can be expected (green solid lines) and at which all scenarios with a metastable stau can be excluded at $95\%\,$C.L. (red dashed lines, with red-shaded regions around the $10^{-7}$ curve indicating the $1\sigma$ and $2\sigma$ probability bands).[^3] The borders of the gray-shaded regions denote the luminosities that correspond to 1, 10, and 100 events. We see that both discovery and exclusion are expected to occur on the basis of very few events.

Observing no events when three are expected by a hypothesis is sufficient to exclude this hypothesis at $95\%\,$C.L. Thus, for luminosities that lead to a sufficiently small background $B\ll1$, the exclusion limit corresponds to the three-event line. This is why for small ${m_{{{\widetilde{\tau}}}_1}}$ the exclusion limits for different ${{r_{\beta}}}$ are degenerate at the three-event line, which coincides with the ${{r_{\beta}}}=10^{-10}$ line in figure \[fig:lum-num-mass-fix\]—the corresponding background is sufficiently suppressed. As it is not possible to obtain less than zero events, the red-shaded regions do not continue beyond the three-event line.

In figure \[fig:lum-num-mass-fix\] we also show the results for the $7{\ensuremath{\,\textnormal{TeV}}}$ run of the LHC. The calculation has been analogous except that we have used a slightly smaller $K$-factor of $1.3$ for the normalization of the stau production 