A Systematic Extended Perturbation Theory for Quantum Chromodynamics

The approximation of Euclidean QCD vertex functions Γ by a double sequence Γ is considered, where p is a perturbative order in g, and r the order of a rational approximation in the QCD scale Λ, non-analytic in g. Self-consistency of Γ in the Dyson-Schwinger equations 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 = 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-analyticity of Γ′s around g = 0, leaves room for a remainder term exponentially small as g → 0, Min {p} ∣∣∣Γ(k; g)− Γ(k; g)∣∣∣ ∝ exp(−const. g2 ) . (1.1) In asymptotically free theories, on the other hand, one knows from renormalization-group (RG) analysis that one class of terms just allowed by this bound is positively there, namely terms involving the RG-invariant mass scale (in a scheme R), (Λ)R = ν 2 exp −2 g(ν) ∫ dg′ β(g′)  R = ν exp { − (4π) 2 β0g(ν) [ 1 +O(g) ]} . (1.2) Here ν is the arbitrary renormalization scale. For QCD, it therefore seems necessary to accommodate such Λ-dependent terms, which the perturbation expansion will miss even when summed to all orders. The present contribution briefly describes some properties of a double approximating sequence for Euclidean QCD vertices, Γ(k; g), r = 0, 1, 2, . . . , p = 0, 1, 2, . . . , (1.3) designed to account for the Λ-dependent "‘missing terms"’, while being