Quarkonia Formation in a Holographic Gravity–Dilaton Background Describing QCD Thermodynamics

Abstract

A holographic model of probe quarkonia is presented, where the dynamical gravity–dilaton background was adjusted to the thermodynamics of 2 + 1 flavor QCD with physical quark masses. The quarkonia action was modified to account for the systematic study of the heavy-quark mass dependence. We focused on the $J/\psi$ and ${\rm Y}$ spectral functions and related our model to heavy quarkonia formation as a special aspect of hadron phenomenology in heavy-ion collisions at LHC.

1. Introduction

Heavy-quark flavor degrees of freedom are currently receiving strong interest as valuable probes of hot and dense strong-interaction matter produced by the heavy-ion collisions at LHC energies. The information encoded, e.g., in quarkonia ($c\overline{c}$, $b\overline{b}$) observables, supplements penetrating electromagnetic probes and hard (jet) probes and the rich flow observables, thus complementing each other in characterizing the dynamics of quarks and gluons until the final hadronic states (cf., contributions in [1,2] for the state of the art). Since heavy quarks essentially emerge in early, hard processes, they witness the course of heavy-ion collision—either as individual entities or subjects of dissociating and regenerating bound states [3,4,5]. Accordingly, the heavy-quark physics addresses such issues as charm (c, $\overline{c}$) and bottom (b, $\overline{b}$) dynamics related to transport coefficients [6,7,8,9,10] in the rapidly evolving and highly anisotropic ambient quark–gluon medium [11,12] as well as $c\overline{c}$ and $b\overline{b}$ states as open quantum systems [13,14,15,16]. The rich body of experimental data from LHC, and also from RHIC, enabled a tremendous refinement of our understanding of heavy-quark dynamics. For a recent survey on quarkonium physics, we refer the interested reader to [17].

The yields of various hadron species, light nuclei and anti-nuclei—even such ones which are only very loosely bound [18]—emerging from heavy-ion collisions at LHC energies are described by the thermo-statistical hadronization model [19] with high accuracy. These yields span an interval of nine orders of magnitude. The final hadrons and nuclear clusters are described by two parameters: the freeze-out temperature $T_{fo}=155$ MeV and a freeze-out volume depending on the system size or centrality of the collision. Due to the near-perfect matter–antimatter symmetry at top LHC energies, the baryo-chemical potential ${\mu }_B$ is exceedingly small, ${\mu }_B/T_{fo}\ll 1$. It is argued in [19] that the freeze-out of color-neutral objects only happens in the demarcation region of hadron matter to quark–gluon plasma, i.e., of confined vs. deconfined strong-interaction matter. In fact, the lattice QCD results report a pseudo-critical temperature of $T_c=(156\pm 1.5)$ MeV [20]—a value agreeing with the disappearance of the chiral condensates and the maximum of some susceptibilities. The key is the adjustment of physical quark masses and the use of 2 + 1 flavors [21,22], in short, QCD_{2 + 1}(phys). Details of the (maybe accidental) coincidence of the deconfinement and chiral symmetry restoration are currently matters of debate [23], as is the formation of color-neutral objects out of the cooling quark–gluon plasma at $T_c$. For instance, [24] advocates flavor-dependent freeze-out temperatures. Note that at $T_c$ no phase transition happens, rather the thermodynamics is characterized by a cross-over accompanied by a pronounced dip in the sound velocity.

Among the tools for describing hadrons as composite strong-interaction systems is holography. Anchored in the famous AdS/CFT correspondence, holographic bottom-up approaches have facilitated a successful description of mass spectra, coupling strengths/decay constants, etc. of various hadron species. While the direct link to QCD by holographic QCD-dual or rigorous top–down formulations are still missing, one has to restrict the accessible observables to explore certain frameworks and scenarios. We consider here a minimal framework within the Einstein–dilaton model class, which merges (i) QCD_{2 + 1}(phys) thermodynamics, described by a dynamical holographic gravity–dilaton background, with (ii) holographic probe quarkonia. We envisaged a scenario which embodies the QCD thermodynamics of QCD_{2 + 1}(phys) and the emergence of hadron states at $T_c$ at the same time. One motivation of our work was the exploration of a holographic model, which is in agreement with the above hadron phenomenology in heavy-ion collisions at LHC energies. Early holographic attempts [25,26,27] of hadrons at non-zero temperatures faced the problem of meson melting at temperatures significantly below the deconfinement temperature $T_c$. Several proposals have been made [28,29,30,31] to determine rescue avenues which accommodate hadrons at and below $T_c$. Otherwise, a series of holographic models of hadron melting without reference to QCD thermodynamics, e.g., [31,32,33,34,35,36,37,38,39,40], finds quarkonia states well above, at and below $T_c$ in agreement with lattice QCD results [41,42,43,44]. It is therefore tempting to account for the proper QCD-related background.

We followed [45,46,47,48] and modelled the holographic background by a gravity–dilaton set-up, i.e., without adding further fundamental degrees of freedom (as done, e.g., in [49,50] and Refs. therein (The influence of the tachyon dual to the quark condensate can be neglected for heavy mesons.)) to the dilaton, which was originally solely related to the gluon degrees of freedom [51]. That is, the dilaton potential was adjusted to QCD_{2 + 1}(phys) lattice data. In the temperature region $T\approx \mathcal{O}\left(T_c\right)$, the impact of the charm and bottom degrees of freedom on the quark–gluon–hadron thermodynamics is minor [52]. Thus, we considered quarkonia as test particles. If the gravity–dilaton background is considered to be unique, it is known that there would be no confinement at zero temperature [48]. As we show, this problem can be efficiently solved by a less intricate generalization of the standard action of the soft-wall model [53] inspired by [54] which evince a principle to bring together QCD thermodynamics and the confinement of test particles at zero temperature. Our emphasis is on the formation of quarkonia in a cooling strong-interaction environment mimicking conditions in heavy-ion collisions at LHC. Thereby, the quarkonia properties are described by spectral functions.

Our paper is organized as follows. In Section 2, the dynamics of the probe quarkonia is formulated and the coupling to the thermodynamics-related background is explained (the recollection of the gravity–dilaton dynamics and the consideration of special features are relegated to Appendix A). Numerical solutions in the charm ($J/\psi$) and bottom (${\rm Y}$) sectors with respect to quarkonium’s spectral functions and systematic formation are dealt with in Section 3. The tested two-parameter Schrödinger potential facilitates bottomonium formation through the rapid squeezing of the spectral function towards a narrow quasi-particle state in a small temperature interval around $T_c$. An analogous behavior is accomplished for charmonium by a three-parameter potential considered in Section 4. The squeezing of the charmonium spectral function extends over a somewhat longer temperature interval and requires a particular parameter setting. We summarize in Section 5.

2. Quarkonia as Probe Vector Mesons

The action of quarkonia as probe vector mesons in string frame is:

$$S_m^V=\frac{1}{k_V}\int {\mathrm{d}}^{4}x\mathrm{d}z\sqrt{g_5}{e}^{-\varphi }{G}_m\left(\varphi \right)F^2,$$

(1)

where the function $G_m\left(\varphi \right)$ carries the flavor (or heavy-quark mass, labeled by m) dependence and $F^2$ is the field strength tensor squared of a $U\left(1\right)$ gauge field $\mathcal{A}$ in 5D asymptotic anti-de Sitter (AdS) space time, with or without a black hole (BH), with the bulk coordinate z and metric fundamental determinant $g_5$; $\varphi$ is the scalar dilatonic field with a zero mass dimension. The gauge field $\mathcal{A}$ in the bulk was sourced by a current operator of the structure $\overline{Q}{\gamma }_{\mu }Q$ at the boundary, where Q stands for the heavy quark field operator. The structure of (1) is that of a field-dependent gauge kinetic term, familiar, e.g., from realizations of a localization mechanism in brane world scenarios [55,56,57].

The action (1) with $G_m=1$, originally put forward in the soft-wall (SW) model for light-quark mesons [53], is also used for describing heavy-quark vector mesons [32,33,34], e.g., charmonium [35,36] or bottomonium [58]. As emphasized, e.g., in [35], the holographic background encoded in $g_5$ and $\varphi$ must be chosen differently to imprint the different mass scales, since (1) with $G_m=1$ as such would be flavor blind. Clearly, the combination $exp\{-\varphi \}{G}_m\left(\varphi \right)$ in (1) with flavor-dependent function $G_m\left(\varphi \right)$ is nothing but effectively introducing a flavor-dependent dilaton profile ${\varphi }_m=\varphi -log{G}_m$, while keeping the thermodynamics-steered hadron-universal dilaton $\varphi$. In fact, many authors use the form $S_m^V=\frac{1}{k_V}\int {\mathrm{d}}^{4}x\mathrm{d}z\sqrt{g_5}{e}^{-{\varphi }_m}{F}^2$ to study the vector meson melting by employing different parameterizations of ${\varphi }_m$ to account for different flavor sectors. Here, we emphasize the use of a unique gravity–dilaton background for all flavors and include the quark mass (or flavor) dependence solely in $G_m$.

Our procedure to determine $G_m$ was based on the import of information from the hadron sector at $T=0$. The action (1) leads via the gauges ${\mathcal{A}}_z=0$ and ${\partial }^{\mu }{\mathcal{A}}_{\mu }=0$ and the ansatz ${\mathcal{A}}_{\mu }={ϵ}_{\mu }\phi \left(z\right)exp\left\{i{p}_{\nu }{x}^{\nu }\right\}$ with $\mu ,\nu =0,\cdots ,3$, which uniformly separates the z dependence of the gauge field by the bulk-to-boundary propagator $\phi$ for all components of $\mathcal{A}$, and the constant polarization vector ${ϵ}_{\mu }$ to the equation of motion:

$${\phi }^{″}+\left[\frac{1}{2}{A}^{\prime }+({\partial }_{\varphi }log{G}_m-1){\varphi }^{\prime }+{(logf)}^{\prime }\right]{\phi }^{\prime }+\frac{p^{\mu }{p}_{\mu }}{f^2}\phi =0,$$

(2)

where $A(z,z_H)$ is the warp factor and $f(z,z_H)$ denotes the blackening function in the AdS + BH metric with a horizon at $z_H$:

$$\mathrm{d}{s}^2=exp\left\{A(z,z_H)\right\}\left[f(z,z_H)\mathrm{d}{t}^2-\mathrm{d}{\overrightarrow{x}}^2-\frac{\mathrm{d}{z}^2}{f(z,z_H)}\right],$$

(3)

and a prime denotes the derivative with respect to the bulk coordinate z. Both $A(z,z_H)$ and $f(z,z_H)$ are solutions of Einstein’s equation with a dilatonic potential adjusted to QCD thermodynamics with physical quark masses in the temperature range 100 MeV $<T<$ 400 MeV (cf., Appendix A in [59] and Appendix A for more details); $\varphi (z,z_H)$ is also dynamically determined and is consistent with the metric coefficients via field equations.

By the transformation $\psi \left(\xi \right)=\phi \left(z\left(\xi \right)\right)exp\left\{\frac{1}{2}{\int }_0^{\xi }\mathrm{d}z{\mathcal{S}}_T\left(\xi \right)\right\}$ one gets the form of a one-dimensional Schrödinger equation with the tortoise coordinate $\xi$:

$$\left[{\partial }_{\xi }^2-(U_T\left(z\left(\xi \right)\right)-m_n^2)\right]{\psi }_n\left(\xi \right)=0,n=0,1,2,\cdots ,$$

(4)

where one has to employ $z\left(\xi \right)$ from solving ${\partial }_{\xi }=(1/f){\partial }_z$. The Schrödinger equivalent potential is:

$$U_T\equiv \left(\frac{1}{2}{\mathcal{S}}_T^{\prime }+\frac{1}{4}{\mathcal{S}}_T^2\right)f^2+\frac{1}{2}{\mathcal{S}}_{T}f{f}^{\prime }$$

(5)

as a function of $\xi \left(z\right)$ with:

$${\mathcal{S}}_T\equiv \frac{1}{2}{A}^{\prime }-{\varphi }^{\prime }+{\partial }_{z}log{G}_m\left(\varphi \left(z\right)\right).$$

(6)

At $T=0$ (label “0”), $f=1$ and $\xi \to z$ and $U_T\to U_0$ with:

$$\begin{array}{ccc}\hfill U_0\left(z\right)& \equiv & \frac{1}{2}{\mathcal{S}}_0^{\prime }+\frac{1}{4}{\mathcal{S}}_0^2,\hfill \end{array}$$

(7)

$$\begin{array}{ccc}\hfill {\mathcal{S}}_0& \equiv & \frac{1}{2}{A}_0^{\prime }\left(z\right)-{\varphi }_0^{\prime }\left(z\right)+{\partial }_{z}log{G}_m\left({\varphi }_0\left(z\right)\right),\hfill \end{array}$$

(8)

and (4) becomes:

$$\left[{\partial }_z^2+(U_0\left(z\right)-m_n^2)\right]{\psi }_n=0$$

(9)

with normalizable solutions ${\psi }_n$ and discrete states with masses squared $m_n^2=p^{\mu }{p}_{\mu }$, $n=0,1,2,\cdots$ for quarkonia at rest. That is, at $T=0$, one has to deal with a suitable Schrödinger equivalent potential $U_0\left(z\right)$ to generate the desired spectrum $m_n$. In such a way, the needed hadron physics information at $T=0$ is imported by parameterizing $U_0$ in a suitable manner (see Section 3 and Section 4).

The next step is solving (7) to obtain ${\mathcal{S}}_0\left(z\right)$ and with (8), then $G_m\left(\varphi \right)$ with $G_m\left(0\right)=1$. This procedure requires the knowledge of the warp factor $A_0\left(z\right)$ and the dilaton profile ${\varphi }_0\left(z\right)$, which are determined by Equations (A2)–(A4) in Appendix A. The governing dilaton potential $V\left(\varphi \right)$ in turn is found by an adjustment to lattice QCD data. On the one hand, at $T<100$ MeV, there are nearly no lattice QCD data of the needed thermodynamic quantities, such as sound velocity and entropy etc., at our disposal. On the other hand, at $T>100$ MeV the rich body of lattice QCD data has, despite the great progress made, still non-negligible error bars. We therefore must assume that the ansatz (A5) delivers an adequate representation of the QCD input (see Figure 5-left in [59] for the sound velocity squared). Leaving a dedicated investigation of the potential impact of variations of $V\left(\varphi \right)$ under constraints of lattice QCD thermodynamics data for follow-up work, we then used the so determined $G_m\left(\varphi \right)=G_m\left(z\left({\varphi }_0\right)\right)$ as universal (i.e., temperature-independent) function.

The equation of motion (2) of $\phi$ can also be employed to compute quarkonia spectral functions, cf., [25,33,34,35,60]. For ${\omega }^2=p^{\mu }{p}_{\mu }>0$ fixed, the asymptotic boundary behavior facilitates two linearly independent solutions by considering the leading order terms on both sides of the interval $[0,z_H]$. (i) For $z\to 0$, one has, due to the AdS asymptotics at the boundary, the general solution is:

$$\phi (z\to 0)\to A\left(\omega \right){\phi }_1+B\left(\omega \right){\phi }_2$$

(10)

with two $\omega$-dependent complex constants A and B, and ${\phi }_1(z\to 0)\to 1$ and ${\phi }_2(z\to 0)\to {(z/z_H)}^2$; (ii) near the horizon, $z\to z_H$, the asymptotic behavior of solutions of (2) is steered by the poles of $1/f$ and $1/f^2$. The two linearly independent solutions are ${\phi }_{\pm }(z\to z_H)\to {(1-\frac{z}{z_H})}^{\pm i\omega /|f^{\prime }\left(z_H\right)|}$, where ${\phi }_{\pm }$ represent out-going and in-falling solutions, respectively. The obvious and commonly used side conditions for the bulk-to-boundary propagator are $\phi \left(0\right)=1$, which means $A\left(\omega \right)=1$, and $\phi (z\to z_H)={\phi }_{-}(z\to z_H)$ (purely in-falling solution at the black hole horizon).

Then, the corresponding retarded Green function ${\mathcal{G}}^{\mathrm{R}}$ of the dual current operator $\overline{Q}{\gamma }_{\mu }Q$, defined within the framework of the holographic dictionary via a generating functional by ${\mathcal{G}}^{\mathrm{R}}=\frac{{\delta }^2}{\delta {\mathcal{A}}^{0\mu }(-\omega )\delta {\mathcal{A}}_{\mu }^0\left(\omega \right)}\langle exp\{i\int {\mathrm{d}}^{4}x{\mathcal{A}}_{\nu }^0\overline{Q}{\gamma }^{\nu }Q\}\rangle$, is given by (cf., [61])

$${\mathcal{G}}^{\mathrm{R}}\left(\omega \right)=\frac{{\delta }^2{S}_m^{V,\mathrm{on}-\mathrm{shell}}}{\delta {\mathcal{A}}^{0\mu }(-\omega )\delta {\mathcal{A}}_{\mu }^0\left(\omega \right)}=k\underset{z\to 0}{lim}\frac{1}{z}{\phi }^{*}\left(z\right){\phi }^{\prime }\left(z\right)=\frac{2k}{z_H^2}B\left(\omega \right)$$

(11)

with $k=\frac{N_c}{24{\pi }^2}$ and ${\mathcal{A}}_{\mu }^0\equiv {ϵ}_{\mu }exp\left\{i{p}_{\nu }{x}^{\nu }\right\}$ for $\mu \in \{1,2,3\}$. The quantity $S_m^{V,\mathrm{on}-\mathrm{shell}}$ denotes here the action (1) with the solution $\phi$ from (2). Finally, the spectral function $\rho$ follows from $\rho \left(\omega \right)=\mathrm{Im}{\mathcal{G}}^{\mathrm{R}}\left(\omega \right)=\frac{2k}{z_H^2}\mathrm{Im}B\left(\omega \right)$.

3. Two-Parameter Potential—Bottomonium Formation

Our setting does not explicitly refer to a certain quark mass m. Instead, an ansatz $U_0(z;\overrightarrow{p})$ with a parameter n-tuple $\left\{\overrightarrow{p}\right\}$ was used such to catch a certain quarkonium mass spectrum. Therefore, m is to be considered as a cumulative label highlighting the dependence of $G_m$ on a parameter set $\left\{\overrightarrow{p}\right\}$ which originally enters $U_0$ and which is to be adjusted to the charmonium and bottomonium masses observed in nature.

As a transparent model, we select the two-parameter potential [33,35]:

$$U_0\left(z\right)=\frac{3}{4}{\left(\frac{1}{z}\right)}^2+a^2{z}^2/L^4+4b/L^2$$

(12)

which is known to deliver via (9) the normalizable functions ${\psi }_n$ with discrete eigenvalues:

$$L^2{m}_n^2=4(a+b+na),n=0,1,2,\cdots .$$

(13)

The potential (12) is a slight modification of the SW model [53]. We choose these parameters as follows. The mass $m_0$ determines the ground state (g.s.) “trajectory” in the a-b plane, $b=\frac{1}{4}{L}^2{m}_0^2-a$, and $m_1$ determines the first excitation (1st) “trajectory” by $b=\frac{1}{4}{L}^2{m}_1^2-2a$. Using the PDG values of $J/\psi ,{\psi }^{\prime }$ and ${\rm Y}(1S,2S)$ adjusts the “trajectories” as solid and dashed lines in Figure 1, where we employ the scale setting with $L^{-1}=1.99$ GeV, which is related to the QCD thermodynamics sector (see Appendix A in [59]). Allowing for a 10% variation of $m_1$, one arrives at the colored bands in Figure 1. By such a parameter choice, one places emphasis on the quarkonia g.s. masses as representatives of the heavy quark masses and less emphasis on the level spacing of excitations and ignores other possible constraints.

As we shall demonstrate below, the ansatz (12) has several drawbacks, and therefore, is to be considered as an illustrative example. For instance, the sequence of radial ${\rm Y}$ excitations in nature does not form a strictly linear Regge trajectory [62]. This prevents an unambiguous mapping of $m_{0,1}\mapsto (a,b)$. While the radial excitations of $J/\psi$ follow quite accurately a linear Regge trajectory in nature [62], the request for accommodating further $J/\psi$ properties in $U_0$ also calls for modifying (12), cf., [35,60]. Despite the mentioned deficits, the appeal of (12) and (13) is nevertheless the simply invertible relation $m_n^2(a,b)$ yielding $a\left(m_{0,1}\right)$ and $b\left(m_{0,1}\right)$. Since we are going to study the systematic, we retain the primary parameters a and b in what follows. Instead of discussing the results at isolated points in parameter space referring to $J/\psi$ and ${\rm Y}$ ground states $m_0$ and first excited states $m_1$, we consider the systematic over the a-b plane.

The black curves in Figure 1 exhibit the contours $T_{melt}^{\mathrm{g}.\mathrm{s}.}(a,b)=150$, 200, and 250 MeV. The melting temperature $T_{melt}^{\mathrm{g}.\mathrm{s}.}$ is determined by the disappearance of the peak of the g.s. spectral function upon the temperature increase. One observes a strong parameter dependence as well, which determines the spectral functions, as can be seen in Figure 2. Changing the parameters $(a,b)$ deforms the potential (12) in a characteristic manner [59], e.g., going on a g.s. trajectory to the right squeezes the excited states to higher energies, as can be identified in Figure 1, in particular for the ${\rm Y}$. Such changes immediately affect the spectral functions.

In Figure 1, it looks like an accidental coincidence that, at the crossing points of the g.s. and first excitation trajectories of $J/\psi$ and ${\rm Y}$, the melting temperature is 150 MeV. In other words, in a cooling system, the formation of the quarkonium ground state seems to start when passing the temperature of 150 MeV. This is consistent with the claim in [19] which advocates the formation of hadron states at $T\approx 155MeV\approx T_c$. Consistency does not necessarily mean perfect agreement. The criteria for “melting” or “onset of formation” are not very precise. For instance, [60] uses the relative high of the spectral function’s peak over the smooth background as a threshold value to define “melting”. The transition to a quasi-particle with sharp spectral function does not happen instantaneously but within some temperature span, as can be seen in the top panels in Figure 3 and Figure 4. Considering the dynamics of the cooling system as a sequence of equilibrium states, the spectral-function contour-plots in Figure 3 and Figure 4 are suggestive: upon cooling, the strength of a hadron state is consecutively concentrated to a narrow energy range, eventually forming the quasi-particle. Displaying a spectral function at a few selected temperatures, as in Figure 2, and bottom panels of Figure 3 and Figure 4 as well, only insufficiently illustrates such a feature but is useful for a more quantitative account.

The inspection of the top panels of Figure 3 and Figure 4 unravels that the temperature difference from $T_{melt}^{\mathrm{g}.\mathrm{s}.}$ until the formation of a sharp quasi-particle state is quite large. Sharp quasi-particles can be identified by the squeezed contour lines which eventually nearly coincide with the peak position of the spectral functions depicted by the red dashed curves in the top panels of Figure 3 and Figure 4. Keeping the quarkonia ground state masses $m_0$ and artificially allowing for a somewhat larger value of the first excited state $m_1$ moves the quarkonia formation temperatures to larger values, in particular for ${\rm Y}$, as can be seen in the right panels in Figure 3. In such a way, the quasi-particle formation temperature $T_{form}^{{\rm Y}\left(1S\right)}\approx T_c$ copes with the claim in [19] of hadron formation at $T_c$. The $J/\psi$, in contrast, would be formed at $T_{form}^{J/\psi }<T_c$ (see Figure 4) in conflict with the advertisement of [19]. Section 4 provides a potential ansatz $U_0(z;\overrightarrow{p})$ which accomplishes $T_{form}^{J/\psi }\approx T_c$.

To understand why the $J/\psi$ (${\rm Y}$) reacts so sluggishly (violently) on a modification of $m_1$ while keeping $m_0$ fixed, we mention that the parameter a in (12) changes by 33% (92%, i.e., a factor of nearly two) upon a 5% increase in $m_1$, (due to the non-linearity of the $J^{PC}=1^{--}$ bottomonium Regge trajectory, the energy of $m_1+5$% is between the $3^3{S}_1/{\rm Y}\left(3S\right)$ and $4^3{S}_1/{\rm Y}\left(4S\right)$ states. For charmonium, in contrast, $m_1+5$% is well below the $3^3{S}_1/\psi \left(4040\right)$ state, cf., [62].) which is to been seen in connection with the curvature $8a^2$ of $U_0$ at the minimum $z_{min}/L={\left(3/\left(4a^2\right)\right)}^{1/4}$. The more the potential $U_0$ is squeezed by parameter variation, e.g., by larger values of $\widehat{a}$, the less sensitive the temperature of is $U_T$, as can be seen in Figure 2 in [59] and Figure 6 below. At the origin of these differences is the ratio of $m_1^2/m_0^2$ which is 1.42 for $J/\psi$ and 1.12 for ${\rm Y}$, respectively. It enters the scaled potential (12) $U_0(\widehat{z}\equiv m_{0}z,\zeta \equiv m_1^2/m_0^2-1)/m_0^2=\frac{3}{4}{\widehat{z}}^{-2}+\frac{1}{16}{\zeta }^2{\widehat{z}}^2-\zeta +1$ solely as a parameter.

A second issue refers to the formation of excited states. It seems to be a generic feature of the holographic model class considered here that higher excited states would form at lower temperatures than the respective g.s.—in particular $T_{form}^{\mathrm{g}.\mathrm{s}.}>T_{form}^{1\mathrm{st}}>T_{form}^{2\mathrm{nd}}\cdots$, as can be seen in the bottom panels in Figure 3 and Figure 4. The conjecture of [19], in contrast, advocates $T_{form}^{\mathrm{g}.\mathrm{s}.}\approx T_{form}^{1\mathrm{st}}$. This feature is to be seen in relation to the considered ansatz of $U_0(z;a,b)$ with the IR behavior $\propto z^2$: a much steeper increase in $U_0$ at larger values of z would concentrate the melting temperatures in a narrow corridor.

In addition to the ansat, (12) facilitates a sequential quarkonium formation upon decreasing temperature, $T_{form}^{\mathrm{g}.\mathrm{s}.}>T_{form}^{1\mathrm{st}}>T_{form}^{2\mathrm{nd}}$ etc., which potentially allows for some flavor dependence, e.g., $T_{form}^{{\rm Y}\left(1S\right)}>T_{form}^{J/\psi }$. The thermal mass shifts have a non-trivial temperature dependence as evidenced in Figure 5. Such thermal mass shifts are employed in [9] to pin down the heavy-quark (HQ) transport coefficient $\gamma$ which can be considered as the dispersive counterpart of the HQ momentum diffusion coefficient $\kappa =2T^3/\left(DT\right)$, where D stands for the HQ spatial diffusion coefficient. In [17], the author seemingly stresses a tension within previous holographic results [36], where positive mass shifts are reported, in contrast to negative shifts, e.g., in [34]. Our set-up qualitatively resolves this issue since, depending on the considered temperature, the thermal mass shift can be negative or positive, see Figure 5. One should note, however, that our thermal mass shifts of $J/\psi$ and ${\rm Y}$ are larger than the lattice QCD-based values quoted in [9,44].

Finally, let us remind that the two-parameter ansatz (12) is appealing since it allows for analytic solutions with respect to the excitation spectrum and an easy overview of the parameter dependencies. However, the authors of [35,60] already promoted (12) a “shift and dip potential” to catch more properties of the $J/\psi$ states rather than only masses.

4. Three-Parameter Potential with Dip—Charmonium Formation

The two-parameter potential $U_0(z;a,b)$ from (12) with the realistic values of $a\left(m_{0,1}\right)$ and $b\left(m_{0,1}\right)$ facilitate $J/\psi$ formation at too low temperatures. This failure can be repaired by turning to more appropriate parameterizations. For instance, [35,60] proposed a four-parameter “dip and shift potential” which allows for $J/\psi$ melting temperatures significantly above $T_c$, as does the construction in [31,32,33,34,36] deploying three parameters. The essence is a dip in $U_0(z;\overrightarrow{p})$ which holds together the spectral strength despite large temperatures. Here, we consider such an option. The difference to previous work is the use of the dynamical background related to QCD, as described in Appendix A.

The construction of a particular three-parameter potential $U_0(z;M,k,\Gamma )$ is as follows. Use $A_0\left(z\right)=-2logz/L$ and ${\varphi }_0(z;M,k,\Gamma )=k^2{z}^2+Mz+tanhx$ [58] with $x\equiv 1/Mz-k/\sqrt{\Gamma }$. Due to $U_0=\frac{1}{2}{(\frac{1}{2}{A}_0^{\prime }-{\varphi }_0^{\prime })}^{\prime }+\frac{1}{4}{(\frac{1}{2}{A}_0^{\prime }-{\varphi }_0^{\prime })}^2$ (see (7) and (8)) the potential is given by

$$\begin{array}{ccc}{U}_0(z;M,k,\Gamma )\hfill & =\hfill & \frac{3}{2z^2}+\frac{1}{4}{M}^2+k^4{z}^2+\frac{M}{2z}+k^{2}Mz\hfill \\ \hfill & -\hfill & \left(\frac{3}{2M{z}^3}+\frac{1}{2z^2}+\frac{k^2}{Mz}\right)\frac{1}{{cosh}^{2}x}\hfill \\ \hfill & +\hfill & \frac{1}{4M^2{z}^4}\left[4sinhxcoshx+1\right]\frac{1}{{cosh}^{4}x}.\hfill \end{array}$$

(14)

The first three terms in the top line suggest a correspondence $M\widehat{=}4\sqrt{b}/L$ and $k\widehat{=}\sqrt{a}/L$ by a comparison with (12), while the next two terms cause some modification of (12) at intermediate values of z. The second line of (14) is essentially responsible for the dip—somewhat modified by terms in the third line. The dip position is determined to a large extent by the $1/{cosh}^2$ term which peaks at $z=\sqrt{\Gamma }/\left(kM\right)$; the sinh term in the third line shifts the dip tip to smaller values of z. The UV and IR asymptotics are the same as for the potential (12). The dip position and the dip depth are interrelated, in contrast to the construction in [35,60].

The potential (14) might exhibit some non-trivial local structures as a function of z for particular parameters. In [58], the author advocates the optimum parameters $M=2.2$ GeV (representing a mass scale of non-hadronic decays); $k=1.2$ GeV (representing the quark mass); and $\sqrt{\Gamma }=0.55$ GeV (representing the $Q\overline{Q}$ string tension) to yield the $J/\psi$ (${\psi }^{\prime }$) mass of 2.943 (3.959) GeV) and the decay constant of 399 (255) MeV (see Table I in [58], the entries displaying “Holographic Results for Charmonium”). Note the resulting overestimated level spacing quantified by $m_1^2/m_0^2=1.81$, in contrast to the PDG value of 1.42, when deploying these parameters.

Completely analog to the two-parameter potential (12), increasing the parameter k at $M\approx const$, the potential (14) is squeezed and becomes deeper. Analogously, decreasing the parameter $\sqrt{\Gamma }$ at constant values of k and M lets drop the absolute minimum of $U_0$. One may select such parameter pairs of $(k,M)$ at a constant $\sqrt{\Gamma }$ to keep the g.s. mass $m_0$ constant, see the horizontal dashed line the in left panel of Figure 6. Due to the squeezing of the potential, the interior (left) part was less influenced when imposing a horizon at $z_H$, where $U_T(z=z_H)=0$ is facilitated according to (5). As a result, the more the potential is squeezed, the smaller the values of $z_H$ are which are allowed to hold the $J/\psi$ prior to melting. This is the very reason which forces us to enlarge the parameter k (or a in (12)) to achieve quarkonium formation at sufficiently high temperatures in agreement with the perspective put forward in [19]. The dip in the potential (14) is useful in that respect since enlarging the parameter a in the flat potential (12) influenced the quarkonium formation in a less effective manner for $J/\psi$. Let us emphasize that we put more weight on the g.s. mass $m_0$ (see the fat solid curve in the right panel of Figure 6) as the representative of the quark mass, while we relaxed the constraint on the excited state $m_1$ to be in a realistic range (see the dashed curve and colored band in the right panel of Figure 6), thus following the rationale in [35].

Having $U_0$ at our disposal, we proceed as in Section 3. The contour plots of the $J/\psi$ spectral function are exhibited in the top panels of Figure 7. One observes again the tendency of charmonium formation as a narrow corridor of contour lines at too low temperatures for parameters delivering exactly the PDG values of $m_{0,1}$, as can be seen in the left top panel. This is quantified by the spectral functions shown in the left bottom panel of Figure 7 which display only a broad peak at $T=150$ MeV. Modifying the parameters as such to catch $m_0$ and $m_1+5$% DPG values significantly improves the approach to charmonium formation near $T_c$, as can be seen in the right panels of Figure 7, despite the yet imperfect squeezing of the contour lines in the right top panel. Nevertheless, the spectral function becomes well peaked at $T=150$ MeV, as can be seen in the right bottom panel of Figure 7.

Finally, we exhibit in Figure 8 the contour plot of the charmonium spectral function (left panel) and the spectral function at selected temperatures (right panel) for the parameter set $(\sqrt{\Gamma },k,M)=(0.55,1.2,2.2)$ GeV favored in [58]. These parameters, albeit with noticeable deviations to the PDG values of $m_{0,1}$, realize the charmonium formation as a transition from the spectral function to a narrow, quasi-particle state at temperatures slightly below $T_c$. While the squeezing of the contour lines near $T_c$ in the left panel of Figure 8 is apparently not so pronounced as in the case of bottomonium (see right top panel in Figure 3), the spectral function displays a sharp peak at $T_c$, see right panel of Figure 8. Insofar, it is justified to speak on charmonium formation at $T_c$ for the given parameter set. We emphasize the QCD-related background employed here, in contrast to the schematic background in [58].

To complete the systematic related to charmonium, we exhibit in Figure 9 the quantity $-log{G}_m$ as a function of $\varphi$. Note the huge variation of $G_m\left(\varphi \right)$. In general, $G_m\left(\varphi \right)$ sensitively depends on the parameters in $U_0$ and is tightly related to the background.

An analog study of the ${\rm Y}$ formation is hampered by some uncomfortable structures of $U_0(z;M,k,\Gamma )$. References [31,32,36] advocate parameters which avoid such obstacles, however, these result in a value of $m_0^2({\rm Y}\left(1S\right))$ being only one half of the PDG value. We therefore did not perform an analysis of the potential ansatz (14) in the QCD-related background since the two-parameter potential (12) was already shown to successfully accomplish bottomonium formation at $T_c$.

5. Summary

In summary, we introduced a modification of the holographic vector meson action for quarkonia such to join (i) the QCD_{2 + 1}(phys) thermodynamics, dynamically and consistently described by a dilaton and the metric coefficients in AdS + BH, with (ii) realistic quarkonia masses at zero temperature. Both pillars, thermodynamics and quarkonium mass spectra, are anchored in QCD as a common footing. The formal holographic construction was based on an effective dilaton ${\varphi }_m=\varphi -log{G}_m$, where $\varphi$ is solely tight to the light-quark–gluon thermodynamics background, while the flavor-dependent quantity $G_m$ is determined by a combination of $\varphi$ and the adopted Schrödinger equivalent potential $U_0$ at zero temperature. $U_0$ encodes the flavor (or quark mass) dependence and can be chosen with much sophistication to accommodate many quarkonia properties. We explore here the systematic of a two-parameter model to demonstrate the features of our scheme, where the thermodynamic background at $T>0$ and meson spectra at $T=0$ serve as a QCD-based input to analyze the quarkonia formation at $T>0$. We tested a scenario where quarkonium formation was considered as an adiabatic process, i.e., a sequence of equilibrium states, and characterized by the shrinking of the respective spectral functions towards narrow quasi-particle states, in qualitative agreement with lattice QCD studies [44]. Realistic values of ${\rm Y}(1S,2S)$ masses in fact allow the formation temperature $T_{form}$ of ${\rm Y}\left(1S\right)$ nearby $T_c$ in line with the claim of [18,19] that hadrons form themselves at temperatures $T_c\approx T_{fo}\approx 155$ MeV. Insofar, the mystery “why $T_{fo}\approx T_c$?” could be resolved by a dynamical process within such a scenario: hadronization is the transit of a broad-to-narrow spectral functions within a few-MeV temperature intervals at $T_c$.

While quite promising, the proposed scenario was hampered by at least three issues. First, the finding of $T_{form}\approx T_c$ looks somewhat accidental and is not explicitly locked to a certain microscopic process; in addition, there is a slight tension due to the tendency of $T_{form}<T_c$ when deploying the exact PDG value of the ${\rm Y}\left(2S\right)$ mass together with the ${\rm Y}\left(1S\right)$ PDG value. Second, the formation of the ${\rm Y}\left(2S\right)$ quasi-particle occurs at $T_{form}^{{\rm Y}\left(2S\right)}<T_c$ due to the sequential formation, which, however, could be an artifact of the two-parameter model of $U_0$. Third, the envisaged scenario quantitatively fails for $J/\psi$ since $T_{form}^{J/\psi }<T_c$ for the two-parameter model. It happens, however, that an improved three-parameter model $U_0$ overcomes such problems to some extent, i.e., charmonium formation at $T_c$ is accomplished. An ideal choice of $U_0$ should deliver the quarkonia mass spectra (and other properties as well) and quarkonia formation as the rapid shrinking of the spectral functions in a narrow temperature interval at $T_c$, including the excited states.

Formally, the hadronization of heavy-flavor probe quarkonia is determined by the potential $U_0$, which governs the crucial function $G_m$, thus partially decoupling it from the holographic background.

The herein proposed bottom-up scenario of quarkonia formation solely accommodates the properties of vector $c\overline{c}$ and $b\overline{b}$ states in the holographic bulk vector field $\mathcal{A}$. This is in contrast to microscopic studies, e.g., in [3,5,63,64,65], where the heavy-quark interaction with constituents of the ambient medium was dealt with in detail. Primordial contributions, early off-equilibrium yields as well as the corresponding feedings were also not accounted for. An important (yet) missing issue of the proposed scenario is a direct relation to observables in relativistic heavy-ion collisions. All this calls for further investigations.