Ladder equation for the three-particle vertex and its approximate solution

We generalize the three two-particle Bethe-Salpeter equations to ten three-particle ladders. These equations are exact and yield the exact three-particle vertex, if we knew the three-particle vertex irreducible in one of the ten channels. However, as we do not have this three-particle irreducible vertex at hand, we approximate this building block for the ladder by the sum of two-particle irreducible vertices each connecting two fermionic lines. The comparison to the exact solution shows that this approximation is only good for rather weak interactions and even than only qualitatively - at least for the non-linear response function analyzed.