Low-energy behaviour of two-point functions of quark currents

We discuss vector, axial-vector, scalar and pseudoscalar two-point functions at low and intermediate energies. We first review what is known from chiral perturbation theory, as well as from a heat kernel expansion within the context of the extended Nambu-Jona-Lasinio (ENJL) model of ref. \cite{12}. In this work we derive then these two-point functions to all orders in the momenta and to leading order in $1/N_c$. We find an improved high-energy behaviour and a general way of parametrizing them that shows relations between some of the two-point functions, which are also valid in the presence of gluonic interactions. The similarity between the shape of the experimentally known spectral functions and the ones we derive, is greatly improved with respect to those predicted by the usual constituent quark like models. We also obtain the scalar mass $M_S = 2 M_Q$ independent of the regularization scheme. In the end, we calculate fully an example of a nonleptonic matrix element in the ENJL--model, the $\pi~+-\pi~0$ electromagnetic mass difference and find good agreement with the measured value.