Finite-Volume Two-Pion Amplitudes in the I=0 Channel

We perform a calculation in one-loop chiral perturbation theory of the two-pion matrix elements and correlation functions of an I=0 scalar operator, in finite and infinite volumes for both full and quenched QCD. We show that major difficulties arise in the quenched theory due to the lack of unitarity. Similar problems are expected for quenched lattice calculations of $K \to \pi \pi$ amplitudes with $\Delta I=1/2$. Our results raise the important question of whether it is consistent to study $K\to\pi\pi$ amplitudes beyond leading order in chiral perturbation theory in quenched or partially quenched QCD.