Asymptotic Analysis of the Local Potential Approximation to the Wetterich Equation

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 Padé techniques. The effective potential thus obtained from the partial differential equation is then used in a Schrödinger-equation setting to study the stability of the ground state. For cubic potentials it is found that this Padé procedure distinguishes between a $PT$-symmetric $ig{\varphi }^3$ theory and a conventional Hermitian $g{\varphi }^3$ theory ($g$ real). For an $ig{\varphi }^3$ theory the effective potential is nonsingular and has a stable ground state but for a conventional $g{\varphi }^3$ theory the effective potential is singular. For a conventional Hermitian $g{\varphi }^4$ theory and a $PT$-symmetric $-g{\varphi }^4$ theory ($g>0$) the results are similar; the effective potentials in both cases are nonsingular and possess stable ground states.

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Cited by in corpus (4)

• $\mathcal{P}\mathcal{T}$-symmetric, non-Hermitian quantum many-body physics -- a methodological perspective
• Unitarity through PT symmetry in Quantum Quadratic Gravity
• Wilsonian approach to the interaction ${\varphi }^2(i\varphi {)}^{\epsilon }$
• Functional renormalization group for non-Hermitian and $\mathcal{P}\mathcal{T}$-symmetric systems