Many body 2nd order energy shift of ground state

This is more of a math question I suppose, but its in the context of calculating the second order energy shift in the ground state energy for a non relativistic collection of electrons.

We end up showing that the energy shift has a finite and divergent piece. The divergent bit is proportional to the following:

$$f(q) = \int_{|k+q|>1} d^3k \int_{|p+q|>1} d^3p \frac{\theta(1-k) \theta(1-p)}{q^2 + \vec{q} \cdot ( \vec{k} + \vec{p} ) }$$

The part I'm getting stuck on is showing that

$$f(q) \approx \frac{2}{3}(2\pi)^2(1-ln2)q$$

as q approaches zero. I've tried a bunch of things, but to no avail. Any suggestions?

Last edited: