Relativistic $N$-particle energy shift in finite volume

TL;DR

This paper develops a method using nonrelativistic effective field theory to calculate the energy shifts of interacting N-particle systems in finite volume, including relativistic corrections, with applications to scalar field theories.

Contribution

It introduces a general approach to derive energy shifts for N-particle systems in finite volume, incorporating relativistic effects up to a specified order in the volume expansion.

Findings

Derived expressions for ground and excited state energy shifts up to order L^{-6}.

Applied the method to analyze energy shifts in the complex φ^4 theory.

Matched low-energy constants to scattering amplitudes for accurate predictions.

Abstract

We present a general method for deriving the energy shift of an interacting system of $N$ spinless particles in a finite volume. To this end, we use the nonrelativistic effective field theory (NREFT), and match the pertinent low-energy constants to the scattering amplitudes. Relativistic corrections are explicitly included up to a given order in the $1/L$ expansion. We apply this method to obtain the ground state of $N$ particles, and the first excited state of two and three particles to order $L^{-6}$ in terms of the threshold parameters of the two- and three-particle relativistic scattering amplitudes. We use these expressions to analyze the $N$-particle ground state energy shift in the complex $\varphi^4$ theory.