Exploring arbitrarily high orders of optimized perturbation theory in QCD with nf -> 16.5

TL;DR

This paper investigates high-order optimized perturbation theory in QCD near the hypothetical limit nf -> 16.5, demonstrating the consistency of the Banks-Zaks expansion across different renormalization schemes and revealing a duality in the fixed-point behavior.

Contribution

It shows that the Principle of Minimal Sensitivity yields consistent Banks-Zaks expansions at all orders, even in irregular schemes, and introduces a master equation for optimization at high orders.

Findings

Optimal renormalization schemes agree with BZ expansion results.

A duality a -> a*^2/a is observed around the fixed point.

The approach enables exploration of perturbation theory at arbitrarily high orders.

Abstract

Perturbative QCD with nf flavours of massless quarks becomes simple in the hypothetical limit nf -> 16.5, where the leading beta-function coefficient vanishes. The Banks-Zaks (BZ) expansion in a0=(8/321)(16.5-nf) is straightforward to obtain from perturbative results in MSbar or any renormalization scheme (RS) whose nf dependence is `regular.' However, `irregular' RS's are perfectly permissible and should ultimately lead to the same BZ results. We show here that the `optimal' RS determined by the Principle of Minimal Sensitivity does yield the same BZ-expansion results when all orders of perturbation theory are taken into account. The BZ limit provides an arena for exploring optimized perturbation theory at arbitrarily high orders. These explorations are facilitated by a `master equation' expressing the optimization conditions in the fixed-point limit. We find an intriguing strong/weak coupling duality a -> a*^2/a about the fixed point a*.