LU TP 99-37, hep-ph/9911392
Scheme Dependence of Weak Matrix Elements in the
$1/N_c$ Expansion
J. Bijnens, J. Prades
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 $0.77\pm0.05\pm0.05$.