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We show that the Pade Approximant (PA) approach for resummation ofperturbative series in QCD provides a systematic method for approximating theflow of momentum in Feynman diagrams. In the large-$\beta_0$ limit, diagonalPA's generalize the Brodsky-Lepage-Mackenzie (BLM) scale-setting method tohigher orders in a renormalization scale- and scheme-invariant manner, usingmultiple scales that represent Neubert's concept of the distribution ofmomentum flow through a virtual gluon. If the distribution is non-negative, thePA's have only real roots, and approximate the distribution function by a sumof delta-functions, whose locations and weights are identical to the optimalchoice provided by the Gaussian quadrature method for numerical integration. Weshow how the first few coefficients in a perturbative series can set rigorousbounds on the all-order momentum distribution function, if it is positive. Weillustrate the method with the vacuum polarization function and the Bjorken sumrule computed in the large-$\beta_0$ limit.Comment: 28 pages, LaTeX, including 6 figures requires epsfig.st

Padé approximants, optimal renormalization scales, and momentum flow in Feynman diagrams