The $\chi ^{2}$ Fit to the $\cos\theta_{\rm W}$ Distribution

The single W SDM elements are normalised to the total number of events in each bin of $\cos\theta_{\rm W}$. This means that single W SDM elements are effectively independent of the shape of the $\cos\theta_{\rm W}$ distribution. The W production angle is sensitive to the TGCs. This means that there is an observable that has been measured, is completely uncorrelated to the single W SDM elements, and as yet has not been used in the TGC fit.

The W production angle, $\cos\theta_{\rm W}$, can be binned into eight bins like the single W SDM elements and normalised to the total number of events. Then a $\chi ^{2}$ fit can be performed in a similar way as for the SDM elements. The reweighted Monte Carlo method can be used to calculate the theoretical distributions in the $\chi ^{2}$ minimisation. The form of the $\chi ^{2}$ would be as in equation 6.10, where $N_{tot}$ is the total number of events and $\sigma_{{N_{k}^{me}}}$ is the error on the number of events measured in bin $k$ of $\cos\theta_{\rm W}$. The error is simply the square-root of the number of events in the bin.

$$\chi^{2} = \sum_{k=1}^{N}\left[\left(\frac{N_{k}^{me}}{N_{tot}^{m... ...{th}}\right) \left(\frac{N_{tot}^{me}}{\sigma_{{N_{k}^{me}}}}\right)\right]^{2}$$ (6.10)