High-precision numerical estimates of the Mellin-Barnes integrals for the structure functions based on the stationary phase contour

Abstract

We present a recipe for constructing the effcient contour which allows one to calculate with high accuracy the Mellin-Barnes integrals, in particular, for the F3 structure function written in terms of its Mellin moments. We have demonstrated that the contour of the stationary phase arising for the F3 structure function tends to the finite limit as Re(z) → –∞. We show that the Q2 evolution of the structure function can be represented as an integral over the contour of the stationary phase within the framework of the Picard-Lefschetz theory. The universality of the asymptotic contour of the stationary phase defined at some fixed value of the momentum transfer square $Q_{0}^{2}$ for calculations with any Q2 is shown.