Fit to Single W SDM Elements

The nine single W SDM elements extracted from the 189 GeV data are shown in figure 7.1. These elements were extracted from the leptonically decaying W bosons in the W-pair only, and CPT-invariance was assumed in all cases so that the results from the leptons and anti-leptons could be combined, as described in chapter 4 by equation 4.6. The errors shown on the plots are purely statistical. A $\chi ^{2}$ fit was performed to these elements. Overlaid on these plots are the predicted SDM elements extracted from fully simulated Monte Carlo.

**Figure 7.1:** Single W SDM elements extracted from the 189 GeV data (points). The solid line is the Standard Model prediction the dotted is the predictions for $\Delta g^{z}_{1}$ and the dashed line is the prediction for $g^{z}_{4}$ .
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The correlations between all the first bins of the nine observables are given in table 7.1. For the calculation of the CP-conserving couplings a fit was made only to the six real single W SDM elements, as these are the only ones sensitive to the CP-conserving couplings.

**Table 7.1:** *The correlations between the $\mbox{$\cos\theta_{\rm W}$}$ bin of all the single W SDM observables extracted from the 189 GeV data.*
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$\displaystyle \Delta\kappa_{\gamma}$	$\displaystyle =$	$\displaystyle -0.170^{+0.461}_{-0.305}$
$\displaystyle \Delta g^{Z}_{1}$	$\displaystyle =$	$\displaystyle +0.047^{+0.117}_{-0.144}$
$\displaystyle \lambda$	$\displaystyle =$	$\displaystyle +0.007^{+0.208}_{-0.226}$

The $\chi ^{2}$ plots for these fits can be seen in figure 7.3, where the dashed line represents the fit to the SDM elements. A similar fit of the couplings was made using the BILGOU reweighting scheme. The fitted values are:

$\displaystyle \Delta\kappa_{\gamma}$	$\displaystyle =$	$\displaystyle -0.229^{+0.496}_{-0.425}$
$\displaystyle \Delta g^{Z}_{1}$	$\displaystyle =$	$\displaystyle +0.040^{+0.116}_{-0.158}$
$\displaystyle \lambda$	$\displaystyle =$	$\displaystyle +0.051^{+0.168}_{-0.177}$

The $\chi ^{2}$ plots for these fits can be seen in figure 7.4, where the dashed line represents the fit the the SDM elements. The results for $\Delta g^{z}_{1}$ and $\lambda$ agree well with those calculated using the WVCXME reweighting scheme. The difference in the fitted values for $\Delta\kappa_{\gamma}$ is slightly greater, but still much less than the statistical error. It will be seen later that this difference has little effect on the overall combined result once information about the W production angle is included in the fit.

The CP-violating couplings were calculated by performing fits to all nine single W SDM elements. The BILGOU reweighting scheme was used for these fits. The fitted values are:

$\displaystyle \tilde{\kappa}_{Z}$	$\displaystyle =$	$\displaystyle -0.191^{+0.078}_{-0.066}$
$\displaystyle \tilde{\lambda}_{Z}$	$\displaystyle =$	$\displaystyle -0.089^{+0.159}_{-0.152}$
$\displaystyle g^{Z}_{4}$	$\displaystyle =$	$\displaystyle +0.029^{+0.211}_{-0.215}$

The $\chi ^{2}$ plots for these fits can be seen in figure 7.5. The plot for the fit of $\tilde{\kappa}_{z}$ to the SDM elements displays a second minimum. This is because the real SDM elements are sensitive only to the magnitude of the coupling and not the sign, and the relative contribution from the real SDM elements in the fit is similar to that from the imaginary parts. A second minimum is not shown for the other CP-violating couplings because the relative contribution from the real parts of the SDM is much less than that of the imaginary for these couplings.