  It is possible to formulate theories with many Lee-Wick particles such that a
limit exists where the low-energy theory approaches the form of a ghost-free
nonlocal theory. Such asymptotically nonlocal quantum field theories have a
derived regulator scale that is hierarchically smaller than the lightest
Lee-Wick resonance; this has been studied previously in the case of
asymptotically nonlocal scalar theories, Abelian and non-Abelian gauge
theories, and linearized gravity. Here we consider the dependence on
center-of-mass energy of scattering cross sections in these theories. While
Lee-Wick resonances can be decoupled from the low-energy theory, scattering
amplitudes nonetheless reflect the emergent nonlocality at the scale where the
quadratic divergences are regulated. This implies observable consequences in
theories designed to address the hierarchy problem, even when the Lee-Wick
resonances are not directly accessible.
