V

. Define the density of states at the Fermi surface, in the normalization of the question, by

ν_{F} = \int_{FS} \frac{d ^{2} k}{( 2 π ) ^{3} ∣ v _{F} ( k ) ∣} .

(1)

In the one-loop Cooper diagram the two internal fermions have opposite momenta. After integrating over internal frequency, the remaining normal-energy integral contains

\int_{∣ ω ∣}^{Λ} \frac{d ξ}{ξ} = lo g \frac{Λ}{∣ ω ∣} .

(2)

Consequently

Γ (ω) = V - ν_{F} V^{2} lo g \frac{Λ}{∣ ω ∣} + O (V^{3} lo g^{2}),

(3)

and summing the leading geometric series gives the running BCS theory coupling

V_{R} (ω) = \frac{V}{1 + ν _{F} V lo g ( Λ/∣ ω ∣ )} .

(4)

For repulsive

V > 0

it flows logarithmically toward zero. For attractive

V < 0

, its denominator vanishes at the Cooper instability scale

∣ ω_{*} ∣ = Λ exp [- \frac{1}{ν _{F} ∣ V ∣}] .

(5)

At this scale the normal Fermi liquid becomes unstable: opposite-momentum fermions form Cooper pairs, a superconducting or superfluid condensate develops, and a gap opens.