Pairing by a Dynamical Interaction in a Metal

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