  We study the singular Landau surfaces of planar diagrams contributing to
scattering of a massless quark and antiquark in 3+1 dimensions. In particular,
we look at singularities which remain after integration with respect to the
various angular degrees of freedom. We derive a general relation between these
singularities and the singularities of quark- antiquark scattering in 1+1
dimensions. We then classify all Landau surfaces of the 1+1 dimensional system.
Combining these results, we deduce that the singular surfaces of the angle-
integrated 3+1 dimensional amplitude must satisfy at least one of three
conditions, which we call the planar light-cone conditions. We discuss the
extension of our results to non-perturbative processes by means of the
non-perturbative operator product expansion. Our findings offer new insights
into the connection between the 't Hooft model and large-N_c mesons in 3+1
dimensions and may prove useful in studies of confinement in relativistic meson
systems.
