Thermal Width of Quarkonium from Holography

From the AdS/CFT correspondence, the effects of charge and finite 't Hooft coupling correction on the thermal width of a heavy quarkonium are investigated. To study the charge effect, we consider Maxwell charge which is interpreted as quark medium. In the case of finite 't Hooft coupling corrections, $\mathcal{R}^4$ terms and Gauss-Bonnet gravity have been considered, respectively. It is shown that these corrections affect the thermal width. It is also argued that by decreasing the 't Hooft coupling, the thermal width becomes effectively smaller. Interestingly, this is similar to analogous calculations in weakly coupled plasma.


Introduction
The experiments of Relativistic Heavy Ion Collisions (RHIC) have produced a stronglycoupled quark−gluon plasma (QGP)(see review [1]). At a qualitative level, the data indicate that the QGP produced at the LHC is comparably strongly-coupled and it is expected to be better approximated as conformal than is the case at RHIC [2]. There are no known quantitative methods to study strong coupling phenomena in QCD which are not visible in perturbation theory (except by lattice simulations). A new method for studying different aspects of QGP is the AdS/CF T correspondence [1,3,4,5,6]. This method has yielded many important insights into the dynamics of strongly-coupled gauge theories. It has been used to study hydrodynamical transport quantities at equilibrium and real time process in the non-equilibrium [7]. Methods based on AdS/CF T relate gravity in AdS 5 space to the conformal field theory on the four-dimensional boundary [5]. It was shown that an AdS space with a black brane is dual to a conformal field theory at finite temperature [6].
In heavy ion collisions at the LHC, heavy quark related observables are becoming increasingly important [8]. In these collisions, one of the main experimental signatures of QGP formation is melting of quarkonium systems, like J/ψ and excited states, in the medium [9]. They are also useful probes for QCD phenomenology [10]. Thermal width of these systems is an important subject in QGP [11]. This quantity emerge from imaginary part of the heavy quark potential which is related to quarkonium decay processes in the QGP [12]. In the effective field theory framework, thermal decay widths have been studied in [13,14]. It was shown that at leading order, two different mechanisms contribute to the decay width as Landau damping and singlet-to-octet thermal breakup. As long as the Debye screening mass is larger than the binding energy, the former mechanism dominates over the latter. Also from the elementary process, the Landau-damping mechanism corresponds to dissociation by inelastic parton scattering and the singlet-to-octet thermal breakup to gluon-dissociation [15]. Beyond leading order, two mechanisms would be the same.
The analytic estimate of the imaginary part of the binding energy and the resultant decay width was studied in [16]. The peak position and its width in the spectral function of heavy quarkonium can be translated into the real and imaginary part of the potential [17]. Using the soft-wall AdS/QCD model the finite-temperature effects on the spectral function in the vector channel has been studied and the similar behavior with lattice QCD results was found for the in-medium mass shift and the width broadening of the vector meson [18].
The effect of the imaginary part of the potential on the thermal widths of the states in both isotropic and anisotropic plasmas has been studied in [19,20]. This study has been done by considering the modifications to the Coulombic wave function of the imaginary part of the potential for J/ψ, Υ and χ b for both an isotropic and anisotropic quark-gluon plasma. The expectation value of the imaginary part of the quarkonium potential gives the thermal width which was obtained analytically in the case of an isotropic plasma in [16]. For the case of Υ it is on the order of 20 MeV-100 MeV which can be compared to the decay width of Υ → e + e − .
The thermal width of heavy quarkonium in a hot strongly coupled isotropic plasma, from holographic point of view, firstly studied in [21]. In this approach, the thermal width of heavy quarkonium states originates from the effect of thermal fluctuations due to the interactions between the heavy quarks and the strongly coupled medium. This is described holographically by integrating out thermal long wavelength fluctuations in the path integral of the Nambu-Goto action in the curved background spacetime. This study was extended to the case of anisotropic plasma in [22,23] and imaginary potential formula in a general curved background was obtained. It was also shown that the thermal width is decreased in presence of anisotropy and bigger decrease happens along the transverse plane. This method was revisited in [24] and general conditions for the existence of an imaginary part for the heavy quark potential were obtained. Also the connection between the imaginary part of the potential and the area law displayed by the Wilson loop in the vacuum of confining gauge theories was studied in detail.
In this paper, we study different effects on the thermal width by considering effects of charge and finite 't Hooft coupling correction on the hot plasma. To study charge effect, we consider Maxwell charge which can be interpreted as quark medium. Melting of a heavy meson like J/ψ exited states like χ c and ψ ′ in the quark medium has been investigated in [47] and it was shown that the excited states melt at higher temperatures. The finite 't Hooft coupling corrections correspond to R 4 corrections and Gauss-Bonnet terms, respectively. Heavy quarks in the presence of higher derivative corrections has been studied in [31,32]. Now we continue with considering these effects on thermal width of quarkoniums.
The thermal width should be calculated using the imaginary part of potential. Because the imaginary part of the potential implies that thermal effects generate a finite width for the quarkonium peak in the dilepton production rate [11]. The static quark-antiquark pairs in a thermal bath in real-time formalism was studdied in [14] and real and imaginary part of the potential were discussed in detail. As it was pointed out, the imaginary potential for a generic background has been found in [22,23,24]. At first, we will give the brief review of main results where one can refer to [23,24] for more details of the calculations.
The article is organized as follows. In the next section, we will present an example for connection between the imaginary potential and confinement. We give the general expressions to study thermal width in section 3. Also in this section, we use the general formulas and investigate thermal width behavior at finite coupling and in the presence of dense medium . In the last section we summarize our results.

Imaginary potential and confinement
As it was argued in [24], the presence of a black brane is necessary in order to have a nonzero imaginary potential. In the other words, in the absence of a black brane it vanishes. In this section, we examine this idea in a theory which exhibits a confinement-deconfinement transition at some finite temperature T c . We are not going to carry out the details of the calculations which prove that the imaginary potential is zero in the confined phase.
One may consider the confining SU(N) gauge theory based on N D4 branes on a circle. In particular, two fundamental parameters are an energy scale in addition to temperature. The vacuum of the theory at zero temperature confines the color charge. One expects the imaginary potential in the low T phase should be vanished.
The model that we consider is the model of Witten [6] which is based on N D4 branes wrapped on a circle. In this theory the field theory is a non-supersymmetric SU(N) Yang-Mills theory that confines at low temperatures [6]. Although the theory is different than pure Yang-Mills theory, it exhibits linear confinement of quarks in the vacuum. The gravitational background dual to the vacuum is known analytically as A typical length scale associated to the D4 brane geometry is l where all dimensionful quantities measure in units of it. The volume of the unit S 4 and the associated volume form are V 4 and ǫ 4 , respectively. Also The second relation follows from demanding absence of a conical singularity at the tip of the cigar u k that is spanned by x 4 and u. At high temperature phase, the black hole geometry is given by Here the blackness function F (u) and the temperature are There is a critical temperature T c = 1 2πR where the theory is confined for T < T c and the geometry is described by (2.1), whereas, for T > T c the theory is deconfined and is given by (2.3). The ratio of u h u k is given by (2.5) Based on the results of [24], If u k > u h then the U-shaped string cannot go past u k and one cannot consider fluctuations beyond u k . From AdS/CFT, one may interpret this condition from (2.5) in terms of the temperature. Then more explicitly, for T < T c the imaginary potential vanishes while for T > T c it is not zero. As a result, in this geometry which shows the confinement-deconfinement transition at T c , the imaginary potential is zero in the confinement phase.

Thermal width from holography
In this section we follow the method proposed in [21] to study the thermal width of heavy quarkonia at strong coupling. In this approach the worldsheet fluctuations of the Nambu-Goto action associated with the heavy quark pair in the gauge/gravity duality are studied. As a result an analytic formula for the imaginary part of the potential can be found in this approach. In [22,23,24], the imaginary potential formula in a general curved background was obtained. To follow the method, we give a brief review of calculation of thermal width of heavy quarkonium from holography. One can refer to [21,23,24] for more details of the calculations.
The general gravity background can be considered as follows: here the metric elements are functions of the radial distance u and x i = x, y, z are the boundary coordinates. In these coordinates, the boundary is located at infinity. We study a static quark antiquark system at the boundary as an open string in the bulk space from the gauge string duality point of view [25]. Using the the usual orthogonal Wilson loop which corresponds to the heavy QQ pair, and assuming to system aligned in the x direction one finds the following generic formulas for the heavy meson. In these formulas u * ≡ u(x = 0) is the deepest point of the U-shaped string. The metric functions appear as • The distance between quark antiquark, L is given by • The real part of heavy quark potential, ReV QQ is as follows: • The imaginary part of the potential is negative and given by The derivatives are respect to u. This formula reduces to the case of isotropic plasma which was studied in [21] when W (U) = 1 but in the case of anisortopic plasma in [22,23] W (U) = 1.
These generic formulas give the related information of the heavy quarkonium in terms of the metric elements of a background (3.1).
In this paper we are mostly interested in the calculation of the imaginary potential in (3.6). It would be noticed that this quantity originates from the fluctuations at the deepest point of the string in the bulk. In the context of AdS/CFT, there are other approaches which can lead to a complex static potential [26,27]. In [26] the extended range of the radial distance u was studied in such a way that the string world-sheet solutions of the Wilson loop corresponding to the static potential, become complex and therefore the corresponding static potential develops an imaginary part. The method in [27] is based on the spectral decomposition of the Euclidean Wilson loop and its analytic continuation to the real-time. The imaginary potential in this method grows linearly with temperature which is qualitatively consistent with that obtained in [21,26].
To find the ImV , one should express it in terms of the length L of the Wilson loop instead of u * using the equation (3.3). There is an important point for long U-shaped strings where there is a possibility of additional configurations contributing to our result according to [28], but here we are interested mostly in distances LT < 1.
We use a first-order non-relativistic expansion as to calculate the thermal width of a heavy quarkonium like Υ meson. The imaginary potential is given by (3.11), also |ψ > is the Coulombic wavefunction of the Coulomb potential of heavy quarkonium. From the holographic point of view, potential between infinitely massive quarks in the pure AdS background is a Coulomb-like potential The original calculation of this result comes from considering the rectangular Wilson loop in the vacuum of strongly-coupled N = 4 SYM theory [29]. The generalization to finite temperature has been done in [30]. The effect of higher derivation corrections on the real part of the potential within the gauge/gravity duality has been done in [31,32] and it was found that the dissociation length becomes shorter with the increase of coupling parameters of higher curvature terms. The rotating heavy quarkonium also studied in [33,34,35,47] Regarding the Coulombic potential in (3.8), applying a potential model may therefore provide qualitatively useful insight. We consider the ground-state wave function of |ψ > in a Coulomb-like potential V = −A/L where A in the case of N=4 SYM reads from (3.8).
In the ground state of energy levels of a bound state of heavy quarks with mass of m Q , the Bohr radius is defined as a 0 = 2/m Q A. The wave function will be (3.9) By considering different effects on the plasma, the real part of the potential is not given by just the Coulombic term. But this term provides the leading contribution for the potential of heavy quarkonia then it justifies the use of Coulomb-like wave function |ψ > in (3.9) to determine the thermal width. Considering finite temperature plasma will not change the coefficient A in the Coulombic potential. However, by studying the charge and finite coupling corrections, we show that the potential must be modified and A changes. As a result the Bohr radius shifted. We will not find the potential analytically and the numerical fitting will be used in this case. The most important effect of changing A is significant changing in the behavior of the thermal width. More details will be found in the next sections where we show behavior of Γ in terms of parameters of the plasma. As a result, by considering different effects on the plasma, we will also find A by fitting methods. This approach was followed in [23] carefully.
To warm up, we calculate Γ in the N = 4 SYM. The metric functions are where horizon is located at u h and the temperature of hot plasma is given by u h = πR 2 T . In this case we do not consider any correction and call imaginary potential as ImV The imaginary potential must be negative which implies that there is a lower bound for deepest point of the U-shaped string, ξ min = 0.76. This minimum value can be found by solving ImV (0) QQ = 0. The maximum value ξ max happens when the distance L gives the maximum value. In the left plot of Fig. 1 we showed explicitly these values. The imaginary potential for this values of ξ is plotted in the right plot of Fig.1. It is clearly seen that the imaginary potential starts generated at L min or ξ min and increased in absolute value with L until a value L max or ξ max . It would be noticed that for ξ very close to the horizon one should consider higher order corrections [28].
By finding the imaginary potential, the thermal width also finds from (3.7) as As it was noted in [23], there is a limitation to calculate the thermal width from holography. As we can see in Fig. 1 the imaginary potential we found in this approach is well-defined for a separation in (L min , L max ). On the other hand from physical point of view we expect that the imaginary potential should exist also for larger separation. However we can fix the limitation by assuming that our calculation in the valid lengths has the same solution for larger lengths, and can be found by extrapolating the curve there. This is the reason that we fitted the straight-line in the right plot of Fig. 1 which covers larger distances to infinity. Briefly, we integrate in (3.7) from (L min , L max ) but because of the physical point of view, the integration is done from (L min , ∞) by using a reasonable extrapolation for imaginary potential. For L < L min the imaginary potential is considered to be zero. Using these methods, one finds that for Υ with parameters as m Q = 4.7GeV, λ = 9, R = 1 and at T = 0.3GeV the thermal width is Γ (0) = .487MeV orΓ = 2148MeV . WhereΓ means that the integration has been done using the extrapolation method 3 . In the next sections we will normalize the thermal widths in the presence of corrections to Γ (0) value, except in the case of λ corrections where we assumed λ = 100 to calculate Γ (0) .

Thermal width at finite coupling
In this section we consider finite coupling corrections on the thermal width. An understanding of how this quantity changes by these corrections may be essential for theoretical predictions in perturbative QCD [36,15].
From the AdS/CFT, the coupling which is denoted as 't Hoof coupling λ is related to the curvature radius of the AdS 5 and S 5 (R), and the string tension A general result of the AdS/CFT correspondence states that the effects of finite but large λ coupling in the boundary field theory are captured by adding higher-derivative interactions in the corresponding gravitational action. In out study, the corrections that will be considered to the AdS-Schwarzschild black brane are R 4 and R 2 corrections. The thermal width in these cases will be called Γ (λ) and Γ (λ GB ) , respectively.
• R 4 corrections: Since AdS/CF T correspondence refers to complete string theory, one should consider the string corrections to the 10D supergravity action. The first correction occurs at order (α ′ ) 3 [37]. In the extremal AdS 5 × S 5 it is clear that the metric does not change [38], conversely this is no longer true in the non-extremal case. Corrections in inverse 't Hooft coupling 1/λ which correspond to α ′ corrections on the string theory side were found in [37]. Functions of the α ′ -corrected metric are given by [39]  where and w = u u h . As before, there is an event horizon at u = u h and the geometry is asymptotically AdS at large u with a radius of curvature R = 1. The expansion parameter k can be expressed in terms of the inverse 't Hooft coupling as The temperature is given by The imaginary potential finds from (3.6). However, in this case it is a lengthy equation, to have some terms we expand the result in terms of expansion parameter k as follows: The first term is the imaginary potential in (3.11). We will not use this expansion and consider the exact expression in the following calculations for finding the thermal width Γ (λ) .
Firstly, we study behavior of the quark-antiquark distance in terms of ξ for different values of λ. It is clearly seen in the left plot of Fig. 2 that by increasing the coupling constant the maximum value of L increases. In the right plot of this figure we show effect of coupling on the imaginary potential. One finds that by increasing the coupling L min also increases. This means that the imaginary part of the potential in the presence of coupling corrections starts generated for larger distances than without such corrections: where λ 1 < λ 2 < ∞. We find that in absolute value the imaginary potential is increased due to finite coupling corrections. It would be important to notice that using extrapolation method is problematic in this case. Because the sign of the imaginary potential for ξ > ξ max is not always negative and changes. Then we can not use straight line approximation to consider the larger values of L like what we did in the right plot of Fig. 1. Using this point we integrate in the thermal width Γ (λ) from L min to L max . We show Γ (λ) relative to the Γ (0) in this interval in Fig.3. As it is clear when λ goes to infinity Γ (λ) Γ (0) goes to 1. It is interesting result that there is a maximum value for this ratio in the presence of finite coupling corrections which happens at λ c .
An estimate of how the thermal width of heavy quarkonium changes with the shear viscosity to entropy density ratio, η/s, at strong coupling was studied in [24]. It was found that in the presence of curvature-squared corrections like Gauss-Bonnet terms, the thermal width decreases as a function of η/s. Note that in the case R 4 corrections, η/s increases when λ is finite [40]. One finds from Fig.3, by decreasing λ from λ c , which means increasing η/s, the thermal width becomes effectively smaller. Interestingly this is similar to analogous calculations in weakly coupled plasma [11]. It was argued that at strong coupling for a quarkonium with a very heavy constituent mass, the thermal width can be ignored [11].
• R 2 corrections: Next, we study R 2 corrections to the thermal width which was called Γ (λ GB ) .
In five dimensions, we consider the theory of gravity with quadratic powers of curvature as Gauss-Bonnet(GB) theory. The exact solutions and thermodynamic properties of the black brane in GB gravity were discussed in [41,42,43]. The metric functions are given by  where In (3.19), N = 1 2 1 + √ 1 − 4λ GB which is an arbitrary constant which specifies the speed of light of the boundary gauge theory and we choose it to be unity. Beyond λ GB < 1/4 there is no vacuum AdS solution and one cannot have a conformal field theory at the boundary. Casuality leads to new bound λ GB < 9/100 [44]. The temperature also is given by Also the 't Hooft coupling of the dual strongly-coupled CFT is λ = N 2 R 4 α ′2 . With using the calculation of imaginary potential in this background, an estimate of how the thermal width of heavy quarkonia changes with the shear viscosity to entropy density ratio, has been done in [24]. Here, we present further details and express the final results in terms of the Gauss-Bonnet coupling constant. As it was pointed out, our extrapolation method is different from [24].
The behavior of L in terms of ξ for different values of λ GB shown in Fig. 4. It is clearly seen in the left plot of Fig. 4 that by increasing the coupling constant the maximum value of L decreases. On the contrary, in the case of R 4 corrections in Fig.2

it increases.
The imaginary potential in the Gauss-Bonnet gravity finds from (3.6). The result can be expressed as follows: , where h = 1 + 4 (−1 + ξ 4 ) λ GB . L min decreases. This means that the imaginary potential in the Gauss-Bonnet gravity starts generated for smaller distances.
whereλ GB >λ GB . We find that in absolute value the imaginary potential is increased due to finite coupling corrections.
In this case also we should do the integration in (3.12) in (L min , L max ) and one can not consider larger lengths by extrapolating the curves in Fig.4. We would like to emphasize that using this method leads to the Fig.5. 4 Also here there is maximum value for the thermal width which happens for λ GB = 0. For coupling constants more and less than this value, thermal width decreases.

Medium effect on the thermal width
In this section, we consider a heavy quark-antiquark (qq) pair in the medium composed of light quarks and gluons as a heavy meson [45]. It was shown that at the high temperature, the gravity dual to the quark-gluon plasma is the Reissner-Nordstrom AdS (RNAdS) black hole and at the low temperature, the dual geometry corresponding to the hadronic phase is the thermal charged AdS (tcAdS) space. The confinement/deconfinement phase transition in the quark medium was discussed in [45] and an influence of matters on the deconfinement temperature, T c was investigated. Using a different normalization for the bulk gauge field, it was shown that the critical baryonic chemical potential becomes 1100MeV which is comparable to the QCD result [46]. Melting of a heavy meson is investigated in [47] and it was found that the melting mechanism in the quark-gluon plasma and in the hadronic phase are the same i.e. the interaction between heavy quarks is screened by the light quarks. The drag force on a moving heavy quark and the jet quenching parameter in the background of RNAdS black hole was studied in [48].
Before calculating the thermal width, we will give a brief review of the background where the density in the dual field theory can be mapped to a bulk gauge field [45]. The Euclidean action describing the five-dimensional asymptotic AdS space with the gauge field is given by Here κ 2 is proportional to the five-dimensional Newton constant and g 2 is a five-dimensional gauge coupling constant. The cosmological constant is given by Λ = −6 R 2 , where R is the radius of the AdS space.
As it was pointed out the quark-gluon plasma and hadronic phase could be described by the tcAdS and the RNAdS black hole, respectively [46]. It was argued in section 2 that the imaginary potential in the confinement phase would be zero, then we focus on the QGP phase. This solution can be considered as follows: where the blackness function f (u) is given by In these coordinates, u denotes the radial coordinate of the black hole geometry and t, x label the directions along the boundary at the spatial infinity. The boundary is located at infinity and the geometry is asymptotically AdS with radius R. The event horizon is located at f (u h ) = 0 where u h is the largest root and it can be found by solving this equation. Then parameters m and q which are the black hole mass and charge are given by m = u 4 h + q 2 u −2 h . The temperature also is (3.28) The time-component of the bulk gauge field is A t (u) = i(2π 2 µ − Q u −2 ) where µ and Q are related to the chemical potential and quark number density in the dual gauge theory. Regarding the Drichlet boundary condition at the horizon, A t (u h ) = 0, one finds Q = 2π 2 µ u h . The black hole charge q and the quark number density Q also are related to each other by this equation where n −1 = g 2 R 2 κ 2 is the color number. Now using the general results in (3.4) and (3.5), we study the behavior of L in terms of ξ for different values of µ. The result is shown in Fig. 6. It is clearly seen in the left plot of Fig. 6 that by increasing the chemical potential the maximum value of L decreases. On the contrary, considering R 4 corrections increases the maximum value of L which can be seen in Fig. 2.
In the right plot of Fig. 6, we show effect of chemical potential on the imaginary potential. Comparing with R 4 corrections, one finds different behavior, i.e. by increasing the chemical potential L min decreases: where µ 1 > µ 2 > µ 3 . As a result, one concludes that the imaginary potential in the presence of light quarks starts generated for smaller distances while at finite λ coupling it starts at larger distance.
To calculate the thermal width, one should do the integration in (3.12) in (L min , L max ) which leads to the Fig.7. In this figure, two different temperatures from top to down are T = .122GeV and T = .3GeV , respectively. It is found that by increasing the temperature, Γ (µ) also increases. At fixed temperature also there is a maximum value for Γ (µ) .
We introduce N f = nN c where N f and N c are the number of flavors and color fields, respectively. Now one can find the behavior of thermal width when flavor number of quarks increases. In the right plot of Fig.7, the effect of increasing flavor color number on the ratio of Γ (µ) Γ (0) has been shown at T = .3GeV . It is seen that the maximum value of thermal width does not change but the rate of it versus the chemical potential goes to zero monotonically.

Conclusion
In this paper, we studied different effects on the thermal width by considering effects of charge and finite 't Hooft coupling correction on the hot plasma. To study charge effect and finite 't Hooft coupling correction, we considered Maxwell charge and GaussBonnet terms, respectively. The effects of finite but large couplings are considered by adding higherderivative corrections in the gravity background. Especially, R 4 terms and Gauss-Bonnet gravity have been studied.
As it was found in [23] the minimum distance of quark anti-quark, L min where the imaginary potential starts depends on the corrections. Our findings in this case can be summarized as follows: • In the presence of R 4 corrections which correspond to finite 't Hooft coupling corrections in the hot plasma, by increasing the coupling L min also increases.
• By considering Gauss-Bonnet corrections, increasing Gauss-Bonnet coupling leads to decreasing of L min .
• In the medium, increasing µ also decreases L min .
The thermal width of a quarkonium also studied in this paper. The result was normalized to Γ (0) which is N = 4 SYM result. It was found that • By turning on the 't Hooft coupling correction in the hot plasma, Γ (λ) Γ (0) increases up to a maximum value and by increasing the coupling it decreases monotonically.
• In the presence of medium also Γ (µ) Γ (0) takes a maximum value which depends on the temperature. The effect of flavor number was shown in the right plot of Fig.7.
In the case of R 4 corrections, relation between the thermal width of heavy quarkonium and the shear viscosity to entropy density ratio, η/s, was discussed. It was found that in the presence of these corrections, by decreasing λ, which means increasing η/s, the thermal width becomes effectively smaller. This is an interesting result which is consistent with the intuition one would get from [11].
It will be very interesting to investigate the thermal width of a quarkonia in more realistic holographic backgrounds, such as [49] and [50]. Finally, comparing holographic results with weakly coupled calculations would be desirable. We would like to report on this study elsewhere.