# Solving the Gap Equation of the NJL Model through Iteration: Unexpected Chaos

## TL;DR

This paper investigates the iterative solutions to the NJL model's gap equation under various regularizations, revealing stable convergence with a hard cut-off and chaotic behavior with other schemes at high coupling.

## Contribution

It demonstrates how different regularization schemes affect the convergence and stability of iterative solutions in the NJL model, highlighting unexpected chaotic behavior.

## Key findings

- Hard cut-off regularization yields stable, accurate solutions.
- Paul-Villars and proper time regularizations can lead to chaos at high coupling.
- Chaotic behavior depends on the regularization scheme used.

## Abstract

We explore the behavior of the iterative procedure to obtain the solution to the gap equation of the Nambu-Jona-Lasinio (NLJ) model for arbitrarily large values of the coupling constant and in the presence of a magnetic field and a thermal bath. We find that the iterative procedure shows a different behavior depending on the regularization scheme used. It is stable and very accurate when a hard cut-off is employed. Nevertheless, for the Paul-Villars and proper time regularization schemes, there exists a value of the coupling constant (different in each case) from where the procedure becomes chaotic and does not converge any longer.