Z→bb̄bb̄ in the light gluino and light sbottom scenario

The light gluino (12\char21{}16 GeV) and light sbottom (2\char21{}6 GeV) scenario has been used to explain the apparent overproduction of b quarks at the Fermilab Tevatron. This scenario also predicts the decay $\stackrel{\ensuremath{\rightarrow}}{Z}b\overline{b}\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{g}\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{g}$ where the gluinos subsequently decay into b quarks and sbottoms. We show that this can contribute to ${\ensuremath{\Gamma}}_{4b}=\ensuremath{\Gamma}(\stackrel{\ensuremath{\rightarrow}}{Z}b\overline{b}b\overline{b})$ since most of the sbottoms and b quarks arising from gluino decay have a small angular separation. We find that while no excess in ${\ensuremath{\Gamma}}_{4b}$ is observable due to large uncertainties in experimental measurements, the ratio $\ensuremath{\Gamma}(\stackrel{\ensuremath{\rightarrow}}{Z}b\overline{b}\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{g}\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{g})/\ensuremath{\Gamma}(\stackrel{\ensuremath{\rightarrow}}{Z}b\overline{b}b\overline{b})$ can be large due to sensitivity to b-quark mass, the sbottom mixing angle and the gluino mass. We calculate it to be in the range 0.05\char21{}0.41 inclusive of the entire parameter space.