**Issue #1:**

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).

**Response:**

Our procedure is to run Pythia Monte Carlo, and to reweight
the p_{T} 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
p_{T} spectrum is calculated by HqT at NNLL+NLO.
We then scale the total yield to the NNLL+NNLO predictions at
μ_{f}=μ_{r}=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 m_{H}=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 m_{H}/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.

**Issue #2:**

The scale uncertainty should vary μ_{r} and μ_{f} 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).

**Response:**

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. C**62** 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.

**Issue #3:**

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.

**Response:**

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.

**Issue #4:**

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).

**Response:**

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 m_{H}
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.

**Issue #5:**

We should use the ABKM PDF set as an alternate prediction to
set the PDF uncertainty.

**Response:**

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.

**Issue #6:**

Combination of scale and
α_{s}+PDF uncertainties -- linear
or quadrature?

**Response:**

We agree that the gluon density in the proton depends on the
factorization and renormalization scale and that the fact that the
PDF+α_{s} predictions vary with the scale choice means we should
include this dependence via the PDF+α_{s} 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+α_{s} 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+α_{s}) uncertainty with the scale at μ=m_{H}/4 and μ=m_{H}. 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+α_{s}error is an error due to the parameterization of the densities which are extracted at a fixed value of Q^{2}= 1 GeV^{2}, and a fixed value of α_{s}at μ=M_{Z}. 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+α_{s}" the piece that's scale
independent, and the uncertainty of the PDF and α_{s} 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.

**Issue #7:**

Remaining uncertainties due to finite-order electroweak and
bottom corrections should be bigger.

**Response:**

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].

**Issue #8:**

Newer calculations of the WH and ZH associated-production
cross sections should be included

**Response:**

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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