Pade-Summation Approach to QCD Beta-Function Infrared Properties

TL;DR

This paper investigates the infrared properties of the QCD beta-function using Padé-summation techniques, finding that for certain flavor ranges, the functions exhibit poles that prevent infrared fixed points, but for higher flavors, stable fixed points emerge.

Contribution

The study applies Padé-summation to the QCD beta-function and reveals the conditions under which infrared fixed points or attractors occur, extending understanding regardless of unknown higher-order terms.

Findings

Padé-summation beta-functions show poles before positive zeros for 6-8 flavors.

Infrared-stable fixed points appear for higher flavor numbers beyond a threshold.

The results are robust against unknown five-loop beta-function contributions.

Abstract

We address whether Pad\'e-summations of the $\bar{MS}$ QCD $\beta$-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 Pad\'e-summation $\beta$-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 $\beta$- function. This result is shown to be true regardless of the magnitude of the presently-unknown five-loop $\beta$-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, Pad\'e-summation $\beta$-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 $\bar{MS}$ $\beta$-function parameter $(\beta_4/\beta_0)$. Above this threshold, all approximants considered exhibit infrared-stable fixed points that decrease in magnitude with increasing flavour number.