Boson-Faddeev in the Unitary Limit and Efimov States

TL;DR

This paper numerically investigates bosonic Faddeev equations near the Unitary limit, analyzing Efimov states and addressing divergence issues through potential modifications, providing insights into three-body quantum systems.

Contribution

It introduces a method to handle divergence in bosonic three-body calculations at the Unitary limit by modifying the off-shell t-matrix with higher-rank potentials.

Findings

Ground state energy E_u = -0.108

Efimov state energy E_u = -1×10^{-4}

Divergence in total energy at Unitary limit

Abstract

A numerical study of the Faddeev equation for bosons is made with two-body interactions at or close to the Unitary limit. Separable interactions are obtained from phase-shifts defined by scattering length and effective range. In EFT-language this would correspond to NLO. Both ground and Efimov state energies are calculated. For effective ranges $r_0 > 0$ and rank-1 potentials the total energy $E_T$ is found to converge with momentum cut-off $\Lambda$ for $\Lambda > \sim 10/r_0$ . In the Unitary limit ($1/a=r_0= 0$) the energy does however diverge. It is shown (analytically) that in this case $E_T=E_u\Lambda^2$. Calculations give $E_u=-0.108$ for the ground state and $E_u=-1.\times10^{-4}$ for the single Efimov state found. The cut-off divergence is remedied by modifying the off-shell t-matrix by replacing the rank-1 by a rank-2 phase-shift equivalent potential. This is somewhat similar to the counterterm method suggested by Bedaque et al. This investigation is exploratory and does not refer to any specific physical system.