Strong coupling asymptotics of the \beta-function in \phi^4 theory and QED

Abstract

The well-known algorithm for summing of divergent series is based on the Borel transformation in combination with the conformal mapping (Le Guillou and Zinn-Justin, 1977). Modification of this algorithm allows to determine a strong coupling asymptotics of the sum of the series through the values of the expansion coefficients. Application of the algorithm to the \beta-function of \phi^4 theory leads to the asymptotics \beta(g)=\beta_\infty g^\alpha at g\to\infty, where \alpha\approx 1 for space dimensions d=2,3,4. The natural hypothesis arises, that asymptotic behavior is \beta(g)\sim g for all d. Consideration of the "toy" zero-dimensional model confirms the hypothesis and reveals the origin of this result: it is related with a zero of a certain functional integral. 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 for QED and gives asymptotics \beta(g)=g, where g is the running fine structure constant. Relation to the "zero charge" problem is discussed.