Current-carrying zero mode for the Nielsen-Olesen string

Zero modes of strings in the abelian Higgs model are analyzed. In spite of the fact that the gauge symmetry is not broken in the string center, the corresponding zero mode is shown to exist and to see it one has to analyze carefully the dependence on transverse coordinates for the excitations. The analysis of this kind is also important for the Witten model of superconducting string. Unusual properties of the zero modes connected with the broken gauge symmetry in the string background are investigated. One of the modes carries the current quite similar to that in the Witten model and gives back reaction to the string profile. It is claimed that the current in the string improves stability of the electroweak string.