A nonperturbative solution to the Dyson-Schwinger-equations of QCD

This is the first of two papers in which we discuss a nonperturbatively modified solution to the Euclidean Dyson-Schwinger equations for the 7 superficially divergent proper verticesΓ of QCD. It takes the formΣ _n g ²ⁿ Γ( ⁿ ) where eachΓ( ⁿ ) approaches its perturbative form at large momenta. At lower momenta, it differs from that form by an additional non-analyticg ² dependence through a dynamical mass scaleb, proportional toΛ _qcd and associated with a pole dependence on the momentum invariants. In the zeroth-order two-point functions, these nonperturbative modifications amount to a generalized Schwinger mechanism, leading to propagators without particle poles. The termsΓ(⁰), representing the Feynman rules of the modified iterative solution, can become self-consistent in the DS equations through a mechanism of “nonperturbative logarithms” which we explain. The mechanism is tied to the presence of divergent loops, and thus represents a pure quantum effect, similar to quantum anomalies. It restricts formation of nonperturbativeΓ(⁰)'s to the 7 primitively divergent vertices, thus escaping the infinite nature of the DS hierarchy. In a given loop order, the self-consistency problem reduces to a finite set of algebraic equations.