Near-mass-shell infrared behavior of non-Abelian gauge theories and the renormalization group

We show the existence of renormalization-group equations that completely determine the infrared structure of near-mass-shell quantum-chromodynamics Green's functions. The formal solutions of these equations are governed completely by the behavior of the long-distance invariant charge g(k²). In the quark-gluon sector, the existence of these equations depends on choosing the particular near-mass-shell renormalization scheme where gluon and fermion momenta q and p are respectively subtracted at the "equal-rate-on-shell" point q² = -λ², γ·p-m = -λ, where m is the physical quark mass. The function that describes the evolution of g(k²), βIR(g(k²)) = lim_k²→0k²(∂∂k²)g(k²), is determined from the infrared structure of the gluon-quark sector alone, and it is shown to equal, explicitly at the one-loop level, βYM(g(k²)), the corresponding function of a pure Yang-Mills theory. We have computed explicitly, up to the two-loop level, the near-mass-shell behavior of the color-singlet form factor. Reorganizing all contributions as suggested by our renormalization-group results, we find that, in the Landau gauge, they sum up to the simple form exp[B₁(p;p^';g²(k²))], where B₁(p;p^';g²(k²)) is the one-loop contribution calculated by replacing the bare vertex (-ig_λ) by the invariant charge [-ig(k)]; where k is the internal soft-gluon momentum. This result is valid to all orders in a near-mass-shell leading-log approximation. The connection of our work with the issue of infrared-driven quark confinement is briefly discussed.