Muon Decay Asymmetries from $K^0_L \to \pi^0 \mu^+ \mu^-$ Decays

We have examined the decay $K^0_L \to \pi^0 \mu^+ \mu^-$ in which the branching ratio, the muon energy asymmetry and the muon decay asymmetry could be measured. In particular, we find that within the Standard Model the longitudinal polarization ($P_L$) of the muon is proportional to the direct CP violating amplitude. On the other hand the energy asymmetry and the out-of-plane polarization ($P_N$) depend on both indirect and direct CP violating amplitudes. Although the branching ratio is small and difficult to measure because of background, the asymmetries could be large $\cal{O}$(1) in the Standard Model. A combined analysis of the energy asymmetry, $P_L$ and $P_N$ could be used to separate indirect CPV, direct CPV, and CP conserving contributions to the decay.