# Four-fermion interactions and the chiral symmetry breaking in an external magnetic field

## TL;DR

This paper studies how external magnetic fields and temperature influence chiral symmetry breaking in four-fermion interaction models, revealing magnetic catalysis and the importance of coupling flow directions.

## Contribution

It introduces a detailed analysis of the renormalization group flows for four-fermion couplings under external magnetic fields and temperature, highlighting the dependence on flow routes and directions.

## Key findings

- Critical temperature increases with magnetic field, indicating magnetic catalysis.
- Flow structure remains unchanged by external parameters, but flow strength varies with coupling directions.
- Chiral symmetry breaking is influenced by the route in the coupling plane.

## Abstract

We investigate the chiral symmetry and its spontaneous breaking at finite temperature and in an external magnetic field with four-fermion interactions of different channels. Quantum and thermal fluctuations are included within the functional renormalization group approach, and properties of the set of flow equations for different couplings, such as its fixed points, are discussed. It is found that external parameters, e.g. the temperature and the external magnetic field and so on, do not change the structure of the renormalization group flows for the couplings. The flow strength is found to be significantly dependent on the route and direction in the plane of couplings of different channels. Therefore, the critical temperature for the chiral phase transition shows a pronounced dependence on the direction as well. Given fixed initial ultraviolet couplings, the critical temperature increases with the increasing magnetic field, viz., the magnetic catalysis is observed with initial couplings fixed.