Aharonov-Bohm problem for spin 1

The basic AB problem is to determine how an unshielded tube of magnetic flux $\Phi$ affects arbitrarily long-wavelength charged particles impinging on it. For spin 1 at almost all $\Phi$ the particles do not penetrate the tube, so the interaction essentially is periodic in $\Phi$ (AB effect). Below-threshold bound states move freely only along the tube axis, and consequent induced vacuum currents supplement rather than screen $\Phi$. For a pure magnetic interaction the tube must be broader than the particle Compton wavelength, i.e., only the nonrelativistic spin-$1$ AB problem exists.