We study the long-distance properties of quantum chromodynamics in an expansion in powers of the three-gluon, four-gluon, and ghost-gluon couplings, but without expanding in the quark-gluon coupling. This is motivated by two observations. First, the gauge sector is well-described by perturbation theory in the context of a phenomenological model with a massive gluon. Second, the quark-gluon coupling is significantly larger than those in the gauge sector at large distances. In order to resum the contributions of the remaining infinite set of QED-like diagrams, we further expand the theory in $1/N_c$, where $N_c$ is the number of colors. At leading order, this double expansion leads to the well-known rainbow approximation for the quark propagator. We take advantage of the systematic expansion to get a renormalization-group improvement of the rainbow resummation. A simple numerical solution of the resulting coupled set of equations reproduces the phenomenology of the spontaneous chiral symmetry breaking: for sufficiently large quark-gluon coupling constant, the constituent quark mass saturates when its valence mass approaches zero. We find very good agreement with lattice data for the scalar part of the propagator and explain why the vectorial part is poorly reproduced.

15 pages, 11 figures