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

TL;DR

This paper investigates two-pion matrix elements in finite and infinite volumes using chiral perturbation theory, highlighting issues in quenched QCD that challenge the study of kaon decay amplitudes beyond leading order.

Contribution

It provides a detailed one-loop analysis of I=0 two-pion amplitudes in finite and infinite volumes, emphasizing problems in quenched QCD affecting kaon decay calculations.

Findings

Quenched QCD exhibits unitarity violations affecting two-pion amplitudes.

Major difficulties arise in quenched calculations of $K o \pi \\pi$ amplitudes.

Questions the validity of beyond-leading-order chiral perturbation theory in quenched QCD.

Abstract

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.