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

TL;DR

This study investigates how a deformation parameter in a modified AdS_5 model affects the entropic force and dissociation of heavy quarkonium, revealing that deformation makes quarkonium harder to dissociate.

Contribution

It introduces a deformed AdS_5 model to analyze heavy quarkonium dissociation, highlighting the impact of deformation on entropic force and inter-distance.

Findings

Deformation increases the inter-distance of quarkonium.

Deformation decreases the entropic force.

Quarkonium dissociates more difficult in deformed AdS background.

Abstract

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.

Equations

$$S=\frac{1}{16\pi G_5}\int d^{5}x\sqrt{g}(\mathcal{R}-\frac{1}{2}(\partial \varphi {)}^2-V(\varphi )-\frac{f(\varphi )}{4}{F}_{MN}{F}^{MN}),$$

$$S=\frac{1}{16\pi G_5}\int d^{5}x\sqrt{g}(\mathcal{R}-2\Lambda -\frac{1}{4}{F}_{MN}{F}^{MN}),$$

$${\mathcal{R}}_{MN}-\frac{1}{2}\mathcal{R}{g}_{MN}+\Lambda g_{MN}=\frac{1}{2}(F_{MA}{F}_N^A-\frac{1}{4}{g}_{MN}{F}_{AB}{F}^{AB}),$$

$${\nabla }_M{F}^{MN}=0.$$

$$d{s}^2=\frac{R^2}{z^2}(-f(z)d{t}^2+d{\vec{x}}^2+\frac{1}{f(z)}d{z}^2),$$

$$f(z)=1-\frac{z^4}{z_h^4},$$

$$d{s}^2=\frac{R^{2}h(z)}{z^2}(-f(z)d{t}^2+d{\vec{x}}^2+\frac{1}{f(z)}d{z}^2).$$

$$d{s}^2=\frac{r^{2}h(r)}{R^2}(-f(r)d{t}^2+d{\vec{x}}^2)+\frac{R^{2}h(r)}{r^{2}f(r)}d{r}^2,$$

$$f(r)=1-\frac{r_h^4}{r^4},$$

$$\mathcal{F}=T\frac{\partial S}{\partial L},$$

$$S=-\frac{\partial F}{\partial T},$$

$$S=T_F\int d\tau d\sigma \mathcal{L}=T_F\int d\tau d\sigma \sqrt{g},$$

$$g_{\alpha \beta }=g_{\mu \nu }\frac{\partial X^{\mu }}{\partial {\sigma }^{\alpha }}\frac{\partial X^{\nu }}{\partial {\sigma }^{\beta }},$$

$$X^{\mu }=(\tau ,\sigma ,0,0,r(\sigma )),$$

$$g_{00}=\frac{r^2}{R^2}h(r)f(r),g_{01}=g_{10}=0,g_{11}=\frac{r^2}{R^2}h(r)+\frac{R^2}{r^2}h(r)f(r{)}^{-1}{\dot{r}}^2,$$

$$\mathcal{L}=\sqrt{a(r)+b(r){\dot{r}}^2}.$$

$$a(r)$$

$$\mathcal{H}=\mathcal{L}-\frac{\partial \mathcal{L}}{\partial \dot{r}}\dot{r}=constant.$$

$$\dot{r}=0,r=r_c,$$

$$\mathcal{H}=\sqrt{\frac{h^2(r_c)f(r_c)r_c^4}{R^4}},$$

$$f(r_c)=1-\frac{r_h^4}{r_c^4},h(r_c)=e^{\frac{c{R}^4}{2r_c^2}},$$

$$\dot{r}=\frac{dr}{d\sigma }=\sqrt{\frac{a^2(r)-a(r)a(r_c)}{a(r_c)b(r)}}.$$

$$L=2{\int }_{r_c}^{\infty }dr\sqrt{\frac{a(r_c)b(r)}{a^2(r)-a(r)a(r_c)}},$$

$$a(r_c)=\frac{h^2(r_c)f(r_c)r_c^4}{R^4}.$$

$$F^{(2)}=2T_F{\int }_{r_h}^{\infty }h(r)dr.$$

$$S^{(2)}\simeq e^{\frac{c}{2r_h^2}}(1-\frac{c}{r_h^2})\sqrt{\lambda }\theta (L-\frac{b}{T}),$$

$$F^{(1)}=\frac{1}{\pi {\alpha }^{\prime }}{\int }_{r_c}^{\infty }dr\sqrt{\frac{a(r)b(r)}{a(r)-a(r_c)}}.$$

$$S^{(1)}=-\frac{\sqrt{\lambda }}{2\pi }{\int }_{r_c}^{\infty }dr\frac{[a^{\prime }(r)b(r)+a(r)b^{\prime }(r)][a(r)-a(r_c)]-a(r)b(r)[a^{\prime }(r)-a^{\prime }(r_c)]}{\sqrt{a(r)b(r)[a(r)-a(r_c){]}^3}},$$

Full text

I Introduction

It is well known that the dissociation of heavy quarknioum can be regarded as an important experimental signal of the formation of strongly coupled quark-gloun plasma (QGP) TMA . 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) AAD ; BBA . Obviously, this is in contradiction with the Debye screening scenario TMA as well as the thermal activation through the impact of gluons DKH ; EV . 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 PBR ; RLT that if a region of deconfined quarks and gluons is formed, the quarkoniums (or bound states) can be formed from a quark and an antiquark which were originally produced in separate incoherent interactions. Recently, Kharzeev DEK argued that this puzzle may be related to the nature of deconfinement and the entropic force would be responsible for melting the quarknium. This argument originated from the Lattice results that a large amount of entropy associated with the heavy quarknioum placed in QGP DKA1 ; DKA2 ; PPE .

AdS/CFT, which maps a $d$-dimensional quantum field theory to its dual gravitational theory, living in $d+1$-dimensional, has yielded many important insights into the dynamics of strongly coupled gauge theories Maldacena:1997re ; Gubser:1998bc ; MadalcenaReview . In this approach, K. Hashimoto et al have studied the entropic destruction of static heavy quarkonium in $\mathcal{N}=4$ SYM theory and a confining YM theory firstly. They found that in both cases the entropy grows as a function of the inter-quark distance giving rise to the entropic force KHA . Soon this studies have been extended to the case of moving quarkonium KBF . It was shown that the velocity has the effect of increasing the entropic force thus making the quarkonium melts easier. Recently, we have analyzed the effect of chemical potential on the entropic force and observed that the moving quarkonium dissociates easier at finite density ZQ .

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 anti-quark interaction, see TSS ; JEE ; AKE ; OA3 ; SH1 ; UGE ; HJ ; SH2 ; OD and references therein. In this paper, we will study the entropic force in the Andreev-Zakharov model OA3 , one of ”soft wall” models. The Andreev-Zakharov model has some properties: (1) The positive quadratic term modification in the deformed warp factor $h(z)=e^{\frac{1}{2}cz^{2}}$ produces linear behavior of heavy flavor potential, namely, it can provide confinement at low temperature. (2) the value of $c$ can be fixed from the $\rho$ 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 OA2 , thermal width NRF ; JSA , and heavy quark potential ZQ1 . 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 $c\neq 0$ with $c=0$ while the ”usual” behavior of meson can be recovered in the limit $c\rightarrow 0$. 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 OD

[TABLE]

where $G_{5}$ is the five-dimensional Newton constant. $g$ denotes the determinant of the metric $g_{MN}$. $\mathcal{R}$ refers to the Ricci scalar. $\phi$ is called the scalar and induces the deformation away from conformality. $f(\phi)$ represents the gauge kinetic function. $F_{MN}$ stands for the field strength associated with an Abelian gauge connection $A_{M}$. $V(\phi)$ is the potential which contains the cosmological constant term 2$\Lambda$ and some other terms.

To obtain an AdS-black hole space-time, one considers a constant scalar field $\phi$ (or called dilaton) and assumes that $V(\phi)=2\Lambda$ as well as $f(\phi)=1$. Then the action of (1) can be simplified as

[TABLE]

with the equations of motion

[TABLE]

[TABLE]

where $\nabla_{M}$ 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

[TABLE]

with

[TABLE]

where $z_{h}$ can be related to the temperature as $T=1/(\pi z_{h})$. Notice that in the limit $z_{h}\rightarrow\infty$ (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, $\phi\propto z^{2}$, similarly to the manipulation mentioned in AKE . To this end, the Andreev-Zakharov model can be defined by the metric of (5) multiplied by a warp factor, $h(z)=e^{\phi}=e^{\frac{1}{2}cz^{2}}$, where $c$ is the deformation parameter whose value can be fixed from the $\rho$ meson trajectory as $c\sim 0.9GeV^{2}$ OA2 . Then the metric of the Andreev-Zakharov model is given by OA3

[TABLE]

If one works with $r=R^{2}/z$ as the radial coordinate, the metric of (7) turns into

[TABLE]

with

[TABLE]

now the wrap factor becomes $h(r)=e^{\frac{cR^{4}}{2r^{2}}}$ and the temperature is $T=r_{h}/(\pi R^{2})$ 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 KH many years ago and proposed to responsible for the gravity EP recently. In a more recent work, D. E. Kharzeev DEK argued that it would be responsible for dissociating the quarkonium.

In DEK , the entropic force is expressed as

[TABLE]

where $T$ is the temperature of the plasma, $L$ represents the inter-distance of $Q\bar{Q}$, $S$ stands for the entropy.

On the other hand, the entropy is given by

[TABLE]

where $F$ is the free energy of $Q\bar{Q}$, 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 JMM ; ABR ; SJR .

We now follow the calculations of KHA to analyze the entropic force with the metric (8). The Nambu-Goto action is

[TABLE]

where $T_{F}=\frac{1}{2\pi\alpha^{\prime}}$ is the fundamental string tension and $\alpha^{\prime}$ can be related to the ’t Hooft coupling constant by $\alpha^{\prime}=\frac{R^{2}}{\sqrt{\lambda}}$. $g$ denotes the determinant of the induced metric with

[TABLE]

where $X^{\mu}$ is the the target space coordinates and $g_{\mu\nu}$ is the metric.

Parameterizing the static string coordinates by

[TABLE]

one finds the induced metric as

[TABLE]

with $\dot{r}=\frac{\partial r}{\partial\sigma}$.

Then the Lagrangian density is found to be

[TABLE]

with

[TABLE]

Note that $\mathcal{L}$ does not depend on $\sigma$ explicitly, so the Hamiltonian density is a constant

[TABLE]

Applying the boundary condition at $\sigma=0$,

[TABLE]

one finds

[TABLE]

with

[TABLE]

where $r_{c}$ is the lowest position of the string in the bulk.

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

[TABLE]

By integrating (22), the inter-quark distance is obtained

[TABLE]

with

[TABLE]

To analyze the effect of the deformation parameter on the inter-distance, we plot $LT$ as a function of $\varepsilon$ with $\varepsilon\equiv r_{h}/r_{c}$ 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 $\varepsilon<\varepsilon_{max}$ but a decreasing one for $\varepsilon>\varepsilon_{max}$. In fact, in the later case some new configurations DB 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 $b\simeq 0.31$ for $c=0.9GeV^{2}$ and $b\simeq 0.27$ for $c=0$.

Next we discuss the free energy. There are two cases:

1). If $L>\frac{b}{T}$, 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 MCH , we here choose a configuration of two disconnected trailing drag strings CPH , that is

[TABLE]

In terms of (11), one finds

[TABLE]

notice that the results of KHA can be reproduced if one neglects the effect of the reformation parameter by plugging $c=0$ in (26).

2). If $L<\frac{b}{T}$, the fundamental string is connected. The free energy of the quark anti-quark pair can be obtained by substituting (22) into (12), that is

[TABLE]

Likewise, using (11) one finds

[TABLE]

where the derivatives are with respect to $r_{h}$ and we have used the relation $\alpha^{\prime}=\frac{R^{2}}{\sqrt{\lambda}}$.

To analyze the effect of the deformation parameter on the entropic force, we plot $S^{(1)}/\sqrt{\lambda}$ 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 JSA 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 KHA . 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 JSA . 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 $c\neq 0$ 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 TSS and the Pirner-Galow model HJ . This will be left as a further investigation.

V Acknowledgments

This research is partly supported by the Ministry of Science and Technology of China (MSTC) under the 973 Project no. 2015CB856904(4). Zi-qiang Zhang and Gang Chen are supported by the NSFC under Grant no. 11475149. De-fu Hou is supported by the NSFC under Grant no. 11375070, 11521064.