Including Systematics

The areas of systematic uncertainties investigated and their individual contributions are described in Chapter 10. The inclusion of these systematic uncertainties occurs before the fit is performed. In this way, not only changes in the overall error, but also a shift in the minimum value due to this uncertainty can be calculated.

A systematic error is calculated on each observable that is measured. This means that all eight bins of each of the nine SDM elements and the eight bins of the $\cos\theta_{\rm W}$ distribution will be assigned an error.

These systematics are then included in the $\chi ^{2}$ fit given by equation 6.6 by including them directly in the covariance matrix as follows:

$$V_{ij}(k) = \omega_{ij}(k)\left(\sigma^{\rm stat}_{i}(k)\sigma^{\rm stat}_{j}(k)+\sigma^{\rm sys}_{i}(k)\sigma^{\rm sys}_{j}(k)\right)$$ (7.1)

Where $\sigma^{\rm stat}_{i}(k)$ is the statistical error and $\sigma^{\rm sys}_{i}(k)$ is the systematic error. The correlation matrix $\omega_{ij}(k)$ is only for statistical correlations, it does not include any extra correlations which may be introduced by the systematic uncertainty. This may not be an exact treatment of the correlations, but will give a reasonable estimate. Calculation of exact correlations for each systematic error would be an extremely difficult and involved task.

All the final fit results for both the CP-conserving and CP-violating couplings, using the BILGOU reweighting scheme, including the systematic uncertainties can be seen in table 7.3. The results for the combined fit are listed below.

Using the WVCXME reweighting scheme the final fitted values for the CP-conserving couplings including systematic uncertainties are:

$$\Delta\kappa_{\gamma} = -0.235^{+0.282}_{-0.232}$$

$$\Delta g^{Z}_{1} = -0.038^{+0.095}_{-0.089}$$

$$\lambda = -0.082^{+0.102}_{-0.095}$$

Using the BILGOU reweighting scheme the final fitted values for the CP-conserving couplings including systematic uncertainties are:

$$\Delta\kappa_{\gamma} = -0.263^{+0.263}_{-0.218}$$

$$\Delta g^{Z}_{1} = -0.029^{+0.092}_{-0.089}$$

$$\lambda = -0.045^{+0.099}_{-0.093}$$

Using the BILGOU reweighting scheme the final fitted values for the CP-violating couplings including systematic uncertainties are:

$$\tilde{\kappa}_{Z} = -0.184^{+0.091}_{-0.065}$$

$$\tilde{\lambda}_{Z} = -0.136^{+0.161}_{-0.194}$$

$$g^{Z}_{4} = +0.070^{+0.263}_{-0.255}$$

The $\chi ^{2}$ plots for the fits of the CP-conserving couplings using the WVCXME and BILGOU reweighting schemes can be seen in figures 7.6 and 7.7 respectively. The $\chi ^{2}$ plots for the fits of the CP-violating couplings are shown in figure 7.8.

Figure 7.6: $\chi ^{2}$ plots of the fit of the CP-conserving couplings, including systematic uncertainties, using the WVCXME reweighting scheme. The dashed line is the fit to single W SDM elements, the dotted line is the fit to the $\cos\theta_{\rm W}$ distribution, and the solid line is the combined fit.

Figure 7.7: $\chi ^{2}$ plots of the fit of the CP-conserving couplings, including systematic uncertainties, using the BILGOU reweighting scheme. The dashed line is the fit to single W SDM elements, the dotted line is the fit to the $\cos\theta_{\rm W}$ distribution, and the solid line is the combined fit.

Figure 7.8: $\chi ^{2}$ plots of the fit of the CP-violating couplings, including systematic uncertainties, using the BILGOU reweighting scheme. The dashed line is the fit to single W SDM elements, the dotted line is the fit to the $\cos\theta_{\rm W}$ distribution, and the solid line is the combined fit.