Quark Schwinger-Dyson equation in temporal Euclidean space

We present an elementary nonperturbative method to obtain Green's functions (GFs) for timelike momenta. We assume there are no singularities in the first and third quadrants of the complex plane of space momentum components and perform a 3d analogue of Wick rotation. This procedure defines Greens functions in a timelike Euclidean space. As an example we consider the quark propagator in QCD. While for weak coupling, this method is obviously equivalent to perturbation theory, for a realistic QCD coupling a complex part of the quark mass and renormalization wave function has been spontaneously generated even below the standard perturbative threshold. Therefore, our method favors a confinement mechanism based on the lack of real poles.