Two- and Three-particle States in a Nonrelativistic Four-fermion Model in the Fine-tuning Renormalization Scheme. Goldstone mode "against" extension theory

TL;DR

This paper investigates a nonrelativistic four-fermion model, demonstrating how specific regularization and fine-tuning yield exact solutions for two-particle states and Goldstone modes, and establishing a connection with self-adjoint extensions and Hamiltonians.

Contribution

It introduces a method to obtain exact solutions in a four-fermion model using regularization and fine-tuning, linking these to self-adjoint extensions and Hamiltonian properties.

Findings

Exact solutions for two-particle sectors and Goldstone modes.

Connection between regularization, fine-tuning, and self-adjoint extensions.

Renormalized Faddeev equations with Fredholm properties.

Abstract

In a nonrelativistic contact four-fermion model we show that simple regularisation prescriptions together with a definite fine-tuning of the cut-off-parameter dependence of ``bare'' quantities give the exact solutions for the two-particle sector and Goldstone modes. Their correspondence with the self-adjoint extension into Pontryagin space is established leading to self-adjoint semi-bounded Hamiltonians in three-particle sectors as well. Renormalized Faddeev equations for the bound states with Fredholm properties are obtained and analysed.