Numerical RG-time integration of the effective potential: Analysis and Benchmark

TL;DR

This paper analyzes and benchmarks numerical methods for RG-time integration of the effective potential in scalar theories, focusing on stability, precision, and handling symmetry breaking and convexity restoration.

Contribution

It provides a comprehensive benchmark and stability analysis of various numerical algorithms for RG-time integration, including reformulations to improve robustness.

Findings

Rosenbrock and implicit Runge-Kutta methods are highly effective.

A logarithmic reformulation helps avoid singularity issues.

Extensive benchmarks demonstrate the stability and precision of selected methods.

Abstract

We investigate the RG-time integration of the effective potential in the functional renormalization group in the presence of spontaneous symmetry breaking and its subsequent convexity restoration on the example of a scalar theory in $d=3$. The features of this setup are common to many physical models and our results are, therefore, directly applicable to a variety of situations. We provide exhaustive work-precision benchmarks and numerical stability analyses by considering the combination of different discrete formulations of the flow equation and a large collection of different algorithms. The results are explained by using the different components entering the RG-time integration process and the eigenvalue structure of the discrete system. Particularly, the combination of Rosenbrock methods, implicit multistep methods or certain (diagonally) implicit Runge-Kutta methods with exact or autodiff Jacobians proves to be very potent. Furthermore, a reformulation in a logarithmic variable circumvents issues related to the singularity bound in the flat regime of the potential.