Systematic Checks of the $\cos\theta_{\rm W}$ Fit Method

To test the $\cos\theta_{\rm W}$ fit method, fits are performed to the $\cos\theta_{\rm W}$ distributions calculated from fully detector simulated Monte Carlo that has been generated with anomalous couplings. The results of these fits for the CP-conserving couplings can be seen in table 6.5 and the bias plots are shown in figures 6.4 and 6.5. The results for the CP-violating couplings can be seen in table 6.6 and the bias plots are shown in figure 6.6.

The fits of the CP-conserving couplings show no obvious bias towards the Standard Model or any other coupling. All fitted values are consistent with the generated values. The measured value at $\mbox{$\Delta\kappa_{\gamma}$}$$=+1$ using the BILGOU reweighting scheme is less consistent with the generated value, but is still consistent within the expected statistical error on the data sample, given in table 6.9.

The fits of the CP-violating couplings show a slight bias towards lower couplings than the generated values. This is expected due to the slight systematic difference between the $\cos\theta_{\rm W}$ distribution extracted from the Standard Model ERATO sample and that calculated from the EXCALIBUR sample that is reweighted to perform the fit. The ERATO $\cos\theta_{\rm W}$ distribution is slightly steeper than the EXCALIBUR one for all CP-violating couplings. This systematic difference is accounted for in the systematic uncertainties described in chapter 10.

The $\cos\theta_{\rm W}$ distribution is much less sensitive to the CP-violating couplings than the CP-conserving ones. This is to be expected as the CP-violating couplings have a large effect on the imaginary observables and a much lesser effect on the real observables, such as cross-sections  [41]. The CP-conserving couplings have a large effect on the real observables, but no effect on any of the imaginary observables.

Table: The bias fits to the $\cos\theta_{\rm W}$ distribution extracted from large Monte Carlo data samples generated with anomalous CP-conserving couplings. Both reweighting techniques were used. The errors shown are the statistical uncertainty on fit to the large samples.

Coupling	Generated Value	Fitted Value
		WVCXME	BILGOU
$\Delta\kappa_{\gamma}$	$-$2.0	$-$1.90 $^{+0.04}_{-0.04}$	$-$1.89 $^{+0.04}_{-0.04}$
$\Delta\kappa_{\gamma}$	$-$1.0	$-$0.98 $^{+0.03}_{-0.03}$	$-$0.95 $^{+0.03}_{-0.04}$
$\Delta\kappa_{\gamma}$	$-$0.5	$-$0.46 $^{+0.04}_{-0.04}$	$-$0.44 $^{+0.04}_{-0.04}$
$\Delta\kappa_{\gamma}$	0.0	0.0 $^{+0.05}_{-0.05}$	$-$0.02 $^{+0.06}_{-0.06}$
$\Delta\kappa_{\gamma}$	$+$0.5	$+$0.55 $^{+0.20}_{-0.12}$	$+$0.43 $^{+0.08}_{-0.08}$
$\Delta\kappa_{\gamma}$	$+$1.0	$+$1.08 $^{+0.14}_{-0.23}$	$+$0.49 $^{+0.10}_{-0.09}$
$\Delta\kappa_{\gamma}$	$+$2.0	$+$1.96 $^{+0.04}_{-0.06}$	$+$2.10 $^{+0.06}_{-0.06}$
$\Delta g^{z}_{1}$	$-$2.0	$-$1.84 $^{+0.05}_{-0.05}$	$-$1.82 $^{+0.05}_{-0.05}$
$\Delta g^{z}_{1}$	$-$1.0	$-$0.97 $^{+0.02}_{-0.01}$	$-$0.96 $^{+0.02}_{-0.02}$
$\Delta g^{z}_{1}$	$-$0.5	$-$0.49 $^{+0.02}_{-0.02}$	$-$0.49 $^{+0.01}_{-0.01}$
$\Delta g^{z}_{1}$	0.0	0.0 $^{+0.02}_{-0.01}$	$-$0.01 $^{+0.01}_{-0.02}$
$\Delta g^{z}_{1}$	$+$0.5	$+$0.47 $^{+0.08}_{-0.05}$	$+$0.41 $^{+0.05}_{-0.04}$
$\Delta g^{z}_{1}$	$+$1.0	$+$0.95 $^{+0.04}_{-0.04}$	$+$0.95 $^{+0.04}_{-0.04}$
$\Delta g^{z}_{1}$	$+$2.0	$+$1.85 $^{+0.03}_{-0.03}$	$+$1.84 $^{+0.03}_{-0.03}$
$\lambda$	$-$2.0	$-$1.92 $^{+0.03}_{-0.03}$	$-$1.91 $^{+0.03}_{-0.03}$
$\lambda$	$-$1.0	$-$0.97 $^{+0.02}_{-0.02}$	$-$0.98 $^{+0.02}_{-0.02}$
$\lambda$	$-$0.5	$-$0.49 $^{+0.02}_{-0.02}$	$-$0.49 $^{+0.02}_{-0.02}$
$\lambda$	0.0	0.0 $^{+0.01}_{-0.01}$	0.0 $^{+0.03}_{-0.03}$
$\lambda$	$+$0.5	$+$0.48 $^{+0.05}_{-0.05}$	$+$0.42 $^{+0.04}_{-0.04}$
$\lambda$	$+$1.0	$+$0.96 $^{+0.04}_{-0.04}$	$+$1.07 $^{+0.04}_{-0.04}$
$\lambda$	$+$2.0	$+$1.92 $^{+0.04}_{-0.03}$	$+$1.99 $^{+0.04}_{-0.04}$

Table: The bias fits to the $\cos\theta_{\rm W}$ distribution extracted from large Monte Carlo data samples generated with anomalous CP-violating couplings. The errors shown are the statistical uncertainty on fit to the large samples.

Coupling	Generated Value	Fitted Value
		WVCXME	BILGOU
$\tilde{\kappa}_{z}$	$-$1.00	-	$\pm$0.89 $^{+0.04}_{-0.04}$
$\tilde{\kappa}_{z}$	$-$0.50	-	$\pm$0.46 $^{+0.04}_{-0.04}$
$\tilde{\kappa}_{z}$	$-$0.25	-	$\pm$0.20 $^{+0.03}_{-0.03}$
$\tilde{\kappa}_{z}$	0.0	-	0.0 $^{+0.03}_{-0.03}$
$\tilde{\kappa}_{z}$	$+$0.25	-	$\pm$0.20 $^{+0.03}_{-0.03}$
$\tilde{\kappa}_{z}$	$+$0.50	-	$\pm$0.46 $^{+0.04}_{-0.04}$
$\tilde{\kappa}_{z}$	$+$1.00	-	$\pm$0.89 $^{+0.04}_{-0.04}$
$\tilde{\lambda}_{z}$	$-$1.00	-	$\pm$0.93 $^{+0.03}_{-0.03}$
$\tilde{\lambda}_{z}$	$-$0.50	-	$\pm$0.44 $^{+0.03}_{-0.03}$
$\tilde{\lambda}_{z}$	$-$0.25	-	$\pm$0.15 $^{+0.05}_{-0.05}$
$\tilde{\lambda}_{z}$	0.0	-	0.0 $^{+0.06}_{-0.06}$
$\tilde{\lambda}_{z}$	$+$0.25	-	$\pm$0.15 $^{+0.05}_{-0.05}$
$\tilde{\lambda}_{z}$	$+$0.50	-	$\pm$0.44 $^{+0.03}_{-0.03}$
$\tilde{\lambda}_{z}$	$+$1.00	-	$\pm$0.93 $^{+0.03}_{-0.03}$
$g^{z}_{4}$	$-$1.00	-	$\pm$0.89 $^{+0.11}_{-0.11}$
$g^{z}_{4}$	$-$0.50	-	$\pm$0.41 $^{+0.20}_{-0.20}$
$g^{z}_{4}$	$-$0.25	-	$\pm$0.13 $^{+0.15}_{-0.15}$
$g^{z}_{4}$	$+$0.00	-	0.0 $^{+0.11}_{-0.11}$
$g^{z}_{4}$	$+$0.25	-	$\pm$0.13 $^{+0.15}_{-0.15}$
$g^{z}_{4}$	$+$0.50	-	$\pm$0.41 $^{+0.20}_{-0.20}$
$g^{z}_{4}$	$+$1.00	-	$\pm$0.89 $^{+0.11}_{-0.11}$