Renormalization Group Functions of the \phi^4 Theory in the Strong Coupling Limit: Analytical Results

TL;DR

This paper analytically investigates the strong coupling limit of the  theory's renormalization group functions, revealing a linear asymptotic behavior of the   function across dimensions and discussing implications for triviality.

Contribution

It provides an analytical demonstration that the   function behaves linearly at strong coupling in all dimensions, supporting the hypothesis of universality in asymptotic behavior.

Findings

  g asymptotically proportional to g in all dimensions

Other renormalization group functions tend to constants at strong coupling

Supports the hypothesis of  triviality and zero-charge problem

Abstract

The previous attempts of reconstructing the Gell-Mann-Low function \beta(g) of the \phi^4 theory by summing perturbation series give the asymptotic behavior \beta(g) = \beta_\infty g^\alpha in the limit g\to \infty, where \alpha \approx 1 for the space dimensions d = 2,3,4. It can be hypothesized that the asymptotic behavior is \beta(g) ~ g for all values of d. The consideration of the zero-dimensional case supports this hypothesis and reveals the mechanism of its appearance: it is associated with a zero of one of the functional integrals. The generalization of the analysis confirms the asymptotic behavior \beta(g)=\beta_\infty g in the general d-dimensional case. The asymptotic behavior of other renormalization group functions is constant. The connection with the zero-charge problem and triviality of the \phi^4 theory is discussed.