Are there any Landau poles in wavelet-based quantum field theory?

TL;DR

This paper explores wavelet-based quantum field theory and demonstrates that, unlike traditional approaches with Landau poles, a non-differentiable scale-dependent coupling can prevent such divergences, ensuring finiteness at all scales.

Contribution

It introduces a wavelet-based renormalization approach where non-differentiable scale dependence of couplings avoids Landau poles in quantum field theories.

Findings

Differentiable coupling functions exhibit Landau poles at one-loop level.

Non-differentiable, scale-dependent couplings remain finite at all scales.

Application to Euclidean ^4 field theory illustrates the approach.

Abstract

Following previous work by one of the authors [M.V.Altaisky, Unifying renormalization group and the continuous wavelet transform, Phys. Rev. D 93, 105043 (2016).], we develop a new approach to the renormalization group, where the effective action functional $\Gamma_A[\phi]$ is a sum of all fluctuations of scales from the size of the system $L$ down to the scale of observation $A$. It is shown that the renormalization flow equation of the type $ \frac{\partial \Gamma_A}{\partial \ln A}=-Y(A) $ is a limiting case of such consideration, when the running coupling constant is assumed to be a differentiable function of scale. In this approximation, the running coupling constant, calculated at one-loop level, suffers from the Landau pole. In general case, when the scale-dependent coupling constant is a non-differentiable function of scale, the Feynman loop expansion results in a difference equation. This keeps the coupling constant finite for any finite value of scale $A$. As an example we consider Euclidean $\phi^4$ field theory.