A Systematic Extended Perturbation Theory for Quantum Chromodynamics

The approximation of Euclidean QCD vertex functions Γ by a double sequence Γ[r,p] is con-sidered, where p is a perturbative order in g2, and r the order of a rational approximation in the QCD scale Λ2, non-analytic in g2. Self-consistency of Γ[r,0] in the Dyson-Schwinger equa-tions comes about by a distinctive mathematical mechanism, which limits the self-consistency problem rigorously to the seven superficially divergent vertices. 1. The Extended Approximating Sequence In a renormalizable but not superrenormalizable field theory, the sequence of partial sums Γ[p]pert(p = 0, 1, 2,...) of the perturbative expansion in the gauge coupling g for an Euclidean proper vertex ΓN(k; g), k = {k1... kN | ∑ ki = 0}, is known to be fundamentally incomplete. Its semi-convergence, combined with the violent non-analyticity1 of Γ′s around g = 0, leaves room for a remainder term exponentially small as g → 0, Mi