Renormalization group flow equations for the sigma model

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.