Light quark mass differences in the $\pi^0 - \eta - \eta'$ system
  A generalized 3 flavor Nambu-Jona--Lasinio Lagrangian including the explicit chiral symmetry breaking interactions which contribute at the same order in the large $1/N_c$ counting as the $U_A(1)$ 't Hooft flavor determinant is considered to obtain the mixing angles in the $\pi^0-\eta-\eta'$ system and related current quark mass ratios in close agreement with phenomenological values. At the same time an accurate ordering and magnitude of the splitting of states in the low lying pseudoscalar nonet is obtained. 
A. A. Osipov, B. Hiller, A. H. Blin, J. Moreira
hep-ph