Physics and the choice of regulators in functional renormalisation group flows

TL;DR

This paper investigates how the choice of regulators in functional renormalisation group flows influences the results, proposing methods for optimisation and analyzing effects in various physical models.

Contribution

It systematically studies regulator dependence in RG flows, introduces a procedure for optimized regularisation, and examines effects in multi-field and non-relativistic models.

Findings

Flow trajectories depend on regulator choice and shape.

Relative cutoff scales significantly affect universality class crossover.

Regulator dependencies can be physically interpreted in models like the Fermi polaron.

Abstract

The Renormalisation Group is a versatile tool for the study of many systems where scale-dependent behaviour is important. Its functional formulation can be cast into the form of an exact flow equation for the scale-dependent effective action in the presence of an infrared regularisation. The functional RG flow for the scale-dependent effective action depends explicitly on the choice of regulator, while the physics does not. In this work, we systematically investigate three key aspects of how the regulator choice affects RG flows: (i) We study flow trajectories along closed loops in the space of action functionals varying both, the regulator scale and shape function. Such a flow does not vanish in the presence of truncations. Based on a definition of the length of an RG trajectory, we suggest a practical procedure for devising optimised regularisation schemes within a truncation. (ii) In systems with various field variables, a choice of relative cutoff scales is required. At the example of relativistic bosonic two-field models, we study the impact of this choice as well as its truncation dependence. We show that a crossover between different universality classes can be induced and conclude that the relative cutoff scale has to be chosen carefully for a reliable description of a physical system. (iii) Non-relativistic continuum models of coupled fermionic and bosonic fields exhibit also dependencies on relative cutoff scales and regulator shapes. At the example of the Fermi polaron problem in three spatial dimensions, we illustrate such dependencies and show how they can be interpreted in physical terms.