Single-scale Renormalisation Group Improvement of Multi-scale Effective Potentials

TL;DR

This paper introduces a novel method for renormalisation group improvement of multi-scalar effective potentials, enabling analysis across energy scales and beyond one-loop order, with applications in stability studies.

Contribution

The authors develop a generalised, single-scale renormalisation group improvement technique for multi-scalar effective potentials applicable at any perturbative order.

Findings

Allows evaluation of the effective potential using tree-level potential with running couplings.

Enables stability analysis of scalar potentials beyond tree level.

Provides a framework for studying multi-field potentials across energy scales.

Abstract

We present a new method for renormalisation group improvement of the effective potential of a quantum field theory with an arbitrary number of scalar fields. The method amounts to solving the renormalisation group equation for the effective potential with the boundary conditions chosen on the hypersurface where quantum corrections vanish. This hypersurface is defined through a suitable choice of a field-dependent value for the renormalisation scale. The method can be applied to any order in perturbation theory and it is a generalisation of the standard procedure valid for the one-field case. In our method, however, the choice of the renormalisation scale does not eliminate individual logarithmic terms but rather the entire loop corrections to the effective potential. It allows us to evaluate the improved effective potential for arbitrary values of the scalar fields using the tree-level potential with running coupling constants as long as they remain perturbative. This opens the possibility of studying various applications which require an analysis of multi-field effective potentials across different energy scales. In particular, the issue of stability of the scalar potential can be easily studied beyond tree level.