Fine-tuning and vacuum stability in the Wilsonian effective action

We have computed the Wilsonian effective action in a simple model with a spontaneously broken chiral parity. The Wilsonian running of relevant parameters makes it possible to discuss in a consistent manner the issues of fine-tuning and the stability of the scalar potential. This has been compared with the standard picture based on a Gell-Mann-Low running. Since the Wilsonian running automatically includes integration of heavy degrees of freedom, the running differs markedly from the Gell-MannLow version. Similar behavior can be observed: the scalar mass-squared parameter and the quartic coupling can change sign from a positive to a negative one, due to the running which causes a 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. Taking the scalar field's amplitude near the cutoff. may cancel the suppression due to the scale, and only the suppression due to small couplings partially justifies the truncation in this region. Also, when taking the cutoff higher, to include larger amplitudes of the fields, the higher-order irrelevant operators, whose coefficients grow with the scale, may affect the conclusion about the stability. The Gell-Mann-Low running allows one to resume relatively easily a class of operators corresponding to large logarithms to the form of the renormalization group equation improved effective potential which is valid over a huge range of scales. In the Wilsonian approach this would correspond to following the running of a large number of irrelevant operators, which is technically problematic. As for the issue of the fine-tuning, since in the Wilsonian approach power-law terms are not subtracted, one can clearly observe the quadratic sensitivity of a fine-tuning measure to the change of the cutoff scale. The Wilsonian version of the radiative symmetry-breaking mechanism has been described.