The MTM2 top quark mass measurement in the lepton+jets channel with 955 pb^-1

The primary event selection requirements for this type of event are as follows: exactly four high energy jets with E_T > 15 GeV in the |eta| < 2 region of the detector (the presumed products of the hadronization of the four daughter quarks in the decay), an isolated electron with E_T > 20 GeV or isolated muon with P_T > 20 GeV, and a missing transverse energy greater than 20 GeV (attributable to the neutrino passing through the detector without interaction). In addition, at least one of the four jets has to be "tagged" as coming from a b quark, using the secondary vertex signature.

A total of 179 events pass these selection criteria. The larger backgrounds to the t-tbar signal are due to other physics processes that pass these criteria as shown in the following table. The most common of these include events in which a W + heavy flavor quarks are produced, events in which a W + light flavor quarks are produced and a jet mistag occurs, and QCD events without a W. As can be seen, approximately 18% of events with one b-tagged and 4% of events with two tags are expected to be background given the primary selection requirements.

Background process	1 tag	2 tags
non-W QCD	5.5 ± 1.1	0.13 ± 0.07
W+light mistag	9.5 ± 1.6	0.65 ± 0.32
W+HF (bbbar, ccbar, c)	7.2 ± 2.6	1.03 ± 0.32
diboson (WW, WZ, ZZ)	1.4 ± 0.3	0.07 ± 0.02
single top	0.6 ± 0.1	0.00 ± 0.00
Total expected	24.1 ± 3.4	1.88 ± 0.48
Events observed	132	47

An additional requirement, see later, reduces the sample to 149 events.

Signal Likelihood Calculation

We define a likelihood function as follows:
For each event we calculate the value of the above likelihood under the assumption that the event is a t-tbar event with the lepton+jets topology. The essential idea is that, assuming a value for the jet energy scale (JES) and the true top mass, we can evaluate the probability that we would observe the measured quantities (represented by the vector y in the equation) in the detector. Here, JES is a constant factor which multiplies the measured jet energies before they are input into the likelihood equation. Given the JES and true top mass, the probability is calculated by an integration over the phase space of quark-level decay kinematics, represented by the vector x. To each possible configuration we assign a weight proportional to the distribution function of the incoming parton energies (the f's), the transfer function (TF) linking the transverse momenta of the four decay quarks to their daughter jet momenta, and the square of the matrix element of the t-tbar decay. This calculation is performed for all 24 possible assignments of the four quarks in the t-tbar decay to the four high energy jets in the event, with an additional weighting (w_i) for the probability of a given assignment.

In principle, this integration could be in 22 dimensions, as there are 6 decay products, each with a four-momentum, but we can take the masses of the neutrino and the muon/electron as known. However, for reasons of practicality, we only integrate over seven dimensions, while taking the quark masses as known, the quark angles to be the same as their resulting jet angles, and the muon/electron momentum to be measured perfectly. Our integration variables are taken to be the squared masses of the top and the W on both sides of the decay, the log of the ratio of the magnitudes of the light quark momenta, and the x- and y-components of the ttbar system's momentum.

The assumptions used to reduce the dimensionality of the integration are not absolutely correct ones - off-shell masses for a given quark type vary from event to event, jet angle resolution is not perfect. For this reason, we attempt to compensate by adjusting the matrix element and the TF's in the integration to take into account the fact that all quarks do indeed have variable masses and their angles are not perfectly measured. To accomplish this, we generate "effective partons" which have the same energies as the tree level partons, the directions of their daughter jets and the masses used to solve the equations (zero for the light quarks and the b quarks on the leptonic side, 4.5 GeV for b quark on the hadronic side). When this is done, the squared W and top mass distributions which can be constructed by taking the invariant masses of these partons are no longer Breit-Wigners; rather they are wider and no longer symmetric, as they have absorbed the error of the assumptions made.

These distributions, "effective propagators", depend on the event kinematics. They are used in place of the standard Breit-Wigner propagators in the matrix element. The partons used in the construction of the TF's are also the "effective partons" obtained as described above. They are made out of the L5 corrected jets, after matching requirements are satisfied.


Background Handling

To extract the top mass we take into account the background contribution to the observed events with the following likelihood: The final likelihood calculated for an event isn't simply the signal likelihood described in the previous section; rather, it can be roughly described as the signal likelihood for that event, minus the average shape of the signal likelihood for a background event weighted by the probability that the event under consideration might be background. Specifically, for a given event, the probability that the event is background, f_bg, is calculated by comparing a carefully chosen discriminant variable for the given event to distributions of that variable in signal and background MC events. Furthermore, the signal likelihood curve is effectively "smoothed" by averaging it with the uniform distribution over the top mass / JES space (the reason for the smoothing will be explained in a moment). Then, a curve which is actually the average of our background MC likelihood curves, and is thus considered to have the characteristic shape of a background curve, is smoothed in the same manner, and its log is subtracted off the log of the smoothed event signal-calculated likelihood curve.

To calculate f_bg, we define a discriminant variable q, as a linear combination of variables which express geometric (as opposed to energy-based) quantities in a given event. This linear combination is chosen such that the distributions are to a good approximmation independent of the true top mass and of JES. Furthermore, q is defined such that it has distinct distributions for MC signal events and background events. For a given event, we calculate q and then calculate the value of the background distribution (B(q)) and signal distribution (S(q)), where the distributions are scaled such that they reflect the expected a priori signal and background fractions for events with 1- or 2-tags, depending on whether the event under question has 1- or 2-tags. Then we take f_bg = B(q)/(B(q) + S(q)). B(q) and S(q) can be seen below:

Finally, the reason for the smoothing is as follows: the characteristic background curve may not actually possess the shape of the sum of the true background curves in the data sample. By smoothing it, we reduce the ability of this problem to hurt our resolution. Of course, to be consistent, we then need to smooth the signal-calculated likelihood curve for the event as well - which will tend to hurt the resolution. By adjusting the value of the kappa function in the above equation, we can balance the competing tendencies of the smoothing to help and hurt our resolution. We take kappa = 1, having found that, at least for the case in which kappa is independent of M_top

Performance of the Method

Two major studies of the performance of the method will be shown below. In the first study, the performance of the method in the ideal case is shown, and in the second study, the performance in the fully realistic case is shown. For each study, at a given input top mass 2000 pseudo-experiments (PEs) were run on the MC events for the relevant input top mass. As far as terminology goes, "reconstructed mass" is taken to be the mean of the PE measurements and the "bias" is simply the difference between the reconstructed mass and the input mass. The "expected error" is taken to be the RMS of the PE measurements, and the pull width is the RMS of the PE pulls, where the pull is the difference in the reconstructed and input mass, divided by the nominal uncertainty of the PE (found by half unit descent on either side of the peak of the PE's log likelihood curve).

In the ideal case study all the events in our PEs are t-tbar signal events in the lepton+jets channel with a good match between the four decay quarks and the four high-energy jets. Furthermore, neither the background handling nor the likelihood probability cut are used, as these are specifically designed to address the type of event which has been left out of these PEs. 179 events/PE were used on average, corresponding to the total number of events found in our data sample.

Clockwise, from top left, the plots are as follows: reconstructed top mass as function of input top mass, bias as function of input top mass, expected error as function of input top mass, and pull width as function of input top mass.

As can be seen from the plots, the bias is quite low and the pull width is just a couple of percent above unity. Furthermore, the expected uncertainty in a measurement using only good signal events is also quite low, ~ 1%.

Of greater interest are the studies corresponding to the fully realistic case; here background is included, along with background handling. Also, studies of our analysis have shown that background events (without a t-tbar pair), as well as "bad" t-tbar signal, hurt our measurement resolution. Here, "bad" means either the t-tbar event doesn't decay in the expected manner, or it does, but the four high energy jets don't match well to the four decay quarks (probably due to misidentification of at least one jet). The estimate from Monte Carlo is that 35% of the t-tbar events belong to this cathegory. Thus, an additional requirement placed on our events is that the value of the peak probability of our individual event's signal likelihood curve must be above a threshold. This peak probability selection requirement has been shown through MC studies to eliminate approximately 40% of background as well as 25% of bad t-tbar signal, while retaining 95% of well measured ("good") signal. The average # of events/PE was 148.8, corresponding to our expectation on the # of events which would survive the likelihood probability cut from the original 179. Results are as follows:

Here, we observe a bias of about -1.2 GeV on the top mass and a pull width of 1.22 . Neither quantity appears to have a dependence on the top mass; consequently, we apply a mass calibration of +1.2 GeV to the data measurement, and multiply the nominal uncertainties by 1.22 in order to compute the final uncertainty on the mass measurement from the two dimensional (2-D) likelihood.

Further studies have been made of the effect of changing the input JES value in our MC events on our analysis. One can get a sense of how sensitive our analysis is to shifts in the JES scale by measuring the mass in PEs run on events with different JES values. Looking at the (uncalibrated) reconstructed masses from MC events at three different mass values (mt = 167.5, 175.0, 182.5) and five different JES values (0.95, 0.97, 1.00, 1.03, 1.05) it can be seen that the dependence of the reconstructed mass on JES shifts of a few percent is consistent with zero, and at most on the order of a few tenths of a GeV:

It can also be seen that the top mass measurement bias is consistent with being independent of JES:

Finally, when comparing the reconstructed JES to the input JES, it appears that the slope is probably a few percent less than unity. Here, for a given PE, the JES is measured simply by taking the profile of the 2-d likelihood along the JES axis, rather than the top mass axis as is the case for mass measurements. The average of the JES linearity slope value at mt = 167.5 and 175.0 was used to calibrate the raw JES measurement in the data, as our mass measurement of 169.8 GeV fell between these two values:

Systematics

Systematic source	Systematic error (GeV/c2)
Residual JES	0.28
PDFs	0.46
ISR	0.75 ± 0.36
FSR	0.67 ± 0.40
MC generator	0.44 ± 0.43
Gluon fraction	0.05
Background fraction	0.20
Background composition	0.39
Background average shape	0.29
Calibration	0.14
b-JES	0.23
b-tag E_T dependence	0.02
Permutation weighting	0.06
Multiple interactions	0.05
Lepton P_T	0.05
Background Q^2	0.30
Total	1.39

Our systematics are summarized in the table above. Here is a brief description of the major systematic sources:

While the 2-D measurement is designed to capture any changes in the JES, we use a constant factor for the jet energy scale, while the jet energy systematic errors depend on the jet transverse momentum and pseudorapidity. To evaluate potential systematics from this, we shift the jet energies by one standard deviation of the systematic uncertainties and measure the resulting top mass.

We evaluate the systematics due to the parton distribution functions (PDFs) used in the matrix element integration by comparing different PDF sets (CTEQ5L and MRST72), varying the characteristic strength of the strong interaction, and varying the eigenvectors of the CTEQ6M PDFs.

Systematics due to initial-state radiation and final-state radiation are evaluated using MC samples where the amount of ISR and FSR has been increased and decreased.

Our analysis uses HERWIG MC for many of its internal workings, so we evaluate a systematic due to the generator by also performing our analysis on a PYTHIA Monte Carlo sample.

There are several uncertainties associated with our background method. First, we vary our overall background fraction by its uncertainty and measure the resulting change in the top mass. Second, we check the background composition by setting our background to 100% W + heavy flavor, W + light, or QCD (non-W) background and measuring the resulting changes. Third, we check the systematics due to our average background likelihood shape by dividing the sample into two subsamples (one with electrons, and one with muons), building the average shape from one subsample, and measuring the top mass on the other subsample. Finally, we use background samples with a different Q² scale used by the Monte Carlo generator to evaluate the systematics due to this source.

Since the jet energy scale may vary differently for b-jets than light jets, we vary the b-jets alone by an additional estimated uncertainty of 0.6% and measure the resulting systematic. We also have small systematics resulting from the dependence of the b-tagging efficiency on jet E_T, the permutation weighting used in our integration, multiple interactions, and uncertainty in the lepton p_T.

Results

In the 955 pb^-1 data sample, 149 events are found to pass our selection cuts: 108 of these are events with one tag, 41 have more than one tag. The final meaurement, after calibration is applied, is M_top = 169.8 ± 2.3 (stat + JES) ± 1.4 (syst) GeV/c², or, taking the statistical error calculated in the 1-d slice at JES = 1 of the 2-d likelihood and then taking the difference between that error and the stat + JES error in quadrature to obtain the additional statistical error due to the JES alone, M_top = 169.8 ± 1.6 (stat) ± 1.7 (JES) ± 1.4 (syst) GeV/c² . Additionally, the JES is measured to be 0.996 ± 0.017. The final 2-d likelihood is seen here:

It should be noted that the effect of using a 2-D likelihood in top mass and JES, rather a 1-D likelihood in top mass alone, is the following: in the 2-D likelihood calculation, as shown above, the JES contributes a 1.7 GeV/c² error in the likelihood itself, as well as a 0.3 GeV/c² "residual JES" systematic. However, if a 1-D likelihood in top mass were used with JES fixed to 1, the systematic on the JES would be approximately 3 GeV/c²! In this sense, by simultaneouslymeasuring JES and M_top through a 2-D likelihood, the total error is noticeably reduced.

Additionally, the total 2-d likelihood can be decomposed into the contributions from one tag events vs. greater than one tag events:

Of additional interest is the distribution of the errors on an ensemble of PE measurements. Below, the nominal uncertainties scaled by the pull widths of 1.22 for PEs run on MC events at 170 GeV/c² are shown; note that only 27% of PEs in the ensemble had a lower error than what was seen in data:

Likewise, a similar study was done for the error on the JES measurements. The errors were corrected by the average of the JES pull widths observed for events at a top mass of 167.5 and 175.0 GeV; this value is 1.14 . Additionally, the errors were corrected due to the non-unit slope found from the dependence of the reconstructed JES on the true JES. Here, 33% of PES had a lower error in JES than was found in data:

Likelihood Curve Cross Check

As a sanity check, we compared the masses taken from the peaks of the individual likelihood curves in our data sample with those of MC events at mt = 170 GeV as well as background, in their expected proportions. We did this for the value of the log likelihood at the peak as well; here all events are shown. The likelihood cut applied for the mass distribution is at 6. Agreement between data and MC in both cases is excellent: