Meson excitation at finite chemical potential
Abstract
We consider a probe stable meson in the holographic quark–gluon plasma at zero temperature and chemical potential. Due to the energy injection into the plasma, the temperature and chemical potential are increased to arbitrary finite values in such a way that the plasma experiences an outofequilibrium process. We then observe that the meson is excited, i.e. the expectation value of Wilson loop oscillates around its static value with a specific angular frequency. By defining excitation time \(t_{ex}\) as a time at which the meson falls into the final excited state, we study the effect of various parameters of theory on the excitation time and observe that for larger values of final temperature and chemical potential the excitation time increases. Furthermore, our outcomes show that the more stable mesons, meaning that the meson with lower static potential, are excited sooner.
1 Introduction
Standard methods applied to calculations in quantum chromodynamics (QCD) are often based on a perturbative expansion. This makes them of limited use if applied to questions about the nonperturbative nature of the theory at low energies and where the coupling constant is large. Strong coupling techniques are in particular needed for understanding the physics of the strongly coupled quark–gluon plasma (QGP) produced at RHIC and at the LHC [1]. One such technique is lattice gauge theory that successfully describes the low energy properties of QCD especially at \(\mu =0\) [2]. Lattice gauge theory, however, is of limited use in describing timedependent quantities and nonequilibrium evolution. Another technique is gaugegravity duality that we apply in this paper to explain the behavior of meson in nonequilibrium plasma in the presence of nonzero chemical potential.
The anti deSitter/conformal field theory (AdS/CFT) correspondence or more generally gaugegravity duality [1, 3, 4, 5] is a conjectured relation between two physical theories. One of them is a strongly coupled gauge theory in d dimensional spacetime and the other one is a classical gravity theory living in an extra dimension of spacetime. In fact, parameters, fields and different processes in the gauge theory are translated into appropriate equivalent on the gravity side. For instance, the (thermal) vacuum state on the gauge theory side corresponds to the (black holeAdS) pure AdS in the gravity theory. Thermalization process, which generally means evolution of a state from zero temperature to a thermal state, is dual to black hole formation in the gravity theory [6]. Moreover, as another example, the meson, quarkantiquark bound state, living in the QGP can be identified with a classical string in the gravity and by using the expectation value of the Wilson loop the static potential between a quark and antiquark has been firstly found in [7]. For more information, the interested reader is referred to [1] and references therein.
In this paper, we consider a probe stable meson in the QGP at zero temperature and chemical potential. Then the temperature and the chemical potential are simultaneously raised from zero to finite values \(T_f\) and \(\mu _f\), respectively. Now the questions we would like to answer are how the stable meson reacts to the energy injection into the system and what the characteristics of the new meson state are? Furthermore, it is instructive to know how much time is needed for the meson to fall into the final excited state, i.e. excitation time, and what the effect of the final values of the temperature and the chemical potential is on the excitation time? By excited state we mean a meson which has a higher energy than the ground state, i.e. the meson in the initial static gauge theory. The holographic dual of the above system is described by the dynamics of the classical string, with appropriate initial and boundary conditions, in the Reissner–Nordström–AdS Vaidya (RN–AdS–Vaidya) background, as we will review in the next section.
2 Review on the backgrounds
The gaugegravity duality proposes a promising approach to investigate different properties of the strongly coupled field theory. Since we want to study the QGP as a strongly coupled system in the presence of nonzero chemical potential, we firstly review its corresponding holographic dual, i.e. RN–AdS background. We then extend our problem to the timedependent case, i.e. RN–AdS–Vaidya, which is dual to the thermalization process in the strongly coupled field theory when the temperature and the chemical potential simultaneously increase.
2.1 RN–AdS black hole background
2.2 RN–AdS–Vaidya background
3 Expectation value of Wilson loop
4 Numerical results

\(\frac{\mu _f}{T_f}>1\) and fixed
In Fig. 1, we show how the expectation value of Wilson loop evolves with time for \(lT_f=0.1012\) (left panel) and \(lT_f=0.1246\) (right panel) for fixed \(\mu _f/T_f=3.15\). We observe that by raising the final value of the temperature of the QGP while the distance between the quark and antiquark is kept fixed, the amplitude of the oscillation increases, also (the energy of the excited pair increases based on the relation between amplitude and energy in simple harmonic oscillator). However, the value of the frequency is independent of the final temperature in agreement with the numerical results of [22]. In Fig. 2, we have \(lT_f=0.0714\) (left panel) and \(lT_f=0.1428\) (right panel) for fixed \(\mu _f/T_f=3.15\). In this figure with increasing the distance between the quark and antiquark while the temperature is kept fixed, the amplitude (frequency) of oscillation increases (decreases).

\(\frac{\mu _f}{T_f}<1\) and fixed
All results for this case are similar to the previous case. In other words, it seems that in the case at hand the value of \(\mu _f/T_f\) does not change the behavior of the expectation value of the Wilson loop, qualitatively.
In Fig. 6, the distance l is increased while the other parameters are kept fixed. In contrast to the chemical potential and the temperature, the distance is an intrinsic characteristic of the meson in the plasma. This figure shows that for larger value of distance l, meaning that the meson is less stable, the excitation time increases. Here, by stability we mean that the value of the static potential is lower at zero temperature and chemical potential, or equivalently before the energy injection, for smaller values of distance l. Although for both quenches the rescaled excitation time behaves similarly, the value of \(k^{1}t_{ex}\) is larger for the fast quench. In other words, this figure indicates that the meson with smaller l, i.e. the more stable quarkantiquark bound states are excited sooner. It may be related to the screening of the force between color charges of the quark and antiquark due to the presence of the medium since they, quark and antiquark, can not communicate, easily. Notice that the distance between quark and antiquark can not be too large because the meson will then dissociate in the plasma.
5 Conclusions

We observe that the expectation value of the timedependent Wilson loop oscillates around the static potential. It may be interpreted as the string connecting quark and antiquark (or flux tube resulted from the gluon fields between the pair) in field theory with Dirichlet boundary conditions on the endpoints. After the energy injection, the string is oscillating in one of the its normal modes or equivalently the quarkantiquark bound state has been excited. As a matter of fact, we investigate the effect of the chemical potential and the temperature on this normal mode.

Making larger each parameter under study in this paper, i.e. final chemical potential, final temperature and the quarkantiquark distance, the energy of the bound state, based on classical harmonic oscillator model, increases since the amplitude of the oscillation becomes larger.

The oscillation frequency is independent of the temperature and approximately of the chemical potential. It is intuitively comprehensible. Similar to the classical harmonic oscillator, the oscillation frequency is an intrinsic characteristic of the meson and is independent of the environmental changes. However, the larger distance between quark and antiquark, the smaller oscillation frequency.

Consider a meson in the plasma with nonzero temperature and chemical potential. For larger values of the chemical potential and higher temperatures, the excitation time of the meson increases. In other words, when the plasma is hotter or denser the meson falls into the final excited state more slowly.

Consider a plasma where its temperature and chemical potential are kept fixed. Then the more stable meson at \(T=\mu =0\), corresponding to the smaller distance between quark and antiquark, falls into the final excited state sooner.

All of the above results are confirmed for the slow and the fast quench. By slow (fast) quench we mean the energy injection into the system is done slowly (rapidly).
In addition, studying nonlocal observables [8, 23] such as the twopoint correlation function and the expectation value of Wilson loop as probes of thermalization process, show that the larger \(\mu /T\) the larger thermalization time. However, by studying the twopoint correlation function, the expectation value of Wilson loop and the entanglement entropy [24], the results indicate that for a fixed small value of lT (\(lT \ll 1\)) and small values of \(\mu /T\) the thermalization time decreases by increasing \(\mu /T\), thus plasma thermalizes faster. But, for larger values of \(\mu /T\) the thermalization time increases with increasing \(\mu /T\). Also, by increasing the value of lT this nonmonotonic behavior becomes less pronounced and finally disappears, so that a monotonic behavior has been observed for \(lT\gg 1\).
More recently, equilibration of a dynamical scalar operator is considered in the charged QGP during its equilibration [25]. The numerical outcomes show that the equilibration time can be a decreasing or increasing function of \(\mu /T\) and depends on the energy injection, so that one cannot report the general behavior. Furthermore, in [26] studying nonhydrodynamic quasinormal modes of a scalar field show that the equilibration time decreases with increasing the chemical potential far from the critical point, while close to the critical point it would increase. On the other hand, the thermalization time always increases with increasing the chemical potential. In summary, although the excitation time increases for larger values of the chemical potential in our case, it should be instructive to check our outcomes in other gauge theories with holographic dual to find a general behavior.
Footnotes
 1.
Holographic models of strongly coupled matter at finite density in equilibrium can display instabilities when T becomes sufficiently small, for instance, see the paper on the RN–AdS black holes in five dimensions where an instability is triggered by a tachyonic field coupled to the sysytem in \({{\mathcal {N}}}\) = 8 gauged supergravity [9, 10].
 2.
One can choose the other relevant numbers and see the final results does not change qualitatively.
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. We also would like to thank the referee for his/her comments which help us to improve presentation of the manuscript.
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