On the Kinematic Reconstruction of $e^+e^-\to W^+W^-\to jj\tau\overline{\nu}_\tau$\ Events

We show that the kinematic reconstruction of the $e^+e^-\rightarrow W^+W^-\rightarrow jj\tau^-\overline{\nu}_\tau$\ events have a one-parameter ambiguity when reconstructed from the momentum of all measured $W^-$\ decay products. We propose a {\em hybrid} method of reconstruction of the $e^+e^-\rightarrow W^+W^-\rightarrow jj\tau^-\overline{\nu}_\tau$\ events. This is based on the observation that the difference between the $\tau^-$\ production angles and the production angles of the sum of its visible decay products is small, whilst the $\tau^-$\ energy is poorly reconstructed. This method consists of taking the $\tau^-$\ production angles from those measured for the sum of the visible $\tau^-$\ decay products and reconstructing the $\tau^-$\ energy from energy-momentum conservation constraints. A reconstruction using this method is found to be well-defined and possess a unique solution for the $\tau^-$\ momentum range at LEP II and NLC.