# Three-body bound states with zero-range interaction in the Bethe-Salpeter approach

## TL;DR

This paper solves the Bethe-Salpeter equation for three bosons with zero-range interactions, revealing regimes of unbound, bound, and unphysical states, and compares results with light-front calculations.

## Contribution

First solution of the three-boson Bethe-Salpeter equation with zero-range interaction, including analysis of different binding regimes and comparison with light-front methods.

## Key findings

- Identification of three interaction regimes: unbound, bound, and unphysical states.
- Discovery of a deeply bound Borromean three-body state.
- Relativistic three-body forces increase binding energy.

## Abstract

The Bethe-Salpeter equation for three bosons with zero-range interaction is solved for the first time. For comparison the light-front equation is also solved. The input is the two-body scattering length and the outputs are the three-body binding energies, Bethe-Salpeter amplitudes and light-front wave functions. Three different regimes are analyzed: ({\it i}) For weak enough two-body interaction the three-body system is unbound. ({\it ii}) For stronger two-body interaction a three-body bound state appears. It provides an interesting example of a deeply bound Borromean system. ({\it iii}) For even stronger two-body interaction this state becomes unphysical with a negative mass squared. However, another physical (excited) state appears, found previously in light-front calculations. The Bethe-Salpeter approach implicitly incorporates three-body forces of relativistic origin, which are attractive and increase the binding energy.