# Meson Excitation Time as a Probe of Holographic Critical Point

Ali<EMAIL_ADDRESS>Steklov Mathematical Institute, Russian
Academy of Sciences, Gubkina str. 8, 119991, Moscow, Russia  School of
Particles and Accelerators, Institute for Research in Fundamental Science
(IPM), P.O. Box 19395-5746, Tehran, Iran  Department of Physics, Shahid
Beheshti University G.C., Evin, Tehran 19839, Iran

###### Abstract

We study the time evolution of expectation value of Wilson loop as a non-local
observable in a strongly coupled field theory with a critical point at finite
temperature and nonzero chemical potential, which is dual to an asymptotically
AdS charged black hole via gauge/gravity duality. Due to inject of energy into
the plasma, the temperature and chemical potential increase to finite values
and the plasma experiences an out-of-equilibrium process. By defining meson
excitation time $t_{ex}$ as a time at which the meson falls into the final
excited state, we investigate the behavior of $t_{ex}$ near the critical point
as the system evolves towards the critical point. We observe that by
increasing the interquark distance the dynamical critical exponent increases
smoothly. Also, we obtain for slow quenches different values of the dynamical
critical exponent, although for fast quenches our result for the dynamical
critical exponent is in agreement with the one that is reported for studying
the quasi-normal modes. Consequently, this indicates that in this model for
fast quenches and small values of interquark distances the gauge invariant
Wilson loop is a good non-local observable to probe the critical point.

###### Contents

  1. I Introduction and results
  2. II  Probing the critical point by a non-local observable
    1. II.1  Background
    2. II.2 Thermodynamics
      1. II.2.1 Temperature and entropy
      2. II.2.2 Free energy and phase transition
    3. II.3  Wilson loop
  3. III Numerical results
  4. IV Discussion and outlook
  5. A Review on the charged black hole backgrounds
    1. A.1  EM-dilaton black hole background
    2. A.2  EM-dilaton-Vaidya background
  6. B Boundary condition
  7. C Initial condition
  8. D Numerical procedure to calculate the critical exponent
  9. E Meson potential at zero temperature
  10. F Time evolution of the string

##  I Introduction and results

Quantum Chromodynamics (QCD) that describes the strong force, is a strongly
coupled gauge theory at low energy. Studying strongly coupled gauge theories
has been attracted much interest. The main motivation comes from the results
of Relativistic Heavy Ion Collider (RHIC) and Large Hadron Collider (LHC) that
a new phase of matter, i.e. Quark Gluon Plasma (QGP) is reported
CasalderreySolana:2011us ; Ammon:2015wua . At the beginning of plasma
formation it is very hot, dense medium and out-of-equilibrium system.
Hydrodynamic simulations show that this plasma is a strongly coupled system,
so that the plasma can be realized as strongly coupled phase of QCD
Shuryak:2003xe ; Shuryak:2004cy . Thermalization process is the process of a
black hole formation in the gravity theory and in the dual gauge theory side
it generally means the evolution of a state from zero temperature to a thermal
state Chesler:2013lia . An example of the thermalization process is the
collapse of a thin-shell matter described by an AdS-Vaidya metric
Balasubramanian:2010ce ; Balasubramanian:2011ur ; Bhattacharyya:2009uu . Due
to the energy injection, using an external source an out-of-equilibrium state
in field theory can be produced Buchel:2014gta . Then, one can probe the out-
of-equilibrium state using local and non-local observables. Although, lattice
gauge theory describes successfully the low energy properties of QCD in
particular at $\mu=0$ Rothkopf:2011db  but the lattice calculations are not
successful when we include the chemical potential. Since the plasma is
strongly coupled, the perturbation theory is not applicable and we need to use
a non-perturbative approach.

Anti de-Sitter/Conformal Field Theory (AdS/CFT) correspondence or more
generally gauge/gravity duality is a non-perturbative approach. According to
this duality, a strongly coupled gauge theory without any gravitational
degrees of freedom living in a $d$-dimensional space-time is in correspondence
with the classical Einstein gravity in a ($d+1$)-dimensional space-time
Maldacena:1997re ; Witten:1998qj ; Gubser:1998bc . In fact, parameters and
different processes in the gauge theory side is translated into the
corresponding equivalent in the gravity side. Utilizing this duality, many
different problems in the strongly coupled regime of QCD are investigated
Gursoy:2007cb ; Gursoy:2007er ; Dudal:2018ztm . For instance, in order to
calculate the static potential energy between a quark and an anti-quark in the
strongly coupled plasma we need to calculate expectation value of a Wilson
loop. In fact, Wilson loop is non-local and gauge invariant observable that
for the first time in Maldacena:1998im  is used to calculate the potential
energy between a pair and its generalization is extensively discussed in
Brandhuber:1998bs ; Rey:1998bq ; Sonnenschein:2000qm ; Finazzo:2013aoa ; Ali-
Akbari:2015ooa ; Dudal:2017max ; Cai:2012xh ; Bohra:2019ebj ; Asadi:2021nbd .
Gauge/gravity duality proposes that the holographic dual of the rectangular
Wilson loop is given by a classical open string suspended from two points on
the boundary of the gravity and hanging down in the gravity bulk with
appropriate boundary conditions. In addition, this duality is used to study
various aspects of strongly coupled systems such as jet quenching parameter
Liu:2006ug ; Cai:2012eh , thermalization process Galante:2012pv ; Ali-
Akbari:2012gku ; Ali-Akbari:2012fzl ; Dey:2015poa  and the features of plasma
in presence of magnetic field Ali-Akbari:2015bha ; Bohra:2020qom ; Ali-
Akbari:2013txa ; Arefeva:2020vae ; Zhou:2020ssi ; Dudal:2021jav ;
Arefeva:2021mag ; Fang:2021ucy ; Dudal:2016joz ; He:2020fdi . For more
details, see CasalderreySolana:2011us  and references therein.

Understanding the phase structure of QCD is received a lot of interest
Qin:2010nq . The endpoint of the line of first-order phase transition is
described as a critical point. In particular, studying the physics of
observables near the critical point in QCD phase diagram is challenging
question. It is very important to note that the investigation of this question
is difficult theoretically because the theory is strongly coupled near the
critical pointDeWolfe:2010he . To do so, the gauge/gravity duality prepared a
new approach in such a way that to investigate a strongly coupled theory near
the critical point one can consider the black hole solution which is
holographically dual to a strongly coupled field theory with a critical point
Ebrahim:2017gvk ; Cai:2022omk .

In this paper, we consider Einstein-Maxwell-dilaton (EM-dilaton) background
and its Vaidya-like solution dual to a field theory with a critical point
Zhang:2015dia ; Ebrahim:2017gvk . In fact, in the gauge theory side we
consider a probe stable meson in the QGP at zero temperature and chemical
potential. Due to the injection of energy the temperature and the chemical
potential are increased from zero to finite values of $T$ and $\mu$,
respectively. A very interesting observation in Hajilou:2018dcb  is the time
that a meson bound state needs to fall into the final excited state is called
excitation time, $t_{ex}$. Now the question we would like to answer is whether
the meson excitation time, $t_{ex}$, can probe the critical point as the
system evolves towards the critical point? Furthermore, what would be the
associated dynamical critical exponent? Holographic critical point and
dynamical critical exponent have been extensively discussed in DeWolfe:2010he
; DeWolfe:2011ts . The holographic dual of the above system in the gravity
side is described by the dynamics of the classical open string with
appropriate initial and boundary conditions in the Einstein-Maxwell-dilaton-
Vaidya (EM-dilaton-Vaidya) background.

Our main findings can be summarized as follows:

  * •

We observed that by moving towards the critical point the excitation time,
$t_{ex}$, gets the finite value thought its slope,
$\frac{dt_{ex}}{d\frac{\mu}{T}}$, diverges at the critical point and its
behavior can be described by
$((\frac{\mu}{T})_{\star}-\frac{\mu}{T})^{-\theta}$ where
$(\frac{\mu}{T})_{\star}$ indicates the valus of $\frac{\mu}{T}$ at the
critical point and $\theta$ is defined as the dynamical critical exponent.

  * •

Our observation showed the dynamical critical exponent $\theta$ depends on the
interquark distance $l$ (or dimensionless quantity $lT$) and the speed of
energy injection $k$ that is whether the quench is fast or slow.

  * •

We observed that increasing both values of $k$ and $l$ the value of $\theta$
increases. Also, it is seen that the effect of changing $k$ on $\theta$ is
much more than the effect of $l$ on $\theta$ and we emphasize that this
feature is our observation. It seems that, except for the non-locality of our
probe some features of our results can be addressed to the non-equilibrium
conditions that is set up in the system.

  * •

A very interesting observation is that for fast quenches ($k\ll 1$) and small
values of interquark distances $l$, the dynamical critical exponent is
$\frac{1}{2}$ which is in good agreement with the result that is obtained from
the investigation of the behavior of scalar quasi-normal modes near the
critical point in Finazzo:2016psx . Therefore, in this model the gauge
invariant Wilson loop is good non-local observable to probe the critical
point.

  * •

Another point is that the dependence of the dynamical critical exponent
$\theta$ to the transition time $k$ is also reported in Ebrahim:2017gvk .
Although, there is an important difference here since our physical observable,
Wilson loop, is a non-local operator versus of the local operator that is used
in Ebrahim:2017gvk . Also, because of non-locality of Wilson loop it is
anticipated that the value of $\theta$ depends to the $l$. This is because the
dependence of meson excitation time, $t_{ex}$, to $l$ is confirmed in
Hajilou:2018dcb  and therefore $\theta$ depends on $l$.

The remainder of this paper is organized as follows. In section II we briefly
review on the EM-dilaton-Vaidya background and then calculate the time
evolution of expectation value of Wilson loop in this background. In Section
III we explain our numerical results. Finally, in Section IV we discuss our
results and compare with previous researches. This work is complemented by
Appendix A, where we review on the EM-dilaton black hole and the EM-dilaton-
Vaidya backgrounds. In Appendices B and C we obtain appropriate boundary and
initial conditions, respectively. In Appendix D we introduce our numerical
procedure to obtain the dynamical critical exponent. In Appendix E we
considered meson potential at zero temperature and in Appendix F we depicted
3-dimensional plot of open string that describes the time evolution of
classical open string in EM-dilaton-Vaidya background.

##  II  Probing the critical point by a non-local observable

Gauge/gravity duality establishes a good framework to study important features
of QGP CasalderreySolana:2011us ; Ammon:2015wua . Studying the phase structure
of gauge theories is a long-standing project and one of the curiosities in
this field is how the system behaves near the critical point. In other words,
utilizing local or non-local observables the critical point can be probed and
one can also calculate the associated dynamical critical exponent. To do so,
in this section we review a charged black hole solution which its holographic
dual is a field theory with a critical point and also a non-local observable,
i.e. Wilson loop that is used to probe the critical point in the field theory.

###  II.1  Background

We want to study the evolution of expectation value of Wilson loop in the
time-dependent and strongly coupled field theory with a critical point. Its
holographic dual is to study the dynamics of an open string in the EM-dilaton-
Vaidya background (for more details see Appendix A). The metric in Eddington-
Finkelstein coordinates is given by Zhang:2015dia :

$\begin{split}ds^{2}=&-N(z)F(\bar{v},z)d\bar{v}^{2}-\frac{2}{z^{2}}\sqrt{\frac{N(z)}{1+b^{2}z^{2}}}d\bar{v}dz+\frac{1+b^{2}z^{2}}{z^{2}}g(z)d\vec{x}^{2},\\\
F(\bar{v},z)=&\frac{1+b^{2}z^{2}}{z^{2}}\Gamma^{2\gamma}-M(\bar{v})\frac{z^{2}}{1+b^{2}z^{2}}\Gamma^{1-\gamma},\end{split}$
(1)

where,

$\begin{split}N(z)=\Gamma^{-\gamma},~{}g(z)=\Gamma^{\gamma},~{}\Gamma(z)=\frac{1}{1+b^{2}z^{2}},~{}\gamma=\frac{\alpha^{2}}{2+\alpha^{2}}~{},\end{split}$
(2)

$\alpha$ is the coupling constant between the dilaton and the gauge field and
$b$ is a constant. The $\bar{v}$-coordinate reduces to the time coordinate of
the gauge theory at the boundary, i.e. $t=\bar{v}|_{z=0}$. $M(\bar{v})$ and
$q(\bar{v})$ are arbitrary functions that are used to show how the mass and
charge of the black hole increase from zero to a finite value. Different
functional forms of $M(\bar{v})$ and $q(\bar{v})$ are discussed in the
literature, for example in Amiri-Sharifi:2016uso . Here, our choice is

$\displaystyle{\zeta}(\bar{v})={\zeta}\left\\{\begin{array}[]{ll}0&{\bar{v}}<0,\\\
k^{-1}\left[{\bar{v}}-\frac{k}{2\pi}\sin(\frac{2\pi{\bar{v}}}{k})\right]&0\mathchar
13366\relax{\bar{v}}\mathchar 13366\relax k,\\\ 1&{\bar{v}}>k~{},\\\
\end{array}\right.$ (6)

where ${\zeta}\in(M,q)$. The time interval that the black hole needs to reach
its final value of the mass and charge, i.e. $M$ and $q$, respectively is the
transition time $k$. In the gauge theory side $k$ is the time that strongly
coupled system needs to reach its final values of temperature and chemical
potential. The fast (slow) quench is attributed to the $k\ll 1$ ($k\gg 1$),
that is small (large) transition time, respectively. The relation between the
final values of mass and charge of the black hole, i.e. $M$ and $q$ and the
constant $b$ are:

$q=\sqrt{\frac{6M}{2+{\alpha}^{2}}}b~{}.$ (7)

It is very important to note that for studying the phase structure of the
gauge theory we consider $\alpha=2$ in this background. After fixing the value
of $\alpha=2$ the equation (7) indicates that this background is parameterized
by three parameters $q$, $M$, $b$, while two of them are independent. The
chemical potential of the strongly coupled field theory, that comes from the
gauge field in the bulk is given by Zhang:2015dia :

$\mu=\frac{b\sqrt{3M}}{(\frac{1}{{z_{h}}^{2}}+b^{2})\sqrt{2(2+{\alpha}^{2})}}~{},$
(8)

Where the $z_{h}$ is the horizon radius and is the smallest root of the
equation $f(z_{h})=0$ (see Appendix A).

###  II.2 Thermodynamics

####  II.2.1 Temperature and entropy

The temperature of the field theory that is correspondence with the Hawking
temperature of the black hole is Zhang:2015dia ; Ebrahim:2017gvk :

$T=\frac{b{\Gamma(z_{h})^{\frac{3\gamma}{2}-1}}}{4\pi\sqrt{1-\Gamma(z_{h})}}[2(3\gamma-1)-3(2\gamma-2)\Gamma(z_{h})]~{}.$
(9)

After simplification of the temperature formula in terms of parameters of the
theory we obtained:

$T=\frac{1}{z_{h}}~{}\frac{3+\sqrt{1+4~{}q^{2}z_{h}^{6}}}{2\pi\sqrt{2}~{}\sqrt{1+\sqrt{1+4~{}q^{2}z_{h}^{6}}}}~{}.$
(10)

Also, the entropy (density) can be written as:

$s=\frac{1}{z_{h}^{3}}~{}\frac{\sqrt{1+\sqrt{1+4~{}q^{2}z_{h}^{6}}}}{4~{}\sqrt{2}}~{}.$
(11)

It is important to note that by choosing the case $\alpha=2$, this background
possesses a critical point at $(\frac{\mu}{T})_{\star}$=1.11072. One can find
more details and its phase diagram in Ebrahim:2017gvk . To do so, utilizing
equations (8) and (9) we have:

$bz_{h}=\frac{1\pm\sqrt{1-\frac{8{\mu}^{2}}{\pi^{2}T^{2}}}}{\frac{2\mu}{\pi
T}}~{},$ (12)

where

$z_{h}=\sqrt{\frac{b^{2}+\sqrt{b^{4}+4m}}{2m}}~{}.$ (13)

Figure 1:  Left: Temperature of the black hole $T$ as a function of the
horizon $z_{h}$ for different values of charges. Right: Entropy of the black
hole $s$ as a function of the temperature $T$ for different values of charges.
For each panel, their values of charges have been fixed as $q=0.01$ (Black),
$q=0.1$ (Magenta), $q=0.3$ (Red), $q=0.6$ (Green), $q=0.9$ (Blue).

It is obvious that for each value of $\frac{\mu}{T}$, there are two distinct
values of $bz_{h}$ which describes stable and unstable branches of solutions.
In other words, the lower sign in equation (12) corresponds to
thermodynamically stable solution and the upper sign, unstable. Consequently,
this indicates that there exist a phase transition in field theory and where
the two branch of solutions merge, there exists a critical point
Ebrahim:2017gvk . In fact, in this theory the black hole solutions possess the
”critical point” at which the thermodynamical stability of a black hole
solution switches. Note also that one can check the thermodynamically stable
or unstable solutions utilizing the Jacobian,
$\mathcal{J}=\frac{\partial(s,\rho)}{\partial(T,\mu)}$ where $s$ is the
entropy and $\rho$ is the charge density

$\begin{split}s&\propto\frac{T^{3}(1+b^{2}z_{h}^{2})^{2}}{(2+b^{2}z_{h}^{2})^{3}}~{},\\\
\rho&\propto\frac{\mu}{T}(2+b^{2}z_{h}^{2})\sqrt{1+b^{2}z_{h}^{2}}~{}.\end{split}$
(14)

In fact, if the Jacobian is positive (negative), there exists a
thermodynamically stable (unstable) solution for the physical system
Finazzo:2016psx . In this research the thermodynamically stable solution is
used.

Studying the thermodynamics of this background, i.e. EM-dilaton black hole can
be helpful to investigate the black hole features and phase transition. In
figure 1, the variation of Hawking temperature $T$ with respect to the horizon
radius $z_{h}$ is depicted for different values of black hole charges, i.e.
$q=0.01$ (Black), $q=0.1$ (Magenta), $q=0.3$ (Red), $q=0.6$ (Green), $q=0.9$
(Blue). We observe that there exists a minimum temperature $T_{min}$ in such a
way that below which there is no black hole solution. However, for
$T>T_{min}$, there are two black hole solutions, large black hole and small
one that are specified in the figure 1. The large black hole that its
temperature increases when $z_{h}$ decreases, is stable, whereas the small
black hole that its temperature increases when $z_{h}$ increases, is unstable
one. This plot also shows that as the charge of the black hole $q$ increases,
the minimum temperature increases. Note also that, the stable-unstable
solutions can be investigated also by studying the entropy of the black hole.
In the right panel of the figure 1, we depicted the behavior of entropy $s$ as
a function of the temperature $T$ for the same values of black hole charges.
As shown in the right panel, there is a minimum of the temperature such that
for the temperature larger than minimum, we have two different black hole
solutions. At the one hand, the solution that its entropy increases when
temperature increases is stable (large black hole) and on the other hand, the
solution that its entropy increases when temperature decreases is unstable one
(small black hole). We have clarified these two branches of solutions in the
right panel of the figure 1. It is important to note that the stable-unstable
nature of the large-small black hole phases can be seen easily from the free
energy behavior that is investigated in the figure 2.

####  II.2.2 Free energy and phase transition

Free energy is very powerful physical quantity to investigate the stability
and unstability of the solutions. According to the first law of thermodynamics
the free energy (density) is defined DeWolfe:2010he ; Arefeva:2020vae ;
He:2013qq :

$F=\epsilon-sT-\mu\rho~{},$ (15)

where $\epsilon$ is energy density and $\rho$ is number density. The
differential of free energy at fixed volume is obtained as $dF=-sdT-\rho
d\mu$. For the fixed values of chemical potential $\mu$, the free energy can
be obtained by the following integral Gursoy:2008za ; Arefeva:2018hyo ;
Gursoy:2018ydr :

$dF=-\int sdT~{}.$ (16)

We expect that at $z_{h}\rightarrow\infty$ the free energy of the black hole
back ground, coincides with the free energy of the zero temperature back
ground (thermal gas) that can be choose to be zero. Therefore, we have
normalised the free energy of the black hole with respect to the thermal gas
case by demanding $F(z_{h}\rightarrow\infty)=0$ Bohra:2019ebj ; He:2013qq  .
Then, one can obtain the following relation for the free energy:

$F=\int_{z_{h}}^{\infty}s~{}\frac{dT}{dz_{h}}dz_{h}~{}.$ (17)

The behavior of free enregy $F$ in terms of horizon radius $z_{h}$ is shown in
the left panel of figure 2 for different values of black hole charges,
$q=0.01$ (Black), $q=0.1$ (Magenta), $q=0.3$ (Red), $q=0.6$ (Green), $q=0.9$
(Blue). As shown in figure 2 we observe that the sign of free energy changes.
In other words, at small $z_{h}$ the free energy gets large negative value and
then gets positive maximum value and finally decreases to zero at
$z_{h}\rightarrow\infty$. The free energy intersecting the horizontal axis
implies that there exists a phase transition from the black hole to the
thermal gas. It is more transparent to investigate the behavior of free energy
$F$ in terms of temperature $T$ as depicted in the right panel of figure 2. We
observe that the free energy of the small black hole phase is always larger
than the large black hole and thermal gas phases. This indicates the unstable
nature of the small black hole phase. It is important to note that, upon
varying the Hawking temperature, a phase transition from the large black hole
phase to thermal AdS phase takes place at the Hawking-Page transition
temperature $T_{HP}$. This is the famous black hole-thermal AdS (Hawking-Page)
phase transition Hawking:1982dh  that is in correspondence with confinement-
deconfinement phase transition and is usually known as a first order phase
transition in the field theory Arefeva:2018hyo ; He:2022amv ; Li:2017tdz .

Figure 2:  Left: Free energy of the black hole $F$ as a function of the
horizon $z_{h}$ for different values of charges. We have normalized the free
energy such that it vanishes when $z_{h}\rightarrow\infty$. Right: Free energy
of the black hole $F$ as a function of the temperature $T$ for different
values of charges. For each panel, values of charges have been fixed as
$q=0.01$ (Black), $q=0.1$ (Magenta), $q=0.3$ (Red), $q=0.6$ (Green), $q=0.9$
(Blue).

The critical point is a point that the Jacobian $\mathcal{J}$ vanishes. The
Jacobian $\mathcal{J}$ is proprtional to the heat capacity $C_{V}$ of the
system, i.e. $\mathcal{J}\sim C_{V}$ DeWolfe:2010he . Note also that, for heat
capacity at constant volume we have DeWolfe:2010he :

$C_{V}=T\left(\frac{\partial s}{\partial T}\right)_{V}~{}.$ (18)

Therefore, we depicted in the left panel of figure 3 the heat capacity of the
system $C_{V}$ (red line) in terms of the horizon radius $z_{h}$ to
investigate where the Jacobian vanishes (changes its sign) He:2013qq ;
Sajadi:2023zke  and the temperature $T$ (blue line) as a function of horizon
radius $z_{h}$. A very interesting observation in the left panel of figure 3
is that the point where the Jacobian changes its sign (changing from stable
solution to the unstable one), is in coincidence with the $T_{min}$ (where we
have changes from stable branch of solution to the unstable one). It is
important to note that, although in this model the free energy just could show
the confinement-deconfinement phase transition and was not capable to
illustrates critical point $T_{crit}$ (since we have no Swallow-tail diagram),
but, using the left panel of figure 3 we could describe the coincidence of the
point where the stable and unstable solutions transforms to each other. In
fact, this observation tells us that Jacobian is not good order parameter to
see Critical End Point (CEP) in this model. Note also that, in the right panel
of figure 3 we plotted the temperature $T$ as a function of chemical potential
$\mu$ via the formula:

$\mu=\frac{\sqrt{1+\pi Tzh(-\pi
Tzh+\sqrt{-1+\pi^{2}T^{2}zh^{2}})}}{\sqrt{2}~{}zh}~{}.$ (19)

This panel shows the first order phase transition line that the end of this
line is the CEP with $(\mu_{c},T_{c})$ where we have
$\frac{\mu_{c}}{T_{c}}\sim 1.1107$.

Figure 3:  Left: Temperature $T$ (blue line) and heat capacity $C_{V}$ (red
line) of the black hole as a function of the horizon radius $z_{h}$. For both
plots, we have fixed the value of charge as $q=0.9$. Right: Temperature of the
black hole $T$ as a function of the chemical potential $\mu$. The critical
point $(\mu_{c},T_{c})$ is shown in this panel such that we have
$\frac{\mu_{c}}{T_{c}}\sim 1.1107$ .

###  II.3  Wilson loop

Wilson loop is a gauge invariant and non-local observable that is very useful
to find the potential energy between quark and anti-quark pair, living in the
plasma. The holographic dual of the rectangular Wilson loop is given by a two-
dimensional world-sheet swept by a classical open string suspended from two
points (corresponding to a quark and an anti-quark), hanging down in the bulk
with appropriate boundary conditions. We used the time-like rectangular Wilson
loop $\cal{C}$, where one side of the loop is the spatial distance between
quark and anti-quark pair, $l$, and the other side is the temporal direction,
$\cal{T}$. If we suppose that ${\cal{T}}\gg l$, that is the world-sheet is
translationally invariant along the time direction, the expectation value of
the Wilson loop is CasalderreySolana:2011us :

$\langle W({\cal{C}})\rangle=e^{-i(2m+V(l)){\cal{T}}},$ (20)

where $m$ is the rest mass of quark (antiquark) that is equal to
$\frac{\sqrt{\lambda}}{2\pi}\int_{\epsilon}^{z_{h}}\,\frac{dz}{z^{2}}$, where
$\epsilon$ is IR regulator in the gravity theory that according to the UV/IR
connection corresponds to the UV cut-off in the gauge theory side
CasalderreySolana:2011us . Also, $V(l)$ represents static potential energy
between the pair. According to AdS/CFT dictionary, the expectation value of
the Wilson loop, in the saddle point approximation, is dual to the on-shell
action of the two-dimensional world-sheet of an open string whose dynamics is
given by Nambu-Goto action. Therefore,

$\langle W({\cal{C}})\rangle=e^{iS({\cal{C}})},$ (21)

where $S({\cal{C}})$ is the Nambu-Goto action:

$S=\frac{-1}{2\pi\alpha^{\prime}}\int d\tau d\sigma\sqrt{-\det(g_{ab})}\,,$
(22)

$\tau$ and $\sigma$ parametrize the two dimensional world-sheet of the string
and ${\alpha}^{\prime}={l_{s}}^{2}$ and ${l}_{s}$ is the fundamental length
scale of string. The induced metric on the string world-sheet is
$g_{ab}=G_{\mu\nu}\frac{\partial X^{\mu}}{\partial\xi^{a}}\frac{\partial
X^{\nu}}{\partial\xi^{b}}$ so that, $G_{\mu\nu}$ is the bulk metric. Here,
$X^{\mu}$ ($\xi^{a}=\tau,\ \sigma$) are the bulk (world-sheet) coordinates.
Using equations (20), (21) and (22) the static potential energy between the
pair can be found.

In order to calculate the Nambu-Goto action in the EM-dilaton black hole
background (37) (see Appendix A), we work in static gauge and choose $\tau=t$,
$\sigma=x_{3}\equiv x$ to parametrize the two-dimensional string world-sheet.
Therefore, except $z$ and $x$, the other bulk coordinates are chosen to be
constant and the function $z=z(x)$ describes the shape of the classical
string. Therefore, the action (22) on background (37) reduces to

$S=-\frac{{\cal{T}}}{2\pi\alpha^{\prime}}\int_{-\frac{l}{2}}^{\frac{l}{2}}\,dx\frac{\sqrt{N(z)}}{z^{2}}\sqrt{\frac{{z^{\prime}}^{2}}{1+b^{2}z^{2}}+z^{2}{f}(z)g(z)(1+b^{2}z^{2})}~{},$
(23)

where $z^{\prime}\\!=\\!dz/dx$. Since the Lagrangian does not depend
explicitly on $x$, the associated Hamiltonian is a constant of motion and will
be used later to obtain initial conditions (see Appendix C). After some simple
algebra, one gets

$z^{\prime}(x)=\pm\sqrt{\frac{N(z)}{N(z_{\ast})}}\frac{g(z)f(z)~{}z_{\ast}}{\sqrt{g(z_{\ast})f(z_{\ast})}}\sqrt{\frac{(1+b^{2}z^{2})^{3}}{(1+b^{2}z_{\ast}^{2})}}~{}\sqrt{1-\frac{z^{2}~{}f(z_{\ast})g(z_{\ast})N(z_{\ast})(1+b^{2}z_{\ast}^{2})}{z_{\ast}^{2}~{}f(z)g(z)N(z)(1+b^{2}z^{2})}}~{},$
(24)

where $z=z_{\ast}$ is where $z^{\prime}(x)=0$.

For studying the evolution of Wilson loop in the time-dependent plasma, one
should inject energy into the plasma. Since the system is time dependent, it
is not translationally invariant along the time direction. Therefore, the
condition ${\cal{T}}\gg l$ will not work as before and therefore the
expectation value of the Wilson loop, (20) in time-dependent case can be
written as Ali-Akbari:2015ooa ; Hajilou:2018dcb ; Hajilou:2017sxf

$\langle W({\cal{C}})\rangle=e^{-i\int dt\ {\cal{W}}(t)}\,.$ (25)

Based on the gauge/gravity duality, ${\cal{W}}(t)$ is the on-shell action of
string where the integration over time coordinate has not been done. In order
to regularize ${\cal{W}}(t)$, we subtracted the infinite mass of the quark and
anti-quark pair. Thus, we have

$\begin{split}{\cal{W}}_{R}(t)={\cal{W}}(t)-2m\equiv\int
d\sigma\left(\sqrt{-\det(g_{ab})}\right)_{\rm{on-shell}}-2m~{},\end{split}$
(26)

where ${\cal{W}}_{R}(t)$ is the regularized form of ${\cal{W}}(t)$ and
describes the time dependence of expectation value of Wilson loop. In order to
calculate ${\cal{W}}_{R}(t)$ in the gauge theory side, we need to calculate
the on-shell action of the string in the EM-dilaton-Vaidya background.

To do so, similar to Ishii:2014paa ; Hajilou:2017sxf ; Hajilou:2018dcb , to
parametrize the two-dimensional world-sheet of the string, we chose the null
coordinates $(u,v)$ on the world-sheet. Therefore, all the background
coordinates on the world-sheet depend on $u$ and $v$ and all coordinates will
be zero except the following ansatz:

$\bar{v}=V(u,v)~{},~{}z=Z(u,v)~{},~{}x_{3}=X(u,v)\,.$ (27)

Substituting this ansatz into the Nambu-Goto action (22), the equations of
motion can be obtained. After some algebra we found

$\begin{split}X_{,uv}&=\left(Z_{,u}X_{,v}+Z_{,v}X_{,u}\right)\frac{3+2b^{2}Z^{2}}{3Z(1+b^{2}Z^{2})}~{},\\\
V_{,uv}&=\left(\frac{2b^{2}Z^{3}F}{3}(1+b^{2}Z^{2})^{\frac{-1}{6}}+\frac{Z^{2}F_{,Z}}{2}(1+b^{2}Z^{2})^{\frac{5}{6}}\right)V_{,u}V_{,v}\\\
&+\left(\frac{1}{Z}(1+b^{2}Z^{2})^{\frac{1}{2}}-\frac{b^{2}Z}{3}(1+b^{2}Z^{2})^{\frac{-1}{2}}\right)X_{,u}X_{,v}~{},\\\
Z_{,uv}&=\left(-\frac{2b^{2}F^{2}Z^{5}}{3}(1+b^{2}Z^{2})^{\frac{2}{3}}-\frac{FZ^{4}F_{,Z}}{2}(1+b^{2}Z^{2})^{\frac{5}{3}}-\frac{Z^{2}F_{,V}}{2}(1+b^{2}Z^{2})^{\frac{5}{6}}\right)V_{,u}V_{,v}\\\
&+\left(-\frac{2b^{2}Z^{3}F}{3}(1+b^{2}Z^{2})^{\frac{-1}{6}}-\frac{Z^{2}F_{,Z}}{2}(1+b^{2}Z^{2})^{\frac{5}{6}}\right)\left(Z_{,u}V_{,v}+Z_{,v}V_{,u}\right)\\\
&+\left(-ZF(1+b^{2}Z^{2})^{\frac{4}{3}}+\frac{b^{2}Z^{3}F}{3}(1+b^{2}Z^{2})^{\frac{1}{3}}\right)X_{,u}X_{,v}+\left(\frac{2}{Z}+\frac{b^{2}Z}{3}(1+b^{2}Z^{2})^{-1}\right)Z_{,u}Z_{,v}~{}.\end{split}$
(28)

Since $u$ and $v$ are null coordinates, two constraint equations corresponding
to $g_{uu}=0$ and $g_{vv}=0$ should be imposed. Therefore, we have

$\begin{split}C_{1}&=Z^{2}F(V,Z)V_{,u}^{2}+2(1+b^{2}Z^{2})^{\frac{-5}{6}}V_{,u}Z_{,u}-(1+b^{2}Z^{2})^{\frac{-1}{3}}X_{,u}^{2}=0~{},\\\
C_{2}&=Z^{2}F(V,Z)V_{,v}^{2}+2(1+b^{2}Z^{2})^{\frac{-5}{6}}V_{,v}Z_{,v}-(1+b^{2}Z^{2})^{\frac{-1}{3}}X_{,v}^{2}=0\,.\end{split}$
(29)

In order to solve the equations of motion (28) and constraint equations (29),
we need to impose suitable boundary and initial conditions. In Appendices B
and C, the appropriate boundary and initial conditions are obtained,
respectively.

##  III Numerical results

Having set up the formalism developed in the previous section, now we are
ready to discuss the numerical results in this section . Consider a probe
stable meson in the QGP at zero temperature and chemical potential. Then,
after the injection of energy into the plasma, the temperature and the
chemical potential raised to final values, i.e. $T$ and $\mu$, respectively.
Before the injection of energy the meson is in its ground state and the
injection of energy puts the meson into a final excited state with specific
frequency and amplitude of oscillation Ali-Akbari:2015ooa ; Hajilou:2018dcb ;
Ishii:2014paa ; Ageev:2016gtl . The time that the meson needs to fall into the
final excited state is called excitation time, $t_{ex}$ Hajilou:2018dcb . In
other words, the excitation time, $t_{ex}$, is the time that the expectation
value of the Wilson loop starts oscillating around the static potential energy
by which we mean the potential energy of the bound state in the plasma with
final values of the temperature and chemical potential $T$ and $\mu$,
respectively. An important point that we would like to emphasize is that since
we work in the probe limit, therefore the energy of the meson does not
dissipate in the plasma and consequently the oscillation of meson remains
unchanged.

Note that the response of the system to the time-dependent change in the
temperature and the chemical potential is described by the time evolution of
the expectation value of Wilson loop ${\cal{W}}_{R}(t)$. Therefore, for better
clarification of the excitation time we depicted the ${\cal{W}}_{R}(t)$ as a
function of boundary time $t$ in figure 4. In this figure we fixed the
interquark distance $l=1$, final value of the chemical potential $\mu=0.0220$,
final value of the temperature $T=0.2200$ and the transition time $k=0.3$. The
value of the static potential is $V(l)=-0.1017$. As shown in figure 4 the
expectation value of the time-dependent Wilson loop oscillates around the
static potential energy. In fact, when the energy injection is started, the
temperature and the chemical potential of the system increase and the quark
and anti-quark pair, or equivalently quark–anti-quark bound state, is excited.
After finishing the energy injection, the pair falls into a final excited
state and starts to oscillate with particular frequency and amplitude of
oscillation.

Figure 4: Evolution of the expectation value of Wilson loop ${\cal{W}}_{R}(t)$
as a function of boundary time $t$. We fixed the interquark distance $l=1$,
final value of the chemical potential $\mu=0.0220$, final value of the
temperature $T=0.2200$ and the transition time $k=0.3$. The dashed red curve
shows the evolution of Wilson loop and the blue sine curve is the fitted
function, i.e. ${\cal{W}}_{f}(t)=A\cos(2\pi\nu t+{\phi})$ where, $A$ is the
amplitude of the oscillation, $\nu$ is the oscillation frequency and $\phi$ is
a phase value. Reprinted from Hajilou:2018dcb .

The dashed red curve is the time evolution of the expectation value of Wilson
loop that is fitted with the blue curve

${\cal{W}}_{f}(t)=A\cos(2\pi\nu t+{\phi})\ ,$ (30)

where $A$, $\nu$ and $\phi$ can be fixed from ${\cal{W}}_{R}(t)$ at asymptotic
times. Consequently, the excitation time, $t_{ex}$, is a time that the
expectation value of Wilson loop oscillates around its static potential energy
with a specific value of frequency $\nu$, amplitude of oscillation $A$ and a
phase value $\phi$. To cover this concept we define a time-dependent function

${\epsilon}(t)=\mid\frac{{\cal{W}}_{R}(t)-{\cal{W}}_{f}(t)}{{\cal{W}}_{R}(t)}\mid\
.$ (31)

Therefore, the excitation time, $t_{ex}$, is defined when the condition
$\epsilon(t_{ex})<5\times 10^{-6}$ satisfies and $\epsilon(t)$ in (31) remains
below afterwards. For more details see Hajilou:2018dcb .

The main important question that we are interested in is to investigate
whether the meson excitation time understands about the phase structure of the
gauge theory. Note that we want to know if the meson excitation time can probe
the critical point. To put in another way, we study the behavior of meson
excitation time, $t_{ex}$, near the critical point when the system moves
towards the critical point.

In figure 5 we plotted the excitation time, $t_{ex}$, as a function of
$\frac{\mu}{T}$ for fixed values of $lT=0.10$ , the temperature $T=0.37$ and
interquark distance $l=0.27$. We fixed the transition time $k=0.3$ and $k=3$
for left and right panel, respectively. In fact, in order to introduce the
critical point and its general features for different quenches, we
investigated the behavior of $t_{ex}$ as a function of $\frac{\mu}{T}$ for
different quenches $k=0.3$ ($k=3$) that corresponds to fast (slow) quench for
the left (right) panel, respectively. In both left and right panels the
magenta dashed line corresponds to the critical point which is at
$(\frac{\mu}{T})_{\star}$=1.11072.

Figure 5:  Excitation time $t_{ex}$ as a function of $\frac{\mu}{T}$. Left: We
fixed the values of interquark distance $l=0.27$, the transition time $k=0.3$
and the temperature $T=0.37$ ($lT=0.10$). Right: We fixed the values of
interquark distance $l=0.27$, the transition time $k=3$ and the temperature
$T=0.37$ ($lT=0.10$). In both left and right panels the magenta dashed line
corresponds to the critical point which is at
$(\frac{\mu}{T})_{\star}$=1.11072.

As shown in figure 5, we observed that at the critical point,
$(\frac{\mu}{T})_{\star}$, the excitation time, $t_{ex}$, gets the finite
value though its slope, $\frac{dt_{ex}}{d\frac{\mu}{T}}$, approaches infinity
at this point. In order to check let’s define the slope

$\frac{dt_{ex}}{d\frac{\mu}{T}}(i)=\frac{t_{ex}(i+1)-t_{ex}(i)}{\frac{\mu}{T}(i+1)-\frac{\mu}{T}(i)}~{},$
(32)

where $i$ represents the ith point of the corresponding data points. It is
important to note that, in our numerical results by varying two independent
input parameters, we plotted the $t_{ex}$ and its slope with respect to the
$\frac{\mu}{T}$. In practice, for $\frac{\mu}{T}$ we utilized

$\frac{\mu}{T}=\frac{\pi\,q\,z_{h}\sqrt{\frac{z_{h}^{4}+\sqrt{z_{h}^{8}+4\,q^{2}\,z_{h}^{2}}}{2}}}{q^{2}+2\,zh^{2}\left(\frac{z_{h}^{4}+\sqrt{z_{h}^{8}+4\,q^{2}\,z_{h}^{2}}}{2}\right)}~{},$
(33)

where, $q$ and $z_{h}$ are two independent input parameters. A very
interesting observation is that the slope of data points near the critical
point can be fitted with the function

$\frac{dt_{ex}}{d\frac{\mu}{T}}=(\frac{\pi}{2\sqrt{2}}-\frac{\mu}{T})^{-\theta}~{},$
(34)

where $\theta$ is a positive number and defined as a dynamical critical
exponent Ebrahim:2017gvk .

As you can see in figure 6 we depicted $\frac{dt_{ex}}{d\frac{\mu}{T}}$ as a
function of $\frac{\mu}{T}$ that is the slope (32) near the critical point. In
the left and right panel, we fixed the values of interquark distance $l=0.27$
and the temperature $T=0.37$ ($lT=0.10$). But, in the left panel we fixed the
transition time $k=0.3$ that corresponds to fast quench and in the right panel
we fixed the transition time $k=3$ that corresponds to slow quench. We
observed that for the fast quench the value of dynamical critical exponent is
$\theta=0.515667$ and for slow quench $\theta=0.572473$. A very interesting
observation is that by increasing the value of transition time from $k=0.3$ to
$k=3$, there is a smooth deviation in dynamical critical exponent that is the
dynamical critical exponent is sensitive to the value of the transition time
$k$, although smoothly. Another point is that our result is in good agreement
with the the result that is obtained from the investigation of the behavior of
scalar quasi-normal modes near the critical point in Finazzo:2016psx .

Figure 6:  $\frac{dt_{ex}}{d\frac{\mu}{T}}$ as a function of $\frac{\mu}{T}$.
Left: We fixed the values of interquark distance $l=0.27$, the transition time
$k=0.3$ and the temperature $T=0.37$ ($lT=0.10$). The blue curve is the
function, $(\frac{\pi}{2\sqrt{2}}-\frac{\mu}{T})^{-\theta}$, fitted with the
data with $\theta=0.515667$. Right: We fixed the values of interquark distance
$l=0.27$, the transition time $k=3$ and the temperature $T=0.37$ ($lT=0.10$).
The red curve is the same function fitted with the data with
$\theta=0.572473$.

Consequently, the gauge invariant and non-local observable, i.e. Wilson loop
is a good observable to probe the critical point of the theory when the system
evolves towards the critical point. Recently, different observables is
investigated to find dynamical critical exponent Ebrahim:2018uky ;
Lezgi:2021qog ; Amrahi:2020jqg ; Ebrahim:2020qif .

To investigate the effect of different values of the transition time on the
dynamical critical exponent the behavior of $\frac{dt_{ex}}{d\frac{\mu}{T}}$
as a function of $\frac{\mu}{T}$ is illustrated in figure 7. We fixed the
values of interquark distance $l=0.45$ and the temperature $T=0.31$
($lT=0.14$) for all cases while for green points the value of the transition
time is $k=0.3$ and for blue (red) points are $k=10$ ($k=20$), respectively.
We obtained $\theta=0.529178$ for fast quench $k=0.3$ and $\theta=0.641450$
($\theta=0.830707$) for slow quenches $k=10$ ($k=20$), respectively. In
comparison with figure 6, a considerable change in the value of the dynamical
critical exponent $\theta$ is observed in the figure 7. The important point is
that the dynamical critical exponent $\theta$ is more sensitive to the more
larger values of the transition time $k$, that is the more slower energy
injection, the more larger deviations in $\theta$. Therefore, it is seen that
just for fast quenches, ($k\ll 1$) the value of the dynamical critical
exponent, $\theta$ can be in good agreement with the result that is reported
in Finazzo:2016psx .

Figure 7:  $\frac{dt_{ex}}{d\frac{\mu}{T}}$ as a function of $\frac{\mu}{T}$.
We fixed the values of interquark distance $l=0.45$ and the temperature
$T=0.31$ ($lT=0.14$) for all cases while for green points the value of the
transition time is $k=0.3$ and for blue (red) points are $k=10$ ($k=20$),
respectively. The green curve is the function,
$(\frac{\pi}{2\sqrt{2}}-\frac{\mu}{T})^{-\theta}$, fitted with the data with
$\theta=0.529178$. The blue and red curves are the same functions fitted with
the data with $\theta=0.641450$ and $\theta=0.830707$, respectively.

Figure 8:  $\frac{dt_{ex}}{d\frac{\mu}{T}}$ as a function of $\frac{\mu}{T}$.
Left: We fixed the values of interquark distance $l=0.27$, the transition time
$k=0.3$ , the temperature $T=0.37$ and $lT=0.10$. The blue curve is the
function, $(\frac{\pi}{2\sqrt{2}}-\frac{\mu}{T})^{-\theta}$, fitted with the
data with $\theta=0.515667$. Middle: We fixed the values of interquark
distance $l=0.45$, the transition time $k=0.3$ , the temperature $T=0.31$ and
$lT=0.14$. The green curve is the same function fitted with the data with
$\theta=0.529178$. Right: We fixed the values of interquark distance $l=1$,
the transition time $k=0.3$ , the temperature $T=0.17$ and $lT=0.17$. The red
curve is the same function fitted with the data with $\theta=0.535088$.

Now we would like to analyze the effect of interquark distance $l$ on the
dynamical critical exponent $\theta$. To do so, the behavior of
$\frac{dt_{ex}}{d\frac{\mu}{T}}$ as a function of $\frac{\mu}{T}$ is
illustrated in figure 8 for fixed values of the transition time $k=0.3$. In
the left panel we fixed the interquark distance $l=0.27$ and the temperature
$T=0.37$ or $lT=0.10$ and obtained the dynamical critical exponent
$\theta=0.515667$. In the middle panel we fixed the interquark distance
$l=0.45$ and the temperature $T=0.31$ or $lT=0.14$ and observed the dynamical
critical exponent $\theta=0.529178$. In the right panel we fixed the
interquark distance $l=1$ and the temperature $T=0.17$ or $lT=0.17$ and
obtained the dynamical critical exponent $\theta=0.535088$. The important
feature is that by fixing the time interval of energy injection, $k$, there is
a low increase in $\theta$ when the value of $lT$ increases. In other words,
the dynamical critical exponent $\theta$ is sensitive to the value of
interquark distance $l$ very smoothly. Therefore, we observed that the value
of dynamical critical exponent $\theta$ is more sensible to the value of
energy injection $k$ rather than the interquark distance $l$.

We would like to emphasize that for all numerical results that is obtained in
this research we chose the value of $lT$ in such a way that $lT\ll 1$. This is
because for the condition $lT\gg 1$ there is no meson bound state and only for
the condition $lT\ll 1$ the meson bound state can exist in the QGP and hence
the meson excitation time, $t_{ex}$, can be defined.

##  IV Discussion and outlook

We have continued the research set out in Hajilou:2018dcb  and utilized the
idea of meson excitation time, i.e. $t_{ex}$ to find out some informations
from the phase structure of the QCD. It is important to note that, the final
results of the paper Hajilou:2018dcb  has two main parts. In the first part,
the effect of the parameters of the theory (such as temperature and chemical
potential) on the characteristics of the oscillation, i.e. frequency and
amplitude have been investigated. Then, in the second part, after defining the
excitation time of the meson, i.e. $t_{ex}$ the effect of various parameters
of the theory on the $t_{ex}$ have been described. But, in this research we
borrowed the idea of $t_{ex}$ from Hajilou:2018dcb  to investigate whether the
meson excitation time, $t_{ex}$, can probe the critical point when the system
evolves towards the critical point? and what would be the associated dynamical
critical exponent? In addition, we study the effect of parameters of the
theory and different quenches on the associated dynamical critical exponent
and compare our results with other papers. Note also that the gravity
background in this research is completely different from Hajilou:2018dcb . In
this work, we consider a particular background that possesses the critical
point in such a way that we could investigate the holographic critical point
in the field theory side.

In this work we studied the dynamics of a open string attached to the AdS
boundary of the Vaidya black hole spacetime in the Einstein-Maxwell dilaton
theory. In this theory the black hole solutions possess the ”critical point”
at which the thermodynamical stability of a black hole solution switches. We
obtained that the relaxation of the string slows down when the background
spacetime is taken to the critical point. We would like to emphasize that, at
each value of the ratio of chemical potential and temperature after the
injection of energy, relaxation of string slows down and we could calculate
the excitation time of meson, $t_{ex}$ to probe the critical point of the
theory and then one can obtain the associated dynamical critical exponent.

An important point that we would like to emphasize is that since we work in
the probe limit, i.e. the effects of the backreaction of meson are not
included, therefore the energy of the meson does not dissipate in the plasma.
Consequently, excitation of the meson will remain without decay and the
meson’s characteristics of oscillation, i.e. frequency and amplitude remain
unchanged. Another point is that, the string dynamics studied in this work
relaxed to the equilibrium states. This is very interesting result that has
been addressed in Ali-Akbari:2015ooa ; Hajilou:2017sxf ; Hajilou:2018dcb .
Let’s consider a bound-state of the stable meson at $t=0$ which is in its
ground state. It is described by the static string hanged from the boundary
into the bulk with the end points on the boundary. Due to the injection of
energy into the plasma, the temperature and the chemical potential are
increased from zero to finite values of $T$ and $\mu$, respectively. According
to the results of Ishii:2014paa ; Ali-Akbari:2015ooa , as the temperature
increases the shape of the brane changes time-dependently. As the energy is
being injected the turning point of the string gets closer to the black hole
horizon. Their results shows that the string oscillates around the string
static solution corresponding to the final temperature of the system after
finishing the quench. These oscillations are described as the oscillations of
the time-dependent Wilson loop in the field theory side. In other words, based
on the Ishii:2014paa  where the authors have studied the dynamics of the shape
of the brane in a time-dependent background, these oscillations can be
interpreted as if the quench puts the stable meson into the final excited
state. The power spectrum of the condensation oscillations gives the excited
mesonic modes in the field theory side. As a matter of fact, after the energy
injection, the string is oscillating in one of the its normal modes, i.e.
relaxes to the equilibrium state or equivalently the quark-antiquark bound
state has been excited and oscillates with specific frequency and amplitude.

It is important to note that, we have chosen the stable branch of background
solution. This is due to the fact that, the thermal stability conditions are
determined by the sign of the heat capacity $C_{V}$. The positivity of the
heat capacity guarantees thermally stable solutions, while the negative heat
capacity determines the unstable solutions. In black hole case, when we
decrease $z_{h}$, i.e. larger horizon, the black hole should be more warmer,
while the unstable branch of the background spacetime gets cooler. Therefore,
we choose the stable (physical) background solution to investigate the physics
of open string as a probe on this background. But, about the solutions of open
string in this background there is one stable and one unstable static solution
for the same boundary condition Hashimoto:2018fkb . Note also that, the
unstable classical configuration is not the minimum energy configuration but
is used in Hashimoto:2018fkb  to probe the horizon.

In a next phase of research, one interesting direction to extend this work and
obtain results of this research from other view point is to consider special
initial conditions for open string. In other words, it could be more
transparent just to study $t_{ex}$ for the motion of the string with initial
(nonlinear) perturbations on a static black hole background, rather than
inducing string motion by using the Vaidya (time-dependent) spacetime as the
background. This approach originates from the conjecture that the behavior of
the relaxation of the string is more or less insensitive to the precise shape
of the string excitation from the equilibrium shape. Another point is that,
the critical point corresponds to the background spacetime that is
thermodynamically marginally stable. Since, it is the property of the
background spacetime, its stability is not directly related to properties of
string dynamics on this background spacetime. Therefore, the physical origin
of the behaviors found in this work about string dynamics near critical point,
could be explained based on the properties of the background geometry and the
classical mechanics of the string dynamics in it.

## Acknowledgement

A. H. would like to thank David Dudal for warm hospitality at KU Leuven Kulak
Campus that some parts of this research are accomplished under a scholarship
that was awarded by the Ministry of Science, Research and Technology
(Department of Scholarship and Students’ s Affairs Abroad) of the Islamic
Republic of Iran. Also, thanks to Mohammad Ali-Akbari, Hajar Ebrahim, Irina
Aref’eva, Keiju Murata, Song He, Subhash Mahapatra, Seyed Naseh Sajadi, Farid
Charmchi, Povel Slepov, and Siddhi Jena for valuable discussions. In addition,
thanks to Leila Shahkarami and Farid Charmchi for supporting the Mathematica
program to solve equations of motion via the Finite Difference Method.

##  Appendix A Review on the charged black hole backgrounds

###  A.1  EM-dilaton black hole background

Consider the action of 5-dimensional Einstein-Maxwell-dilaton gravity
Zhang:2015dia :

$S=-\frac{1}{16\pi}\int
d^{5}x\sqrt{-g}[{\cal{R}}-\frac{4}{3}(\nabla\Phi)^{2}-V(\Phi)-e^{-\frac{4\alpha}{3}\Phi}\hat{F}_{\mu\nu}\hat{F}^{\mu\nu}]~{}.$
(35)

where ${\cal{R}}$ is Ricci scalar, $\hat{F}_{\mu\nu}$ is field strength of the
$U(1)$ gauge field, $\Phi$ is the scalar dilaton field, $\alpha$ is the
coupling constant between the dilaton and the Maxwell field and $V(\Phi)$ is
dilaton potential. For more details see Zhang:2015dia . The equations of
motion are:

$\begin{split}{\cal{R}}_{\mu\nu}-\frac{1}{2}g_{\mu\nu}{\cal{R}}-\frac{4}{3}\left({\partial}_{\mu}\Phi{\partial}_{\nu}\Phi-\frac{1}{2}g_{\mu\nu}(\nabla\Phi)^{2}-\frac{3}{8}g_{\mu\nu}V(\Phi)\right)-2e^{\frac{-4{\alpha}\Phi}{3}}\left(\hat{F}_{\mu\rho}\hat{F}_{\nu}^{\rho}-\frac{1}{4}g_{\mu\nu}\hat{F}_{\rho{\sigma}}\hat{F}^{\rho{\sigma}}\right)&=0~{},\\\
{\nabla}^{2}{\Phi}-\frac{3}{8}\frac{{\partial
V}}{\partial\Phi}+\frac{\alpha}{2}e^{\frac{-4{\alpha}\Phi}{3}}\hat{F}_{\rho{\sigma}}\hat{F}^{\rho{\sigma}}&=0~{},\\\
{\nabla}_{\mu}\left(e^{\frac{-4{\alpha}\Phi}{3}}\hat{F}^{\mu\nu}\right)&=0~{}.\end{split}$
(36)

The solution for the metric obtaining from the above action is:

$\begin{split}ds^{2}=&-N(z)f(z)dt^{2}+\frac{1}{z^{4}}\frac{dz^{2}}{(1+b^{2}z^{2})f(z)}+\frac{1+b^{2}z^{2}}{z^{2}}g(z)d\vec{x}^{2}~{},\\\
f(z)=&\frac{1+b^{2}z^{2}}{z^{2}}\Gamma^{2\gamma}-M\frac{z^{2}}{1+b^{2}z^{2}}\Gamma^{1-\gamma}~{},\end{split}$
(37)

where $z$ is the radial coordinate in bulk gravity, so that $z=0$ is boundary
of the gravity theory where the field theory lives and
$\vec{x}\equiv(x_{1},x_{2},x_{3})$. Also, the other functions and parameters
are already defined in equation (2). This solution is asymptotically AdS5. We
set the AdS radius to be one. When we set $\alpha=0$ in equation (36), the
solution reduces to the well-known Reissner-Nordström-AdS black hole.

###  A.2  EM-dilaton-Vaidya background

The generalization of the static background (37) to the time-dependent case
can be achieved by adding external source terms to the action (35). To do so,
we need introduce external matter sources and then the equations of motion
are:

$\begin{split}{\cal{R}}_{\mu\nu}-\frac{1}{2}g_{\mu\nu}{\cal{R}}-\frac{4}{3}\left({\partial}_{\mu}\Phi{\partial}_{\nu}\Phi-\frac{1}{2}g_{\mu\nu}(\nabla\Phi)^{2}-\frac{3}{8}g_{\mu\nu}V(\Phi)\right)-2e^{\frac{-4{\alpha}\Phi}{3}}\left(\hat{F}_{\mu\rho}\hat{F}_{\nu}^{\rho}-\frac{1}{4}g_{\mu\nu}\hat{F}_{\rho{\sigma}}\hat{F}^{\rho{\sigma}}\right)&=8\pi
T_{\mu\nu}^{ext},\\\
{\nabla}_{\mu}\left(e^{\frac{-4{\alpha}\Phi}{3}}\hat{F}^{\mu\nu}\right)&=8\pi
J_{ext}^{\nu}~{}.\end{split}$ (38)

The equation of motion of the dilaton field will not change and are the same
as equation (36). The solution for the EM-dilaton-Vaidya metric in Eddington-
Finkelstein coordinates is equation(1) provided that the external matter
source satisfies

$\begin{split}8\pi
T_{\mu\nu}^{ext}&=\left[\frac{z^{3}(6+4b^{2}z^{2})}{4(1+b^{2}z^{2})^{\frac{3}{2}}}\dot{M}(\bar{v})\right]{\delta}_{\mu}^{\bar{v}}{\delta}_{\nu}^{\bar{v}}~{},\\\
8\pi
J_{ext}^{\nu}&=\left[\frac{bz^{5}}{2}\left(\frac{1}{1+b^{2}z^{2}}\right)^{\frac{1}{3}}\frac{\dot{M}(\bar{v})}{\sqrt{M(\bar{v})}}\right]{\delta}_{z}^{\bar{v}}~{},\end{split}$
(39)

where $\dot{M}(\bar{v})=dM/d\bar{v}$. For more details see Zhang:2015dia .

##  Appendix B Boundary condition

The boundary conditions for solving the equations of motion (28) can be
obtained by fixing the diffeomorphism invariance on the two-dimensional world-
sheet of the string as done in Ishii:2014paa . Therefore, on the boundary one
can choose $u=v$ for fixing one of the endpoints of string and $u=v+L$ for
fixing the other one. Therefore, the boundary condition at the AdS boundary
for $Z$ and $X$ are:

$\begin{split}Z|_{u=v}&=0~{};~{}~{}~{}~{}X|_{u=v}=\frac{-l}{2}~{},\cr
Z|_{u=v+L}&=0~{};~{}~{}~{}~{}X|_{u=v+L}=\frac{l}{2}~{}.\end{split}$ (40)

By applying these boundary conditions, during the injection of energy, the
distance between the quark and antiquark will not change. One can get the rest
of the boundary conditions by expanding $V(u,v)$, $Z(u,v)$ and $X(u,v)$ about
the point $u=v$ at the boundary. As follows

$\displaystyle V(u,v)$ $\displaystyle=$ $\displaystyle
V_{0}(v)+V_{1}(v)(u-v)+...~{},~{}~{}~{}~{}~{}~{}~{}~{}$ (41) $\displaystyle
Z(u,v)$ $\displaystyle=$ $\displaystyle
Z_{1}(v)(u-v)+Z_{2}(v)(u-v)^{2}+...~{},~{}~{}~{}~{}~{}~{}~{}~{}$ (42)
$\displaystyle X(u,v)$ $\displaystyle=$
$\displaystyle\frac{-l}{2}+X_{1}(v)(u-v)+...~{}.~{}~{}~{}~{}~{}~{}~{}~{}$ (43)

By putting the above equations into the dynamical equations (28) and demanding
the regularity condition at $u=v$, the other boundary conditions one can
obtained. The consistency with the constraint equations (29) should be
checked. For the other point, $u=v+L$, the mentioned procedure should be
followed. Finally, the result of the expansion for the point $u=v$ are:

$\displaystyle V(u,v)$
$\displaystyle=V_{0}(v)+{\cal{O}}\left((u-v)^{5}\right),$ (44a) $\displaystyle
Z(u,v)$
$\displaystyle=\frac{\dot{V}_{0}(v)}{2}(u-v)+\frac{\ddot{V}_{0}(v)}{4}(u-v)^{2}+\frac{\dddot{V}_{0}(v)}{12}(u-v)^{3}+{\cal{O}}\left((u-v)^{4}\right),$
(44b) $\displaystyle X(u,v)$
$\displaystyle=\frac{-l}{2}+{\cal{O}}\left((u-v)^{3}\right).$ (44c)

From the above one can see that

$Z_{,uv}|_{u=v}=0,\ 2Z_{,u}|_{u=v}=\dot{V}_{0}(v)~{},$ (45)

where $\dot{V}(v)=\frac{dV(v)}{dv}$. It is easy to check that the same results
for the other point at the boundary, i.e. $u=v+L$ can be obtained. The
interested reader can refer to Ishii:2014paa  for more details.

##  Appendix C Initial condition

The initial condition for $Z$, $V$ and $X$ can be obtained by using the static
equation (24) and constraint equations (29). In the equation (24) we replace
$x$ and $z$ with the capital ones and set $b=0$, $N(z)=1$, $g(z)=1$ and
$f(z)=\frac{1}{z^{2}}$. Using (44a) , (44b) and considering $V_{,v}>0$ at the
boundary, the other conditions $Z_{,u}>0$ and $Z_{,v}<0$ can be found. By
imposing $X_{,u}|_{Z=0}=X_{,v}|_{Z=0}=0$ and using the constraint equations
(29), we get

$\displaystyle V_{,u}$
$\displaystyle=Z_{,u}\bigg{(}-1+\sqrt{1+\big{(}\frac{dX}{dZ}\big{)}^{2}}\bigg{)}~{},~{}$
(46) $\displaystyle V_{,v}$
$\displaystyle=Z_{,v}\bigg{(}-1-\sqrt{1+\big{(}\frac{dX}{dZ}\big{)}^{2}}\bigg{)}~{}.$
(47)

Now by taking the derivative of equation (46) with respect to $v$ and equation
(47) with respect to $u$ and considering that $V_{,uv}=V_{,vu}$ the result is

$\bigg{(}Z_{,u}\sqrt{1+\big{(}\frac{dX}{dZ}\big{)}^{2}}\bigg{)}_{,v}=0~{}.$
(48)

Using equation (24) and substituting $\frac{dX}{dZ}$ into the above equation
one can get the initial condition for $Z(u,v)$

$Z\,_{2}F_{1}\bigg{(}\frac{1}{2},\frac{1}{4};\frac{5}{4};\frac{Z^{4}}{{Z_{*}}^{4}}\bigg{)}=\phi(u)-\phi(v)~{},$
(49)

where $\phi(y)$ is an arbitrary function and our choice is $\phi(y)=y$
Ishii:2014paa . One can get the initial condition for $X(u,v)$ by integrating
the equation (24) as follows

$X(u,v)=\frac{l}{2}-\frac{Z^{3}}{3{Z_{*}}^{2}}\,_{2}F_{1}\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};\frac{Z^{4}}{Z_{*}^{4}}\right)~{}.$
(50)

Where, $Z_{*}$ is the turning point of the string and since at $Z=Z_{*}$ the
equation $X(u,v)=0$, we have

$Z_{*}=\frac{3~{}l}{2~{}_{2}F_{1}\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};1\right)}~{}.$
(51)

Also, using the equations (46) and (47) one can obtain the initial condition
for $V(u,v)$:

$\displaystyle V(u,v)$
$\displaystyle=-Z\left(1-\,_{2}F_{1}\left(\frac{1}{2},\frac{1}{4};\frac{5}{4};\frac{Z^{4}}{Z_{*}^{4}}\right)\right)+\chi(v)~{},~{}~{}~{}~{}~{}~{}~{}$
(52) $\displaystyle V(u,v)$
$\displaystyle=-Z\left(1+\,_{2}F_{1}\left(\frac{1}{2},\frac{1}{4};\frac{5}{4};\frac{Z^{4}}{Z_{*}^{4}}\right)\right)+{\tilde{\chi}}(u)~{},~{}~{}~{}~{}~{}~{}~{}$
(53)

where $\chi$ and $\tilde{\chi}$ are arbitrary functions. Moreover, by
equalizing the above equations and using (49), we would have

$\chi(v)=2\phi(v)~{},~{}~{}~{}~{}~{}{\tilde{\chi}}(u)=2\phi(u)~{}.$ (54)

For more details see Ishii:2014paa .

##  Appendix D Numerical procedure to calculate the critical exponent

In this section we discuss the numerical procedure that is used above to
determine the dynamical critical exponent $\theta$ when the physical system
moves towards the critical point $(\frac{\mu}{T})_{\star}=1.11072$. At first,
for producing the $t_{ex}$ for each point, i.e. the finite value of
$\frac{\mu}{T}$, we used equation (31). In addition, to calculate the
dynamical critical exponent $\theta$ let’s take an specific example which the
parameters of our physical system are as the transition time $k=0.3$,
interquark distance $l=0.45$ and $lT=0.14$. Therefore, when our system is
moving towards the critical point from point $\frac{\mu}{T}=1.0243$ to
$(\frac{\mu}{T})_{\star}=1.11072$ we split this interval into $22$
subintervals with different step sizes ($\Delta(\frac{\mu}{T})$ that are
between $10^{-2}$ and $10^{-5}$). Also, for computing the numerical
derivatives we used the slope (32). In Table 1 we specified the subintervals
utilized to determine the dynamical critical exponent $\theta$. The result
corresponding with above fixed parameters is shown in the middle panel of the
figure 8 and (after using Wolfram’s Mathematica) the associated critical
exponent $\theta=0.529178$ is obtained.

point |  Value of $t_{ex}$ |  Value of $\frac{\mu}{T}$ |  Value of $\frac{dt_{ex}}{d\frac{\mu}{T}}$
---|---|---|---
1 | 0.7101 | 1.0243 | 0.64
2 | 0.7234 | 1.0451 | 1.64
3 | 0.7561 | 1.0651 | 2.12
4 | 0.7794 | 1.0761 | 2.20
5 | 0.7974 | 1.0843 | 3.10
6 | 0.8225 | 1.0924 | 3.34
7 | 0.8489 | 1.1003 | 4.40
8 | 0.8621 | 1.1033 | 6.67
9 | 0.8701 | 1.1045 | 9.45
10 | 0.8805 | 1.1056 | 12.85
11 | 0.8972 | 1.1069 | 16.15
12 | 0.9295 | 1.1089 | 23.33
13 | 0.9365 | 1.1092 | 31.60
14 | 0.9523 | 1.1097 | 40.29
15 | 0.9664 | 1.11005 | 50.00
16 | 0.9714 | 1.11015 | 72.50
17 | 0.9772 | 1.11023 | 82.50
18 | 0.9871 | 1.11035 | 87.00
19 | 0.9958 | 1.11045 | 102.31
20 | 1.0091 | 1.11058 | 112.50
21 | 1.0136 | 1.11062 | 143.28
22 | 1.0232 | 1.11069 | 206.06
23 | 1.0300 | 1.11072 |
Table 1: Values of $t_{ex}$, $\frac{\mu}{T}$ and
$\frac{dt_{ex}}{d\frac{\mu}{T}}$ used for the fit procedure for the transition
time $k=0.3$, interquark distance $l=0.45$ and $lT=0.14$ with the fitted
critical exponent $\theta=0.529178$.

##  Appendix E Meson potential at zero temperature

In this research, before the injection of energy the spacetime background is
pure AdS5. Utilizing the equations (20), (21) and (22) one can find the static
potential of meson in the AdS5 background that is correspondence with zero
temperature field theory. After some algebra one can find:

$\displaystyle V(l)=\frac{1}{\pi\alpha^{\prime}}\bigg{[}\frac{1}{z_{\ast}}$
$\displaystyle\int_{1}^{\infty}\,dy\left(\sqrt{\frac{y^{4}-y_{h}^{4}}{y^{4}-1}}-1\right)-\left(\frac{1}{z_{\ast}}-\frac{1}{z_{h}}\right)\bigg{]}~{},$
(55)

where, $y=z/z_{h}$ and $y_{h}=z_{\star}/z_{h}$. The static potential of the
meson at zero temperature is depicted in figure 9. By considering the Cornell
potential Eichten:1978tg ; Andreev:2006ct ; Bruni:2018dqm ; Yang:2015aia :

$V(l)=-\frac{\kappa}{l}+\sigma_{s}l+C~{},$ (56)

where, $\kappa$ is a Coulomb strength parameter, $\sigma_{s}$ is QCD string
tension and $C$ is a constant. We observe that here at zero temperature, since
the spacetime is pure AdS5 then surely when we calculate the potential of
meson via Wilson loop, we will obtain just Coulomb potential part
($-\frac{\kappa}{l}$) of the Cornell Potential. This is because that the
linear regime of the Cornell potential can be obtained when the geometry is
ended in the holographic direction $z$ Witten:1998zw ; Polchinski:2001tt ;
CasalderreySolana:2011us . But, in the case of pure AdS5 the spacetime does
not end in the direction of $z$ in such a way that Wilson loop potential does
not possess the linear regime of the Cornell potential.

Figure 9:  Potential of meson in the $AdS_{5}$ background that corresponds
with zero temperature in field theory. Utilizing the Wilson loop the Coloumb
potential can be obtained.

##  Appendix F Time evolution of the string

We depicted the time evolution of the classical open string in the EM-dilaton-
Vaidya background, in figure (10) for fixed values of the interquark distance
$l=0.27$, transition time $k=0.3$ and the temperature $T=0.37~{}(lT=0.10)$. As
shown in in figure (10), in the gravity side we see that the open string
oscillates around the static configuration correspondence with the static
potential of the meson after injection of energy in the field theory side. In
the figure (10), we have chosen the $\frac{\mu}{T}=1.11071$, but it is
important to note that, this behavior, i.e oscillation around static
configuration is observed for all values of the ratio of $\frac{\mu}{T}$ as
well as critical point. For each values of $\frac{\mu}{T}$ we can obtain the
excitation time of the meson and then the dynamical critical exponent $\theta$
can be calculated.

Figure 10:  Time evolution of the classical open string in the EM-dilaton-
Vaidya background. We have fixed the interquark distance $l=0.27$, transition
time $k=0.3$ and the temperature $T=0.37~{}(lT=0.10)$. Here, we have
$\frac{\mu}{T}=1.11071$

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