Quarkonia formation in a holographic gravity-dilaton background describing QCD thermodynamics

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


Introduction
Heavy-quark flavor degrees of freedom receive currently some interest as valuable probes of hot and dense strong-interaction matter produced in heavy-ion collisions at LHC energies. The information encoded, e.g. in quarkonia (cc, bb) 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 up to the final hadronic states (cf. contributions in [1] for the state of the art). Since heavy quarks emerge essentially in early, hard processes, they witness the course of a heavy-ion collision -either as individual entities or subjects of dissociating and regenerating bound states [2][3][4]. Accordingly, the heavy-quark physics addresses such issues as charm (c,c) and bottom (b,b) dynamics related to transport coefficients [5][6][7][8][9] in the rapidly evolving and highly anisotropic ambient quark-gluon medium [10,11] as well as cc and bb states as open quantum systems [12][13][14][15]. The rich body of experimental data from LHC, and also from RHIC, enabled a tremendous refinement of our understanding heavy-quark dynamics. For a recent survey on the quarkonium physics we refer the interested reader to [16].
The yields of various hadron species, light nuclei and anti-nuclei -even such ones which are only very loosely bound [17] -emerging from heavy-ion collisions at LHC energies are described by the thermo-statistical hadronization model [18] 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 f o = 155 MeV and a freezeout volume depending on the system size or centrality of the collision. Due to the nearperfect matter-antimatter symmetry at top LHC energies the baryo-chemical potential µ B is exceedingly small, µ B /T f o 1. It is argued in [18] that the freeze-out of color-neutral objects happens just in the demarcation region of hadron matter to quark-gluon plasma, i.e. confined vs. deconfined strong-interaction matter. In fact, lattice QCD results report a pseudo-critical temperature of T c = (156 ± 1.5) MeV [19] -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 [20,21], in short QCD 2+1 (phys). Details of the (may be accidental) coincidence of deconfinement and chiral symmetry restoration are matter of debate [22], as also the formation of color-neutral objects out of the cooling quark-gluon plasma at T c . For instance, Reference [23] 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 a 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 framework which merges (i) QCD 2+1 (phys) thermodynamics described by a dynamical holographic gravity-dilaton background and (ii) holographic probe quarkonia. We envisage a scenario which embodies 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 is the exploration of a holographic model which is in agreement with the above hadron phenomenology in heavyion collisions at LHC energies. Early holographic attempts [24][25][26] to 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 [27][28][29][30] to find 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. [30][31][32][33][34][35][36][37][38][39], finds quarkonia states well above, at and below T c in agreement with lattice QCD results [40][41][42][43]. It is therefore tempting to account for the proper QCD-related background.
In the temperature region T ≈ O(T c ), the impact of charm and bottom degrees of freedom on the quark-gluon-hadron thermodynamics is minor [44]. Thus, we consider quarkonia as test particles. We follow [45][46][47][48] and model 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][51]) to the dilaton, which was originally related solely to gluon degrees of freedom [52]. That is, the dilaton potential is adjusted to QCD 2+1 (phys) lattice data. Our emphasis is on the formation of quarkonia in a cooling strong-interaction environment. Thereby, the quarkonia properties are described by spectral functions. We restrict ourselves to equilibrium and leave non-equilibrium effects, e.g. [53,54], for future work.
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/ψ) and bottom (Υ) sectors w.r.t. quarkonium spectral functions and the quarkonium formation systematic are dealt with in section 3. The tested two-parameter Schrödinger potential facilitates bottomonium formation as 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.

Quarkonia as probe vector mesons
The action of quarkonia as probe vector mesons in string frame is where the function G m (φ) carries the flavor (or heavy-quark mass, labeled by m) dependence and F 2 is the field strength tensor squared of a U (1) gauge field A in 5D asymptotic anti-de Sitter (AdS) space time, with or without black hole (BH), with the bulk coordinate z and metric fundamental determinant g 5 ; φ is the scalar dilatonic field with zero mass dimension. The gauge field A in the bulk is sourced by a current operator of the structurē Qγ µ Q at the boundary, where Q stands for the heavy quark field operator. The structure of (2.1) 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]. In holographic Einstein-Maxwell-dilaton models (cf. [58]), often employed in including a conserved charge density (e.g. [59,60]), such a term refers to the gauge coupling. The action (2.1) with G m = 1, originally put forward in the soft-wall (SW) model for light-quark mesons [61], is also used for describing heavy-quark vector mesons [31][32][33], e.g. charmonium [34,35] or bottomonium [66]. As emphasized, e.g. in [34], the holographic background encoded in g 5 and φ must be chosen differently to imprint the different mass scales, since (2.1) with G m = 1 as such would be flavor blind. Clearly, the combination exp{−φ}G m (φ) in (2.1) with flavor dependent function G m (φ) is nothing but introducing effectively a flavor dependent dilaton profile φ m = φ − log G m , while keeping the thermodynamics-steered hadron-universal dilaton φ. In fact, many authors use the form √ g 5 e −φm F 2 to study the vector meson melting by employing different parameterizations of φ 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 is based on the import of information from the hadron sector at T = 0. The action (2.1) leads via the gauges A z = 0 and ∂ µ A µ = 0 and the ansatz A µ = µ ϕ(z) exp{ip ν x ν } with µ, ν = 0, · · · , 3, which uniformly separates the z dependence of the gauge field by the bulk-to-boundary propagator ϕ for all components of A, and the constant polarization vector µ to the equation of motion where A(z, z H ) is the warp factor and f (z, z H ) denotes the blackening function in the AdS + BH metric with horizon at z H , and a prime denotes the derivative w.r.t. 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 [62] and appendix A for details); also φ(z, z H ) is determined dynamically and is consistent with the metric coefficients via field equations. By the transformation ψ(ξ) = ϕ(z(ξ)) exp{ 1 2 ξ 0 dz S T (ξ)} one gets the form of a onedimensional Schrödinger equation with the tortoise coordinate ξ where one has to employ z(ξ) from solving ∂ ξ = (1/f )∂ z . The Schrödinger equivalent potential is as a function of ξ(z) with At T = 0 (label "0"), f = 1 and ξ → z and U T → U 0 with and (2.4) becomes with normalizable solutions ψ n and discrete states with masses squared m 2 n = p µ p µ , n = 0, 1, 2, · · · for quarkonia at rest. That is, at T = 0 one has to deal with a suitable Schrödinger equivalent potential U 0 (z) 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 suitable manner (see sections 3 and 4). The next step is solving (2.7) to obtain S 0 (z) and, with (2.8), then G m (φ) with G m (0) = 1. This needs A 0 (z) and φ 0 (z), which follow from the thermodynamics sector (see Appendix A) via A 0 = A(z) = lim z H →∞ A(z, z H ) and φ 0 = φ(z) = lim z H →∞ φ(z, z H ). One has to suppose that these limits are meaningful. The limited information from lattice QCD thermodynamics w.r.t. the finite temperature range and the data accuracy may pose here a problem. Ignoring such a potential obstacle we use then G m (φ) = G m (z(φ 0 )) as universal (i.e. temperature independent) function.
The equation of motion (2.2) of ϕ can also be employed to compute quarkonia spectral functions, cf. [24,[32][33][34]64]. For ω 2 = p µ p µ > 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 → 0, one has, due to the AdS asymptotics at the boundary, the general solution with two ω-dependent complex constants A and B, and Near the horizon, z → z H , the asymptotic behavior of solutions of (2.2) is steered by the poles of 1/f and 1/f 2 . The two linearly independent solutions are where ϕ ± represent out-going and in-falling solutions, respectively. Then, the general near-horizon solution is given by again with complex constants C and D which depend on ω. The obvious and commonly used side conditions for the bulk-to-boundary propagator are ϕ(0) = 1, which means In more detail, the first integration starts with some sufficiently small ε, thus setting the near-boundary initial conditions for ϕ equal to ϕ 1 , i.e. ϕ(z H ε) = 1 and ϕ (z H ε) = 0. Near the horizon, at z = z H (1 − ε), the obtained value y 1 of this solution and the value y 1 of its derivative can be written as superposition of ϕ + and ϕ − : (2.12) The constants C 1 , D 1 are determined by solving this linear system. The second integration works analogously, now based on ϕ 2 for near-boundary initial values of ϕ and its derivative, i.e. ϕ(z H ε) = ε 2 and ϕ (z H ε) = 2ε/z H , thus yielding another solution, which is decomposed as ). This, together with y 2 , determines C 2 and D 2 . A straightforward calculation using the above mentioned linearity shows that the particular value B = −C 1 /C 2 eliminates the out-going part of the general solution near the horizon [65]. Then, the corresponding retarded Green function G R of the dual current operator Qγ µ Q, defined within the framework of the holographic dictionary via a generating func- The quantity S V, on-shell m denotes here the action (2.1) with the solution ϕ from (2.2). Finally, the spectral function ρ follows from ρ(ω) = Im G R (ω) = 2k z 2 H Im B(ω). It has the dimension of energy squared.

Two-parameter potential -bottomonium formation
Our setting does not explicitly refer to a certain quark mass m. Instead, an ansatz U 0 (z; p ) with parameter n-tuple { p } is used such to catch a certain quarkonium mass spectrum. Insofar, m is to be considered as cumulative label highlighting the dependence of G m on a parameter set { p } which originally enters U 0 and which is to be adjusted to charmonium and bottomonium masses.
As a transparent model we select the two-parameter potential [32,34] which is known to deliver via (2.9) the normalizable functions ψ n with discrete eigenvalues The potential (3.1) is a slight modification of the SW model [61] with 3/(4z 2 ) stemming from the near-boundary warp factor A(z) and the term ∝ z 2 emerging originally from a quadratic dilaton profile ansatz. Note the Regge type excitation spectrum m 2 n = m 2 0 + nâ with intercept and slope to be steered by two independent parametersâ andb. We choose these parameters as follows. The mass m 0 determines the ground state (g.s.) "trajectory" in the a-b plane,b = 1 4 m 2 0 −â, and m 1 determines the first excitation (1st) "trajectory" bŷ b = 1 4 m 2 1 − 2â. Using the PDG values of J/ψ, ψ and Υ(1S, 2S) adjusts the "trajectories" as solid and dashed lines in figure 1, where we employ the scale setting viaâ = a/L 2 and b = b/L 2 with L −1 = 1.99 GeV, which is related to the QCD thermodynamics sector (see appendix A in [62]). Allowing for a 10% variation of m 1 one arrives at the colored bands in figure 1. By such a parameter choice one puts 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 (3.1) has several drawbacks and, therefore, is to be considered as an illustrative example. For instance, the sequence of radial Υ excitations in nature does not form a strictly linear Regge trajectory [63]. This prevents an unambiguous mapping of m 0,1 → (a, b). While the radial excitations of J/ψ follow quite accurately a linear Regge trajectory in nature [63], the request of accommodating further J/ψ properties in U 0 calls also for modifying (3.1), cf. [34,64]. Despite the mentioned deficits, the appeal of (3.1, 3.2) is nevertheless the simply invertible relation m 2 n (a, b) yielding a(m 0,1 ) and b(m 0,1 ). Since we are going to study the systematic, we keep the primary parameters a and b in what follows. Instead of discussing results at isolated points in parameter space referring to J/ψ and Υ ground states m 0 and first excited states m 1 , we consider the systematic over the a-b plane. Υ J/ψ melt is determined by the disappearance of the peak of the g.s. spectral function upon temperature increase. One observes a strong parameter dependence as well, which determines the spectral functions, see figure 2. Changing the parameters (a, b) deforms the potential (3.1) in a characteristic manner [62], e.g. going on a g.s. trajectory to right squeezes the excited states to higher energies, as can be identified in figure 1, in particular for the Υ. Such changes affect immediately 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/ψ and Υ, 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 [18] which advocates the formation of hadron states at T ≈ 155 MeV ≈ T c . Consistency does not necessarily mean perfect agreement: The criteria for "melting" or "onset of formation" are not very sharp. For instance, [64] uses as threshold value the relative high of the spectral function's peak over the smooth background for defining "melting". The transition to a 1 An analog figure in [62] exhibits the contour plot of the dissociation temperature T dis (a, b) which has been determined by the disappearance of normalizable solutions of the Schrödinger equation (2.4) in the interval z = [ 0,zH ] with boundary conditions ψn(z = 0) ∝ 2 0 and ψn(z =zH ) = 0.zH = zH (1 − H ) with H = 10 −2 sets a convenient cut-off which suppresses the highly oscillating solutions towards the horizon at zH . In contrast, the limit 0 → 0 is well defined. We find in general T melt (a, b) > T dis (a, b). quasi-particle with sharp spectral function does not happen instantaneously but within some temperature span, see top panels in figures 3 and 4. Considering the dynamics of the cooling system as a sequence of equilibrium states, the spectral-function contour-plots in figures 3 and 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 figures 3 and 4 as well, illustrates such a feature only insufficiently but is useful for a more quantitative account. Inspection of the top panels of figures 3 and 4 unravels that the temperature difference from T g.s.
melt until the formation of a sharp quasi-particle state is quite large. Sharp quasiparticles can be identified by the squeezed contour lines which eventually coincide nearly with the peak position of the spectral functions depicted by the red dashed curves in top panels of figures 3 and 4. Keeping the quarkonia ground state masses m 0 and allowing artificially for a somewhat larger value of the first excited state m 1 moves the quarkonia formation temperatures to larger values, in particular for Υ, see right panels in figure 3. In such a way, the quasi-particle formation temperature T

Υ(1S)
f orm ≈ T c copes with the claim in [18] of hadron formation at T c . The J/ψ, in contrast, would be formed at T J/ψ f orm < T c (see figure 4) in conflict with the advertisement of [18]. Section 4 provides a potential ansatz U 0 (z; p ) which accomplishes T To understand why the J/ψ (Υ) reacts so sluggishly (violently) on a modification of m 1 while keeping m 0 fixed, we mention that the parameterâ in (3.1) changes by 33% (92%, i.e. a factor of nearly two) upon a 5% increase of m 1 , 2 which is to been seen in connection with the curvature 8â 2 of U 0 at the minimum z min = 3/(4â 2 ) 1/4 . The more the potential 2 Due to the non-linearity of the J P C = 1 −− bottomonium Regge trajectory, the energy of m1 + 5% is between the 3 3 S1/Υ(3S) and 4 3 S1/Υ(4S) states. For charmonium, in contrast, m1 + 5% is well below the 3 3 S1/ψ(4040) state, cf. [63].  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 g.s.
f orm ≈ Υ J/ψ T 1st f orm . This feature is to be seen in relation to the considered ansatz of U 0 (z; a, b) with the IR behavior ∝ z 2 : a much steeper increase of U 0 at larger values of z would concentrate the melting temperatures in a narrow corridor.
Besides the ansatz (3.1) facilitates a sequential quarkonium formation upon decreasing temperature, T g.s.
f orm > T 1st f orm > T 2nd f orm etc., it allows potentially for a some flavor dependence, e.g. T

Υ(1S)
f orm > T J/ψ f orm . The thermal mass shifts have a non-trivial temperature dependence as evidenced in figure 5. Such thermal mass shifts are employed in [8] to pin down the heavy-quark (HQ) transport coefficient γ which can be considered as the dispersive counterpart of the HQ momentum diffusion coefficient κ = 2T 3 /(DT ), where D stands for the HQ spatial diffusion coefficient. Reference [16] stresses a seemingly tension within previous holographic results [35], where positive mass shifts are reported, in contrast to negative shifts, e.g. in [33]. Our set-up resolves qualitatively that 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/ψ and Υ are larger than the lattice QCD-based values quoted in [8,43].
Finally, let us remind that the two-parameter ansatz (3.1) is appealing since it allows for analytic solutions w.r.t. the excitation spectrum and an easy overview on the parameter dependencies. However, already the authors of [34,64] promoted (3.1) to a "shift and dip potential" to catch more properties of the J/ψ states than only masses.

Three-parameter potential with dip -charmonium formation
The two-parameter potential U 0 (z; a, b) from (3.1) with realistic values of a(m 0,1 ) and b(m 0,1) facilitates J/ψ formation at too low temperatures. This failure can be repaired by turning to more appropriate parameterizations. For instance, [34,64] proposed a fourparameter "dip and shift potential" which allows for J/ψ melting temperatures significantly above T c , as also the construction in [30][31][32][33]35] deploying three parameters. The essence is a dip in U 0 (z; 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, Γ) is as follows. (2.7, 2.8)) the potential is given by The first three terms in the top line suggest a correspondence M =4 √ b/L and k = √ a/L by a comparison with (3.1), while the next two terms cause some modification of (3.1) at intermediate values of z. The second line of (4.1) is essentially responsible for the dipsomewhat 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 = √ Γ/(kM ); 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 (3.1). The dip position and the dip depth are interrelated, in contrast to the construction in [34,64].
The potential (4.1) might exhibit some non-trivial local structures as a function of z for particular parameters. Reference [66] 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 √ Γ = 0.55 GeV (representing the QQ string tension) to yield the J/ψ (ψ ) mass of 2.943 (3.959) GeV) and the decay width of 399 (255) MeV. Note the resulting overestimated level spacing quantified by m 2 1 /m 2 0 = 1.81, in contrast to the PDG value of 1.42, when deploying these parameters.
Completely analog to the two-parameter potential (3.1), increasing the parameter k at M ≈ const, the potential (4.1) is squeezed and becomes deeper. Analogously, decreasing the parameter √ Γ 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 constant √ Γ to keep the g.s. mass m 0 constant, see the horizontal dashed line in left panel of figure 6. Due to the squeezing of the potential, the interior (left) part is less influenced when imposing a horizon at z H , where U T (z = z H ) = 0 is facilitated according to (2.5). As a result, the more the potential is squeezed the smaller values of z H are allowed to hold the J/ψ prior to melting. That is the very reason which forces us to enlarge the parameter k (or a in (3.1)) to achieve quarkonium formation at sufficiently high temperatures in agreement with the perspective put forward in [18]. The dip in the potential (4.1) is useful in that respect since enlarging the parameter a in the flat potential (3.1) influenced the quarkonium formation in a less effective manner for J/ψ. Let us emphasize that we put more weight on the g.s. mass m 0 (see 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 dashed curve and colored band in the right panel of figure 6), thus following the rationale in [34].
Having U 0 at our disposal we proceed as in Section 3. Contour plots of the J/ψ spectral function are exhibited in the top panels of figure 7. One observes again the tendency of charmonium formation as narrow corridor of contour lines at too low temperatures for parameters delivering exactly the PDG values of 2) GeV favored in [66]. These parameters, albeit with noticeably deviations to the PDG values of m 0,1 , realize the charmonium formation as transition of 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 [66].
To complete the systematic related to charmonium we exhibit in figure 9 the quantity  − log G m as a function of φ. Note the huge variation of G m (φ). In general, G m (φ) depends sensitively on the parameters in U 0 and is tightly related to the background.
An analog study of the Υ formation is hampered by some uncomfortable structures of U 0 (z; M, k, Γ). References [30,31,35] advocate parameters which avoid such obstacles, however, result in a value of m 2 0 (Υ(1S)) being only one half of the PDG value. We therefore do not perform an analysis of the potential ansatz (4.1) in the QCD-related background since the two-parameter potential (3.1) was already shown to accomplish successfully bottomonium formation at T c .

Summary
In summary we introduce a modification of the holographic vector meson action for quarkonia such to join (i) the QCD 2+1 (phys) thermodynamics, described dynamically consistently 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 is based on an effective dilaton φ m = φ − log G m , where φ is solely tight to the light-quark-gluon thermodynamics background, while the flavor dependent quantity G m is determined by a combination of φ 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 features of our scheme, where the thermodynamic background at T > 0 and meson spectra at T = 0 serve as QCD-based input to analyze the quarkonia formation at T > 0. We test a scenario where quarkonium formation is 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 [43]. Realistic values of Υ(1S, 2S) masses allow in fact the formation temperature T f orm of Υ(1S) nearby T c in line with the claim of [17,18] that hadrons form themselves at temperatures T c ≈ T f o ≈ 155 MeV. Insofar, the mystery "why T f o ≈ T c ?" could be resolved by a dynamical process within such a scenario: Hadronization is the transit of broad to narrow spectral functions within a few-MeV temperature interval at T c .
While quite promising, the proposed scenario is hampered by three issues, at least. First, the finding of T f orm ≈ T c looks somewhat accidental and is not locked explicitly to a certain microscopic process; in addition, there is a slight tension due to the tendency of T f orm < T c when deploying the exact PDG value of the Υ(2S) mass together with the Υ(1S) PDG value. Second, the formation of the Υ(2S) quasi-particle occurs at T Υ(2S) f orm < 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 fails quantitatively for J/ψ since T J/ψ f orm < 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 rapid shrinking of the spectral functions in a narrow temperature interval at T c , including the excited states.
Formally, 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 here proposed bottom-up scenario of quarkonia formation solely accommodates properties of vector cc and bb states in the holographic bulk vector field A. This is in contrast to microscopic studies, e.g. in [2,4,54,[67][68][69], where the heavy-quark interaction with constituents of the ambient medium is dealt with in detail. Also primordial contributions and early off-equilibrium yields as well as corresponding feedings are 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.  [44]. The quoted temperature values bracket the pseudo-critical temperature T c = (156 ± 1.5) MeV which is determined by a peak of the chiral susceptibility [19]. We focus here on the local minimum of the sound velocity and its mapping onto the gravity-dilaton background.
Deforming the AdS metric by putting a black hole with horizon at z H yields the metric for the infinitesimal line elements squared (2.3) where f (z, z H )| z=z H = 0 is a simple zero. Identifying the Hawking temperature T (z H ) = −∂ z f (z, z H )| z=z H /4π with the temperature of the system at bulk boundary z → 0 and the attributed Bekenstein-Hawking entropy density s(z H ) = 2π κ exp{ 3 2 A(z, z H )| z=z H }, one describes holographically the thermodynamics. f = 1 at T = 0 refers to the vacuum.
The gravity-dilaton background is determined by the action in the Einstein frame where R stands for the curvature invariant and κ = 8πG 5 . (For our purposes, the numerical values of κ and G 5 as well as k V in (2.1) are irrelevant.) The field equations and equation of motion for the metric coefficients and the dilaton follow from (A.1) as to be solved with boundary conditions A(z → 0) → −2 log(z/L), φ(0) = 0, φ (0) = 0, f (0) = 1, f (z H ) = 0; the prime means differentation w.r.t. z. The dilaton potential V (φ) is the central quantity [48]. Imposing certain conditions one can describe the QCDrelevant cross-over (instead of phase transitions of first or second order or a Hawking-Page transition). A necessary condition for a cross-over is (i) ∂ φ V /V , as a function of φ, has a local maximum and (ii) ∂ φ V /V < 2/3 (for refinements, cf. [48]). The three-parameter ansatz − L 2 V = 12 cosh(γφ) + φ 2 φ 2 + φ 4 φ 4 (A.5) is sufficient for a satisfactory description of the lattice QCD 2+1 (phys) data [20,21] 3 by coefficients (γ, φ 2 , φ 4 ) = (0.568, −1.92, −0.04) together with L −1 = 1.99 GeV, see figure 5left in [62]. In fact, the above mentioned conditions are met: maximum of ∂ φ V /V = 0.58 at φ = 1.84. In general, the sound velocity squared, v 2 s = d log T d log s , acquires a local minimum if s(T ), or s(T )/T 4 , has an inflection point. Surprisingly, neither T (z H ) nor s(z H ) display such a feature. Instead, both T (z H ) and s(z H ) are monotonous functions of z H , see figure 10. That means, the minimum of the sound velocity is caused by a subtle interplay of derivatives of T (z H ) and s(z H ). Displaying the sound velocity squared by v 2 s (z H ) = ∂ z H log T /∂ z H log s, the local minimum is determined by These individual terms are exhibited in figure 11. It turns out that the actually chosen parameters facilitate the minimum of sound velocity at the crossing of the fat solid and thin solid curves at z H /L = 5.17, corresponding to T = 152 MeV, i.e. nearby T c and thus T f o . In contrast to T (z H ) and A(z, z H )| z=z H , the dilaton profile φ(z, z H ) has a marked imprint of the QCD specifics: it exhibits inflection points in both z direction and z H direction, see figure 12. This is a remarkable property which makes the use of the QCD-related gravity-dilaton background distinct in comparison with schematic ansätze, which additionally miss the consistent interrelations of the quantities A, f and φ via field equations. Note that the dilaton enters explicitly the quarkonium action (2.1), thus leaving directly its imprints related to quarkonium formation.