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Advances in High Energy Physics

Volume 2017 (2017), Article ID 8910210, 6 pages

https://doi.org/10.1155/2017/8910210

## Entropic Destruction of Heavy Quarkonium from a Deformed AdS_{5} Model

^{1}School of Mathematics and Physics, China University of Geosciences, Wuhan 430074, China^{2}Key Laboratory of Quark and Lepton Physics (MOE), Central China Normal University, Wuhan 430079, China

Correspondence should be addressed to Zi-qiang Zhang

Received 11 February 2017; Accepted 9 April 2017; Published 30 April 2017

Academic Editor: Song He

Copyright © 2017 Zi-qiang Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

We study the destruction of heavy quarkonium due to the entropic force in a deformed AdS_{5} model. The effects of the deformation parameter on the interdistance and the entropic force are investigated. The influence of the deformation parameter on the quarkonium dissociation is analyzed. It is shown that the interdistance increases in the presence of the deformation parameter. In addition, the deformation parameter has the effect of decreasing the entropic force. These results imply that the quarkonium dissociates harder in a deformed AdS background than in a usual AdS background, in agreement with earlier findings.

#### 1. Introduction

It is well-known that the dissociation of heavy quarkonium can be regarded as an important experimental signal of the formation of strongly coupled quark-gluon plasma (QGP) [1]. It was argued earlier that the quarkonium suppression is due to the Debye screening effects induced by the high density of color charges in QGP. But the recent experimental research showed a puzzle: the charmonium suppression at RHIC (lower energy density) is stronger than that at LHC (larger energy density) [2, 3]. Obviously, this is in contradiction to the Debye screening scenario [1] as well as the thermal activation through the impact of gluons [4, 5]. To explain this puzzle, some authors suggested that the recombination of the produced charm quarks into charmonia may be a solution. This argument was based on the results [6, 7] that if a region of deconfined quarks and gluons is formed, the quarkonia (or bound states) can be formed from a quark and an antiquark which were originally produced in separate incoherent interactions. Recently, Kharzeev [8] argued that this puzzle may be related to the nature of deconfinement and the entropic force would be responsible for melting the quarkonium. This argument originated from the Lattice results that a large amount of entropy associated with the heavy quarkonium placed in QGP [9–11].

AdS/CFT, which maps a -dimensional quantum field theory to its dual gravitational theory, living in -dimensional, has yielded many important insights into the dynamics of strongly coupled gauge theories [12–14]. In this approach, Hashimoto and Kharzeev have studied the entropic destruction of static heavy quarkonium in SYM theory and a confining YM theory firstly. They found that in both cases the entropy grows as a function of the interquark distance giving rise to the entropic force [15]. Recently, these studies have been extended to the case of moving quarkonium [16]. It was shown that the velocity has the effect of increasing the entropic force, thus making the quarkonium melt easier. In a more recent work, we have analyzed the effect of chemical potential on the entropic force and observed that the moving quarkonium dissociates easier at finite density [17].

Now, we would like to give such analyses from AdS/QCD. The motivation is that AdS/QCD models can provide a nice phenomenological description of hadronic properties as well as quark-antiquark interaction; see [18–26] and references therein. In this paper, we will study the entropic force in the Andreev-Zakharov model [21], one of “soft wall” models. The Andreev-Zakharov model has some properties: the positive quadratic term modification in the deformed warp factor produces linear behavior of heavy flavor potential; namely, it can provide confinement at low temperature. The value of can be fixed from the meson trajectory, so that the metric contains no free parameter. Actually, this model has been used to investigate some quantities, such as thermal phase transition [27], thermal width [28, 29], and heavy quark potential [30]. Likewise, it is of interest to study the entropic force in this model. Besides that, we have several other reasons: first, we want to know what will happen if we have meson in a deformed AdS background or how the deformation parameter affects the quarkonium dissociation? Moreover, evaluation of the entropic force helps us to understand the “usual” or “unusual” behavior of meson, because one can compare the results of with , while the “usual” behavior of meson can be recovered in the limit . On the other hand, such an investigation can be regarded as a good test of AdS/QCD.

The paper is organized as follows. In the next section, we briefly review the action of holographic models and then introduce the Andreev-Zakharov model. In Section 3, we study the effects of the deformation parameter on the interdistance as well as the entropic force and then analyze how the deformation parameter affects the quarkonium dissociation. The last part is devoted to discussion and conclusion.

#### 2. The Andreev-Zakharov Model

Before reviewing the Andreev-Zakharov model, let us briefly introduce the holographic models in terms of the action [26]:where is the five-dimensional Newton constant. denotes the determinant of the metric . refers to the Ricci scalar. is called the scalar and induces the deformation away from conformality. represents the gauge kinetic function. stands for the field strength associated with an Abelian gauge connection . is the potential which contains the cosmological constant term 2 and some other terms.

To obtain an AdS-black hole space-time, one considers a constant scalar field (or called dilaton) and assumes that as well as . Then the action of (1) can be simplified aswith the equations of motionwhere is the Levi-Civita covariant derivative with respect to the metric .

Supposing that the horizon function vanishes at the point , then the solution of (3) (with vanishing right-hand side) becomes the -Schwarzschild metric:withwhere can be related to the temperature as . Notice that, in the limit (correspond to zero temperature), the metric of (5) reduces to the metric, as expected.

To emulate confinement in the boundary theory, one can introduce a quadratic dilaton, , similar to the manipulation mentioned in [20]. To this end, the Andreev-Zakharov model can be defined by the metric of (5) multiplied by a warp factor, , where is the deformation parameter whose value can be fixed from the meson trajectory as [27]. Then the metric of the Andreev-Zakharov model is given by [21]

If one works with as the radial coordinate, the metric of (7) turns intowithNow, the wrap factor becomes and the temperature is with as the horizon. Note that the two metrics (7) and (8) are equal but only with different coordinate systems.

#### 3. The Entropic Force

The entropic force is an emergent force. According to the second law of thermodynamics, it stems from multiple interactions which drive the system toward the state with a larger entropy. This force was originally introduced in [31] many years ago and proposed to responsible for the gravity [32] recently. In a more recent work, Kharzeev [8] argued that it would be responsible for dissociating the quarkonium.

In [8], the entropic force is expressed aswhere is the temperature of the plasma, represents the interdistance of , and stands for the entropy.

On the other hand, the entropy is given bywhere is the free energy of , which is equal to the on-shell action of the fundamental string in the dual geometry from the holographic point of view. In fact, the free energy has been studied, for example, in [33–35].

We now follow the calculations of [15] to analyze the entropic force with the metric (8). The Nambu-Goto action iswhere is the fundamental string tension and can be related to the ’t Hooft coupling constant by . denotes the determinant of the induced metric withwhere is the target space coordinates and is the metric.

Parameterizing the static string coordinates byone finds the induced metric aswith .

Then the Lagrangian density is found to bewith

Note that does not depend on explicitly, so the Hamiltonian density is a constant:

Applying the boundary condition at ,one findswithwhere is the lowest position of the string in the bulk.

From (16), (18), and (20), one gets

By integrating (22), the interquark distance is obtained: with

To analyze the effect of the deformation parameter on the interdistance, we plot as a function of with for and GeV^{2} in Figure 1. From the figures, one can see that increases in the presence of . Namely, the deformation parameter has the effect of increasing the interdistance.