Asymptotic Analysis of the Local Potential Approximation to the Wetterich Equation

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.