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Issue #1:
Baglio and Djouadi suggest that we should use
central cross section values taken from NNLO calculations
rather than those from the NNLL+NNLO calculations used
currently. They argue that the additional contributions
to the total cross section that arise in NNLL+NNLO will
not survive experimental cuts and should therefore be
ignored.
Response:
We disagree with the suggested approach.
Our analyses incorporate a more accurate modeling of the
effects of experimental cuts on the signal acceptance
than what is obtainable via a parton-level calculation.
Our acceptances are obtained using events generated with
PYTHIA, which uses matrix element calculations followed
by a parton shower to model higher-order radiative effects.
In particular, the parton shower models "resummed" higher-
order radiative corrections due to multiple emission of
of soft and collinear partons. Although these higher-order contributions do not exactly match those included
in the analytic resummed calculations, the parton shower
model introduces soft and collinear effects beyond those
contained within the customary fixed-order perturbative
calculations. These simulated events are then re-weighted
in order to match the PYTHIA generator-level Higgs pT
spectrum, both in shape and normalization, with that
obtained directly from the NNLL+NNLO prediction. The
Higgs pT is the most important variable to model properly
as the boost of the dilepton system directly translates
into the efficiency of the selection cuts used in defining
our search samples (lepton pT, isolation, and missing ET).
The generated and re-weighted events are then passed
through a full simulation of the detector response. Based
on this approach, any changes in predicted yields as a
function of the reconstructed jet multiplicity are modeled
in a precise way, taking into account both physics and
instrumental effects. We have also checked the effect an
additional, subsequent re-weighting of our PYTHIA event
sample to match the rapidity distribution for the Higgs
obtained from a NNLO calculation and find that this results
in negligible changes to the measured signal acceptances.
Issue #2:
A wider variation over scales should be used to
determine uncertainties on the cross section.
Response:
There does not appear to be a consensus within the
theoretical community on this issue. We have consulted the
authors of the articles from which we take our current gluon
cross section values and uncertainties. Their responses
are included below. They disagree with the assertion that
their recommended choices of scale variations for determining
uncertainties is insufficient.
Response from Massimiliano Grazzini [S. Catani, D. de Florian, M. Grazzini, and P. Nason, J. High Energy Phys. 0307, 028, (2003)]
The uncertainty from missing higher order contributions is usually evaluated by varying the scales around the central value. This is certainly somewhat arbitrary, and one can of course increase the range of scale variations to be more conservative. We think that the uncertainty from scale variations (whatever way it is defined) should be contrasted with the difference between your reference prediction and the previous order. For example, if you vary the scales by a factor of 2 around μf=μr=MH, and you compare the LO and NLO uncertainty bands obtained in this way, you will see that the bands do not overlap. This is a consequence of the well known fact that NLO corrections are very large. At this order, one is forced to conclude that scale variations performed in the usual way underestimate the true uncertainty. However if you compare the NNLO with the NLO bands obtained in the same way, they do overlap. This means that already at NNLO the usual way of estimating the uncertainty leads to a result that is consistent with the difference with the previous order. These conclusions are made stronger if you add the effect of resummation. The NNLL+NNLO result has a smaller scale dependence with respect to the NNLO result. More importantly, the NNLL impact on the NNLO result is smaller with respect to the NNLO impact on the NLO result. All this points to a nice convergence of the perturbative series, and such a conclusion is confirmed by the approximate N^3LO computations based on soft approximations that have appeared in the last few years. In conclusion, we think that varying the scales by a factor of 3 as Djouadi and Baglio suggest, is exaggerated, or simply too conservative.
Response from Frank Petriello [C. Anastasiou, R. Boughezal, and F. Petriello, J. High Energy Phys. 0904, 003, (2009)]
Part of the motivation for performing higher-order calculations is to gain intuition into what contributions are important at higher orders. We can then use that intuition to do things such as select scales appropriately, in order to give the best possible prediction. With this in mind, we disagree with Baglio and Djouadi's choice of μ=mH for the default scale in their calculation. The structure of the logarithms (as discussed in NPB646 200, for example) suggests that μ ~ mH/2 is the more appropriate choice in that it minimizes their size. Recent work in SCET by Becher and Neubert also suggests based on other arguments that the scale choice should be less than mH. The better convergence of the perturbative series is further evidence that μ ~ mH/2. Note that these arguments hold not only for the inclusive cross section calculations but also for fixed-order perturbative calculations that attempt to incorporate experimental cuts. We disagree with Baglio and Djouadi's argument that the scale used in our calculation is simply chosen to mimic threshold resummation. With respect to assigning uncertainties, this can again be based on the experience obtained by studying the higher-order calculation. We know what the dominant contributions to the inclusive cross section are at higher-orders: the CAπ2 pieces that Becher and Neubert have recently re-emphasized and the threshold logs. We are fairly certain that no new surprises occur at N3LO; there is no parametric dependence that we are missing. In fact, we know from studies by Becher and Neubert that there is an additional 5% increase in the cross section occurs at N3LO from the CAπ2 terms. We also disagree with the philosophy of allowing large separations between μR and μF for determining scale uncertainties. In using this approach one artificially introduces a large ratio of scales in logs that we know does not appear in the all-orders result. From LHC studies of W/Z production (for which the perturbative expansion is claimed to be well-understood), we know that this approach increases the scale uncertainty by factors of 3 or 4. Looking at the numbers, it's clear to see that allowing a large separation between μR and μF is not correct. For these reasons, we disagree with the wide range that Baglio and Djoudai use for estimating scale uncertainties.
Issue #3:
Need to incorporate uncertainties on αs in
conjunction with our current PDF model uncertainties.
Response:
We agree with this statement and are appreciative
that the appropriate tools for incorporating uncertainties on
αs are now available from the various PDF fitting groups.
We plan to incorporate these uncertainties in the next analysis
round. Preliminary estimates indicate that PDF uncertainties
on the gluon fusion cross section will increase from about 8%
to around 13%. Our studies show that this increase will not
have a visible effect on our current exclusion region.
Update (July 2010)
As of ICHEP 2010, the results of CDF and D0 and the Tevatron
combination include the uncertainties on
αs in conjunction with the current PDF model uncertainties.
Issue #4:
Need to reconsider how we combine uncertainties
on the gluon fusion cross sections originating from scale
and PDF choices.
Response:
We believe that our current procedure, which
considers scale and PDF uncertainties as uncorrelated is
a reasonable approximation. Since PDF uncertainties come
primarily from experimental sources, we believe that these
should be mostly uncorrelated with choice of scale for the
perturbative cross section calculation. Studies have been
made to evaluate the procedure advocated by the authors.
The change in the predicted cross section due to each
MSTW eigenvector obtained from the combined PDF+αs
treatment at the low and high ends of the scale variation
was evaluated. We found that variations originating from
each of the eigenvectors had a negligible dependence on
the scale choice and concluded that our current approach
is sufficient.
The CDF and DØ Collaborations,
and the Tevatron New Physics and Higgs Working Group
May 13, 2010.
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