Renormalization group functions of the φ4 theory in the strong coupling limit: Analytical results

The previous attempts of reconstructing the Gell-Mann-Low function β(g) of the φ4 theory by summing perturbation series give the asymptotic behavior β(g) = β∞ g in the limit g→∞, where α = 1 for the space dimensions d = 2, 3, 4. It can be hypothesized that the asymptotic behavior is β(g) ∼ g for all d values. The consideration of the zero-dimensional case supports this hypothesis and reveals the mechanism of its appearance: it is associated with vanishing of one of the functional integrals. The generalization of the analysis confirms the asymptotic behavior β(g) ∼ g in the general d-dimensional case. The asymptotic behaviors of other renormalization group functions are constant. The connection with the zero-charge problem and triviality of the φ4 theory is discussed.