Contents

• Abstract
• Event selection
• Signal likelihood calculation
• Background handling
• Method validation
• Data results
• Systematics
• Conclusion

• Public note (CDF-9025, September 2007)
• CDF internal page (password protected)

Note: to download high-resolution Encapsulated PostScript (.eps) versions of the plots and figures, click on the plot. There are also .eps versions as well as LaTeX source available for the tables.

Abstract

We report a measurement of the top quark mass obtained from proton-antiproton collisions at a center-of-mass energy of 1.96 TeV at the Fermilab Tevatron using the CDF II detector. Our method uses a matrix element integration method for the signal and a neural network discriminant to identify background events. We employ an "effective propagator" in the matrix element to take into account incorrect assumptions about some kinematic variables. We compute a 2D likelihood as a function of m_top and JES, an overall factor that scales all jet energies. We apply a cut to the likelihood in order to reduce the effect of badly reconstructed events. We then extract a value for the top mass. This version of the analysis uses 1.7 fb^-1 of integrated luminosity, and requires events with a lepton and four high energy jets in the |η| ≤ 2 region with at least one jet tagged as coming from a b quark. We observe 293 events passing all of our cuts and obtain a final value of m_t = 172.7 ± 1.3 (stat.) ± 1.2 (JES) ± 1.2 (syst) GeV/c² = 172.7 ± 2.1 (total) GeV/c².

Event selection

In our analysis, we look for events in which ttbar pairs are produced, each decays into a W boson and a b quark, and then one W decays into a neutrino and a lepton (meaning, in this paper, an electron or muon) and the other W decays into a quark-antiquark pair; this is called the "lepton + jets" channel."

We identify top mass candidates in this channel by requiring four high energy jets from the four quarks and a W decay into a lepton and a neutrino. Specifically, for the lepton we require either an identified electron with E_T > 20 GeV or an identified muon with p_T > 20 GeV/c in the central region of the detector. The neutrino is identified by requiring a missing E_T > 20 GeV in the event. For the jets, we require exactly 4 jets with E_T > 20 GeV and |η| ≤ 2, where the jet energies have been corrected for non-uniform detector response, calorimeter stability, and nonlinear response to particle momenta. The missing E_T is also corrected for muons and jet response. In addition, at least one of the jets must be tagged as a b-jet using a secondary vertex tagging algorithm. With these selection criteria, we observe a total of 343 events in the data.

The background to this signal consists of three main sources: events where a W is produced in conjunction with heavy flavor quarks, events where a W is produced with light flavor quarks which are mistagged, and QCD events where a jet is misidentified as an electron. There are also smaller contributions from diboson (WW, WZ, or ZZ) production, events where a Z produces a lepton pair, and single-top production; we do not consider these events directly, but rather increase the contribution from W+light (for diboson and Z) and W+heavy flavor (for single-top) to include the contributions from these sources.

We use a variety of Monte Carlo samples to test and calibrate our method and evaluate the backgrounds to ttbar production. For signal events, we use ttbar events generated at a variety of top masses from 152 GeV/c² to 190 GeV/c² by the PYTHIA generator. The non-W QCD background is derived from data with non-isolated leptons, while the other backgrounds are generated using the ALPGEN generator with parton showering by PYTHIA. Overlaps in the W+partons samples are removed using the ALPGEN jet-parton matching along with a jet-based heavy flavor overlap removal algorithm.

The background estimate is shown in the table below:

The transfer functions, acceptance, and normalization described below are obtained from ttbar events generated by the HERWIG generator, so we also cross-check against HERWIG events as well.

Signal likelihood calculation

Our signal likelihood calculation is performed by integrating over the matrix element using the following formula:

eps version | LaTeX source

This likelihood gives us the probability that we observe in our detector an event with kinematic variables y as a function of the true top mass m_t and the jet energy scale JES by integrating over the unknown parton-level quantities x. Specifically:

• M_eff represents the matrix element for ttbar production and decay, modified by the effective propagators discussed below.
• f(z) represents the parton distribution functions (PDFs) for the momenta of the two incoming particles. The flux factor (FF) is a normalization for these terms.
• The transfer functions (TF) connect the parton level momentum x to the measured jets y; they give the probability of observing a jet with a given momentum as a function of the parton energy E. We build these from Monte Carlo simulated events.
• The normalization factor N(m_t) is obtained by integrating the matrix element together with the PDFs and flux factor over the phase space.
• The acceptance factor A(m_t, JES) corrects for the changing detector acceptance as a function of m_t and JES. It is also obtained from Monte Carlo.
• Φ represents the parton-level phase space integrated over, including all Jacobians.
• The integral is performed over all 24 possible permutations of assigning partons to jets. Each permutation is weighted by the probability w_i that it is consistent with the tagging information.

In order to make the integration computationally tractable, we make a few simplifying assumptions. Specifically, we assume the lepton angle and momentum are perfectly measured, the four parton angles are measured perfectly by the jet angles, the b quark from the hadronically decaying top is on mass shell, and the other three quark masses are 0. This reduces the number of integration variables from 22 to 7. However, of course, it also introduces some imperfection. To compensate for this imperfection, we alter the distributions of M_W² and M_t² corresponding to the propagator terms in the matrix element so as to take into account these effects; we call the resulting distributions "effective propagators". These effective propagators, as shown below, are more smeared than the Breit-Wigners they would be without these effects.

Hadronic-side effective propagator

Leptonic-side effective propagator

Background handling

In order to distinguish between signal and background events, we employ a neural network discriminant. Our discriminant uses ten variables: the P_T of the 4 leading jets, the missing E_T, the lepton E_T, H_T (the scalar sum of the jet transverse energies, missing energy, and lepton energy), aplanarity, D_R, and H_TZ. The neural network is trained on Monte Carlo events with a signal mass of 170 against a W+bbbar background and then checked to see that the output does not change significantly with different signal masses and background types. The result is shown below.

Neural network discriminant. The solid lines show the output for different signal masses, while the dashed lines show the output for different background types.

Overall our discriminant shows good stability with respect to signal mass and background types. For a given event, we calculate the background fraction for that event f_bg(q) = B(q)/(S(q)+B(q)), where q is the neural network output for that event. Note that the distributions for B(q) and S(q) are normalized to the overall expected background and signal fractions.

Our background handling proceeds in two ways. Our method does not include an explicit background likelihood, as we do not integrate over the background matrix elements. Instead, we treat all events under the assumption that they are signal. Thus, we expect that when we add all of the likelihoods for the observed events, the events will contain signal and background in their expected fractions. Thus, to recover the likelihood for the signal events, we subtract off the expected contribution from the background events:

where the L_i are the individual likelihoods for each event and L_avg is the average likelihood for background events, as obtained from Monte Carlo. We can rewrite this slightly using the individual background fraction for each event:

The second step in our background handling is to de-weight events which have been identified by our discriminant as being more likely to be background. We perform this de-weighting by averaging these curves with a uniform distribution U. So, for a single event, we have:

In this equation, κ is an adjustable parameter that can be tuned to perform the optimum amount of de-weighting. However, in our studies, we determined that the optimum value of κ in the current analysis is 0; that is, the benefits of de-weighting background events are never greater than the penalties from accidentally de-weighting signal events in our case. Thus, for this version of the analysis, we do not perform the above de-weighting, only the subtraction of the expected background.

Finally, in addition to background events, there is another class of events not handled well by our signal integration. These are events which contain a true ttbar pair, but where the four observed jets do not correspond to the four quarks produced in the ttbar decay; we call these events "bad signal" events. These can occur due to a variety of possibilities (extra jets from radiated gluons, misidentified dilepton or all-hadronic events, W → τ decay, etc.) and overall comprise roughly 35% of our total signal. In order to deal with these events, we implement a cut on the log of the peak value of the likelihood curve; studies have shown that the optimal value of this cut is 6. We find that such a cut eliminates a good percentage of bad signal and background events while retaining nearly all signal events. The below table shows the efficiency for "good signal", "bad signal", and background events. For a signal mass of 170 GeV/c², 63.9% of 1-tag and 69.4% of >1-tag events are "good signal".

Method validation

To test and calibrate our method, we perform our integration on Monte Carlo samples at a variety of signal top masses with background events included in the expected fraction. For a given top mass, 2000 pseudo-experiments (PEs) are performed, where each pseudo-experiment includes 302.8 events (the expected number of observed events after applying the likelihood cut) randomly drawn from the signal and background pools according to their expected fraction; the number of events for each pool is fluctuated around its average by a Poisson fluctuation.

For a given pseudo-experiment, we combine the individual event likelihoods, subtract off the expected background contribution as described above, and then extract the overall top mass using the "profile likelihood" method; that is, for each value along the m_t axis, we select the value along the JES axis where the likelihood is maximized:

We then extract our result and statistical uncertainty from the resulting 1-D likelihood curve. For an ensemble of 2000 PEs, we then compute the measured mass (determined by the mean of the ensemble), bias, expected statistical uncertainty, and pull.

The plots below show the results of this test. The upper-left plot shows the measured mass as a function of input mass, while the upper-right plot shows the measured bias as a function of input mass. The lower-left plot shows the pull widths as a function of input mass, while the lower-right plot shows the expected uncertainty as a function of input mass.

Due to some concern about the stability of our measured pull width with respect to m_t, we fit our pull width in the mass range 160 to 180 only. The resulting constant is somewhat higher than the fit over the full range; we adopt this as our pull correction.

From the results of the above test, we adopt the following calibration constants:

• Bias: -1.217 GeV/c² (correction applied to measured mass)
• Pull: 1.245 (correction applied to measured uncertainty)
• Slope: 0.988 (correction applied to both mass and uncertainty)

To check the stability of our method with respect to JES, we also perform tests on samples where the input JES has been shifted to five different values: 0.95, 0.97, 1.00, 1.03, and 1.05. (The nominal uncertainty on the JES is approximately 3%.) The plot below shows the measured mass as a function of the input JES:

A flat line is fit to the result; the constants are 160.71 ± 0.18, 168.67 ± 0.17, and 174.46 ± 0.18, respectively, with fit probabilities of 0.946, 0.978, and 0.918, respectively. This shows that our method exhibits excellent stability with respect to input JES. We can also look at the output mass vs. input mass as a function of JES, or the output JES vs. input JES as a function of mass:

We can see from the plot on the left that the calibration constants are not strongly affected by the input JES; conversely, the right plot shows that the JES linearity is largely independent of the input top mass.

Data results

In the 1.7 fb^-1 data sample, we find a total of 343 events passing all of our selection cuts before the likelihood cut, 263 single-tag and 80 multiple-tag events. After applying the likelihood cut, we have 219 single-tag events and 74 multiple-tag events for a total of 293 events in our final likelihood. Applying the background subtraction, taking the profile likelihood, and applying the above calibration factors, we obtain a measurement of:

We can separate this uncertainty into a statistical uncertainty and uncertainty due to JES by comparing this with the 1-dimensional result, which yields a result of:

We can also perform separate measurements on the 1-tag and >1-tag samples, which yield m_t = 171.9 ± 2.2 GeV/c² and m_t = 174.5 ± 3.0 GeV/c², respectively.

The plots below show the overall likelihood in data events. The plot on the left shows the likelihood over most of the range used in our integration. The right plot shows the contours corresponding to a 1-sigma, 2-sigma, and 3-sigma uncertainty around the peak. Calibration has been applied to the m_t axis; since we do not make a JES measurement, we do not calibrate the JES axis.

We can also compare the observed uncertainty with the expected uncertainty from pseudo-experiments. The below plot shows this comparison for PEs at a signal mass of 172 GeV/c². 34% of pseudo-experiments had a smaller uncertainty than our uncertainty measured in data.

Another comparison of interest is to compare the likelihoods observed in data with the likelihoods observed in Monte Carlo, to check the validity of our likelihood cut as applied to data. The plot on the left shows the top mass value at the peak of the likelihood curve for events which pass the likelihood cut. The plot on the right shows the value of the log-likelihood at the peak of the curve for all events; the cut at 6 is shown as the dashed line on the plot. In both cases the Monte Carlo is normalized to the number of data events (343 pre-cut and 293 post-cut).

Systematics

Our systematics are summarized in the table below.

Here is a brief summary of the systematic uncertainties:

• Calibration: The uncertainty on our calibration constants translates directly into a systematic uncertainty.
• MC generator: While our analysis is tested and calibrated on PYTHIA Monte Carlo, we test it using a HERWIG sample and take the difference as a systematic uncertainty. Note that since these are independent samples, we also have to consider the error on the measured difference. Since the error is larger than the measured difference, we use the error value in computing our total systematic uncertainty.
• ISR/FSR: To assess effects due to initial state and final state radiation, we run on samples in which the amount of ISR and FSR has been increased and decreased. We take the largest resulting difference as a systematic uncertainty.
• Residual JES: Our method assumes that the JES is a flat scale factor applying to all jets, but in reality the jet energy uncertainty is a function of jet P_T and η. To evaluate systematics due to this, we shift the jet energies by the one-sigma uncertainty for each of the jet correction factors and add the resulting uncertainties in quadrature to get a total estimate.
• b-JES: Jets produced by b quarks have an additional uncertainty of 0.6% in their jet energy scale due to uncertainties in the b fragmentation and semileptonic decay. To evaluate a systematic from this source, we shift the b-jets by the 0.6% uncertainty and measure the resulting difference.
• Lepton P_T: We determine the systematic resulting from the 1% uncertainty on the lepton P_T.
• Permutation weighting: Our permutation weights are determined from the expected tagging efficiencies. We vary these by one sigma and compute the resulting systematic.
• Multiple interactions: We use the 50 MeV systematic uncertainty assigned for multiple interactions.
• PDFs: We reweight our signal events to correspond to different sets of parton distribution functions from the nominal CTEQ5L: we compare CTEQ5L with MRST72 (a different set), MRST72 with MRST75 (sets with two different values of α_s), and CTEQ6M with sets where the eigenvectors of CTEQ6M have been varied according to their uncertainty. We add up the resulting uncertainties to get a total systematic.
• Background fraction: We vary the background fraction by its uncertainty and examine the uncertainty in the result.
• Background composition: We run pseudo-experiments in which the background is entirely W+bbar, W+ccbar/c, W+light, or QCD, to see how the background composition affects the result. We also consider the effect of neglecting the single-top contribution here.
• Background average shape: To assess potential systematic effects resulting from our background subtraction, we divide our sample into two subsamples (one with electron-only events, and one with muon-only events), building our average background curve from one subsample, and then measuring the top mass on the other subsample.
• Background Q²: We use background samples in which the Q² scale used to generate events has been increased or decreased by a factor of 2 and measure the resulting systematic.
• Gluon fraction: PYTHIA and HERWIG are leading-order MC generators, so the fraction of gg → ttbar events is approximately 5%. At the Tevatron we expect that the fraction of gg → ttbar events is closer to 15%, so we reweight our events to correspond to this fraction and measure the resulting difference.
• b-tag E_T dependence: We introduce a weighting factor to tagged jets depending on the uncertainty from the dependence of the tagging efficiency on E_T and measure the resulting systematic.

Conclusion

We have measured the mass of the top quark on a total of 1.7 fb^-1 of integrated luminosity, and found a total of 293 events, from which we extract a measurement of: