ar X iv : h ep - p h / 98 09 47 1 v 1 2 1 Se p 19 98 FIXED POINT STRUCTURE OF PADÉ - SUMMATION APPROXIMATIONS TO THE QCD β - FUNCTION

Padé-improvement of four-loop β-functions in massive φ 4 scalar field theory is shown to predict the known five-loop contribution with astonishing (0.2%) accuracy, supporting the applicability of Padé-summations for approximating all-orders MS QCD β-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 β-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 β-function that is decoupled entirely from the infrared region, similar to the β-function of SUSY gluodynamics. Higher order terms of the QCD MS β-function µ 2 dx dµ 2 ≡ β(x), (1a) β(x) = − ∞ i=0 β i x i+2 , (1b) x ≡ α s (µ)/π are, upon truncation, known to permit the occurrence of fixed points other than the ultraviolet fixed point at x = 0; e.g. the positive infrared fixed point (IRFP) which occurs for 9 ≤ n f ≤ 16 when the series for β(x) in (1) is truncated after two terms [β 0 = (11 − 2n f /3)/4; β 1 = (102 − 38n f /3)/16; x IRF P = −β 0 /β 1 ]. However, the fixed points arising from such truncation are likely to be spurious, as the candidate-value for x IRF P is sufficiently large for the highest-order term in the series β(x IRF P) to be comparable in magnitude to lower terms [e.g.|β 1 x 3 | = |β 0 x 2 |]. In a recent paper, 1 Ellis, Karliner and Samuel predicted the coefficient β 3 via Padé approximant methods, and claimed that β 0−2 and their prediction for β 3 yield a Padé summation of the β-function with a nonzero IRFP consistent with an earlier prediction by Mattingly and Stevenson. 2 This Mattingly-Stevenson scenario leads to the freezing-out of the coupling to a constant value in the infrared region, as shown schematically in Fig. 1. Padé summation of the M S β-function identifies the infinte series