Probing HQCD
Sara<EMAIL_ADDRESS>and Razieh
<EMAIL_ADDRESS>
a Department of Engineering Sciences and Physics, Buein Zahra Technical
University, Buein Zahra, Qazvin, Iran
b UNESCO UNISA ITL/NRF Africa Chair in Nanosciences/Nanotechnology, College of
Graduate Studies, University of South Africa (UNISA), South Africa
c Nanosciences African Network, Materials Research Department, iThemba LABS,
National Research Foundation, South Africa

In this paper, we employ the gauge/gravity duality to study some features of
the quark-gluon plasma. For this purpose, we implement a holographic QCD model
constructed from an Einstein-Maxwell-Dilaton gravity at finite temperature and
finite chemical potential. The model captures both the confinement and
deconfinement phases of QCD and we use it to study the effect of temperature
and chemical potential on a heavy quark moving through the plasma. We
calculate the drag force, Langevin diffusion coefficients and also the jet
quenching parameter, and our results align with other hQCD models and the
experimental data.

###### Contents

  1. 1 Introduction
  2. 2 Einstein-Maxwell-dilaton gravity
  3. 3 Drag Force
  4. 4 Langevin equation
  5. 5 Jet Quenching Parameter
  6. 6 Conclusion

##  1 Introduction

Quark-gluon plasma (QGP) produced at the Relativistic Heavy Ion Collider
(RHIC) and the Large Hadron Collider (LHC) is a strongly coupled plasma whose
dynamics is dominated by non-perturbative effects [1, 2]. Since the
perturbative QCD is applicable only in the weak coupling regime, the lattice
QCD techniques could be hired for understanding the static equilibrium
thermodynamics of such matter. Alternatively, the ”AdS/CFT correspondence”
[3]-[7] provides a non-perturbative tool to examine the dynamical quantities
of the strongly coupled QGP. The ”AdS/CFT correspondence” indicates a duality
between the $\mathcal{N}=4\,SU(N_{c})$ super-Yang-Mills theory and type IIB
string theory on $AdS_{5}\times S^{5}$, which is a powerful tool to study the
strongly coupled gauge theory in the large $N_{c}$ limit and large ’t Hooft
coupling. The original duality maps an asymptotically AdS space to a conformal
gauge theory at zero temperature. On the other hand, due to the fact that the
physical quantities of the quark-gluon plasma are temperature dependent, many
attempts were made to extend the original duality to holographic models
describing the QGP at finite temperature using top-down [8]-[10] or bottom-up
models [11]-[39].

One of the interesting features of QCD is the confinement-deconfinement phase
transition where the QCD coupling constant becomes very large. This phenomenon
leads to a strong suppression of quarkonium near the phase transition
temperature, as the confined phase of QCD is at a lower temperature and
density compared to the deconfined phase [40]-[42]. Lattice QCD calculations
demonstrate that the entropy of the quark-antiquark pair develops a peak
around the phase transition area [43, 44] which indicates the strong
interaction between the quark-antiquark pair in this region. The quark-
antiquark entropy in both phases and the confinement-deconfinement phase
transition have both been thoroughly studied utilizing the AdS/CFT
correspondence and generalized dual holographic models [45]-[50].

Among holographic models of QCD, the model constructed from a black hole
solution using the Einstein-Maxwell-Dilaton gravity is beneficial as it
incorporates the temperature dependency of the entropy of the quark-antiquark
pair in both confined and deconfined phases [51, 52]. On the other hand, this
holographic model includes chemical potential as the QCD equation of state and
phase diagram depends on the chemical potential. On the gravity side,
increasing temperature leads to a phase transition from thermal AdS to a black
hole using a specific choice of an arbitrary function $A(z)$. In the boundary
theory, these are dual to phase transitions from confined to deconfined
phases. The results of this model for the quark-antiquark entropy are in good
agreement with those from Lattice QCD and therefore, this holographic model
could be a capable model to study the QCD feature in confined and deconfined
phases.

Jets of quarks propagating through the medium are the most interesting objects
produced in QGP and their interaction with the medium is one of the
challenging problems in new physics. The transport coefficient, also known as
the jet quenching parameter, is one of the most interesting experimental
observables associated with quark energy loss in the hot medium observed at
RHIC and LHC [54]-[61]. It is defined as the average squared transverse
momentum transferred from a traversing parton to the medium, per unit mean
free path [62]. The parton energy loss through the medium can also be
investigated using the drag force experienced by heavy quarks moving in the
medium. Due to the strong nature of the interaction, holographic models of QCD
are utilized as powerful tools to investigate the dynamics of this phenomena
and characterize properties of jets and QGP medium as well as their
interactions which leads to the quark energy loss. Also, heavy quarks moving
in QGP undergo a Brownian-like motion which could be studied by the Langevin
diffusion coefficients [63, 64].

According to the AdS/CFT correspondence, a heavy quark is illustrated as a
fundamental string attached to a flavor brane. The string endpoint could be
considered as a quark in the boundary theory and the string itself can be
considered as a gluonic cloud surrounding the quark. The drag force of a quark
moving in the medium can be measured from the corresponding momentum flowing
of an open trailing string to the bulk [65]-[83]. The quantum fluctuations of
this trailing string are also dual to the momentum broadening of a heavy quark
moving in the QGP and can be computed along the quark direction of the motion
as well as the transverse plane. The stochastic Brownian motion is associated
with the correlators of the string fluctuations and can be expressed in terms
of the Langevin coefficients [81]-[96]. According to this duality, the jet
quenching parameter is related to the thermal expectation value of the light-
like Wilson loop operator created from the trajectory of two string endpoints,
and many attempts were made to compute this parameter in the context of the
AdS/CFT correspondence [97]-[123].

In this paper, we study the QGP aspects such as the drag force, Langevin
coefficients, and the jet quenching parameter for a heavy quark moving through
the plasma using the gauge/gravity duality. For this purpose, we implement the
dynamical holographic QCD model of ref.[52] as the dual boundary theory
involves temperature in both confined and deconfined phases as well as the
chemical potential and has a qualitative agreement with Lattice results for
the quark-antiquark entropy in both confined and deconfined phases, therefore
it would be an appropriate model to examine the QGP features.

This paper is organized as follows: In section 2 we briefly review the
holographic QCD model constructed from the Eintein-Maxwell-Dilaton gravity
introduced in [52]. In section 3 we discuss the drag force on heavy quark
moving in the dynamical holographic model of QCD. In section 4, the Langevin
coefficients are calculated. In section 5, the jet quenching parameter is
studied, and in section 6, a summary and conclusion are presented.

##  2 Einstein-Maxwell-dilaton gravity

In this section, we review the Einstein-Maxwell-dilaton gravity (EMD) model at
finite and zero temperature [51, 52] which is described by the following
action in 5 dimensions,

$\displaystyle S_{EM}=-\frac{1}{16\pi G_{5}}\int\mathrm{d^{5}}x\sqrt{-g}\ \
\bigl{[}R-\frac{f(\phi)}{4}F_{MN}F^{MN}-\frac{1}{2}\partial_{M}\phi\partial^{M}\phi-V(\phi)\bigr{]}\,.$
(1)

Here $G_{5}$ is the Newton constant, $V(\phi)$ is the potential of the dilaton
field and $f(\phi)$ is the gauge kinetic function representing the coupling
between the dilaton and the gauge field $A_{M}$.

By assuming the following ansatz for the metric, gauge field, and dilaton
field,

$\displaystyle
ds^{2}=\frac{L^{2}e^{2A(z)}}{z^{2}}\biggl{(}-g(z)dt^{2}+dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}+\frac{dz^{2}}{g(z)}\biggr{)}\,,$
$\displaystyle A_{M}=A_{t}(z),\ \ \ \ \phi=\phi(z),$ (2)

where L is the AdS length scale and fields are assumed to be functions of
extra radial coordinate $z$, one can solve the equations of motion
analytically in terms of $A(z)$ and $f(z)$ [11, 12]

$\displaystyle g(z)=1-\frac{\int_{0}^{z}dx\
x^{3}e^{-3A(x)}\int_{x_{c}}^{x}dx_{1}\
\frac{x_{1}e^{-A(x_{1})}}{f(x_{1})}}{\int_{0}^{z_{h}}dx\
x^{3}e^{-3A(x)}\int_{x_{c}}^{x}dx_{1}\ \frac{x_{1}e^{-A(x_{1})}}{f(x_{1})}},$
$\displaystyle\phi^{\prime}(z)=\sqrt{6(A^{\prime
2}-A^{\prime\prime}-2A^{\prime}/z)},$ $\displaystyle
A_{t}(z)=\sqrt{\frac{-1}{\int_{0}^{z_{h}}dx\
x^{3}e^{-3A(x)}\int_{x_{c}}^{x}dx_{1}\
\frac{x_{1}e^{-A(x_{1})}}{f(x_{1})}}}\int_{z_{h}}^{z}dx\
\frac{xe^{-A(x)}}{f(x)},$ $\displaystyle
V(z)=-\frac{3z^{2}ge^{-2A}}{L^{2}}\bigl{[}A^{\prime\prime}+A^{\prime}\bigl{(}3A^{\prime}-\frac{6}{z}+\frac{3g^{\prime}}{2g}\bigr{)}-\frac{1}{z}\bigl{(}-\frac{4}{z}+\frac{3g^{\prime}}{2g}\bigr{)}+\frac{g^{\prime\prime}}{6g}\bigr{]}\,.$
(3)

The spacetime asymptotic boundary is located at $z=0$ where $g(z)$ goes to 1,
and $z=z_{h}$ is the position of the horizon with the boundary condition of
$g(z_{h})=0$.

It is shown that by considering the simple form of $f(z)=e^{cz^{2}-A(z)}$, one
can reproduce the linear Regge trajectory of the discrete spectrum of the
mesons. The value of $c=1.16\ \text{GeV}^{2}$ is fixed by matching the
holographic meson mass spectrum to the lowest-lying heavy meson states [11].

The gravity solution corresponds to a black hole with horizon located at
$z_{h}$, is then given by

$\displaystyle g(z)=1-\frac{1}{\int_{0}^{z_{h}}dx\
x^{3}e^{-3A(x)}}\biggl{[}\int_{0}^{z}dx\
x^{3}e^{-3A(x)}+\frac{2c\mu^{2}}{(1-e^{-cz_{h}^{2}})^{2}}\det\mathcal{G}\biggr{]},$
$\displaystyle\phi^{\prime}(z)=\sqrt{6(A^{\prime
2}-A^{\prime\prime}-2A^{\prime}/z)},$ $\displaystyle
A_{t}(z)=\mu\frac{e^{-cz^{2}}-e^{-cz_{h}^{2}}}{1-e^{-cz_{h}^{2}}},$
$\displaystyle
V(z)=-\frac{3z^{2}ge^{-2A}}{L^{2}}\left[A^{\prime\prime}+A^{\prime}\left(3A^{\prime}-\frac{6}{z}+\frac{3g^{\prime}}{2g}\right)-\frac{1}{z}\left(-\frac{4}{z}+\frac{3g^{\prime}}{2g}\right)+\frac{g^{\prime\prime}}{6g}\right]\,,$
(4)

where $\mu$ is the chemical potential and

$\det\mathcal{G}=\begin{vmatrix}\int_{0}^{z_{h}}dx\
x^{3}e^{-3A(x)}&\int_{0}^{z_{h}}dx\ x^{3}e^{-3A(x)-cx^{2}}\\\
\int_{z_{h}}^{z}dx\ x^{3}e^{-3A(x)}&\int_{z_{h}}^{z}dx\
x^{3}e^{-3A(x)-cx^{2}}\end{vmatrix}\,.$

The Hawking temperature and entropy of this solution are

$\displaystyle T$ $\displaystyle=$
$\displaystyle\frac{z_{h}^{3}e^{-3A(z_{h})}}{4\pi\int_{0}^{z_{h}}dx\
x^{3}e^{-3A(x)}}\biggl{[}1+\frac{2c\mu^{2}\bigl{(}e^{-cz_{h}^{2}}\int_{0}^{z_{h}}dx\
x^{3}e^{-3A(x)}-\int_{0}^{z_{h}}dx\
x^{3}e^{-3A(x)}e^{-cx^{2}}\bigr{)}}{(1-e^{-cz_{h}^{2}})^{2}}\biggr{]}$
$\displaystyle S_{BH}$ $\displaystyle=$
$\displaystyle\frac{L^{3}e^{3A(z_{h})}}{4G_{5}z_{h}^{3}}\,.$ (5)

There is another solution for the Einstein-Maxwell-dilaton equations which
corresponds to a thermal-AdS space. This solution is obtained by taking the
$z_{h}\rightarrow\infty$ limit of the solution 4 which results in $g(z)=1$.
The chemical potential goes to zero in this limit and therefore, the thermal
gas solution has zero chemical potential. This thermal solution is
asymptotically AdS. Nevertheless, it can have a non-trivial structure in the
bulk due to the scale factor $A(z)$.

For the holographic model to describe the confinement-deconfinement phases in
the boundary theory, the following form of A(z) is being considered

$\displaystyle A(z)=A_{2}(z)=-\bar{a}z^{2}\,,$ (6)

which vanishes at the boundary. The value of $\bar{a}=c/8\simeq 0.145$ is
determined such that the Hawking/Page phase transition occurs at $T\simeq
270MeV$ at zero chemical potential.

With this choice of $A(z)$, the thermodynamic properties of the solution are
shown in Figs.1 and 2. In Fig. 1(a), the temperature of the solution is
plotted in terms of $z_{h}$. At each temperature, for small values of $\mu$,
there are two solutions: large black hole (small $z_{h}$) and small black hole
(large $z_{h}$). The small black hole solution is thermodynamically unstable.
Also the black hole solution only exist above a minimum temperature, $T_{min}$
indicating there is a first order phase transition from black hole to thermal-
gas phase. Although, this transition happens at some critical temperature
$T_{c}$ above the $T_{min}$. In Fig. 1(b), the ratio of critical temperature
to the minimum temperature is plotted for different values of $\mu$. This
ratio is always larger than one and enhanced by increasing the chemical
potential.

Increasing the value of chemical potential in Fig. 1(a), the branch with
negative slope (the small black hole solution) becomes smaller and completely
vanishes at some critical chemical potential $\mu=\mu_{c}=0.673$. At
$\mu>\mu_{c}$, we have a single black hole solution which remains stable at
all temperatures, see Fig. 1(a).

(a)

(b)

Figure 1: (Color online) (1(a)) The temperature of the solution versus $z_{h}$
for different values of $\mu$.(1(b)) The ratio of critical temperature to the
minimum temperature for different values of $\mu$. Stable solutions are
represented by solid lines while dashed lines characterize the unstable
solutions.

In Fig. 2(a), the normalized entropy of the system is plotted in terms of
temperature. By increasing temperature, this ratio reaches the value of one.
In Fig. 2(b), the free energy of the solution is plotted in terms of the
temperature. The free energy is normalized such that the free energy of
thermal-AdS is zero. Therefore, changing the sign of free energy means a
first-order phase transition from AdS black hole to thermal-AdS as the
temperature decreases which takes place at $T=T_{c}$. This is the famous
Hawking–Page phase transition. In [52], this Hawking-Page phase transition on
the gravity side is used to determine the confinement/deconfinement phase
transition on the dual boundary side. They used the phase for which the free
energy of the probe quark-antiquark pair varies linearly with respect to their
separation length, as confinement. Here, we explore this
confinement/deconfinement phase transition using the external probes on the
gravity side.

(a)

(b)

Figure 2: (Color online)(2(a)) The ratio of black hole entropy to the AdS-Sch
entropy versus temperature for different values of $\mu$. (2(b)) The free
energy versus temperature for different values of chemical potential between
zero (red) and $\mu=\mu_{c}$ (purple). For each value of $\mu$, there is a
phase transition between the AdS-BH and thermal gas solution. Here,
$\mu_{c}\,=\,0.673\,GeV$ is the critical value for $\mu$ for which there is no
phase transition and the large black hole solution exists from zero
temperature to high temperature. Solid (Dashed) lines show the stable
(unstable) solutions.

In order to study the holographic probes we are interested in, it is more
convenient to use the string frame metric which can be obtained using the
following standard transformations,

$\displaystyle(g_{s})_{MN}=e^{\sqrt{\frac{2}{3}}\phi}g_{MN},$ $\displaystyle
ds_{s}^{2}=\frac{L^{2}e^{2A_{s}(z)}}{z^{2}}\biggl{(}-g(z)dt^{2}+\frac{dz^{2}}{g(z)}+dx_{1}^{2}+dx_{3}^{2}+dx_{3}^{2}\biggr{)}\,,$
(7)

where $A_{s}(z)=A(z)+\sqrt{\frac{1}{6}}\phi(z)$.

The open string tension $T_{s}$ in units of the AdS length scale $L^{2}$ is
considered to be $T_{s}L^{2}\simeq 0.1$ [52], which is obtained by comparing
the numerical results with the lattice QCD estimate of the string tension,
i.e. $\sigma_{s}\approx 1/(2.34)^{2}GeV^{2}$ [53].

##  3 Drag Force

In this section, we consider a heavy quark moves with a constant velocity in
one of the spatial directions denoted by x. Therefore, the embedding function
of a heavy quark in the static gauge $\tau=t\,,\sigma=z$ is
$X=\\{t,x(t,z),z\\}$. The action of a fundamental string is given by the
Nambu- Goto action as,

$\displaystyle S$ $\displaystyle=$
$\displaystyle-\frac{1}{2\pi\alpha^{\prime}}\int\,d\sigma\,d\tau\,\sqrt{-\gamma}\,$
(8) $\displaystyle=$
$\displaystyle-\frac{1}{2\pi\alpha^{\prime}}\int\,d\sigma\,d\tau\,\sqrt{-G_{tt}G_{zz}-G_{tt}G_{xx}x^{\prime\,2}-G_{xx}G_{zz}\dot{x}^{2}}\,,$

where dot and prime are derivatives with respect to $\tau$ and $\sigma$ and
$G$ represent the components of the background metric. In order to study the
dynamics of string, we use the metric of Eq.7. In the static gauge, the
corresponding Lagrangian density takes the following form,

$\displaystyle\mathcal{L}=-\frac{1}{2\pi\alpha^{\prime}}\frac{e^{2A_{s}(z)}}{z^{2}}\sqrt{1-\frac{\dot{x}^{2}}{g(z)}+g(z)x^{\prime}(z)^{2}}\,.$
(9)

Then the equation of motion for x is,

$\displaystyle\partial_{t}\left(\frac{\dot{x}}{\sqrt{-\gamma^{\prime}}}\right)-\frac{z^{2}\,g(z)}{e^{2A_{s}(z)}}\,\partial_{z}\left(\frac{e^{2A_{s}(z)}g(z)x^{\prime}}{z^{2}\sqrt{-\gamma^{\prime}}}\right)=0\,.$
(10)

The static string stretching from the boundary to the horizon,
$x(t,z)=Constant$ is a trivial solution of this equation. For a string whose
endpoint at the boundary moves with a constant velocity $v$, the following
ansatz is chosen,

$X=\\{t,vt+\xi(z),z\\}\,,$ (11)

which leads to the following equation of motion,

$\frac{e^{2A_{s}(z)}g(z)\xi^{\prime}(z)}{\sqrt{-\gamma}z^{2}}\,=\,const\,\equiv\pi_{x}\,,$
(12)

where $\pi_{x}$ is the worldsheet conserved quantity. One can obtain the
equation for $\xi$ as follows,

$\xi^{\prime}(z)=\pm\frac{\pi_{x}}{g(z)}\sqrt{\frac{g(z)-v^{2}}{\frac{e^{4A_{s}(z)}g(z)}{z^{4}}-\pi_{x}^{2}}}\,.$
(13)

Requiring the real values for the function under the square root fixes the
constant of the equation of motion as,

$\displaystyle g(z_{s})-v^{2}=0\,,$
$\displaystyle\frac{e^{4A_{s}(z_{s})}g(z_{s})}{z_{s}^{4}}-\pi_{x}^{2}=0\,\,\,\Rightarrow\,\,\,\pi_{x}=\frac{e^{2A_{s}(z_{s})}v}{z_{s}^{2}}\,.$
(14)

Finally, substituting the Eq.3 into the Eq.13, one can solve the string
solution,

$\xi^{\prime}(z)=-\frac{e^{2A_{s}(z_{s})}v}{z_{s}^{2}}\frac{1}{g(z)}\sqrt{\frac{g(z)-g(z_{s})}{\frac{e^{4A_{s}(z)}g(z)}{z^{4}}-\frac{e^{4A_{s}(z_{s})}g(z_{s})}{z_{s}^{4}}}}\,.$
(15)

The canonical momentum densities associated to the string are,

$\displaystyle\left(\begin{array}[]{c}\pi^{0}_{t}\\\
\pi^{0}_{x}\end{array}\right)$ $\displaystyle=$
$\displaystyle\frac{T_{0}L^{4}}{\sqrt{-\gamma}}\,\frac{e^{4A_{s}(z)}}{z^{4}g(z)}\left(\begin{array}[]{c}-g(z)\,\left(g(z)\xi^{\prime}(z)^{2}+1\right)\\\
v\end{array}\right),$ (20)
$\displaystyle\left(\begin{array}[]{c}\pi^{1}_{t}\\\
\pi^{1}_{x}\end{array}\right)$ $\displaystyle=$
$\displaystyle\frac{T_{0}L^{4}}{\sqrt{-\gamma}}\,\frac{e^{4A_{s}(z)}}{z^{4}}\left(\begin{array}[]{c}-v\,g(z)\,\xi^{\prime}(z)\\\
g(z)\,\xi^{\prime}(z)\end{array}\right).$ (25)

Integrating the $\pi^{0}_{t}$ and $\pi^{0}_{x}$ along the string gives us the
total energy and the total momentum in the direction of motion of the string,
respectively. While, $\pi^{1}_{t}$ and $\pi^{1}_{x}$ are the energy and
momentum flow down along the string. It is straightforward to show that
$\pi^{1}_{x}$ is exactly the constant of the equations of motions, $\pi_{x}$
in Eq.3. Also, just similar to the case of $\mathcal{N}=4$ SYM plasma,
$\pi^{1}_{t}=-v\,\pi^{1}_{x}$. It means that if we pull the quark with the
constant velocity, the fraction of energy flow at a given point along the
string, $\pi^{1}_{t}$ is constant. This is the energy dissipating into the
surrounding medium by the quark. Therefore, the drag force is obtained as,

$F_{drag}=-\pi^{1}_{x}=-\frac{1}{2\pi\alpha^{\prime}}\frac{e^{2A_{s}(z_{s})}v}{z_{s}^{2}}$
(26)

The drag force of the AdS/Sch background can be obtained analytically as [76,
77],

$F_{drag}^{SYM}=-\frac{\pi\,T^{2}\,\sqrt{\lambda}}{2}\frac{v}{\sqrt{1-v^{2}}}\,,$
(27)

while in this background, we first need to solve the Eq.3 numerically to
obtain the $z_{s}$, and then calculate the drag force using the Eq.26.

In Fig. 3, the ratio of drag force to its conformal value is plotted in terms
of temperature for two different quark velocities and different values of
$\mu$ from zero to the critical value of $0.673$. The solid curves represent
the drag force ratio of the deconfined phase starting from the critical
temperature $T_{c}$ while the non-physical dashed curves correspond to the
confined phase drag ratio. As mentioned in section 2, a first-order phase
transition from thermal AdS to black hole phase occurs by decreasing $z_{h}$,
and for each chemical potential value, the critical temperature is the
temperature in which the free energy sign changes. Therefore, there exist two
solutions at each temperature, one for the thermal AdS solution and the other
for the AdS Black hole solution. The drag ratio develops a peak around the
critical temperature that moves towards higher temperatures for smaller values
of chemical potential and higher quark velocities, as shown in red and orange
curves inside Fig.3(a). For the deconfined phase, the drag force ratio
increases by increasing the chemical potential value and decreasing the quark
velocity. The same behavior in temperature and chemical potential has been
reported in the hQCD model of [84] for the Einstein-Maxwell-scalar gravity
system (also in [98] the same behavior on baryon chemical potential has been
shown). From this figure, one could also find that at high temperatures, the
curves of all chemical potential values converge and the drag force would be
less than the value of $\mathcal{N}=4\,SYM$ theory due to the smaller ’t Hooft
coupling of the holographic QCD model 111In this paper, we have considered
$T_{s}L^{2}=0.1$ [52] which corresponds to a small ’t Hooft coupling for the
holographic QCD model compared to the $\mathcal{N}=4SYM$ coupling that is
considered to be $6\pi$.. At lower temperatures, the drag force ratio is more
sensitive to the chemical potential value, and the drag force of the
holographic model is larger than the drag force of the $\mathcal{N}=4SYM$
theory for larger $\mu$ values. For lower quark velocities, the drag force
ratio is less sensitive to the chemical potential (Fig.3(b)).

(a)

(b)

Figure 3: (Color online) The ratio of drag force in the holographic QCD model
to its conformal value in terms of temperature for $\mu=0,0.2,0.4,0.6$ and
$0.673$ (red, orange, green, blue and violet respectively). The solid curves
correspond to the deconfined phase and the dashed curves represent the data of
the confined phase. (3(a)) The velocity of quark is $v=0.99$. (3(b)) The
velocity of quark is $v=0.3$.

The world-sheet coordinate can be reparametrized as,

$\displaystyle\tau\,=\,t+K(z),$ $\displaystyle x\,=\,v\,t+v\,K(z)+\xi(z)\,\,.$
(28)

Since, the dynamic of the string is independent from $K(z)$, one can choose
the following ansatz,

$\displaystyle
K^{\prime}(z)\,=\,-\frac{G_{xx}\,v}{G_{tt}\,+\,G_{xx}\,v^{2}}\,\xi^{\prime}(z)\,.$
(29)

By substituting the background metric components and using Eqs.3, and 15, the
induced metric simplifies as,

$\displaystyle
h_{\alpha,\beta}\,=\,\frac{L^{2}\,e^{2A_{s}(z)}}{z^{2}}\left(\begin{array}[]{cc}-(g(z)\,-v^{2})&0\\\
0&\frac{z_{s}^{4}\,e^{4A_{s}(z)}}{e^{4A_{s}(z)}\,g(z)\,z_{s}^{4}\,-\,e^{4A_{s}(z_{s})\,g(z_{s})\,z^{4}}}\end{array}\right)$
(32)

which can be considered as the metric of a two-dimensional world-sheet black
hole with the horizon radius of $z_{s}$. The local speed of light at the
worldsheet horizon corresponds to the speed of a quark at the boundary. The
associated Hawking temperature of the worldsheet is calculated numerically and
its ratio to its conformal value is plotted in Fig. 4 for two different quark
velocities. Similar to the drag force, the behavior of this ratio at lower
temperatures, near the critical temperature, is very dependent on the chemical
potential and quark velocity, such that for $\mu$ close to $\mu_{c}$ the
worldsheet temperature in this background can be larger than its conformal
value. However, at large temperatures, this ratio tends to one independent of
the $\mu$ and quark velocity values. Similar to the drag force ratio plot of
Fig.3, for each chemical potential value, the solid curve starts from its
critical temperature and the solid and dashed curves correspond to the
deconfined and confined phases respectively.

(a)

(b)

Figure 4: (Color online) The ratio of the worldsheet temperature of the
holographic QCD model to its conformal value in terms of temperature for
$\mu=0,0.2,0.4,0.6$ and $0.673$ (red, orange, green, blue and violet
respectively). The solid curves correspond to the deconfined phase and the
dashed curves represent the ratio in the confined phase. (3(a)) The velocity
of quark is $v=0.99$. (3(b)) The velocity of quark is $v=0.3$.

##  4 Langevin equation

Heavy quarks moving at a constant velocity in the plasma experience a Brownian
motion which can be described by the effective equation of motion [76]

$\frac{dp}{dt}\,=\,-\,\eta_{D}\,p\,+\,\xi(t)$ (33)

where $\eta_{D}$ is the drag coefficient, $p$ is the relativistic expression
of the momentum of the quark, and $\xi$ is a random force, expressing the
interaction of the medium with the heavy quark. This random force causes the
momentum broadening of the quark which can be extracted by analyzing small
fluctuations in the path of the Wilson line. This is dual to perturbing the
location of the classical string endpoint on the boundary which yields
fluctuations on the string world sheet dragging behind the quark.

Therefore, we consider the quadratic fluctuations around the classical
trailing string solution obtained in the previous section. In the static
gauge, $\tau=t$, and $\sigma=z$, we generalise the embedding function of the
string as

$x_{L}(t,z)=v\,t\,+\xi(z)+\delta\,x(t,z)\,\,\,\,,\,\,\,x_{T}=\delta\,x_{T}(t,z)\,,$
(34)

where $L$ and $T$ denote to the parallel and transverse to the direction of
motion. Now, we rewrite the Nambu-Goto action Eq.8 and expand it to second
order in terms of fluctuations around the string solution, Eq.15

$\displaystyle
S=-\frac{1}{2\pi\alpha^{\prime}}\int\,dz\,dt\,\sqrt{-\gamma}\frac{\gamma^{\alpha\beta}}{2}\,\left[N(z)\,\partial_{\alpha}\delta
x_{L}\,\partial_{\beta}\delta x_{L}+G_{xx}\,\partial_{\alpha}\delta
x_{T}\,\partial_{\beta}\delta x_{T}\right],$ (35)

where

$\displaystyle\gamma=\frac{L^{4}z_{s}^{4}}{z^{4}}\frac{e^{8A_{s}(z)}(g(z)-v^{2})}{e^{4A_{s}(z)}g(z)z_{s}^{4}-e^{4A_{s}(z_{s})}g(z_{s})z^{4}}\,$
(36)

and

$\displaystyle
N(z)=\frac{L^{2}}{e^{2As(z)}z^{2}}\frac{e^{4A_{s}(z)}g(z)z_{s}^{4}-e^{4A_{s}(z_{s})}g(z_{s})z^{4}}{z_{s}^{4}(g(z)-v^{2})}\,$
(37)

The above action can be rewritten in terms of the worldsheet coordinate Eq.3
and ansatz Eq.29 which diagonalized the induced metric, Eq.32

$\displaystyle
S=-\frac{1}{2\pi\alpha^{\prime}}\int\,dz\,dt\,\frac{H^{\alpha\beta}}{2}\,\left[N(z)\,\partial_{\alpha}\delta
x_{L}\,\partial_{\beta}\delta x_{L}+G_{TT}\,\partial_{\alpha}\delta
x_{T}\,\partial_{\beta}\delta x_{T}\right],$ (38)

where $H^{\alpha\beta}$ is defined in terms of inverse of the diagonalized
induced metric Eq.32 as

$H^{\alpha\beta}=\sqrt{h}h^{\alpha\beta}\,,$ (39)

One can read the transport coefficient directly from the above action
according to the membrane paradigm [91]. The quadratic effective action for a
massless scalar field $\phi$ has the form of

$\displaystyle
S=-\frac{1}{2}\int\,dz\,dt\,\sqrt{-g}\,q(z)\,g^{MN}\partial_{M}\phi\,\partial_{N}\phi\,$
(40)

The momentum broadening coefficients can be read directly from the above
action as

$\displaystyle\kappa=\lim\limits_{\omega\to
0}\left(-\frac{2T_{wsh}}{\omega}\mathrm{Im}\hat{G}_{R}(\omega)\right)=2\,T_{wsh}\,q(z)\,$
(41)

where $\mathrm{Im}\hat{G}_{R}$ is the imaginary part of retarded correlation
function at the world-sheet horizon. By comparing the action 38 with Eq.40 and
implying the relation 41, the Langevin coefficients can be read as [7, 92, 93]

$\displaystyle\kappa_{T}$ $\displaystyle=$
$\displaystyle\frac{1}{\pi\alpha^{\prime}}\,T_{wsh}\,\left.G_{xx}\right|_{z\rightarrow
z_{s}}\,$ $\displaystyle\kappa_{L}$ $\displaystyle=$
$\displaystyle\frac{1}{\pi\alpha^{\prime}}\,T_{wsh}\,\left.N(z)\right|_{z\rightarrow
z_{s}}$ (42)

The final results for the ratio of the longitudinal to transverse transport
coefficients in this model is

$\frac{\kappa_{L}}{\kappa_{T}}=1+\frac{4g(z_{s})\left(z_{s}A_{s}^{\prime}(z_{s})-1\right)}{z_{s}g^{\prime}(z_{s})}$
(43)

where the L’ Hospital’s rule were employed since our functions are continuous
and $z_{s}$ is calculated using Eq.3.

In Fig.5, we have plotted the transverse and longitudinal transport
coefficients versus temperature for two different quark velocities. In these
plots, the black solid curves represent the $\kappa_{T}$ and $\kappa_{L}$ of
the $\mathcal{N}=4\,SYM$ theory, and the dashed lines represent the data of
the confined phase of the holographic model. From the deconfined curves, we
find that increasing the temperature and chemical potential, increases the
transport coefficients and for high quark velocity, the dependency on $\mu$ is
more relevant. Also, one could observe that at lower temperatures (available
for higher chemical potential values) the transverse and longitudinal
transport coefficients are larger than the $\mathcal{N}=4\,SYM$ while for
smaller $\mu$ values, they are smaller than $\mathcal{N}=4\,SYM$. Our
numerical results for the transport coefficients hold the expected inequality
in isotropic backgrounds $\kappa_{L}>\kappa_{T}$ [93].

(a)

(b)

(c)

(d)

Figure 5: (Color online) The longitudinal and transverse transport
coefficients in the holographic model in terms of the temperature for
$\mu=0,0.2,0.4,0.6$ and $0.673$ (red, orange, green, blue and violet
respectively). (5(a)) $\kappa_{T}$ for $v=0.99$, (5(b)) $\kappa_{L}$ for
$v=0.99$, (5(c)) $\kappa_{T}$ for $v=0.3$ and (5(d)) $\kappa_{L}$ for $v=0.3$.

The ratios of $\kappa_{L}/\kappa_{L(SYM)}$ and $\kappa_{T}/\kappa_{T(SYM)}$ in
terms of the temperature for two quark velocities are shown in Fig.6. At lower
temperatures, the dependency on $\mu$ is more evident while at high
temperatures, all the curves converge to a single value smaller than one. The
data of each $\mu$, exhibit a peak around the critical temperature that moves
towards higher temperatures for smaller values of chemical potential (except
for the small $\mu$ curves of Fig.6(b) in which no peaks exist). From these
plots, one could find that the transport coefficients increase by increasing
the chemical potential similar to the transport coefficients in a medium with
baryon density [98].

(a)

(b)

(c)

(d)

Figure 6: (Color online) The ratio of the longitudinal to transverse transport
coefficients in the holographic model in terms of the temperature for
$\mu=0,0.2,0.4,0.6$ and $0.673$ (red, orange, green, blue and violet
respectively). (6(a)) The transverse ratio for $v=0.99$, (6(b)) The
longitudinal for $v=0.99$, (6(c)) The transverse ratio for $v=0.3$ and (6(d))
the longitudinal ratio for $v=0.3$.

##  5 Jet Quenching Parameter

In this section, we study the light quark energy loss by calculating the jet
quenching parameter. For this purpose, we follow the method of [97] to compute
this parameter in the holographic QCD model of [51, 52] reviewed in section 2.

Starting from the string frame metric of Eq.7 and using the light-cone
coordinates $x^{\pm}=(x_{1}\pm t)/\sqrt{2}$, the bulk metric takes the
following form,

$\displaystyle
ds^{2}=\frac{L^{2}\,e^{2A_{s}(z)}}{z^{2}}\left(d{x_{2}}^{2}+d{x_{3}}^{2}+\frac{1}{2}\left(1-g(z)\right)\left(d{x^{+}}^{2}+d{x^{-}}^{2}\right)+\left(1+g(z)\right)dx^{+}dx^{-}+\frac{dz^{2}}{g(z)}\right).$
(44)

To calculate the jet quenching parameter, the thermal expectation value of a
closed rectangular Wilson loop is being used as [97],

$\langle{W^{A}({\cal
C})}\rangle\approx\exp\left[-\frac{1}{4\sqrt{2}}\hat{q}\,L^{-}\,L^{2}\right]\,,$
(45)

where $L^{-}$ is the distance (conjugate to partons with relativistic
velocities) and $L$ is the transverse distance (conjugate to the transverse
momentum of the radiated gluons). This equation is valid for $L^{-}\gg L$.
On the other hand, Thermal expectation value of the Wilson loop
$\langle{W^{F}({\cal C})}\rangle$ is obtained by using the extremal surface
action as [45, 46, 47, 100, 101]

$\langle{W^{F}({\cal C})}\rangle=\exp\left[-S_{I}({\cal C})\right]\,,$ (46)

where $S_{I}$ is the normalized action of a hanging string from Wilson loop
$\cal C$ joining two light-like lines. to the bulk (after subtracting the self
energy of the $q\bar{q}$ pair from the Nambu-Goto action of the string
worldsheet).

In the large $N_{c}$ limit, one can use the relation ${\rm Tr}_{(Adj.)}={\rm
Tr^{2}}_{(Fund.)}$ and compare the equations 45 and 46 to obtain,

$\hat{q}=\frac{8\sqrt{2}S_{I}}{L^{-}\,L^{2}}.$ (47)

By the string parametrization $x^{\mu}(\tau,\sigma)$, the Nambu-Goto action of
the string is written as,

$S_{NG}={1\over 2\pi\alpha^{\prime}}\int d\sigma
d\tau\,\sqrt{-\det\gamma_{\alpha\beta}},$ (48)

where $\gamma_{\alpha\beta}$ is the induced metric on the string worldsheet
and the worldsheet coordinates $\sigma^{\alpha}=(\tau,\sigma)$ are set to be
$(x^{-},x_{2})$. The contour length along $x_{2}$-direction is defined by $L$
and the length along $\tau$-direction by $L^{-}$. The boundary conditions are
$\phi(\pm\frac{L}{2})=0$ and $x_{3}(\sigma)$ and $x^{+}(\sigma)$ coordinates
are constant. Then the action of Eq.48 reads,

$S_{NG}=\frac{L^{-}}{\sqrt{2}\pi\alpha^{\prime}}\int_{0}^{{L\over
2}}d{\sigma}\,\frac{e^{2A_{s}(z)}}{z^{2}}\sqrt{\frac{1-g(z)}{2}\left(1+\frac{z^{\prime
2}}{g(z)}\right)}\ ,$ (49)

where prime denotes the derivative with respect to $x_{2}$. Since the
Lagrangian density is time independent, the Hamiltonian of the system is
constant,

$\mathcal{L}-z^{\prime}\frac{\partial\mathcal{L}}{\partial
z^{\prime}}=\frac{\Pi_{z}}{\sqrt{2}}.$ (50)

From the above equation, one can obtain $z^{\prime}$ as,

$z^{\prime}=\sqrt{g(z)\left(\frac{e^{4A_{s}(z)}(1-g(z))}{{\Pi_{z}}^{2}z^{4}}-1\right)}.$
(51)

Integrating equation 51 leads to,

$\frac{L}{2}=a_{0}{\Pi_{z}}+\mathcal{O}({\Pi_{z}}^{3}),$ (52)

where,

$a_{0}=\int_{z_{h}}^{{0}}dz\frac{z^{2}e^{-2A_{s}(z)}}{\sqrt{g(z)\left(1-g(z)\right)}}\,.$
(53)

Here, we have considered that for small length $L$, the constant $\Pi_{z}$ is
small and its higher order terms are negligible. Substituting Eq.51 into Eq.49
yields,

$S_{NG}=\frac{L^{-}}{\pi\alpha^{\prime}}\int_{z_{H}}^{{0}}dz\frac{e^{4A_{s}(z)}(1-g(z))}{z^{2}\sqrt{2g(z)(e^{4A_{s}(z)}(1-g(z))+\Pi_{z}^{2}z^{4})}},$
(54)

where we have used $z^{\prime}=\frac{\partial{z}}{\partial{\sigma}}$.
Expanding this equation for small $\Pi_{z}$, leads to,

$S_{NG}=\frac{L^{-}}{\pi\alpha^{\prime}}\int_{z_{H}}^{{0}}dz\frac{e^{2A_{s}(z)}}{z^{2}}\sqrt{\frac{1-g(z)}{2g(z)}}\left(1+\frac{e^{-4A_{s}(z)}\Pi_{z}^{2}z^{4}}{2(1-g(z))}+...\right),$
(55)

This action diverges and one should subtract the self energy of two
disconnected strings whose worldsheets are located at $x_{2}=\pm\frac{L}{2}$
and extended from boundary to horizon as,

$S_{0}=\frac{L^{-}}{2\pi\alpha^{\prime}}\int_{z_{H}}^{{0}}dz\sqrt{g_{--}g_{zz}}=\frac{L^{-}}{2\pi\alpha^{\prime}}\int_{z_{H}}^{{0}}dz\,\frac{e^{2A_{s}(z)}}{z^{2}}\sqrt{\frac{1-g(z)}{2g(z)}}\,.$
(56)

The normalized action is therefore,

$S_{I}=S_{NG}-2S_{0}\equiv\frac{L^{-}\Pi_{z}^{2}a_{0}}{2\sqrt{2}\pi\alpha^{\prime}}\,.$
(57)

Inserting Eq.57 into Eq.47 leads to the following expression for the jet
quenching parameter of holographic model,

$\hat{q}=\frac{1}{\pi\alpha^{\prime}a_{0}},$ (58)

where we have used Eq.52 for $\Pi_{z}$ and $a_{0}$ is the numerical integral
defined in Eq.53.

For $\mathcal{N}=4$ supersymmetric Yang-Mills theory in the large $N_{c}$ and
large $\lambda$ limits, Eq.58 leads to the following analytical equation [97],

$\hat{q}_{SYM}=\frac{\pi^{3/2}\Gamma(\frac{3}{4})}{\Gamma(\frac{5}{4})}\sqrt{\lambda}\,T^{3},$
(59)

In order to obtain the jet quenching parameter for the holographic QCD model
of [52], we have solved the equation of 58 numerically for different values of
temperature ($z_{h}$) and chemical potential. The resulting curves are plotted
in Fig.7. In this figure, the numerical values of the jet quenching parameter
are shown for the holographic QCD model (red, orange, green, blue, and purple
curves with $\mu=0,\,0.2,\,0.4,\,0.6$ and $\mu_{c}$ respectively). Solid
curves correspond to the deconfined phase of the holographic model and dashed
curves represent the confined phase data. For each chemical potential value,
the solid curve starts from the critical temperature $T_{c}$, at which the
confinement/deconfinement phase transition occurs. In this figure, the solid
black curve displays the $\hat{q}_{SYM}$ and the black circles with error bars
are the absolute values of $\hat{q}$ for a 10 GeV quark jet in the most
central Au-Au collisions at RHIC with the highest temperature $T=0.37GeV$ and
Pb-Pb collisions at LHC with the highest temperature $T=0.47GeV$ [61]222In
this plot, these two temperatures are rescaled as $T\approx
T_{SYM}=3^{-1/3}T_{QCD}$ since for the holographic QCD theories, the number of
degrees of freedom is more than those of 3 favor QCD [61].. From this figure,
one can observe that increasing the chemical potential and temperature leads
to an increase of the jet quenching parameter. At lower temperatures
associated with larger values of chemical potential, $\hat{q}>\hat{q}_{SYM}$
and at higher temperatures, $\hat{q}<\hat{q}_{SYM}$. Experimental values of
the jet quenching parameter, are closer to the higher chemical potential
curves.

Figure 7: (Color online) Jet quenching parameter versus temperature for the
holographic model with different chemical potentials. The black curve is
$\hat{q}_{SYM}$ and the circles indicate the experimental values from RHIC and
LHC. The dashed curves represent the jet quenching parameter for the confined
phase and the colored solid curves stand for the deconfined phase of the
holographic model.

The ratio of the jet quenching parameter in the holographic QCD model to the
$\hat{q}_{SYM}$ is plotted in Fig.8(a) in terms of temperature for
$\mu=0,\,0.2,\,0.4,\,0.6$ and $\mu_{c}$. From this figure, one could observe
that at higher temperatures, the jet quenching curves converge to a value
lower than the corresponding value of the $\mathcal{N}=4$ SYM theory similar
to the curves of the drag force and Langevine coefficients in Figs.3 and 6.
However, at lower temperatures, the jet quenching parameter of the holographic
QCD model is larger than the $\hat{q}_{SYM}$. Similar to Figs.3(a) and 5(a)
for a high-velocity heavy quark, increasing the chemical potential value,
increases this ratio, and the $\hat{q}/\hat{q}_{SYM}$ plot indicates a smooth
confinement/deconfinement phase transition that results in a peak around the
critical temperature for $\mu<\mu_{c}$ (for smaller $\mu$’s, the peak shifts
to higher temperatures). The same behavior has been observed for
$\hat{q}/T^{3}$ in the dynamical hQCD model of [99] suggesting the rapid
changing in the system degrees of freedom during the phase transition [98,
99].

Finally, the logarithmic plot of the jet quenching parameter over the entropy
density is plotted in terms of the temperature in Fig. 8(b) (in units of
$G_{5}$) for different values of chemical potential. The dashed line
represents the logarithmic value of $\hat{q}/G_{5}s$ for the $\mathcal{N}=4$
SYM theory and each curve has a sharp rise with a finite peak at the crossover
temperature. The peak increases by increasing the chemical potential and is
higher than the $\hat{q}/s$ curves of [99]. It is worth mentioning that,
despite the different concept and calculation methods, the jet quenching
parameter of a light quark has the same behavior as the transverse transport
coefficient of a high-velocity heavy quark (and consequently, to the Langevin
jet quenching parameter $\hat{q}_{T}=2\kappa_{T}/v$).

(a)

(b)

Figure 8: (Color online) (8(a)) $\hat{q}/\hat{q}_{SYM}$ and (8(b))
$Log(\hat{q}/G_{5}s)$ versus temperature for the holographic QCD model. Red,
orange, green, blue and violet curves correspond to $\mu=0,0.2,0.4,0.6$ and
$0.673$ respectively).

##  6 Conclusion

In this paper, we studied the drag force and the Langevin diffusion
coefficients for a heavy quark and also, the jet quenching parameter for a
light quark moving through the QGP. For this purpose, we have implemented the
holographic QCD model of [52] as a realistic framework for the QCD
confinement/deconfinement phase transition with thermodynamic properties
consistent with QCD and lattice data.

Our results demonstrate a nontrivial behavior on temperature and chemical
potential, especially near the phase transition temperature. For a high-
velocity heavy quark, the drag force and the transverse transport coefficient
ratios increase by increasing the chemical potential. For each chemical
potential value, these ratios develop a peak around the critical temperature
that shifts to higher temperatures for higher quark velocities and smaller
values of chemical potential. The longitudinal transport coefficients exhibit
the same behavior except for the lower values of the chemical potential where
no peaks exist. For a low-velocity quark, the drag force and transport
coefficients ratios, monotonically decrease by increasing temperature and are
less sensitive to the chemical potential values.

Also, we have calculated the jet quenching parameter for a light-like
trajectory in terms of temperature for different values of chemical potential.
This parameter, increases by increasing the chemical potential and
temperature. Our results are in good agreement with the experimental data from
RHIC and LHC. The ratios of $\hat{q}/\hat{q}_{SYM}$ are more than one at lower
temperatures (accessible to the higher values of chemical potential). Similar
to the drag force and transport coefficients, the curves develop peaks around
the crossover temperature and at high $T$’s, converge to a single value less
than one due to the small ’t Hooft coupling of the hQCD model we have
implemented here. The jet quenching parameter of a light quark and the
transverse transport coefficient of a high-velocity heavy quark, display the
same qualitative behavior despite the differences. In the end, the ratio of
$\hat{q}/G_{5}s$ has been computed in terms of temperature for different
values of chemical potential. The curves have sharp rising by decreasing the
temperature and develop finite peaks at the crossover temperature, indicating
that the system degrees of freedom change rapidly during the phase transition.

## Acknowledgement

The authors wish to thank Hesam Soltanpanahi for the useful discussions and
comments.

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