Susceptibility of a spinon Fermi surface coupled to a U(1) gauge field

We study the theory of a U(1) gauge field coupled to a spinon Fermi surface. Recently this model has been proposed as a possible description of the organic compound $\kappa-(BEDT-TTF)_2 Cu_2 (CN)_3$. We calculate the susceptibility of this system and in particular examine the effect of pairing of the underlying spin liquid. We show that this proposed theory is consistent with the observed susceptibility measurements.