  We present a nonperturbative renormalization group solution of the
Gell-Mann--Levy $\sigma$-model which was originally proposed as a
phenomenological description of the dynamics of nucleons and mesons. In our
version of the model the fermions are interpreted as quarks which interact via
the $\sigma$ and $\pi$ mesons. We derive and numerically solve renormalization
group (RG) flow equations to leading order in a derivative expansion to study
the behavior of the model as it evolves from high to low momentum scales. We
develop an expansion in chiral-symmetry-breaking which enables us to track this
symmetry breaking with the evolution of the scale. We use infrared observables
to constrain the phenomenology allowing predictions of other quantities such as
$\pi-\pi$ scattering lengths. The results show improvement over the tree level
calculation and are consistent with experiment and the results of alternate
theoretical approaches such as chiral perturbation theory and lattice gauge
theory.
