Correcting for Detector Effects

The correction of angular resolution, finite selection efficiency and detector acceptance effects can be made by introducing a correction factor into the calculation of the SDM elements. This correction factor is calculated from the ratio of the number of fully detector simulated, selected Monte Carlo events, to the number of generated Monte Carlo events.

This is done in a bin-wise manner, so a correction factor is calculated for a certain volume of phase space as the ratio of the number of events in that phase space, before simulation and selection, to the number of events after full simulation and selection.

The relevant SDM elements for calculating the W-pair polarised cross-sections and the individual W polarised cross-sections are only functions of the W production angle $\cos\theta_{\rm W}$, and the polar angle of the W decay product, $\cos\theta^{*}$, so the correction factor only has to be calculated as a function of these variables.

The correction factor will then be calculated as:

$$f = \left(\frac{{\rm d}\sigma}{{\rm d}\cos\theta_{\rm W}{\rm d}\c... ...d}\cos\theta^{*}_{\ell},{\rm d}\cos\theta^{*}_{\rm j}}\right)^{\rm true}\right.$$ (8.1)

Where rec denotes the selected, reconstructed cross-section and true denotes the generated cross-section. $\theta^{*}_{\rm j}$ is the folded polar angle of the decay hadron. The hadronic jet with $\theta^{*}_{\rm j} > 0$ is the angle used. $\theta^{*}_{\ell}$ is the polar angle of the decay lepton. If the $\cos\theta_{\rm W}$ distribution is separated into $k$ bins, then there will be a set of correction factors for each bin of $\cos\theta_{\rm W}$, each of which will only be a function of the polar angles of the W decay products. This correction factor can be applied to the calculation of the appropriate SDM element as follows:

$$\rho^{k}_{\tau\tau\tau^{\prime}\tau^{\prime}} = \frac{1}{N^{\rm c... ...{\ell}^{*})_{i}\Lambda_{\tau^{\prime}\tau^{\prime}}(\cos\theta_{\rm j}^{*})_{i}$$ (8.2)

where $N^{cor}_{k}$ is the corrected number of events in bin $k$ given by equation (8.3).

$$N^{\rm cor}_{k} = \sum_{i=1}^{N_{k}}\frac{1}{\{f^{k}(\theta^{*}_{\ell},\theta^{*}_{\rm j})_{i}\}}$$ (8.3)

The single W SDM elements are only a function of the polar decay angle of one of the W bosons. For these the correction factor is only calculated as a function of the one relevant angle. The value of $N^{cor}_{k}$ therefore also changes. The full set of equations needed to calculate SDM element combinations for all the W-pair and individual W cross-sections, including the correction factor would then be:

$${\rm TT} \equiv \mbox{$\rho_{++++}$} + \mbox{$\rho_{++--}$} + \mbox{$\rho_{--++}$} + \mbox{$\rho_{----}$}$$ (8.4)

$$= \frac{1}{N^{\rm cor}_{k}}\sum_{i=1}^{N_{k}}\frac{1}{\{f^{k}(\thet... ...j})_{i}\}}(5\cos^{2}\theta^{*}_{\ell}-1)_{i}(5\cos^{2}\theta^{*}_{\rm j}-1)_{i}$$

$${\rm LL} \equiv \mbox{$\rho_{0000}$}$$ (8.5)

$$= \frac{1}{N^{\rm cor}_{k}}\sum_{i=1}^{N_{k}}\frac{1}{\{f^{k}(\thet... ...j})_{i}\}}(2-5\cos^{2}\theta^{*}_{\ell})_{i}(2-5\cos^{2}\theta^{*}_{\rm j})_{i}$$

$${\rm TL} \equiv \mbox{$\rho_{++00}$} + \mbox{$\rho_{--00}$} + \mbox{$\rho_{00++}$} + \mbox{$\rho_{00--}$}$$ (8.6)

$$= \frac{1}{N^{\rm cor}_{k}}\sum_{i=1}^{N_{k}}\frac{1}{\{f^{k}(\thet... ...j})_{i}\}}(5\cos^{2}\theta^{*}_{\ell}-1)_{i}(2-5\cos^{2}\theta^{*}_{\rm j})_{i}$$

$$+ \frac{1}{N^{\rm cor}_{k}}\sum_{i=1}^{N_{k}}\frac{1}{\{f^{k}(\thet... ...j})_{i}\}}(2-5\cos^{2}\theta^{*}_{\ell})_{i}(5\cos^{2}\theta^{*}_{\rm j}-1)_{i}$$

$${\rm T} \equiv \mbox{$\rho_{++}$} + \mbox{$\rho_{--}$}$$ (8.7)

$$= \frac{1}{N^{\rm cor}_{k}}\sum_{i=1}^{N_{k}}\frac{1}{\{f^{k}(\theta^{*}_{i})\}}(5\cos^{2}\theta^{*}_{i}-1)$$

$${\rm L} \equiv \mbox{$\rho_{00}$}$$ (8.8)

$$= \frac{1}{N^{\rm cor}_{k}}\sum_{i=1}^{N_{k}}\frac{1}{\{f^{k}(\theta^{*}_{i})\}}(2-5\cos^{2}\theta^{*}_{i})$$

The correction factors are calculated from Monte Carlo data. The width of the bins used has to be small enough to give reasonable results, but its width is limited by the resolution of the angular variables. Although the correction is designed to account for events migrating between bins, if the bin width is much less than the resolution, then the correction will be less reliable due to the very large numbers of events migrating between bins. The angular resolutions of the W production and polar decay angles are shown in figure 8.1.

Figure 8.1: The resolution of the angular variables $\cos\theta_{\rm W}$, $\cos\theta^{*}_{\ell}$ and $\cos\theta^{*}_{\rm j}$ used in calculating the polarised cross-sections. These were calculated from Monte Carlo data as $x_{\rm measured} - x_{\rm true}$.

The resolution of $\cos\theta_{\rm W}$ is 0.04, the resolution of $\cos\theta^{*}_{l}$ is 0.06 and the resolution of of $\cos\theta^{*}_{\rm j}$ is 0.08, so the bin widths have to be larger than these. If the $\cos\theta_{\rm W}$ distribution is once again divided into 8 equal bins, the width of each will be 0.25, which is much larger than the resolution. For the polar angles it is important to have more bins as it is these that are most sensitive to the W polarisation. For the lepton the bin width is chosen as 0.1, which means that the $\cos\theta^{*}_{\ell}$ distribution is split into 20 equal bins in the range [$- 1$,$+1$]. For $\cos\theta^{*}_{\rm j}$ a bin width of 0.1 is used in the folded range of [0,$+1$], thus it is divided into 10 equal bins when calculating the correction factor. Examples of some of the correction factor distributions are shown in figures 8.2 and 8.3. For the $\cos\theta_{\rm W}$ range [$-$0.25,0.0] it is obvious that the correction is limited by Monte Carlo statistics.

Figure 8.2: Distributions of the correction factors calculated from EXCALIBUR Monte Carlo. The first four are a function of $\cos\theta_{\ell}$ for one bin of $\cos\theta_{\rm j}$ and the last four are a function of $\cos\theta_{\rm j}$ for one bin of $\cos\theta_{\ell}$. All eight are for $-0.25<$ $\cos\theta_{\rm W}$$<0.0$.

Figure 8.3: Distributions of the correction factors calculated from EXCALIBUR Monte Carlo. The first four are a function of $\cos\theta_{\ell}$ for one bin of $\cos\theta_{\rm j}$ and the last four are a function of $\cos\theta_{\rm j}$ for one bin of $\cos\theta_{\ell}$. All eight are for $0.75<$ $\cos\theta_{\rm W}$$<1.0$.

The correction can be tested by applying it to fully simulated Monte Carlo samples. It will obviously correct the EXCALIBUR sample used to calculate the correction factors to extremely good accuracy, but as figure 8.4 demonstrates, it also gives a good approximation of the true SDM elements when applied to a sample of fully simulated PYTHIA Monte Carlo. Figure 8.4 shows the combinations of SDM elements needed to calculate the polarised cross-sections.

Figure 8.4: Comparison of the SDM elements extracted from a fully detector simulated sample of Standard Model PYTHIA Monte Carlo. The solid circles are the measured and the open circles are the corrected. The histogram shows the true SDM elements extracted from the generator level Monte Carlo. Shown, are only the combinations of SDM elements needed to calculate the polarised cross-sections. $\rho_{\rm TT}$ indicates that it is the combination needed to calculated $\sigma_{\rm TT}$, i.e. $\rho_{++++}$+ $\rho_{++--}$+ $\rho_{--++}$+ $\rho_{----}$. The same applies for $\rho_{\rm LL}$ and $\rho_{\rm TL}$.

Using these corrected SDM elements to calculate the individual W and W-pair polarised differential cross-sections for the PYTHIA sample, as described by equations 3.38 and 3.33, the plots in figure 8.5 are obtained. Once again, the correction produces a good approximation of the true cross-sections.

Figure 8.5: Comparison of the differential cross-sections extracted from a fully detector simulated sample of Standard Model PYTHIA Monte Carlo. The solid circles are the measured and the open circles are the corrected. The histogram shows the true differential cross-sections extracted from the generator level Monte Carlo.

Integrating over $\cos\theta_{\rm W}$ on the polarised differential cross-sections gives the total polarised cross-section. The total polarised cross-section divided by the total cross-section gives the fraction of each polarisation state. Calculating these numbers from generator level Monte Carlo and from the corrected polarised cross-sections will give a quantitative check of the detector correction. Figure 8.6 shows the calculated fractions from generator level Monte Carlo and also fully simulated Monte Carlo, both before and after the detector correction. Shown, are the results for a number of Standard Model Monte Carlo samples and also some non-Standard Model samples. It is obvious that the detector simulation has a large effect on the measured polarised fraction. In all cases, for both Standard and non-Standard Model Monte Carlo the detector correction gives results that are within one standard deviation of the true polarised fractions. It is interesting to note that for all Monte Carlo samples the detector simulation has a similar effect. The fraction of TT pairs is decreased and thus the fraction of other polarised W-pairs is enhanced. This is due to the detector simulation having the largest effect in the high $\cos\theta_{\rm W}$ region and this is where most of the TT W-pairs are found, as can be seen for example in figure 8.5.

Figure 8.6: The fraction of each polarisation state calculated from different Monte Carlo samples. The black circle is for generator level Monte Carlo, the red for fully detector simulated and the green is the result after the fully simulated has been corrected for detector effects. Descriptions of each Monte Carlo sample can be found in table 5.1.

The correction has also been tested on the polarised cross-sections extracted from Monte Carlo subsamples with the same statistics to the data sample. Figure 8.7 shows an example of one of these tests. The correction appears to give a good approximation of the true cross-sections.

Figure 8.7: The polarised differential cross-sections extracted from a small Standard Model Monte Carlo sample. The histogram represents the true cross-sections extracted from the generator level Monte Carlo. The solid circles are those extracted from the fully detector simulated Monte Carlo and the open circles are as for the solid circles except the detector correction has been implemented. The errors represented are purely statistical and only for the corrected results.

The fractional polarised cross-sections have also been extracted from the subsamples. They were calculated for both the generator level Monte Carlo and the fully simulated Monte Carlo. Figure 8.8 shows the difference between the true (generator level) and measured (fully simulated, corrected) values of the fraction of W bosons with longitudinal polarisation. Results for both the leptonically and hadronically decaying W boson as well as the combined result are shown. Good agreement is seen between the measured and true values. Also shown on figure 8.8 are the pull distributions for these results. The widths are all close to unity, although the width for the hadronically decaying W boson is slightly low, suggesting that the statistical error may be slightly overestimated.

Figure 8.8: Distribution of the comparison of the true to the measured fractions of longitudinal W bosons calculated from Monte Carlo subsamples. The true values were calculated from generator level Monte Carlo. The measured values were calculated from fully detector simulated Monte Carlo and have been corrected for detector effects. The first two plots are for leptonically decaying W bosons. The next two plots are for the hadronically decaying W boson. The last two plots are the combined results. The three plots on the left show the difference between the measured and true results. The three plots on the right are the pull distributions.

Similar plots for the W-pair polarised cross-sections can be seen in figure 8.9. Good agreement between the true and measured values is once again seen. The pull distributions all have widths close to unity except that for $\sigma_{\rm TT}$, which is slightly low, again suggesting a slight overestimation in the statistical error.

Figure 8.9: Distribution of the comparison of the true to the measured fractions of each polarisation state calculated from Monte Carlo subsamples. The true values were calculated from generator level Monte Carlo. The measured values were calculated from fully detector simulated Monte Carlo and have been corrected for detector effects. The three plots on the left show the difference between the measured and true results. The three plots on the right are the pull distributions.

The distributions of the measured values of both the fraction of longitudinal W bosons and the W-pair polarised cross-section fractions can be seen in figure 8.10. The width of these distributions can be taken as the expected statistical error on the measured values.

Figure 8.10: Distribution of the measured values of the fraction of each polarisation state from fully simulated Monte Carlo. All measured values have been corrected for detector effects. The three plots on the left are the fraction of longitudinally polarised W bosons from the leptonically, hadronically decaying W bosons and all W bosons respectively. The three plots on the right are for the W-pair polarised cross-sections.