In view of the measured Higgs mass of 125 GeV, the perturbative
renormalization group evolution of the Standard Model suggests that our Higgs
vacuum might not be stable. We connect the usual perturbative approach and the
functional renormalization group which allows for a straightforward inclusion
of higher-dimensional operators in the presence of an ultraviolet cutoff. In
the latter framework we study vacuum stability in the presence of
higher-dimensional operators. We find that their presence can have a sizable
influence on the maximum ultraviolet scale of the Standard Model and the
existence of instabilities. Finally, we discuss how such operators can be
generated in specific models and study the relation between the instability
scale of the potential and the scale of new physics required to avoid
instabilities.Comment: 27 pages, 12 figure