  Just because the propagator of some field obeys a de Sitter invariant
equation does not mean it possesses a de Sitter invariant solution. The classic
example is the propagator of a massless, minimally coupled scalar. We show that
the same thing happens for massive scalars with $M_S^2 < 0$, and for massive
transverse vectors with $M_V^2 \leq -2 (D-1) H^2$, where $D$ is the dimension
of spacetime and $H$ is the Hubble parameter. Although all masses in these
ranges give infrared divergent mode sums, using dimensional regularization (or
any other analytic continuation technique) to define the mode sums leads to the
incorrect conclusion that de Sitter invariant solutions exist except at
discrete values of the masses.
