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

Abstract

The previous attempts of reconstructing the Gell-Mann-Low function β(g) of the ϕ^4 theory by summing perturbation series give the asymptotic behavior β(g) = β_\infty g^αin the limit g\to \infty, where α\approx 1 for the space dimensions d = 2,3,4. It can be hypothesized that the asymptotic behavior is β(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 β(g)=β_\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 ϕ^4 theory is discussed.

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