$K_L \ra \mu^\pm e^\mp \nu \overline{\nu}$ as background to $K_L \ra \mu^\pm e^\mp$

We consider the process $K_L \ra \mu^\pm e^\mp \nu \overline{\nu}$ at next to leading order in chiral perturbation theory. This process occurs in the standard model at second order in the weak interaction and constitutes a potential background in searches for new physics through the modes $K_L \ra \mu^\pm e^\mp$. We find that the same cut, $M_{\mu e}>489$~MeV, used to remove the sequential decays $K_{l3}\ra \pi_{l2}$ pushes the $B(K_L \ra \mu^\pm e^\mp \nu \overline{\nu})$ to the $10^{-23}$ level, effectively removing it as a background.