Efficiency vs E T measured by fitting M ee

The 2010 statistics are fully sufficient to fit theZ invariant mass spectrum integrated over all ET bins. The distribution is fitted using RooFit with a Breit-Wigner con-voluted with a Crystal Ball⁵ as signal probability density function (PDF). Since the majority of the electrons are at lower ET the background is well described by an exponential function. The invariant mass distribution is then simultaneously fitted with the signal and background function to obtain the global PDF:

PDF(Mee) = NSig·PDF(Mee)Sig+NBkg ·PDF(Mee)Bkg, (5.7) where, N_Sig and N_Bkg are the number of signal and background events found by the fit. Figure 5.13 shows the invariant mass together with the fit for the different identification levels (container, loose, medium and tight) for probes outside the crack region and with opposite sign as the tag. All four fits have a χ²/N dof of about 1.4.

The fraction of background estimated by the fitting method can then be assessed for different invariant mass ranges and the case of 80< Mee <100 GeV is presented in Table 5.12. The uncertainty of the background fraction is propagated from the un-certainty of the fit. Due to the very small background at medium and especially tight level, the uncertainty is large compared to the estimated background. Comparing the amount of background at medium and tight level in Table 5.12 shows that the back-ground fraction at tight level is clearly overestimated, since the fit finds compatible background for medium and tight although the tight background rejection is notably higher than for medium. Due to the small values, however, this has little effect on the final efficiencies. The uncertainty of the background estimated at container level has a much higher impact on the efficiency measurements due to the significantly

5A Crystal Ball function is a Gaussian PDF with an exponential tail

(a) ^Mee

Figure 5.13: The fitted invariant mass distributions in data for opposite sign tag and probe pairs integrated over all ET and η regions (except the crack) at (a) container, (b) loose, (c) medium and (d) tight level. The red line shows the signal fit, the green line shows the background fit and the black line shows the global fit. χ²/N dof = 1.4 for all four fits.

larger background. The values obtained at container, medium and tight level can be compared with the overall background estimated with the OS-SS sideband method, shown in Table 5.8. The results agree within the uncertainties.

ID level Background [%]

Container 9.92 ± 0.76 Loose 1.40 ± 0.78 Medium 0.64 ± 0.81 Tight 0.60 ± 0.89

Table 5.12: Resulting background fraction and statistical uncertainty obtained through fitting the invariant mass at the different identification levels for opposite signed tag and probe pairs within 80< Mee <100 GeV.

Subtracting the estimated background at each level allows for the overall iden-tification efficiencies and data/MC scale factors to be calculated. The results are documented in Table 5.13. Comparing the values with the overall scale factors es-timated through the OS-SS sideband background subtraction method (Table 5.10), shows an overall agreement within the statistical uncertainties between the two meth-ods. Note, however, that the scale factors obtained with the fitting method are for efficiencies with respect to container while Table 5.10 shows results with respect to track quality cuts. This effect on the scale factors is however small since the track quality cut itself has been estimated to be well described in MC. Another observation that can be made from the tables is that the uncertainties from the fitting method are 3-4 times larger than for the OS-SS sideband method, originating from the large uncertainty of the fit.

[%] Efficiency Data/MC scale factor

Loose ID Data 97.27 ± 1.16

98.80 ± 1.18 MC truth 98.45± 0.01

Medium ID Data 92.00 ± 1.09

98.34 ± 1.16 MC truth 93.56± 0.02

Tight ID Data 78.69 ± 0.91

102.52 ± 1.19 MC truth 76.75± 0.03

Table 5.13: Loose, medium and tight identification efficiencies, integrated over ET, with respect to container for Z tag-and-probe applied to data, MC truth and their data/MC scale factor. The data-driven efficiencies are obtained with fitting the invariant mass spectrum for all tag and probe pairs with opposite sign, within 80< Mee <100 GeV and outside the crack region.

To estimate an efficiency binned in ET, the invariant mass distribution must be fitted for each bin in order to assess the background for each bin. The shape of the background changes for higher ET bins and the exponential distribution is no longer satisfactory. A second order Chebychev polynomial is thus used for ET > 40 GeV.

Figure 5.14 shows two example fits for container level probes for the [25,30] GeV and the [45,50] GeV range. It is clear from the figure that both the amount of background and the shape of the background strongly depends on the ET range of the probes.

The fit is problematic in some E_T regions with little statistics, especially for medium and tight level with miniscule background. For lower ET bins, the number of events in the tail at high invariant mass are very scarce, which imposes difficulties on the background fit. This affects the stability of the results and the fit does often not converge at tight level.

The largest uncertainty, however, originates from the denominator level, where background is significantly increased. Figure 5.15 shows the estimated background

(a) ^Mee

Figure 5.14: Fitted invariant mass distribution for tag and probe pairs with the probe at container level in data for opposite sign pairs, for (a) 25 < ET < 30 GeV and (b) 45 < ET < 50 GeV. The red line shows the signal fit, the green line shows the background fit and the black line the global fit. The background shape for the invariant mass distribution changes throughout the probe ET spectrum.

fraction from the fit, for the different ET bins, at container and loose level. The estimated background fractions for medium and tight identification level are negli-gible compared to container and loose level. The background fractions with errors estimated from the fit at all identification levels are documented in Table B.5 in Appendix B.

After the estimated background has been subtracted in each bin, theETdependent efficiencies are calculated. These are displayed in Figure 5.16 together with MC true efficiencies. The data-driven efficiencies show an increase of the efficiency for the lowestET bin. There is a large uncertainty in this region due to the large background and low signal statistics. The method overestimates the amount of background at denominator level and consequently also the efficiency of this bin.

The data-driven efficiencies as well as the data/MC scale factors, obtained with the method of fitting the invariant mass, are explicitly stated in Table B.6 in Appendix B for the different ET ranges.