A Classical String in Lifshitz–Vaidya Geometry
Abstract
We study the time evolution of the expectation value of a rectangular Wilson loop in strongly anisotropic timedependent plasma using gaugegravity duality. The corresponding gravity theory is given by describing time evolution of a classical string in the Lifshitz–Vaidya background. We show that the expectation value of the Wilson loop oscillates about the value of the static potential with the same parameters after the energy injection is over. We discuss how the amplitude and frequency of the oscillation depend on the parameters of the theory. In particular, for the transverse case, by raising the anisotropy parameter, we observe that the amplitude and frequency of the oscillation increase. In the longitudinal case, although the amplitude of the oscillation increases for larger values of anisotropy parameter, the frequency is independent of anisotropy parameter.
1 Introduction and result
Quarkgluon plasma, as a new phase of matter, is produced at Relativistic Heavy Ion Collider (RHIC) and Large Hadron Collider (LHC) by colliding two pancakes of heavy nuclei such as Gold (Au) or lead (Pb) at a relativistic speed [1]. Through viscous hydrodynamical simulations, it is realized that viscosity over entropy density is small, i.e. \(\eta /s=1/4\pi \) [2, 3], and it is then a strong indication that the plasma is strongly coupled. Furthermore, at early times, the plasma is far from equilibrium and after a certain time viscous hydrodynamic description can be applied. Although the viscous hydrodynamics is applicable during most of the time evolution, a significantly different pressure between longitudinal and transverse directions exists indicating that the plasma created in the heavy ion collision is anisotropic.
Since the plasma is strongly coupled, it is not reliable to describe various properties of the plasma by applying perturbation method. As a results, as a nonperturbative method, gaugegravity duality provides a novel approach for studying the strong coupling limit of a large class of nonabelian quantum gauge theories [4]. According to this duality, a strongly coupled gauge theory defined in a ddimensional spacetime corresponds to a classical gravity in a \(d+1\)dimensional spacetime [1, 4, 5, 6]. Therefore, different questions in the strongly coupled gauge theory can be translated into corresponding problems in the classical gravity. This duality has been frequently applied to study various aspects of the strongly coupled systems such as static potential energy between a quark and antiquark pair [7], jet quenching parameter [8], thermalization and isotropization process [9, 10, 11, 12, 13]. For more details, see [1] and references therein.
Finding static potential energy between a quark and antiquark pair, or equivalently quark–antiquark bound state, living in the plasma is an interesting problem that has been attracted a lot of attention. This problem has been firstly addressed in [7] and then its generalization has been widely discussed in the literature. Concisely, in order to calculate the static potential energy between the pair we need to compute expectation value of a rectangular Wilson loop in the strongly coupled plasma. The holographic dual of the rectangular Wilson loop is given by a classical string suspended from two points (corresponding to quark and antiquark), hanging down in extra dimension with appropriate boundary conditions. Using this idea, static potential energy is studied in different gauge theories with holographic duals and it recently generalizes to a timedependent case in [14]. In the timedependent case, the time evolution of the expectation value of the Wilson loop during the energy injection into the gauge theory is investigated. Holographically, the mentioned system corresponds to the time evolution of the classical string in the AdSVaidya background. As a toy model, the AdSVaidya background is dual to thermalization process in the gauge theory [15].
The Lifshitzlike background, which is holographically dual to an anisotropic plasma, is applied to investigate different properties of the anisotropic plasma. One of the things that makes the Lifshitzlike background interesting is that holographic estimates of the total multiplicity can fit the experimental data at high energy for certain values of critical exponent [16]. It was also shown that the Lifshitzlike background can be considered as the IR limit of the 10dimensional IIb supergravity anisotropic background suggested in [18]. Vaidya solutions in the Lifshitzlike background have been found in [19].

The quark–antiquark bound state is excited in the anisotropic plasma due to the energy injection. The characteristic of the excited bound state does depend on anisotropy parameter which, in our case, is given by the critical exponent of the Lifshitzlike metric. In fact, for larger values of the anisotropy parameter, when the other parameters have been fixed, the excited bound state oscillates with larger oscillation frequency f and amplitude A.

To compare with the real plasma produced at RHIC or LHC, the case of \(\nu =4\) is more reasonable^{2} [16]. Our numerical results show that \(\frac{f_{\nu =4}}{f_{\nu =1}}\simeq 8.1\) and \(\frac{A_{\nu =4}}{A_{\nu =1}}\simeq 2.1\) for the same values of the transition time, final temperature and distance l between quark and antiquark. By transition time we mean how slow or fast the energy has been injected into the system under study. As a matter of fact, the anisotropy of the system substantially influences the bound state living in the plasma.

We observe that the oscillation frequency of the excited bound state depends on the transition time. In other words, for fast (slow) energy injection the bound state is excited with larger (smaller) oscillation frequency for a fixed value of anisotropy parameter at fixed temperature. Larger values of the anisotropy parameter have larger oscillation frequencies. Similarly, the amplitude of the oscillation increases for smaller transition time k.

Our numerical calculations show that the final temperature and the oscillation frequency are independent. It happens for all cases with or without anisotropy parameter. At fixed temperature, we observe that the anisotropy parameter, the amplitude and frequency of oscillation increase together. However, for given anisotropy parameter, the amplitude of the oscillation and temperature increase together while the oscillation frequency does not change.

Another result is that the frequency and amplitude of the oscillation depend on the distance between quark and antiquark. By raising the distance, both frequency and amplitude increase.

For the longitudinal case, when the classical string is located transverse to the anisotropic directions, opposite to the transverse case, the oscillation frequency is independent of the anisotropy parameter and the bound state is less stable for larger values of the anisotropy parameter. Similar to the transverse case, the amplitude of the oscillation increases when \(\nu \) is larger.
2 Review on the static and timedependent backgrounds
The transition time k plays a central role in energy injection into the system. As a matter of fact, for small values of k a universal behavior is observed [24, 25, 26]. By universal behavior we mean the rescaled equilibration time \(k^{1}t_{eq}\) is independent of the final value of the temperature. This result is perhaps common to all strongly coupled gauge theory with gravity dual. For this reason, we are particularly interested in studying two different regimes for the transition time, that is \(k<1\) and \(k>1\). Thus, in the following we choose \(k=0.3\) and 3.
3 Probe classical string
In this section, using gaugegravity duality, we will obtain the static potential energy between a quark and an antiquark (or equivalently expectation value of the Wilson loop) in the anisotropic plasma describing by the background (2.1). Then the expectation value of the Wilson loop is numerically computed in the Lifshitz–Vaidya metric (2.3). In fact, timedependent solution oscillates about the static one found in the background (2.1).
3.1 On the static Lifshitz black hole background
3.2 In the Lifshitz–Vaidya background
Appropriate numbers for boundary conditions
\(\nu \)  1  2  3  4 

m  5  4  6  8 
n  4  4  4  4 
r  3  2  6  7 
 Boundary condition Based on the discussions in [27, 28, 29], by fixing the diffeomorphism on the string worldsheet one may choose the AdS boundary to be at \(u = v\) for one of the endpoints and \(u =v+L\) for the other one. Then, on the AdS boundary, the appropriate boundary conditions on Z and X areOne can find the rest of the boundary conditions by expanding the fields near the boundary \(u=v\) as follows$$\begin{aligned} \begin{aligned} Z_{u=v}&=0,\quad X_{u=v}=\frac{l}{2},\\ Z_{u=v+L}&=0,\quad X_{u=v+L}=\frac{l}{2}. \end{aligned} \end{aligned}$$(3.15)$$\begin{aligned} V(u,v)= & {} V_0(v) + V_1(v) (uv) + \cdots , \end{aligned}$$(3.16)$$\begin{aligned} Z(u,v)= & {} Z_1(v) (uv) +Z_2(v) (uv)^2+ \cdots , \nonumber \\ \end{aligned}$$(3.17)Then demanding the regularity condition at \(u\!=\!v\) and \(u\!=\!v+L\) the rest of the boundary condition can be found. Furthermore, the consistency of the results with the constraint equations (3.14) must be checked. The final results for boundary conditions in terms of anisotropic parameter \(\nu \) at \(u=v\) are:$$\begin{aligned} X(u,v)= & {} \frac{l}{2} + X_1(v) (uv) + \cdots . \end{aligned}$$(3.18)$$\begin{aligned} V(u,v)&=V_0(v) + {{\mathcal {O}}}\left( (uv)^m\right) , \end{aligned}$$(3.19a)$$\begin{aligned} Z(u,v)&=\frac{{\dot{V}}_0(v)}{2} (uv) + \frac{{\ddot{V}}_0(v)}{4} (uv)^2\nonumber \\&\quad +\frac{\dddot{V}_0(v)}{12} (uv)^3 + {{\mathcal {O}}}\left( (uv)^n\right) , \end{aligned}$$(3.19b)where \({\dot{V}}(v)=\frac{dV(v)}{dv}\) and so on. m, n and r are listed in Table 1. The above equations then imply that$$\begin{aligned} X(u,v)&=\frac{l}{2} + {{\mathcal {O}}}\left( (uv)^r\right) , \end{aligned}$$(3.19c)One can easily check that for the another boundary \(u=v+L\) the results are the same. We refer interested reader to [27, 28, 29] for more details.$$\begin{aligned} Z_{,uv}_{u=v} = 0,\ 2 Z_{,u}_{u=v} = {\dot{V}}_0(v). \end{aligned}$$(3.20)
 Initial condition To obtain the initial conditions for the variables V, Z and X we use the constraint equations and the static solution (3.7). Notice that in this equation we replace z and x with the capital ones and \(f(z)=1\). Since \(V_{,v}>0\) at the boundary, therefore by using the boundary conditions (3.19a) and (3.19b), \(Z_{,u}>0\) and \(Z_{,v}<0\). Applying these conditions on Z and V derivatives and using \(X_{,u}_{Z=0}=X_{,v}_{Z=0}=0\), the constraint equations (3.14) lead to$$\begin{aligned} V_{,u}&= Z_{,u} \left( 1+\sqrt{1+Z^{2\frac{2}{\nu }}\left( \frac{dX}{dZ} \right) ^2}\right) ,~ \end{aligned}$$(3.21)By taking the derivative of the Eq. (3.21) with respect to v and of the Eq. (3.22) with respect to u, we obtain$$\begin{aligned} V_{,v}&= Z_{,v} \left( 1\sqrt{1+Z^{2\frac{2}{\nu }}\left( \frac{dX}{dZ}\right) ^2}\right) . \end{aligned}$$(3.22)and it can be then written as$$\begin{aligned}&Z_{,uv} \left( \sqrt{1+Z^{2\frac{2}{\nu }}\left( \frac{dX}{dZ}\right) ^2}\right) \nonumber \\&\quad +\, Z_{,v} Z_{,u} \left( \sqrt{1+Z^{2\frac{2}{\nu }}\left( \frac{dX}{dZ} \right) ^2} \right) _{,Z}=0, \end{aligned}$$(3.23)One can substitute \(\frac{dX}{dZ}\) into (3.24) by using (3.7) and we then have$$\begin{aligned} \left( Z_{,u} \sqrt{1+Z^{2\frac{2}{\nu }} \left( \frac{dX}{dZ}\right) ^2}\right) _{,v}=0. \end{aligned}$$(3.24)where \(\phi (y)\) is an arbitrary function. The form of the right hand side of the above equation is fixed by applying the fact that the left hand side is zero at \(u=v\). By integrating (3.7), we get the initial configuration for X(u, v) as follows$$\begin{aligned}&Z \, _2F_1 \left( \frac{1}{2},\frac{\nu }{2+2\nu }; \frac{2+3\nu }{2+2\nu };\frac{Z^{2+\frac{2}{\nu }}}{Z_*^{2+\frac{2}{\nu }}} \right) \nonumber \\&\quad = \phi (u)  \phi (v), \end{aligned}$$(3.25)where \(Z_*\) is the turning point of the string. Since \(X(u,v)=0\) at \(Z=Z_*\), we have$$\begin{aligned} X(u,v)= & {} \frac{l}{2}  \frac{Z^{1+\frac{2}{\nu }}}{(1+\frac{2}{\nu }) Z_*^{1+\frac{1}{\nu }}} \,_2F_1 \nonumber \\&\times \, \left( \frac{1}{2},\frac{2+\nu }{2+2\nu };\frac{4+3\nu }{2+2\nu };\frac{Z^{2+\frac{2}{\nu }}}{Z_*^{2+\frac{2}{\nu }}}\right) , \end{aligned}$$(3.26)Also, the initial condition on V(u, v) is obtained from (3.21) and (3.22)$$\begin{aligned} Z_*=\left( \frac{2+\nu }{2\nu }\frac{l}{ _2F_1\left( \frac{1}{2},\frac{2+\nu }{2+2\nu };\frac{4+3\nu }{2+2\nu };1\right) }\right) ^{\nu }. \end{aligned}$$(3.27)$$\begin{aligned} V(u,v)= & {} \, Z \left( 1\, _2F_1\left( \frac{1}{2},\frac{\nu }{2+2\nu }; \frac{2+3\nu }{2+2\nu };\frac{Z^{2+\frac{2}{\nu }}}{Z_*^{2+ \frac{2}{\nu }}}\right) \right) \nonumber \\&+\,\chi (v), \end{aligned}$$(3.28)where \(\chi \) and \({\tilde{\chi }}\) are arbitrary functions. To have better clarification of \(\chi \) and \({\tilde{\chi }}\), let’s equalize the above equations and use (3.25), we get$$\begin{aligned} V(u,v)= & {}  Z \left( 1+\,_2F_1\left( \frac{1}{2}, \frac{\nu }{2+2\nu };\frac{2+3\nu }{2+2\nu };\frac{Z^{2+ \frac{2}{\nu }}}{Z_*^{2+\frac{2}{\nu }}}\right) \right) \nonumber \\&+\,{{\tilde{\chi }}}(u), \end{aligned}$$(3.29)An appropriate choice of the arbitrary function, introduced in (3.25), is \(\phi (y)=y\). Our calculations in this paper are done by the same choice. For more details, see [27, 28, 29].$$\begin{aligned} \chi (v) = 2 \phi (v),\quad {{\tilde{\chi }}}(u) = 2 \phi (u). \end{aligned}$$(3.30)
4 Numerical results
Based on the gaugegravity duality, classical string in the anisotropic background is dual to a quark–antiquark bound state in the anisotropic gauge theory. When the boundary time is smaller than zero, that is \(t<0\), the meson is stable and in its ground state. As the energy injection is started, or equivalently the temperature is raised on the gauge theory side, the shape of the string changes timedependently. In fact, during the energy injection, the turning point of the string goes closer to the horizon in the background. Our numerical results show that the string oscillates about the static string solution corresponding to the final temperature of the energy injection. In the gauge theory, this is the reason why expectation value of the Wilson loop oscillates about static potential by which we mean the potential of the bound state in the anisotropic plasma with finite temperature \(T_f\). These oscillations indicate that bound state is excited by energy injection. This result is in agreement with the similar one reported in [14]. Note that since there is no energy dissipation, the excited meson is stable.
Understanding how anisotropy parameter \(\nu \) affects on the time evolution of the expectation value of the Wilson loop is an interesting issue to investigate. In Fig. 2, \({{\mathcal {W}}}_R(l,t)\) has been plotted for various anisotropy parameters at fixed values of transition time k, final temperature \(T_f\) and distance l. It is clearly seen that the excited bound state is different for each anisotropy parameter. More precisely, the larger anisotropy parameter, the larger frequency. Furthermore, independent of the anisotropy parameter, the timedependent expectation value starts oscillating around the negative equilibrium value of the static potential almost at the same time, i.e. \(t \simeq 0.30\). In Fig. 4, we show that the amplitude of the oscillation increases for larger \(\nu \), too.
The oscillation frequency of Fig. 4 for various anisotropy parameters
\(\nu \)  1  2  3  4 

f  0.31  0.50  1.25  2.50 
In order to have better estimate of the dependence of the frequency on the anisotropy parameter, we have plotted the expectation value of the Wilson loop in term of the boundary time in the region of \(t=3\)–9 in Fig. 4. As it is clearly seen from this figure and confirmed by Table 2, the oscillation frequency substantially increases when the anisotropy parameter is raised in the plasma. It is important to notice that although the frequency of the excited bound state is larger, it is deeply bounded due to the anisotropy in the system.
We would also like to investigate the effect of the temperature on the oscillation frequency. To do so, we plot the expectation value of the Wilson loop in terms of boundary time for two different final temperatures in Fig. 5 for fixed values of anisotropy parameter, i.e. \(\nu =4\), and distance l. The temperature in the left graph is less than the right one. Interestingly, the frequency is the same for both cases. It means that the oscillation frequency is independent of final temperature. However, this figure indicates that the amplitude of the oscillation depends on the final temperature and they increase together. The same behavior is also observed in case with \(k=3\).
Finally in the Fig. 6, we have plotted the time evolution of the expectation value of the Wilson loop in terms of boundary time for two different values of distance l. At larger distances, the amplitude of the oscillation increases while the oscillation frequency decreases. Therefore, the oscillation characteristics depend on the distance l, too.
To summarize, we find that the oscillation frequency is independent of time and final temperature, i.e. \(f(l,k,\nu )\). However, the amplitude of oscillation depends on the all parameters in the theory, that is \(A(l,T_f,k,\nu )\). Notice that neither the frequency nor the amplitude of the oscillation does not change with the time since the bound state is living in the plasma without dissipation. From our results, one can conclude that the amplitude of the oscillation increases when each parameter of the problem at hand raises. It is then interesting to compare our results with harmonic oscillator model. If we consider M(V), corresponding to the timedependent temperature in the gauge theory, as an external force, the (average) energy of the bound state increases due to energy injection. This enhancement is more substantial in the presence of the anisotropy as well as at higher final temperatures.
Footnotes
 1.
Note that this background can be considered as the IR limit of a 10dimensional solution of IIb supergravity similar to the case suggested in [18].
 2.
In [17] it states that \(\nu =4.45\) is better fitted with experimental data. For this value of anisotropy parameter we have \(\frac{f_{\nu =4.45}}{f_{\nu =1}}\simeq 1.7\) and \(\frac{A_{\nu =4.45}}{A_{\nu =1}}\simeq 9.5\).
 3.
Note that since there is a U(1) symmetry in \((x_2,x_3)\) plane, we expect similar results for the cases in which classical string is located in \(x_3\), \(x_2\) or an arbitrary orientation in this plane. Moreover, the longitudinal case is considered in the Appendix A.
Notes
Acknowledgements
M. A. would like to thank School of Physics of Institute for research in fundamental sciences (IPM) for the research facilities and environment.
Supplementary material
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