Low temperature condensation and scattering data

TL;DR

This paper investigates low temperature condensation in lattice field theory, relating particle condensation thresholds to scattering data, and demonstrates how scattering information can predict multi-particle energies.

Contribution

It introduces a method to extract scattering data from condensation thresholds in lattice field theory, connecting finite volume energies to scattering parameters in two and four dimensions.

Findings

Scattering length in 4D determined from 2-particle threshold.

Full phase shift obtained in 2D from condensation data.

Predicted 3-particle energies agree with direct calculations.

Abstract

We study $\phi^4$ lattice field theory at finite chemical potential $\mu$ in two and four dimensions, using a worldline representation that overcomes the complex action problem. We compute the particle number at very low temperature as a function of $\mu$ and determine the first three condensation thresholds, where the system condenses 1, 2 and 3 particles. The corresponding critical values of the chemical potential can be related to the 1-, 2- and 3-particle energies of the system, and we check this relation with a direct spectroscopy determination of the $n$-particle energies from $2n$-point functions. We analyze the thresholds as a function of the spatial size of the system and use the known finite volume results for the $n$-particle energies to relate the thresholds to scattering data. For four dimensions we determine the scattering length from the 2-particle threshold, while in two dimensions the full scattering phase shift can be determined. In both cases the scattering data computed from the 2-particle threshold already allow one to determine the 3-particle energy. In both, two and four dimensions we find very good agreement of this ''prediction'' with direct determinations of the 3-particle energy from either the thresholds or the 6-point functions. The results show that low temperature condensation is indeed governed by scattering data.