  Recently, diagrammatic extensions of dynamical mean field theory (DMFT) have
been proposed for including short- and long-range correlations beyond DMFT on
an equal footing. We employ one of these, the dynamical vertex approximation
(D$\Gamma$A), and study the two-dimensional Hubbard model on a square lattice.
We define two transition lines in the phase diagram which correspond,
respectively, to the opening of the gap in the nodal direction and throughout
the Fermi surface. Our self-energy data show that the evolution between the two
regimes occurs in a gradual way (crossover) and also that at low enough
temperatures the whole Fermi surface is always gapped. Furthermore, we present
a comparison of our D$\Gamma$A calculations at a parameter set where data
obtained by other techniques are available.
