  We discuss the matching between long-distance and short-distance at
next-to-leading in $1/N_c$ and show how the scheme-dependence from the two-loop
renormalization group running can be treated. We then use this method to study
the three $O(p^2)$ terms contributing to non-leptonic kaon decays, namely the
usual octet and 27-plet derivative terms as well as the weak mass term using
the Extended Nambu--Jona-Lasinio model as the low energy approximation. We also
discuss subtleties in the momentum routing in the low energy theory and a
problem in separating factorizable and non-factorizable contributions from the
$Q_6$ operator in the chiral limit. We update our earlier results on the $B_K$
parameter as well.
