Evidence for Production of Single Top Quarks
using Matrix Elements

The DØ Collaboration

December 2006



[Analysis Method]   [ME Performance]   [Cross Checks]   [ME Outputs]   [Cross Sections]


Abstract

The DØ Collaboration presents first evidence for the production of single top quarks at the Fermilab Tevatron ppbar collider. Using a 0.9 fb-1 dataset, we use leading order matrix elements to separate signal from background and measure σ(ppbar→ tb + X, tqb + X) = 4.6+1.8-1.5 pb. The probability to measure a cross section at this value or higher in the absence of signal is 0.21%, corresponding to a 2.9 standard deviation significance.



Analysis Method

Matrix Element Discriminants

The matrix-element-based single top quark search attempts to make use of all the available kinematic information in the event. Therefore, x (the set of quantities with which the analysis depends) represents the set of reconstructed four-momenta for all selected final state objects in the event. For instance, in the case of "lepton+ 2 jet" events, x=(plepton, pleading jet, psecond jet). The reconstructed missing ET is not explicitly used, as it results from imposing momentum conservation in the transverse plane, and thus it does not represent an independent observable. The method is general enough that information related to fragmentation or b tagging, can in principle also be incorporated. In fact, information about the latter is already being used in the current analysis. In order to achieve optimal signal-to-background discrimination, given x, an event discriminant is defined as:

Prob

where the signal hypothesis `S` can be: s-channel, tb or t-channel single top quark production. The signal and background probability density functions as a function of x are computed numerically based on the normalized differential cross section for signal (PS) and background processes (PB), respectively. Since the differential cross section for the process of interest is proportional to the matrix element squared, we call this method the "Matrix Element (ME) Method." We shall refer to the above equation as the "ME discriminant."

Event Probability Density: PS|B(x)

The ME event probability density function is defined as the properly normalized differential cross section for an event characterized by the reconstructed four-momenta x:

Prob

where the differential cross section, shown below, is defined as the integration of the proton-antiproton differential cross section over the phase space (y) of the final state particles.

Prob

The function W(x,y) represents the probability with which an event y that is produced by a hard scatter collision is reconstructed as event x in the detector.

For events with two jets, the background probability is defined as the weighted sum of the probabilities to produce a Wbb, a Wcg, and Wgg event. For events with three jets, the background probability is defined as the probability of originating from the Wbbg process. The s-channel signal uses a 2->2 ud->tb matrix element for events with two jets and a 2->3 ud->tbg matrix element for three jet events. The t-channel signal uses a 2->2 ub->td matrix element for events with two jets and a 2->3 ug->tdb~ matrix element for three jet events. Representative Feynman diagrams for all these processes are shown below.

Feynman diagrams for single top signal processes

s-channel for 2-jet events s-channel for 3-jet events t-channel for 2-jet events t-channel for 3-jet events

Feynman diagrams for background processes

Wbb for 2-jet events Wcg for 2-jet events Wgg for 2-jet events Wbbg for 3-jet events

Note: Clicking on a plot will give the .eps version. Right click and "View Image" will get the full resolution .png version.



Discriminant Performance

This section contains overlayed plots of the one-dimensional (1D) tb and tqb discriminants for signal and background events. The events in the plots come from the combined e+mu with 1 and 2 tags and 2 and 3 jets channel. All plots are normalized to unit area to emphasize differences in shape for signal and background events.

s-channel discriminants

s-channel discriminant for
tb Monte Carlo (cyan)
and Wbb Monte Carlo (green)
s-channel discriminant for
tb Monte Carlo (cyan)
and Wjj Monte Carlo (light green)
s-channel discriminant for
tb Monte Carlo (cyan)
and ttbar->l+jets Monte Carlo (red)

t-channel discriminants

t-channel discriminant for
tqb Monte Carlo (blue)
and Wbb Monte Carlo (green)
t-channel discriminant for
tqb Monte Carlo (blue)
and Wjj Monte Carlo (light green)
t-channel discriminant for
tqb Monte Carlo (blue)
and ttbar->l+jets Monte Carlo (red)


Cross Checks

Two cross check samples were created for each discriminant tb and tqb: (i) a soft W+jets-enriched sample that has exactly two jets, one of them b-tagged, and HT (scalar sum of jet pT, lepton pT and MET) less than 175 GeV, and (ii) a hard W+jets-enriched sample with exactly two jets, one of them b-tagged, and HT greater than 300 GeV. Data was compared with the expected backgrounds for several variable distributions. Below the ME output distributions are showed for these cross-check samples.

Wjets cross check
		    sample for tb discriminant Wjets cross check
		    sample for tb discriminant
ME tb discriminant for the
soft W+jets-enhanced cross-check sample
for events with two jets
ME tqb discriminant for the
soft W+jets-enhanced cross-check sample
for events with two jets
TTbar cross check
		    sample for tb discriminant TTbar cross check
		    sample for tb discriminant
ME tb discriminant for the
hard W+jets-enhanced cross-check sample
for events with two jets
ME tqb discriminant for the
hard W+jets-enhanced cross-check sample
for events with two jets

Note: The KS shown in the cross check and ME output plots is the Kolmogorov-Smirnov test statistic for the data with respect to the sum of the signal and background yields.


Matrix Element Outputs

The following plots shows a comparison of data with the background+signal expectation for the tb and tqb discriminants for events with two jets and three jets combined.

ME tb discriminant for
events with 2 or 3 jets
ME tqb discriminant for
events with 2 or 3 jets
High-end of the ME tb discriminant
for events with 2 or 3 jets
High-end of the ME tqb discriminant
for events with 2 or 3 jets


Cross Section Measurements

We apply a Bayesian approach to measure the single top quark production cross section. We form a binned likelihood from a 2D histogram of the s-channel discriminant versus the t-channel discriminant for all channels (lepton flavor, jet multiplicity, and tag multiplicity) of the matrix element discriminant for the tb+tqb analysis. We assume a Poisson distribution for the observed counts and flat nonnegative prior probabilities for the signal cross sections. Systematic uncertainties and their correlations are taken into account by integrating over the signal acceptances, background yields, and integrated luminosity with Gaussian priors for each systematic uncertainty. The final posterior probability density is computed as a function of the production cross section. For each analysis, we measure the cross section using the position of the posterior density peak and we take the 68% asymmetric interval about the peak as the uncertainty on the measurement. The tb+tqb posterior for the expected result (Standard Model cross section) and observed result is shown below:

Expected Bayesian posterior probability density Measured Bayesian posterior probability density

We obtain the following results:

s+t channels  σ(ppbar → tb + X, tqb + X)  = 4.6+1.8-1.5 pb

which is consistent with the SM next-to-leading-order prediction of 2.9 pb.

Significance of the Result

A large ensemble of pseudo-datasets (~40,000 entries) with all systematic uncertainties included, has been generated with zero signal content, i.e., they contain only events from the background. We have performed the matrix elements analysis on each of these datasets, including full systematic error treatment, and have measured the cross section for tb+tqb in each set. We measure the probability that data containing no single top quark events could fluctuate to give us at least our measured cross section value. This is the so-called "p-value" and is widely used to estimate the significance of a measurement.

We find that the probability that the background fluctuates up to produce the measured cross section of 4.6 pb or greater is 0.21%, corresponding to a significance for our result of 2.9 Gaussian-equivalent standard deviations. Using a second ensemble of pseudo-datasets which includes a SM tb+tqb signal with 2.9 pb cross section, with all systematic uncertainties included, we find the probability to measure a cross section of at least 4.6 pb to be 21%.

Single top significance (p-value)
Measured cross section in a large ensemble of pseudo-datasets without single top content.
The significance of the the measurement is given by the fraction of pseudo-datasets with
measured cross section greater than the measured cross section in the real data.

E-mail the single top subgroup leaders: Arán García-Bellido, Ann Heinson

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