  We construct a theory of real scalar fields that interpolates between two
different theories: a Lee-Wick theory with $N$ propagator poles, including
$N-1$ Lee-Wick partners, and a nonlocal infinite-derivative theory with kinetic
terms modified by an entire function of derivatives with only one propagator
pole. Since the latter description arises when taking the $N\rightarrow\infty$
limit, we refer to the theory as "asymptotically nonlocal." Introducing an
auxiliary-field formulation of the theory allows one to recover either the
higher-derivative form (for any $N$) or the Lee-Wick form of the Lagrangian,
depending on which auxiliary fields are integrated out. The effective scale
that regulates quadratic divergences in the large-$N$ theory is the would-be
nonlocal scale, which can be hierarchically lower than the mass of the lightest
Lee-Wick resonance. We comment on the possible utility of this construction in
addressing the hierarchy problem.
