limit based on our treatment of quantum gases in [20], which appears to be well suited to the problem since it allows an expansion in the inverse coupling. Let us motivate this approach as follows. In one spatial dimension, the fixed point occurs for repulsive interactions. The model is integrable[21] and its thermodynamics determined exactly by the so-called thermodynamic Bethe ansatz (TBA)[22]. In the TBA, the free energy is expressed in terms of a pseudo-energy which is a solution to an integral equation with a kernel related to the logarithm of the S-matrix. In the unitary limit the coupling goes to ∞ and the S-matrix S = − 1. The TBA is then identical to a free gas of fermionic particles. The formalism developed in [20] was modeled after the TBA: the free energy is expressed as a sum of diagrams where the vertices represent matrix elements of the logarithm of the (zero temperature) S-matrix. However since generally the N-body S-matrix does not factorize into 2-body scattering, the series cannot be summed exactly as in the TBA. Nevertheless, a consistent resummation of an infinite number of diagrams involving only 2-body scattering, the so-called foam diagrams, can serve as a useful approximation if the gas is not too dense. The result of summing these diagrams leads to an integral equation for a pseudo-energy, as in the TBA; in fact in 1 spatial dimension the TBA is recovered to lowest order in the kernel. Since the formalism is based on the S-matrix, it can be very useful for studying the unitary limit where S = − 1. In this paper we present the main formulas for the 3-dimensional case, however we mainly analyze the 2-dimensional case; analysis of the 3d case will be published separately[23]. Phase transitions in two-dimensional Fermi gases were studied in e.g. [25]. The fixed point separating the BEC and BCS regimes goes to zero coupling when d = 2, thus it is not obvious whether a unitary limit exists at strong coupling. As we will argue, there is a scale-invariant limit at infinite coupling g = ±∞ where the S-matrix S = − 1. This is a meaningful unitary limit at very low ( g = −∞ ) or very high ( g = + ∞ ) energy, although it does not correspond to a fixed point in the 3