  We compute the vacuum polarization energy of kink solitons in the $\phi^{8}$
model in one space and one time dimensions. There are three possible field
potentials that have eight powers of $\phi$ and that possess kink solitons. For
these different field potentials we investigate whether the vacuum polarization
destabilizes thesolitons. This may particularly be the case for those
potentials that have degenerate ground states with different curvatures in
field space yielding different thresholds for the quantum fluctuations about
the solitons at negative and positive spatial infinity. We find that
destabilization occurs in some cases, but this is not purely a matter of the
field potential but also depends on the realized soliton solution for that
potential. One of the possible field potentials has solitons with different
topological charges. In that case the classical mass approximately scales like
the topological charge. Even though destabilization precludes robust
statements, there are indications that the vacuum polarization energy does not
scale as the topological charge.
