Pairing by a dynamical interaction in a metal

We consider pairing of itinerant fermions in a metal near a quantum-critical point (QCP) towards some form of particle-hole order (nematic, spin-density-wave, charge-density-wave, etc). At a QCP, the dominant interaction between fermions comes from exchanging massless fluctuations of a critical order parameter. At low energies, this physics can be described by an effective model with the dynamical electron-electron interaction $V(\Omega_m) \propto 1/|\Omega_m|^\gamma$,uptosomeuppercutoff$\Lambda$\mathrm{.}Thecase$\gamma =0$ corresponds to BCS theory, and can be solved by summing up geometric series of Cooper logarithms. We show that for a finite $\gamma$, the pairing problem is still marginal (i.e., perturbation series are logarithmic), but one needs to go beyond logarithmic approximation to find the pairing instability. We discuss specifics of the pairing at $\gamma >0$ in some detail and also analyze the marginal case $\gamma =0+$, when $V(\Omega_m) = \lambda \log{(\Lambda/|\Omega_m|)}$. We show that in this case the summation of Cooper logarithms does yield the pairing instability at $\lambda \log^2{(\Lambda/T_c)} = O(1)$, but the logarithmic series are not geometrical. We reformulate the pairing problem in terms of a renormalization group (RG) flow of the coupling, and show that the RG equation is different in the cases $\gamma =0$, $\gamma = 0+$,and$\gamma >0$.