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For the two-component fermion defined by the action (3), spin up particles scatter with spin down, and G$_{↑↓}$ = G$_{↓↑}$ . Thus, if the chemical potentials are equal, µ$_{↑}$ = µ$_{↓}$ = µ , then ε$_{↑}$ = ε$_{↓}$ , and the thermodynamics is just two copies of the single component fermion described above. In particular, the density and c are doubled, but the energy per particle scaling functions are the same. 

## X. CONCLUSIONS 

We have shown that the S-matrix based approach to quantum gases developed in [20] leads to a new treatment of the scale-invariant unitary limit, where all of the thermodynamic scaling functions can be computed as a function of µ/T . Though our 

