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 β-function of φ4 theory leads to the asymptotics β(g)=β∞gα at g→∞, where α≈1 for space dimensions d=2,3,4. The natural hypothesis arises, that the asymptotic behavior is β(g)∼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 β(g) for all d. The same idea can be applied to QED and gives the asymptotics β(g)=g, where g is the running fine structure constant. A relation to the “zero charge” problem is discussed.