Rare $\Lambda_{b}\to \Lambda $ l$^{+}$ l$^{-}$ Decays with Polarized $\Lambda $

We investigate the rare baryonic exclusive decays of $\Lambda_b\to \Lambda l^+ l^- (l=e,\mu,\tau)$ with polarized $\Lambda$. Under the approximation of the heavy quark effective theory (HQET), in the standard model we derive the differential decay rates and various polarization asymmetries by including lepton mass effects. We find that with the long-distance effects the decay branching ratios are $5.3\times 10^{-5}$ for $\Lambda_b\to \Lambda l^+ l^-\ (l=e,\mu)$ and $1.1\times 10^{-5}$ for $\Lambda_b\to \Lambda \tau^+\tau^-$. The effects of new physics in the decay rates are also discussed. The integrated longitudinal $\Lambda$ polarizations are -0.31 and -0.12, while that of the normal ones 0.02 and 0.01, for di-muon and tau modes, respectively. The CP-odd transverse polarization of $\Lambda$ is zero in the standard model but it is expected to be sizable in some theories with new physics.