  A new analytic treatment of the two-dimensional Hubbard model at finite
temperature and chemical potential is presented. A next nearest neighbor
hopping term of strength t' is included. This analysis is based upon a
formulation of the statistical mechanics of particles in terms of the S-matrix.
In the 2-body scattering approximation, the S-matrix allows a systematic
expansion in t/U. We show that for U/t large enough, a region of attractive
interactions exists near the Fermi surface due to 1-loop renormalization
effects. For t'/t = -0.3, attractive interactions exist for U/t > 6.4. Our
analysis suggests that superconductivity may not exist for t'=0. Based on the
existence of solutions of the integral equation for the pseudo-energy, we
provide evidence for the superconducting phase and estimate Tc/t = 0.02.
