The Cornell black hole

The Cornell potential can be derived from a recently proposed non-local extension of Abelian electrodynamics. Non-locality can be alternatively described by an extended charge distributions in Maxwell electrodynamics. We state that in these models the energy momentum tensor necessarily requires the presence of the interaction term between the field and the charge itself. We show that this extended form of energy momentum tensor leads to an exact solution of the Einstein equations describing a charged AdS black hole. We refer to it as the"Cornell black hole"(CBH). Identifying the effective cosmological constant with the pressure of Van der Waals fluid, we study the gas-liquid phase transition and determine the critical parameters.


Introduction
The Cornell potential is a phenomenological potential exhibiting long distance confinement.It was originally introduced to reproduce charmonium spectrum [1].To obtain such a potential out of a Yang-Mills gauge theory is still challenging since perturbation approach is not applicable in the strong coupling regime.On the other hand, it has long been acknowledged that confinement is, in fact, an " Abelian " long distance phenomenon [2].However, obtaining an effective abelian approximation of QCD turns out to be a lengthy and non-trivial task [3,4,5].It has been recently noticed that, for a special class of null SU(N) gauge potentials A a µ = c a φ(x)l µ , where l µ l µ = 0 and c a is constant vector in color space, Yang-mills field equations reduce to a system of decoupled Maxwell equations.This particular choice for the gauge field allows to establish an intriguing duality between Yang-Mills theory and gravity ( see [6] for a general review of this topic.).As an outcome, the dual of the Coulomb potential turns out to be a Schwarzschild black hole.Following this line of reasoning, we recently described the Cornell potential as a static solution of the field equations of a non-local Abelian gauge theory [7,8].We shall refer to this Lagrangian model as Cornell electrodynamics(CE).The null Yang-Mills fields, quoted above, are the electromagnetic analogue of the Kerr-Schild decomposition of the metric tensor given by where, the null condition l µ l µ = 0 holds both with respect to Minkowski and the complete metric tensor.This analogy is referred to as " Kerr-Schild double copy " [9,10,11,12,13].
The advantage of Kerr-Schild metric is to reduce Einstein field equations to a single Poisson equation for the unknown scalar function φ ( x ): The physical meaning of φ ( x ) as the relativistic gravitational potential becomes clear once the metric is written in the spherical gauge: In summary, once T µν is chosen, finding the corresponding curved metric is reduced to solving equation (2) in Minkowski space.When the electromagnetic part of energy-momentum tensor T µν is taken into account in the Einstein equation (2) the result is proportional to ∇ 2 ( A 0 ) 2 , where A 0 is the time-like component of the gauge potential.Therefore, we conclude that the electromagnetic contribution to the metric tensor is of the form A 2 0 .For example, in the case of the Einstein equations coupled to a static, point-like, Coulomb potential, one finds the usual Reissner-Nordström form of metric: where q is the electric charge.φ m is the contribution from the mass, i.e. the Schwarzild term.
Our purpose in this Letter is to solve Einstein equations coupled to Cornell electrodynamics and obtain the Kerr-Schild double copy of the confinement potential.We expect the metric to keep the form of ( 4), but with the Coulomb potential replaced by the Cornell one.
The paper is organized as follows.In Sect.(2) we discuss the equivalence between non-local Cornell electrodynamics and Maxwell gauge theory with an extended source.The energy momentum tensor, including the interaction energy between the field and the extended charge, is described.Then, we solve the Einstein equations and find the Kerr-Schild double copy of the Cornell potential, i.e. the Cornell black hole.In Sect.(3) the thermodynamic properties of the CBH are described.Identifying the effective cosmological constant with the pressure of Van der Waals fluid we study the gas-liquid phase transition and determine the critical parameters in terms of the mass and charge in the Cornell effective Lagrangian.In Sect.(4) we summarize the main results obtained.Finally, in the Appendix we present the computation of the energy momentum tensor part coming from the interaction term J µ A µ .

Cornell effective Lagrangian.
The phenomenological potential between a quark anti-quark pair is describes by the Cornell formula where α s ( µ 2 ) is the strong running coupling constant; σ is the " string tension " between a quark anti-quark pair and the renormalization scale µ is chosen to be with m Q and m Q the masses of quark anti-quark, respectively.Finally, n f is the number of quark flavors and Λ QCD ≃ 0.15GeV is the QCD energy scale.
In a recent paper we introduced a novel way to generate a confining linear potential in the framework of an Abelian gauge theory.Since the gauge potential ( 5) is a sum of the Coulomb and a linear term one has to modify Maxwell electrodynamics by adding an inverse Lee-Wick term [7].The result is the non-local Lagrangian g is the usual gauge coupling constant.Variation of ( 8) with respect A µ gives the field equations (9) are a non-local version of Maxwell equations.
QCD By introducing the following definitions one reproduces the Cornell potential (10).
The solution (10) can be also recovered from the equivalent Lagrangian The new version of the model turns out to be ordinary Maxwell electrodynamics in the presence of a non-local current J µ .Thus, the original point-like charge is replaced by a non-local charge density.As already mentioned in the introduction, in the case of distributed charges the interaction energy between the field and its source cannot neglected.It must be included in the Lagrangian.Following the procedure discussed in the Appendix, one finds the following energy-momentum tensor: Notice that T µν is no more traceless due to the presence of an extended source In the electrostatic case It is appropriate to recall that in classical electrodynamics E 2 /2 is the field " kinetic term " while eJ 0 A 0 /2 represents the " interaction " energy between the field and the source.The final step is to rewrite (16) in a convenient way: Thus, the Poisson equation ( 2) has a general solution of the form φ 0 = c 0 +c 1 /r is the solution of the homogeneous equation ∇ 2 φ 0 = 0.The two integration constants are determined by the specific type of physical problem under consideration.One assumes that the source, apart from carrying charge, has also mass.Thus, c 1 is chosen to be c 1 = −mG N to reproduce the Schwarzschild gravitational potential.The remaining constant, c 0 , is fixed by the boundary conditions.We are going to determine c 0 in a while.Inserting the solution (10) in (18) we find Now, we shall determine constant c 0 in such a way to obtain an AdS spacetime at large distance5 The final form of the metric is Equation ( 20) describes an electrically charged Anti-deSitter geometry.We can establish the correspondence between gauge and gravitational parameters as follows: We remark that, in our approach, the result is neither a conjectured duality between Yang-Mills theory and higher dimensional gravity [14,15,16], nor an " ad hoc " identification between gauge and gravitational coupling constants as it is done in the double copy framework.Rather than a conjectured duality, we obtained an exact connection between a confining gauge model and 4D Einstein gravity.Let us notice that the identification ( 23) Λ ∝ α s ( µ ) leads to a " running cosmological constant " rather than a fixed one.This property makes it reasonable to identify Λ as the pressure in the thermodynamical description of QCD .For γ > 2.1 there two horizons.They merge into an extremal black hole for γ = 2.1 (dashed curve).Finally, if the mass is too small, i.e. γ < 2.1 there are no horizons.
a CBH.The plot of f (r) for various m is shown in Fig. (1).QCD .The minimum of the curves correspond to extremal configuration where the two horizons merge.To investigate the existence of horizons we consider the equation f (r H ) = 0 and express mass as a function of horizon r H For parameter β ≡ G s κ 2 > 0 there are two horizons which merge into a degenerate one in the case of an extremal CBH.Let us note that as the cosmological constant is g-dependent.Thus, the g → 0 limit of f (r) is not a neutral AdS metric but a simpler Schwarzschild geometry.

Thermodynamics analysis of CBH
The thermodynamical description of AdS black holes has been given due attention in several papers, e.g.[17,18,19,20].In the original work by Hawking and Page [21], a phase transition between a gas of particles and a Schwarzschild-AdS black hole was introduced.The main difficulty in this approach is the proper identification of ( thermodynamical ) canonical variables.While fluid temperature can be naturally related to the Hawking temperature, the identification of other quantities such as pressure, volume, enthalpy, etc., is not so straightforward [22,23,24,25,26,27].As an illustration of this difficulty, consider the cosmological constant.By definition is a " constant ".However, once it is identified with the fluid pressure, it becomes a function of the temperature and the volume [28,29] .This relation is provided by the der Waals equation.
To give a thermodynamical description of the CBH, we start from the black hole temperature equation [28] and define the pressure and the specific volume as follows: v ≡ 2G N r H , " specific volume " .In this way, Eq. ( 25) turns into a Van der Waals equation ( κ B ≡ 1 ): The phase transition occurs at the critical temperature T = T c , where the isotherm shows an inflexion point: The values of the critical parameters are: Once the CBH is reformulated as a Van der Waals fluid, then the critical parameters satisfy the following relations There is a characteristic value, T 0 , of the temperature where: For any temperature T < T 0 there is unphysical region where the isotherms show a negative pressure over some interval.To avoid this problem, isotherms must be replaced by isobars determined by the Maxwell' s area law [30,31].and the action is where the Dirac delta is defined by the normalization condition In the case of non-point like charges the Dirac delta is replaced by a charge distribution ρ(x) normalized by the same condition (42).The current density becomes It follows that the variation of the action is now given by and the contribution of the interaction term to the energy-momentum tensor is In (45) world indices are symmetrized.

2 Figure 1 :
Figure 1: Plot of the metric function for various values of the mass m at fixed g and Λ 2QCD .For γ > 2.1 there two horizons.They merge into an extremal black hole for γ = 2.1 (dashed curve).Finally, if the mass is too small, i.e. γ < 2.1 there are no horizons.

Figure 2 :
Figure 2: Plot of the horizon equation for various values of β = G s Λ 2QCD .The minimum of the curves correspond to extremal configuration where the two horizons merge.

Figure 3 :
Figure 3: Plot of the horizon equation for β ≡ G s Λ 2 QCD = 0.4.The minimum of the curve corresponds to an extremal BH.For y > y extr there are two horizons; for y < y extr there are no horizons.

Figure 4 :
Figure 4: Plot of the horizon temperature as a function of the horizon radius r H .The curves are obtained for different values of the parameter β ≡ G s Λ.The dotted curve represent the limiting case of a neutral Schwarzschild AdS black hole.

Figure 5 :
Figure 5: PV diagram for G N /2π = 4, and 2G s G 2N /2π = 1.The region T > T c corresponds to an " ideal gas ".The dotted line is critical isotherm T = T c .The oscillating part of the isotherms for T < T c must be replaced by isobars according with the Maxwell's area law.For T < T 0 isotherms can show negative pressure intervals.This part of the diagram is unphysical and is eliminated by applying Maxwell's construction.