Exact Renormalization Relation and Binding Energies for Three Identical Bosons

TL;DR

This paper derives an exact renormalization relation linking the three-body parameter to Efimov state energies in a universal three-boson system, validated by numerical simulations.

Contribution

It provides the first exact relation between the three-body parameter, coupling constants, and Efimov binding energies for identical bosons.

Findings

Established an exact renormalization relation for the three-body parameter.

Connected the three-body parameter to Efimov bound state energies.

Validated the relation through high-precision numerical simulations.

Abstract

In the low-energy limit, non-relativistic particles with short-range interactions exhibit universal behavior that is largely independent of microscopic details. This universality is typically described by effective field theory, in which the two-body interaction is renormalized to a single parameter-the scattering length. For systems of identical bosons, the three-body problem reveals the Efimov effect, a novel phenomenon proposed that necessitates the introduction of an additional three-body parameter. However, the exact relation between this three-body parameter, the coupling constants in the effective field theory, and the binding energies of Efimov states remains unresolved. In this Letter, we address this question through a comprehensive analysis of the Skorniakov-Ter-Martirosian equation with a finite cutoff. We establish an exact renormalization relation for the three-body parameter and determine its connection to the energies of Efimov bound states. These results are validated through high-precision numerical simulations. We expect our findings to be of fundamental interest across various fields, including atomic, nuclear, condensed matter, and particle physics, and to have broad applications in both few-body and many-body physics.