brackets contain $\hat{s}$ as the only kinematic variable, and even this dependence becomes negligible when exceeding a few times $M_{{Z}}$. The left panel of figure \[fig:theta-stau\] shows the ratio between the first term and all terms in (\[eq:DYcross-section\]) as a function of $\hat{s}$ (arbitrarily normalized). From a few hundred GeV on the ratio is almost constant. Thus, in this region a change in ${\theta_{{{\widetilde{\tau}}}}}$ will only shift the overall cross section without any impact on the kinematic distributions. In fact, the dependence of the cross section on ${\theta_{{{\widetilde{\tau}}}}}$ shown in the right panel of figure \[fig:theta-stau\] is applicable to all stau masses considered in the following.

For the plots in figures \[fig:sigma-mass\] and \[fig:distau-obs\] we consider the case ${\theta_{{{\widetilde{\tau}}}}}=0$. However, the minimum of $\sigma_{\rm ew}({\theta_{{{\widetilde{\tau}}}}})$ is at ${\theta_{{{\widetilde{\tau}}}}}^{\rm min}\neq0$ and it is about $7\%$ lower than the value at $\sigma_{\rm ew}(0)$. Therefore, when estimating the discovery potential and exclusion limits we conservatively choose ${\theta_{{{\widetilde{\tau}}}}}={\theta_{{{\widetilde{\tau}}}}}^{\rm min}$. The limits for other values of ${\theta_{{{\widetilde{\tau}}}}}$ can easily be obtained from the displayed curves. Since it turns out that we can achieve a (almost) clean signal region, the required luminosity is, to a very good approximation, simply proportional to the inverse of the cross 