Applicability of the two-particle quantization condition to partially-quenched theories

TL;DR

This paper examines the validity of the two-particle quantization condition in partially-quenched theories, showing it fails in some cases due to unitarity issues, with implications for lattice QCD analyses.

Contribution

The study demonstrates that the Lüscher quantization condition does not universally apply to partially-quenched correlation functions, especially when one-loop amplitudes violate unitarity.

Findings

Fails for correlation functions violating s-wave unitarity.

Possibly valid for unphysical amplitudes satisfying unitarity.

Impacts interpretation of previous lattice QCD results.

Abstract

Partial quenching allows one to consider correlation functions and amplitudes that do not arise in the corresponding unquenched theory. For example, physical $s$-wave pion scattering can be decomposed into $I=0$ and $2$ amplitudes, while, in a partially-quenched extension, the larger symmetry group implies that there are more than two independent scattering amplitudes. It has been proposed that the finite-volume quantization condition of L\"uscher holds for the correlation functions associated with each of the two-particle amplitudes that arise in partially-quenched theories. Using partially-quenched chiral perturbation theory, we show that this proposal fails for those correlation functions for which the corresponding one-loop amplitudes do not satisfy $s$-wave unitarity. For partially-quenched amplitudes that, while being unphysical, do satisfy one-loop $s$-wave unitarity, we argue that the proposal is plausible. Implications for previous work are discussed.