Spherically symmetric Yang-Mills solutions in a 5-dimensional (Anti-) de Sitter space-time

We consider an Einstein-Yang-Mills Lagrangian in a five dimensional space-time including a cosmological constant. Assuming all fields to be independent of the extra coordinate, a dimensional reduction leads to an effective (3+1)-dimensional Einstein-Yang-Mills-Higgs-dilaton model where the cosmological constant induces a Liouville potential in the dilaton field. We construct spherically symmetric solutions analytically in specific limits and study the generic solutions for vanishing dilaton coupling numerically. We find that in this latter case the solutions bifurcate with the branch of (Anti-) de Sitter-Reissner-Nordstrom ((A)dSRN) solutions.


Introduction
The scalar dilaton field arised as companion of the metric tensor in (super)string theories and is associated with the scale invariance of these theories [1]. Thus it is interesting to study classical field theory solutions coupled to a dilaton. In most studies, the dilaton was assumed to be massless while, however, from the viewpoint of a realistic theory the dilaton should be massive in order to avoid long-range scalar forces. In [2] a dilaton potenial of Liouville type was introduced to take into account the effects of a specific symmetry-breaking mechanism which gives mass to the dilaton. This type of potential has a constant prefactor which in the limit of vanishing dilaton coupling reduces to a cosmological constant. It was found that there exist no asymptotically flat/ de Sitter/ Anti-de Sitter solutions for non-vanishing potential [3]. Rotating generalisations of the black hole solutions found in [2] have been constructed in [4].
Volkov argued recently [5] that if ∂ ∂x 4 is a symmetry of the Einstein-Yang-Mills (EYM) system in (4 + 1) dimensions, where x 4 is the coordinate associated with the 5th dimensions, than the (4 + 1)-dimensional EYM system reduces effectively to a (3 + 1)-dimensional EYMHD system with a specific coupling between the dilaton field and the Higgs field. The generalisation of this (3 + 1)-dimensional EYMHD model was consequently studied in [6].
In this paper, we study spherically symmetric solutions of the (3 + 1)-dimensional EYMHD model deduced from the (4+1)-dimensional EYM system including a cosmological constant. The dimensional reduction then leads to a Liouville-type potential in the (3 + 1)-dimensional model. In Section 2, we present both the five-dimensional model and the from this deduced and then generalised (3 + 1)-dimensional EYMHD model. In Section 3, we discuss the solutions for the case of vanishing dilaton coupling, especially, we present our numerical results for the generic solutions in this case. In section 4, we discuss possible solutions for the generic case of non-vanishing dilaton coupling. Our conclusions are presented in Section 5.

The model
We start with the Einstein-Yang-Mills Lagrangian in five dimensions including a cosmological constant given by: the gauge index a = 1, 2, 3 and the space-time index M = 0, 1, 2, 3, 4. G 5 andẽ denote respectively the 5-dimensional Newton's constant and the coupling constant of the gauge field theory. G 5 is related to the 5-dimensional Planck scale M P l(5) by G 5 = M −3 P l (5) . Λ (5) is the 5-dimensional cosmological constant.
If both the matter functions and the metric functions are independent on x 4 , the 5-dimensional fields can be parametrized as follows [5]: (4) µν dx µ dx ν + e 2ζ (dx 4 ) 2 , µ, ν = 0, 1, 2, 3 and where g (4) is the 4-dimensional metric tensor and ζ plays the role of the dilaton. In [5] it was shown that for Λ (5) = 0 the classical equations are equivalent to those of a four-dimensional Einstein-Yang-Mills-Higgs dilaton theory. In this paper, we consider the case with a cosmological constant. We then choose the generalised (3 + 1)-dimensional action to be: with the gravity Lagrangian: and G denoting the 4-dimensional Newton's constant. The matter Lagrangian L M reads: with the Higgs potential the non-abelian field strength tensor and the covariant derivative of the Higgs field in the adjoint representation The gauge field coupling constant is denoted e, λ is the Higgs field coupling constant and v the vacuum expectation value of the Higgs field. Note that we have introduced a coupling κ specific to the dilaton field by setting ζ = 2κΨ. This will allow to study the influence of the dilaton systematically. We remark that the 5-dimensional cosmological constant has through dimensional reduction led to a Liouville potential in the dilaton field with coupling constantΛ. For κ = 0,Λ is proportional to the four-dimensional cosmological constant.

The Ansatz
For the metric the spherically symmetric Ansatz in Schwarzschild-like coordinates reads [7,8]: In these coordinates, m(∞) denotes the (dimensionful) mass of the field configuration.
For the gauge and Higgs fields, we use the purely magnetic hedgehog ansatz [11] A r a = A t a = 0 , The dilaton is a scalar field depending only on r Inserting the Ansatz into the Lagrangian and varying with respect to the matter fields yields the Euler-Lagrange equations, while variation with respect to the metric yields the Einstein equations.

Classical field equations
With the introduction of dimensionless coordinates and fields the Lagrangian and the resulting set of differential equations depend on the following coupling constants: where With the rescalings (16) and (17), the dimensionless mass of the solution is given by µ(∞) α 2 . Note that we have rescaled the cosmological constant in order to obtain the equations of a conventional (3+1)dimensional Einstein-Yang-Mills-Higgs model including a cosmological constant in the limit of vanishing dilaton coupling.
With (16) and (17) the Euler-Lagrange equations read: where the prime denotes the derivative with respect to x, while we use the following combination of the Einstein equations to obtain two differential equations for the two metric functions: Note that the equations of the original five dimensional theory are recovered by using the following specific choice of the coupling constants: The case Λ = 0 was previously studied in [5,6]. If in addition γ = 0, the equations of the Einstein-Yang-Mills-Higgs equations are recovered [7,8]. Choosing Λ = α = 0 (assuming Λ/α 2 = 0 as well), the model reduces to the Yang-Mills-Higgs-dilaton system studied in [9].
3 Spherically symmetric solutions for γ = 0 We will first discuss the solutions in the case γ = 0. The equation of the dilaton field can then be decoupled and ψ(x) ≡ 0. We will study solutions of this system which are regular at the origin this implies the following conditions Finiteness of the ADM mass requires that the fields approach particular values asymptotically, namely: For Λ > 0 the metric function N(x) has a zero at a finite value of x, say x = x c . This is the so-called "cosmological horizon". The value x c depends on the actual values of the coupling constants.

(Anti-) De Sitter-Reissner-Nordström ((A)dSRN) solutions
Setting γ = 0 the system admits embedded abelian solutions, the so-called (Anti-) de Sitter-Reissner-Nordström solutions: The metric function N(x) has a physical singularity at the origin x = 0 which is evident from the Kretschmann scalar K = R αβγδ R αβγδ : This is solved by: and For Λ > 0, the solution with the plus sign is the outer, cosmological horizon x c , while the inner, event horizon x h is the solution with the minus sign. Obviously, the appearance of horizons in dS space is restricted by α 2 ≤ 1 2Λ . The corresponding mass of the extremal solution is given by: Apparently, the Λ = 0 limit is ill-defined. However, for 0 < Λ ≪ 1 we find which for Λ → 0 obviously leads to the corresponding values of the well-known asymptotically flat Reissner-Nordström solution.

de Sitter (dS) gravitating monopoles
Since gravitating monopoles in Anti-de Sitter space have been studied previously [10], we concentrate here on monopoles in de Sitter space. To our knowledge, these type of solutions have not been studied previously.
In the absence of a cosmological constant, the flat space magnetic monopole [11] is deformed by gravity and exist up to a critical value of α = α cr where the solution bifurcates with the branch of extremal Reissner-Nordström solutions [7]. For instance in the BPS limit (β = 0) the gravitating monopole bifurcates with this branch at α cr ≈ 1.386. Now analysing the equations in the presence of a cosmological constant, we were able to construct dS-gravitating monopoles. They are characterised by a cosmological horizon at x = x c with N(x = x c ) = 0. The behaviour of the function N(x) is illustrated in Fig. 1 for α = 0.8 and different values of Λ. We find that x c is decreasing with the increase of Λ: x c ∼ 108 for Λ ∼ 0.0005 and x c ∼ 77 for Λ = 0.001. As is obvious from the figure, the solutions have a local minimum at some value of the radial coordinate x = x min (Λ).
The main aim of this study was to determine the domain of coupling constants in which dS-gravitating monopoles exist. Fixing β and Λ our analysis demonstrates that dS-gravitating monopoles bifurcate with the branch of extremal dSRN solutions described in the previous section at a critical value of α. Since we limited our analysis to small values of Λ the critical value of α where the bifurcation occurs hardly differs from the corresponding one in the asymptotically flat case.
The way how the extremal dSRN solution is approached is illustrated in Fig. 2 for Λ = 0.001 and β = 0.1. This clearly shows that the value of the local minimum of the function N(x) decreases while α increases. We find that solutions exist up to a maximal value of the gravitational coupling α = α max ≈ 1.382. There another branch of non abelian solutions exist which bifurcates with the branch of dSRN solutions at a critical value of α = α cr ≈ 1.378. At this point, a degenerate horizon forms at x = x h . The critical solution can be described by the dS-RN solution with horizons (31) for x ≥ x h , while for x h > x ≥ 0, it is non-singular and non-trivial. Compared to the case Λ = 0 [7], the values of α max and α cr are smaller when Λ > 0. Moreover, the interval of α on which two solutions exist decreases. This can be related to the increased cosmological expansion for Λ > 0.

Spherically symmetric solutions for γ = 0
In the case of Einstein-Maxwell-dilaton theory, the Liouville potential leads to the fact that the solutions are neither asymptotically flat nor de Sitter nor Anti-de Sitter [2]. As far as our numerical simulations suggest, this holds also true for the case of non-abelian gauge fields, since we were not able to construct asymptotically flat/ de Sitter/ Anti-de Sitter solutions. However, in a specific limit, namely the embedded abelian case, analytic solutions are available.

The case H(x) ≡ 1, K(x) ≡ 0
Setting H ≡ 1 and K ≡ 0 for all x, we find the following solutions of the system of equations: and The cosmological constant is given by: This solution has a single event horizon for n 1 > 0. Moreover, it can be seen, that this solution is ill-defined for α = γ. Note that these are generalisations of the solutions constructed in [2]. For α = 1, the above solution corresponds to one of the solutions found in [2]. In Fig.3, we show qualitative profiles of the functions for the choice of parameters which corresponds to the 5-dimensional limit (25). In addition, we choose γ = ψ 0 = a 0 = n 1 = 1. It is obvious from this figure that the solution has a horizon (here at x = x h ≈ 0.655) and thus represents a black hole. If we choose instead the limit α = 0, the function A(x) becomes constant = a 0 . The metric function N(x) = 1 − n 1 x −1 in this limit.

Conclusions
In a previous paper [5], it was shown that a Einstein-Yang-Mills model in 5 dimensions can be reduced to an effective (3 + 1)-dimensional Einstein-Yang-Mills-Higgs-dilaton model under certain symmetry conditions -spherical symmetry and independence on the coordinate associated with the 5th dimension. One of the main results of the present paper shows that the reduction of a 5 dimensional de Sitter (dS)/Anti-de Sitter (AdS) Einstein-Yang-Mills system to an effective (3 + 1)-dimensional action (with the same symmetry assumptions as in [5]) leads to a self-interaction of the dilaton field via a Liouville potential.
Previous considerations of an Einstein-Maxwell-dilaton model including a Liouville potential [2] have revealed that no asymptotically flat/ de Sitter/ Anti-de Sitter solutions can be constructed [4]. All our attempts to construct numerically solutions of the non-abelian counterpart have failed. Thus, we believe that the absence of asymptotically flat/ de Sitter/ Anti-de Sitter solutions holds also true in the case of non-abelian gauge fields. However, considering the limit of vanishing dilaton coupling, we were able to recover the AdS gravitating monopoles studied previously [10] and to produce previously not studied solutions, namely the dS gravitating monopoles.
We show for the latter solutions that they bifurcate with the branch of dS-Reissner-Nordström (dSRN) solutions at a critical value of the gravitational coupling. Finally, considering the limit K(x) ≡ 0 and H(x) ≡ 1 for non-vanishing Liouville potential, we were able to construct generalisations of the solutions found in [2].