We discuss the implementation and properties of the quenched approximation in the calculation of the left-right, strong penguin contributions (i.e. $Q_6$) to ${ϵ}^{\prime }/ϵ$. The coefficient of the new chiral logarithm, discovered by Golterman and Pallante, which appears at leading order in quenched chiral perturbation theory is evaluated using both the method proposed by those authors and by an improved approach which is free of power divergent corrections. The result implies a large quenching artifact in the contribution of $Q_6$ to ${ϵ}^{\prime }/ϵ$. This failure of the quenched approximation affects only the strong penguin operators and so does not affect the $Q_8$ contribution to ${ϵ}^{\prime }/ϵ$ nor $\mathrm{Re}{A}_0$, $\mathrm{Re}{A}_2$ and thus, the $\Delta I=1/2$ rule at tree level in chiral perturbation theory.

• Received 11 May 2006

DOI:https://doi.org/10.1103/PhysRevD.74.034510