Application of AdS/CFT in Quark-Gluon Plasma

Abstract

We review some important applications of AdS/CFT correspondence to gain insight into properties of the quark-gluon plasma. We study some important quantities such as drag force, screening length, and jet-quenching parameter of an external probe quark and also quark-antiquark configuration. In particular, we focus on the STU background and compare our results with other important backgrounds.

1. Introduction

The relation between gauge theories and string theory has been the subject of many important studies in the last three decades. First, Maldacena [1] proposed the AdS/CFT correspondence therefore the AdS/CFT correspondence, sometimes called Maldacena duality. According to this conjecture there is a relation between a conformal field theory (CFT) in 𝑑-dimensional space and a supergravity theory in (𝑑+1)-dimensional anti-de Sitter (AdS) space. Maldacena suggests that a quantum string in (𝑑+1)-dimensional AdS space, mathematically, is equivalent to the ordinary quantum field theory with conformal invariance in 𝑑-dimensional space-time which lives on the boundary of AdS_𝑑+1 space. The preliminary formulation of Maldacena is developed and completed by independent works of Witten [2] and Gubser et al. [3]. The famous example of AdS/CFT correspondence is the relation between type IIB string theory in AdS₅×𝑆⁵ space and 𝒩=4 super Yang-Mills gauge theory on the 4-dimensional boundary of AdS₅ space. For more studying about the AdS/CFT correspondence and its applications see [4–9]. One of the most interesting applications of the AdS/CFT correspondence is to study the quark-gluon plasma (QGP). The study of the QGP is a testing ground for finite temperature field theory. Such studies are important to understand the early evolution of our universe. Already, there are many attempts to study QCD by using gauge/gravity duality, where the 𝒩=4 super Yang-Mills (SYM) plasma is considered. The most important quantities of QGP are the shear viscosity, drag force, and jet-quenching parameter. The shear viscosity is one of the important hydrodynamical quantities of QGP which relates to the important thermodynamical quantity so-called entropy; in particular it is found that the ratio of shear viscosity 𝜂 to the entropy density 𝑠 had a universal value: 𝜂/𝑠=1/4𝜋 [10–39]. However, for the several cases, this value may be modified [28–39].

The calculation of energy loss of moving heavy charged particle through a thermal medium known as the drag force [40–45]. The moving heavy quark in context of QCD has dual picture in the string theory where an open string was attached to the 𝐷-brane and stretched to the horizon of the black hole. Already the issue of the drag force was considered in the 𝒩=4 super Yang-Mills thermal plasma with several interesting backgrounds [40–50].

Another important property of the QGP is called the jet-quenching parameter (𝑞). The knowledge about this parameter increases our understanding about the QGP. In that case the jet-quenching parameter was obtained by calculating the expectation value of a closed light-like Wilson loop and using the dipole approximation. In order to calculate this parameter in QCD one needs to use perturbation theory. But, by using AdS/CFT correspondence, the jet-quenching parameter was calculated in nonperturbative quantum field theory. This calculations were already performed in the several interesting backgrounds [51–58]. In this paper we shall investigate some important properties of the QGP, specially in the STU background. The STU model is a special example of 𝐷=5, 𝒩=2 gauged supergravity theory which is dual to the 𝒩=4 SYM theory with finite chemical potential. Also the solutions of 𝒩=2 supergravity may be solutions of supergravity theory with more supersymmetry. Already the duality between gravity and 𝒩=2 gauged theory was investigated and found that 𝒩=2 supergravity is an ideal laboratory [59–62]. Therefore, we may consider the STU model as a gravity dual of a strongly coupled plasma. Indeed, we review some of the previous results and also add some new things and collect all of them in this review article.

2. General Solutions

One can consider the following general line element [45]: 𝑑𝑠²=𝑔_𝑡𝑡𝑑𝑡²+𝑔_𝑟𝑟𝑑𝑟²+𝑔_𝑥𝑥𝛿_𝑖𝑗𝑑𝑥^𝑖𝑑𝑥^𝑗, (1) where 𝑖=1,2,…,𝑑−1. The geometry of this general space-time is asymptotically AdS_𝑑+1. Also, it is assumed to have a horizon at 𝑟=𝑟_ℎ, and 𝑔_𝑥𝑥 is finite at horizon. Another important assumption is that the metric components depend only on the radial coordinate. The Hawking temperature of this black hole space-time, which is dual to the temperature of the field theory, is given by 𝑇=−𝑔_𝑡𝑡^′𝑔^𝑟𝑟′4𝜋_𝑟=𝑟ℎ, (2) where prime denotes derivative with respect to the radius 𝑟. The entropy density of the field theory is given by 𝑔𝑠=^{𝑥𝑥(𝑑−1)/2}4𝐺_{𝑁𝑟=𝑟ℎ}, (3) where 𝐺_𝑁 is gravitational Newton constant.

The open string is described by the following Nambu-Goto action: 𝑆=−𝑇₀𝑑𝜏𝑑𝜎√−𝐺, (4) where 𝑇₀ is the string tension. The coordinates 𝜏 and 𝜎 are corresponding to the string world-sheet. Also, 𝐺 is determinant of the world-sheet metric 𝐺_𝑎𝑏, which is obtained as the following: −𝐺=𝑋̇⋅𝑋^′2−𝑋^′2𝑋̇²=−𝑔_𝑟𝑟𝑔_𝑡𝑡−𝑔_𝑥𝑥𝑔_𝑡𝑡𝑥^′2−𝑔_𝑟𝑟𝑔_𝑥𝑥(𝑥̇)², (5) where it is assumed that the string only extends in one direction 𝑥(𝜏,𝜎) and therefore string profile described by the 𝑋(𝑡,𝑟,𝑥). Hence, the equation of motion reads 𝜕_𝑟−𝑔_𝑥𝑥𝑔_𝑡𝑡𝑥^′√−𝐺−𝑔_𝑟𝑟𝑔_𝑥𝑥𝜕_𝑡𝑥̇√−𝐺=0, (6) which yields to the following canonical momentum densities associated to the string 𝜋⁰_𝜇=−𝑇₀𝑔_𝜇𝜈𝑋̇𝑋^′𝑋^𝜈′−𝑋^′2𝑋̇^𝜈√,𝜋−𝐺_1𝜇=−𝑇₀𝑔_𝜇𝜈𝑋̇𝑋^′𝑋̇^𝜈−𝑋̇²𝑋^𝜈′√.−𝐺 (7) These components are used to obtain the total energy and momentum of string by using the following integrals: 𝐸=−^𝑟𝑚_𝑟ℎ𝜋⁰_𝑡𝑑𝑟,𝑃=^𝑟𝑚_𝑟ℎ𝜋⁰_𝑥𝑑𝑟, (8) where 𝑟_𝑚 is location of 𝐷7-brane.

We will use above relations to obtain the drag force and the jet-quenching parameter in various backgrounds. Specially we focus on STU background introduced in the next section.

3. STU Background

In this paper we are interested in the special form of the 𝒩=2 supergravity in five dimensions. This model, which is called STU model, has generally 8-charged (4 electric and 4 magnetic) nonextremal black holes. However, there are many situations with less charges such as four-charged and three-charged black holes. In 5 dimensions the situation is different and actually much simpler; there is no distinction between BPS and non-BPS branches. So, in 5 dimensions, the three-charged configurations are the most interesting ones. The STU model admits a chemical potential for the 𝑈(1)³ symmetry, and this makes it more interesting. For instance, presence of a baryon number chemical potential for heavy quark in the context of AdS/CFT correspondence yields to introducing a macroscopic density of heavy quark baryons. Indeed the STU model, which contains a nonextremal charged black hole, was obtained from five-dimensional gauged supergravity theory. Therefore, we may to consider the STU model as a gravity dual of a strongly coupled plasma. So, we begin with the three-charged nonextremal black hole solution in 𝒩=2 gauged supergravity, described by the following solution [63]: 𝑑𝑠²𝑓=−^𝑘ℋ^2/3𝑑𝑡²+ℋ^1/3𝑑𝑟²𝑓_𝑘+𝑟²𝑅²𝑑Ω²_3,𝑘, (9) where 𝑓_𝑘𝜇=𝑘−𝑟²+𝑟²𝑅²ℋ,ℋ=³_𝑖=1𝐻_𝑖,𝐻_𝑖𝑞=1+^𝑖𝑟²𝐴,𝑖=1,2,3,_𝑖𝑡=𝑘𝑞_𝑖+𝜇𝑞_𝑖1−𝐻_𝑖⁻¹, (10) where 𝑅 is the constant AdS radius and relates to the coupling constant via 𝑅=1/𝑔 (also, coupling constant relates to the cosmological constant via Λ=−6𝑔²) and 𝑟 is the radial coordinate along the black hole, so the boundary of AdS space was located at 𝑟→∞ (or 𝑟=𝑟_𝑚 on the 𝐷7-brane). The black hole horizon was specified by 𝑟=𝑟_ℎ which is obtained from 𝑓_𝑘=0. In this background there are three real scalar fields, which is also solution of the metric (1), as 𝑋^𝑖=ℋ^1/3/𝐻_𝑖, which satisfy the following condition: ∏³_𝑖=1𝑋^𝑖=1. In another word, if we set 𝑋¹=𝑆, 𝑋²=𝑇, and 𝑋³=𝑈, then there is the STU=1 condition. For the three 𝑅-charges 𝑞_𝑖, in (2), there is an overall factor such as 𝑞_𝑖=𝜇sinh²𝛽_𝑖, where 𝜇 is called nonextremality parameter and 𝛽_𝑖 are related to the three independent electrical charges of the black hole. Finally, the factor of 𝑘 indicates the space curvature, so the metric (9) includes a 𝑆³ (three-dimensional sphere) for 𝑘=1, a pseudosphere for 𝑘=−1, and a flat space for 𝑘=0. So, for 𝑘=1, 𝑘=0, and 𝑘=−1, one can write, respectively, 𝑑Ω²_3,𝑘≡𝑅²𝑑𝜌²+sin²𝜌𝑑𝜃²+sin²𝜌sin²𝜃𝑑𝜙²𝑑𝑥²+𝑑𝑦²+𝑑𝑧²𝑅²𝑑𝜌²+sinh²𝜌𝑑𝜃²+sinh²𝜌sin²𝜃𝑑𝜙². (11) These are general properties of our interesting background which are used to study QGP by using AdS/CFT correspondence.

Here, we study thermodynamics of STU black hole and extract some important thermodynamical quantities such as temperature and entropy. Also we discuss dual picture of the STU model which is 𝒩=4 SYM with finite chemical potential.

First of all we compute some thermodynamical quantities in the STU model with three different black hole charges for the arbitrary spaces. Some of these quantities such as temperature and entropy will be useful to study QGP in the next sections. According to the previous works, the Hawking temperature of the black hole solution (9) will be as follows [63]: 𝑟𝑇=^ℎ2𝜋𝑅²2+1/𝑟²_ℎ∑³_𝑖=1𝑞_𝑖−1/𝑟⁶_ℎ∏³_𝑖=1𝑞_𝑖∏³_𝑖=11+𝑞_𝑖/𝑟²_ℎ. (12) There is also a chemical potential which is given by the following relation: 𝜙²_𝑖=𝑞_𝑖𝑟²_ℎ+𝑞_𝑖1𝑅²𝑟²_{ℎ𝑗≠𝑖}𝑟²_ℎ+𝑞_𝑗+𝑘. (13) Also, the entropy density in 𝑑=4 dimension is given by the following expression, which is valid for 𝑘=±1 and 𝑘=0: 1𝑠=4𝐺𝑅³𝑟³_ℎℋ𝑟_ℎ, (14) where 𝐺 is Newton’s constant and relates to the AdS curvature as 𝐺=𝜋𝑅³/2𝑁², where 𝑁 is the number of colors.

As we mentioned already, the STU solution (9) is dual to the 𝒩=4 SYM with finite chemical potential in Minkowski space. It can be shown by the following rescaling [63]: 𝑟⟶𝜆^1/4𝑡𝑟,𝑡⟶𝜆^1/4𝑞,𝜇⟶𝜆𝜇,_𝑖⟶𝜆^1/2𝑞_𝑖, (15) and taking 𝜆→∞ limit while 𝑑Ω²_3,𝑘⟶1𝑅²𝜆^1/2𝑑𝑥²+𝑑𝑦²+𝑑𝑧², (16) and also we set 𝑟⁴₀≡𝜇𝑅². Then, solution (9) reduces to the following: 𝑑𝑠²=𝑒^2𝐴(𝑟)−𝑓ℋ^2/3𝑑𝑡²+ℋ^1/3𝑑𝑋²+ℋ^1/3𝑓𝑑𝑟²,𝑟𝑓=ℋ−⁴⁰𝑟⁴,ℋ=_𝑖𝑞1+^𝑖𝑟², (17) where the geometric function 𝐴(𝑟) is defined as 𝐴(𝑟)≡ln(𝑟/𝐿) and 𝑟₀ is the horizon radius in the 𝒩=4 SYM theory. In that case the chemical potential conjugate to the physical charge for the 𝑈(1)  𝑅-charges is given by 𝜙_𝑖=𝑟²_ℎ𝑅²2𝑞_𝑖𝑟²_ℎ+𝑞_𝑖𝑗𝑞1+^𝑗𝑟²_ℎ. (18) This is dual expression of the chemical potential which is given by the relation (13). For the special case of 𝑞₁=𝑞₂=𝑞₃=𝑞 the Hawking temperature reads as 𝑇_𝐻=𝑞+2𝑟²_ℎ2𝜋𝑅²𝑞+𝑟²_ℎ, (19) where the radius of the horizon (root of 𝑓=0) is given by 𝑟²_ℎ=124𝑟⁴₀+𝑞²−𝑞. (20)

4. Other Backgrounds

In this section we review some important backgrounds where the drag force and jet-quenching parameter are calculated by previous papers.

4.1. AdS Black Brane

It is possible to study QGP by using the metric of the AdS black brane solution in 𝑑+1 dimensions which is given by [44] 𝑑𝑠²=𝑑𝑟²𝑅²𝑓(𝑟)−𝑅²𝑓(𝑟)𝑑𝑡²+𝑟²𝑅²𝛿_𝑖𝑗𝑑𝑥^𝑖𝑑𝑥^𝑗, (21) where 𝑓(𝑟)=𝑟²𝑅⁴𝑟1−^ℎ𝑟^𝑑. (22) We can see that the black hole horizon is located at 𝑟=𝑟_ℎ, where 𝑓(𝑟) vanishes. It is clear that the metric (22) in 𝑑=4 is indeed similar to the metric (17) with 𝑞=0. This solution is corresponding to 𝒩=4 super Yang-Mills theory [46]. The Hawking temperature is given by 𝑑𝑇=4𝜋𝑅²𝑟_ℎ. (23) Also the entropy is obtained as the following: 1𝑠=𝑟4𝐺𝑅^𝑑−1. (24)

4.2. NR NC Yang-Mills

Another interesting background is the nonrelativistic, noncommutative Yang-Mills theory which is described by the following metric [64]: 𝑑𝑠²=𝑟²𝐾𝑅²1−𝛽²𝑟²𝑓(𝑑𝑥⁻)²−1+𝛽²𝑟²×𝑓𝑑𝑥⁺²+2𝛽²𝑟²𝑓𝑑𝑥⁻𝑑𝑥⁺+ℎ𝑟²𝑅²𝑑𝑥²²+𝑑𝑥³²+𝑅²𝑓𝑟²𝑑𝑟², (25) where the 5-sphere (𝑆⁵) part of the metric is neglected. Indeed, the metric (25) represents the AdS₅ space, and 𝑟 denotes vertical direction to 𝐷-branes. In the above solution 𝐾≡1+𝛽²(𝑟⁴_ℎ/𝑟²), and 𝑅²=𝑟²_ℎsinh𝜑; also 𝑟_ℎ denotes the horizon radius, and 𝜑 is called the boost parameter; also 𝛽 is a physical parameter related to the chemical potential of the Yang-Mills theory on the boundary. Moreover, 𝑟𝑓=1−^4ℎ𝑟⁴,1ℎ=1+𝑎⁴𝑟⁴, (26) where 𝑎⁴=1𝑟⁴_ℎsinh²𝜑cos²𝜃. (27)𝐷3-branes are lying along 𝑥¹,𝑥², and 𝑥³, and 𝐷1-branes are lying along 𝑥¹. The angle 𝜃 in the relation (26) measures the relative numbers of 𝐷-branes. So, for 𝑁𝐷3-branes, and 𝑀𝐷1-branes one can write cos𝜃=𝑁/𝑁²+𝑀².

In this configuration there is a large magnetic field in the 𝑥²−𝑥³ directions, so these directions satisfy the non-commutativity relation [𝑥²,𝑥³]=𝑖Θ, where Θ is called the noncommutativity parameter [65]. It has shown that 𝑎⁴𝑟⁴_ℎ∼Θ² [64], so the parameter 𝑎 measures the non-commutativity. Also 𝛽→0 limit recovered the relativistic cases, so the parameter 𝛽 specifies the nonrelativistic feature. The temperature of the nonrelativistic, non-commutative Yang-Mills theory is given by 𝑟𝑇=^ℎ𝜋𝑅²=𝑟_ℎ2𝜆𝜋𝛼^′, (28) where 𝛼^′ is the slop parameter (𝛼^′=1/2𝜋𝑇₀) and 𝜆 is the ’t Hooft coupling of the nonrelativistic, non-commutative theory, which is related to the ’t Hooft coupling of the ordinary Yang-Mills theory by the relation 𝜆=(𝛼^′/Θ)𝜆.

There are also other backgrounds which we can not represent here; we just try to recall results of them in the next sections.

5. Drag Force

Study of drag force on a moving heavy quark through a thermal plasma is interesting point to understand physics of charm and bottom quark at RHIC [66–68].

5.1. Single Quark

It is known that a moving quark in the 𝒩=2 thermal plasma corresponds to the stretched string from 𝑟=𝑟_𝑚 on the 𝐷-brane to the black hole horizon. So, calculating the energy loss of a heavy quark is reduced to find components of momentum density along the string. Detailed calculations of drag force for special case of STU model moved to Appendix A. These calculations are similar for other backgrounds. Therefore, here we just represent results of previous studies.

Drag force is obtained by using momentum components (see Appendix A) which in STU model is obtained as the following: 𝑑𝑃𝑑𝑡=𝜋¹_{𝑥𝑟=𝑟𝑚}=−𝑇₀𝐶𝑣,𝑑𝐸𝑑𝑡=𝜋¹_{𝑡𝑟=𝑟𝑚}=𝑇₀𝐶𝑣². (29)

In order to find 𝐶 we use reality condition for 𝑥^′2 and √−𝑔; in that case one can fix the constant 𝐶 as the following: 𝐶=³_𝑖=1𝑞1+^𝑖𝑟²_𝑐^1/3𝑟²_𝑐𝑅², (30) where 𝑟_𝑐=𝑟_ℎ+𝑟²𝑣²ℋ×2𝑅²𝜇𝑟³+𝑟ℋ𝑅²−𝑞₁𝐻₂𝐻₃+𝑞₂𝐻₁𝐻₃+𝑞₃𝐻₁𝐻₂×𝑟𝑅²⁻¹⁻¹_𝑟=𝑟ℎ+𝒪𝑣⁴. (31) It is important to note that this result is independent of curvature parameter, 𝑘; however, we should set 𝑘=1 in the relation (10) to have AdS₅×𝑆⁵ space. Therefore the drag force may be written as 𝑑𝑃𝑑𝑡=−𝑇₀𝑣³_𝑖=1𝑞1+^𝑖𝑟²_ℎ^1/3×𝑟²_ℎ𝑅²1+𝒪𝑣². (32) It is clear that the black hole charge increases the value of drag force.

The drag force corresponding to the solution (21) was calculated by [44–46] and found that 𝑑𝑃𝑑𝑡=−𝑇₀𝑟²_ℎ𝑅²𝑣1−𝑣^22/𝑑. (33) It is clear that both expressions (32) and (33) are the same at 𝑞=0 limit.

The drag force corresponding to the solution (25) was calculated by [64] and found that 𝑑𝑃𝑑𝑡=−𝑇₀𝑟²_ℎ𝑅²𝑣1+𝑎⁴𝑟⁴_ℎ=−𝑇₀𝑟²_ℎ𝑟²_𝑐𝑣sinh𝜑1+𝑎⁴𝑟⁴_ℎ, (34) where the critical radius 𝑟_𝑐 is the root of the following: 𝑟⁴−𝑟⁴_ℎ1+𝑎⁴𝑟⁴1+𝛽²𝑟²−𝑣²𝑟⁴+𝛽²𝑟⁴_ℎ𝑟²=0. (35) It is easy to check that the special case of 𝛽=𝑎=0 with sinh𝜑=−√1−𝑣² was reduced to (33).

5.2. Quark-Antiquark Configuration

Now, we consider a moving quark-antiquark pair which may be interpreted as a meson. Indeed there is a moving meson with the constant speed 𝑣 in the 𝒩=2 supergravity thermal plasma. Already the energy of a moving quark-antiquark pair in 𝒩=4 SYM plasma was calculated [69]. Now, we would like to repeat same calculations in the STU background (see Appendix A). The quark-antiquark pair in the thermal QGP corresponds to an open string in AdS₅ space with two endpoints on the 𝐷-brane in the (𝑋,𝑌) plan. Two end points of string on the 𝐷-brane represent quark and antiquark which are separated from each other by a constant 𝑙. We assume that at the 𝑡=0 string is straight and two endpoints of string move with the constant velocity 𝑣 along the 𝑋 direction. The dynamics of such configuration is discussed in detail in [69] for the 𝒩=4 SYM plasma.

Figure 1 shows the general configuration of string. The points 𝐴 and 𝐵 in Figure 1 represent quark and antiquark with separating length 𝑙. The radial coordinate 𝑟 varies from 𝑟_ℎ (black hole horizon radius) to 𝑟=𝑟_𝑚 on 𝐷-brane. 𝑟_𝑐 is a critical radius, obtained for single quark solution, which the string cannot penetrate beyond it, and 𝑟_min≥𝑟_𝑐. 𝑟_min=𝑟_𝑐 is satisfied if points 𝐴 and 𝐵 are located at origin (𝑙=0); in that case there is the straight string which is dual picture of the single static quark. 𝜃 is assumed to be the angle with 𝑌-axis and the string center of mass moves along 𝑋-axis with velocity 𝑣. Solutions of this configuration satisfy boundary conditions 𝑥(∞,𝑡)=𝑣𝑡±(𝑙/2)sin𝜃 and 𝑦(∞,𝑡)=±(𝑙/2)cos𝜃, which reduced to the boundary condition of the case without rotational motion for 𝜃=0.

Figure 1 

A rotating ∩-shape string dual to a 𝑞𝑞 pair which can be interpreted as a meson.

By using calculations given by Appendix A one can obtain 𝜋¹_𝑥²=1±2𝑎𝑏²,𝜋−4𝑎𝑐−𝑏_1𝑦2=12𝑎^′±𝑏^′2−4𝑎^′𝑐^′−𝑏^′, (36) where 𝑎=_𝑖𝑞1+^𝑖𝑟²_min^1/3cos²𝜃ℛ²_min𝜍+𝜉𝜒,𝑏=𝑇²_0𝑖𝑞1+^𝑖𝑟²_min^2/3𝜒×2ℛ⁴_min𝜍+cos²𝜃ℛ²_min𝜉𝜒−𝜍,𝑐=𝑇⁴_0𝑖𝑞1+^𝑖𝑟²_minℛ⁶_minsin²𝜃𝜍𝜒²,𝑎^′=_𝑖𝑞1+^𝑖𝑟²_min^2/3ℛ²_min,𝑏𝜍+𝜉𝜒_′=𝑇²₀ℛ²_min𝜉ℛ⁴_min𝜍_𝑖𝑞1+^𝑖𝑟²_min^2/3−2ℛ²_min𝜍−_𝑖𝑞1+^𝑖𝑟²_min^2/3,𝑐𝜉𝜒_′=𝑇²_0𝑖𝑞1+^𝑖𝑟²_min^2/3ℛ⁶_min𝜍𝜉²×_𝑖𝑞1+^𝑖𝑟²_min^4/3−𝑇²₀, (37) with 𝜍=_𝑖𝑞1+^𝑖𝑟²_min𝑦²𝜃̇²ℛ²_minsin²𝜃𝑣+𝑥𝜃̇cos𝜃²,𝑓𝜉=^𝑘𝑟_min∏_𝑖1+𝑞_𝑖/𝑟²_min^1/3−_𝑖𝑞1+^𝑖𝑟²_min^2/3×𝑣+𝑥𝜃̇cos𝜃²ℛ²_min,𝜒=_𝑖𝑞1+^𝑖𝑟²_min^1/3𝑦²𝜃̇²𝑅²_minsin²𝜃−𝑓_𝑘𝑟_min∏_𝑖1+𝑞_𝑖/𝑟²_min^2/3, (38) where ℛ_min=𝑟_min/𝑅 and 𝑟_min is the turning point. The direct consequence of rotational motion is that drag force is no longer constant. But, this result is not appropriate description of a meson. According to previous works [69] the 𝑞𝑞 pair should be close enough together and not moving too quickly. The presence of functions 𝑥(𝑟) and 𝑦(𝑟) is consequence of relativistic motion, which is not acceptable. On the other hand, because of nonvanishing drag forces, it is expected that the velocity of a 𝑞𝑞 pair decreases. So, we consider a moving heavy 𝑞𝑞 pair with non-relativistic speed, which rotates by angel 𝜃=𝜔𝑡 around the center of mass. Indeed this situation is corresponding to the motion of the heavy meson with large spin. Actually, in the very large angular momentum limit, a classical approximation is reliable. In this case, the angular velocity of the string is very small. Therefore, we are going to discuss the case of non-relativistic motion (𝜃̇²→0 and 𝜃̇𝑣→0). In that case 𝜍=𝑐=𝑐^′=0 and we have 𝜋¹_𝑥²=𝑟²_min𝑅²𝑇²₀𝑓_𝑘𝑟_minℋ^−1/3𝑟_min,𝜋¹_𝑦²=𝑟²_min𝑅²𝑇²₀𝑓_𝑘𝑟_min−ℋ𝑟_min𝑟²_min𝑣²𝑅²ℋ^−1/3𝑟_min. (39) Now, we assume that 𝑣²→0 and angular velocity is infinitesimal constant (𝜃̇=𝜔≪1), and the quark-antiquark pair rotates around origin. In that case we neglect 𝜔⁴ terms and obtain values of momentum densities as the following: 𝜋¹_𝑥=𝜋¹_𝑦=𝑇₀𝑟_min𝑅𝜇𝑘−𝑟²_min+𝑟²_min𝑅²×_𝑖𝑞1+^𝑖𝑟²_min^1/2×_𝑖𝑞1+^𝑖𝑟²_min^1/6−1. (40)

6. Screening Length

Also it is possible to calculate screening length [70–74] for a quark-antiquark pair by using the following relations: 𝐿_𝑥=2^∞_𝑟min𝑑𝑟𝑥^′,𝐿_𝑦=2^∞_𝑟min𝑑𝑟𝑦^′. (41) In the case of 𝑞=0 limit one can obtain 𝐿_𝑥=2^∞₁𝑑𝑧𝛼𝜌1−𝑣²𝑧⁴−𝜌⁴𝜋𝑇𝑟_minsin𝜃𝑧⁴𝑧⁴−𝜌⁴𝑓(𝑧),𝐿_𝑦=2^∞₁𝑑𝑧𝛽𝜌1−𝑣²𝑧⁴−𝜌⁴𝜋𝑇𝑟_mincos𝜃𝑧⁴𝑓(𝑧), (42) where 𝑓(𝑧)=𝑧⁴𝑇²₀𝑧⁴−𝜌⁴1−𝑣²𝑧⁴−𝜌⁴+𝜌⁴𝛼²+𝛽²−𝑧⁴𝛼²+𝛽²−𝛼²𝑣^2−1/2, (43) and we defined the following variables: 𝜋𝛼=^1𝑥𝑙²𝑟²_min,𝜋𝛽=^1𝑦𝑙²𝑟²_min,𝑟𝑧=𝑟_min,𝑟𝜌=^ℎ𝑟_min. (44)

In [73] the screening length of a heavy quark-antiquark pair in strongly coupled gauge theory plasmas has been studied by using the AdS/CFT correspondence and found that the screening length is proportional to inverse of boosted energy density with power 1/𝑑, and some examples in (𝑑+1)-dimensions were studied. We find that our results agree with the previous works.

7. Jet-Quenching Parameter

One of the interesting properties of the stronglycoupled plasma at RHIC is the jet quenching of partons, produced with high transverse momentum [75]. This parameter controls the description of relativistic partons and it is possible to employ the gauge/gravity duality and determine this quantity in the finite temperature gauge theories. In order to obtain the jet-quenching parameter one needs to rewrite the metric background in the light-cone coordinates. In Appendix B we give detailed calculation of the jet-quenching parameter in STU background. So, we can specify the jet-quenching parameter as the following [51]: 𝐼𝑞𝑞=⁻¹𝜋𝛼^′, (45) where 𝐼𝑞=𝑅^2∞_𝑟ℎ𝑑𝑟ℋ^2/3𝑟²/𝑅²−𝑓_𝑘/ℋ^1/3𝑓_𝑘𝑟⁴. (46) In order to obtain the explicit expression of the jet-quenching parameter we set 𝑘=1 and consider three special cases of one-, two-, and three-charged black hole.

In the case of one-charged black hole we set 𝑞₁=𝑞,𝑞₂=𝑞₃=0 and obtain 𝐼𝑞₁=𝑅^4∞_𝑟ℎ1+𝑞/𝑟^21/3𝑟²−𝜇𝑟⁴+𝑞+𝑅²𝑟²−𝜇𝑅²𝑑𝑟. (47) In the similar work the jet-quenching parameter in medium with chemical potential has been studied [54].

In the case of metric background (21) it is found that [58] 𝜋𝑞=^3/2Γ(3/4)Γ()√5/4𝜆𝑇³. (48) From (48), one finds 𝑞=4.5,10.6, and 20.7GeV²/fm for 𝑇=300,400, and 500MeV. Comparing (45) and (48) tells that the jet-quenching parameter was enhanced for STU background. For example, by choosing 𝑅²=𝛼^′√𝜆, 𝛼^′=0.5, 𝜆=6𝜋, 𝑞=10⁶, and 𝑇=300MeV one can obtain 𝑞=42GeV²/fm in STU model. It means that the black hole charge increases the jet-quenching parameter.

Numerically, we give plots of the jet-quenching parameter in terms of the black hole charges and the temperature in Figures 2 and 3, respectively. These plots show that the jet-quenching parameter of the 𝒩=2 theory is larger than the jet-quenching parameter of the 𝒩=4 theory. Also we find that the jet-quenching parameter of the three-charged black hole is larger than the jet-quenching parameter of the one-charged and two-charged black holes.

Figure 2 

Plot of the jet-quenching parameter in terms of the black hole charge. We fixed our parameters as 𝛼′=0.5, 𝜆=6𝜋, and 𝑇=300MeV. The solid line represents the case of 𝑞1=𝑞,𝑞2=𝑞3=0. The dotted line represents the case of 𝑞1=𝑞2=𝑞,𝑞3=0. The dashed line represents the case of 𝑞1=𝑞2=𝑞3=𝑞. It shows that increasing the number of black hole charges increases the value of the jet-quenching parameter.

Figure 3 

Plot of the jet-quenching parameter in terms of the temperature for small black hole charge. We fixed the parameters as 𝛼′=0.5, 𝜆=6𝜋. In that case three different cases of one-, two-, and three-charged black holes have similar behaviors.

Also the jet-quenching parameter corresponding to metric background (17) is given by 𝑟𝑞=²⁰𝜋𝛼^′𝑅^4∞_𝑟ℎ𝑑𝑟𝑟²𝑓/𝐻⁻¹, (49) where 𝑓=𝐻³−𝑟⁴₀𝑟⁴,𝑞𝐻=1+𝑟², (50) which agree with the results of the previous works, where the jet-quenching parameter is calculated with the chemical potential [56]. The horizon radius 𝑟₀ is obtained for the case of zero-charged black hole. For the black hole with nonvanishing charges, it is clear that the horizon radius decreases (𝑟_ℎ<𝑟₀). We know that the 𝑞=0 limit is equal to 𝜙=0 limit, and one can say that the jet-quenching parameter from the 𝒩=2 supergravity theory with zero chemical potential is equal to the jet-quenching parameter from the 𝒩=4 SYM theory.

On the other hand by using the metric background (25) we find the following expression of the jet-quenching parameter [76]: 𝑞=2√2𝑇₀^∞_𝑟ℎ𝑑𝑟𝑅⁴𝑟⁴𝐾𝑓ℎ²𝑟⁴_ℎ/2𝑟⁴−2𝑟²𝛽²𝑓⁻¹. (51) It is found that for the large chemical potential (for both infinitesimal and large non-commutativity parameters) the jet-quenching parameter is obtained as (48), but for the case of infinitesimal non-commutativity parameter and large 𝛽 the jet-quenching parameter is proportional to temperature. We have found that the presence of non-commutativity is necessary to obtain the jet-quenching parameter in the experimental range.

8. Conclusion

In this paper we used AdS/CFT correspondence and reviewed some important quantities to understand the nature of QGP more exactly. Indeed, we considered thermal QGP includes a chemical potential. This chemical potential comes from 𝒩=2 supergravity in 5 dimensions. This theory contains a nonextremal black hole with three electrical charges and is well known as STU model. First of all we introduced STU model and wrote their important properties. We studied thermodynamics of STU background and extracted the Hawking temperature and entropy density. In order to compare our results with the 𝒩=4 SYM plasma we used special rescaling that bring the spherical-horizon metric to the flat-horizon metric.

We considered problem of the drag force and found energy loss of single quark and quark-antiquark pair. We showed that the value of the drag force enhanced due to the black hole charges. Also we calculated screening length for the special case of 𝑞=0.

Finally we studied the jet-quenching parameter and found that the jet-quenching parameter enhanced due to the black hole charges. It means that the energy of the string in 𝒩=2 thermal plasma is larger than the string in 𝒩=4 thermal plasma hence the string in 𝒩=2 thermal plasma loses more energy than the string in 𝒩=4 thermal plasma. We examine our solution for three special cases of one-, two-, and three-charged black holes. All cases yield to the same value of the jet-quenching parameter for the small black hole charge. However, thermodynamical stability allows choosing the black hole charge of order 10⁶. In that case we found 𝑞=42,49, and 58GeV²/fm for one-, two-, and three-charged black hole, respectively. These values of the jet-quenching parameter are far from experiments of RHIC; experimental data tell us that (5<𝑞<25). There is no worry for this statement because the temperature of the 𝒩=2 supergravity theory should be smaller than that of the 𝒩=4 SYM theory. In that case with the temperature about 155MeV we obtained the jet-quenching parameter in the experimental range.

Appendices

A. Drag Force

It is assumed that the string moves along 𝑥 direction and uses static gauge, where 𝜏=𝑡 and 𝜎=𝑟. Therefore, the string world-sheet is described by 𝑥(𝑟,𝑡).

By using Euler-Lagrange equation one can obtain the string equation of motion as the following expression: 𝜕𝑓𝜕𝑟^𝑘𝑟²ℋ^1/3√𝑥−𝑔^′=ℋ^2/3𝑟²𝑓_𝑘𝜕𝑥̇𝜕𝑡√−𝑔. (A.1) In order to obtain the total energy and momentum, drag force, or energy loss of particle in the thermal plasma, we have to calculate the canonical momentum densities. In that case one can obtain the following expressions: 𝜋⁰_𝑥𝜋¹_𝑥𝜋⁰_𝑟𝜋¹_𝑟𝜋⁰_𝑡𝜋¹_𝑡𝑇=−⁰ℋ^1/3√×−−𝑔ℋ𝑟²𝑓_𝑘𝑅²𝑥̇𝑓_𝑘𝑟²𝑅²𝑥^′ℋ𝑟²𝑓_𝑘𝑅²𝑥̇𝑥^′1−ℋ𝑟²𝑓_𝑘𝑅²𝑥̇²1+𝑓_𝑘𝑟²𝑅²𝑥^′2−𝑓_𝑘𝑟²𝑅²𝑥̇𝑥^′. (A.2) Corresponding to the single quark, in CFT side, we have an open string in AdS space which stretched from 𝑟=𝑟_𝑚 on 𝐷7-brane to 𝑟=𝑟_ℎ at the horizon. An ansatz made in order to describe the system may be 𝑥(𝑟,𝑡)=𝑥(𝑟)+𝑣𝑡, (A.3) where 𝑣 is the constant velocity of the single quark. In that case by using equation of motion (15) one can find 𝑓_𝑘𝑟²𝑅²𝑣ℋ^1/3√𝑥−𝑔^′=𝐶, (A.4) where 𝐶 is an integration constant and √−𝑔 is obtained by using the following: 1−𝑔=ℋ^1/31−ℋ𝑟²𝑓_𝑘𝑅²𝑣²+𝑓_𝑘𝑟²𝑅²𝑥^′2. (A.5) By using these results in the canonical momentum densities (15) we find 𝜋¹_𝑥=−𝑇₀𝜋𝐶𝑣,_1𝑡=𝑇₀𝐶𝑣². (A.6) In the case of quark-antiquark, one can obtain the following equations, respectively, 𝜕1𝜕𝑟√𝑟−𝑔𝑅⁴²ℋ^2/3𝑦̇²𝑥^′−𝑥̇𝑦𝑦̇^′+𝑓_𝑘𝑟²𝑥^′ℋ^1/3+𝑟²ℋ^2/3𝜕1𝜕𝑡√𝑥̇−𝑔𝑓_𝑘+𝑟²𝑅²𝑦^′2𝑥̇−𝑥^′𝑦𝑦̇^′𝜕=0,1𝜕𝑟√𝑟−𝑔𝑅⁴²ℋ^2/3𝑥̇²𝑦^′−𝑥̇𝑦𝑥̇^′−𝑓_𝑘𝑟²𝑦^′ℋ^1/3+𝑟²ℋ^2/3𝜕1𝜕𝑡√̇−𝑔𝑦𝑅²𝑓_𝑘+𝑟²𝑥^′2𝑦̇−𝑥^′𝑥̇𝑦^′=0, (A.7) and the momentum densities are obtained by the following:𝜋⁰_𝑥𝜋¹_𝑥𝜋⁰_𝑦𝜋¹_𝑦𝜋⁰_𝑟𝜋¹_𝑟𝜋⁰_𝑡𝜋¹_𝑡=−𝑇₀𝑟²ℋ^1/3𝑅²√×𝑟−𝑔₂𝑅²ℋ^1/3𝑥^′𝑦𝑦̇^′−ℋ^1/3𝑓_𝑘+𝑟²𝑅²ℋ^1/3𝑦^′2𝑥̇𝑟²𝑅²ℋ^1/3𝑦^′𝑦̇̇ℋ𝑥−^1/3𝑟²𝑅²𝑦̇²−𝑓_𝑘ℋ^2/3𝑥^′𝑟²𝑅²ℋ^1/3𝑦^′𝑥̇𝑥^′−ℋ^1/3𝑓_𝑘+𝑟²𝑅²ℋ^1/3𝑥^′2𝑦̇𝑟²𝑅²ℋ^1/3𝑥^′𝑦̇̇ℋ𝑥−^1/3𝑟²𝑅²𝑥̇²−𝑓_𝑘ℋ^2/3𝑦^′ℋ^1/3𝑓_𝑘𝑥̇𝑥^′+𝑦𝑦̇^′𝑅²ℋ^2/3𝑟²−ℋ^1/3𝑓_𝑘𝑥̇²+𝑦̇²𝑅²ℋ^1/3𝑟²+𝑓_𝑘ℋ^1/3𝑥^′2+𝑦^′2−𝑓_𝑘ℋ^2/3𝑥̇𝑥^′+𝑦𝑦̇^′. (A.8)There are two interesting motions for the meson. The first one is the moving quark-antiquark pair with constant speed 𝑣. The second case is the rotational motion of the quark-antiquark pair.

The first system may be described by the following ansatz: 𝑥()(𝑟),𝑦()(𝑟).𝑟,𝑡=𝑣𝑡+𝑥𝑟,𝑡=𝑦 (A.9) These solutions satisfy boundary conditions as 𝑥(∞,𝑡)=𝑣𝑡 and 𝑦(∞)=±𝑙/2. In order to obtain drag force, we calculate 𝜋¹_𝑥 and 𝜋¹_𝑦 components, solve them for 𝑥^′ and 𝑦^′, respectively, and obtain 𝑥^′(𝑟)=𝜋¹_𝑥𝑅𝑟1−ℋ𝑟²𝑣²𝑓_𝑘𝑅²×𝑓_𝑘ℋ−𝑟²𝑣²𝑅²𝑇²₀𝑟²𝑅²𝑓_𝑘ℋ^2/3−ℋ𝜋¹_𝑥²−𝑓_𝑘(𝜋¹_𝑦)^2−1/2,𝑦^′(𝑟)=𝜋¹_𝑦𝑅𝑟𝑓_𝑘ℋ−𝑟²𝑣²𝑅²×𝑇²₀𝑟²𝑅²𝑓_𝑘ℋ^2/3−ℋ𝜋¹_𝑥²−𝑓_𝑘𝜋¹_𝑦^2−1/2. (A.10) As before, by using reality condition one can obtain 𝜋¹_𝑦²=𝑓_𝑘ℋ−𝑟²𝑣²𝑅²×𝑇²₀𝑟²𝑅²ℋ^2/3−ℋ𝑓_𝑘𝜋¹_𝑥²_𝑟=𝑟min, (A.11) where 𝑟_min is turning point of string. If 𝜋¹_𝑦=0, then 𝑟_min=𝑟_𝑐 and above solutions are similar to single quark solution (𝑙=0). Here, as the string have a chance of turning around smoothly, it requires that 𝜕𝑦/𝜕𝑥=𝑦^′/𝑥^′=∞ at 𝑟_min. So, it is necessary to have 𝜋¹_𝑥=0. Therefore, one can find the momentum component as 𝜋¹_𝑦=𝑇₀𝑅𝑟_minℋ^1/3𝑟_min×𝑓_𝑘𝑟_minℋ𝑟_min−𝑟²_min𝑣²𝑅². (A.12) In the second case we add a rotational motion with angular velocity 𝜃̇ to the motion of meson. Therefore, the string may be described by the 𝑥(𝑟,𝑡)=𝑣𝑡+𝑥(𝑟)sin𝜃 and 𝑦(𝑟,𝑡)=𝑦(𝑟)cos𝜃 profiles.

Also there is another condition due to our conjecture, 𝑦^′/𝑥^′=cot𝜃, which reduces to 𝑦^′/𝑥^′→∞ at the 𝜃→0 limit, which agrees with the first case.

These boundary conditions can also satisfy with two separated string which move at velocity 𝑣 along 𝑋-axis and simultaneously swing a circle with radius 𝑙/2. Specifying these boundary conditions does not lead to a unique solution for equation of motion, so we should specify additional conditions for this motion.

Here, we assume that the string is initially upright, moves at velocity 𝑣, and rotates around its center of mass.

Now, one can obtain the following: 𝐴𝑥^′2+𝐵𝑦^′2+𝐶𝑥^′𝑦^′𝐴+𝐷=0,_′𝑥^′2+𝐵^′𝑦^′2+𝐶^′𝑥^′𝑦^′+𝐷^′=0, (A.13) where 𝐴=ℛ²sin²𝜃𝜋¹_𝑥²𝑓_𝑘ℋ^1/3−ℋ^2/3𝑦²𝜃̇²ℛ²sin²𝜃−𝑇²₀ℋ^2/3ℛ²ℋ^1/3𝑦²𝜃̇²ℛ²sin²𝜃−𝑓_𝑘ℋ^2/32,𝐵=ℛ²cos²𝜃𝜋¹_𝑥²𝑓_𝑘ℋ^1/3−ℋ^2/3𝑣+𝑥𝜃̇cos𝜃²ℛ²−𝑇²₀ℋ^4/3𝑦²𝜃̇²ℛ⁶sin²𝜃𝑣+𝑥𝜃̇cos𝜃²,𝐶=−2𝑦𝜃̇ℛ⁴sin²𝜃𝜋¹_𝑥²ℋ^2/3cos𝜃+𝑇²₀ℋℛ²×ℋ^1/3𝑦²𝜃̇²ℛ²sin²𝑓𝜃−^𝑘ℋ^2/3×𝑣+𝑥𝜃̇cos𝜃,𝐷=ℛ²𝜋¹_𝑥²ℋ^2/3𝑓_𝑘𝑓_𝑘ℛ²ℋ−𝑦²𝜃̇²sin²𝜃−𝑣+𝑥𝜃̇cos𝜃²,𝐴^′=ℛ²sin²𝜃𝜋¹_𝑦²𝑓_𝑘ℋ^1/3−ℋ^2/3𝑦²𝜃̇²ℛ²sin²𝜃−𝑇²₀ℋ^4/3𝑦²𝜃̇²ℛ⁶sin²𝜃𝑣+𝑥𝜃̇cos𝜃²,𝐵^′=ℛ²cos²𝜃𝜋¹_𝑦²𝑓_𝑘ℋ^1/3−ℋ^2/3𝑣+𝑥𝜃̇cos𝜃²ℛ²−𝑇²₀ℋ^2/3ℛ²ℛ²ℋ^1/3𝑣+𝑥𝜃̇cos𝜃²−𝑓_𝑘ℋ^2/32,𝐶^′=−2𝑦𝜃̇ℛ⁴sin²×𝜋𝜃cos𝜃_1𝑦2ℋ^2/3+𝑇²₀ℋℛ²ℛ²ℋ^1/3𝑣+𝑥𝜃̇cos𝜃²−𝑓(𝑟)ℋ^2/3̇,𝐷𝑣+𝑥𝜃cos𝜃_′=ℛ²𝜋¹_𝑦²ℋ^2/3𝑓_𝑘𝑓_𝑘ℛ²ℋ−𝑦²𝜃̇²sin²𝜃−𝑣+𝑥𝜃̇cos𝜃², (A.14) and we set 𝑟/𝑅≡ℛ. Therefore, from (A.13), one can obtain 𝑥^′(𝑟)𝐷𝜋=2𝐵−¹_𝑦²/𝜋¹_𝑥²𝐵^′𝐶²−𝐶^′2−4𝐵𝐴−𝐵^′𝐴^′1/2,𝑦^′(𝑟)𝐷𝜋=2𝐴−¹_𝑦²/𝜋¹_𝑥²𝐴^′𝐶²−𝐶^′2−4𝐵𝐴−𝐵^′𝐴^′1/2. (A.15) Here, if the rotational motion vanishes (𝜃̇=0), one can see that coefficients of 𝑥^′𝑦^′ vanish (𝐶=𝐶^′=0) and our solutions recover the motion of quark-antiquark pair without rotation. In order to obtain drag force we use reality condition and find 𝐴𝐴^′=𝐵𝐵^′=𝐶𝐶^′=𝐷𝐷^′=𝜋¹_𝑦𝜋¹_𝑥². (A.16) Then one can find two equations as 𝐶²−4𝐴𝐵=0 and 𝐶^′2−4𝐴^′𝐵^′=0. These equations specify 𝜋¹_𝑥 and 𝜋¹_𝑦, respectively.

B. Jet-Quenching

First of all we introduce light-cone coordinates 𝑥^±=(𝑡±𝑥¹)/√2 and rewrite the metric (1) in the following form: 𝑑𝑠²=12ℋ^1/3𝑟²𝑅²−𝑓_𝑘ℋ^2/3𝑑𝑥⁺²+(𝑑𝑥⁻)²−ℋ^1/3𝑟²𝑅²+𝑓_𝑘ℋ^2/3𝑑𝑥⁺𝑑𝑥⁻+ℋ^1/3𝑟²𝑅²𝑑𝑥²₂+𝑑𝑥²₃+𝑑𝑟²𝑓_𝑘. (B.1) We begin with the general relation for the jet-quenching parameter, 𝑞≡8√2𝑆_𝐼𝐿⁻𝐿², (B.2) where 𝑆_𝐼=𝑆−𝑆₀ (𝑆 denotes 𝑞𝑞 pair action and 𝑆₀ denotes the action of isolated 𝑞 and 𝑞). It means that the jet-quenching parameter is proportional to energy of the string, so we expect that this quantity will be opposite to the drag force which is indeed energy loss of the string. Therefore, calculation of the jet-quenching parameter reduces to obtain actions 𝑆 and 𝑆₀.

One can image the situation with an open string whose endpoints lie on the brane. In the light-cone coordinates, the string may be described by 𝑟(𝜏,𝜎). We use the static gauge where 𝜏=𝑥⁻ and 𝜎=𝑥²≡𝑦, and all other coordinates are considered as constants. In that case −𝐿/2≤𝑦≤𝐿/2 and 𝐿⁻≤𝑥⁻≤0, and because of 𝐿⁻≫𝐿 one can assume that the world-sheet is invariant along the 𝑥⁻ direction. Therefore, the string may be described by the function 𝑟(𝑦), so the boundary condition is 𝑟(±𝐿/2)=∞. Then the Nambu-Goto action is given by √𝑆=2𝐿⁻2𝜋𝛼^′₀^𝐿/2𝑑𝑦ℋ^2/3𝑟²𝑅²−𝑓_𝑘ℋ^1/3𝑟²𝑅²+1𝑓_𝑘𝑟^′2. (B.3) One can remove the 𝑟^′ by using the equation of motion. For the low energy limit (𝐸→0) one can obtain 𝐿𝑆=⁻2𝜋𝛼^′∞_𝑟ℎ𝑑𝑟2ℋ^1/3𝑓_𝑘ℋ^1/3𝑟²𝑅²−𝑓_𝑘ℋ^2/3×𝑅1+²ℋ^2/3𝑟²/𝑅²−𝑓_𝑘/ℋ^1/3𝐵𝑟². (B.4) Now, one can extract action 𝑆₀ which can be interpreted as the self-energy of the isolated quark and the isolated antiquark. In that case one can obtain 𝑆₀=𝐿⁻2𝜋𝛼^′∞_𝑟ℎ𝑑𝑟2ℋ^1/3𝑓_𝑘ℋ^1/3𝑟²𝑅²−𝑓_𝑘ℋ^2/3. (B.5) Therefore, we can extract 𝑆_𝐼 as the following: 𝑆_𝐼=1√𝐵𝐿⁻2𝜋𝛼^′∞_𝑟ℎ𝑑𝑟2𝑅⁴ℋ^2/3𝑟²/𝑅²−𝑓_𝑘/ℋ^1/3𝐵𝑓_𝑘𝑟⁴. (B.6) These relations give us the jet-quenching parameter.