Quasi normal modes for Heavy Vector Mesons in a Finite Density Plasma

Plasma Nelson R. F. Braga∗ and Rodrigo da Mata† Instituto de F́ısica, Universidade Federal do Rio de Janeiro, Caixa Postal 68528, RJ 21941-972 – Brazil Abstract There are strong indications that ultra-relativistic heavy ion collisions, produced in accelarators, lead to the formation of a new state of matter: the quark gluon plasma (QGP). This deconfined QCD matter is expected to exist just for very short times after the collision. All the information one can get about the plasma is obtained from the particles that reach the detectors. Among them, heavy vector mesons are particularly important. The abundance of cc̄ and bb̄ states produced in a heavy ion collision is a source of information about the plasma. In contrast to the light mesons, that completely dissociate when the plasma is formed, heavy mesons presumably undergo a partial thermal dissociation. The dissociation degree depends on the temperature and also on the presence of magnetic fields and on the density (chemical potential). So, in order to get information about the plasma out of the quarkonium abundance data, one needs to resort to models that provide the dependence of the dissociation degree on these factors. Holographic phenomenological models provide a nice description for charmonium and bottomonium quasi-states in a plasma. In particular, quasi-normal modes associated with quarkonia states have been studied recently for a plasma with magnetic fields. Here we extend this analysis of quasinormal models to the case when charmonium and bottomonium are inside a plasma with finite density. The complex frequencies obtained are then compared with a Breit Wigner approximation for the peaks of the corresponding thermal spectral functions, in order to investigate the quantitative agreement of the different descriptions of quarkonium quasi-states.

I.

INTRODUCTION
One of the most fascinating challenges currently faced by physicists is to build a detailed picture of the quark gluon plasma (QGP) from the particles that reach the detectors in heavy ion collisions. Hadronic matter, at the very high energy densities produced in these processes, apparently produce the QGP, a state of matter where color is not confined inside bound states. It is not possible to observe the plasma directly. All the available information comes from particles created after this very short-lived state of matter hadronizes and disappears.
An important source of indirect information about the QGP is the abundance of heavy vector mesons. In contrast to light mesons, that dissociate once the plasma is formed, mesons made of cc or bb quarks may survive in the thermal medium. They undergo a partial dissociation process that depends on the flavor (charm or bottom), on the temperature and density of the plasma and on the presence of magnetic fields produced by the motion of charged particles. It is important to understand the thermal behavior of heavy vector mesons in a plasma in order to relate their relative abundance with the properties of the medium.
In recent years, it has been shown that holographic phenomenological models are a fruitful framework for describing heavy vector mesons inside a thermal medium. In ref. [4] a holographic AdS/QCD model was proposed in order to represent the spectra of masses and decay constants of charmonium and bottomonium states in the vacuum. This model has two energy parameters, fixed by the criteria of the best fit to the experimental data.
The extension to finite temperature and density appeared in refs. [5,6] where the spectral functions representing the quasi-states of heavy vector mesons were calculated.
Then, it was shown that a better fit for the spectra is obtained from an improved model involving three energy parameters [7,8]. These parameters represent: the quark mass, the string tension and an ultraviolet (UV) energy scale. This UV energy parameter is related to the large mass change that happens in non-hadronic decays and is necessary in order to fit the decay constant spectra. A very nice picture of the dissociation process of charmonium and bottomnium states was obtained in these references by calculating the thermal spectral functions.
A complementary tool for studying the dissociation process using holographic models is the determination of the quasinormal modes. The spectral function is obtained from real frequency solutions of the equations of motion of gravity fields dual to the hadrons.
In contrast, quasinormal modes are solutions with complex frequency and more restrictive boundary conditions. The real part of the frequency is related to the thermal mass and the imaginary part to the thermal width of the quasi-states. In ref. [9], quasi-normal modes for heavy vector mesons were studied using the holographic model of [7,8]. This analysis was then extended, for the case when a magnetic field is present, in ref. [10]. The dissociation process corresponds to an increase in the imaginary part of the frequency of the quasinormal modes.
Here we present an extension of the analysis of heavy vector meson quasi-normal modes to the case when the medium has a finite density. We investigate the dependence on the density of the complex frequencies for bottomonium and charmonium quasi-states at different temperatures. Then we compare the results obtained for the quasinormal frequencies with the results form the spectral function approach. The comparison is performed by considering a Breit Wigner approximation for the spectral function peaks. Such an approximation provides estimates for the real and imaginary parts of the quasinormal frequencies that are in a very nice agreement with the results obtained directly. This shows that the two approaches are consistent not only qualitatively but also from a quantitative point of view.
The article is organized as follows: in section II we review the holographic model for heavy vector mesons. In section III we present the calculation of quasinormal modes at finite temperature and density. Then in section IV we compare the results obtained with a Breit Wigner approximation of the first spectral funcion peaks. Section V is devoted to conclusions and final comments.

II. HOLOGRAPHIC MODEL FOR HEAVY VECTOR MESONS
In the phenomenological model developed in Refs. [7,8], vector mesons are described by a vector field V m = (V µ , V z )(µ = 0, 1, 2, 3), representing the gauge theory current J µ =ψγ µ ψ in the gravity side of the duality. The fields live in a curved five dimensional space, that is anti-de Sitter space for the case when the mesons are in the vacuum. Additionally, there is a scalar background. The action for the vector field reads where F mn = ∂ m V n −∂ n V m . The model contains three energy parameters that are introduced through the background scalar field φ(z): The parameters represent effectively: the string tension, the mass of the heavy quarks and a large mass scale associated with the decay of the heavy meson to a non hadronic-state.
The values that provide the best fit to charmonium and bottomonium spectra of masses and decay constants at zero temperature are respectively where all quantities are expressed in GeV. The geometry that corresponds to a thermal medium with a finite density µ was studied in refs. [11][12][13]. It is a 5-d anti-de Sitter charged black hole space-time with the metric where and z h is the horizon position where f (z h ) = 0. The relation between z h and the temperature T of the black hole, assumed to be the same as the gauge theory temperature, comes from the condition of absence of conical singularity at the horizon: The parameter q is proportional to the black hole charge. In the dual gauge theory, q is related to the density of the medium, or quark chemical potential, µ. In the gauge theory Lagrangean, µ would appear multiplying the quark densityψγ 0 ψ . So it would work as the source of correlators for this operator. In the dual supergravity description the time component V 0 of the vector field plays this role. So, one considers a particular solution for the vector field V m with only one non-vanishing component: Assuming that the relation between q and µ is the same as in the case of no background, that means φ(z) = 0, the solution for the time component of the vector field is: where c is a a constant. Imposing A 0 (0) = µ and A 0 (z h ) = 0 one finds: By eqs. (6) and (7) it becomes clear that specifying both z h and q, the values of the temperature and the chemical potential are fixed and contained into the metric (4).

A. Physical interpretation
In the context of gauge/gravity duality, quasinormal modes are normalizable gravity solutions representing the quasi-particle states in a thermal medium, with complex frequencies ω. The real part, Re(ω), is related to the thermal mass and the imaginary part, Im(ω), is related to the thermal width. In the zero temperature limit one recovers the vacuum hadronic states, corresponding to solutions that are called normal modes. The normalizability condition requires that, either in the zero temperature case or in the finite temperature one, the fields must vanish at the boundary z=0. The main difference in that at finite T there is an event horizon where one additionally have to impose infalling boundary conditions.
These two types of boundary conditions are simultaneously satisfied in general for complex frequencies.

B. Equations of motion and boundary conditions
In the radial gauge V z = 0, and considering solutions of the type that corresponds to plane waves propagating in the x 3 direction with wave vector p µ = (−ω, 0, 0, k). In terms of the electric field components E 1 = ωV 1 , E 2 = ωV 2 and E 3 = ωV 3 + kV t the equations of motion coming from action (1) with the metric(4) take the form: where (') represents derivatives with respect to the radial z coordinate.
One has to impose the normalizability condition at z = 0 and the infalling condition at z = z h . The idea, in order to impose the boundary conditions at the horizon, is to re-write the field equations in such a way that they separate into a combination of infalling and outgoing waves. It is convenient to introduce the Regge-Wheeler tortoise coordinate r * .
This coordinate is defined by the relation ∂ r * = −f (z)∂ z with z in the interval 0 ≤ z ≤ zh and can be expressed explicitly by integrating the former relation and imposing r * (0) = 0.
In order to have a Schrödinger like equation, one can perform a Bogoliubov transformation on the electric fields, introducing (ψ j = e − B j (z) 2 E j ). Then, Eqs. (8) and (9) reduce to the form: with For j = (1, 2, 3) these potentials diverge at z = 0 so we must have ψ j (z = 0) = 0. At the horizon we have for both potentials that U j (z = z h ) = 0 in the limit z → z h one expects to find infalling ψ j = e −iωr * and outgoing ψ j = e +iωr * wave solutions for equation (10).
Only the first kind of solutions are physically allowed. The Schödinger like equation can be expanded near the horizon leading to the following expansion the field solution: One can solve recursively for a (n) j . The first coefficients are: Eqs. (8) and (9) are solved numerically for complex frequencies using a method that consists in imposing infalling boundary conditions for the electric field at the horizon. These conditions are obtained by expressing the expansion (12) in terms of the field E near the horizon: This expansion leads to the following boundary conditions for the field and it's derivative at the horizon: Then one searches for the complex frequencies that provide solutions vanishing on the boundary: E j (z = 0) = 0. These are the quasinormal frequencies and the correspondingnormalizable -solutions are the quasinormal modes, that represent the heavy meson quasistates in the thermal medium.

C. Results
Let us start with heavy mesons at rest in the medium. Figures 1 and 2  dissociation degree increases with the momentum k while for longitudinal motion it decreases with the momentum. This effect is more noticeable for higher chemical potentials, when the dissociation caused by the density is partially compensated by longitudinal motion.

IV. COMPARISON WITH SPECTRAL FUNCTION RESULTS
Quasinormal modes (QNM) and spectral functions are complementary approaches for studying the thermal behavior of heavy vector mesons in a plasma. It is interesting to check if the results obtained here using QNM are consistent with the spectral function description, using the same holographic approach, presented in ref. [7]. It is known that an increase in the imaginary part of the quasinormal frequencies corresponds to a broadening of the corresponding peak of the thermal spectral function. Such a qualitative analysis is trivial and indicates consistency between the two approaches. Much more interestingly, it is possible to make a quantitative test of consistency. In the vicinity of the peaks, the spectral function calculated from the imaginary part of the current-current retarded propagator has the approximate Breit Wigner (BW) form: where the height of the peak a, the thermal mass M and the width Γ can be determined by the numerical adjust of the spectral function ρ(ω) (see for example [14] for a discussion). details on how to determine these spectral functions, see ref. [7].
It is clear from these plots that the heigth of the peaks decrease with the chemical potential. In order to illustrate how the BW approximation works, we show in figure 6 the BW fit and the actual peak for charmonium at T = 125 MeV and µ = 150 MeV One notices that near the peak the BW approximation works very well.
Performing a numerical fit of spectral functions, near the first peak, to the Breit Wigner Tables I and II summarize the results for QNM frequencies calculated using the method presented at section III and the results for the corresponding quantities obtained from a Breit-Wigner fit of the spectral functions peaks. One notices that the relative discrepancies are very small and then concludes that the results of quasinormal modes and spectral functions are consistent. They represent complementary approaches to study the thermal behavior of quarkonium inside a plasma.

V. CONCLUSIONS
In this article the behavior of heavy vector mesons inside a plasma with finite chemical potential was studied through the determination of the quasinormal modes. A holographic model was used, in order to describe the quasi-states of cc and bb in terms of normalizable field solutions in a five dimensional dual space. This dual geometry contains a black hole with a Hawking temperature assumed to be the same as the gauge theory temperature. The chemical potential, or density, of the medium is represented in the holographic description, by the charge of the black hole.
The results obtained show how the density affects the partial thermal dissociation and the thermal mass of the quarkonia in the medium. In particular, the higher the density, the higher the dissociation degree. This fact holds for heavy mesons at rest or in motion relative to the medium. However, there is a non trivial aspect. For motion in the direction perpendicular to the polarization (transverse motion) the dissociation degree increases with momentum. In contrast, for motion in the direction of polarization (longitudinal motion), the dissociation degree decreases with momentum. So, for this longitudinal case, motion and density have opposite effects.
Regarding the thermal mass, there is also a non trivial behavior. For low temperatures (T > ∼ 200M eV for J/ψ and T > ∼ 400M eV for Υ) the thermal mass decreases with the chemical potential. For higher temperatures, the mass increases slightly with the chemical potential. It is important to note that the plasma at T = 125 MeV considered here would be a super-cooled plasma phase if the critical deconfinement temperature is above this value.