Evidence for production of single top quarks

We present first evidence for the production of single top quarks in the D0 detector at the Fermilab Tevatron $p\overline{p}$ collider. The standard model predicts that the electroweak interaction can produce a top quark together with an antibottom quark or light quark, without the antiparticle top-quark partner that is always produced from strong-coupling processes. Top quarks were first observed in pair production in 1995, and since then, single top-quark production has been searched for in ever larger data sets. In this analysis, we select events from a $0.9{\mathrm{fb}}^{-1}$ data set that have an electron or muon and missing transverse energy from the decay of a $W$ boson from the top-quark decay, and two, three, or four jets, with one or two of the jets identified as originating from a $b$ hadron decay. The selected events are mostly backgrounds such as $W+\mathrm{\text{jets}}$ and $t\overline{t}$ events, which we separate from the expected signals using three multivariate analysis techniques: boosted decision trees, Bayesian neural networks, and matrix-element calculations. A binned likelihood fit of the signal cross section plus background to the data from the combination of the results from the three analysis methods gives a cross section for single top-quark production of $\sigma (p\overline{p}\to tb+X,tqb+X)=4.7\pm 1.3\mathrm{pb}$. The probability to measure a cross section at this value or higher in the absence of signal is 0.014%, corresponding to a 3.6 standard deviation significance. The measured cross section value is compatible at the 10% level with the standard model prediction for electroweak top-quark production. We use the cross section measurement to directly determine the Cabibbo-Kobayashi-Maskawa quark mixing matrix element that describes the $Wtb$ coupling and find $|V_{tb}{f}_1^L|={1.31}_{-0.21}^{+0.25}$, where $f_1^L$ is a generic vector coupling. This model-independent measurement translates into $0.68<|V_{tb}|\le 1$ at the 95% C.L. in the standard model.

Figure 1

Main tree-level Feynman diagrams for (a) $s$-channel single top-quark production, and (b) $t$-channel production.

Figure 2

Measurements of the scale factor $\alpha$ used to convert the fraction of $Wb\overline{b}$ and $Wc\overline{c}$ events in the $W+\mathrm{\text{jets}}$ background model from leading order to higher order. The points are the measured correction factor in each data set. The solid line is the average of these values. The dot-dashed inner band shows the uncertainty from the fit to the eight data points. The dashed outer line shows the uncertainty on $\alpha$ used in the analysis to allow for the assumption that the scale factor should be the same for $Wb\overline{b}$ and $Wc\overline{c}$, and for small differences in the shapes of distributions between the $W+\mathrm{\text{heavy}}$ flavor and $W+\mathrm{\text{light}}$ flavor jets.

Figure 3

The tag-rate functions (TRFs) used to weight the MC events according to the probability that they should be $b$ tagged. In plots (a)–(d), the points show the neural network $b$ tagging algorithm (the “tagger”) applied directly to the MC events. The upper line that passes through the points is the result of the tag-rate functions, before scaling-to-data, being applied to the MC events to reproduce the result from the tagger. The lower line, with dotted error band, shows the tag-rate functions after they have been scaled to match the efficiency of the NN $b$ tagging algorithm applied to data. In plot (e), the lines show the (scaled) tag-rate functions that are applied to MC events.

Figure 4

The factors used to normalize the $W+\mathrm{\text{jets}}$ background model to pretagged data in each analysis channel.

Figure 5

The first row shows pretagged distributions for the $p_T$ of the electron, the $p_T$ of the leading jet, and the reconstructed $W$ boson transverse mass. The second row shows the same distributions after tagging for events with exactly one $b$-tagged jet. The hatched area is the $\pm 1\sigma$ uncertainty on the total background prediction.

Figure 6

The first row shows pretagged distributions for the $p_T$ of the muon, the $p_T$ of the leading jet, and the reconstructed $W$ boson transverse mass. The second row shows the same distributions after tagging for events with exactly one $b$-tagged jet. The hatched area is the $\pm 1\sigma$ uncertainty on the total background prediction.

Figure 7

Comparison of SM signal, backgrounds, and data after selection and requiring at least one $b$-tagged jet for six discriminating individual object variables. Electron and muon channels are combined. The plots show (a) the lepton transverse momentum and (b) pseudorapidity, (c) the leading jet transverse momentum and (d) pseudorapidity, (e) the second leading jet transverse momentum and (f) pseudorapidity. The hatched area is the $\pm 1\sigma$ uncertainty on the total background prediction.

Figure 8

Comparison of SM signal, backgrounds, and data after selection and requiring at least one $b$-tagged jet for six discriminating event kinematic variables. Electron and muon channels are combined. Shown are (a) the invariant transverse mass of the reconstructed $W$ boson, (b) the invariant mass of the $b$-tagged jet and the $W$ boson, (c) the invariant mass of all jets, (d) the missing transverse energy, (e) the scalar sum of the transverse momenta of jets, lepton, and neutrino, (f) the invariant mass of all final-state objects. The hatched area is the $\pm 1\sigma$ uncertainty on the total background prediction.

Figure 9

Comparison of SM signal, backgrounds, and data after selection and requiring at least one $b$-tagged jet for three discriminating angular correlation variables. Electron and muon channels are combined. Shown are (a) the angular separation between the two leading jets, (b) the cosine of the angle between the reconstructed $b$-tagged top quark in the center-of-mass rest frame and the lepton in the $b$-tagged top-quark rest frame, and (c) the charge of the lepton multiplied by the pseudorapidity of the leading untagged jet. The hatched area is the $\pm 1\sigma$ uncertainty on the total background prediction.

Figure 10

Shape comparison between the $s$- and $t$-channel signals and the backgrounds in the most discriminating variables for the decision tree analysis chosen from the $e+2\mathrm{\text{jets}}/1\mathrm{\text{tag}}$ channel. Shown are (a) the invariant mass of all jets, (b) the invariant mass of the $b$-tagged jet and the $W$ boson, (c) the invariant mass of the two leading jets and the $W$ boson, (d) the cosine of the $b$-tagged jet and the lepton in the reconstructed $b$-tagged top-quark rest frame, (e) the charge of the lepton multiplied by the pseudorapidity of the leading untagged jet, (f) the scalar sum of the transverse momenta of all the jets, (g) the invariant mass of all jets minus the best jet, (h) the invariant transverse mass of the reconstructed $W$ boson, and (i) the cosine of the angle between the reconstructed $b$-tagged top quark in the center-of-mass rest frame and the lepton in the $b$-tagged top-quark rest frame. All histograms are normalized to unit area.

Figure 11

Graphical representation of a decision tree. Nodes with their associated splitting test are shown as circles and terminal nodes with their purity values are shown as leaves. An event defined by variables ${\mathbf{x}}_{\mathbf{i}}$ of which $H_T<242\mathrm{GeV}$ and $m_{\mathrm{top}}>162\mathrm{GeV}$ will return $D({\mathbf{x}}_{\mathbf{i}})=0.82$, and an event with variables ${\mathbf{x}}_{\mathbf{j}}$ of which $H_T\ge 242\mathrm{GeV}$ and $p_T\ge 27.6\mathrm{GeV}$ will have $D({\mathbf{x}}_{\mathbf{j}})=0.12$. All nodes continue to be split until they become leaves.

Figure 12

BNN input variables according to their RuleFit ranking for the $\mathrm{\text{electron}}+2\mathrm{\text{jets}}/1\mathrm{\text{tag}}$ channel.

Figure 13

For plots (a)–(d), DT and BNN discriminant outputs for $tb+tqb$ in the $e+\mathrm{\text{jets}}$ channel (left column) and $\mu +\mathrm{\text{jets}}$ channel (right column) for events with two jets of which one is $b$ tagged. Plots (e) and (f) show the ME discriminant outputs for $tb$ in the $e+\mathrm{\text{jets}}$ channel, for two-jet and three-jet events, respectively. Plots (g) and (h) show the ME discriminants for $tqb$ in the $b$ tagged $e+\mathrm{\text{jets}}$ channel, for two-jet and three-jet events, respectively. All histograms are normalized to unity.

Figure 14

For plots (a)–(d), DT and BNN $tb+tqb$ signal efficiency versus background efficiency in the $e+\mathrm{\text{jets}}$ channel (left column) and $\mu +\mathrm{\text{jets}}$ channel (right column) for events with two jets of which one is $b$ tagged. Plots (e)–(h) show the ME signal versus background efficiency for $tb$ signal (third row) and $tqb$ signal (fourth row), for $b$ tagged $e+\mathrm{\text{jets}}$ events with two jets (left column) and three jets (right column). These curves are derived from the discriminants shown in Fig. 13.

Figure 15

The discriminant outputs of the three multivariate discriminants: (a) DT, (b) BNN, (c) ME $s$-channel, and (d) ME $t$-channel discriminants. The signal components are normalized to the expected standard model cross sections of 0.88 pb and 1.98 pb for the $s$- and $t$-channels, respectively. The hatched bands show the $1\sigma$ uncertainty on the background.

Figure 16

Ensemble average of measured cross section as a function of the input single top-quark cross section for the (a) DT, (b) BNN, and (c) ME analyses.

Figure 17

DT outputs from the $W+\mathrm{\text{jets}}$ (upper row) and $t\overline{t}$ (lower row) cross-check samples for $e+\mathrm{\text{jets}}$ events (left column) and $\mu +\mathrm{\text{jets}}$ events (right column).

Figure 18

BNN outputs from $W+\mathrm{\text{jets}}$ (upper row) and $t\overline{t}$ (lower row) cross-check samples for $e+\mathrm{\text{jets}}$ events (left column) and $\mu +\mathrm{\text{jets}}$ events (right column).

Figure 19

$H_T<175\mathrm{GeV}$ cross-check plots in two-jet (upper row) and three-jet (lower row) events for the $s$-channel ME discriminant (left column) and the $t$-channel ME discriminant (right column). The plots have electrons and muons, one and two $b$ tags combined.

Figure 20

$H_T>300\mathrm{GeV}$ cross-check plots in two-jet (upper row) and three-jet (lower row) events for the $s$-channel ME discriminant (left column) and the $t$-channel ME discriminant (right column). The plots have electrons and muons, one and two $b$ tags combined.

Figure 21

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.

Figure 22

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.

Figure 23

Zooms of the high-discriminant output regions of the three multivariate discriminants: (a) DT, (b) BNN, (c) ME $s$-channel, and (d) ME $t$-channel discriminants. The signal component is normalized to the cross section measured from data in each case. The hatched bands show the $1\sigma$ uncertainty on the background.

Figure 24

The $b$-tagged top-quark mass (top row), $W$ boson transverse mass (second row), and $Q(\mathrm{\text{lepton}})\times \eta (\mathrm{\text{untag}}1)$ (third row) for the $tb+tqb$ analysis with a low ($<0.3$, left column), high ($>0.55$, middle column), and very high ($>0.65$, right column) DT output, for lepton flavor $(e,\mu )$, number of $b$-tagged jets (1,2), and jet multiplicity (2,3,4) combined. Hatched areas represent the systematic and statistical uncertainties on the background model. The signal cross section is the measured one (4.9 pb).

Figure 25

Posterior probability density as a function of ${\sigma }_{tb}$ and ${\sigma }_{tqb}$, when both cross sections are allowed to float in the fit of the $tb+tqb$ DT analysis. Shown are the contours of equal probability density corresponding to 1, 2, and 3 standard deviations and the location of the most probable value, together with the SM expectation.

Figure 26

Distributions of the measured cross sections from (a) the individual analyses, and (b) the combined analysis, using the SM ensemble with systematics.

Figure 27

The measured single top-quark cross sections from the individual analyses and their combination.

Figure 28

Distributions of the cross sections measured from data by the three analyses and their combination, using the background-only ensemble. The arrow shows the combined cross section measurement, 4.7 pb.

Figure 29

The $p$-value computed from the $\mathrm{SM}\mathrm{\text{−}}\mathrm{\text{signal}}+\mathrm{\text{background}}$ ensemble versus the $p$-value from the background-only ensemble for reference cross sections varying monotonically from 0–10 pb. For a given significance, that is, the probability to reject the background-only hypothesis if true, the power is the probability to accept the $\mathrm{\text{signal}}+\mathrm{\text{background}}$ hypothesis if it is true. For a given significance, one wants the power to be a large as possible.

Figure 30

The posterior density distributions for $|V_{tb}{|}^2$ for (a) a nonnegative flat prior, and (b) a flat prior restricted to the region [0,1] and assuming $f_1^L=1$. The dashed lines show the positions of the 1, 2, and 3 standard deviation distances away from the peak of each curve.