A Simultaneous Measurement of the Strong Coupling Constant and the QCD Colour Factors, based on an Improved Theoretical Prediction for the Two-Jet Rate in $e^+e^-$ Annihilations

In the first part of this thesis we show that an Ansatz to resum all leading and next-to-leading logarithms in the theoretical prediction for the two-jet rate in $e^+e^-\rightarrow hadrons$, where jets are defined with the Durham algorithm, is consistent with a full ${\cal O}(\alpha_s^2)$ calculation done by Monte Carlo integration. From the asymptotic behaviour of the full ${\cal O}(\alpha_s^2)$ calculation we extract the subleading coefficient $G_{21}$ and the constant $C_2$. Then, using data collected from 1992 to 1995 with the ALEPH detector at LEP, a measurement of the colour factor ratios $C_A/C_F$ and $T_F/C_F$ and the strong coupling constant $\overline{\alpha}_s = C_F \alpha_s(M_Z^2)/(2 \pi)$ has been performed by fitting theoretical predictions simultaneously to the measured differential two-jet rate and angular distributions in four-jet events. In the case of the two-jet rate, the improved resummation of next-to-leading logarithms has been used. The result is found to be in excellent agreement with QCD, \benn \frac{C_A}{C_F} = 2.20 \pm 0.09\, (stat) \pm 0.13\, (syst) \quad,\quad \frac{T_F}{C_F} = 0.29 \pm 0.05\, (stat) \pm 0.06\, (syst) . \eenn Fixing $C_A/C_F$ and $T_F/C_F$ to the QCD values permits a determination of $\alpha_s(M_Z^2)$ and $n_f$, the number of active flavours. With this measurement the existence of a gluiNO1with mass below $6.3$ GeV/$c^2$ is excluded at $95 \%$ confidence level. For $n_f = 5$ the strong coupling constant is determined as $\alpha_s(M_Z^2) = 0.1195 \pm 0.0002 (stat) \pm 0.0038 (syst)$.