Padé Approximants, Optimal Renormalization Scales, and Momentum Flow in Feynman Diagrams

We show that the Padé Approximant (PA) approach for resummation of perturbative series in QCD provides a systematic method for approximating the flow of momentum in Feynman diagrams. In the large-β0 limit, diagonal PA’s generalize the Brodsky-LepageMackenzie (BLM) scale-setting method to higher orders in a renormalization scaleand scheme-invariant manner, using multiple scales that represent Neubert’s concept of the distribution of momentum flow through a virtual gluon. If the distribution is non-negative, the PA’s have only real roots, and approximate the distribution function by a sum of δ-functions, whose locations and weights are identical to the optimal choice provided by the Gaussian quadrature method for numerical integration. We show how the first few coefficients in a perturbative series can set rigorous bounds on the all-order momentum distribution function, if it is positive. We illustrate the method with the vacuum polarization function and the Bjorken sum rule computed in the large-β0 limit. ∗ Work supported in part by the Department of Energy, contract DE–AC03–76SF00515.