Properties of scalar hairy black holes and scalarons with asymmetric potential

Gravitating Scalarons with Inverted Higgs Potential

“…Meanwhile, Figure 4b,c depict the profiles of functions σ(x) and ϕ(x) in the compactified coordinate x, where they are also regular at the origin but decrease monotonically to zero at infinity. On the other hand, we find that the profiles of gravitating scalaron solutions behave quite similar to the hairy black holes and gravitating scalarons, supported by V(ϕ) with asymmetric vacua [36]. In addition, our solutions are completely regular everywhere, but other classes of gravitating scalarons are not; for instance, only the scalar field of ultra-compact gravitating scalarons diverges at the origin in EsGB theory [38,39], but other functions and effective stress-energy tensors are regular everywhere.…”

“…Figure 2 exhibits the ADM mass M of gravitating scalarons versus the scalar field at the origin ϕ c . Analogous to the gravitating scalaron supported by V(ϕ) with asymmetric vacua [36], our gravitating scalarons also bifurcate from the Minkowski space by gaining the mass M when ϕ c increases where M is positive. Hence, we demonstrate that the construction of globally-regular gravitating scalarons with a phantom field can be avoided [28], since the profile of V(ϕ) is more crucial to determining the existence of gravitating scalarons.…”

“…with k 0 = 0 such that ϕ = 0 at the spatial infinity. V(ϕ) could take the asymmetric profile with two asymmetric vacua as shown in Figure 1a when k 3 ̸ = 0, such that V(ϕ) has been utilized to obtain solutions for hairy black holes [35,36] and for the fermionic solitonic star [46]. When the coefficients of terms with odd powers become zero, i.e., k 1 = k 3 = 0, then V(ϕ) can be symmetric with two degenerate minima that look like Higgs-like potentials, 4 such that V(ϕ) has been considered with a phantom field to construct the globally-regular gravitating scalaron with negative ADM mass [28].…”

“…with asymmetric vacua (ϕ = 0 and ϕ = ϕ 1 ) has been utilized in constructing hairy black holes and gravitating scalaron [36]. (b) The inverted Higgs-like V(ϕ), which contains two degenerate global maxima at ϕ max = ± µ/(2Λ), has been considered in our recent work for constructing hairy black holes [32].…”

“…Furthermore, the existence of hairy black holes could be connected smoothly by the existence of the gravitating scalarons in the small horizon limit. For instance, the hairy black hole can be connected smoothly with the gravitating scalaron in the EKG with V(ϕ) containing two asymmetric vacua [35,36]. Analogously, we also want to study the possible connection of the hairy black hole in the small horizon limit from our recent work [32] along with the gravitating scalaron in this paper.…”