The Question of Abelian Higgs Hair Expulsion from Extremal Dilaton Black Holes

It has been argued that the extremal dilaton black holes exhibit a flux expulsion of Abelian-Higgs vortices. We re-examine carefully the problem and give analytic proofs for the flux expulsion always takes place. We also conduct numerical analysis of the problem using three initial data sets on the horizon of an extreme dilatonic black hole, namely, core, vacuum and sinusoidal initial conditions. We also show that an Abelian-Higgs vortex can end on the extremal dilaton black hole. Concluding, we calculate the backreaction of the Abelian-Higgs vortex on the geometry of the extremal black hole and drew a conclusion that a straight cosmic string and the extreme dilaton black hole hardly knew their presence.