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arXiv · hep-ph/9706467

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

Stanley J. Brodsky·John Ellis·Einan Gardi·Marek Karliner·Mark. A. Samuel
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Abstract

We show that the Pade 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-$β_0$ limit, diagonal PA's generalize the Brodsky-Lepage-Mackenzie (BLM) scale-setting method to higher orders in a renormalization scale- and 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 delta-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-$β_0$ limit.

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Stanley J. Brodsky, John Ellis, Einan Gardi, Marek Karliner, Mark. A. Samuel. 1997-06-23. Pade Approximants, Optimal Renormalization Scales, and Momentum Flow in Feynman Diagrams. https://doi.org/10.1103/physrevd.56.6980

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