Strong coupling asymptotics of the β-function in φ4 theory and QED

The well-known algorithm for summing divergent series is based on the Borel transformation in combination with the conformal mapping. A modification of this algorithm allows one to determine a strong coupling asymptotics of the sum of the series through the values of the expansion coefficients. An application of the algorithm to the beta-function of phi(4) theory leads to the asymptotics beta(g) = beta(infinity)g(alpha) at g -> infinity, where alpha approximate to 1 for space dimensions d = 2, 3, 4. The natural hypothesis arises, that the asymptotic behavior is beta(g) similar to g for all d. Consideration of the "toy" zero-dimensional model confirms the hypothesis and reveals the origin of this result: it is related to a zero of a certain functional integral. A generalization of this mechanism to the arbitrary space dimensionality leads to the linear asymptotics of beta(g) for all d. The same idea can be applied to QED and gives the asymptotics beta(g)= g, where g is the running fine structure constant. A relation to the "zero charge" problem is discussed. (C) 2010 IMACS. Published by Elsevier B.V. All rights reserved.