The decays $K \to \pi \pi \nu \overline\nu$ within the standard model

We study the reactions $K \rightarrow \pi \pi \nu \overline{\nu}$ within the minimal standard model. We use isospin symmetry to relate the matrix elements to the form factors measured in $K_{\ell 4}$. We argue that these modes are short distance dominated and can be used for precise determinations of the CKM parameters $\rho$ and $\eta$. Depending on the value of the CKM angles we find branching ratios in the following ranges: $B(K_L \rightarrow \pi^+ \pi^- \nu \overline{\nu}) = [2-5] \times 10^{-13}$; $B(K_L \rightarrow \pi^0 \pi^0 \nu \overline{\nu}) = [1-3] \times 10^{-13}$; $B(K^+ \rightarrow \pi^+ \pi^0 \nu \overline{\nu}) = [1-2] \times 10^{-14}$. We also discuss a possible $CP$-odd observable.