Minlos-Faddeev regularization of zero-range interactions in the three-body problem

TL;DR

This paper analyzes Minlos-Faddeev regularization in the three-body problem, identifying different regimes based on a regularization parameter and their implications for phenomena like Efimov and Thomas effects.

Contribution

It explicitly characterizes the effects of Minlos-Faddeev regularization on three-body systems, determining critical parameter values and boundary conditions for various regimes.

Findings

Efimov and Thomas effects persist for certain regularization parameters.

Different boundary conditions are required depending on the regularization parameter.

Critical values of the regularization parameter are identified for various three-body configurations.

Abstract

To regularize the three-body problem, Minlos and Faddeev suggested a modification of zero-range model, which diminishes interaction at the triple-collision point. The analysis reveals that this regularization results in four alternatives depending on the regularization parameter $ \sigma $. Explicitly, Efimov or Thomas effects remain for $ \sigma < \sigma_c $, the additional boundary conditions of two types should be imposed at the triple-collision point for $ \sigma_c \le \sigma < \sigma_e $ and $ \sigma_e < \sigma < \sigma_r $, and the problem is regularized for $ \sigma \ge \sigma_r $. Critical values $ \sigma_c < \sigma_e < \sigma_r $ separating different alternatives are determined both for a two-component three-body system and for three identical bosons.