Asymptotic Analysis of the Local Potential Approximation to the Wetterich Equation

TL;DR

This paper analyzes the asymptotic behavior of the local potential approximation to the Wetterich equation, revealing how the nature of the resulting heat equation changes with space-time dimension and exploring the stability of ground states in related quantum theories.

Contribution

It introduces a perturbative asymptotic analysis of the Wetterich equation's local potential approximation, including a novel application of Padé techniques to extrapolate solutions and study ground state stability.

Findings

For D<2, the equation is a well-posed forward heat equation.

For D>2, the equation becomes an ill-posed backward heat equation.

Padé extrapolation distinguishes PT-symmetric and Hermitian cubic theories, revealing differences in potential stability.

Abstract

This paper reports a study of the nonlinear partial differential equation that arises in the local potential approximation to the Wetterich formulation of the functional renormalization group equation. A cut-off-dependent shift of the potential in this partial differential equation is performed. This shift allows a perturbative asymptotic treatment of the differential equation for large values of the infrared cut-off. To leading order in perturbation theory the differential equation becomes a heat equation, where the sign of the diffusion constant changes as the space-time dimension $D$ passes through $2$. When $D<2$, one obtains a forward heat equation whose initial-value problem is well-posed. However, for $D>2$ one obtains a backward heat equation whose initial-value problem is ill-posed. For the special case $D=1$ the asymptotic series for cubic and quartic models is extrapolated to the small infrared-cut-off limit by using Pad\'e techniques. The effective potential thus obtained from the partial differential equation is then used in a Schr\"odinger-equation setting to study the stability of the ground state. For cubic potentials it is found that this Pad\'e procedure distinguishes between a $PT$-symmetric $ig\phi^3$ theory and a conventional Hermitian $g\phi^3$ theory ($g$ real). For an $ig\phi^3$ theory the effective potential is nonsingular and has a stable ground state but for a conventional $g\phi^3$ theory the effective potential is singular. For a conventional Hermitian $g\phi^4$ theory and a $PT$-symmetric $-g\phi^4$ theory ($g>0$) the results are similar; the effective potentials in both cases are nonsingular and possess stable ground states.