Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 312 / 3 / ii / Solution

The ultraviolet softness of the second-order density kernel requiresfor fixed $\mathbf{k}$ as $q\to \infty$. In this limit the two hard vectors are opposite,The constant part of the stated kernel is thereforeIt must vanish, givingThis recovers the standard standard perturbation theory density kernel $F_2$.