
process proc_Z_mumu = Z => e2, E2
process proc_H_bb = H => b, B
unstable Z (proc_Z_mumu)
unstable H (proc_H_bb)

2)

process proc_ee_4b2mu = e1, E2 => b, B, b, B, e2, E2 {\$restrictions = "3+4~H && 5+6~H && 7+8~Z"}

How are these handled differently with regards to the cross-section value whizard outputs. One thing I noticed was that the branching ratios of H and Z to the respective particles as defined in 1, are not included in the value of the cross-sections but it is included in method 2. This does not seem to explain all the differences. It seems that maybe a symmetry factor of 2 due to there being two H's in 1) is also not included.

Are there any other differences and are the differences I mention correct?

Thanks,
Nikhil

## Question information

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WHIZARD Edit question
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Juergen Reuter Edit question
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Juergen Reuter
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## This question was reopened

 Revision history for this message Juergen Reuter (j.r.reuter) said on 2020-06-06: #1

Dear Nikhil,
several remarks here, hope they are helpful: you are right, using the unstable option only affects event samples, the cross section that is displayed is always the production cross section. The branching ratio is taken to be one if there is only a single decay. Partial widths can be reset by the user e.g. to insert the most precise values as given by the Higgs Cross Section WG, cf. e.g. the file br_redef_1.sin in share/tests/functional_tests.
Clearly there are several symmetry factors to be considered here: there is a symmetry factor of 1/2 for the double Higgs production while there is a symmetry factor of 1/4 from two b and two anti-b quarks in the final state of the second process.
Note that using restrictions is a dangerous business as it does break (electroweak) gauge invariance. In this case I don't expect this to be problematic.
Also note that especially for H->bb using the tree-level coupling mb/vev is not a very good description as there are large QCD corrections which effectively lead to using a running mass in the bottom Yukawa coupling in the MSbar scheme at the scale of the decaying Higgs boson. This yealds a bottom mass of the order of 2.1 GeV roughly, leading to a bottom Yukawa coupling half the tree-level value.
But I think what you are seeing is most likely combinarotics: doing restrictions on 3+4~H && 5+6~H gives you only part of the cross section. There is also a piece where 3+6~H && 4+5~H . I would guess that this leads to a factor of approximately 2.
Cheers,
JRR (Juergen)

 Revision history for this message Nikhil Bachhawat (nikhilb1997) said on 2020-06-06: #2

That clears a lot up.

Thanks,
Nikhil

 Revision history for this message Nikhil Bachhawat (nikhilb1997) said on 2020-06-06: #3

Thanks Juergen Reuter, that solved my question.

 Revision history for this message Nikhil Bachhawat (nikhilb1997) said on 2020-06-08: #4

Hi, I tried adding the second combination in the restrictions to try and see what happens but I realised that actually the value from the restrictions (method 2) is larger than that from the unstable decay option (method 1). So approximately multiplying the cross section value from method 2 by two will only increase the difference.

This might have to do with the other points you raise in this case.

Thanks,
Nikhil

 Revision history for this message Juergen Reuter (j.r.reuter) said on 2020-06-08: #5

Yes, I expect the normalization for the off-shell process to be off. I suppose you inserted the default width for the Higgs, something of the order of 4 MeV. However, in method 2 you are using the tree-level coupling of the Higgs to b bbar, so this will lead approx. to
(Gamma(H->bb)/Gamma_tot)**2. Now, Gamma_tot is the Higgs XSWG value, while Gamma(H->bb) is calculated with the tree-level values. So the partial widths is too large by roughly a factor of 4 or so. This explains the difference.

 Revision history for this message Nikhil Bachhawat (nikhilb1997) said on 2020-06-08: #6

Thanks Juergen Reuter, that solved my question.

 Revision history for this message Nikhil Bachhawat (nikhilb1997) said on 2020-06-12: #7

Hi,

I am really sorry but I will open this question up once again to just make sure I understand the difference in the coupling values used.

So, if I run the process:
process proc_3h = e1, E1 => b, B, b, B, Z
process proc_zmumu = Z => e2, E2
unstable Z (proc_zmumu)

I was hoping to generate a "background" sample using this.

Then I wished to see how the di-higgs process described in method 1 in the main question compared to this "background". Does this "background" process use tree level couplings (especially the Hbb coupling) in its calculation? Because then as you mention, different couplings might be used between the two samples. Or does that apply only to the Hbb decay when restrictions are used.

Nikhil

 Revision history for this message Juergen Reuter (j.r.reuter) said on 2020-06-12: #8

Hi Nikhil,
WHIZARD always uses tree-level couplings for Hbb (unless you ask for an NLO QCD process, but then you get only a fixed-order
corrections which is just alpha_s/pi, so 3 per cent). This applies to the signal as well as to the background sample, though both would contain the Higgs, so the names are maybe not ideal. Higgs exchange can be excluded using the "!H" restriction.
Cheers,
JRR

 Revision history for this message Nikhil Bachhawat (nikhilb1997) said on 2020-06-12: #9

Yes, you are right the names are not a great choice for this. I am looking at it as if I have an inclusive sample and I am trying to understand its compositions, i.e. the different processes that contribute and by how much, specifically the di-higgs process.