Long distance expansion for the NJL model with SU(3) and U_A(1) breaking
  This work is a follow up of recent investigations, where we study the implications of a generalized heat kernel expansion, constructed to incorporate non-perturbatively the effects of a non-commutative quark mass matrix in a fully covariant way at each order of the expansion. As underlying Lagrangian we use the Nambu -- Jona-Lasinio model of QCD, with $SU_f(3)$ and $U_A(1)$ breaking, the latter generated by the 't Hooft flavour determinant interaction. The associated bosonized Lagrangian is derived in leading stationary phase approximation (SPA) and up to second order in the generalized heat kernel expansion. Its symmetry breaking pattern is shown to have a complex structure, involving all powers of the mesonic fields allowed by symmetry. The considered Lagrangian yields a reliable playground for the study of the implications of symmetry and vacuum structure on the mesonic spectra, which we evaluate for the scalar and pseudoscalar meson nonets and compare with other approaches and experiment. 
A.A. Osipov, H. Hansen, and B. Hiller