  We show that the one-loop self-energy at finite temperature has a unique
limit as the external momentum $p_\mu\rightarrow 0$ {\it if} the loop involves
propagators with distinct masses. This naturally arises in theories involving
particles with different masses as is demonstrated for a toy model of two
scalars as well as in a $U(1)$ Higgs theory. We show that, in spontaneously
broken gauge theories, this observation nonetheless does not affect the
difference between the Debye and plasmon masses, which are often thought of as
the $(p_0=0, \vec{p}\to 0)$ and $(p_0\to 0,\vec{p}=0)$ limits of the
self-energy.
