The fate of non-polynomial interactions in scalar field theory

Abstract

We present an exact RG (renormalization group) analysis of $O(N)$-invariant scalar field theory about the Gaussian fixed point. We prove a series of statements that taken together show that the non-polynomial eigen-perturbations found in the LPA (local potential approximation) at the linearised level, do not lead to new interactions, \textit{i.e.} enlarge the universality class, neither in the LPA or treated exactly. Non-perturbatively, their RG flow does not emanate from the fixed point. For the equivalent Wilsonian effective action they can be re-expressed in terms of the usual couplings to polynomial interactions, which can furthermore be tuned to be as small as desired for all finite RG time. For the infrared cutoff Legendre effective action, this can also be done for the infrared evolution. We explain why this is nevertheless consistent with the fact that the large field behaviour is fixed by these perturbations.

32 pages, 1 figure; extended to exact treatment

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• Nonabelian Higgs models: paving the way for asymptotic freedom
• Momentum dependence of quantum critical Dirac systems
• Quantum gravity, renormalizability and diffeomorphism invariance
• Provable properties of asymptotic safety in $f(R)$ approximation
• Scheme dependence of asymptotically free solutions
• Renormalization group flows and fixed points for a scalar field in curved space with nonminimal $F(\varphi )R$ coupling
• Critical Wess-Zumino models with four supercharges from the functional renormalization group
• Perturbative versus Non-Perturbative Renormalization
• Universal scaling dimensions for highly irrelevant operators in the Local Potential Approximation