Zero Modes of Rotationally Symmetric Generalized Vor- tices and Vortex Scattering

Zero modes of rotationally symmetric vortices in a hierarchy of generalized Abelian Higgs models are studied. Under the finite-energy and the smoothness condition, it is shown, that in all models, n self-dual vortices superimposed at the origin have 2n modes. The relevance of these modes for vortex scattering is discussed, first in the context of the slow-motion approximation. Then a corresponding Cauchy problem for an all head-on collision of n vortices is formulated. It is shown that the solution of this Cauchy problem has a π n symmetry.