Measurement of the top quark mass in the $\mathrm{\text{lepton}}+\mathrm{\text{jets}}$ final state with the matrix element method

We present a measurement of the top quark mass with the matrix element method in the $\mathrm{\text{lepton}}+\mathrm{\text{jets}}$ final state. As the energy scale for calorimeter jets represents the dominant source of systematic uncertainty, the matrix element likelihood is extended by an additional parameter, which is defined as a global multiplicative factor applied to the standard energy scale. The top quark mass is obtained from a fit that yields the combined statistical and systematic jet energy scale uncertainty. Using a data set of $0.4{\mathrm{fb}}^{-1}$ taken with the D0 experiment at Run II of the Fermilab Tevatron Collider, the mass of the top quark is measured using topological information to be: $m_{\mathrm{top}}^{\ell +\mathrm{jets}}(\mathrm{topo})={169.2}_{-7.4}^{+5.0}(\mathrm{stat}+JES{)}_{-1.4}^{+1.5}(\mathrm{syst})\mathrm{GeV}$, and when information about identified $b$ jets is included: $m_{\mathrm{top}}^{\ell +\mathrm{jets}}(b\mathrm{\text{-tag}})={170.3}_{-4.5}^{+4.1}(\mathrm{stat}+JES{)}_{-1.8}^{+1.2}(\mathrm{syst})\mathrm{GeV}$. The measurements yield a jet energy scale consistent with the reference scale.

Figure 1

Distribution of the topological likelihood for the $0.4{\mathrm{fb}}^{-1}$ D0 Run II data sample. The distribution for $e+\mathrm{\text{jets}}$ events is shown in plot (a) and for $\mu +\mathrm{\text{jets}}$ events in plot (b). The points with error bars indicate the data, and the fitted fractions of $t\overline{t}$ events (open area), $W+\mathrm{\text{jets}}$ events (diagonally hatched area), and QCD multijet events (horizontally hatched area) are superimposed.

Figure 2

Monte Carlo study of the effect of charm-jet tagging on the signal to background probability ratio in the $b$-tagging analysis, for $t\overline{t}$ events generated with $m_{\mathrm{top}}=175\mathrm{GeV}$ that contain two $b$-tagged jets. The $P_{\mathrm{sig}}$ values are calculated for the assumption $m_{\mathrm{top}}=175\mathrm{GeV}$. (a) Only the two jet-parton assignments in which tagged jets are assigned to $b$ quarks are considered. (b) All weighted jet-parton assignments enter the probability calculation. In both plots, the hatched histogram corresponds to those cases where the two $b$-tagged jets are correctly assigned to $b$ quarks, which happens 84% of the time in the double tag sample.

Figure 3

Jet transfer functions for light quark jets, $|\eta |<0.5$, for parton energies $E_p=30\mathrm{GeV}$ (solid curve), 60 GeV (dashed curve ), and 90 GeV (dash-dotted curve). The parametrization corresponds to the reference jet energy scale, $JES=1.0$.

Figure 4

Observed $t\overline{t}$ cross section computed with the leading-order matrix element for (a) $e+\mathrm{\text{jets}}$ and (b) $\mu +\mathrm{\text{jets}}$ events as a function of the top quark mass $m_{\mathrm{top}}$ for different choices of the $JES$ scale factor: $JES=1.12$ (dash-dotted line), $JES=1.0$ (solid line), and $JES=0.88$ (dotted line).

Figure 5

Distributions of ${log}_{10}(P_{\mathrm{sig}}/P_{\mathrm{bkg}})$ for $t\overline{t}$ events with $m_{\mathrm{top}}=175\mathrm{GeV}$ (solid line) and $W+\mathrm{\text{jets}}$ events (dashed line) for (a) $e+\mathrm{\text{jets}}$ events and (b) $\mu +\mathrm{\text{jets}}$ events. The $P_{\mathrm{sig}}$ values are calculated for the assumption $m_{\mathrm{top}}=175\mathrm{GeV}$. The distributions for signal and background events are normalized individually. The distributions for those $t\overline{t}$ events that fail the requirement of jets matched to partons are shown separately (dash-dotted line).

Figure 6

Calibration of the matrix element mass fitting procedure for the topological analysis, using ensembles with a top quark mass of 175 GeV and $JES=1.0$. The 68% confidence interval distributions for (a) the measured top quark mass and (b) the jet energy scale is given by the solid, the upper and lower error bands by the dashed histograms. A value of 0.68 as indicated by the dash-dotted line would mean that the corresponding $m_{\mathrm{top}}$ ($JES$) value is included in the fitted 68% confidence interval in 68% of the ensembles.

Figure 7

Calibration of the matrix element mass fitting procedure for the topological analysis. The first two plots show the reconstructed top mass (a) and the measured jet energy scale (b) as a function of the input top mass. The third and fourth plot show the reconstructed top mass (c) and the measured jet energy scale (d) as a function of the input jet energy scale. The solid lines show the results of linear fits to the points, which are used to calibrate the measurement technique. The dashed lines would be obtained for equal fitted and true values of $m_{\mathrm{top}}$ and $JES$.

Figure 8

Calibration of the matrix element mass fitting procedure for the topological analysis. The first two plots show the widths of the pull distributions for the top mass (a) and jet energy scale (b) as a function of the input top mass. The third and fourth plot show the widths of the pull distributions for the top mass (c) and jet energy scale (d) as a function of the input jet energy scale. The solid lines show the mean pull width, while the dashed lines indicate a pull width of 1.0.

Figure 9

Application of the topological matrix element method to the data. The $m_{\mathrm{top}}$ and $JES$ axes correspond to the calibrated values. Plot (a) shows the probability as a function of assumed top quark mass. The correlation with the jet energy scale is taken into account. The fitted curve is shown, as well as the most likely value and the 68% confidence level region. The corresponding plot for the $JES$ parameter is shown in (b).

Figure 10

Uncertainties on $m_{\mathrm{top}}$ (a) and $JES$ (b) in the topological analysis. The distributions of fitted uncertainties obtained from ensemble tests are shown by the histograms. Both upper and lower uncertainties are shown; their distributions are very similar. The upper (lower) uncertainty in the data is indicated by the solid (dashed) arrow. The probability for a lower uncertainty on $m_{\mathrm{top}}$ with a magnitude larger than that observed in the data is 2%.

Figure 11

Application of the topological matrix element method to the data. Fit of a two-dimensional fourth-order polynomial to the $-\ln L$ values as a function of both $m_{\mathrm{top}}$ and $JES$. Shown are the contours corresponding to $\Delta \ln L=0.5$, 2.0, 4.5, and 8.0 relative to the minimum.

Figure 12

Calibration of the matrix element mass fitting procedure for the $b$-tagging analysis, using ensembles with a top quark mass of 175 GeV and $JES=1.0$. The 68% confidence interval distributions for (a) the measured top quark mass and (b) the jet energy scale is given by the solid, the upper and lower error bands by the dashed histograms. A value of 0.68 as indicated by the dash-dotted line would mean that the corresponding $m_{\mathrm{top}}$ ($JES$) value is included in the fitted 68% confidence interval in 68% of the ensembles.

Figure 13

Calibration of the matrix element mass fitting procedure for the $b$-tagging analysis. The first two plots show the reconstructed top mass (a) and the measured jet energy scale (b) as a function of the input top mass. The third and fourth plot show the reconstructed top mass (c) and the measured jet energy scale (d) as a function of the input jet energy scale. The solid lines show the results of linear fits to the points, which are used to calibrate the measurement technique. The dashed lines would be obtained for equal fitted and true values of $m_{\mathrm{top}}$ and $JES$.

Figure 14

Calibration of the matrix element mass fitting procedure for the $b$-tagging analysis. The first two plots show the widths of the pull distributions for the top mass (a) and jet energy scale (b) as a function of the input top mass. The third and fourth plot show the widths of the pull distributions for the top mass (c) and jet energy scale (d) as a function of the input jet energy scale. The solid lines show the mean pull width, while the dashed lines indicate a pull width of 1.0.

Figure 15

Application of the matrix element $b$-tagging method to the data. The fitted $m_{\mathrm{top}}$ and $JES$ likelihoods for each of the 3 tag categories: 0-tag ((a) and (b)), 1-tag ((c) and (d)), and $\ge 2$-tag ((e) and (f)). The 68% confidence level interval around the most likely value is shown by the hatched region under the fitted curve.

Figure 16

Application of the matrix element $b$-tagging method to the data. The final results of the fitted $m_{\mathrm{top}}$ (a) and $JES$ (b) likelihoods for the combined event sample are shown. The 68% confidence level interval around the most likely value is indicated by the hatched region under the fitted curve.

Figure 17

Uncertainties on $m_{\mathrm{top}}$ (a) and $JES$ (b) obtained in the $b$-tagging analysis with the combined sample. The distributions of fitted uncertainties obtained from ensemble tests are shown by the histograms. Both upper and lower uncertainties are shown; their distributions are very similar. The upper (lower) uncertainty in the data is indicated by the solid (dashed) arrow.

Figure 18

Application of the matrix element $b$-tagging method to the data. Fit of a two-dimensional fourth-order polynomial to the $-\ln L$ values as a function of both $m_{\mathrm{top}}$ and $JES$. Shown are the contours corresponding to $\Delta \ln L=0.5$, 2.0, 4.5, and 8.0 relative to the minimum.