  Asymptotically nonlocal field theories represent a sequence of
higher-derivative theories whose limit point is a ghost-free,
infinite-derivative theory. Here we extend this framework, developed previously
in a theory of real scalar fields, to gauge theories. We focus primarily on
asymptotically nonlocal scalar electrodynamics, first identifying equivalent
gauge-invariant formulations of the Lagrangian, one with higher-derivative
terms and the other with auxiliary fields instead. We then study mass
renormalization of the complex scalar field in each formulation, showing that
an emergent nonlocal scale (i.e., one that does not appear as a fundamental
parameter in the Lagrangian of the finite-derivative theories) regulates loop
integrals as the limiting theory is approached, so that quadratic divergences
can be hierarchically smaller than the lightest Lee-Wick partner. We conclude
by making preliminary remarks on the generalization of our approach to
non-Abelian theories, including an asymptotically nonlocal standard model.
