  The kinetic mixing of two U(1) gauge theories can result in a massless photon
that has perturbative couplings to both electric and magnetic charges. This
framework can be used to perturbatively calculate in a quantum field theory
with both kinds of charge. Here we re-examine the running of the magnetic
charge, where the calculations of Schwinger and Coleman sharply disagree. We
calculate the running of both electric and magnetic couplings and show that the
disagreement between Schwinger and Coleman is due to an incomplete summation of
topological terms in the perturbation series. We present a momentum space
prescription for calculating the loop corrections in which the topological
terms can be systematically separated for resummation. Somewhat in the spirit
of modern amplitude methods we avoid using a vector potential and use the field
strength itself, thereby trading gauge redundancy for the geometric redundancy
of Stokes surfaces. The resulting running of the couplings demonstrates that
Dirac charge quantization is independent of renormalization scale, as Coleman
predicted. As a simple application we also bound the parameter space of
magnetically charged states through the experimental measurement of the running
of electromagnetic coupling.
