Chiral symmetry breaking in the presence of a confining interaction

We study chiral symmetry breaking, making use of a generalized Nambu–Jona-Lasinio (NJL) model that includes a description of confinement. The Schwinger-Dyson and Bethe-Salpeter equations are solved for our model of the self-energy of a quark. We show that our analysis is consistent with the Goldstone theorem. That is, the pion has zero mass, if the current quark masses are zero. We use a confining interaction with a (Dirac) matrix structure that leads to simple equations for the self-energy and for a vertex function that serves to sum a ladder of confining interactions. We consider spacelike values of $q^2$, and carry out our analysis in a Euclidean momentum space. For timelike $q^2$, we use calculational procedures that we have developed in our earlier work in order to exhibit properties of the confining vertex. For the spacelike values of $q^2$ considered here, we see that the effects due to the introduction of our model of confinement are small. (However, such effects are very important for timelike $q^2$. Their consideration is essential, if we wish to study mesons, such as the rho and omega, in our model.)