ar X iv : h ep - p h / 00 10 05 3 v 1 6 O ct 2 00 0 Probing the Infrared Structure of Gauge Theories : A Padé - Approximant Approach

Padé-approximant treatments of the known terms of the QCD β-function are seen to develop possible infrared fixed point structure only if the number of fermion flavours is sufficiently large. This flavour threshold is seen to be between six and nine flavours, depending upon both the specific choice of approximant as well as on the presently-unknown five-loop β-function contribution. Below this flavour threshold, Padé approxi-mants based upon the QCD β-function manifest the same infrared attractor structure as that which characterizes the exact NSVZ β-function of supersymmetric gluodynamics. Such infrared attractor structure is also seen to characterize Padé-approximant treatments of vector SU (N) gauge theory in the large N limit, suggesting common infrared dynamics for the strong and weak phases of this theory. Although the ultraviolet properties of the QCD couplant constant are now quite precisely determined via knowledge of the four-loop-order MS β-function, 1 the in-frared behaviour of the QCD couplant remains a mystery. The simplest possibility would be for the existence of an infrared-stable fixed point (IRFP) in the couplant. However, since QCD with a small number of fermion flavours n f is a confining theory, one should anticipate that the relevant degrees of freedom for QCD are essentially different in the infrared (hadrons) and ultraviolet (quarks and gluons) regions. Such expectations would argue against the existence of an IRFP. Indeed, a lattice study has indicated that for three colours, an infrared fixed point is not possible until n f ≥ 7, 2 a result qualitatively similar to the n f threshold anticipated from β-functions truncated after two-loop order. 3 The present work (which is presented in detail elsewhere 4) utilizes Padé approx-imants constructed from the known terms of the N c = 3 QCD MS β-function for various numbers of flavours in order to extract infrared properties that are common to all such approximants. Consider for example the general case of a k-loop β-function series β (k) (x) = −β 0 x 2