BCS superconductivity in quantum critical metals

We consider the superconducting transition in fermionic quantum critical systems. Assuming the validity of Migdal theorem, the gap equation can be written in terms of the retarded pair susceptibility. Scale invariance leads to an algebraic gap equation, instead of the BCS exponential form. With reasonably small glue strength, we can get very large transition temperatures comparable to those observed in cuprates. We get nice dome structures around the QCP for a wide range of parameters. With a non-Lorentzian dynamical exponent, the upper critical field is greatly enhanced when approaching the critical point, while the transition temperature only changes modestly, in agreement with recent experiments on heavy fermions.