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

Abstract

Using non-relativistic effective Lagrangians in the particle-dimer picture, we rederive the expression for the energy shift of a loosely bound three-particle bound state of identical bosons in the unitary limit. The effective field theory formalism allows us to explicitly investigate the role of the three-particle force. Moreover, we discuss relaxing the unitary limit of infinite scattering length and demonstrate a smooth transition from the weakly bound three-particle state to a two-particle bound state of a particle and a deeply bound dimer.

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

“…As a further illustration, note that this approach was successfully used [69] to reproduce the result of ref. [27] on the three-body bound state energy shift in a finite volume, as well as to study the role of the three-particle force and the generalization of the above result beyond the unitary limit.…”

Three-body spectrum in a finite volume: The role of cubic symmetry

“…Two important remarks are in order. First, the dimer picture is not an approximation, but an equivalent description of the three-particle system, if one allows for dimers of any spin and for a generic two-particle-dimer vertex [3,4]. This corresponds to the partial-wave expansion in the two-body amplitude.…”

“…Definition and implementation of such an expansion for the three-body quantization condition is the aim of the current study. We shall in particular demonstrate that, owing to the octahedral symmetry, the three-body quantization condition [3][4][5][6][10][11][12][14][15][16][17] can be split into different independent equations, whose solutions determine the finite-volume energy spectrum in different irreducible representations (irreps) of the octahedral group. This is extremely convenient, since the source and sink operators in lattice calculations are usually chosen to transform as irreducible tensor operators under the octahedral group, allowing one to determine the energy spectra in different irreps independently.…”

“…In the last few years, considerable progress has been made in understanding three-body observables in a finite volume and in deriving the three-particle quantization condition-that is, an equation that determines the finitevolume spectrum of the three-particle system through the parameters of the three-particle S matrix in the infinite volume [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]; see Refs. [18,19] for recent reviews.…”

“…[3][4][5], simple and transparent procedures have been designed, which allow one to extract the three-body observables in the infinite volume from the measured finitevolume spectrum. The complications arising from coupled two-and three-body channels in a finite volume have also been addressed recently [6].…”