Perturbative results for two and three particle threshold energies in finite volume

TL;DR

This paper computes finite-volume energy shifts for two and three identical particles in relativistic quantum field theory, providing perturbative results up to order λ³ and validating existing quantization conditions.

Contribution

It offers explicit perturbative calculations of threshold energies in finite volume for two and three particles, including subleading terms, and compares with non-relativistic and existing formalism results.

Findings

Energy shifts start at order 1/L^3.

Subleading terms up to 1/L^6 are calculated.

Results validate Lüscher's and our three-particle quantization conditions.

Abstract

We calculate the energy of the state closest to threshold for two and three identical, spinless particles confined to a cubic spatial volume with periodic boundary conditions and with zero total momentum in the finite-volume frame. The calculation is performed in relativistic quantum field theory with particles coupled via a $\lambda \phi^4$ interaction, and we work through order $\lambda^3$. The energy shifts begin at ${\cal O}(1/L^3)$, and we keep subleading terms proportional to $1/L^4$, $1/L^5$ and $1/L^6$. These terms allow a non-trivial check of the results obtained from quantization conditions that hold for arbitrary interactions, namely that of L\"uscher for two particles and our recently developed formalism for three particles. We also compare to previously obtained results based on non-relativistic quantum mechanics.