We study the problem of vortex solutions in the background of an electrically
charged black string. We show numerically that the Abelian Higgs field
equations in the background of a four-dimensional black string have vortex
solutions. These solutions which have axial symmetry, show that the black
string can support the Abelian Higgs field as hair. This situation holds also
in the case of the extremal black string. We also consider the self-gravity of
the Abelian Higgs field and show that the effect of the vortex is to induce a
deficit angle in the metric under consideration.Comment: REVTEX4, 12 pages, 6 figures, The version to be appeared in Phys.
Rev.