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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 Kmasses. We determine the self-consistent set of one-quark-loop diagrams bothfor the Schwinger-Dyson quark mass gap equation and for the Bethe-Salpeterquark-antiquark boundstate equation. We identify the dominant diagrams, and wefocus on the boundstate equation. We detail the Bethe-Salpeter equation, addingthe dominant new diagram to the BCS kernel. The relevant numerical techniquesare also discussed. The ideal cases of one, two and three light flavors,relevant to lattice QCD are also explored, together with the case of realisticcurrent quark masses.

Microscopic quark study of the eta and eta' masses