Generalized proper-time approach for the case of broken isospin symmetry

Abstract

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 $\mathrm{SU}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(2)$ chiral symmetry of the NJL Lagrangian is broken by the current quark mass matrix with ${m}_{u}\ensuremath{\ne}{m}_{d}.$ 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.