Background subtraction

The background subtraction is a crucial constituent of the tag-and-probe methodology and the largest source of uncertainty of the measurement. As was mentioned above, the background at each level of the efficiency estimation must be properly subtracted in order to estimate the electron efficiency in the data. There are different methods of subtracting the background, where most of the methods involve using the peak structure of the invariant mass (Mee) distribution. Since it is desirable to measure the efficiencies as a function of electronηandET, the background subtraction must be carried out in eachη orET bin. The poor statistics in certain bins, together with the fact that the signal and background invariant mass distribution changes significantly for different ET bins, impose difficulties.

Figure 5.2 shows the invariant mass distribution for the tag and the probe at track quality level as well asET,ηandφfor the probe after a requirement of 80< Mee<100 GeV for the tag and the probe has been imposed. Energy scaling and smearing, which is further discussed in section 6.3.5, is not applied. The plots contain signal MC only and the comparison with data provides some indication of the background distributions. The regions of lower ET and lower Mee have an evident excess of

background events in data. Figure 5.3 displays the equivalent distributions for the probe at medium level. Here the data and signal MC distributions agree well and no background is visible.

Figure 5.2: Distributions for the probe at track quality level in data and signal MC:

(a) Mee of the tag and the probe, (b) ET of the probe, (c) η of the probe and (d) φ of the probe. An 80< Mee<100 GeV cut is imposed on the tag and the probe in all distributions except for (a) and the probe is required to be outside the crack region in all distributions except for (c). The MC is normalized to integrated luminosity (39 pb⁻¹). The distributions are not well described by the signal MC due to significant background present in certain kinematic regions.

Sideband methods

Sideband methods are robust against low statistics, since these methods count events rather than performing a fit on a distribution. Several different sideband methods have been tested in data and MC (for more explicit details see [81]). The method that has been proven the most accurate, called the “OS-SS sideband method”, divides the invariant mass distribution into three regions; a signal region B (80 < Mee < 100 GeV) and two background sideband regions A (60 < Mee <80 GeV) and C (100 <

(a) ^Mee

Figure 5.3: Distributions for the probe at medium level in data and signal MC: (a) Mee of the tag and the probe, (b) ET of the probe, (c) η of the probe and (d) φ of the probe. An 80 < Mee < 100 GeV cut is imposed on the tag and the probe in all distributions but (a) and the probe is required to be outside the crack region in all distributions except for (c). The MC is normalized to integrated luminosity (39 pb⁻¹). Background contamination is almost negligible and cannot be identified in the plots.

Mee <120 GeV). An opposite sign requirement is imposed for the tag and the probe in the signal region and due to a non-negligible amount of signal in the sideband regions, only same sign events are considered as background. A linear behavior of the background is assumed and the average number of same sign events in the background regions is subtracted from the opposite sign events in the signal region.

Figure 5.4 shows the invariant mass distribution for the tag and the probe at container, loose, medium and tight level with 2010 data. The opposite sign events are overlaid with the same sign events to illustrate the amount of background being subtracted in the OS-SS sideband method. There is also a larger number of OS events than SS events outside the signal region, arising from the tails of theMeedistribution of the signal. It is therefore not correct to classify all OS events in the sideband regions as background. A Z mass peak is also visible in the signal region for the SS

events, especially from loose level on where the background is much reduced. These events arise from Z events where the charge of one of the two electrons has been mis-identified. SS events in the signal region can thus not be used for background estimation. What remains is to use OS events in the signal region and SS events in the background region, which corresponds to the OS-SS sideband method previously described.

Figure 5.4: The invariant mass distribution of tight tag and the probe at container (not including track quality cuts), loose, medium and tight level in logarithmic scale, with 2010 data. The opposite sign events in white are overlaid with the same sign events in red. The dotted lines show the separation between signal and background regions for the OS-SS sideband background subtraction method.

Fitting methods

A crucial disadvantage of the OS-SS sideband method is that it relies on the linearity of the background shape as a function of the invariant mass. This is sufficient for the overall efficiency calculation as well as for the different η regions, but for higher ET

ranges, the background rather peaks in the signal region and the method is no longer

valid. This is illustrated in section 5.4, where the ET dependent efficiency method and results are further discussed.

The primary background subtraction method for measuring the ET dependent ef-ficiencies is fitting the invariant mass distribution, which is presented in section 5.4.1.

A secondary method of fitting the E_T spectrum directly has also been investigated and is discussed in section 5.4.2. The fitting methods have the disadvantage of diffi-culties with low statistics, leading to issues of finding the correct level of background, especially at the medium and tight selection levels.

Statistical uncertainty

Binomial or Bayesian methods are nominally used for calculating the statistical un-certainty of an efficiency where the numerator is a subset of the denominator. The tag-and-probe efficiency measured in data, however, includes a background subtrac-tion, which must be taken into account in the uncertainty calculation. The methods used here to estimate the statistical uncertainty are described in [86], which addresses background subtraction methods using both sidebands and fits.

For sideband methods, the efficiency after background subtraction is defined as += Nnum−αNnum,SB

Ndenom−αNdenom,SB

, (5.1)

where α is the weight of the sidebands, which is 1/2 in the default OS-SS sideband method described above. The uncertainty for sideband methods is then calculated according to

(∆+)² = (1−2+)(Nnum+αNnum,SB) ++²(Ndenom+αNdenom,SB)

(N_denom−αN_denom,SB)² . (5.2)

If a fit is used to estimate the amount of background, the efficiency is defined as += Nnum,f it

Ndenom,f it

, (5.3)

where Nnum,f it is the number of signal events that the fit returns for the numerator and Ndenom,f it the equivalent for the denominator. The uncertainty of the efficiency is then calculated according to:

(∆+)² = (1−2+)(∆Nnum,f it)²++²(∆Ndenom,f it)²

N_{denom,f it}² . (5.4)