Schwinger vs Coleman: Magnetic Charge Renormalization

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.