Dilatonic dyon black hole solutions in the model with two Abelian gauge fields

Dilatonic black hole dyon-like solutions in the gravitational 4d model with a scalar field, two 2-forms, two dilatonic coupling constants λi 0, i = 1, 2, obeying λ1 −λ2 and sign parameter ε = ±1 for scalar field kinetic term are considered. Here ε = −1 corresponds to ghost scalar field. These solutions are defined up to solutions of two master equations for two moduli functions, when λ2i 1/2 for ε = −1. A set of bounds on gravitational mass and scalar charge are presented by using a certain conjecture on parameters of solutions, when 1 + 2λ2i ε > 0, i = 1, 2.


Introduction
Here we give a brief extension of our previous work [1] devoted to dilatonic dyon black hole solutions.We consider a subclass of dilatonic black hole solutions with electric and magnetic charges Q 1 and Q 2 , respectively in 4d model with metric g, scalar field ϕ, two 2-forms F (1) and F (2) , corresponding to two dilatonic coupling constants λ 1 and λ 2 , respectively.For coinciding dilatonic couplings λ 1 = λ 2 = λ we get a trivial non-composite generalisation of dilatonic dyon black hole solutions in the model with one 2-form which was considered in ref. [1].The dilatonic scalar field may be either an ordinary one or a phantom (or ghost) one.
Here we present lower bounds on gravitational mass M and scalar charge Q ϕ .As in ref. [1] this problem is solved here up to a conjecture, which states one to one (smooth) correspondence between the pair (Q , where Q 1 is electric charge and Q 2 is magnetic charge, and the pair of positive parameters (P 1 , P 2 ), which appear in decomposition of moduli functions at large distances.This conjecture is believed to be valid for all λ i 0 in the case of ordinary scalar field and for 0 < λ 2 i < 1/2 for the case of phantom scalar field (in both cases the inequality λ 1 −λ 2 is assumed).

Black hole dyon solutions
Let us consider a model governed by the action where g = g µν (x)dx µ ⊗ dx ν is metric, ϕ is the scalar field, We consider a family of dyonic-like black hole solutions to the field equations corresponding to the action (2.1) which are defined on the manifold and have the following form Here Q 1 and Q 2 are (colored) charges -electric and magnetic, respectively, µ > 0 is the extremality parameter, dΩ 2 2 = dθ 2 +sin 2 θdφ 2 is the canonical metric on the unit sphere S 2 (0 < θ < π, 0 < φ < 2π), τ = sin θdθ ∧ dφ is the standard volume form on S 2 , i = 1, 2, and Functions H s > 0 obey the equations In (2.9) we denote where A 12 is defined in (2.8) and (2.13)These solutions may be obtained just by using general formulae for non-extremal (intersecting) black brane solutions from [2].The composite analogs of the solutions with one 2-form and λ 1 = λ 2 were presented in ref. [1].
The first boundary condition (2.10) guarantees (up to a possible additional demand on analicity of H s (R) in the vicinity of R = 2µ) the existence of (regular) horizon at R = 2µ for the metric (2.3).The second condition (2.11) ensures an asymptotical (for R → +∞) flatness of the metric.

Bounds on mass and scalar charge
For ADM gravitational mass we get from (2.3) where the parameters P s appear in asymptotical relations H s = 1 + P s /R + o(1/R), as R → +∞.The scalar charge just follows from (2.4) Here we outline the following hypothesis, which is supported by certain numerical calculations [1,3].For h 1 = h 2 this conjecture was proposed in ref. [1].

Conclusions
Here a family of non-extremal black hole dyon-like solutions in a 4d gravitational model with a scalar field and two Abelian vector fields is considered.The scalar field is either ordinary (ε = +1) or phantom one (ε = −1).The model contain two dilatonic coupling constants λ s 0, s = 1, 2, obeying The solutions are defined up to two moduli functions H 1 (R) and H 2 (R), which obey two differential equations of second order with boundary conditions imposed.For ε = +1 these equations are integrable for four cases, corresponding to Lie algebras A 1 + A 1 , A 2 , B 2 = C 2 and G 2 .In the first case (A 1 + A 1 ) we have λ 1 λ 2 = 1/2, while in the second one (A 2 ) we get λ 1 = λ 2 = λ and λ 2 = 3/2.
Here we have presented lower bounds on the gravitational mass and upper bounds on the scalar charge for 1 + 2λ 2 s ε > 0, which are based on the conjectur on the parameters of solutions 2 ).For ε = +1 the lower bound on the gravitational mass is in agreement for λ 1 = λ 2 with that obtained earlier by Gibbons et al. by using certain spinor techniques.