  We show that it is necessary to go beyond the BCS (rainbow-ladder)
approximation to split the $\eta$ and $\eta'$ masses from the $\pi$ and K
masses. We determine the self-consistent set of one-quark-loop diagrams both
for the Schwinger-Dyson quark mass gap equation and for the Bethe-Salpeter
quark-antiquark boundstate equation. We identify the dominant diagrams, and we
focus on the boundstate equation. We detail the Bethe-Salpeter equation, adding
the dominant new diagram to the BCS kernel. The relevant numerical techniques
are also discussed. The ideal cases of one, two and three light flavors,
relevant to lattice QCD are also explored, together with the case of realistic
current quark masses.
