Renormalization-group analysis for low-energy scattering of charged particles

TL;DR

This paper applies a renormalization group approach to analyze low-energy charged particle scattering, revealing a marginally unstable solution linked to zero-energy bound states and clarifying power counting for effective potentials.

Contribution

It introduces a renormalization group framework with dimensional regularization for charged particle scattering, connecting it to Coulomb-distorted effective-range expansion and off-shell effects.

Findings

Identifies a nontrivial marginally unstable solution with zero-energy bound state.

Shows energy-dependent perturbations align with Wilsonian power counting.

Demonstrates off-shell momentum dependence affects only off-shell scattering matrix form.

Abstract

The low-energy scattering of two charged particles is analyzed using a renormalization group approach based on dimensional regularization with power-divergence subtraction. A nontrivial solution with a marginally unstable direction is found, corresponding to a system with a bound state at zero energy. For purely energy-dependent perturbations around this solution, the power counting agrees with that from Wilsonian methods. These terms in the effective potential are in direct correspondence with the the terms in the Coulomb-distorted effective-range expansion. We also study perturbations that depend on off-shell momenta as well as energy, and we show that these affect only the off-shell form of the scattering matrix. These terms are of higher order that the corresponding energy-dependent ones and so terms in the potential that depend only on the off-shell momenta do not have definite orders in power counting.