Validity of the method

The stability of the fits was checked on Monte Carlo, using 1000 random ”fake-data” samples or ”pseudo-experiments” each with an integrated luminosity of 3 pb−1.

These fake data-samples are constructed sampling randomly a data-like mix of signal and background events, computed using high statistics Monte Carlo samples. Signal and electroweak backgrounds contributions are obtained from the theoretical predictions of their cross-section and the desired integrated luminosity. Since we only have a LO prediction for the QCD cross- section, and in data we observe a hadronic contribution almost a 30% larger than this, the contribution of QCD is enhanced to match more closely the observations.

Each of these 1000 samples was fitted with the method presented in the previous section. A first test of the method was done using templates derived directly from Monte Carlo. An example of one of these fits is shown in Figure 7.1 (left). The agreement between the fake-data and the fitted function is, as expected, very good over the full range of MT. This figure shows

each one of the fitted templates for the different components (signal and background processes). Introducing templates to model both signal and background and repeating the fit we obtain Figure 7.1 (centre). In this case once again the fit agrees very well with the fake-data, but the W cross-section obtained is slightly lower (−0.7%). Both fits are compared to the fake-data and

7. CROSS-SECTION MEASUREMENT

Figure 7.1: Example of a fit to a pseudo-experiment with an equivalent statistic of 3 pb−1. The

left panel shows the result of the fit using Monte Carlo to model signal and background. The central panel shows the result of the fit using templates to describe W and QCD. In both cases points represent the fake data, and lines the different components of the fit. Finally, the right panel shows the comparison of both fits to the data.

shown in the same Figure, in the right panel. Although both results are almost identical, the fit with the templates is slightly lower overall.

The same procedure is repeated over the remaining 999 sets ofLint= 3 pb−1luminosity, and

the results are summarized in Figure 7.2. The results of these fits using the original Monte Carlo distributions as shapes are shown in blue; and the results of the fits using the templates described in Chapter 6 for signal and for QCD background in red. The use of templates instead of Monte Carlo shapes deviates the output parameters obtained slightly towards lower cross-sections and higher ratios.

These differences between the fits are systematic effects, to be taken into account (and re-computed) in data. Table 7.1 summarises the results, comparing fits with signal modelled directly from Monte Carlo as well as from a data-like template, and with QCD modelled with Monte Carlo and with the hadronic template described in Chapter 6 (both for different normal- izations for positive and negative muons and for a single normalisation).

The input cross-section and ratio are recovered within 0.7% and 0.8% respectively, due to the modelling of QCD background. This bias is not a systematic effect to be directly ported to the measurement on data. E_Tmiss behaviour in data and in Monte Carlo is different, and therefore the corrections applied to the QCD template change. As seen in Chapter 6, in Monte Carlo this very simple correction had room left for improvement (using a more complicated parametrization) - while in data further corrections had no effect.

Figure 7.2: Results from the fit, in 1000 pseudo-experiments. Blue: fitting with the original Monte Carlo shapes, and using two different normalizations for negative and positive QCD contributions. Red: fitting with templates for signal and background, with a single normalization of QCD back- ground.

7. CROSS-SECTION MEASUREMENT

Table 7.1: Summary of the means values of the fits performed over the 1000 fake-data samples, using Monte Carlo distributions to model the shapes (MC), modelling the signal from a template but background from Monte Carlo (W Temp) and using templates both for signal and QCD background. The fits have been performed both under the assumption that QCD background is symmetric with respect to charge (1xQCD) and using independent normalizations for positive and negative muons (2xQCD).

W QCD σW (pb) Pulls σW Ratio Yield (W+) Yield (W−)

W MC MC 10436 ± 3 −0.03 ± 0.03 1.4363 ± 0.0007 8795 ± 3 5672 ± 2 2xQCD RMS=88 RMS=1.02 ± 0.02 RMS=0.0236 RMS=92 RMS=75 MC 10435 ± 3 −0.04 ± 0.03 1.4429 ± 0.0007 8810 ± 3 5656 ± 2 1xQCD RMS=87 RMS=1.02 ± 0.02 RMS=0.0236 RMS=92 RMS=75 Temp 10361 ± 3 −0.91 ± 0.03 1.4400 ± 0.0008 8741 ± 3 5622 ± 2 2xQCD RMS=88 RMS=1.02 ± 0.02 RMS=0.0239 RMS=92 RMS=75 Temp 10360 ± 3 −0.91 ± 0.03 1.4460 ± 0.0008 8755 ± 3 5607 ± 2 1xQCD RMS=88 RMS=1.03 ± 0.02 RMS=0.0238 RMS=92 RMS=74 W T em plate MC 10440 ± 3 0.01 ± 0.03 1.4364 ± 0.0008 8798 ± 3 5674 ± 2 2xQCD RMS=92 RMS=1.01 ± 0.02 RMS=0.0253 RMS=96 RMS=77 MC 10438 ± 3 0.01 ± 0.03 1.4423 ± 0.0008 8815 ± 3 5656 ± 2 1xQCD RMS=87 RMS=1.01 ± 0.02 RMS=0.0256 RMS=92 RMS=74 Temp 10363 ± 3 −0.88 ± 0.03 1.4413 ± 0.0008 8746 ± 3 5621 ± 2 2xQCD RMS=88 RMS=1.02 ± 0.02 RMS=0.0239 RMS=92 RMS=75 Temp 10362 ± 3 −0.89 ± 0.03 1.4474 ± 0.0008 8760 ± 3 5607 ± 2 1xQCD RMS=88 RMS=1.02 ± 0.02 RMS=0.0238 RMS=92 RMS=74 Input 10438 - 1.4354 8795 5675

modelling of the MT distribution, but also to the assumption that QCD background contains

the same proportion of positive and negative muons. If the fit is performed with independent normalizations for the QCD positive and negative contributions, this bias is reduced to 0.4%. This implies that QCD in Monte Carlo is not completely symmetric with respect to charge. However, since the modelling of QCD in Monte Carlo is not reliable enough, this asymmetry does not necessarily have to be the same in data. This effect will have to be verified later, directly on the experimental data-sample.

On the other hand, the variation due to the modelling of signal is negligible (< 0.1%), as expected. In data the difference between the signal template and Monte Carlo truth will therefore provide insight on the uncertainty associated to the E_Tmiss scale.

Table 7.2: Summary of numbers needed for the extraction of the cross section values.

W → µν W+_{→ µ}+_{ν W}−_{→ µ}−_ν¯

AW 46.20 ± 0.07 47.65 ± 0.07 44.13 ± 0.06

ρef f (92.83 ± 1.4)%

L 2.88pb−1

In summary, we conclude that in ideal conditions the method works perfectly well (< 0.1%). In more realistic conditions where both signal and background are modelled from template, the fit method still works very well, recovering the signal cross-section within 0.7% of its input value, and the ratio of cross-sections within 0.8%, due to well understood sources of bias.