  In this work we present a comprehensive analysis of collective electronic
fluctuations and their effect on single-particle properties of the Hubbard
model. Our approach is based on a standard dual fermion/boson scheme with the
interaction truncated at the two-particle level. Within this framework we
compare various approximations that differ in the set of diagrams (ladder vs
exact diagrammatic Monte Carlo), and/or in the form of the four-point
interaction vertex (exact vs partially bosonized). This allows to evaluate the
effect of all components of the four-point vertex function on the electronic
self-energy. In particular, we observe that contributions that are not
accounted for by the partially bosonized approximation for the vertex have only
a minor effect on electronic degrees of freedom in a broad range of model
parameters. In addition, we find that in the regime, where the ladder dual
fermion approximation provides an accurate solution of the problem, the leading
contribution to the self-energy is given by the longitudional bosonic modes.
This can be explained by the fact that contributions of transverse
particle-hole and particle-particle modes partially cancel each other. Our
findings justify applicability of the recently introduced dual triply
irreducible local expansion (D-TRILEX) method that provides one of the simplest
consistent diagrammatic extensions of the dynamical mean-field theory.
