  Pad\'e-improvement of four-loop beta-functions in massive phi^4 scalar field
theory is shown to predict the known five-loop contribution with astonishing
(0.2%) accuracy, supporting the applicability of Pade-summations for
approximating all-orders MS-bar QCD beta-functions, as suggested by Ellis,
Karliner, and Samuel. Surprisingly, the most general set of [2|2] approximants
consistent with known two-, three-, and four-loop contributions to the QCD
beta-function with up to six flavours fail to exhibit any zeros that could be
interpreted as positive infrared fixed points, regardless of the unknown
five-loop term. When they occur, positive zeros of such [2|2] approximants are
preceded by singularities, leading to a double-valued beta-function that is
decoupled entirely from the infrared region, similar to the beta-function of
SUSY gluodynamics.
