Theta Dependence in the Presence of Massless Fermions

We show that if there are conserved flavor symmetries then some properties of a monopole can depend on $\theta$ even when a fermion is massless. The quantized nature of global symmetries and the fractional nature of the Witten effect can lead to interesting structure. Seen from another point of view, aside from possibly breaking baryon and lepton flavor symmetries (the Callan-Rubakov effect), monopole boundary conditions can also break the axial symmetry that otherwise could have been used to remove $\theta$ from the Lagrangian. As an example, in a toy model, we calculate the $\theta$ dependence of the mass of the monopole and properties of the non-zero charge density surrounding the monopole.