W-Pair Polarised Cross-Sections

Figure 9.2 shows the total differential cross-section for the production of W-pairs and the differential cross-section for the production of two transversely polarised W bosons, two longitudinally polarised W bosons and the production of W bosons of opposite polarisation. All these were obtained from the 189 GeV data by the procedure described in the previous chapter.

**Figure 9.2:** The differential cross-sections for producing W-pairs (all), transversely polarised W-pairs (TT), longitudinally polarised W-pairs (LL) and pairs of W bosons with opposite polarisation (TL+LT) calculated from the 189 GeV data (points). Overlaid are the prediction for the Standard Model (solid line), $\tilde{\kappa}_{z}$ (dotted line) and $\tilde{\lambda}_{z}$ (dashed line). The dotted-dashed line shows the zero.
$\begin{figure}\begin{center} \epsfig{file=figs/final_joint_xsec.eps,width=1.0\linewidth}\end{center}\end{figure}$

Integrating over $\cos\theta_{\rm W}$ and dividing by the total cross-section will once again give the fraction of each helicity state. The results for this are shown in table 9.2. The areas of systematic uncertainty are described in chapter 10 and the overall systematic uncertainty is added in quadrature with the statistical error. The fractions are highly correlated and the correlations are obtained directly from the data by a statistical analysis on an event by event basis. The following correlations are obtained for the 189 GeV data:

$\displaystyle {\rm TT}:{\rm LL}$	$\displaystyle =$	$\displaystyle +0.648 \pm 0.018$
$\displaystyle {\rm TT}:{\rm TL}$	$\displaystyle =$	$\displaystyle -0.890 \pm 0.006$
$\displaystyle {\rm LL}:{\rm TL}$	$\displaystyle =$	$\displaystyle -0.867 \pm 0.008$

Using the results in table 9.2 and taking into consideration the above correlations, the $\chi ^{2}$ value of the three measurements compared to the Standard Model expectations, including systematic uncertainties, is 4.7. This corresponds to a $\chi ^{2}$ probability of 10%. Also shown in table 9.2 are the total cross-sections for the production of the different W-pair polarisation states. These were calculated using the total cross-section for W-pair production at 189 GeV measured at OPAL.

Table 9.2: The fraction of two transversely polarised W bosons (TT), two longitudinal W bosons (LL) and one of each polarisation (TL) calculated from the 189 GeV data. The first error is statistical and the second error is systematic. Also shown are the total cross-sections for the production of each polarisation state. In all cases the Standard Model expectations are calculated from EXCALIBUR Monte Carlo.

	Fraction	Cross-section (pb)
Data
TT	0.768 $\pm$ 0.090 $\pm$ 0.032	5.41 $\pm$ 0.65 $\pm$ 0.24
LL	0.206 $\pm$ 0.072 $\pm$ 0.018	1.45 $\pm$ 0.35 $\pm$ 0.09
TL	0.026 $\pm$ 0.147 $\pm$ 0.038	0.18 $\pm$ 1.02 $\pm$ 0.26
Standard Model Expectation
TT	0.570 $\pm$ 0.010	4.17 $\pm$ 0.07
LL	0.090 $\pm$ 0.008	0.66 $\pm$ 0.06
TL	0.340 $\pm$ 0.016	2.49 $\pm$ 0.12