Scattering in the quenched approximation

TL;DR

This paper investigates how pion scattering lengths relate to finite-volume energies in the quenched approximation, revealing significant deviations from the full theory due to unique properties of the $ a'$ particle.

Contribution

It demonstrates that quenched approximation introduces drastically different finite-volume effects, including enhanced corrections, in the relation between scattering lengths and energy levels.

Findings

Quenched approximation alters Luescher's relation significantly.

Enhanced finite-volume corrections of order 1 and $L^{-2}$ are identified.

Numerical examples indicate these effects can be substantial.

Abstract

We study, in the quenched approximation, Luescher's relation between pion scattering lengths and the finite-volume energy of two pions at rest. The quenched relation is drastically different from the full theory one; in particular, ``enhanced finite-volume corrections" of order $L^0=1$ and $L^{-2}$ occur at one loop ($L$ is the linear size of the box), due to the special properties of the $\eta'$ in the quenched approximation. Numerical examples show that the size of these effects can be substantial.