$K \to \pi\pi$ Decays in a Finite Volume

TL;DR

This paper extends the theoretical framework for calculating $K o \pi\pi$ decay amplitudes in finite-volume lattice QCD simulations, addressing elastic states below inelastic thresholds and comparing different extraction methods.

Contribution

It generalizes the Lellouch-Lüscher relation to all elastic states below inelastic thresholds and analyzes alternative methods for decay amplitude extraction.

Findings

Extended the Lellouch-Lüscher relation to all elastic states below inelastic thresholds.

Identified finite-volume corrections that decay as inverse powers of volume in alternative methods.

Compared different approaches for extracting decay amplitudes and discussed their limitations.

Abstract

We discuss finite-volume computations of two-body hadronic decays below the inelastic threshold (e.g. $K\to\pi\pi$ decays). The relation between finite-volume matrix elements and physical amplitudes, recently derived by Lellouch and L\"uscher, is extended to all elastic states under the inelastic threshold. We present a detailed comparison of our approach with that of Lellouch and L\"uscher and discuss the possible limitations of the method which could arise due to the presence of inelastic thresholds. We also examine a standard alternative method which can be used to extract the real part of the decay amplitude from correlators of the form $< 0 |T[\pi\pi{\cal H}_WK ]| 0 >$. We show that in this case there are finite-volume corrections which vanish as inverse powers of the volume, which cannot be removed by a multiplicative factor.