Abstract
We study the existence of hairy black holes in the generalized Einstein-Skyrme model. It is proven that in the BPS model limit there are no hairy black hole solutions, although the model admits gravitating (and flat space) solitons. Furthermore, we find strong evidence that a necessary condition for the existence of black holes with Skyrmionic hair is the inclusion of the Skyrme term $\mathcal{L}_4$. As an example, we show that there are no hairy black holes in the $\mathcal{L}_2+\mathcal{L}_6+\mathcal{L}_0$ model and present a new kind of black hole solutions with compact Skyrmion hair in the $\mathcal{L}_4+\mathcal{L}_6+\mathcal{L}_0$ model.
Highlights
The Skyrme model [1] is a candidate for the low-energy effective theory of Quantum Chromodynamics in the non-perturbative regime
The submodel L6 + L0, known as the BPS Skyrme model [6], provides a physically well-motivated idealization of nuclear matter, which is a perfect fluid with SDiff symmetry and zero binding energies and clearly contributes to the bulk properties of nuclei
The main result of the paper is the observation that there are Skyrme type models with the usual S3 target space which do not possess hairy black hole solutions, even though they support solitons (Skyrmions) both in flat and dynamically curved spacetime
Summary
Let us consider the BPS part of the full model i.e., the BPS Skyrme model [6] (for recent results concerning the model see [8]). We prove that this is a general property of the BPS Skyrme model for any single vacuum potential U To show this surprising result, it is enough to analyze the field equation for the profile function of a Skyrmion of charge B. where we extended the hedgehog ansatz to an axially symmetric generalization n = (sin θ cos Bφ, sin θ sin Bφ, cos θ). Assuming that the baryon density is a regular function everywhere we get that at the horizon, sin f f ′(rh) = ∞, and at the boundary (where f (R) = 0), sin f f ′(R) = ∞. This implies that the l.h.s. is 0. We want to underline that it is not the non-existence of hairy black holes by itself which is interesting, but the fact that this happens in a solitonic model with topologically nontrivial solitons
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