In the BCS theory of superconductivity, an instability towards pairing develops at arbitrarily weak dimensionless coupling $\lambda$ due to a divergence of logarithmic perturbative series for the pairing susceptibility (Cooper logarithms) at $T_c\sim {\omega }_0{e}^{-1/\lambda }$, where ${\omega }_0$ is an energy cutoff. On the contrary, in many models of superconductivity out of a non-Fermi-liquid, the Cooper logarithm is absent, and superconductivity emerges only when $\lambda$ exceeds a certain threshold. We argue that there are situations when there is no threshold, and at weak coupling the formula for $T_c$ 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 $(\pi ,\pi )$ spin-density-wave order. Earlier studies of this problem found no threshold and a BCS-like expression for $T_c$ at weak coupling. We argue that, despite this, the pairing is not caused by the Cooper logarithm, and in many respects it is qualitatively similar to that in non-Fermi-liquids.

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• Superconductivity
• Unconventional superconductors
• BCS theory
• Non-Fermi-liquid theory