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At zero temperature and weak coupling, the BCS gap equation gives $\Delta \simeq 2\Lambda e^{-1/({\nu }_F|V|)}$. Its exponential scale agrees with the renormalization-group scale of the Cooper instability.

The tree-level Cooper-pair scattering amplitude is simply $V$. Define the density of states at the Fermi surface, in the normalization of the question, byIn the one-loop Cooper diagram the two internal fermions have opposite momenta. After integrating over internal frequency, the remaining normal-energy integral containsConsequentlyand summing the leading geometric series gives the running BCS theory couplingFor repulsive $V>0$ it flows logarithmically toward zero. For attractive $V<0$, its denominator vanishes at the Cooper instability scaleAt this scale the normal Fermi liquid becomes unstable: opposite-momentum fermions form Cooper pairs, a superconducting or superfluid condensate develops, and a gap opens.