Head-on collision ofnvortices

We show that the result of the head-on collision on $n$ vortices in the Abelian-Higgs model is the scattering by an angle of $\frac{\pi }{n}$ even when there is another vortex of arbitrary topological index at the center of scattering. This is obtained by analysis of both the geodesic flow on the moduli space and of the string approximation to the short range interactions of vortices. We also provide formal tools for an analysis of more general scattering events with $n$ vortices by diagonalization of the moduli space metrics around the point where the $n$ vortices coincide. Arguments after a certain degree of equivalence of the string and moduli space approaches are presented.