Meson loop corrections to the NJL model

Abstract

We present a symmetry preserving approach t o 1 =N N being the number of colors corrections in the Nambu-Jona-Lasinio model. This is achieved through an explicit 1=N expansion where N is considered as a free parameter. Veri cation and use of the pertinent W ard identity to the next to leading order in 1=N is made in obtaining the Goldberger-Treiman relation. Although the present results are not new, the approach m a y be useful for deriving higher than one loop corrections to the NJL model in a systematic way. I IntroductionThe Nambu-Jona-Lasinio NJL model 1 has proven to be a valuable tool for our understanding of pion chiral dynamics and several important aspects of chiral symmetry in the domain of low energy hadron physics. Also of theoretical relevance is the fact that nonperturbative methods devised to treat the NJL model turned out to be adequate to study spontaneous symmetry breaking in con ning quark models 2 and QCD 3 .This suggests that the study of chirally invariant nonperturbative approximations has considerable interest in its own right. In the context of the Nambu-JonaLasinio NJL 1 model the best known nonperturbative method is the one quark loop approximation. It has been applied to study numerous features of strong interaction physics at low energies for recent reviews see 5 with striking success. Much less work, however, has been comparatively devoted to implementations of higher order approximations. Among these we can quote 6-10 . These approaches are based on a 1=N expansion, where N denotes the number of colors. As discussed in 10 , symmetry properties of the system can be easily violated by an inappropriate choice of diagrams. Methods which preserve the symmetry properties at higher orders deserve our special attention, refs. 9,10 .The purpose of this paper is to present a mathematical implementation of a 1=N expansion where N is considered as a free parameter. We make use of the Functional Integral Approach and show that a completely systematic expansion can be constructed in such a w ay that it automatically yields all necessary Feynman graphs to a given order in 1=N, being therefore self consistent. Our results coincide with those of refs. 9,10 , which are symmetry preserving, to their calculation order. Higher order contributions can be computed in the present s c heme. Moreover, we implement the calculation of the pion decay constant within the very same approximation scheme and corrections to it are given by the same means and are on the same footing as corrections to the meson and fermion propagators. II The ModelLet us start by considering the two a vor NJL model Lagrangian:

Low energy constants ofχPTfrom the instanton vacuum model

“…in agreement with the above-given estimates (2,3). Note that though the evaluation of the meson loop corrections in the instanton vacuum model is similar to the earlier meson loop evaluations [15,16,17,18] in the NJL model, there are a few differences which should be mentioned:…”

QCD in infrared region and spontaneous breaking of the chiral symmetry

“…Though the evaluation of the meson loop corrections in the instanton vacuum model is similar to the earlier meson loop evaluations in the NJL model [18,19,20,21] important differences should be mentioned:…”