It is inconsistent to use NNLL resummed cross section calculations for the total gg→H cross section but not for the individual pieces (0j, 1j, and 2+j).
Our procedure is to run Pythia Monte Carlo, and to reweight the pT spectrum of the generated Higgs bosons to that predicted by HqT. Code and citations for HqT are available on Massimiliano Grazzini's Higgs Production Tools website. The pT spectrum is calculated by HqT at NNLL+NLO. We then scale the total yield to the NNLL+NNLO predictions at μf=μr=mh/2. The Monte Carlo events are input to detailed detector simulation programs for CDF and D0, and subject to our trigger and event selection requirements. This sample then is divided into the 0j, 1j, and 2+j subsets, based on the reconstructed jet counts. We do not use directly the theoretical predictions for the jet counts as detector acceptance and resolution are not included in the theoretical predictions. The parton shower and hadronization models in Pythia are important to predict the impacts of detector response and analysis cuts, such a lepton isolation.
We apply the uncertainties on the theoretical predictions in each jet category from Anastasiou, Dissertori, Grazzini, and Webber (JHEP 0908 099, 2009) [arXiv:0905.3529].
As previously clarified in our responses to arXiv:1003.4266v1, this procedure takes soft-gluon emission into account in all three jet samples, and that the large uncertainties split by the jet category cover the fact that the cross sections with different numbers of jets are known at different orders.
We would like to emphasize here that one of the main reasons experimentalists separate the dilepton+MET events into the different Njet categories is to separate different sources of signal from the different sources of background. For the zero-jet bin, for example, the main signal source is gg→H, and the main background source is WW. In the two-jet bin, on the other hand, there is as much WH signal as there is gg→H, and yet again as much ZH+VBF: In CDF, the expected signal yields after all detector effects, reconstruction, and analysis requirements in 5.9 fb-1 for mH=165 GeV are 2.6 events of gg→H, 2.5 events of WH, 1.3 events of ZH, and 1.4 events of VBF. By far the largest background is ttbar in the two-jet bin. It is therefore in our best interest to analyze these events separately as we can train our neural networks to discriminate these signals and backgrounds more optimally if they can focus on the relevant subset within the jet category. In the one-jet bin, it's a mixture of the different cases: 8 events of gg→H are expected, 1.13 from WH, 0.44 from ZH, and 0.7 from VBF, with most of the background coming from DY and WW, and a bit from ttbar.
More information about the CDF channels and their relative sensitivities can be found at http://www-cdf.fnal.gov/physics/new/hdg//Results_files/results/hwwmenn_100618/ and from D0 at http://www-d0.fnal.gov/Run2Physics/WWW/results/prelim/HIGGS/H94/.
We are happy to note also that for a scale choice of mH/2, the ambiguity in the gg→H production cross section is minimized. Not only are the NLO and NNLO predictions comparable, but also the NNLL addition is very small at that choice of scale, and so this point becomes less important.
The scale uncertainty should vary μr and μf by a factor of 3 instead of a factor of 2. The justification given on pages 3 and 4, and in Figure 2, is for the LO production cross section prediction to encompass the NNLO prediction within its scale uncertainty (where the two scales are set equal in order to maximize the variation in the prediction).
We have discussed this point with several theorists. Massimiliano Grazzini and Robert Harlander have given us detailed answers regarding the discrepancy of the LO calculation with its scale uncertainty and the higher-order calculations. The text of these answers can be found under Issue 2 in our May 2010 responses.
We were concerned that the scale uncertainty was an underestimate, which is why we asked. The answer is that the scale uncertainty on the LO prediction indeed is underestimated, due to the fact that the LO process lacks important scale-dependent pieces. In particular, there is no emission of jets at LO along with the H. Adding in important pieces makes the cross section calculation more realistic and also makes the scale dependence more realistic.
Indeed, this feature is well known from other processes where the LO calculation
is even less scale dependent than the gg→H calculation. For inclusive W and Z boson
production at the Tevatron, for example, Anastasiou, Dixon, Melnikov, and Petriello
compare LO, NLO, and NNLO calculations in Figs. 5 and 10 of Phys. Rev. D
The recent paper of Ahrens, Becher, Neubert, and Yang, Eur. Phys. J. C62 333-353 (2009) [arXiv:0809.4283], and updated in arXiv:1008.3162, shows that renormalization-group-improved predictions of the gg→H production cross section converge much more rapidly and the scale uncertainty does a much better job of covering the differences between the calculations at the different orders. The uncertianties quoted by Ahrens et al. are even smaller than the ones we are currently using. We believe that the factor of two variation in the scale, as customarily calculated by most theorists, covers the higher-order difference at NLO and NNLO, but not for gg→H at LO.
The variation of scale over a factor of 3 appears to give a total uncertainty similar to that quoted by CDF and D0 when weighting uncertainties for the various jet cross sections, where CDF and D0 use a factor of 2 variation.
An initial point of our confusion about the jet-bin by jet-bin uncertainties in Anastasiou, Dissertori, Grazzini, and Webber's article is that uncertainties were quoted for the inclusive cross section before a simulation of experimental cuts and for the jet categories after simulation of experimental cuts. We use their uncertainties after the simulation of the cuts which are larger, and may coincidentally be closer when the weighted average is formed to the scale variation over a factor of 3. Anastasiou, Grazzini, Dissertori, Stockli, Webber, Boughezal, and Petriello, all recommend a factor of two for the scale variation.
An additional uncertainty of 7.5% due to jet acceptance is assessed by CDF and D0 -- is it a theory error or an experimental error (if it is a theory error it should be collected in with the scale error).
CDF evalulates and includes in the final result uncertainites on the acceptance via the Higgs boson Pt and Eta distributions from the scale and PDF uncertainties. D0 also includes an uncertainty from the reweighting of the signal distributions as well, which covers this effect. Currently these uncertainties are treated as uncorrelated, but in the next iteration we will correlate them since they come from the same scale choice.
At this point we should again make clear that we are treating the scale and PDF uncertainties in the same way as we treat other sources of systematic uncertainty in our analysis. The Bayesian prescription, which we use because there is no statistical interpretation of the scale uncertainties or the differences between separate PDF parameterizations, is to integrate all probability distributions over the possible values of the uncertain parameters within their prior distributions. This means that our belief in any parameter, in particular, the scale factor R, is summed over all values of the uncertain parameters, given how much we believe in each value of the uncertain parameters. We choose to set limits on R, a common scale factor on all signals (WH, ZH, VBF, gg→H) scaled together with fixed SM branching fractions, for convenience. It allows us to show easily which values of mH are excluded within the SM, and it allows us to easily calculate how much additional data or analysis improvement we need in order to test the SM predictions. We also set limits on the gg→H production cross section times the decay branching ratio in arXiv:1005.3216, but this approach does not allow us to combine the searches for different signal modes which are all present in the SM and which can be used to test the SM with the most sensitivity.
The dependence of the Higgs boson rapidity and Pt distributions on the scale choice, as well as the dependence of the total cross section on the scale choice, is another reason why we must use a Bayesian approach. We cannot simply subtract one sigma from the cross section prediction and draw the line on our limit plot. Instead we must also distort the kinematic shapes and change the cross section predictions in each jet bin separately. Furthermore, the impact on our limit isn't the full cross section change since we combine the WH, ZH, and VBF pieces as well. We therefore take the Bayesian procedure of integrating over all uncertain parameters to be the most consistent way to handle parameters which have multiple, correlated impacts on the predictions we are testing.
We should use the ABKM PDF set as an alternate prediction to set the PDF uncertainty.
The ABKM09 PDF set includes only DIS and fixed-target DY data. The other main sets, MSTW, CTEQ, and NNPDF also include Tevatron jet and vector boson data and other data. The Tevatron jet data in particular have an effect on the high-x gluon distributions. The high-x gluon distributions do not agree between ABKM09 and the other three sets within the quoted uncertainties.
We choose to follow the recommendation of the PDF4LHC group which is to take the envelope of global sets, MSTW, CTEQ, and NNPDF. See also slide 41 of Robert's talk at the Higgs Hunting 2010 workshop.
A new preprint from S. Alekhin, J. Blümlein, P. Jiminez-Delgado, S. Moch, and E. Reya, http://arxiv.org/abs/arXiv:1011.6259 provides calcluations of the gg→H production cross section using the ABM10 PDF set, which includes the latest combined HERA data and make similar predicitons as the ones using ABKM09 PDFs.
We have been in contact with many of the members of the several PDF groups as well as the QCD groups within the Tevatron collaborations and will continue to explore the issue.
Combination of scale and αs+PDF uncertainties -- linear or quadrature?
We agree that the gluon density in the proton depends on the factorization and renormalization scale and that the fact that the PDF+αs predictions vary with the scale choice means we should include this dependence via the PDF+αs as part of the scale uncertainty. We checked with Babis Anastasiou and his collaborators, who assured us that in their calculations of the scale variation of the gg→H cross section, the impact of the scale choice on the PDF is included as part of the scale uncertainty.
On the subject of evaluating the PDF+αs uncertainty at the top and bottom scale choices, we have asked Babis and collaborators to perform this calculation, for the scale choices mh/4 and mh. Here is what he and his group find:
------------- e-mail from Babis: --------------- We've done a study calculating the (PDF+αs) uncertainty with the scale at μ=mH/4 and μ=mH. We find that for any value of μ, dσ(pdf @ μ)/σ(μ) is independent of μ within 3 per mille. This is anticipated: the pdf+αs error is an error due to the parameterization of the densities which are extracted at a fixed value of Q2 = 1 GeV2, and a fixed value of αs at μ=MZ. The evolution to the choice of mu in the Higgs cross-section has a very small effect in the propagation of this error, for all reasonable scale choices. ------------------------------------------------
Our approach is to collect uncertainties by source and to treat all uncertainties that come from the same source as correlated, and thus added linearly. Uncertainties from different sources are added in quadrature. If sources are partially correlated, we seek to decompose them into correlated and uncorrelated parts (the PDF uncertainty is an example of this). The reason for this treatment is that if a parameter is varied (such as the scale choice) and it has an impact on a prediction in more than one way (via the PDF and the matrix element), then the sum of the impacts is the total impact of the scale variation on the result. If, on the other hand, the Tevatron high-Et jet data fluctuate, then the impact of this, via the PDF's, is uncorrelated with the scale choice, and this piece should be added in quadrature. arXiv:1009.1363v1 contains text indicating a preference to add uncorrelated uncertainties linearly, although no statement is made about adding anything in quadrature. Indeed whether to add uncertainties linearly or in quadrature depends only on whether the uncertainties are correlated or not, and correlated ones need to be added linearly and uncorrelated ones in quadrature, not the other way around. We have to be careful with this treatment not only for these theoretical uncertainties but for other sources of uncertainty within the Higgs searches, such as luminosity, b-tagging efficiencies, triggers, and W+jets cross sections for example. If we add uncertainties linearly, we make the assumption of correlation. When we fit the data to the model, which is mostly background, of which we have more than enough, we start to be able to measure the uncertain parameters. The assumption of correlation means we can interpret a measurement of one uncertain parameter as a measurement of another uncertain parameter, which may not be fair.
Thus we call the uncertainty "PDF+αs" the piece that's scale independent, and the uncertainty of the PDF and αs due to the scale uncertainty is collected in linearly with the rest of the scale uncertainty.
The choice of a flat prior in the scale uncertainty does not affect the prescription of whether to add uncertainties linearly or not. We integrate our posteriors over all the prior distributions for the uncertain parameters. For uncorrelated uncertainties with flat priors, the probability distribution for their sum is triangular (if they have the same magnitude of their uncertainties), and a trapezoid if they have unequal magnitudies. We perform the integrations explicitly instead of adding uncertainties in quadrature. For uncertainties with Gaussian priors, the integrals are the same as adding in quadrature, and for flat priors, they are similar but not exactly the same.
Remaining uncertainties due to finite-order electroweak and bottom corrections should be bigger.
We asked Babis about the remaining electroweak and bottom-quark uncertainties and he replies that the paper he wrote with Radja Boughezal and Frank Petriello, JHEP 0904 003 (2009) [arXiv:0811.3458] already includes a 1-2% uncertainty at NNLO. Given that the entire effect is 8% for a light Higgs, a 20% uncertainty on it has a small effect on the total cross section uncertainty.
Top-bottom interference effects are computed in JHEP 0910 068 (2009) [arXiv:0907.2362] and the earlier paper JHEP 0701 082 (2007) [arXiv:hep-ph/0611236].
Newer calculations of the WH and ZH associated-production cross sections should be included
These became available at a time too late for the summer 2010 Tevatron results. We will update our cross section inputs for the associated production mechanisms in our next update of the low-mass channels. We thank the authors for the updated calculations.
The CDF and DØ Collaborations,
and the Tevatron New Physics and Higgs Working Group
October 5, 2010.
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