arXiv2000
We address whether Padé-summations of the $\bar{MS}$ QCD $β$-function for a given number of flavours exhibit an infrared-stable fixed point, or alternatively, an infrared attractor of a double valued couplant as noted by Kogan and Shifman for the case of supersymmetric gluodynamics. Below an approximant-dependent flavour threshold $(6 \leq n_f \leq 8)$, we find that Padé-summation $β$-functions incorporating $[2|1], [1|2], [2|2], [1|3]$, and $[3|1]$ approximants always exhibit a positive pole prior to the occurrence of their first positive zero, precluding any identification of this first positive zero as an infrared-stable fixed point of the $β$- function. This result is shown to be true regardless of the magnitude of the presently-unknown five-loop $β$-function contribution. Moreover, the pole in question suggests the occurrence of dynamics in which both a strong and an asymptotically-free phase share a common infrared attractor. We briefly discuss the possible relevance of infrared-attractor dynamics to the success of recent calculations of the glueball mass spectra in QCD with $N_c \to \infty$ via supergravity. As $n_f$ increases above an approximant-dependent flavour threshold, Padé-summation $β$-functions incorporating $[2|2], [1|3]$, and $[3|1]$ approximants exhibit dynamics controlled by an infrared-stable fixed point over a widening domain of the five-loop $\bar{MS}$ $β$-function parameter $(β_4/β_0)$. Above this threshold, all approximants considered exhibit infrared-stable fixed points that decrease in magnitude with increasing flavour number.