  Extensions of dynamical-mean-field-theory (DMFT) make use of quantum impurity
models as non-perturbative and exactly solvable reference systems which are
essential to treat the strong electronic correlations. Through the introduction
of retarded interactions on the impurity, these approximations can be made
two-particle self-consistent. This is of interest for the Hubbard model,
because it allows to suppress the antiferromagnetic phase transition in
two-dimensions in accordance with the Mermin-Wagner theorem, and to include the
effects of bosonic fluctuations. For a physically sound description of the
latter, the approximation should be conserving. In this paper we show that the
mutual requirements of two-particle self-consistency and conservation lead to
fundamental problems. For an approximation that is two-particle self-consistent
in the charge- and longitudinal spin channel, the double occupancy of the
lattice and the impurity are no longer consistent when computed from
single-particle properties. For the case of self-consistency in the charge- and
longitudinal as well as transversal spin channels, these requirements are even
mutually exclusive so that no conserving approximation can exist. We illustrate
these findings for a two-particle self-consistent and conserving DMFT
approximation.
