On-shell renormalization of the〈q¯q〉contribution to theΔI=12s−dself-energy transition

The dimension-3 quark condensate contribution to the renormalized $s-d$ self-energy transition is evaluated within the framework of on-shell renormalization with respect to the (Lagrangian) quark mass matrix. This transition is then related to $\Delta I=\frac{1}{2}$ matrix elements for nonleptonic kaon decay rates. The contributions to such matrix elements from the $〈\overline{q}q〉$ component of the $s-d$ self-energy transition, when considered in isolation, are seen to generate rates of the same order of magnitude as the experimental values for $K_L\to \gamma \gamma$ and $K_S\to \pi \pi$ decays.