Summing Divergent Perturbative Series in a Strong Coupling Limit. The Gell-Mann - Low Function of the ϕ^4 Theory

Abstract

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. Operation of the algorithm is illustrated by test examples, method for estimating errors is developed, and an optimization procedure is described. Application of the algorithm to the ${\varphi }^4$ theory gives a behavior $\beta (g)\approx 7.4g^{0.96}$ at large $g$ for its Gell-Mann -- Low function. The fact that the exponent is close to unity can be interpreted as a manifestation of the logarithmic branching of the type $\beta (g)\sim g(\ln g{)}^{-\gamma }$ (with $\gamma \approx 0.14$), which is confirmed by independent evidence. In any case, the ${\varphi }^4$ theory is internally consistent. The procedure of summing perturbartive series with arbitrary values of expansion parameter is discussed.

23 pages, PDF

References in corpus (2)

• New approach to Borel summation of divergent series and critical exponent estimates for an N-vector cubic model in three dimensions from five-loop εexpansions
• A lattice test of alternative interpretations of ``triviality'' in $(\lambda {\Phi }^4{)}_4$ theory

Cited by in corpus (5)

• Short-time dynamics and critical behavior of three-dimensional site-diluted Ising model
• Weak-Coupling, Strong-Coupling and Large-Order Parametrization of the Hypergeometric-Meijer Approximants
• Triviality problem and the high-temperature expansions of the higher susceptibilities for the Ising and the scalar field models on four-, five- and six-dimensional lattices
• Landau poles in condensed matter systems
• Continuation of the Stieltjes Series to the Large Regime by Finite-part Integration