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 $g\left(k^2\right)$. 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 $q^2 = -{\lambda }^2$, $\gamma ·p-m = -\lambda$, where $m$ is the physical quark mass. The function that describes the evolution of $g\left(k^2\right)$, ${\beta }_{\mathrm{IR}}\left(g\right(k^2\left)\right) = {\lim }_{k^2\to 0}{k}^2\left(\frac{\partial }{\partial k^2}\right)g\left(k^2\right)$, is determined from the infrared structure of the gluon-quark sector alone, and it is shown to equal, explicitly at the one-loop level, ${\beta }_{\mathrm{YM}}\left(g\right(k^2\left)\right)$, 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 $\exp \left[B_1\right(p;p^{\prime };g^2\left(k^2\right)\left)\right]$, where $B_1(p;p^{\prime };g^2(k^2\left)\right)$ is the one-loop contribution calculated by replacing the bare vertex ($-i{g}_{\lambda }$) by the invariant charge [$-ig\left(k\right)\mathrm{}$]; 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.