Entropic destruction of heavy quarkonium from a deformed $AdS_5$ model

In this paper, 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 inter-distance and the entropic force are investigated. The influence of the deformation parameter on the quarkonium dissociation is analyzed. It is shown that the inter-distance increases in the presence of the deformation parameter. In addition, the deformation parameter has the effect of decreasing the entropic force. This results imply that the quarkonium dissociates harder in a deformed AdS background than that in an usual AdS background, in agreement with earlier findings.

one can compare the results of c = 0 with c = 0 while the "usual" behavior of meson can be recovered in the limit c → 0. 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 inter-distance 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.

II. THE ANDREEV-ZAKHAROV MODEL
Before reviewing the Andreev-Zakharov model, let us briefly introduce the holographic models in terms of the action [26] where G 5 is the five-dimensional Newton constant. g denotes the determinant of the metric g MN . R refers to the Ricci scalar. φ is called the scalar and induces the deformation away from conformality. f (φ) represents the gauge kinetic function. F MN stands for the field strength associated with an Abelian gauge connection A M . V (φ) is the potential which contains the cosmological constant term 2Λ and some other terms.
To obtain an AdS-black hole space-time, one considers a constant scalar field φ (or called dilaton) and assumes that V (φ) = 2Λ as well as f (φ) = 1. Then the action of (1) can be simplified as with the equations of motion where ∇ M is the Levi-Civita covariant derivative with respect to the metric g MN . Supposing that the horizon function f (z) vanishes at the point z = z h , then the solution of (3) (with vanishing right-hand side) becomes the AdS 5 -Schwarzschild metric with where z h can be related to the temperature as T = 1/(πz h ). Notice that in the limit z h → ∞ (correspond to zero temperature), the metric of (5) reduces to the AdS 5 metric, as expected.
To emulate confinement in the boundary theory, one can introduce a quadratic dilaton, φ ∝ z 2 , similarly 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, h(z) = e φ = e 1 2 cz 2 , where c is the deformation parameter whose value can be fixed from the ρ meson trajectory as c ∼ 0.9GeV 2 [27]. Then the metric of the Andreev-Zakharov model is given by [21] If one works with r = R 2 /z as the radial coordinate, the metric of (7) turns into with now the wrap factor becomes h(r) = e cR 4 2r 2 and the temperature is T = r h /(πR 2 ) with r = r h the horizon. Note that the two metrics (7) and (8) are equal but only with different coordinate systems.

III. 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, D. E. Kharzeev [8] argued that it would be responsible for dissociating the quarkonium.
In [8], the entropic force is expressed as where T is the temperature of the plasma, L represents the inter-distance of QQ, S stands for the entropy.
On the other hand, the entropy is given by where F is the free energy of QQ, 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][34][35].
We now follow the calculations of [15] to analyze the entropic force with the metric (8). The Nambu-Goto action is where T F = 1 2πα ′ is the fundamental string tension and α ′ can be related to the 't Hooft coupling constant by α ′ = R 2 √ λ . g denotes the determinant of the induced metric with where X µ is the the target space coordinates and g µν is the metric.
Parameterizing the static string coordinates by one finds the induced metric as g 00 = r 2 R 2 h(r)f (r), g 01 = g 10 = 0, withṙ = ∂r ∂σ . Then the Lagrangian density is found to be with Note that L does not depend on σ explicitly, so the Hamiltonian density is a constant Applying the boundary condition at σ = 0,ṙ one finds where r c is the lowest position of the string in the bulk. From (16), (18) and (20), one getsṙ By integrating (22), the inter-quark distance is obtained with To analyze the effect of the deformation parameter on the inter-distance, we plot LT as a function of ε with ε ≡ r h /r c for c = 0 and c = 0.9GeV 2 in Fig.1. From the figures, one can see that LT increases in the presence of c. Namely, the deformation parameter has the effect of increasing the inter-distance.
Moreover, one finds that for each plot LT is a increasing function for ε < ε max but a decreasing one for ε > ε max . In fact, in the later case some new configurations [36] should be taken into account. However, these configurations are not solutions of the Nambu-Goto action so that the range of LT > LT max is not trusted. In other words, we have more interest in the range of LT < LT max . For convenience, we write b = LT max . With numerical methods, we find b ≃ 0.31 for c = 0.9GeV 2 and b ≃ 0.27 for c = 0.
Next we discuss the free energy. There are two cases: 1). If L > b T , the fundamental string will break in two pieces implying the quarks are completely screened. For this case, the choice of the free energy F (2) is not unique [37], we here choose a configuration of two disconnected trailing drag strings [38], that is In terms of (11), one finds notice that the results of [15] can be reproduced if one neglects the effect of the reformation parameter by plugging c = 0 in (26).
2). If L < b T , the fundamental string is connected. The free energy of the quark anti-quark pair can be obtained by substituting (22) into (12), that is Likewise, using (11) one finds where the derivatives are with respect to r h and we have used the relation α ′ = R 2 √ λ . To analyze the effect of the deformation parameter on the entropic force, we plot S (1) / √ λ as a function of LT for c = 0 and c = 0.9GeV 2 in Fig 2, respectively. One can see that increasing c leads to smaller entropy at small distances. In addition, from (10) one knows that the entropic force is related to the growth of the entropy with the distance, so one finds that increasing c leads to decreasing the entropic force. On the other hand, the entropic force is responsible for melting the quarkonium. Thus, one concludes that the presence of the deformation parameter tends to decrease the entropic force thus making the heavy quarkonium dissociates harder. This results can be understood as follows. Increase of the inter-distance can be regarded as decrease of r h or decrease of the system temperature. Since the deformation parameter has the effect of increasing the inter-distance, it will cool the system temperature thus making the quarkonium dissociates harder. Interestingly, it was argued [29] that the deformation parameter has the effect of increasing the thermal thus increasing the dissociation length, in agreement with our findings.

IV. SUMMARY AND DISCUSSIONS
In heavy ion collisions, the dissociation of heavy quarknioum is an important experimental signal for QGP formation. Recently, the destruction of heavy quarkonium due to the entropic force has been discussed in the context of AdS/CFT [15]. It was shown that a sharp peak of the entropy exists near the deconfinement transition and the growth of the entropy with the distance is responsible for the entropic force.
In this paper, we have investigated the destruction of heavy quarkonium in a deformed AdS 5 model. The effect of the deformation parameter on the inter-distance was analyzed. The influence of the deformation parameter on the entropic force was also studied. It is shown that the inter-quark distance increases in the presence of the deformation parameter. Moreover, the deformation parameter has the effect of decreasing the entropic force. Since the entropic force is responsible for destroying the bound quarkonium states, therefore, we conclude that the presence of the deformation parameter tends to decrease the entropic force thus making the quarkonium melts harder, consistently with the findings of [29]. Also, we have presented a possible understanding to this result: Increase of the inter-distance is equivalent to decrease of r h or decrease of the system temperature. As the deformation parameter can increase the inter-distance, it will cool the system temperature thus making the quarkonium dissociates harder.
In addition, to understand the "usual" or "unusual" behavior of meson, we have compared the results between c = 0 and c = 0. It is found that the quarkonium dissociates harder in a deformed AdS background than that in an usual AdS background.
Finally, it would be interesting to study the entropic force in some other holographic QCD models, such as the Sakai-Sugimoto model [18] and the Pirner-Galow model [24]. This will be left as a further investigation.