Holographic Screening Length in a Hot Plasma of Two Sphere

We study the screening length of a quark-antiquark pair moving in a hot plasma living in two sphere $S^2$ manifold using AdS/CFT correspondence where the background metric is four dimensional Schwarzschild-AdS black hole. The geodesic solution of the string ends at the boundary is given by a stationary motion in the equatorial plane as such the separation length $L$ of quark-antiquark pair is parallel to the angular velocity $\omega$. The screening length and the bound energy are computed numerically using Mathematica. We find that the plots are bounded from below by some functions related to the momentum transfer $P_c$ of the drag force configuration. We compare the result by computing the screening length in the quark-antiquark reference frame where the gravity dual are"Boost-AdS"and Kerr-AdS black holes. Finding relations of the parameters of both black holes, we argue that the relation between mass parameters $M_{Sch}$ of the Schwarzschild-AdS black hole and $M_{Kerr}$ of the Kerr-AdS black hole in high temperature is given by $M_{Kerr}=M_{Sch}(1-a^2l^2)^{3/2}$, where $a$ is the angular momentum parameter.


Introduction
One of the important signatures of Quark Gluon Plasma produced by heavy ion collision's experiment at RHIC and the new LHC is suppression of J/ψ meson production, cc pair. This phenomena is understood qualitatively as the temperature of the plasma reaches above Hagedorn temperature the potential interaction of cc pair would not able to hold them anymore therefore J/ψ meson will dissociate and be screened in Quark Gluon Plasma [1]. The screening potential of cc pair has a dependence on separation length between c andc, L. The length L could have value up to some maximum length L s called screening length where beyond this length the screening potential becomes flat.
In string theory, heavy quarks pair described by Wilson loop is defined as an open string where both ends attached on the probe brane. Evaluating the Wilson loop will tell us information about screening length dependent of quark-antiquark potential. The procedure to evaluate the Wilson loop goes by extremizing the action of corresponding gravity dual theory [2,3,4].
A rigorous calculation on the screening length of a moving quark-antiquark pair at finite temperature four dimensional N = 4 Supersymmetric Yang-Mills theory from AdS/CFT correspondence was pioneered by Liu, Rajagopal, and Wiedemann [5]. It was then generalized to arbitrary dimension for conformal and nonconformal gauge theories by Cáceres, Natsuume, and Okamura [6]. The calculations there were done by going to reference frame of the moving quark-antiquark pair or explicitly by boosting the background metric to direction of quark-antiquark's velocity. A different approach was done in the reference frame of the plasma by Chernicoff, Garcia, and Guijosa [7] and furthermore they also calculated the energy from both approaches. One interesting result of these calculations is the present of boost factor, (1 − v 2 ), in the screening length. In [6], it was shown that the scale of this factor depends on the dimension of black hole backgrounds. However, this scaling factor is valid in ultrarelativistic limit and disagrees with numerical fitting that was found in [7] 1 . For the case considered in this article both scallings coincide.
In this article, we are interested to compute the screening length for the case where the gauge theory lives in a compact two dimensional sphere S 2 or the corresponding background background metric is four dimensional Schwarzschild-AdS Black Hole in Global coordinate. In Poincare coordinate case we can compute the screening length for arbitrary angle θ L between the separation length of quarkantiquark pair and its velocity direction or hot wind plasma direction. Unfortunately the situation is quite restricted for Global coordinate case. Main reason is because the geodesic of external quarks at the boundary must follow the great circle solutions. Since we want to keep the screening length to be fixed, both quark and antiquark must stay on the same great circle plane. In that way, the only possible angle is at θ L = 0 or the separation length or quark-antiquark pair to be parallel with its angular velocity or the hot wind plasma. Furthermore, using SO(3) symmetry we can rotate arbitrary great circle planes to equatorial plane.
In the next section 2, we will compute the screening length in the reference frame of the plasma. We plot numerically the screening length for various fixed angular velocities as a function of momentum conjugate of angular coordinate φ in space direction, π σ φ ≡ P . In section 3, we proceed the computation in the reference frame of the quark-antiquark pair. Unlike the Poincare case we have more than one background metrics; we classify these into "Boost-AdS" and Kerr-AdS Black Holes. A priori we can not compare the results from these different backgrounds since there is no clear relation between some parameters of the background metrics. Using the fact that angular velocity ω and rotation parameter a are bounded, taking the scalar curvature is given by the same parameter l, and assuming the screening length is invariant under different coordinate representations, we are able to find relation between mass parameter M Sch in Schwarzschild AdS Black Hole and mass parameter M Kerr in Kerr-AdS Black Hole which is given by M Kerr = M Sch (1 − a 2 l 2 ) 3/2 . We will plot the screening length as a function of angular velocity for these backgrounds and show that this mass parameters relation is in accordance with physical situation compared to the naive relation, M Sch = M Kerr . In the last section 4, we discuss the results in more detail.

Screening Length in Plasma Reference Frame
In the plasma reference frame the background metric is static and is given by four dimensional Schwarzschild-AdS Black Hole [8] where l is the radius of curvature of AdS space, M Sch is proportional to the mass of the Schwarzschild-AdS Black Hole, T H is the Hawking temperature, and r H is the even horizon given by the most positive real root of h(r), h(r H ) = 0. The Schwarzschild-AdS Black Hole has a minimum temperature and two branches of high temperature. We choose the branch for where σ α is the worldsheet coordinate, with σ α ≡ (τ, σ), X µ (σ α ) is the space-time coordinate where the string worldsheet is embedded, and G µν is the space-time metric (2.1). The equation of motion derived from action (2.4) is simply written as where π α µ is the cannonical worldsheet momentum with g = det g αβ . In deriving the equation of motion we will use the gauge τ = t while for σ will be determined later for convenient. As it was explained in the Introduction we will consider the solution on equatorial plane θ = π/2. Taking the ansatz for a moving quark-antiquark with angular velocity ω, the equations of motion are given by: for r, and for φ, with π σ φ is constant and ≡ ∂ ∂σ . For quark-antiquark system, the solution for r must satisfy conditions where the length of string is L and −L/2 < σ p < L/2 is the turning point where r takes the minimum value r(σ p ) = r p > r H . Since we still have one degree of freedom to be gauged, we can take φ = σ and one can easily check that equations (2.8) and (2.9) are equivalent; it is so because we only have one function r to solve 2 .
We are going to use equation (2.9) to find the separation length, which is now precisely L, of quark-antiquark pair. The solution for r is given by where P ≡ π σ φ is a constant denoting amount of φ-component of momentum flow or transfer on the string. Our convention is that P ≥ 0 and positive sign in (2.11) corresponds to the momentum transfer from boundary to r p and negative sign corresponds to the momentum transfer from r p to boundary. The quark-antiquark pair can be built of two configurations (a) and (b) in Figure 1. It is necessary to take these two configurations in order for string to join up at r = r p . Note that since φ coordinate is periodic, 0 ≤ φ < 2π, there are two possible configurations of string for quark-antiquark pair with the separation length 0 ≤ L ≤ π/2 and π/2 < L < 2π. For our purpose, it is natural to take string configuration with the shortest separation length.
Using symmetry, φ → −φ, it is enough just to find solution of 2.11 with positive sign which corresponds to configuration (b) in Figure 1; this sets φ p ≡ σ p = 0. At σ p , we must have r 4 p h(r p ) − P 2 = 0 and r 4 p h(r p ) − ω 2 r 4 p > 0 which implies that h(r p ) > 0 or r p > r H for P = 0. There is a critical radius r c defined as r 4 c h(r c ) = ω 2 r 4 c and it's also r c > r H for ω = 0. If r p = r c then the condition (2.10) could not be satisfied and the string configuration would just be a single quark at the boundary where one end of string at the boundary and the other end will be at the horizon. Therefore for quark-antiquark pair we must have r p > r c or P 2 > ω 2 r 4 c . Notice that this r c equals to r Sch appears in the drag force computation [12,13]. We also interpret the constant π σ φ in [12,13] equals to P because both are given by the same formula and a constant should be invariant under any gauge choice. So for some fixed angular velocity ω, or r c , the string configuration is characterized by amount of momentum transfer P to horizon with P 2 ≥ ω 2 r 4 c . At P = ωr 2 c , the string tends to represent a single quark and if we increase the momentum transfer P the quark-antiquark pair is formed. Figure 2 shows schematic pictures of string configuration describing quark-antiquark pair and drag force of a single quark.  Figure 2: Picture on the left shows at critical momentum transfer P 2 = ω 2 r 4 c the string describe a single quark while picture on the right is if we increase P such that P 2 > ω 2 r 4 c .
From equation (3.8), we obtain an integral formula for computing separation length as below (2.12) The integral above can not be solved analytically. So we plots numerically the separation length for various value of ω/l in Figure 3. As one can see, we produce similar behaviour of screening length L s as in Poincare case. Our plots do not start from P l = 0 which is different from the plots produced in [5], for θ L = π/2, since P is bounded from below P > ωr 2 c .

Screening Length in Quark-Antiquark Reference Frame
A different way to compute screening length is by going to the reference frame of the moving quark-antiquark pair. In this case the metric will be seen as it is rotating at angular velocity proportional to angular velocity of the quark-antiquark pair. A usual procedure, as in Poincare case, of how to find the rotating metric is by boosting the static metric (2.1), at θ = π/2, along φ-direction.

"Boost-AdS" Black Hole
In order to get the rotating metric at equatorial, we boost the static metric (2.1) with the following transformation is the boost factor, and the resulting metric is Taking the same gauge, t = τ and φ = σ, the static solution for radial coordinates, r ≡ r(σ), is given by where P is again the φ-component of momentum transfer as in previous static black hole case. One can immediately see from (3.3) that separation length will be different with the static case by a boost factor γ and hence the screening length as well, L Boost = γL Sch .

Kerr-AdS Black Hole in Boyer-Linquist coordinate
The "Boost-AdS" metric in (3.2) is only valid at equatorial plane. In general there exist a solution for the full geometry namely Kerr-AdS Black Holes. There are many coordinates representation for Kerr-AdS [14]. The simplest one is using Boyer-Linquist coordinate. We assume the equatorial plane in Kerr-AdS is similar to equatorial plane in Schwarzschild-AdS. The four dimensional Kerr-AdS Black Hole metric in Boyer-Linquist coordinates at equatorial is given by [15,16] Using the gauge, t = τ and φ = σ, the radial solution is now given by The momentum transfer P here, although a constant, has a different formula with the one in Schwarzschild-AdS or "Boost-AdS" case and this could give us a different separation length function of P compare to the previous one. However since we only want to compute the screening length, maximum of separation length which is an invariant scalar under coordinate transformations, it would be natural to expect the same result as in the "Boost-AdS" case.

Kerr-AdS Black Hole in Asymptotically AdS coordinate
Another representation of Kerr-AdS metric is using Asymptotically AdS (AAdS) coordinate. Unlike the previous two rotating black hole metrics, the Kerr-AdS metric in AAdS coordinate is AdS asymptoticallly, Y → ∞, and hence is preferred by AdS/CFT correspondence prescription [17,18]. In general, to get the full geometry of AAdS coordinate is quite involved, yet it is still possible for particular case such as at equatorial. The four dimensional Kerr-AdS metric in AAdS coordinates at equatorial can be obtained by the following coordinate transformation from metric (3.4) [15,16]: The resulting metric is Using similar gauge as before, the radial solution, Y ≡ Y (σ), is given by (3.8) Now, we are going to compute numerically and compare the screening length for all metrics (3.2), (3.4), and (3.7) as functions of angular velocity ω or a. To compare the results we need to find relations between all the black hole parameters. One immediately notice that all these metrics have the same curvature radius of AdS space which is l. Angular velocity ω in (3.2) is bounded at boundary by l 2 ≥ ω 2 while a is also bounded by a 2 l 2 ≤ 1 and so we are tempted to take ω 2 = a 2 l 4 . Remaining parameter that needs to be related is the mass parameter. There is no clear relation between mass parameter of Schwarzschild-AdS Black Hole, M Sch , and mass parameter of Kerr-AdS Black Hole, M Kerr . One could naively take M Sch = M Kerr and try to see if this is in accordance with our physical picture. We would expect the screening length calculated in the reference frame of quark-antiquark pair should be different by a factor of compared to the screening length calculated in the reference frame of hot plasma. Furthermore, all screening length in reference frame of quark-antiquark pair should be given by the same plot.
As we can see from Figure 4, the screening length of Boyer-Linquist and AAdS Black Holes are much closer to the screening length of Schwarzschild-AdS Black Hole rather than to the screening length of "Boost-AdS" Black Hole. Hence its is unlikely that M Sch equals to M Kerr and so we need to find how they are related exactly. Suppose that Y p , which is solution to ∆ Y (Y p ) = P 2 Ξ 2 , is very large such that with ω 2 = a 2 l 4 . Comparing both equations (3.10) and (3.11), we can extract a relation The conditions for large r p and Y p can be satisfied if we take a limit of large M Sch and M Kerr . Recall that this is also the same limit if one would like to go from Global coordinate to Poincare coordinate of AdS [8]. Using this relation we obtain the numerical plots in Figure 5 where now the screening length of Kerr-AdS Black Holes closer to the screening length of "Boost-AdS" Black Hole which is in accordance with the physical picture we expected before.

Discussion
In section 2, we have computed the separation length as a function of momentum transfer P and plotted it for various angular velocity ω. In the plots of Figure 3, there is minumum value of separation length for low momentum transfer P . This is because P is bounded from below where its value has to be bigger than the drag force momentum transfer, P = ωr 2 c . Below this value there is no physical solution for string configuration of quark-antiquark pair. Therefore right after the momentum transfer P passes this bound the quar-antiquark pair is formed at the boundary with non-zero separation length but we suspect that this quark-antiquark pair is unstable. The reason is because the tip of the string in this configuration, located at r p , is very close to r c ; we identify r c as a "shifted" horizon where at ω = 0, r c = r H . However, there is another configuration of string, which is much more stable then the previous one, with the same separation length at high momentum transfer P where the tip of string, r p , is far away from r c . We then speculate that momentum transfer P in the string configuration of quark-antiquark pair is related to the strength of potential interaction between quark and anti-quark. It is interesting to study in more detail how exactly the relation is.
We also computed the screening length in the reference frame of quark-antiquark pair. Although there are more than one background metrics, we argued that the resulting screening length should not be different significantly. To compare the result for these backgrounds, we found a relation between parameters of this black hole backgrounds. At the end, we found that M Kerr = M Sch (1 − a 2 ) 3/2 is more suitable for our physical picture at large mass limit.
In the literature, we have not found so far the relation between mass parameter of Schwarzschild-AdS Black Hole and mass parameter of Kerr-AdS Black Hole. A general procedure for constructing the metric of Kerr-AdS Black Holes in [14] was built of AdS metric plus the mass term, scaled with mass parameter of Kerr-AdS, in the Kerr-Schild form. Thus it has no direct connection with the mass parameter of Schwarzschild-AdS Black Hole. It would be interesting to find a procedure for constructing Kerr-AdS Black Hole from Schwarzschild-AdS Black Hole using the mass relation obtained in this article.