The third virial coefficient of a two-component unitary Fermi gas across an Efimov-effect threshold

We consider a mixture of two single-spin-state fermions with an interaction of negligible range and infinite $s$-wave scattering length. By varying the mass ratio $\alpha$ across $\alpha_c\simeq 13.6069$ one can switch on-and-off the Efimov effect. We determine analytically the third cluster coefficient of the gas. We show that it is a smooth function of $\alpha$ across $\alpha_c$ since, unexpectedly, the three-body parameter characterizing the interaction is relevant even on the non-Efimovian side $\alpha<\alpha_c$.