NNLO uncertainties

[mcremone]

In principle, composed by three pieces: epsilonQCD, epsilonEW, and epsilonMIX.

epsilonQCD

It describes uncertainties related to variations of the renormalization and factorization scales which are performed to estimate the uncertainty of the theoretical prediction due to missing higher-order contributions.

The uncertainty is split into three parts:

• epsilonQCD1, a normalization uncertainty
• epsilonQCD2, a shape uncertainty
• epsilonQCD3, an additional uncertainty estimating unknown correlations between the QCD uncertainties of the different vector boson plus jets processes.

All three nuisance parameters need to be treated uncorrelated, however each parameter is correlated for all V+jets processes and all bins of boson pT.

Questions:

• Is epsilonQCD3 a shape or a normalization uncertainty?
• The fact that each parameter is correlated for all V+jets processes means that is goes up for one process, it should go up for all of them. Does this mean that it goes up/down by the same quantity?

epsilonEW

It describes uncertainties related to missing even higher-order(NNLO) contributions:

• epsilon1, a universal effect of the missing higher-order corrections and can therefore be used correlated across all V+jets processes.
• epsilon2 and epsilon3 describe subleading higher-order effects with unknown process correlation. Therefore, these parameters are used uncorrelated for the different processes.

Questions:

• Is there any correlation between epsilon1, epsilon2, and epsilon3
• Are there shape or normalization factors?
• regarding epsilon1 as correlated across all V+jets processes, does it vary by the same quantity for all processes?

epsilonMIX

It is an additional uncertainty regarding mixed contributions that cannot be described with the multiplicative or factorized approach. This nuisance parameter is chosen to be completely uncorrelated between the different processes.

Questions

• is epsilonMIX a shape or a normalization factor?

So far I see at least three nuisances that for sure need to be accounted for in the fit, that are epsilon2, epsilon3, and epsilonMIX. It needs to be clarified if they are shape or normalization effects. We are going to have a different nuisance per V+jets process, therefore we will have: epsilon2^{V}, epsilon3^{V}, and epsilonMIX^{V}, where V can be W, Z, or gamma.
The remaining nuisances, epsilonQCD1, epsilonQCD2, epsilonQCD3, and epsilon1, if they not only move in correlated fashion for all V+jets processes, but they also move by the same quantity, they are going to cancel out in the transfer factor ratio and therefore they can be excluded from the pool of uncertainties. If they do not, we are going to have one nuisance for all processes.

[mcremone]

Answers from Michael:

• "I would treat epsilonQCD3 as a shape uncertainty. Actually, I would just do that for all uncertainties."
• "As far as I know epsilonEW1,2,3 should be used uncorrelated."
• "As the plots show, most uncertainties have a comparable effects, there seem to be some (small) differences between the different vector bosons."

It turns out (see slides) that epsilonEW1, epsilonEW2, epsilonEW3, epsilonMIX, and epsilonQCD3 are actually very small effects, of the order of a fraction of percent. Therefore the can be neglected.

The only two to consider are epsilonQCD1 and epsilonQCD2. Looking at the slides, it appears that:

• epsilonQCD1 can be treated as a rate uncertainty
• epsilonQCD2 is a shape uncertainty

This should be confirmed looking at the following ratios:

• QCD1up^Z/QCD1up^W
• QCD1down^Z/QCD1down^W
• QCD1up^Z/QCD1up^gamma
• QCD1down^Z/QCD1down^gamma
• QCD2up^Z/QCD2up^W
• QCD2down^Z/QCD2down^W
• QCD2up^Z/QCD2up^gamma
• QCD2down^Z/QCD2down^gamma

since QCD1 and QCD2 are correlated between all processes, we need to check only the ratios of up variations for Z with up variations for W and gamma, and similarly and for down variations.

Three things can happen:

• the ratios are reasonably ~1 over the full bosons pT spectrum, in that case we can get rid of these uncertainties.
• the ratio is ~constant around a value different from 1, in that case we can treat them as rate uncertainties in the transfer factor. This will happen for sure for epsilonQCD1 if it will not satisfy the requirement at the first bullet.
• the ratio varies all over the bosons pT spectrum, in that case we would need to introduce them as shape uncertainties (correlated between all different processes). This will happen for sure for epsilonQCD2 if it will not satisfy the requirement at the first two bullets.

[mcremone]

Renormalization and Factorization Scale Uncertainties

Need to take into account these too. We will label renormalization as muR and factorization as mF. In the files, alternate shapes are defined as:

• muR_down: *_NNLO_NLO_nnn_nnn_n_Weight_scale_variation_muR_0p5_muF_1p0
• muR_up: *_NNLO_NLO_nnn_nnn_n_Weight_scale_variation_muR_2p0_muF_1p0
• muF_down: *_NNLO_NLO_nnn_nnn_n_Weight_scale_variation_muR_1p0_muF_0p5
• muF_up: *_NNLO_NLO_nnn_nnn_n_Weight_scale_variation_muR_1p0_muF_2p0

Obtained by multiplying by 0.5 muR and muF scales for the down variations, and multiplying by 2 muR and muF scales for the up variations.

First to check would be the absolute scale of the variations with respect to the nominal:

• muR_down/muR_nominal
• muR_up/muR_nominal
• muF_down/muF_nominal
• muF_up/muF_nominal

where muR_nominal=muF_nominal=*_NNLO_NLO_nnn_nnn_n. These ratios should be computed per process, that means that the same process should appear in both the numerator and the denominator. For example, muR_down/muR_nominal for W+jets will be:

evj_NNLO_NLO_nnn_nnn_n_Weight_scale_variation_muR_0p5_muF_1p0 / evj_NNLO_NLO_nnn_nnn_n

Second, we need to compute ratios as done for the other uncertainties:

• (muR_down^Znunu/muR_down^W) / (muR_nominal^Znunu/muR_nominal^W)
• (muR_up^Znunu/muR_up^W) / (muR_nominal^Znunu/muR_nominal^W)
• (muR_down^Znunu/muR_down^gamma) / (muR_nominal^Znunu/muR_nominal^gamma)
• (muR_up^Znunu/muR_up^gamma) / (muR_nominal^Znunu/muR_nominal^gamma)
• (muR_down^Znunu/muR_down^Zll) / (muR_nominal^Znunu/muR_nominal^Zll)
• (muR_up^Znunu/muR_up^Zll) / (muR_nominal^Znunu/muR_nominal^Zll)

and same for muF.