On the modification of the Efimov spectrum in a finite cubic box

Abstract

Three particles with large scattering length display a universal spectrum of three-body bound states called "Efimov trimers". We calculate the modification of the Efimov trimers of three identical bosons in a finite cubic box and compute the dependence of their energies on the box size using effective field theory. Previous calculations for positive scattering length that were perturbative in the finite volume energy shift are extended to arbitrarily large shifts and negative scattering lengths. The renormalization of the effective field theory in the finite volume is explicitly verified. We investigate the effects of partial wave mixing and study the behavior of shallow trimers near the dimer energy. Moreover, we provide numerical evidence for universal scaling of the finite volume corrections.

Three-particle quantization condition in a finite volume: 1. The role of the three-particle force

“…At the same time, in refs. [30][31][32][33], the volume dependence of the discrete three-body spectrum has been investigated for bosons as well as nucleons by solving the bound state equations in a finite volume numerically. An effective field theory in the dimer formalism has been used to derive the finite volume bound state equations and relate the bound state properties to scattering parameters in the infinite volume, which greatly simplifies the handling of the three-body problem.…”

Three-particle quantization condition in a finite volume: 2. General formalism and the analysis of data

“…Finally, we mention refs. [37][38][39][40], which addressed the three-body problem in a finite volume numerically using an effective field theory in the particle-dimer picture. Their numerical results for the finite volume spectrum strongly support the statement that the spectrum does not depend on the off-shell behavior of the three-body amplitudes, which was first proven in ref.…”

“…Note also that the equation (3.1) was used earlier in refs. [37][38][39][40] to calculate the spectrum of the three-body bound states in a finite volume numerically. Now, our strategy for the analysis of data in the three-particle sector can be formulated as follows:…”

Three-particle bound states in a finite volume: Unequal masses and higher partial waves

“…Despite the significant effort, the progress has been slow so far. Namely, the finite-volume spectrum of the threeparticle system in some simple models has been calculated only very recently [12,27] (see also earlier work [18][19][20][21], where exclusively the three-body bound-state sector was addressed). Such calculations are very useful since, at this stage, one does not yet have enough insight into the problem and lacks intuition to predict the behavior of the three-particle finite-volume energy levels.…”