Ultraviolet softness of the second-order density kernel (source code)

= Ultraviolet softness of the second-order density kernel

At fixed external momentum $\mathbf k$, mass and momentum conservation imply
$$
F_2(\mathbf q,\mathbf k-\mathbf q)=O(k^2/q^2)
$$
as $q\to\infty$. In particular, the constant term in the hard-momentum expansion must cancel.