A Model Study of Discrete Scale Invariance and Long-Range Interactions

TL;DR

This paper explores how long-range Coulomb interactions modify the discrete scale invariance and bound state spectrum in a one-dimensional inverse square potential model, relevant for nuclear physics phenomena like the Efimov effect.

Contribution

It demonstrates that the same counterterm renormalizes both the inverse square potential and the combined system with Coulomb interaction, revealing how discrete scale invariance is broken.

Findings

Coulomb interaction modifies the bound state spectrum.

Deep bound states are dominated by the inverse square potential.

Shallow bound states are significantly affected by Coulomb forces.

Abstract

We investigate the modification of discrete scale invariance in the bound state spectrum by long-range interactions. This problem is relevant for effective field theory descriptions of nuclear cluster states and manifestations of the Efimov effect in nuclei. As a model system, we choose a one dimensional inverse square potential supplemented with a long-range Coulomb interaction. We study the renormalization and bound-state spectrum of the system as a function of the Coulomb interaction strength. Our results indicate, that the counterterm required to renormalize the inverse square potential alone is sufficient to renormalize the full problem. However, the breaking of the discrete scale invariance through the Coulomb interaction leads to a modified bound state spectrum. The shallow bound states are strongly influenced by the Coulomb interaction while the deep bound states are dominated by the inverse square potential.