Naturalness of effective theories in Wilsonian approach

TL;DR

This paper computes the Wilsonian effective action for a scalar-fermion model, analyzing how parameters evolve with scale and discussing implications for fine-tuning and stability of the scalar potential.

Contribution

It provides an explicit calculation of Wilsonian running in a simple model, highlighting differences from Gell-Mann-Low running and clarifying fine-tuning issues.

Findings

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

Scalar mass squared and quartic coupling can change sign, leading to symmetry breaking or potential instability.

Quadratic sensitivity of fine-tuning to the cutoff scale is clearly observed in the 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 behavior 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. 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.