Fitting two nucleons inside a box: exponentially suppressed corrections to the Luscher's formula

TL;DR

This paper estimates exponentially suppressed finite volume corrections to Luscher's formula for two nucleons, providing guidance on lattice sizes needed for accurate scattering calculations in lattice field theory.

Contribution

It offers a quantitative estimate of these corrections, highlighting their dependence on the potential and pion effects, and specifies minimal lattice volumes for reliable results.

Findings

Corrections are proportional to the square of the potential.

Pion effects around the world are negligible.

Lattice volume should exceed (5 fm)^3 for <1° phase correction.

Abstract

Scattering observables can be computed in lattice field theory by measuring the volume dependence of energy levels of two particle states. The dominant volume dependence, proportional to inverse powers of the volume, is determined by the phase shifts. This universal relation (\Lu's formula) between energy levels and phase shifts is distorted by corrections which, in the large volume limit, are exponentially suppressed. They may be sizable, however, for the volumes used in practice and they set a limit on how small the lattice can be in these studies. We estimate these corrections, mostly in the case of two nucleons. Qualitatively, we find that the exponentially suppressed corrections are proportional to the {\it square} of the potential (or to terms suppressed in the chiral expansion) and the effect due to pions going ``around the world'' vanishes. Quantitatively, the size of the lattice should be greater than $\approx(5 {fm})^3$ in order to keep finite volume corrections to the phase less than $1^\circ$ for realistic pion mass.