"No-hair" theorems for the Abelian Higgs and Goldstone models

We examine the question of whether black holes can have associated external massive vector and/or scalar fields, when the masses are produced by spontaneous symmetry breaking. Working throughout in the spherically symmetric case, we show that "no-hair" theorems can be proved for the vector field in the Abelian Higgs model, for an arbitrary $\xi R{\left|\phi \right|}^2$ term in the Higgs Lagrangian, and for the Goldstone scalar field model with $\xi =0\mathrm{}$. We also show that a Minkowski-space analog problem does have nontrivial screened charge solutions, indicating that the "no-hair" theorems which we prove are consequences of the stringent conditions at the assumed horizon in the general-relativistic case, not of the interacting field or spontaneous-symmetry-breaking aspects of the problem.