Measurement of the top quark mass in the lepton+jets channel using the template method with 1.7 fb^-1

Candidate events are selected for the lepton+jets channel of the ttbar system, ie where one W from the tops decays to a pair of hadrons, and the other W decays to a charged lepton (electron or muon) plus a neutrino. We require a well-identified electron or muon, large missing transverse energy and 4, at least one of which is identified tagged as arising from a b quark. We take advantage of different signal-to-background (S:B) and event shapes by splitting our sample into two non-overlapping subsamples, based on the number of jets with a b-tag (using CDF's secondary vertex tagger, SECVTX). Events with exactly one tag are required to have exactly 4 jets. In events with two or more tags, which have a higher S:B and more statistiacl power, we loosen the cut on the 4th jet and allow more than 4 jets. The event selection is summarized in the following table:

	2-tag	1-tag
Number of b-tags	>= 2	1
Jets 1-3 Et threshold (GeV)	>20	>20
4th jet Et threshold (GeV)	>12	>20
Extra jets (GeV)	Any	< 20

Top mass (m_t) reconstruction:

A chi-2 minimization is performed to reconstruct a top quark mass for each event. The fitter is based on the hypothesis that the event is ttbar: it contains W mass constraints on the hadronic and leptonic side and requires the two top masses in the event to be equal. Only the leading 4 jets are assigned to the four quark daughters from the top quark decay. The jet-parton assignment that yields the lowest chi2 after minimization is kept for further analysis, and the corresponding top mass (m_t) is used in our templates (see below). To improve the S:B and remove poorly reconstructed events, we make a cut at minimum chi2 < 9. M_top for the 2 subsamples are shown below.




W_jj reconstruction:

To measure the JES, mass templates of the W boson decaying hadronically (m_jj) are also constructed in addition to the top mass templates. The chi2 fitter used for m_t reconstructio is not used, though events failing the chi2 cut are also not used to measuring the JES. In 2-tag events, there is only one dijet mass from among the leading 4 jets consistent with b-tagging (ie not tagged as a b). In 1-tag events, there are 3 dijet masses consistent with b-tagging. We take the single dijet mass closest to the well known W mass as the single value of m_jj per event. M_jj for the 2 subsamples for 3 different values of the JES in the detector are shown below.


Kernel Density Estimation:

We use a non-parametric Kernel Density Estimate-based approach to forming probability density functions from fully simulated Pythia MC. The probability for an event with an observale x is given by the linear sum of contributions from all entries in the MC:

Here, f(x) is the probability to observe x given some MC sample with known mass and JES (or the background). The kernel function K is a normalized function that adds varying probability to a measurement at x depending on its distance from xi. The smoothing parameter h is a number that determines the width of the kernel. Larger values of h smooth out the density estimate, and smaller values of h keep most of the probability weight near xi. We use an adaptive method in which the value of h = h(f(x_i)). The peak of the distribution, we use smaller smoothing. In the tails of the distribution, where statistics are poor and we are sensitive to statistiacl fluctuation, we use a larger amount of smoothing. The figures below show the effect of using adaptive smoothing on both the reconstructed masses and the dijet masses. The peak of the distribution changes shape (as can be seen on the linear scale), and we smooth out fluctuations in the tails (as can be seem on the log scale):

KDE can be expanded to two dimensions by multiplying together the two kernels:


The two-dimensional density estimates for an input signal mass of 170 GeV/c2 are shown here:
In order to propertly normalize our density estimates, we define hard boundary cuts on our density estimates and make the same cut data, ensuring that the integral of our estimates is 1.0. For both categories, we remove events with mtreco < 110 or mtreco > 350 (GeV/c2). For the 1-tag sample, we also remove any events with the dijet mass < 50 or the dijet mass > 115 (GeV/c2). This cut is loosened for the 2-tag category, in which we remove any events with wjj < 30 or wjj > 120.

Background

The background sources and their expected fraction of the total background are given in the table below. The backgrounds are dominated by real W boson production in association with high-pt jets. The absolute normalization of W+jets is determined from the data, but the relative normalization between the different flavor samples is taken from MC. The expected number of events for single-top and diboson background are taken from theoretical cross-sections and MC predictions. We expect a total of 44.6 +/- 8.7 2-tag background events and 12.3 +/- 3.3 1-tag background events after all boundary cuts, but before chi2 cuts. This a priori information on the expected number of background events is used as a gaussian constraint in the likelihood fit for each subsample. The fractions that go into these numbers (after boundary cuts but before chi2 cuts) are as follows:

	2-tag	1-tag
W+LF (mistags + QCD)	0.28	0.49
W+bb+jets	0.44	0.06
W+cc+jets	0.06	0.12
W+c+jets	0.02	0.08
s-channel single-top	0.09	0.02
t-channel single-top	0.06	0.03
WW	0.02	0.04
WZ	0.03	0.01
ZZ	0.002	0.001

	2-tag	1-tag
Observed number of events after all cuts	89	218
Expected background after all cuts	6.4 +/- 1.7	36.6 +/- 7.1

The background templates are independent of Mtop and JES (possible JES dependence is taken as a systematic), and are shown here:

Technique and validation

With 2d density functions in place, we compare the data (or pseudodata) to every one of 2002 points in a grid of top quark masses and shifts in JES. At each point, MINUIT is used to fit for the expected number of signal background events separately in each of the two subsamples. The expected number of backgrounds events is subject to a Gaussian constraint to the numbers given above. The JES is also subject to a prior, constraining shifts in JES to be near the nominal CDF value (0.0) within its uncertainy (1 sigma_c). The grid of values of -ln(L) is fit with a 2-d parabola that includes a cross-term to account for the correlation between the top quark mass and the JES. We profile out the JES to measure a value for the top quark mass that includes an uncertainty due to the JES. Before examining the data, we perform a variety of checks to ensure that our method is unbiased and performs as expected over an a priori determined range of top masses from 165-180 GeV/c2 and JES from -1 to 1 sigma_c. We run pseudoexperiments using the observed number of datas, with the background fluctuating around the expectation. We find no significant bias in the top mass as a function of true mass. In the plot below, we fit only to points with JES = 0.0.

Systematic uncertainties

We have estimated the largest sources of systematic uncertainty and summarize these below. The jet energy scale is treated as a nuisance parameter in this analysis, and is constrained both by the a priori calorimeter calibration and the statistical information from the W mass. It is thus not considered a traditional source of systematic uncertainty, but included in the uncertainty arising from the likelihood fit.

Systematic Source	Delta Mtop (GeV/c2)
b-jet energy scale	0.6
Residual JES	0.6
Background JES	0.4
ISR	0.4
FSR	0.2
Parton Distribution Functions	0.2
Generators	0.3
Background Shape	0.2
Background composition	0.2
QCD modeling	0.1
Monte Carlo statistics	0.1
TOTAL	1.1

Fit and results

The likelihood fit to the data returns Mtop = 171.6 +/- 2.0 GeV/c2 (stat+JES). Delta ln(L) contours that can be thought of as roughly corresponding to 1, 2 and 3 sigma contours are shown below, along with the distribution of returned stat+JES errors from pseudoexperiments with the observed numer of events. 23.4% of pseudoexperiments report a smaller error than the observed value.



We check the coverage of our the pulls out of our machinery by examining differential pull distributions, which show the pull width as a function of reported error. Using 3 input masses close to the measured mass to increase statistics, we find the following distribution:



The value of the pull width on the best-fit line at our measured error is 1.047, which we use to inflate our statistical uncertainty.

We run a variety of cross-checks and fits on the data. We fit removing the JES prior constraint and also without the background constraint, showing that the priors do not significantly change our result. A 1d fit for only the top quark mass allows us to disentable the statistical part of the stat+JES error (1.5 GeV/c2) from the 1st-order JES error (also 1.5 GeV/c2). We also fit the 1-tag and 2-tag samples separately, and perform measurements dividing by lepton type. Results of cross-checks, with all errors uncorrected, are here:

	Mass (GeV/c2)
Nominal measurement	171.6 +/- 2.0
No background prior constraint	171.7 +/- 2.0
No JES prior constraint	171.6 +/- 2.1
1-tag only	167.5 +/- 3.1
2-tag only	175.0 +/- 2.6
Electron-only (no bkgd prior constraint)	170.6 +/- 2.9
Muon-only (no bkgd prior constraint)	172.3 +/- 3.0
1d-only fit (no JES)	171.7 +/- 1.5

1-d projections of the templates at JES = 0.0 with the best-fit global mass and constrained background are shown here:

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Jahred Adelman for the TMT group

Last modified August 9, 2007