Winding number dependence of quantum vortex energies at one-loop

We compute the one-loop vacuum polarization energies of Abrikosov-Nielsen-Olesen (ANO) vortices with topological charge $n$ in scalar electrodynamics, for the BPS case of equal gauge and scalar masses. This calculation allows us to investigate the relationship between the winding number and the quantum-corrected vortex energy, which in turn determines the stability of higher winding configurations against decay into configurations with unit winding. While the classical energy is proportional to $n$, we find that the vacuum polarization energy is negative and approximately proportional to $n-1$ with a small constant offset.