  We study three dimensional Bose and Fermi gases in the upper branch, a phase
defined by the absence of bound states in the repulsive interaction regime,
within an approximation that considers only two-body interactions. Employing a
formalism based on the S-matrix, we derive useful analytic expressions that
hold on the upper branch in the weak coupling limit. We determine upper branch
phase diagrams for both bosons and fermions with techniques valid for arbitrary
positive scattering length.
