Top Mass Measurement on 1 fb-1 using the 3 Best Combinations Method |
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| Michele Giunta (Universita di Siena and INFN Pisa), Giorgio Bellettini (Universita di Pisa and INFN, Pisa), Guram Chlachidze (FNAL), Fedor Prokoshin (JINR Dubna), George Velev (FNAL) |
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| Abstract | ||
| 1. Introduction | ||
| 2. Procedure | ||
| 3. Results | ||
| References | ||
| Public Note (8669) (abstract) | ||
| 1. Introduction |
| We measured the Top quark mass using the Template method and events without btag information corresponding to 1 fb-1 of data. In doing this, we applied a statistical method to improve the resolution due to the statistical error. The reconstruction of each event, when using no b-tag information for the jets, can be done a priori in 24 different ways. Each of the 24 reconstructions can be associated to a χ² value which is smaller for better agreements of data and MC kinematics. Once the 24 combinations are ordered by increasing χ² values, the first is commonly chosen when applying the standard Template Method. Figure 1 shows how many times each χ² rank is the correct assignment. The plot deals with events where the four leading jets are associated with the four ttbar decay quarks. We notice that the χ² rank =1 point corresponds to the correct association in less than 50% of times. The (2n)th bins are less populated than the (2n - 1)th ones because their entries are often rejected to avoid double counting. |
Figure 1: The plot shows for the Herwig MC simulation with Mtop = 175 GeV how many times the χ² rank corresponds to the correct jet-to-parton association. |
| This happens when the 2nd degree equation for the neutrino longitudinal momentum determines appoximatively the same Top mass value. We reject the second solution whenever it differs less than 100 MeV from the first one. We used the three best reconstructions and combined them together. In order to take into account the correlation between the three best combinations, we used the Best Linear Unbiased Estimate (BLUE) method to combine them and compute the BLUE-combined mass and error. This method is not expected to improve the systematic error. |
| 2. Procedure |
| We selected, out of 1 fb-1 of data, 645 events passing the CDF standard kinematic cuts for high Pt physics and having a χ² < 9 as quality factor for the best reconstructed combination. A number of relevant kinematic quantities are represented in figure 2 to compare the selection operated on the data and the closer MC sample to the most recent measurements. |
Figure 2: Left: transverse energy distributions of the five leading jets in the selected data and MC events. Right: distributions of lepton transverse momentum, E/T and number of jets in the selected data and MC samples. |
| The Templates we used are the probability density functions obtained from 21 Herwig MC samples having as input 21 Top masses from 150 to 200 GeV. Those signal templates have been parametrized using 30 parameters. The BG samples have been obtained using the four leading BG contributions: W+light jets (63.3 %), W+heavy jets (13.9 %), QCD (14.6 %), diboson (8.2 %). The BG shapes have been combined using the estimated relative ratios as weights. All signal templates and BG samples have been obtained for each of the the three best reconstructions so to run three mass measurements independently using the first, the second and the third best reconstructions. We built a large number of experiments using the MC samples, each experiment modeling the data sample in composition and amount of events. In the Template method, each experiment is treated as it was the actual data sample and fitted with a likelihood fit procedure providing a mass measure. In our BLUE method, this happens once for each of the three best reconstructions, so that we obtain three measures for each experiment. Each of the three measures has been tested to check for biases: figure 3 shows the pull distribution means and widths as a function of Mtop for the three reconstructed best combinations (left). We can see that, inside the errors, no appreciable bias is present. Figure 4 shows a number of reconstructed masses compared to the input masses (left) and the BLUE pull distribution means and widths as a function of the input masses compared with the three best combinations pulls (right). |
Figure 3: Left: pull distribution means and widths. The rows correspond to the three combinations, in order up to down. The relatively large error bars are due to the limited statistics. The red horizontal lines show the fits to a constant. Right: Values of the weights used to combine the three measurements as in the text, to obtain the BLUE mass. |
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Figure 4: Left: reconstructed masses versus input masses. Right: pull means and widths of the three best combinations and the BLUE over the studied mass range. |
| By studying the experiments, we computed the correlations between the first and second, first and third, second and third combinations. Making use of the correlation factors we computed then [1] the weights α1, α2, α3 we assign to each measured mass to obtain the combined mass. 
Equation 1 shows the expression for the BLUE variance to be associated to the BLUE mass Mcombined = α1M1+α2M2+α3M3. The α1, α2, α3 factors are computed for each experiment by minimizing the BLUE variance using the constrain α1+α2+α3 = 1. Figure 3 on the right shows how the weight values depend on the Mtop value. In figure 5 we show the distribution of the reconstructed masses (left) and errors (right) relative to the experiments run for Mtop = 175 GeV . |
Figure 5: Left: The three combinations and the BLUE mass distributions for Mtop = 175 GeV . Right: Error distributions for the three combinations and for the BLUE combined. Although the errors of the second and third combination are larger, the information provided by these combinations reduce the BLUE errors below those of the first combination. |
| 3. Results | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| We report in figure 6 on the left the data fitted histograms relative to the three best combinations and the relative likelihood shapes. The data mass measure is reported in table 1 where the statistical error only is reported. The unconstrained fit on the same data sample is reported in the right column as comparison. The BLUE-combined data measure is reported in the same table and allowed an improvement of the statistical error by 5.1 % with respect to the standard choice of the best reconstructed mass. | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Figure 6: Left: likelihood fit of the three best combinations and relative likelihood functions which minimum determines the mass value. Right: distribution of the BLUE improvements with respect to the best combination. This study is based on 2000 MC experiments. | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
In figure 6 (right) we report the distribution obtained in a MC study relative to Mtop = 175 GeV of the BLUE improvements while running 2000 experiments. The mean of this distribution is about 10 %. Basing on the MC study, the probability to obtain a BLUE improvement larger than 5.1 % is 77 %. We estimated the systematic error relative to our mass measure using the same BLUE technique. The relevant contribution and their quadratic sum are reported in table 2.
Our final measure of the Top quark mass in the semileptonic channel using no b-tag information and applying the BLUE technique is: | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| References |
| [1] L. Lyons, D. Gibaut; NIMA A270 1998 110-117. |
| maintained by Fedor Prokoshin | ||