NLO EW ZH production at FCCee

[arossia94]

Implementation of the NLO EW results for ZH production at FCCee from this paper.
This has to be done for all the ZH FCCee datasets, including the ones for each Higgs decay channel.

• Implementation of Op sensitivity
• Implementation of all other NLO corrections in Higgs inclusive datasets.
• Extend the former to the CEPC files.
• Implementation of all other NLO corrections for each Higgs decay channel.
• Extend the former to the CEPC files.

Optional task:

• Implement files with only LO predictions so one can remove NLO EW easily without removing all the NLO QCD datasets.

[jacoterh]

Adding @TommasoArmadillo to this PR as he already started working on implementing ZH at NLO independently (with Fabio and Eleni)

[LucaMantani]

Hi Alejo, thanks for this!
I just skimmed it and I have a bunch of comments. There are also come conflicts for one of the OOS file, not exactly clear why they are modified at all in this PR, maybe it's a remnant of some previous state of main when you branched out.

[TommasoArmadillo]

Hello! I went through the PR, in particular the file ZH_NLO_EW_2409.11466.nb and I have some comments. I already discussed a bit with Alejo, but I report here for keeping record.

• In the paper from Dawson et al. they report the Wilson coefficient for the operator $$O_{\varphi\Box}=(\varphi^\dagger \varphi)\Box (\varphi^\dagger \varphi)$$ in the Warsaw basis. However, in smefit, we use the operator $$O_{\varphi d}=\partial^\mu(\varphi^\dagger \varphi)\partial_\mu(\varphi^\dagger \varphi)$$. The two operators are related by integration by parts identities $$\partial_\mu((\varphi^\dagger \varphi)\partial^\mu(\varphi^\dagger \varphi))=O_{\varphi\Box}+O_{\varphi d}$$. So there should be a minus sign between $$C_{\varphi \Box}$$ and $$C_{\varphi d}$$;
• In the paper they introduce the coefficient $$C_2^{4f}=C_{ll}[1,2,2,1]+C_{ll}[2,1,1,2]$$ in eq. A.3. In smefit we use the operator $$O_{ll}=(\bar{\ell_1} \gamma_\mu \ell_2)(\bar{\ell_2} \gamma_\mu \ell_1)$$. So I think there should be a factor 2 between the two. Alejo had a doubt that the factor 2 might already be taken into account in smefit. He proposed to check it by computing the interference of some process that includes $$C_{ll}$$ at LO and is included in SMEFiT (ee>mumu) with SMEFTsim in MG5 turning on only one of them and relating it with the results already in SMEFiT. I am currently working on it.

[TommasoArmadillo]

I have news concerning the $$C_{ll}$$ coefficient.

I generated $$e^+ e^- \to \mu^+ \mu^-$$ at LO with SMEFTsim in MG5. I considered only the contribution due to $$C_{ll}[1,2,2,1]$$. The number I obtain agrees with the one for $$C_{ll}$$ that is already present in SMEFit in FCCee_mumu_240GeV.json.

This means that we need to take into account an extra factor 2 for the WC $$C_{2}^{4f}$$ in ZH_NLO_EW_2409.11466.nb.

[jacoterh]

That's great news! Have you also checked whether the other flavour combination actually gives the same contribution as we expect?

[jacoterh]

@arossia94 are we sure

implements correctly the definition of ctZ from here?

I would expect cu3W to also enter the linear combination to get our ctZ.

[arossia94]

@arossia94 are we sure implements correctly the definition of ctZ from here? I would expect cu3W to also enter the linear combination to get our ctZ.

I'm quite sure it does, I did the computation by hand at the level of operators and WCs in the Lagrangian. I don't have my handwritten notes at hand now, I can share them next week. The transformation is quite non intuitive.

[jacoterh]

@arossia94 are we sure implements correctly the definition of ctZ from here? I would expect cu3W to also enter the linear combination to get our ctZ.

I'm quite sure it does, I did the computation by hand at the level of operators and WCs in the Lagrangian. I don't have my handwritten notes at hand now, I can share them next week. The transformation is quite non intuitive.

I already managed to convince myself you did everything correctly! It's the same transformation that confuses me sometimes with O3pq and OpqMi. One needs to check at the Lagrangian level what it means to turn on ctW = 1 in SMEFiT. This turns on OtW and ObW weighted by cW/sW.

[jacoterh]

Given PR#99 on smefit_release we can remove all files in nlo_ew_implementation and put them all in theory instead, can't we?

[LucaMantani]

There are three files in the interface_oos that I do not understand why they are there, but for the rest it looks good to me.