  We show how the scheme- and scale-dependence of the short-distance operator
product expansion with four-quark operators can be correctly accounted for in
the framework of the $1/N_c$-expansion once the hadronization of two-quark
currents has been fixed. We show formulas explicitly in the case of the
$B_K$-parameter. We then use them with our earlier estimates of the
long-distance effects. We compare Chiral Perturbation Theory at Leading- and
Next-to-Leading-Order with the ENJL model results in the chiral limit. Good
matching between the long- and short-distance regimes is obtained and our final
value for the physical scheme independent $\hat B_K$ is $0.77\pm0.05\pm0.05$.
