Metal-SDW phase transition in $3- \epsilon$ dimensions

The quantum phase transition associated with a spin density wave (SDW) instability in two dimensional metals is relevant to various strongly correlated electron systems including high-$T_c$ superconductors. However, the critical point associated with the phase transition has remained largely inaccessible due to the absence of a small parameter. In this work we use the recently developed dimensional regularization scheme, where the co-dimension of a Fermi surface is extended to general values while its dimension is held fixed, to perturbatively access the quantum critical point in $3-\epsilon$ space dimensions. We derive the beta functions and compute the critical exponents to the one loop order.