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We show how the scheme- and scale-dependence of the short-distance operatorproduct expansion with four-quark operators can be correctly accounted for inthe framework of the $1/N_c$-expansion once the hadronization of two-quarkcurrents has been fixed. We show formulas explicitly in the case of the$B_K$-parameter. We then use them with our earlier estimates of thelong-distance effects. We compare Chiral Perturbation Theory at Leading- andNext-to-Leading-Order with the ENJL model results in the chiral limit. Goodmatching between the long- and short-distance regimes is obtained and our finalvalue for the physical scheme independent $\hat B_K$ is $0.77\pm0.05\pm0.05$.Comment: 20 pages, use JHEP.cls and axodraw.sty; small error detected,  numerical results changed with about quoted erro

Scheme dependence of weak matrix elements in the 1/Nc expansion

Scheme dependence of weak matrix elements in the 1/N_c expansion