Systematic Checks of the SDM Fit Method

Reweighting methods are prone to introduce biases. Therefore, the possible scale of any biases has been measured by applying the $\chi ^{2}$ fit method to samples of fully simulated four-fermion Monte Carlo data that have been generated with non-Standard Model couplings.

For the CP-conserving couplings, fully simulated samples of four-fermion EXCALIBUR are available with anomalous couplings; $\Delta\kappa_{\gamma}$, $\Delta g^{z}_{1}$ and $\lambda$, with values of of $\pm$2, $\pm$1 and $\pm\frac{1}{2}$. All these were generated with the ${\rm SU}(2)_{L}\times {\rm U}(1)_{Y}$ constraints on the couplings, equation 3.16. Fits were performed to all the samples using both reweighting techniques. Table 6.3 shows all the results and figures 6.4 and 6.5 show the bias plots. The values in table 6.3 show no bias towards the Standard Model for any particular coupling value. The differences between the measured and the generated values are all less than the statistical accuracy expected on the data sample shown in table 6.9. Both reweighting techniques give similar results.

Table 6.3: The bias fits to the single W SDM elements 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 the fit to the large sample.

Coupling	Generated Value	Fitted Value
		WVCXME	BILGOU
$\Delta\kappa_{\gamma}$	$-$2.0	$-$2.00 $^{+0.04}_{-0.04}$	$-$1.99 $^{+0.04}_{-0.04}$
$\Delta\kappa_{\gamma}$	$-$1.0	$-$0.98 $^{+0.04}_{-0.04}$	$-$0.98 $^{+0.04}_{-0.03}$
$\Delta\kappa_{\gamma}$	$-$0.5	$-$0.44 $^{+0.04}_{-0.04}$	$-$0.45 $^{+0.05}_{-0.04}$
$\Delta\kappa_{\gamma}$	0.0	$+$0.02 $^{+0.05}_{-0.07}$	$-$0.01 $^{+0.07}_{-0.07}$
$\Delta\kappa_{\gamma}$	$+$0.5	$+$0.19 $^{+0.09}_{-0.09}$	$+$0.35 $^{+0.15}_{-0.13}$
$\Delta\kappa_{\gamma}$	$+$1.0	$+$0.87 $^{+0.05}_{-0.06}$	$+$0.88 $^{+0.05}_{-0.06}$
$\Delta\kappa_{\gamma}$	$+$2.0	$+$1.96 $^{+0.04}_{-0.04}$	$+$1.87 $^{+0.05}_{-0.03}$
$\Delta g^{z}_{1}$	$-$2.0	$-$1.99 $^{+0.05}_{-0.05}$	$-$1.94 $^{+0.05}_{-0.05}$
$\Delta g^{z}_{1}$	$-$1.0	$-$0.95 $^{+0.04}_{-0.04}$	$-$0.93 $^{+0.04}_{-0.04}$
$\Delta g^{z}_{1}$	$-$0.5	$-$0.49 $^{+0.04}_{-0.04}$	$-$0.51 $^{+0.04}_{-0.04}$
$\Delta g^{z}_{1}$	0.0	0.0 $^{+0.02}_{-0.02}$	$-$0.02 $^{+0.03}_{-0.02}$
$\Delta g^{z}_{1}$	$+$0.5	$+$0.45 $^{+0.03}_{-0.03}$	$+$0.43 $^{+0.03}_{-0.03}$
$\Delta g^{z}_{1}$	$+$1.0	$+$0.96 $^{+0.02}_{-0.02}$	$+$0.96 $^{+0.02}_{-0.02}$
$\Delta g^{z}_{1}$	$+$2.0	$+$1.97 $^{+0.06}_{-0.06}$	$+$2.11 $^{+0.07}_{-0.07}$
$\lambda$	$-$2.0	$-$1.90 $^{+0.07}_{-0.07}$	$-$1.89 $^{+0.07}_{-0.07}$
$\lambda$	$-$1.0	$-$0.86 $^{+0.05}_{-0.05}$	$-$0.91 $^{+0.06}_{-0.06}$
$\lambda$	$-$0.5	$-$0.42 $^{+0.05}_{-0.05}$	$-$0.44 $^{+0.05}_{-0.05}$
$\lambda$	0.0	$+$0.01 $^{+0.04}_{-0.02}$	$+$0.02 $^{+0.03}_{-0.03}$
$\lambda$	$+$0.5	$+$0.48 $^{+0.03}_{-0.03}$	$+$0.49 $^{+0.03}_{-0.03}$
$\lambda$	$+$1.0	$+$0.93 $^{+0.01}_{-0.01}$	$+$1.03 $^{+0.02}_{-0.02}$
$\lambda$	$+$2.0	$+$1.92 $^{+0.07}_{-0.07}$	$+$1.93 $^{+0.07}_{-0.07}$

For the CP-violating couplings only the BILGOU reweighting technique can be used. The ERATO generator was used to generate the Monte Carlo samples with CP-violating anomalous couplings. Standard Model EXCALIBUR is still used as the sample that is reweighted to calculate the coupling values. Samples of fully detector simulated ERATO with $\tilde{\kappa}_{z}$, $\tilde{\lambda}_{z}$ and $g^{z}_{4}$ values from -1 to +1 in nine equal increments were available. Table 6.4 shows a sample of the fit results and figure 6.6 shows the bias plots for the samples with these couplings. All results are consistent within the generated coupling.

Table 6.4: The bias fits to the single W SDM elements 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	-	$-$1.04 $^{+0.05}_{-0.05}$
$\tilde{\kappa}_{z}$	$-$0.50	-	$-$0.51 $^{+0.04}_{-0.04}$
$\tilde{\kappa}_{z}$	$-$0.25	-	$-$0.29 $^{+0.03}_{-0.03}$
$\tilde{\kappa}_{z}$	0.0	-	$-$0.01 $^{+0.03}_{-0.03}$
$\tilde{\kappa}_{z}$	$+$0.25	-	$+$0.27 $^{+0.03}_{-0.03}$
$\tilde{\kappa}_{z}$	$+$0.50	-	$+$0.50 $^{+0.03}_{-0.03}$
$\tilde{\kappa}_{z}$	$+$1.00	-	$+$1.02 $^{+0.05}_{-0.05}$
$\tilde{\lambda}_{z}$	$-$1.00	-	$-$0.93 $^{+0.06}_{-0.05}$
$\tilde{\lambda}_{z}$	$-$0.50	-	$-$0.53 $^{+0.05}_{-0.05}$
$\tilde{\lambda}_{z}$	$-$0.25	-	$-$0.29 $^{+0.04}_{-0.04}$
$\tilde{\lambda}_{z}$	0.0	-	$-$0.02 $^{+0.03}_{-0.03}$
$\tilde{\lambda}_{z}$	$+$0.25	-	$+$0.21 $^{+0.04}_{-0.04}$
$\tilde{\lambda}_{z}$	$+$0.50	-	$+$0.46 $^{+0.05}_{-0.04}$
$\tilde{\lambda}_{z}$	$+$1.00	-	$+$0.92 $^{+0.06}_{-0.06}$
$g^{z}_{4}$	$-$1.00	-	$-$0.97 $^{+0.06}_{-0.06}$
$g^{z}_{4}$	$-$0.50	-	$-$0.56 $^{+0.05}_{-0.05}$
$g^{z}_{4}$	$-$0.25	-	$-$0.21 $^{+0.04}_{-0.04}$
$g^{z}_{4}$	0.0	-	$-$0.02 $^{+0.04}_{-0.04}$
$g^{z}_{4}$	$+$0.25	-	$+$0.21 $^{+0.04}_{-0.04}$
$g^{z}_{4}$	$+$0.50	-	$+$0.53 $^{+0.05}_{-0.05}$
$g^{z}_{4}$	$+$1.00	-	$+$1.02 $^{+0.06}_{-0.06}$