Finite-volume two-pion energies and scattering in the quenched approximation

TL;DR

This paper explores how the relation between finite-volume two-pion energies and scattering lengths is altered in quenched QCD, revealing significant finite-volume corrections and divergences in the chiral limit.

Contribution

It demonstrates the drastic modifications to L"uscher's relation in quenched QCD and introduces the concept of quenched pion scattering lengths with their divergent behavior.

Findings

Enhanced finite-volume corrections of order L^0 and L^{-2} occur at one loop.

Quenched pion scattering lengths are linearly divergent in the chiral limit.

Effects can be substantial in numerical estimates.

Abstract

We investigate how L\"uscher's relation between the finite-volume energy of two pions at rest and pion scattering lengths has to be modified in quenched QCD. We find that this relation changes drastically, and in particular, that ``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. We define quenched pion scattering lengths, and show that they are linearly divergent in the chiral limit. We estimate the size of these various effects in some numerical examples, and find that they can be substantial.