I. INTRODUCTION In three spatial dimensions, in the unitary limit of a quantum Bose or Fermi gas with point-like interactions, the scattering length diverges. “Unitary” here refers to the limit on the cross section imposed by unitarity. These systems provide intriguing examples of interacting, scaling invariant theories with dynamical exponent z = 2, i.e. non-relativistic. The infinite scattering length occurs at a fixed point of the renormalization group in the zero temperature theory, thus the models are quantum critical. The only energy scales in the problem are the temperature and chemical potential, and thermodynamic properties are expected to reveal universal behavior. They can be realized experimentally by tuning the scattering length to ±∞ using a Feshbach resonance. (See for instance [1, 2] and references therein.) They are also thought to occur at the surface of neutron stars. The systems have attracted much theoretical interest, and remain challenging problems due to the lack of small parameter for a perturbative expansion, such as n 1 $^{/d}$a or n 1 $^{/d}$r where a is the scattering length, and r the range of the 2-body potential. Early works were done by Leggett and Nozi` eres and Schmitt-Rink[3, 4]. The universal scaling behavior was studied in [5, 6]. In 3 dimensions, this is the physics of the BCS/BEC crossover: since the fixed point occurs for an attractive coupling, the fermions may form a bosonic bound state which can subsequently undergo BEC. This cross-over was studied analytically in [7] by considering a model of 2-component fermions coupled to the bosonic bound state. Monte-Carlo studies were performed in [8, 11–13]. The models can be studied in spatial dimension 2 < d < 4[9, 10] and an epsilon expansion carried out[14, 15]. There has also been some attempts to apply the AdS/CFT correspondence to these non-relativistic systems[16– 19]. In the present work, we describe a new analytic approach to studying the unitary 2