Tau polarizations in the three-body slepton decays with stau as the next lightest supersymmetric particle

In the gauge-mediated supersymmetry breaking models with a scalar tau as the next-to-lightest supersymmetric particle, a scalar lepton may decay dominantly into its superpartner, the tau lepton, and the lightest scalar tau particle through $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{l}\ensuremath{\rightarrow}l{\ensuremath{\tau}}^{\ifmmode\pm\else\textpm\fi{}}{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\tau}}}_{1}^{\ensuremath{\mp}}.$ We give detailed formulas for the three-body decay amplitudes and the polarization asymmetry of the outgoing tau lepton. We find that the tau polarizations are sensitive to the model parameters such as the stau mixing angle, the neutralino to slepton mass ratio, and the neutralino mixing effect.