The Canonical Nuclear Many-Body Problem as a Rigorous Effective Theory

TL;DR

This paper proposes an exact, efficient approach to the nuclear many-body problem using a Green's function expansion, contrasting with traditional shell model approximations, and demonstrates its application to light nuclei with realistic interactions.

Contribution

It introduces a self-consistent solution method for the Bloch-Horowitz equation using Lanczos-based Green's function expansion, providing a new effective theory for nuclear systems.

Findings

Demonstrates the method on deuteron and helium-3 nuclei

Shows a simple renormalization group behavior different from EFT

Analyzes wave function normalization and operator evolution

Abstract

The shell model is the standard tool for addressing the canonical nuclear many-body problem of nonrelativistic nucleons interacting through a static potential. We discuss several of the uncontrolled approximations that are made in this model to motivate a different approach, one based on an exact solution of the Bloch-Horowitz equation. We argue that the necessary self-consistent solutions of this equation can be obtained efficiently by a Green's function expansion based on the Lanczos algorithm. The resulting effective theory is carried out for the simplest nuclei, d and 3He, using realistic NN interactions such as the Argonne v18 and Reid93 potentials, in order to contrast the results with the shell model. We discuss the wave function normalization, the evolution of the wave function as the "shell model" space is varied, and the magnetic elastic effective operator. The numerical results show a simple renormalization group behavior that differs from effective field theory treatments of the two- and three-body problems. The likely origin of this scaling is discussed.