  Vortices in spin-$0$ superfluids generically carry magnetic fields inside
their cores, so that even neutral superfluid vortices may be thought of as
magnetic flux tubes. We give a systematic analysis of this `vortex magnetic
effect' using effective field theory, clarifying earlier literature on the
subject. Our analysis shows that in superfluid Helium-$4$ the vortex magnetic
effect may be large enough to be experimentally detectable.
