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 SU (2) × SU (2) chiral symmetry of the NJL Lagrangian is broken by the current quark mass matrix withm u =m d . The non-degeneracy of d and u masses destroys one of the most crucial features of the proper-time expansionthe 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. PACS number(s): 12.39.Fe, 11.30.Rd.

π0−η−η′mixing in a generalized multiquark interaction scheme

“…These coefficients are totally defined in terms of h a and the parameters of the model. Now to the fermion determinant related to the integration over the fermion fields: We expand it using a heat-kernel technique that takes appropriately into account the quark mass differences, being chiral covariant at each order of the expansion [55][56][57],…”

Light quark masses in multi-quark interactions

“…Fortunately, the result is known. One can find all necessary details of such calculations, for instance, in [38], where we used the modified heat kernel technique [39][40][41] developed for the case of explicit chiral symmetry breaking. Here we quote the main outcome.…”

“…For that we must exclude quark degrees of freedom in (13), e.g., by integrating them out from the corresponding generating functional. The standard Gaussian path integral leads us to the fermion determinant, which we expand by using a heat-kernel technique [38][39][40][41]. The remaining part of the Lagrangian, L aux , depends on auxiliary fields which do not have kinetic terms.…”

Large mass invariant asymptotics of the effective action

“…In QCD the nondegenerate mass matrix of heavy (constituent) quarks results from the spontaneous breakdown of chiral symmetry, as it is the case, for instance, in the Nambu -Jona-Lasinio model [4], or from the specially added invariant term to the QCD Lagrangian that regulates the infrared behaviour of the effective action [5]. We do not consider here the manifest chiral symmetry breaking effect on the effective action [6]. A careful analysis of this problem would lead us too far away from the subject, leaving the present result without changes.…”

“…The mathematical formalism being presented in this letter is a necessary element of the approach which faithfully mirrors the vacuum structure of such a theory. This approach was formulated for the theory with an explicit and spontaneous breakdown of the global SU (2) × SU (2) chiral symmetry in [6] and generalized to all orders of the asymptotic expansion in [7]. Here we extend this scheme to the SU (3) × SU (3) chiral theory where the mass matrix has a form m = diag(m 1 , m 2 , m 3 ).…”