$K_L \to \pi^0 \nu \bar \nu$ Beyond the Standard Model

We analyze the decay $K_L \to \pi^0 \nu \bar \nu$ in a model independent way. If lepton flavor is conserved the final state is (to a good approximation) purely CP even. In that case this decay mode goes mainly through CP violating interference between mixing and decay. Consequently, a theoretically clean relation between the measured rate and electroweak parameters holds in any given model. Specifically, $\Gamma(K_L \to \pi^0 \nu \bar \nu)/\Gamma(K^+ \to \pi^+ \nu \bar \nu)= \sin^2\theta$ (up to known isospin corrections), where $\theta$ is the relative CP violating phase between the $K-\bar K$ mixing amplitude and the $s\to d\nu\bar\nu$ decay amplitude. The experimental bound on $BR(K^+ \to \pi^+ \nu \bar \nu)$ provides a model independent upper bound: $BR(K_L \to \pi^0 \nu \bar \nu) < 1.1 \times 10^{-8}$. In models with lepton flavor violation, the final state is not necessarily a CP eigenstate. Then CP conserving contributions can dominate the decay rate.