BCS-like formula for Tc does not necessarily imply BCS pairing mechanism: the case of magnetically-mediated quantum-critical pairing

In the BCS theory of superconductivity, an instability towards pairing develops at arbitrary weak dimensionless coupling $λ$ due to a divergence of logarithmic perturbative series for the pairing susceptibility (Cooper logarithms) at Tc ~ ω₀ e-1/λ, where ω₀ is an energy cutoff. On the contrary, in many models of superconductivity out of a non-Fermi liquid, Cooper logarithm is absent and superconductivity emerges only when $λ$ exceeds a certain threshold. We argue that there are situations when there is no threshold and at weak coupling the formula for Tc is BCS-like, yet the origin of the pairing instability is fundamentally different from that in the BCS scenario. As an example, we revisit superconductivity in a metal at the onset of $(π,π)$ spin-density-wave order. Earlier studies of this problem found no threshold and a BCS-like expression for Tc at weak coupling. We argue that, despite this, the pairing is not caused by the Cooper logarithm and in many respects is qualitatively similar to that in non-Fermi liquids.