We consider a general Einstein-scalar-Gauss-Bonnet theory with a coupling function fðϕÞ. We demonstrate that black-hole solutions appear as a generic feature of this theory since a regular horizon and an asymptotically flat solution may be easily constructed under mild assumptions for fðϕÞ. We show that the existing no-hair theorems are easily evaded, and a large number of regular black-hole solutions with scalar hair are then presented for a plethora of coupling functions fðϕÞ. DOI: 10.1103/PhysRevLett.120.131102 Introduction.-The existence or not of black holes associated with a nontrivial scalar field in the exterior region has attracted the attention of researchers over a period of many decades. Early on, a no-hair theorem [1] appeared that excluded static black holes with a scalar field, but this was soon outdated by the discovery of black holes with Yang-Mills [2] or Skyrme fields [3]. The emergence of additional solutions where the scalar field had a conformal coupling to gravity [4] led to the formulation of a novel nohair theorem [5] (for a review, see [6]). Recently, this argument was extended to the case of standard scalar-tensor theories [7], and a new form was proposed that covers the case of Galileon fields [8].However, both novel forms of the no-hair theorem [5,8] were shown to be evaded: the former in the context of the Einstein-dilaton-Gauss-Bonnet theory [9] and the latter in a special case of shift-symmetric Galileon theories [10][11][12]. A common feature of the above theories was the presence of the quadratic Gauss-Bonnet (GB) term defined as R 2 GB ¼ R μνρσ R μνρσ − 4R μν R μν þ R 2 , in terms of the Riemann tensor R μνρσ , the Ricci tensor R μν , and the Ricci scalar R. In both cases, basic requirements of the no-hair theorems were invalidated, and this paved the way for the construction of the counterexamples.Here, we consider a general class of scalar-GB theories, of which the cases [9,11] constitute particular examples. We demonstrate that black-hole solutions, with a regular horizon and an asymptotically flat limit, may in fact be constructed for a large class of such theories under mild only constraints on the coupling function fðϕÞ between the scalar field and the GB term. We address the requirements of both the old and novel no-hair theorems, and we show

In the context of the Einstein-scalar-Gauss-Bonnet theory, with a general coupling function between the scalar field and the quadratic Gauss-Bonnet term, we investigate the existence of regular black-hole solutions with scalar hair. Based on a previous theoretical analysis, which studied the evasion of the old and novel no-hair theorems, we consider a variety of forms for the coupling function (exponential, even and odd polynomial, inverse polynomial, and logarithmic) that, in conjunction with the profile of the scalar field, satisfy a basic constraint. Our numerical analysis then always leads to families of regular, asymptotically flat black-hole solutions with nontrivial scalar hair. The solution for the scalar field and the profile of the corresponding energy-momentum tensor, depending on the value of the coupling constant, may exhibit a nonmonotonic behavior, an unusual feature that highlights the limitations of the existing no-hair theorems. We also determine and study in detail the scalar charge, horizon area, and entropy of our solutions.

Novel wormholes are obtained in Einstein-scalar-Gauss-Bonnet theory for several coupling functions. The wormholes may feature a single-throat or a double-throat geometry and do not demand any exotic matter. The scalar field may asymptotically vanish or be finite, and it may possess radial excitations. The domain of existence is fully mapped out for various forms of the coupling function.

We consider the Einstein-scalar-Gauss-Bonnet theory in the presence of a cosmological constant Λ, either positive or negative, and look for novel, regular black-hole solutions with a non-trivial scalar hair. We first perform an analytic study in the near-horizon asymptotic regime, and demonstrate that a regular black-hole horizon with a non-trivial hair may be always formed, for either sign of Λ and for arbitrary choices of the coupling function between the scalar field and the Gauss-Bonnet term. At the far-away regime, the sign of Λ determines the form of the asymptotic gravitational background leading either to a Schwarzschild-Anti-de Sitter-type background (Λ < 0) or a regular cosmological horizon (Λ > 0), with a non-trivial scalar field in both cases. We demonstrate that families of novel black-hole solutions with scalar hair emerge for Λ < 0, for every choice of the coupling function between the scalar field and the Gauss-Bonnet term, whereas for Λ > 0, no such solutions may be found. In the former case, we perform a comprehensive study of the physical properties of the solutions found such as the temperature, entropy, horizon area and asymptotic behaviour of the scalar field. 1