# Anatomy of the magnetic catalysis by renormalization-group method

## TL;DR

This paper analyzes magnetic catalysis using renormalization-group methods, emphasizing the role of screening effects and scale dependence in determining mass gaps in quantum field theories under strong magnetic fields.

## Contribution

It develops a scale-dependent RG framework including screening effects and demonstrates its equivalence with Schwinger-Dyson equations for analyzing magnetic catalysis.

## Key findings

- RG method with screening effects accurately predicts mass gaps
- Equivalence established between RG and Schwinger-Dyson approaches
- Intrinsic energy-scale dependence is crucial for understanding magnetic catalysis

## Abstract

We first examine the scaling argument for a renormalization-group (RG) analysis applied to a system subject to the dimensional reduction in strong magnetic fields, and discuss the fact that a four-Fermi operator of the low-energy excitations is marginal irrespective of the strength of the coupling constant in underlying theories. We then construct a scale-dependent effective four-Fermi interaction as a result of screened photon exchanges at weak coupling, and establish the RG method appropriately including the screening effect, in which the RG evolution from ultraviolet to infrared scales is separated into two stages by the screening-mass scale. Based on a precise agreement between the dynamical mass gaps obtained from the solutions of the RG and Schwinger-Dyson equations, we discuss an equivalence between these two approaches. Focusing on QED and Nambu--Jona-Lasinio model, we clarify how the properties of the interactions manifest themselves in the mass gap, and point out an importance of respecting the intrinsic energy-scale dependences in underlying theories for the determination of the mass gap. These studies are expected to be useful for a diagnosis of the magnetic catalysis in QCD.