  The dynamical susceptibility of strongly correlated electronic systems can be
calculated within the framework of the dynamical mean-field theory (DMFT). The
required measurement of the four-point vertex of the auxiliary impurity model
is however costly and restricted to a finite grid of Matsubara frequencies,
leading to a cutoff error. It is shown that the propagation of this error to
the lattice response function can be minimized by virtue of an exact
decomposition of the DMFT polarization function into local and nonlocal parts.
The former is measured directly by the impurity solver, while the latter is
given in terms of a ladder equation for the Hedin vertex that features an
unprecedentedly fast decay of frequency summations compared to previous
calculation schemes, such as the one of the dual boson approach. At strong
coupling the local approximation of the TRILEX approach is viable, but vertex
corrections to the polarization should be dropped on equal footing to recover
the correct prefactor of the effective exchange. In finite dimensions the DMFT
susceptibility exhibits spurious mean-field criticality, therefore, a
two-particle self-consistent and frequency-dependent correction term is
introduced, similar to the Moriya-$\lambda$ correction of the dynamical vertex
approximation. Applications to the two- and three-dimensional Hubbard models on
the square and cubic lattices show that the expected critical behavior near an
antiferromagnetic instability is recovered.
