Summing divergent perturbative series in a strong coupling limit. The Gell-Mann-Low function of the ϕ4 theory

An algorithm is proposed for determining asymptotics of the sum of a perturbative series in the strong coupling limit using given values of the expansion coefficients. Application of the algorithm is illustrated, methods for estimating errors are developed, and an optimization procedure is described. Applied to the ϕ 4 theory, the algorithm yields the Gell-Mann-Low function asymptotics of the type β( g )≈7.4 g 0.96 for large g . The fact that the exponent is close to unity can be interpreted as a manifestation of the logarithmic branching of the type β( g )∼ g (ln g ) −γ (with γ≈0.14), which is confirmed by independent evidence. In any case, the ϕ 4 theory is self-consistent. The procedure of summing perturbative series with arbitrary values of the expansion parameter is discussed.