Fine-tuning and vacuum stability in Wilsonian effective action

TL;DR

This paper computes the Wilsonian effective action for a scalar-fermion model, analyzing fine-tuning and stability issues, and compares Wilsonian and Gell-Mann-Low running, highlighting differences and implications for symmetry breaking.

Contribution

It provides explicit Wilsonian running calculations in a scalar-fermion model, illustrating differences from Gell-Mann-Low running and discussing implications for fine-tuning and vacuum stability.

Findings

Wilsonian running differs from Gell-Mann-Low due to automatic integration of heavy modes.

Scalar mass squared and quartic coupling can change sign, indicating symmetry breaking or instability.

Quadratic sensitivity of fine-tuning to cutoff scale is evident in Wilsonian approach.

Abstract

We have computed Wilsonian effective action in a simple model containing scalar field with quartic self-coupling which interacts via Yukawa coupling with a Dirac fermion. The model is invariant under a chiral parity operation, which can be spontaneously broken by a vev of the scalar field. We have computed explicitly Wilsonian running of relevant parameters which makes it possible to discuss in a consistent manner the issue of fine-tuning and stability of the scalar potential. This has been compared with the typical picture based on Gell-Mann-Low running. Since Wilsonian running includes automatically integration out of heavy degrees of freedom, the running differs markedly from the Gell-Mann-Low version. However, similar behaviour can be observed: scalar mass squared parameter and the quartic coupling can change sign from a positive to a negative one due to running which causes spontaneous symmetry breaking or an instability in the renormalizable part of the potential for a given range of scales. However, care must be taken when drawing conclusions, because of the truncation of higher dimension operators. As for the issue of fine-tuning, since in the Wilsonian approach power-law terms are not subtracted, one can clearly observe the quadratic sensitivity of fine-tuning measure to the change of the cut-off scale.