K0↔K¯0transition amplitude in the MIT bag model

Abstract

Estimates of the $K^0\leftrightarrow {\overline{K}}^0$ transition amplitude and the resulting $K_L{K}_S$ mass difference are often based on an effective-Lagrangian treatment, where ${\mathrm{L}}_{\mathrm{eff}}$ is taken to be the four-quark operator extracted from a lowest-order calculation of the free-quark $\overline{s}+d\to s+\overline{d}$ scattering amplitude. For conventional SU(2) × U(1) models this effective Lagrangian has the form of a ($V-A$) current-current interaction ${\mathrm{O}}_{\mathrm{JJ}}$, with coefficient $C$ parametrized by quark masses and mixing angles. An important question has to do with gluonic and other corrections to the coefficient, but, independent of this, it is also necessary to estimate the matrix element of ${\mathrm{O}}_{\mathrm{JJ}}$ between physical $K^0$ and ${\overline{K}}^0$ states. We focus on this aspect of the problem. Earlier treatments have relied on a "vacuum-insertion" approximation. For an alternative approach, we employ the static MIT bag model. With the standard values for the bag parameters the matrix element is smaller than that obtained with vacuum insertion by about a factor of 2. This result is reasonably stable for small variations of the bag parameters.