  In this paper, we revisit the antiferromagnetic (AF) phase diagram of the
single-band three-dimensional half-filled Hubbard model on a simple cubic
lattice studied within the dynamical mean field theory (DMFT). Although this
problem has been investigated extensively in the literature, a comprehensive
understanding of the impact of the different one- and, in particular,
two-particle local correlation functions of DMFT on the AF transition
temperature is still missing. We have, hence, performed a "fluctuation
diagnostics" of $T_N$ with respect to different local bosonic fluctuations
(charge, spin, particle-particle) contained in the two-particle vertex of DMFT.
Our results indicate that, beyond weak coupling, the screening of the DMFT
vertex by local fluctuations leads to an enhancement of $T_N$ with respect to a
random phase approximation (RPA) like calculation where this vertex is replaced
by the bare interaction. The overall suppression of $T_N$ in DMFT with respect
to RPA is then solely due to the incoherence introduced by the DMFT self-energy
in the one-particle Green's functions. This illustrates the Janus faced role of
the local moment formation in the DMFT solution of the Hubbard model which
leads to completely opposite effects in the one- and two-particle correlations
functions.
