Leading all-loop quantum contribution to the effective potential in general scalar field theory

TL;DR

This paper develops a leading log approximation RG equation for the effective potential in scalar field theories, summing contributions to all orders and revealing a characteristic discontinuity replacing the Landau pole.

Contribution

It derives a general RG equation valid for any 4D scalar field theory and provides solutions that include all leading log contributions, including numerical solutions for specific cases.

Findings

The solution exhibits a discontinuity replacing the Landau pole.

Numerical solutions constructed for specific potentials.

No new minima appear in power-like potentials due to Coleman-Weinberg mechanism.

Abstract

The RG equation for the effective potential in the leading log (LL) approximation is constructed which is valid for an arbitrary scalar field theory in 4 dimensions. The solution to this equation sums up the leading $\log\phi$ contributions to all orders of perturbation theory. In general, this is the second order nonlinear partial differential equation, but in some cases it can be reduced to the ordinary one. For particular examples, this equation is solved numerically and the LL effective potential is constructed. The solution has a characteristic discontinuity replacing the Landau pole typical for the $\phi^4$ theory. For a power-like potential no new minima appear due to the Coleman-Weinberg mechanism.