Lina Galtieri, Paul Lujan, Jeremy Lys (LBNL), John Freeman (FNAL), Jason Nielsen (UC Santa Cruz), Igor Volobouev (Texas Tech)

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**Measured value:**
**m _{t} = 172.6 ± 0.9 (stat.) ± 0.7 (JES) ± 1.1 (syst.) GeV/c^{2} = 172.6 ± 1.6 (total) GeV/c^{2}**

- Abstract
- Event selection
- Signal likelihood calculation
- Background handling
- Method validation and calibration
- Data results
- Systematics
- Conclusion

- Public note (CDF-9880, August 2009)
- CDF internal page (password protected)

Our previous measurements:

- MTM3 with 3.2 fb
^{-1}, February 2009, CDF-9692, m_{t}= 172.1 ± 1.6 GeV/c^{2}(used in Winter '09 CDF mass combination) - MTM3 with 2.7 fb
^{-1}, July 2008, CDF-9427, m_{t}= 172.2 ± 1.7 GeV/c^{2}(used in Summer '08 CDF mass combination) - MTM3 with 1.9 fb
^{-1}, April 2008, CDF-9301, m_{t}= 171.4 ± 1.8 GeV/c^{2} - MTM2.5 with 1.9 fb
^{-1}, February 2008, CDF-9196, m_{t}= 172.7 ± 2.1 GeV/c^{2}(used in Winter '08 CDF mass combination) - MTM2.5 with 1.7 fb
^{-1}, September 2007, CDF-9025, m_{t}= 172.7 ± 2.1 GeV/c^{2} - MTM2 with 955 pb
^{-1}, March 2007, CDF-8780, m_{t}= 169.8 ± 2.7 GeV/c^{2}

*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.*

We report an updated 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. We calculate a
signal likelihood using a matrix element integration method with a
Quasi-Monte Carlo integration to take into account finite detector
resolution and quark mass effects. We use a neural network
discriminant to distinguish signal from backgrounds. Our overall
signal probability is a 2-D function of m_{t} and
Δ_{JES}, where Δ_{JES} is a shift applied
to all jet energies in units of the jet-dependent systematic error. We
apply a cut to the peak value of individual event likelihoods in order
to reduce the effect of badly reconstructed events. This measurement
updates our previous 1.9 fb^{-1}, 2.7 fb^{-1}, and 3.2
fb^{-1} measurements to use a dataset corresponding to 4.3
fb^{-1} of integrated luminosity, requiring events with a
lepton, large missing E_{T}, and exactly four high-energy jets
in the pseudorapidity range |η| ≤ 2. We require that at least
one of the jets is tagged as coming from a b quark, and observe 738
total events before and 630 events after applying our likelihood
cut. We find m_{t} = 172.6 ± 0.9 (stat.) ± 0.7
(JES) ± 1.1 (syst.) GeV/c^{2}, or m_{t} = 172.6
± 1.6 (tot.) GeV/c^{2}.

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 738 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 single-top production, which we include in our background shape, and also from diboson (WW, WZ, or ZZ) production and events where a Z produces a lepton pair; we do not consider these events directly, but rather increase the contribution from W+light to include them.

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
160 GeV/c^{2} to 184 GeV/c^{2} 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, except for the
single top samples which are generated using MadEvent with parton
showering by PYTHIA at a top mass of m_{t} = 175
GeV/c^{2}. Overlaps in the W+partons samples are removed using
the ALPGEN jet-parton matching along with a jet-based heavy flavor
overlap removal algorithm. We also use events generated with the
HERWIG generator as a cross-check.

The background estimate is shown in the table below:

Background | 1 tag | ≥ 2 tags |

non-W QCD | 34.6 ± 29.8 | 2.2 ± 2.8 |

W+light mistag | 29.4 ± 10.7 | 0.6 ± 0.3 |

diboson (WW, WZ, ZZ) | 7.5 ± 0.8 | 0.7 ± 0.1 |

Z → ℓℓ + jets | 5.4 ± 0.8 | 0.5 ± 0.1 |

Sum of above 3 | 42.2 ± 10.7 | 1.7 ± 0.3 |

W+bb | 39.0 ± 16.3 | 7.5 ± 2.7 |

W+cc | 24.0 ± 9.0 | 1.1 ± 0.4 |

W+c | 12.1 ± 4.5 | 0.6 ± 0.2 |

Single top s-channel | 2.7 ± 0.3 | 0.9 ± 0.1 |

Single top t-channel | 3.5 ± 0.4 | 0.9 ± 0.1 |

Sum of above 5 | 81.4 ± 27.3 | 11.0 ± 3.2 |

Total background | 158.2 ± 49.6 | 15.0 ± 5.7 |

Events observed | 591 | 148 |

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 shift
parameter Δ_{JES} by integrating over the unknown
parton-level quantities
**x**. Specifically:

- Δ
_{JES}is a parameter we introduce to account for the uncertainty in the jet energy scale. For a given value of Δ_{JES}, the p_{T}of the jets are scaled by (1 + Δ_{JES}× σ_{jet}), where σ_{jet}represents the relative jet energy scale uncertainty for that jet. - M represents the matrix element for ttbar production and decay.
- 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 for a given parton momentum. They are discussed in more detail below. - 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}, Δ_{JES}) corrects for the changing detector acceptance as a function of m_{t}and Δ_{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.

We assume that the lepton momentum is well-measured, leaving us 19 dimensions in our phase space Φ. In the past, we made further assumptions to reduce the dimensionality of this phase space. However, in this analysis, we use Quasi-Monte Carlo integration, which allows us to integrate over all 19 variables. Quasi-Monte Carlo integration differs from standard Monte Carlo integration in that it uses quasi-random sequences to generate points. Formally, a quasi-random sequence is one with a low discrepancy, where the discrepancy is a measure of the nonuniformity of the sequence. This allows an improvement in the convergence rate of the integral over the 1/√N convergence of standard Monte Carlo integration.

The transfer functions relate the observed jets to the parton-level
quantities. We construct our transfer functions from Monte Carlo
events and matching the simulated jets to their parent partons (or
"proto-jets"). In our analysis, we factorize the transfer functions
into separate momentum and angular parts. The momentum transfer
functions are built as probability distributions of the ratio of the
p_{T} of the jet to the p_{T} of the parton, while the
angular transfer functions are built as probability distributions of
Δη and Δφ, the differences between the η and
φ of the jet and the parton. Both the momentum and angular
transfer functions are built with dependence on the proto-jet
p_{T} and mass, and there are separate transfer functions
built for each of 4 separate bins of jet η as well as for b and
light quarks. A sample momentum (left) and angular (right) transfer
function are shown below.

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.

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. The actual
templates used in the calculation of f_{bg}(q) are shown
below.

Our background handling is relatively simple. 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:

These two expressions are equivalent if the events follow their
expected distributions. However, the advantage to using f_{bg}
per event rather than the total of n_{bg} is that if there are
more or fewer background events in our sample than expected,
f_{bg} should be able to capture some of this change.

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 tight jets
and/or lepton are not produced directly from 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 of 10. 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 172
GeV/c^{2}, 63.8% of 1-tag and 69.1% of >1-tag events are
"good signal".

Type of event |
Total |
1-tag |
>1-tag |

Good signal | 96.8% ± 0.2% | 96.5% ± 0.2% | 97.5% ± 0.3% |

Bad signal | 79.7% ± 0.5% | 80.1% ± 0.6% | 78.5% ± 1.1% |

Background | 73.5% ± 0.4% | 73.6% ± 0.4% | 73.6% ± 1.3% |

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 640.5 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.

We also examine samples in which the Δ_{JES} has been
shifted from its nominal value of 0. The below plots show the results
of these tests. The top two show the output JES and JES pull width for
m_{t} = 172 GeV/c^{2}, which we use to calibrate our
Δ_{JES} measurement. The bottom left plot shows the
output mass vs. Δ_{JES} for different input top
masses. There is a small dependence of output mass on input JES, as
the lower right plot shows; we account for this dependence in our
final calibration.

From the results of the above tests, we calibrate our final
measurement for the top mass and Δ_{JES}. First we apply
the bias and slope for the mass and Δ_{JES} measurement,
and then we use the measured slope of the output m_{t} as a
function of Δ_{JES} as a final correction. This gives us
the following formulas, where Δm = m_{t} - 172, since
our fits are centered around 172:

Δm_{calib}= (Δm_{meas}+ 0.597)/0.957 - 0.28 ⋅ (Δ_{JES})_{calib}

(Δ_{JES})_{calib}= ((Δ_{JES})_{meas}+ 0.303)/0.893

We also correct the measured uncertainties using the slope and pull widths obtained:

(σ_{m})_{calib}= (σ_{m})_{meas}× 1.165/0.957

(σ_{ΔJES})_{calib}= (σ_{ΔJES})_{meas}× 1.089/0.893

In the 4.3 fb^{-1} data sample, we find a total of 738
events passing all of our selection cuts before the likelihood cut,
590 single-tag and 148 multiple-tag events. After applying the
likelihood cut, we have 493 single-tag events and 137 multiple-tag
events for a total of 630 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 extract a measured value for Δ_{JES}, which is:

We can also perform separate measurements on the 1-tag and
>1-tag samples, which yield m_{t} = 171.8 ± 1.3
GeV/c^{2} and m_{t} = 174.4 ± 2.4
GeV/c^{2}, 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. The full 2-D calibration has been applied to both axes.

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/c^{2}. 52% 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 below shows the value of the log-likelihood at the peak of the curve for all events; the cut at 10 is shown as the dashed line on the plot. The Monte Carlo is normalized to the number of data events. Performing a Kolmogorov-Smirnov (K-S) test to check the consistency of the data and Monte Carlo indicates a confidence level of 0.73, showing good agreement.

Our systematics are summarized in the table below. Note that most
of our systematics are retained unchanged from our 3.2 fb^{-1}
analysis; only the calibration, generator, ISR/FSR, multiple hadron
interaction, and color reconnection uncertainties have been updated.

Systematic source | Systematic uncertainty
(GeV/c^{2}) |

Calibration | 0.1 |

MC generator | 0.6 |

ISR and FSR | 0.3 |

Residual JES | 0.5 |

b-JES | 0.4 |

Lepton P_{T} | 0.2 |

Multiple hadron interactions | 0.1 |

PDFs | 0.2 |

Background modeling | 0.5 |

Color reconnection | 0.3 |

Total | 1.1 |

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.
- ISR and 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 half the difference between the increased and decreased samples as our uncertainty.
- Residual JES: While the 2-D measurement is designed to capture any
changes in the Δ
_{JES}, the jet energy systematics are derived from several separate sources of uncertainty, which may exhibit different behavior with respect to jet p_{T}and η than the overall uncertainty. We thus vary the jet energies by one σ for each of the separate sources. The resulting shifts are added in quadrature to obtain our residual JES systematic uncertainty. - b-JES: We evaluate three potential sources of uncertainty for jets produced by b quarks. First, we vary the semileptonic decay fraction by its expected uncertainty and measure the resulting shift in the top mass. Second, we reweight our events to correspond to two different b fragmentation models and measure the resulting shift. Finally, we account for differing calorimeter response by shifting b-jets by the estimated uncertainty of 0.2% and measuring the resulting difference.
- Lepton P
_{T}: We determine the systematic resulting from the 1% uncertainty on the lepton P_{T}. - Multiple hadron interactions: Because the Monte Carlo samples are generated with a lower instantaneous luminosity and hence have fewer multiple hadron interactions (pileup) per event than the most recent data, we evaluate a potential systematic by measuring the top mass as a function of the number of interactions in the event and multiplying the resulting slope by the difference in the average number of interactions between Monte Carlo and data.
- 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. This also includes a systematic on the fraction of gg → ttbar vs. qqbar → ttbar events. - Background: There are four sources of uncertainty in our
background. First, we vary the background fraction by its uncertainty
and examine the uncertainty in the result. Second, we run
pseudo-experiments in which the background is entirely W+bbar,
W+ccbar, W+c, W+light, single top, or QCD, to see how the background
composition affects the result. Third, to assess potential systematic
effects resulting from our background subtraction, we divide our
sample into two subsamples (one with odd-numbered events, and one with
even-numbered events), build our average background curve from one
subsample, and then measure the top mass on the other
subsample. Finally, we use background samples in which the
Q
^{2}scale used to generate events has been increased or decreased by a factor of 2 and measure the resulting systematic. We also include an uncertainty for limited background statistics in the MC. - Color reconnection: For this analysis, we also include an updated estimate of the systematic uncertainty due to color reconnection effects, which we measure by taking the difference between two PYTHIA 6.4 tunes with and without color reconnection enabled.

We have measured the mass of the top quark on a total of 4.3
fb^{-1} of integrated luminosity, and found a total of 630
events passing all of our cuts, from which we extract a measurement of: