Strong coupling asymptotics of the β-function in 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 φ 4 theory leads to the asymptotics β ( g ) = β ∞ g α 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.