Generalized Proper-Time Approach For The Case Of Broken Isospin Symmetry

We present a derivation of the low-energy effective meson Lagrangian of the Nambu -- Jona-Lasinio (NJL) model on the basis of Schwinger's proper-time regularization of the one-loop fermion determinant. We consider the case in which the $SU(2)\times SU(2)$ chiral symmetry of the NJL Lagrangian is broken by the current quark mass matrix with $\hat{m}_u\ne\hat{m}_d$. The non-degeneracy of $d$ and $u$ masses destroys one of the most crucial features of the proper-time expansion -- the chiral-invariant structure of Seeley -- DeWitt coefficients. We show however that systematic resummations inside the proper-time expansion are still possible and derive a result which is in full agreement with the chiral Ward -- Takahashi identities.