  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.
