Quenched penguin operators and theΔI=1/2rule

The transformation properties of strong penguin operators under the action of the flavor group change when they are considered as operators in (partially) quenched QCD instead of the unquenched theory. As a result, additional operators and new low-energy constants appear in the effective theory describing nonleptonic kaon decay matrix elements in the partially quenched setting. These new low-energy constants do not have a counterpart in the unquenched theory, and should thus be considered as an artifact of the quenched approximation. Here we consider strong penguin operators consisting of products of two left-handed flavor currents, and give a complete one-loop analysis in the effective theory for $K^0$ to vacuum and $K^{+}$ to ${\pi }^{+}$ matrix elements. We find that the new low-energy constants already appear in these matrix elements at leading order. This implies that (partially) quenched lattice computations of for instance the $\Delta I=1/2$ rule are affected by ambiguities intrinsic to the use of the quenched approximation at leading order. The only exception is the partially quenched case with three light sea quarks, consistent with general expectations. Our results are also relevant when the charm quark is kept in the theory.