Bethe-Salpeter equation in QCD in a Wilson loop context

We give a nonperturbative derivation of the Bethe-Salpeter equation in QCD based on the Feynman-Schwinger path integral representation of the one particle propagator in an external field. We obtain a path integral representation for a second order quark-antiquark amplitude in which the gauge field appears only through an appropriate Wilson loop integral W. Then, for such a quantity we derive a q (q) over bar BS equation assuming that ilnW can be written as the sum of a perturbative contribution and an area term as in the derivation of the heavy quark potential. We also show that, by standard approximations, an effective meson mass operator can be obtained from our BS kernel. From this the corresponding Wilson loop potential is recovered, by 1/m(2) expansion, spin-dependent and velocity-dependent terms included. On the contrary, neglecting spin-dependent terms, the relativistic flux tube model is reproduced. The method is illustrated also on the simplified case of two spinless particles interacting via a scalar field and on a one-dimensional potential model.