Perturbative non-Fermi liquids from dimensional regularization

We devise a dimensional regularization scheme for quantum field theories with Fermi surface to study scaling behaviour of non-Fermi liquid states in a controlled approximation. Starting from a Fermi surface in two space dimensions, the co-dimension of Fermi surface is extended to a general value while the dimension of Fermi surface is fixed. When Fermi surface is coupled with a critical boson centered at zero momentum, the interaction becomes marginal at a critical space dimension d_c=5/2. A deviation from the critical dimension is used as a small parameter for a systematic expansion. We apply this method to the theory where two patches of Fermi surface is coupled with a critical boson, and show that the Ising-nematic critical point is described by a stable non-Fermi liquid state slightly below the critical dimension. Critical exponents are computed upto the two-loop order.