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A new analytic treatment of three-dimensional homogeneous Bose and Fermigases in the unitary limit of negative infinite scattering length is presented,based on the S-matrix approach to statistical mechanics we recently developed.The unitary limit occurs at a fixed point of the renormalization group withdynamical exponent z=2 where the S-matrix equals -1. For fermions we find T_c/T_F is approximately 0.1. For bosons we present evidence that the gas does notcollapse, but rather has a critical point that is a strongly interacting formof Bose-Einstein condensation. This bosonic critical point occurs at n lambda^3approximately 1.3 where n is the density and lambda the thermal wavelength,which is lower than the ideal gas value of 2.61.

S-matrix approach to quantum gases in the unitary limit II: the three-dimensional case