Scheme Dependence of Weak Matrix Elements in the $1/N_c$ Expansion

TL;DR

This paper investigates how scheme and scale dependencies in the operator product expansion affect weak matrix elements within the $1/N_c$ expansion, providing explicit formulas and comparing different theoretical approaches to achieve a consistent estimate of $ark$.

Contribution

It introduces a method to correctly incorporate scheme and scale dependence in the $1/N_c$ expansion for weak matrix elements, with explicit formulas and comparisons to other models.

Findings

Good matching between long- and short-distance regimes.

Final $ark$ value of 0.77 with uncertainties.

Comparison of Chiral Perturbation Theory and ENJL model results.

Abstract

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 $0.77\pm0.05\pm0.05$.