Quasinormal modes and dispersion relations for quarkonium in a plasma

Recent investigations show that the thermal spectral function of heavy $ {b \bar b } $ and $ {c \bar c} $ vector mesons can be described using holography. These studies consider a bottom up model that captures the heavy flavour spectroscopy of masses and decay constants in the vacuum and is consistently extended to finite temperature. The corresponding spectral functions provide a picture of the dissociation process in terms of the decrease of the quasi-state peaks with temperature. Another related tool that provides important information about the thermal behaviour is the analysis of the quasinormal modes. They are field solutions in a curved background assumed to represent, in gauge/gravity duality, quasi-particle states in a thermal medium. The associated complex frequencies are related to the thermal mass and width. We present here the calculation of quasinormal modes for charmonium and bottomonium using the holographic approach. The temperature dependence of mass and thermal width are investigated. Solutions corresponding to heavy mesons moving into the plasma are also studied. They provide the dependence of the real and imaginary parts of the frequency with the quasi-particle momenta, the so called dispersion relations.

I.

INTRODUCTION
The fraction of heavy vector mesons produced in a heavy ion collision provides important information about the possible formation of a quark gluon plasma [1,2]. This is so because the presence of a thermal medium leads to the partial dissociation of these hadronic states.
That is why it is important to understand the thermal behaviour of heavy mesons. In particular, the dependence of the dissociation degree on the temperature of the medium and on the state of motion of the meson.
The thermal behaviour of heavy vector mesons can be described using holographic bottom up models. A holographic model for bb (bottomonium) and cc (charmonium), involving two dimensionfull parameters, was proposed in Refs. [3][4][5]. An improved model containing three energy parameters appeared then in Refs. [6,7]. These parameters have a simple physical interpretation. They represent: the quark mass and the string tension, that are related to the mass spectra, and an ultraviolet (UV) energy scale, necessary in order to fit the decay constant spectra. This UV energy parameter is related to the large mass change that occurs in a non hadronic decay, when a heavy vector meson transforms into light leptons.
The thermal spectral function for heavy vector mesons was constructed using this holographic approach in references [6,7]. Quasi-Particle states appear as peaks that decrease in height when the temperature increases. Important information about the behaviour of hadrons inside a thermal medium can be obtained also from the so called quasinormal modes. They are gravitational field solutions that play the role of gravity duals to quasi-particle states. The associated frequencies are complex and the imaginary parts are related to the thermal width of the quasi-state. Quasinormal modes are the finite temperature version of the normalized solutions, that describe particle states at zero temperature in holography.
We will develop here the calculation of quasinormal modes using the bottom up holographic model of references [6,7]. The dependence of the real and imaginary part of the frequency on the temperature will be investigated for charmonium and bottomonium states.
Then the dispersion relations will be considered. The dependence of the complex frequencies on the linear momentum, for hadrons moving inside a plasma, will be analysed for different temperatures. This type of analysis provides a detailed picture of the thermal behaviour of the heavy vector mesons in a medium like the quark gluon plasma.
The article is organized in the following way. In section 2 we review the holographic model for charmonium and bottomonium. Then in section 3 we study the spectral functions for charmonium and bottomonium. The quasinormal modes are calculated in section 4 and conclusions and final comments are shown in section 5.

II. HOLOGRAPHIC MODEL FOR HEAVY VECTOR MESONS
The bottom up holographic model for heavy vector mesons of refs. [6,7] is defined, in the zero temperature case, in 5-d anti-de Sitter space-time, with metric The gravity duals of heavy vector mesons are fields V m = (V µ , V z ) (µ = 0, 1, 2, 3), that are assumed to represent the gauge theory currents J µ =ψγ µ ψ . The action integral is: where F mn = ∂ m V n − ∂ n V m . The energy parameters of this phenomenological model are introduced through the background scalar field φ(z), with the form The parameter k is related to the (heavy) quark mass, while parameter Γ, with dimension of energy squared, is related to the string tension of the strong quark anti-quark interaction.
The third parameter M is a large mass scale associated with non hadronic decays. In such processes the heavy meson decays into light leptons, therefore there is a very large mass change.
The simplest way to realise gauge gravity duality is to gauge away the z component: The normalizability requirement for solutions of equation of motion (4) corresponds to the boundary condition V (p, z = 0) = 0. The corresponding solutions V (p, z) = Ψ n (z) show up for a discrete spectrum of p 2 = −m 2 n where m n are interpreted as the masses of the corresponding meson states.
Decay constants are proportional to the transition matrix from the vector meson n excited state to the vacuum: 0| J µ (0) |n = µ f n m n . In the present holographic model they are given by [6] f n = 1 The values of the parameters that describe charmonium and bottomonium are respectively: As shown in [6] this model provides a nice fit of masses and decay constants (see in Table   1 and 2 the results for charmonium and bottomonium, respectively. For comparison, the experimental data from Ref. [8] is shown inside parentheses. The extension to finite temperature is obtained by replacing the AdS space of eq. (1) by an AdS black hole geometry version of the metric. In this imaginary time formulation, the time variable is periodic, with period 0 ≤ t ≤ β = 1/T , where T is the temperature. The regularity of the metric leads to III. THERMAL SPECTRAL FUNCTION

A. Equations of motion
The equations of motion for the holographic model come from action (2) with the metric (8). One chooses again the radial gauge V z = 0 and consider plane wave solutions of the propagating in the x 3 direction with the wave vector p µ = (−ω, 0, 0, q). The equations have the following form V where the prime ( ) denotes the derivative with respect to z. The corresponding equations in terms of the electric field components: E 1 = ωV 1 , E 2 = ωV 2 and E 3 = ωV 3 + qV t , are given by B.

Retarded Green's function
In the four dimensional vector gauge theory we define a retarded Green's functions of the currents J ν as Current conservation implies that p µ G R µν (p) = 0. So, the structure of the Greens function at zero temperature can be written in terms of a projector that makes explicit this property: where At finite temperature, in thermal equilibrium, it is interesting to separate this projector into transverse and longitudinal parts [9] introducing Then the retarded Green's function can be writen in the finite temperature case when there is rotation invariance as where Π T (p 0 , p 2 ) and Π L (p 0 , p 2 ) are independent scalar functions.
Choosing, as in the previous section, the wave vector with the form p µ = (−ω, 0, 0, q), corresponding to propagation in the z direction, the relevant (non-vanishing) components of the Green's function can all be written in terms of the longitudinal and transversal scalar functions Considering now the holographic approach, the gauge theory current correlators are represented in terms of the vector fields living in the five dimensional space and described by the action integral of eq. (2) with metric (8). In momentum space the on shell action takes the form It is convenient to express this action in terms of the electric field components The z dependent part of the field, the so called bulk to boundary propagator, can be separated from the boundary value E 0 j as where the functions E j (z, p) are defined to satisfy E j (0, p) = 1 and the superscript (−) indicates that E (−) j (z, p) satisfies the infalling condition at the horizon, as required by the Lorentzian form of Son-Starinets prescription [10]. Then, substituting (25) in the action (24) one finds In terms of the boundary values of the potential this action reads Finally, using the Son-Starinets [10] prescription for the vector field case one finds

C. Spectral Function
In order to calculate the spectral function it is convenient write the equations of motion for the electric fields (14) and (15) in terms of the bulk to boundary propagator E j (z, p) = Now the spectral function can be extracted using the relation (28) presented in the last section. Particularly, in the case of G R x 3 x 3 and G R αα , the corresponding spectral functions in terms of the function E j are where E is the solution of the eqs. (29) and (30) satisfying the infalling boundary condition and the bulk to boundary condition

D. Numerical Results
We performed the numerical analysis of the spectral functions for the transverse and longitudinal sectors solving the equations (29) and (30), for the charmonium and bottomonium cases, using the boundary conditions (33),(34) for the fields. Then, relations (31) and (32) were used to find the spectral functions. The model parameters are the zero temperature ones, presented in section 2.
Let us start with the spectral functions for the vector mesons at rest. In this case the spectral function is the same for the transverse and longitudinal directions. In figure 1 we show the dissociation process at finite temperature for charmonium and in figure 2 the  bottomonium case. Note that for both flavours the peaks decrease as the temperature increases. In particular, the peak decrease faster for charmonium than for bottomonium.
We also studied the spectral function at non vanishing momentum. In figure 3

IV. QUASINORMAL MODES
The peaks shown in figure 1 and 2 indicate that the corresponding retarded Green's functions present poles. These poles are related to the frequencies of the electromagnetic quasinormal modes of the black brane. The quasinormal modes correspond, in the dual gauge theory, to quasi-particle states of vector mesons. The frequencies of these quasinormal modes present real (ω R ) and imaginary (ω I ) parts. The real part is related to the mass of the vector mesons when q = 0, while the imaginary part to the decay rate of the quasi-particle states formed near the confining/deconfining transition. One observes that when the temperature increases, the widths of the peaks increase and the mean life τ = 2π/ω I decrease. In contrast to the zero temperature case, where there are particle states, represented by normalized solutions, in the finite temperature case we have quasi-particles, described by quasinormal modes. They are field solutions in the curved background, subject infalling condition at the horizon. Previous studies of quasinormal modes in the context of gauge/gravity duality can be found, for example, in Refs. [9,[11][12][13][14][15][16][17][18][19][20].
In this section we are going to obtain the quasinormal modes for electromagnetic perturbations by solving the equations of motion using numerical methods. The shooting method of refs. [21][22][23] is of particular interest for the present work since it is suited to find quasinormal modes of space-times that are only know numerically. The method consist in specify two boundary conditions at the horizon, and then adjust the free parameter given by the frequency. In our case we need solve the equations (29) and (30) using the boundary conditions given by the infalling boundary conditions at the horizon near horizon expansion (37) for higher temperatures in order to find the quasinormal modes.
This issue is discussed in [22].

A. Quasi-particles at rest in the medium
Using the shooting method the quasinormal frequencies were determined as a function of the temperature, for the case of zero momentum q = 0.
We show in figures 5 and 6 the results for the real and imaginary parts of the frequencies for the first three modes n = 0, 1, 2 with q = 0 for charmonium and bottomonium, respectively. From these figures one notes that in the region of high temperatures, the frequencies show a linear behaviour that is in agreement with ref. [9].
On the other hand, one can note that in the zero temperature limit, the real part of the quasinormal frequencies coincide with the corresponding mass spectrum of Table 1 and 2 and the behaviour at low temperature is not linear as in the high temperature. One also notes that the width for the charmonium grow faster than for bottomonium, showing that

Transverse Perturbations
The results of the quasinormal modes for the transverse perturbations are presented in figures 9 and 10 as a function of the temperature. On one hand, the real part of the quasinormal frequencies have a similar behaviour to that of the longitudinal sector. On the other hand, the imaginary part of ω increases with momentum, in contrast to the longitudinal case. These results indicate that dissociation effect increases due to motion relative to the medium in the direction transverse to the polarization of the heavy vector mesons.

V. CONCLUSIONS
Using a holographic bottom up model, we have studied the real and imaginary parts of the quasinormal modes frequencies corresponding to charmonium and bottomonium states.
The dependence with the temperature for quasi-particles at rest, or moving with respect to the medium, was investigated. For comparison, the spectral function for these heavy mesons was also obtained. The outcomes from the quasinormal modes are consistent with the spectral function behaviour, as we now discuss. For mesons at rest, the real part of the frequency shows a tendency to increase with the temperature. In terms of the spectral function this is translated into the increase in the position of the location of the peak. The imaginary part of the frequencies monotonically increase, a behaviour that is translated into the decrease in the height and an increase in the width of the peaks.
For mesons in motion with respect to the medium, the real part of the quasinormal modes frequencies increase with the value of the momentum, for both longitudinal and transverse motion. This is translated in terms of the spectral function into the increase in the value of the frequency where the peaks are located. For the imaginary part of the frequency, the behaviour is different if the motion is in the direction of the polarization (longitudinal case) or transverse to the polarization. In the longitudinal case the imaginary part decreases with the momentum. This behaviour is consistently reproduced in the spectral function as an increase in the height of the peaks. For transverse motion, the behaviour is the opposite.
The imaginary part of the frequency increases with the momentum and, correspondingly, the height of the peaks of the spectral function decrease.