Two-Loop Approximations to Propagators in theσModel

A simple and systematic procedure is established for directly generating perturbation theory for the $\sigma$ model order by order in the number of loops rather than in powers of the bare coupling constant. This self-consistent Green's-function method is demonstrated by deriving the already known tree and one-loop approximations, and then is applied to the much more complex problem of the two-loop approximations to the propagators. The explicit form is displayed for these, and it is demonstrated that in the limit of no symmetry breaking and zero pion bare mass, the renormalized pion mass remains at zero as is required by the Goldstone theorem. These are the essential first steps toward calculating a unitary pion-pion scattering amplitude with this model to the two-loop order.