Exploring arbitrarily high orders of optimized perturbation theory in QCD with nf→1612

Perturbative QCD with nfnf flavours of massless quarks becomes simple in the hypothetical limit nf→1612, where the leading β  -function coefficient vanishes. The Banks–Zaks (BZ) expansion in a0≡8321(1612−nf) is straightforward to obtain from perturbative results in MS‾ or any renormalization scheme (RS) whose nfnf 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/aa→a⁎2/a about the fixed point a⁎a⁎.