Analytical properties of the quark propagator from truncated Dyson-Schwinger equation in complex Euclidean space

In view of the mass spectrum of heavy mesons in vacuum the analytical properties of the solutions of the truncated Dyson-Schwinger equatio for the quark propagator within the rainbow approximation are analysed in some detail. In Euclidean space, the quark propagator is not an analytical function possessing, in general, an infinite number of singularities (poles) which hamper to solve the Bethe-Salpeter equation. However, for light mesons (with masses M_{q\bar q} <= 1 GeV) all singularities are located outside the region within which the Bethe-Salpeter equation is defined. With an increase of the considered meson masses this region enlarges and already at masses >= 1 GeV, the poles of propagators of u,d and s quarks fall within the integration domain of the Bethe-Salpeter equation. Nevertheless, it is established that for meson masses up to M_{q\bar q}~=3 GeV only the first, mutually complex conjugated, poles contribute to the solution. We argue that, by knowing the position of the poles and their residues, a reliable parametrisation of the quark propagators can be found and used in numerical procedures of solving the Bethe-Salpeter equation. Our analysis is directly related to the future physics programme at FAIR with respect to open charm degrees of freedom.