The rare decay KL -> pi0 mu+ mu- within the SM
  We present an updated analysis of the rare decay $K_{L}\to\pi^{0}\mu^{+}\mu^{-}$ within the Standard Model. In particular, we present the first complete calculation of the two-photon CP-conserving amplitude within Chiral Perturbation Theory, at the lowest non-trivial order. Our results confirm previous findings that the CP-conserving contribution to the decay rate cannot be neglected. By means of an explicit two-loop calculation, we show that this contribution can be estimated with sufficient accuracy compared to the CP-violating terms. We predict $\cB(K_L\to\pi^0\mu^+\mu^-)_{\rm SM}=(1.5\pm0.3)\times10^{-11}$, with approximately equal contributions from the CP-conserving component, the indirect-CP-violating term, and the interesting direct-CP-violating amplitude. The error of this prediction is mainly of parametric nature and could be substantially reduced with better data on the $K_{S}\to\pi^{0}\ell^{+}\ell^{-}$ modes. The Standard Model predictions for various differential distributions and the sensitivity to possible new-physics effects are also briefly discussed. 
Gino Isidori, Christopher Smith, and Rene Unterdorfer