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

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.