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 matrixelementbased 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 fourmomenta for all
selected final state objects in the event. For instance, in the case
of "lepton+ 2 jet" events,
x=(p_{lepton}, p_{leading jet},
p_{second jet}). The reconstructed
missing E_{T} 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
signaltobackground discrimination, given x, an event
discriminant is defined as:
where the signal hypothesis `S` can be: schannel, tb or
tchannel 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 (P_{S}) and background
processes (P_{B}), 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: P_{SB}(x)
The ME event probability density function is defined as the properly
normalized differential cross section for an event characterized by
the reconstructed fourmomenta x:
where the differential cross section, shown below, is defined
as the integration of the protonantiproton differential cross
section
over the phase space (y) of the final state
particles.
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 schannel 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 tchannel
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




schannel for 2jet events

schannel for 3jet events

tchannel for 2jet events

tchannel for 3jet events

Feynman diagrams for background processes




Wbb for 2jet events

Wcg for 2jet events

Wgg for 2jet events

Wbbg for 3jet 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 onedimensional (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.
schannel discriminants



schannel discriminant for
tb Monte Carlo (cyan)
and Wbb Monte Carlo (green)

schannel discriminant for
tb Monte Carlo (cyan)
and Wjj Monte Carlo (light green)

schannel discriminant for
tb Monte Carlo (cyan)
and ttbar>l+jets Monte Carlo (red)

tchannel discriminants



tchannel discriminant for
tqb Monte Carlo (blue)
and Wbb Monte Carlo (green)

tchannel discriminant for
tqb Monte Carlo (blue)
and Wjj Monte Carlo (light green)

tchannel 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+jetsenriched sample that has exactly two jets, one of them btagged, and
H_{T}
(scalar sum of jet p_{T}, lepton p_{T} and MET)
less than 175 GeV, and (ii) a hard W+jetsenriched sample with exactly two jets, one of them btagged, and H_{T} greater than 300 GeV.
Data was compared with the expected backgrounds for several variable distributions.
Below the ME output distributions are showed for these crosscheck samples.


ME tb discriminant for the
soft W+jetsenhanced crosscheck sample
for events with two jets 
ME tqb discriminant for the
soft W+jetsenhanced crosscheck sample
for events with two jets 


ME tb discriminant for the
hard W+jetsenhanced crosscheck sample
for events with two jets 
ME tqb discriminant for the
hard W+jetsenhanced crosscheck sample
for events with two jets 
Note: The KS shown in the cross check and ME output plots is the
KolmogorovSmirnov 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



Highend of the ME tb discriminant
for events with 2 or 3 jets

Highend 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 schannel discriminant versus the
tchannel 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 nexttoleadingorder prediction of 2.9 pb.
Significance of the Result
A large ensemble of pseudodatasets (~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 socalled "pvalue" 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 Gaussianequivalent
standard deviations. Using a second ensemble of
pseudodatasets 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%.
Measured cross section in a large ensemble of pseudodatasets without single top content.
The significance of the the measurement is given by the fraction of pseudodatasets with
measured cross section greater than the measured cross section in the real data.
Email the single top subgroup leaders: Arán GarcíaBellido, Ann Heinson
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