Before making a measurement using data, it is useful to calculate the expected sensitivity of these analyses.
Furthermore, this expected sensitivity is used to optimize the choice of parameters in the analyses. For each case under consideration we calculate an expected Bayes ratio as defined in Sec. XVIII C. The highest Bayes ratio corre- sponds to the optimal parameter choice.
Table XV shows the expected Bayes ratio for each possible combination of analysis channels in the DT analy- sis. It can be seen from the numbers in the table that combining the two single top-quark signals (i.e., searching for tbþtqb together) results in the best expected sensi- tivity. The single-tag two-jet channel contributes the most to this sensitivity, as expected from the high signal accep- tance and reasonable signal-to-background ratio, but the addition of the other channels does improve the result; including the poorer ones does not degrade it. While the result from this table refers specifically to the DT analysis, the conclusions hold for all three multivariate techniques. Therefore, from this point onward, the 2–4 jets 1–2 tags result using electrons and muons in the tbþtqbchannel will be considered as default (2–3 jets for the ME analysis).
B. Expected cross sections
We measure the expected cross sections for the various channels by setting the number of data events in each channel equal to the (noninteger) expected number of background events plus the expected number of signal events (using the SM cross section of 2.86 pb at mtop¼ 175 GeV), and obtain the following results:
expðpp !tbþX; tqbþXÞ ¼2:7þ1:6 1:4 pbðDTÞ ¼2:7þ1:5 1:5 pbðBNNÞ ¼2:8þ1:6 1:4 pbðMEÞ:
The expected cross sections agree with the input cross section. The small deviation, less than 10%, is from the nonsymmetric nature of several of the systematic uncer- tainties, in particular, the jet energy scale and btagging. This effect is also observed in the pseudo–data sets.
The linearity of the methods to measure the appropriate signal cross section was discussed in Sec. XVI and Fig.16, and no calibration is necessary based on those results.
TABLE XV. Expected Bayes ratios from the decision tree analysis, including systematic uncertainties, for many combinations of analysis channels. The best values from all channels combined are shown in bold type.
Expected Bayes Ratios
1–2tags, 2–4jets eþ, 2–4jets eþ, 1–2tags All e-chan -chan 1 tag 2 tags 2 jets 3 jets 4 jets channels
tb 1.1 1.1 1.1 1.1 1.2 1.0 1.0 1:2
tqb 2.5 1.8 4.5 1.1 3.1 1.5 1.1 4:7
C. Measured cross sections
The cross sections measured using data with the three multivariate techniques are shown in Fig.21 where each measurement represents an independent subset of the data, for example, the 2-jet sample with 1btag in the electron channel.
The full combination of available channels (the most sensitive case) yields the Bayesian posterior density func-
tions shown in Fig.22and cross sections of
obsðpp !tbþX; tqbþXÞ ¼4:9þ1:4 1:4 pbðDTÞ ¼4:4þ1:6 1:4 pbðBNNÞ ¼4:8þ1:6 1:4 pbðMEÞ:
Figure23shows the high-discriminant regions for each of the multivariate methods, with the signal component
FIG. 22 (color online). Expected SM and observed Bayesian posterior density distributions for the DT, BNN, and ME analyses. The shaded regions indicate 1 standard deviation above and below the peak positions.
FIG. 21 (color online). Summaries of the cross section measurements using data from each multivariate technique. The left plot is DT, the middle one is BNN, and the right plot is ME.
normalized to the cross section measured from data. Clearly, a model including a signal contribution fits the data better than does a background-only model.
To further illustrate the excess of data events over back- ground in the high-discriminant region, Fig.24shows three variables that are inputs to the DT analysis: invariant mass of leptonþbtagged jetþneutrino,Wtransverse mass, and so-called ‘‘Q’’ (lepton charge times of the leading untagged jet). They are each shown for low dis- criminant output, high output, and very high output. The excess of data over a background-only model clearly in- creases as the discriminant cut is increased.
The DT analysis has also measured thes- andt-channel cross sections separately. The cross sections are found to be
obsðpp !tbþXÞ ¼1:00:9 pb;
obsðpp !tqbþXÞ ¼4:2þ1:8
1:4 pb:
These measurements each assume the standard model value of the single top-quark cross sections not being measured, since thes-channel measurement considers the t-channel process as a background and vice versa.
We can remove the constraint of the standard model ratio and form the posterior probability density as a func- tion of the tb and tqb cross sections. This model- independent posterior is shown in Fig. 25 for the DT analysis, using thetbþtqbdiscriminant. The most prob-
able value corresponds to cross sections ofðtbÞ ¼0:9 pb
and ðtqbÞ ¼3:8 pb. Also shown are the 1, 2, and 3 standard deviation contours. While this result favors a higher value for the t-channel contribution than the SM expectation, the difference is not statistically significant. Several models of new physics that are also consistent with this result are shown in Ref. [89].