Efimov states of three unequal bosons in non-integer dimensions

TL;DR

This paper explores the existence and properties of Efimov states in three-boson systems with unequal masses in non-integer dimensions, introducing a novel mathematical approach to analyze how these states depend on dimension, mass ratio, and scattering lengths.

Contribution

It develops a new mathematical technique to study Efimov states in non-integer dimensions and analyzes their conditions and properties for systems with unequal bosons.

Findings

Efimov states disappear below a critical non-integer dimension.

The number and scaling of Efimov states depend on mass ratio and scattering length.

A qualitative relation between dimension parameter and external squeezing field is proposed.

Abstract

The Efimov effect for three bosons in three dimensions requires two infinitely large $s$-wave scattering lengths. We assume two identical particles with very large scattering lengths interacting with a third particle. We use a novel mathematical technique where the centrifugal barrier contains an effective dimension parameter, which allows efficient calculations precisely as in ordinary three spatial dimensions. We investigate properties and occurrence conditions of Efimov states for such systems as functions of the third scattering length, the non-integer dimension parameter, mass ratio between unequal particles, and total angular momentum. We focus on the practical interest of the existence, number of Efimov states and their scaling properties. Decreasing the dimension parameter from $3$ towards $2$ the Efimov effect and states disappear for critical values of mass ratio, angular momentum and scattering length parameter. We investigate the relations between the four variables and extract details of where and how the states disappear. Finally, we supply a qualitative relation between the dimension parameter and an external field used to squeeze a genuine three dimensional system.