Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 337 / 2 / c

Put $\xi =v_F(k)\ell$ and use the same ${\nu }_F$. With a symmetric cutoff $|\xi |<\Lambda$, the BCS gap equation becomesThe frequency integral is $1/[2\sqrt{{\xi }^2+{\Delta }^2}]$, soThusThe gap and the one-loop strong-coupling scale have the same nonperturbative exponential dependence; their order-one prefactors depend on the cutoff convention and microscopic completion. The renormalization-group divergence is therefore the normal-state signal of the paired, gapped phase found from the self-consistent gap equation.