Structure of High Order Corrections to Lipatov's Asymptotics

Abstract

High orders of perturbation theory can be calculated by the Lipatov method, whereby they are determined by saddle-point configurations (instantons) of the corresponding functional integrals. For most field theories, the Lipatov asymptotics has the functional form c a^N Γ(N+b) (N is the order of perturbation theory), and the relative corrections to it have a form of a series in powers of 1/N. It is shown that this series diverges factorially and its high-order coefficients can be calculated using a procedure similar to Lipatov's one. The K-th expansion coefficient has the form const(\ln(S1/S0))^{-K}Γ(K+(r1-r0)/2), where S0 and S1 are the values of the action for the first and the second instanton of the field theory under consideration, while r0 and r1 are the corresponding numbers of zero modes. The instantons satisfy the same equation as in the Lipatov method and are assumed to be renumbered in order of increasing of their action. This result has the universal character and is valid in any field theory for which the Lipatov asymptotic form is as specified above.