Selection rules for splitting strings

It has been pointed out that Nielsen-Olesen vortices may be able to decay by pair production of black holes. We show that when the abelian Higgs model is embedded in a larger theory, the additional fields may lead to selection rules for this process - even in the absence of fermions - due to the failure of a charge quantization condition. We show that, when there is topology change, the criterion based on the charge quantization condition supplements the usual criterion based on $\pi_0(H)$. In particular, we find that, unless $2\sin^2\theta_W$ is a rational number, the thermal splitting of electroweak Z-strings by magnetically neutral black holes is impossible, even though $\pi_0(H)$ is trivial.