Thermal entropy of a quark-antiquark pair above and below deconfinement from a dynamical holographic QCD model
TThermal entropy of a quark-antiquark pair aboveand below deconfinement from a dynamicalholographic QCD model
David Dudal a,b ∗ , Subhash Mahapatra a † a KU Leuven Kulak, Department of Physics, Etienne Sabbelaan 53 bus 7657,8500 Kortrijk, Belgium b Ghent University, Department of Physics and Astronomy, Krijgslaan 281-S9, 9000 Gent, Belgium
Abstract
We discuss the entropy carried by a quark-antiquark pair, in particular across thedeconfinement transition. We therefore rely on a self-consistent solution to Einstein-Maxwell-dilaton gravity, capable of mimicking essential features of QCD. In particular weintroduce a novel model that still captures well the QCD confinement and deconfinementphases, while allowing the introduction of a temperature in a phase which resembles theconfined phase, this thanks to it being dual to a small black hole. We pay due attentionto some subtleties of such model. We confirm the lattice picture of a strong build-up ofthermal entropy towards the critical temperature T c , both coming from below or above T c . We also include a chemical potential, confirming this entropic picture and we considerits effect on the speed of sound. Moreover, the temperature dependent confinement phasefrom the holography side allows us to find a string tension that does not vanish at T c , afinding also supported by lattice QCD. The investigation of QCD at finite temperature is extremely important for various rea-sons, ranging from heavy ion physics to cosmology. It is expected that studies of heavyquarkonium at finite temperature may enhance our understanding of the deconfinementtransition and that these may shed new light onto the (creation of the) QCD plasmaphase. A lot of new results, both from experimental as well as from theoretical side, areappearing. For example, recent experimental results from RHIC and LHC have indicateda strong suppression of charmonium near the deconfinement transition [1, 2]. The non-trivial experimental result that the charmonium suppression is stronger at lower energydensity than at larger energy density has led to an intense investigation into the natureof this suppression and many theoretical scenarios such as the renown colour screeningmechanism [3], a recombination of the produced charm quarks into charmonia [4, 5], theimaginary potential mechanism for heavy quark dissociation [6–8], etc have been sug-gested for its explanation. ∗ [email protected] † [email protected] a r X i v : . [ h e p - t h ] N ov igure 1: Two flavour lattice QCD result for theentropy of a qq pair as function of quark-antiquarkseparation at temperature T (cid:39) . T c . The figure istaken from [12]. Figure 2:
Lattice QCD result for the entropy of a qq pair as function of temperature T /T c for large qq separation. The figure is taken from [12].One such mechanism which attracted interest of late, and a topic of this paper, wasproposed in [9]: it was suggested that the suppressed production of heavy quarkoniummay be related to the nature of the deconfinement transition. In [9], based on latticeQCD results [10–12], the notion of an “emergent entropic force”, F = T ∂S∂(cid:96) , was introduced. Here S denotes the entropy, T is the temperature and (cid:96) is the inter-quarkseparation. The lattice prediction that the entropy of a quark-antiquark ( qq ) pair increaseswith distance, shown in Figure 1, leads to a positive repulsive force and can promote theself-destruction of the bound state. It was shown that the destruction of a bound stateto a delocalized state was maximal near the deconfinement transition temperature, givinga possible explanation for the strong suppression of quarkonium around that temperature.The emergent entropic force mechanism of [9] is thus based on lattice QCD simulationswhich suggested a considerable amount of entropy is associated with a qq pair in the hotQCD plasma, indicating a strong entanglement between the heavy bound states with therest of QCD plasma. The lattice QCD data are summarized in Figures 1 and 2. Someof the main lattice observations are i) prediction of a large amount of entropy associatedwith a heavy qq pair near the deconfinement transition temperature ii) away from thedeconfinement temperature the entropy decreases with temperature in both confined anddeconfined phases and iii) the entropy grows as the separation between the quarks in-creases. This entropy first increases gradually and then saturates to a constant value atlarge separations.As emphasized in [13, 14], the lattice data for the qq entropy at the peak near the transi-tion temperature also indicates the breakdown of the weak coupling approximation. Theentropy near the transition temperature computed from a mean field Debye screeningapproximation was found to be an order of magnitude lower than that of lattice results.This suggests the need of a method which is more suitable for entropy calculations in thestrongly coupled regime. One such non-perturbative method is the gauge/gravity dual-ity [15–17]. Using this duality, a lot of new insights into the regime of strongly coupled CD, which agrees qualitatively with the lattice QCD results, have been obtained bothfrom “top-down” and phenomenological “bottom-up” models [18–43]. Since gauge/gravityduality provides a valuable and unique technique to investigate strongly coupled gaugetheory, therefore it would be interesting to see whether this duality can also provide newinsights into the entropy of the qq pair.The entropy of the qq pair and the idea of an entropic force was first discussed holo-graphically in [14] for top-down models. Two models — N = 4 supersymmetric Yang-Millstheory (SYM) and Pure 4D Yang-Mills theory obtained from compactification of 5D SYMon a circle— were considered and the holographic qq thermal entropy was calculated fromthe free energy of an open string hanging from the asymptotic boundary into the bulkspacetime [44–46]. Albeit not exactly corresponding to the (lattice) QCD setup, similarfeatures were seen in these dual black hole phases: the growth of the entropy with inter-quark distance, saturation to a constant value at large separations and in the second case,a divergence at the deconfinement temperature, reflecting the sharp peak observed ingenuine lattice QCD. These calculations were then extended to other holographic modelsin [47–49]. However, since these two holographic models do not accurately describe realQCD, there were some discrepancies too. For example, the higher temperature asymp-totics of the entropy do not reconcile with lattice QCD results. This discrepancy was laterrectified in [47] using the improved holographic bottom-up models of [26, 27]. Anotherimportant lattice result which could not be reproduced by holographic models in [14, 47]was the temperature dependence of the entropy in the confined phase. This is preciselydue to the fact that in the gauge/gravity duality, the confined phase is generally dual toan asymptotically AdS space (without a horizon), which is thence independent of tem-perature. Subsequently all the physical quantities in the confined phase are independentof temperature. In order to study temperature dependence in this phase one would needto consider 1 /N corrections in the holographic models, something extremely difficult tocalculate.However there might be another way to introduce temperature dependence in theconfined phase, without having to compute 1 /N corrections: we can construct a blackhole solution whose dual boundary theory satisfies all the necessary properties of confine-ment. By confinement we simply mean a phase that satisfies area law behaviour for the(expectation value of the) Wilson loop and a Polyakov loop with vanishing expectationvalue . Indeed, one can see from Figure 2 that even in the confined phase a large amountof entropy is associated with the qq pair and therefore it is important to try to analyzethis observation from the holography viewpoint. The black hole solution will naturallyintroduce a notion of temperature in the confined phase and it would be interesting tosee whether such solution, if constructed, can also capture lattice results for the entropyof the qq pair. We will show in section 3 that it is indeed possible to construct sucha close cousin of the confined phase, which we call specious-confined phase, for whichthe dual gravity side contains a black hole. Interestingly, the entropy of the qq pair inthis holographic specious-confined phase turns out to be in qualitative agreement withlattice QCD confined phase results. The specious-confined model will also allow us tostudy, to our knowledge for the first time from holography, the temperature dependenceof the ensuing QCD string tension, finding qualitative agreement with quenched latticeQCD as well, in the sense that the string tension does not vanish at deconfinement [50–52].It is also well known that the QCD phase diagram strongly depends on the chemical We work in the holographic analogue of quenched QCD, i.e. with non-propagating quark flavour degreesof freedom. As such, QCD enjoys an explicit Z N symmetry in the confined phase, with the Polyakov loopexpectation value serving as order parameter. otential (i.e. a nonzero quark density). In particular, the confinement/deconfinementtransition temperature and QCD equation of state are sensitive to the value of the chem-ical potential. Accordingly, one might think that the entropy of the qq pair may alsodepend non-trivially on it. However such results are are not available in lattice QCD yetdue to the sign problem plaguing finite density simulations. In the absence of lattice QCDresults, therefore, it is of great importance to investigate the entropy holographically andsee the effects of chemical potential on it.In this work our aim is to construct one such holographic QCD model that displaysqualitative agreement with lattice results for the qq entropy, not only for the deconfinedphase but in particular for the confined phase as well. We will also include effects of achemical potential. For this purpose we consider a phenomenological bottom-up Einstein-Maxwell-dilaton gravity model [53–63]. The gravity system is analytically solvable interms of an arbitrary scale function A ( z ) (see eq. (2.14)), depending on which there canbe various kinds of phase transitions on the gravity side. In this paper, we consider twodifferent forms for A ( z ). The first form, discussed in eqs. (3.1), leads to a first order phasetransition in terms of a small to a large black hole phase. These small and large black holephases correspond to specious-confinement and deconfinement phases in the dual bound-ary theory respectively. Notably, the boundary dual of the small black hole phase doesnot exactly correspond to confinement as it has a non-zero (albeit exponentially small)Polyakov loop expectation value while showing linear confinement for larger distances atlow temperatures only. For this reason we called this dual phase specious-confined insteadof confined. Furthermore, the specious-confinement /deconfinement transition tempera-ture decreases with the chemical potential and the corresponding first order small/largeblack hole phase transition line terminates at a second order critical point, as predicted bythe well-known lattice study [64]. Similarly, the second form of A ( z ), shown in eqs. (4.1),leads to a first order Hawking/Page phase transition from thermal-AdS to black hole onthe gravity side. These thermal-AdS/black hole phases correspond to the standard con-finement/deconfinement phases in the dual boundary QCD theory.Having constructed a black hole solution (using the first form of A ( z )) for both specious-confinement and deconfinement phases, we then study temperature dependenceof the qq entropy in these phases. We find that our phenomenological model does qualita-tively reproduce the lattice QCD results of the qq entropy. Our main results are shown inFigures 12, 13 and 14. In the deconfined phase, we find that our gravity model displays thegrowth of the entropy with inter-quark distance, corroborating the entropic force scenarioof [9] for the self-destruction of the bound state. This entropy saturates to a constantvalue at large distances, as also predicted by lattice QCD. Our model further reproduces asharp peak in the entropy near the critical temperature. The sharp peak near the criticaltemperature can be appreciated from both deconfined as well as from specious-confined phases. Interestingly, even though specious-confined is strictly speaking not referring to agenuine confined phase, the entropy of the qq pair matches qualitatively with the latticeQCD predictions. We further provide a holographic estimate for the qq entropy withchemical potential which could be tested using lattice QCD in the near future. Similarly,the results for the qq entropy using the second form of A ( z ) are again in agreement withlattice QCD. The qualitative results in the deconfined phase remains the same, however,since the confined phase is now dual to thermal-AdS, the entropy is zero in that phase.We also study the speed of sound ( C s ) in our holographic models and find results whichagain are in qualitative agreement with lattice QCD. In particular, as in lattice QCD, C s is greatly suppressed near the transition temperature in both models. Importantly, C s is suppressed in the specious-confined phase side too. We find that C s approaches itsconformal value 1 / igher values of chemical potential try to suppress C s more near the transition tempera-ture. However for extremely low (in the specious-confined phase) and high temperatures, C s approaches a chemical potential independent constant value.The paper is organized as follows. In the next section, we describe our gravity modeland obtain the solution analytically. In section 3, using a particular form of A ( z ), wefirst study the thermodynamics of the gravity solution and then discuss the free energyand entropy of a qq pair with and compare it with lattice QCD results. We repeat thecalculations of section 3 with a different A ( z ) Ansatz in section 4. In section 5, we studythe speed of sound in the constructed specious-confined , confined and deconfined phases.Finally, we conclude this paper with some discussions and an outlook to future researchin section 6. We start with the Einstein-Maxwell-dilaton action in five dimensions, S EM = − πG (cid:90) d x √− g (cid:2) R − f ( φ )4 F MN F MN − ∂ M φ∂ M φ − V ( φ ) (cid:3) (2.1)where G is the corresponding Newton constant, V ( φ ) is the potential of the dilaton fieldand f ( φ ) is a gauge kinetic function which represents the coupling between dilaton andgauge field A M . The Einstein, Maxwell and dilaton equations of motion derived fromeq. (2.1) are, respectively, R MN − g MN R − T MN = 0 , (2.2) ∇ M [ f ( φ ) F MN ] = 0 , (2.3) ∂ M (cid:2) √− g∂ M φ (cid:3) − √− g (cid:18) ∂V∂φ + F ∂f∂φ (cid:19) = 0 (2.4)where T MN = 12 (cid:18) ∂ M φ∂ M φ − g MN ( ∂φ ) − g MN V ( φ ) (cid:19) + f ( φ )2 (cid:18) F MP F PN − g MN F (cid:19) . In order to simultaneously solve eqs. (2.2), (2.3) and (2.4), we consider the followingAns¨atze for metric, gauge and dilaton fields, ds = L e A ( z ) z (cid:18) − g ( z ) dt + dz g ( z ) + dy + dy + dy (cid:19) ,A M = A t ( z ) , φ = φ ( z ) (2.5)where we have assumed that the various fields depend only on the extra radial coordinate z . Here L is the AdS length scale and in our notation, z = 0 corresponds to the asymp-totic boundary of the spacetime, i.e. where the strongly coupled gauge theories is located.Plugging the above Ans¨atze into eqs. (2.2)-(2.4), we get the following equations φ (cid:48)(cid:48) + φ (cid:48) (cid:18) − z + g (cid:48) g + 3 A (cid:48) (cid:19) − L e A z g ∂V∂φ + z e − A A (cid:48) t L g ∂f∂φ = 0 , (2.6) (cid:48)(cid:48) t + A (cid:48) t (cid:18) − z + f (cid:48) f + A (cid:48) (cid:19) = 0 , (2.7) g (cid:48)(cid:48) + g (cid:48) (cid:18) − z + 3 A (cid:48) (cid:19) − e − A A (cid:48) t z fL = 0 , (2.8) A (cid:48)(cid:48) + g (cid:48)(cid:48) g + A (cid:48) (cid:18) − z + 3 g (cid:48) g (cid:19) − z (cid:18) − z + 3 g (cid:48) g (cid:19) + 3 A (cid:48) + L e A V z g = 0 , (2.9) A (cid:48)(cid:48) − A (cid:48) (cid:18) − z + A (cid:48) (cid:19) + φ (cid:48) A ( z ) and f ( z ) i.e. in terms of a scaling factor and a kinetic gauge function [60, 61]. Interm of A ( z ) and f ( z ), the solutions are g ( z ) = 1 − (cid:82) z dx x e − A ( x ) (cid:82) xx c dx x e − A ( x f ( x ) (cid:82) z h dx x e − A ( x ) (cid:82) xx c dx x e − A ( x f ( x ) ,φ (cid:48) ( z ) = (cid:112) A (cid:48) − A (cid:48)(cid:48) − A (cid:48) /z ) ,A t ( z ) = (cid:115) − (cid:82) z h dx x e − A ( x ) (cid:82) xx c dx x e − A ( x f ( x ) (cid:90) zz h dx xe − A ( x ) f ( x ) ,V ( z ) = − z ge − A L (cid:2) A (cid:48)(cid:48) + A (cid:48) (cid:0) A (cid:48) − z + 3 g (cid:48) g (cid:1) − z (cid:0) − z + 3 g (cid:48) g (cid:1) + g (cid:48)(cid:48) g (cid:3) (2.11)where we have used the boundary condition that at the horizon g ( z h ) = 0 and g ( z )goes to 1 at the asymptotic boundary. It can be explicitly verified, with the help of thepotential expression in eq. (2.11), that the Einstein, Maxwell and dilaton equations areindeed satisfied. The undetermined integration constant x c in eq. (2.11) can be fixed interms of a chemical potential ( µ ) present in the boundary theory. Expanding A t near theasymptotic boundary z = 0 and using the gauge/gravity mapping, we get µ = − (cid:115) − (cid:82) z h dx x e − A ( x ) (cid:82) xx c dx x e − A ( x f ( x ) (cid:90) z h dx xe − A ( x ) f ( x ) . (2.12)The only non-trivial inputs which remain to be fixed are A ( z ) and f ( z ). Indeed eq. (2.11) isa gravity solution for the Einstein-Maxwell-dilaton system for any A ( z ) and f ( z ). There-fore, we have a freedom of choosing any A ( z ) and f ( z ). We can use this freedom toconstrain the arbitrary factors A ( z ) and f ( z ) by matching the properties of the bound-ary gauge theory with real QCD. For example, f ( z ) can be fixed by demanding the dualboundary theory to satisfy linear Regge trajectories, i.e. the squared mass of the mesonsvary linearly with respect to the radial excitation number. In this paper, we will considerthe following simple form for f ( z ) f ( z ) = e − cz − A ( z ) . (2.13)Using this form of f ( z ), one can easily show that the discrete spectrum of the mesonsindeed lie on a linear Regge trajectory. One can fix the magnitude of c by matching the olographic meson mass spectrum to that of lowest lying heavy meson states [60]. Bydoing that, one gets c = 1 .
16 GeV . With this choice of f ( z ), the gravity solution is then given by g ( z ) = 1 − (cid:82) z h dx x e − A ( x ) (cid:20)(cid:90) z dx x e − A ( x ) + 2 cµ (1 − e − cz h ) det G (cid:21) ,φ (cid:48) ( z ) = (cid:112) A (cid:48) − A (cid:48)(cid:48) − A (cid:48) /z ) ,A t ( z ) = µ e − cz − e − cz h − e − cz h ,V ( z ) = − z ge − A L (cid:20) A (cid:48)(cid:48) + A (cid:48) (cid:18) A (cid:48) − z + 3 g (cid:48) g (cid:19) − z (cid:18) − z + 3 g (cid:48) g (cid:19) + g (cid:48)(cid:48) g (cid:21) (2.14)where det G = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:82) z h dx x e − A ( x ) (cid:82) z h dx x e − A ( x ) − cx (cid:82) zz h dx x e − A ( x ) (cid:82) zz h dx x e − A ( x ) − cx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . We now have a complete solution of Einstein-Maxwell-dilaton gravity system. Thesolution in eq. (2.14) corresponds to a black hole with horizon at z h . The Hawkingtemperature and entropy of this black hole solution are given by, T = z h e − A ( z h ) π (cid:82) z h dx x e − A ( x ) (cid:20)
1+ 2 cµ (cid:0) e − cz h (cid:82) z h dx x e − A ( x ) − (cid:82) z h dx x e − A ( x ) e − cx (cid:1) (1 − e − cz h ) (cid:21) S BH = L e A ( z h ) G z h . (2.15)Let us also mention that there is another solution to the Einstein-Maxwell-dilaton equa-tions, i.e. one without a horizon corresponding to a thermal-AdS space. The thermal-AdSsolution can be obtained by taking the z h → ∞ limit of the black hole solution presentedabove, i.e. g ( z ) = 1. Although this thermal solution again asymptotes to AdS at theboundary, however depending on the scale factor A ( z ), it can have a non-trivial structurein the bulk. This solution will play a significant role in our analysis in section 4.The only function that now remains to be fixed is the scale factor A ( z ). Indeed, inorder to proceed further we now need to specify a specific form for A ( z ). In the nexttwo sections, we consider two different forms of A ( z ) and study the thermodynamicsof the gravity system, the free energy and entropy of the qq pair, speed of sound etc for each form separately. As we will show, depending on the form of A ( z ), there canbe various kinds of phase transitions between the gravity solutions, and correspondingly,the interpretation of these gravity solutions in the boundary gauge theory can be different.Above we have presented a gravity solution in the Einstein frame, useful for investigat-ing thermodynamical properties and equations of state of the system. However, in orderto study some confinement/deconfinement properties and in particular the free energy of The factor c can also be fixed by matching with the ρ meson mass instead, as done in e.g. [65]. However,since we are more interested in the properties of heavy bound states, we find it more appropriate to comparewith the J/ Ψ and Ψ (cid:48) state, following [60]. qq pair, it is more convenient to use the string frame metric. It can be obtained fromthe Einstein frame metric by the following standard transformation via the dilaton,( g s ) MN = e √ φ g MN ,ds s = L e A s ( z ) z (cid:18) − g ( z ) dt + dz g ( z ) + dy + dy + dy (cid:19) (2.16)where A s ( z ) = A ( z ) + (cid:113) φ ( z ).For completeness, we should mention here that we treat the scalar field φ as a genuinedilaton, although our models are lacking, as any bottom-up AdS/QCD model, a properembedding into an underlying string theory. If φ was just another scalar field other than adilaton so that ( g s ) MN ≡ g MN , then confinement and all of its related features are absentand thus any qualitative comparing with QCD becomes obsolete, both from the Case 1and Case 2 model we are going to discuss below. This finding is analogous to another self-consistent model used to connect to magnetized QCD, as introduced and discussed in [69].Before ending this section, we like to mention that the Gubser criterion of [70] forEinstein-scalar gravity theory is satisfied in our gravity model as well . More precisely,the dilaton potential at any point in the bulk is bounded from above by its value at theUV boundary, i.e. V (0) ≥ V ( z ). Let us first consider the following simple form of A ( z ), A ( z ) = A ( z ) = −
34 ln ( az + 1) + 12 ln ( bz + 1) −
34 ln ( az + 1) . (3.1)It is straightforward to see that A ( z ) → z = 0. This is nothing but thestatement that bulk spacetime asymptotes to AdS at the boundary. Moreover, expandingthe dilaton field and potential near the asymptotic boundary and rewriting the dilatonpotential in terms of dilaton field, we get V ( φ ) = − L + ∆(∆ − φ ( z ) + . . . , ∆ = 3 . (3.2)The above equation indicates that the dilaton mass m meets a well known result ingauge/gravity duality, i.e. m = ∆(∆ − < ∆ < V ( z ) | z → = − /L = 2Λ, where Λ is the negative cosmolog-ical constant in five dimensions, as expected.The parameters a and b in A ( z ) can be determined by comparing the holographicresults with lattice QCD results at zero chemical potential. For example, as we will showshortly, the gravity solution with the above form of A ( z ) undergoes a small to large blackhole phase transition. This phase transition has been suggested to be dual to confinement-deconfinement phase transition in the boundary side [60, 61]. Demanding the critical See for example [66–68] exploring this possibility. Other, stronger, criteria for the physical reliability of generic naked singularities are suggested in [68, 71],but our potentials fall beyond the class discussed there. emperature of this transition at zero chemical potential to be around 270 MeV fixes theparameters a and b as a = c , b = 5 c . We again like to emphasize that eq. (2.14) is a solution of Einstein-Maxwell-dilaton systemfor any A ( z ), and eq. (3.1) is just a particular form. We chose this expression to reproducesome of the QCD results holographically, after which new predictions for other quantitiescan be made without further introduction of new parameters. Let us first discuss the thermodynamics of the gravity system with A ( z ) as in eq. (3.1).All the results in this section should be understood to be derived from this form of A ( z ). T crit z h Figure 3: T as a function of z h for various valuesof the chemical potential µ . Here red, green, blue,brown and cyan curves correspond to µ = 0, 0 . .
2, 0 .
312 and 0 .
35 respectively. In units GeV. T crit - - Figure 4: F as a function of T for various valuesof the chemical potential µ . Here red, green, blue,brown and cyan curves correspond to µ = 0, 0 . .
2, 0 .
312 and 0 .
35 respectively. In units GeV.The variation of Hawking temperature with respect to horizon radius z h for variousvalues of chemical potential is shown in Figure 3. We find that for small chemical po-tential there are three branches in the ( T, z h ) plane, out of which one branch is stable,one metastable and the last branch is unstable. The stable and metastable branches,for which the slope in ( T, z h ) plane is negative, correspond to small and large black holephases. For the unstable branch, the slope is positive. This suggest a phase transitionform a small black hole (large z h ) to a large black hole (small z h ) as we steadily increasethe Hawking temperature. This is indeed the case as can be observed from the free energybehaviour which is shown in Figure 4. In Figure 4, we used the normalization such thatthe free energy of thermal-AdS is zero. We see that the free energy has a swallowtail likestructure — a characteristic feature of a first order phase transition — for small chemicalpotential. The unstable branch corresponds to the base of the swallowtail and a phasetransition occurs at the kink. The corresponding temperature at the kink around whichthe free energy of the large black hole becomes larger than that of the small black holedefines the critical temperature T crit . For µ = 0, we find T crit = 0 .
276 GeV. Importantly, See [74] for a lattice determination of the deconfinement temperature for pure gauge theories for severalvalues of N . Only a mild dependence on (growing) N is to be reported. mall BH phaseLarge BH phaseCritical point μ T crit Figure 5: T crit as a function of µ . For small µ , there is a first order phase transition line betweenlarge and small black hole phases. This first order phase transition line terminates at a second ordercritical point. In units GeV.the free energy is always negative in stable branches which indicates that the black holephase always has a lower free energy than thermal-AdS and hence is more stable.However, the above pattern changes as we increase the value of µ . For higher valuesof µ , the size of the swallowtail starts to decrease and it completely disappears above acertain critical chemical potential µ c . For A ( z ) = A ( z ), we find µ c = 0 .
312 GeV. At µ c ,the large and small black hole branches merge together and form a single stable black holebranch which is stable at all temperature. This can be seen from cyan curve for which µ = 0 .
35 GeV is considered. The overall dependence of T crit on µ is shown in Figure 5.We find that T crit decreases with µ . The solid line in Figure 5 which separates the smalland large black hole phases terminates at the second order critical point. This behaviouris analogous to the Van der Waal like liquid-gas phase transition and has been thoroughlydiscussed in context of charged AdS black holes in many cases [75–80].The above small-large black hole phase transition on the gravity side has been sug-gested to correspond to the confinement-deconfinement phase transition in the dual bound-ary side in [60, 61] . In particular, the small black hole phase was suggested to be dualto confinement whereas the large black hole phase was suggested to be dual to decon-finement. However, as we will show shortly, this interpretation is not entirely correct.Especially, there are a few non-trivial issues in interpreting the small black hole phase asdual to confinement. Again by confinement we simply mean a phase for which the Wilsonloop expectation value satisfies the area law while the expectation value of the Polyakovloop is zero. We will show in the next section that by taking the correct expression ofthe qq free energy neither of these conditions are strictly satisfied in the boundary theorydual to the small black hole phase. However interestingly, despite of these limitations ininterpreting the small black hole phase as being dual to genuine confinement, the entropyof the qq pair and speed of sound in this phase turn out to be in good qualitative agree-ment with lattice QCD in its confined phase. In [60, 61], a different expression for A ( z ) was used. .2 Free energy of the qq pair In order to study the Wilson and Polyakov loops, we first need to access the free energy of a qq pair which can be easily computed holographically from the world sheet on-shell action.The gauge/gravity correspondence relates the free energy F of a qq pair to the on-shellaction of a fundamental string in the dual gravity background. In particular, the F of a qq pair separated by a distance (cid:96) and evolving over time t can be calculated by the on-shellaction of the fundamental string, with the boundary condition that at the asymptoticallyAdS boundary, the string world sheet shares the rectangular boundary of sides (cid:96) and T .We have F ( T, (cid:96) ) =
T S on − shellNG (3.3)where T is the temperature and S on − shellNG is the open string on-shell Nambu-Goto action, S NG = 12 π(cid:96) s (cid:90) dτ dσ (cid:112) − det g s , ( g s ) αβ = ( g s ) MN ∂ α X M ∂ β X N . (3.4)Here, g s denotes the gravity background in string frame , X M ( τ, σ ) denote the openstring embedding, T s = 1 / π(cid:96) s is the open string tension and ( τ, σ ) are the world sheetcoordinates. Depending upon on the background geometry there can be multiple openstring solutions and below we will show that each solution provides a unique result whichcan mimic the results of lattice QCD.Here we will work in the static gauge with τ = t and σ = y . In this case there can betwo world sheet configurations that minimize the Nambu-Goto action: a connected anda disconnected one. The connected world sheet is a ∪ -shape configuration which extendsfrom the boundary ( z = 0) into the bulk. In this case, we have the following expressionfor the free energy of the qq pair F con = L π(cid:96) s (cid:90) z ∗ dz z ∗ z (cid:112) g ( z ) e A s ( z ) − A s ( z ∗ ) (cid:112) g ( z ) z ∗ e − A s ( z ∗ ) − g ( z ∗ ) z e − A s ( z ) (3.5)where z ∗ is the turning point of the connected world sheet. This turning point is relatedto the length of the qq pair as (cid:96) = 2 (cid:90) z ∗ dz z (cid:115) g ( z ∗ ) g ( z ) e − A s ( z ) (cid:112) g ( z ) z ∗ e − A s ( z ∗ ) − g ( z ∗ ) z e − A s ( z ) . (3.6)On the other hand, the disconnected configuration consists of two lines which are separatedby distance (cid:96) and are extended from the boundary to the horizon, F discon = L π(cid:96) s (cid:90) ¯ z dz e A s ( z ) z . (3.7) F discon is independent of z ∗ and therefore of the qq separation length (cid:96) as well. Here,¯ z = z h for the black background and ¯ z = ∞ for thermal-AdS. It is important to mentionthat both F con and F discon are divergent quantities. The divergence arises from the z = 0part of the integral. In order to regularize the free energy, we will use the temperatureindependent renormalization scheme suggested in [81], which amounts to minimally sub-tracting the pole when cutting at the integral at z = ε (cid:28)
1. However, in most part of thispaper we will deal with the difference in free energy anyhow where the diverging partstrivially cancel out. In the following, a subscript “ s ” is used to denote quantities in the string frame. .2 0.4 0.6 0.8 1.0 1.2 z * Figure 6: (cid:96) as a function of z ∗ in the large blackhole background for various values of z h . Here µ =0 and red, green and blue curves correspond to z h =1 .
2, 0 . . - - Δℱ T s L Figure 7: ∆ F = F con −F discon as a function of (cid:96) inthe large black hole background for various valuesof z h . Here µ = 0 and red, green and blue curvescorrespond to z h = 1 .
2, 0 . . z h ) as a gravity background. InFigure 6, the length (cid:96) as a function of z ∗ for various values of the horizon radius is plotted.We observe that for every horizon radius there exists an (cid:96) max above which the connectedstring configuration does not exist. We further find that there are two solutions for agiven (cid:96) : one for small z ∗ (solid lines) and one for large z ∗ (dotted lines). As we will seeshortly, the one with smaller z ∗ corresponds to an actual minimum of the string actionwhereas the one with larger z ∗ corresponds to a saddle point.In Figure 7, we have shown the difference in free energy between the connected anddisconnected string solutions. In this Figure, solid and dotted lines correspond to smallerand larger branches of (cid:96) respectively. We observe that the former branch always has a lowerfree energy than the latter branch, indicating that it is a true minimum. However, we alsoobserve that ∆ F can be greater or less than zero depending on the value of (cid:96) . This suggestsa phase transition from a connected to a disconnected string solution as we increase the(QCD) string length (cid:96) . The string length at which ∆ F turns from negative to positivevalue defines the critical length (cid:96) crit . We find that (cid:96) crit increases with increasing z h . Thissuggests that for larger size black holes (small z h ), the connected string configurationconfines closer to the AdS boundary. The behaviour that (cid:96) crit decreases with temperatureis consistent with the physical expectation that at higher and higher temperatures theboundary meson state would eventually melt to a free quark and antiquark (deconfinedphase), a situation which is on the dual gravity side described by the disconnected stringconfiguration. Since for large separations, this disconnected string configuration which isindependent of separation length (cid:96) is more favourable, therefore the corresponding freeenergy of the qq pair is also independent of (cid:96) . It implies that the QCD string tension iszero and that there is no linear law confinement in the boundary theory dual to the largeblack hole phase.We now discuss the free energy of a qq pair in the small black hole phase (large z h ).We find that for a small black hole background, (cid:96) max does not exist and (cid:96) continuouslyincreases with z ∗ . This is shown in Figure 8. The string world sheet does not penetratedeep into the bulk and saturates near z (cid:39) . − , suggesting some kind of an “imagi- .2 0.4 0.6 0.8 1.0 1.2 z * Figure 8: (cid:96) as a function of z ∗ in the small blackhole background for various values of z h . Here µ =0 and red, green and blue curves correspond to z h =3 .
0, 3 . . - ℱ con T s L Figure 9: F con as a function of (cid:96) in the small blackhole background for various values of z h . Here µ =0 and red, green and blue curves correspond to z h =3 .
0, 3 . . (cid:96) also increases rapidly. At first sight this naively suggests that in the smallblack hole background, the qq pair is always connected by an open string and forms aconfined state. However, we need to be careful with this interpretation as we will showshortly. The regularized part of the corresponding free energy curve is shown in Figure 9.We find that F con is negative for small (cid:96) and increases linearly for large (cid:96) . In fact, it canbe shown analytically that F con ∝ − /(cid:96) for small (cid:96) exhibiting a Coulomb potential, and F con = σ s (cid:96) for large (cid:96) suggesting confinement [60]. Here σ s is the QCD string tension.Therefore, from the connected string configuration in the small black hole background weget the famous Cornell expression [82] F con T s L = − κ(cid:96) + σ s (cid:96) + . . . for the energy of a qq pair. By comparing the lattice QCD estimate of the string ten-sion, σ s ≈ / (2 . GeV [83], with our numerical results, we can further fix the valueof the open string tension T s in units of the AdS length scale L . By doing that we find T s L (cid:39) .
1. This small value of T s L , corresponding to a small ’t Hooft coupling, gener-ally indicates the breakdown of the classical gravity approximation and it suggests thatthe higher derivative α (cid:48) correction terms might become important. Although the smallvalue of T s L is certainly a drawback of our model, however, we also like to point out thatmost of the phenomenological bottom-up gauge/gravity models suffer from this ambiguity(see for example [58], based on the self-consistent model of [53,54] or [84,85], to name onlya few.), and other models which do exhibit a large T s L generally have additional scalingsymmetries in the dilaton potential (or additional parameters) which makes a large T s L arguable as it depends non-trivially on those additional scaling (as well as on the choiceof dilaton normalization) [30]. See [66] for further details on certain conceptual difficultiesassociated with the magnitude of T s L in bottom-up holographic models. Clearly it isan unsettled relevant question and more work is needed. Currently we do not have aresolution for this problem. Indeed, the size of T s L is only determined quantitativelyupon selecting a “stringy” (loop related) QCD observable. In practice, this is usuallyachieved by matching on the string tension entering the Wilson loop, or, in the presenceof dynamical quarks, the Polyakov-loop related free energy. If we refrain from matching o the precise QCD values, our qualitative findings stand as they are and are trustwor-thy. Keeping this ambiguity in mind, in the current work we will always express the freeenergy and entropy of the quark pair in (unspecified) units of T s L . This is reasonableas our main aim here is to find a qualitative picture for the free energy and entropy ofthe quark pair from holography. Such approach is not uncommon, notice that also in [32]thermodynamics of unquenched holographic QCD was discussed without actually fixingthe string tension. Though, it must be kept in mind that quantitative holographic QCDloop-related results will only be trustworthy at the expense of having these quantities anorder of magnitude smaller than expected from comparing with genuine QCD.It is clear from the linear dependence of F con on the inter-quark separation lengththat the Wilson loop of the dual boundary theory exhibits an area law, as one expectsfor a confined gauge theory, and that the boundary theory dual to the small black holephase seems to satisfy the typical properties of confinement. This behaviour of F con wasused in [60,61] to motivate that the small black hole phase is actually dual to confinement.However, as mentioned before, we need to be careful with the above interpretation.In particular, the connected string is not the only world sheet solution of the Nambu-Goto action and one needs to compare with the free energy of the disconnected stringas well. Generally, in the gauge/gravity correspondence, the confined phase is dual toa thermal-AdS space. In that case the upper limit in the integral of F discon is at ¯ z = ∞ , which makes F discon divergent. Therefore, in that case the connected string alwaysgives a lower free energy than the disconnected one. Consequently, one gets a linear lawconfinement for the qq pair from the connected string solution using the dual thermal-AdS background. However, with a black hole background, the disconnected strings extendfrom the asymptotic boundary to maximally the horizon at ¯ z = z h , which makes F discon finite (of course after removing the usual UV divergences), and hence its free energy couldbecome less than F con as well. For this reason, it is now important to compare F con and F discon when one uses black hole backgrounds (like we did in the large black hole phase inFigure 7) to get the true minimum of Nambu-Goto action and correspondingly to studythe properties of confinement and deconfinement.The comparison between F con and F discon in the small black hole background is shownin Figure 10, where ∆ F = F con − F discon is plotted as a function of (cid:96) for the same param-eter values as in Figure 9. We see that ∆ F can indeed be greater than zero, especially forlarge qq separation length. This implies that F discon is actually the true minimum for thestring action for large (cid:96) , and therefore, one should consider F discon to study the (correct)properties of the qq pair. Accordingly, since F discon is independent of (cid:96) , the string tensionis zero and hence there is no more linear law confinement in the boundary theory dual tosmall black hole phase. Therefore, although F con does indeed show an area law for theWilson loop, however, it is F discon that is relevant for large (cid:96) , not leading to an area law.Further details are presented in Figure 11, where flattening of inter-quark energy at large (cid:96) is explicitly shown.In Figure 11, the behaviour of the qq free energy as a function of (cid:96) for various tem-peratures is shown. For T < T crit (lines above the upper black line), we are in the smallblack hole phase; and for
T > T crit , (lines below the 2nd black line), we are in the largeblack hole phase. The two black lines correspond to T = T crit at which the small andthe large black hole phases (having different horizon radius) coexist. We find that thestring tension for T ≤ T crit , calculated from the slope of the linear part of F , is almosttemperature-independent and remains non-zero even at T = T crit , a result supported bylattice QCD as well [50, 51]. Moreover, for T > T crit the size of the linear part in F starts to decrease and so is the string tension. However, as opposed to its lattice QCD crit - - - - Δℱ T s L Figure 10: ∆ F = F con − F discon as a functionof (cid:96) in the small black hole background for variousvalues of z h . Here µ = 0 and red, green and bluecurves correspond to z h = 3 .
0, 3 . . ℓ crit T - ℱ T s L Figure 11: F as a function of (cid:96) for various values of T . Here µ = 0 and red, green, blue, brown, black,cyan and magenta curves correspond to T /T crit =0 .
6, 0 .
7, 0 .
8, 0 .
9, 1 .
0, 1 . . T has been indicated. In unitsGeV.counterpart, there exists a small discontinuity in the string tension at T = T crit . Thiscan again be traced back to the first order phase transition between the small and largeblack hole phases at T = T crit , where both phases coexist.For completeness, let us notice that according to Figure 11, the critical length (cid:96) crit atwhich the string tension vanishes in the specious-confined phase, decreases with increasingtemperature. A similar signal is obtained from the entanglement entropy using the samemodel [86].To appreciate the length scales (cid:96) crit in Figure 11, let us remind here the typicalscale of heavy quark bound states, like charmonia, having a binding size of the orderof 0 . (cid:39) . − [87]. In [51], the linear potential at T = 0 was considered up to (cid:96) (cid:39) σ (cid:39) − . This strongly suggests that the values of (cid:96) crit obtained here fortemperatures not too far from T crit , are well compatible with QCD phenomenologicallyrelevant length scales.We further like to add that these results are valid irrespective of the value of the chem-ical potential. For example, for finite µ < µ c we again find ∆ F > (cid:96) . Moreover,at a fixed temperature, (cid:96) crit is found to decrease with µ in the small black hole phase.Further evidence for the problem of correctly interpreting the small black hole phaseas the gravity dual of confined phase can be inferred from the Polyakov loop expectationvalue. As said before, and well known in the literature, the Polyakov loop acts as an orderparameter for the confinement/deconfinement phase transition, vanishing in the confinedphase and attaining a non-zero value in the deconfined phase. Therefore it is of impor-tance to find out how the expectation value of Polyakov loop behaves in the current model.The expectation value of the Polyakov loop can be calculated holographically fromthe heavy quark free energy. The regularized form of the latter (after removing the UV σ is the string tension at T = 0. ivergence) is given by F reg = F discon , (3.8)from which one can extract the Polyakov loop P = e − F reg /T . We see that P can be zero in confined phase when either T is zero or F reg is divergent.In the small black hole phase, none of these conditions is met. For example, T is finite forthe small black hole phase and, as discussed above, F reg does not have an IR divergence.Therefore, the Polyakov loop expectation value is strictly non-zero for the boundary the-ory dual to the small black hole phase and consequently this dual boundary phase is notexactly corresponding to a confined phase.We have followed the holograpic literature here and considered a single Polyakov loop,even in the presence of a non-vanishing chemical potential [32,57,88]. We should howevernotice that the latter breaks the charge conjugation symmetry C between the (unequalnumber of) quarks and anti-quarks, and as such the Polyakov loop expectation valuesfor an isolated quark or anti-quark are not supposed to be equal [89–91]. In the currentapproximation of the gauge-gravity duality to access loop quantities as the Polyakov loop,we are however unable to probe this C breaking since the Nambu-Goto action is blindto C . It would require to take into account stringy corrections to the classical Nambu-Goto worldsheet action (including back reaction corrections related to introducing eithera quark or antiquark located at the end point of the string considered dual to the loop), atask far beyond this and most –not to say all– other works on holographic QCD models.We also like to point out another important issue here. Although, P is non-zero inthe small black hole phase in a strict sense, however it is extremely small. For instance, P ∼ − at T = 0 . T c , and its magnitude is even smaller at lower temperatures.Moreover, it is also clear from Figure 10 that the value of (cid:96) crit at which ∆ F changesits sign, increases with decreasing temperature (or as z h increases). This suggest thatfor smaller and smaller temperature, the quark and antiquark form a bound state up tolarger distances. For example, (cid:96) crit at T = 0 . T c is an order of magnitude larger than at T = 1 . T c . For these reasons, keeping in mind various technical issues in interpreting thesmall black hole phase as the gravity dual of the confined phase, we may call this dualphase the specious-confined phase, to distinguish it from the “standard” confined phase.We will see in the following sections that the entropy of the qq pair as well as the speedof sound in this specious-confined phase is also rather similar to that of the lattice QCDconfined phase.Strangely, our specious-confined phase resembles the behaviour of full (unquenched)QCD at the level of non-local observables such as Wilson or Polyakov loops expectationvalues, while at the level of thermodynamics, it becomes closer to quenched QCD. In fullQCD, the Polyakov loop is not an exact order parameter because of the explicit breakingof the Z N symmetry due to the dynamical quarks in the fundamental representation.Moreover, quark pair creation prevents the linear behaviour of the confining potentialin full QCD, rather one observes a flattening (the so-called “string breaking”) at largervalues of (cid:96) [92], not unlike what we find here. Indeed, as shown in Figure 11, one canclearly observe the flattening of inter-quark energy for large (cid:96) . Moreover, just like in full(unquenched) QCD, the inter-quark energy in our holographic model also approaches atemperature dependent constant value at larger values of (cid:96) . This temperature dependentconstant value decreases with temperature (from specious-confined to deconfined), which gain is in line with full QCD results (see for example Figure 1 of [12]). Although we donot have a clear understanding where this peculiar “mixture” of quenched vs. unquenchedQCD-like behaviour is caused by precisely, it is not related to having a too small ’t Hooftcoupling (or small T s L ) that would cloud the dual gauge-gravity interpretation: as wenoticed before, the qualitative features of our models persist, independent on how oneultimately decides to choose by hand or fix on an external QCD variable the string tension,or similar quantity.Another possibility might be that it is related to a wrongful identification of the scalarfield φ with the dilaton, see discussion below eq. (2.16), but as discussed there, not makingthis identification would lead to a loss of all confining-related properties of our models. qq pair We now move on to discuss the thermal entropy S of a qq pair in the above specious-confined /deconfined phases. This entropy can be calculated from the qq free energy F via the relation, S = − ∂ F ∂T . (3.9)For black hole backgrounds, we have two choices for S , corresponding to the two differentbehaviours of F with respect to the qq separation length. For large separation, we have S discon ( (cid:96) > (cid:96) crit ) = − ∂ F discon ∂T . (3.10)On the other hand for small separation, we have S con ( (cid:96) < (cid:96) crit ) = − ∂ F con ∂T . (3.11)We will see that the two distinct behaviours of F as a function of the qq separation lengthprecisely capture the QCD results for the entropy in their respective regimes.We now discuss a few silent features of the holographic calculations for the qq entropyand compare with the lattice QCD results: • The variation of S con with respect to temperature in the small black hole phase isshown in Figure 12. Here, we have fixed the qq separation length to (cid:96) = 2 GeV − .The essential features of our analysis remain unchanged for other values of (cid:96) , and (cid:96) = 2 GeV − is just a particular choice. We find that in this specious-confined phasethe entropy initially varies slowly at low temperatures, and then increases rapidlytowards the critical temperature. It indicates a large amount of entropy associatedwith the qq pair near the critical temperature, as also observed in lattice QCD. Wesee that the holographic results for the qq entropy in this specious-confined phaseare qualitatively similar to those of lattice QCD. It also indicates the main differencecompared to [47], where the entropy in the confined phase was reported to be zero.We again like to point out that the non-zero entropy in the specious-confined phasehere arises precisely due to the fact that the dual gravity background is a (small)black hole, which depends on temperature. In the usual AdS/CFT correspondence,the confined phase is generally dual to thermal-AdS (without horizon and temper-ature) and therefore the entropy of quark pair is inherently zero in those confinedphases. • The entropy of the qq pair as a function of temperature in the large black hole phaseat large separation is shown in Figure 13. For this purpose we use eq. (3.10) for the .85 0.90 0.95 1.00 TT crit T ST s L Figure 12:
Entropy of the qq pair as a functionof temperature in the specious-confined phase forvarious values of the chemical potential µ . Herered, green, blue and brown curves correspond to µ = 0 .
10, 0 .
15, 0 .
20 and 0 .
25 respectively. In unitsGeV. TT crit T ST s L Figure 13:
Entropy of the qq pair as a functionof temperature in the deconfined phase for variousvalues of the chemical potential µ . Here red, green,blue and brown curves correspond to µ = 0 . .
15, 0 .
20 and 0 .
25 respectively. In units GeV.entropy. We observe that similarly to the specious-confined phase a large amount ofentropy is associated with the qq pair near the critical temperature. Similar resultsin the deconfined phase were found in [47] using the improved holographic modelof [26, 27] . We further find that the higher temperature asymptotics of the qq entropy in our model is similar to those of [47]. In particular, for T (cid:38) T c , wefind a tendency of a rise of T S with temperature as also observed in lattice QCD.Moreover, our analysis further predicts similar asymptotic behavior of
T S in thepresence of chemical potential too. • Another important lattice QCD result which the holographic model considered herecorrectly describes, is the increase in the entropy of the qq pair as a function ofdistance between them, see Figure 14. Here we have shown results in deconfinedphase for three different temperatures. We see that for each case, S increases with (cid:96) .Moreover, for large (cid:96) , S saturates to a constant value and becomes independent ofit . This is due to the fact that the disconnected string configuration has lower freeenergy at large separations, while it is independent of (cid:96) . Therefore, the correspondingentropy is also independent of (cid:96) . We see that these results qualitatively match withthe results predicted by lattice QCD (shown in Figure 1). However, as opposed to thelatter, the entropy here does not smoothly go to saturation. There is a discontinuityin the entropy at (cid:96) crit (denoted by dotted lines in Figure 14). This discontinuity inthe entropy arises precisely due to the first order transition between the differentstring configurations at (cid:96) crit . • In Figure 15, the variation of the entropy as a function of the inter-quark distance forvarious values of chemical potential is shown. We do not have lattice results here to The temperature dependence of the entropy of a qq pair and of the string tension were also discussed ina dual phenomenological model in [85]. Unlike for our current model, the metric of [85] was chosen by handand it displays no clear phase transition picture on the gravity side. Rather it relied on an effective potentialapproach. In order to make it more readable, the magnitude of S has been suppressed by a factor of 2 in (cid:96) > (cid:96) crit region of Figure 14. crit T ST s L Figure 14:
Entropy of the qq pair as a func-tion of distance in the deconfined phase for varioustemperatures. Here µ = 0 and red, green and bluecurves correspond to T /T crit = 1 .
1, 1 . . ℓ crit T ST s L Figure 15:
Entropy of the qq pair as a function ofdistance in the deconfined phase for various chemi-cal potentials. Here T = 1 . T crit and red, green,blue and brown curves correspond to µ = 0 . .
15, 0 .
20 and 0 .
25 respectively. In units GeV.compare with. In this regard, these entropic results can be thought of as a originalprediction from holography. We find that essential features of the entropy remainthe same even with a chemical potential present. The entropy again saturates toa constant value at large separations. Also, (cid:96) crit increases with chemical potential.This suggest that the distance around which entropy saturates, increases with thechemical potential.We end this section by summarising our main results obtained so far. Till now we showedthat by considering the form of A ( z ) as in eq. (3.1), a small/large black hole phase transi-tion in the gravity side appears which on the dual boundary side corresponds to a specious-confined /deconfined phase. We showed that although there is always a ∪ -shape connectedstring configuration in the specious-confined phase, it is however the disconnected stringconfiguration which is more favourable for large qq separation length. Interestingly, wefind that in spite of the reported differences between the specious-confined and standardconfined phase, the entropy of the quark pair in the former phase is qualitatively similarto lattice QCD confined phase results. Similarly, in the large black hole phase when thetemperature is sufficiently high, it is the disconnected string configuration which is morefavourable at larger qq separation. The two disconnected strings describe the dissociationof a bound state into a quark and antiquark, thereby corresponding to the deconfinedphase in the dual boundary theory. The small/large black hole phases therefore cor-respond to specious-confinement /deconfinement phases, and the corresponding criticaltemperature is found to decrease with chemical potential. Moreover, the entropy of the qq pair in this phase is in qualitative agreement with lattice QCD results. It is instructive to also study the entropy by considering a different form of A ( z ). In par-ticular, in order to check the universal nature of the results presented for the qq entropy,especially for the deconfined phase, we will now study the entropy in a different holo-graphic model where the gravity dual genuinely describes the confinement/deconfinement hases in the boundary theory. For this purpose we need to consider a different gravitysolution. As mentioned earlier, the same can be achieved by considering different form of A ( z ).Let us now consider another simple form of A ( z ), such as A ( z ) = A ( z ) = − ¯ az . (4.1)It is again straightforward to see that A ( z ) → z = 0, imply-ing that the bulk spacetime asymptotes to AdS at the boundary. Near the boundary V ( z ) | z → = − /L = 2Λ. Further, it can be explicitly checked that V ( z ) has the sameasymptotic form for both A ( z ) and A ( z ). The parameter ¯ a = c/ (cid:39) .
145 in eq. (4.1)is fixed by demanding the critical temperature of the Hawking/Page (or the dual confine-ment/deconfinement) phase transition to be around 270 MeV at zero chemical potential.Importantly, for A ( z ) = − ¯ az , the equation for dilaton field in (2.14) can be solvedexplicitly φ ( z ) = z (cid:112) a (3 + 2¯ az ) + 3 (cid:114)
32 sinh − (cid:20)(cid:114) a z (cid:21) . (4.2)Its near boundary expansion is φ ( z ) = 6 √ ¯ az + 2 / a / z + . . . (4.3)and in terms of φ ( z ) the dilaton potential can again be written as V ( φ ) = − L + ∆(∆ − φ ( z ) + . . . , ∆ = 3 (4.4)We repeat here once more that the phenomenological bottom-up holographic models weare considering are based on the (implicit) assumption that these phenomenological mod-els are derivable from a consistent truncation of a higher dimensional string theory. There-fore, the dilaton might be expected to be massless in a 5D truncated gravity theory (likeours) as well. Though, as frequently done, see e.g. [93–96], the scalar dilaton can be as-sociated to the dual scalar glueball state [97] described by F µν , which is massive in QCD(or even pure Yang-Mills gauge theory). In fact, whatever the dual of the dilaton wouldbe, QCD(-like) theories are not expected to have any massless physical degrees of freedomin their spectrum, at least away from the chiral limit. As such, it is no surprise that, toour knowledge, phenomenological holographic QCD models are frequently incorporatinga massive dilaton as an easy access to a massive scalar glueball, let us refer to [96] for arecent discussion and relevant references. Another example with a massive dilaton is [98].Seminal works using bottom-up massive dilatons are [53, 54], there more in relation toQCD(-like) thermodynamics than to glueball spectra. The mass was related to the (UV)anomalous dimension of F µν . One of the considered dilaton potentials in [53, 54] wasa simple exponential, as also discussed in [68], which evidently include a φ term uponexpansion. A more refined version of the models considered here in our paper would beto ensure that the quadratic term in the expansion of V ( φ ) is related to the anomalousdimension of F µν instead of to the rather simple (and crude) value of 3 reported now.For completeness, also the V ( φ ) of Case 1 gives rise a dilaton mass term in the potentialthat is consistent with the BF bound for a scalar degree of freedom, see eq. (3.2). In fact,up to order φ , the dilaton potentials of both Cases are identical. z h Figure 16: T as a function of z h for various valuesof the chemical potential µ . Here red, green, blue,brown, cyan and magenta curves correspond to µ =0, 0 .
2, 0 .
4, 0 .
5, 0 . .
673 respectively. In unitsGeV. - - - - × - × - Figure 17: F as a function of T for various valuesof the chemical potential µ . Here red, green, blue,brown and cyan curves correspond to µ = 0, 0 . .
4, 0 .
5, 0 . .
673 respectively. In units GeV.
The thermodynamic results of the gravity solution with A ( z ) as in eq. (4.1) are shownin Figures 16 and 17. In this case no small to large black hole phase transition takesplace. There are now only two branches in the ( T, z h ) plane. The branch with negativeslope (small z h ) is stable whereas the branch with positive slope (large z h ) is unstable.The appearance of this unstable branch however depends on the value of the chemicalpotential and in particular it ceases to exist for higher chemical potential. This defines acritical chemical potential µ c = 0 .
673 GeV. Moreover, the black hole solution does notexist below a certain minimal temperature T min . This suggests a phase transition fromblack hole to thermal-AdS as we decrease the Hawking temperature. The phase transi-tion can be observed from the free energy behaviour, shown in Figure 17, where the samenormalization as in the previous section is used. We see that the free energy is positive forthe unstable branch and becomes negative after some critical temperature T crit along thestable branch, implying a first order Hawking/Page phase transition from thermal-AdSto AdS black hole as the temperature increases. The overall dependence of T crit on µ isthe same as in Figure 5, however with larger µ c = 0 .
673 GeV .As we will show in the next subsection, this thermal-AdS/black hole phase transition inthe gravity side corresponds to the standard confinement/deconfinement phase transitionon the dual boundary side. In particular, the thermal-AdS phase corresponds to confine-ment whereas the black hole phase corresponds to deconfinement. Correspondingly, T crit defines the dual transition temperature. By construction, our result T crit = 0 .
264 GeVat zero chemical potential agrees with its lattice estimate. Moreover, we find that thetransition temperature decreases with the chemical potential which is again in qualitativeagreement with the lattice results. A similar kind of Hawking/Page phase transition with a planar horizon has been reported recently in [99]. .2 Free energy of the qq pair We now discuss the free energy of the qq pair with A ( z ) as in eq. (4.1). Again, depend-ing upon the background geometry, there can be both connected as well as disconnectedstring solutions. The equations for the connected (eqs. (3.5) and (3.6)) and disconnectedstring (eq. (3.7)) remain the same, except A ( z ) is replaced by eq. (4.1). z * Figure 18: (cid:96) as a function of z ∗ in the thermal-AdS background. In units GeV. - ℱ con T s L Figure 19: F con as a function of (cid:96) in the thermal-AdS background. In units GeV.Let us first examine the free energy in the thermal-AdS background. The results areshown in Figures 18 and 19. Again, just as for the small black hole background of theprevious section, an “imaginary wall” appears and the connected string world sheet doesnot go deep into the bulk and tries to stay away from the singularity. Importantly, thefree energy of the qq pair is again found to be of the Cornell type, F con = − κ(cid:96) + σ s (cid:96) . This suggests that the qq pair is connected by the string and forms a confined state inthe dual boundary theory. However, as opposed to the small black hole background, here F con is actually always the true minimum of the Nambu-Goto action, which can be easilyverified by noticing that the disconnected string solution has an additional IR divergenceand therefore is never a relevant solution of Nambu-Goto action. In order to explicitlysee this, let us first note the IR (large z ) expansions of φ ( z ) and A s ( z ), φ ( z ) = √ az + 3 √
62 ln z + . . . , (4.5) A s ( z ) = A ( z ) + 1 √ φ ( z ) = − ¯ az + 1 √ (cid:20) √ az + 3 √
62 ln z + . . . (cid:21) , = 32 ln z + . . . . (4.6)Substituting these expressions into F discon expression, we have F discon = L π(cid:96) s (cid:90) z h = ∞ dz e A s ( z ) z = L π(cid:96) s (cid:90) z h = ∞ dz (cid:2) z + . . . (cid:3) (4.7) Again, one can fix the value of open string tension T s by comparing the lattice QCD result σ s ≈ / (2 . GeV with our numerical results. By doing that we find T s L (cid:39) . here z h = ∞ for thermal-AdS, which makes F discon divergent. Therefore, F con is indeedthe relevant quantity for the qq free energy and since F con shows linear confinementfor large qq separation length it implies that the boundary theory dual to thermal-AdScorresponds to confinement. Moreover, the Polyakov loop expectation value is now zero bydefault. Of course, this discussion is just a straightforward generalization of known factsin the soft wall AdS/QCD models, however, now with a consistent solution of Einstein-Maxwell-dilaton gravity, also displaying the area law for the Wilson loop, a feature missedby the original soft wall model [36]. z * Figure 20: (cid:96) as a function of z ∗ in the AdS blackhole background for various values of z h . Here µ =0 and red, green and blue curves correspond to z h =1 .
5, 1 . . - Δℱ T s L Figure 21: ∆ F = F con − F discon as a function of (cid:96) in the AdS black hole background for various valuesof z h . Here µ = 0 and red, green and blue curvescorrespond to z h = 1 .
5, 1 . . qq free energy in the AdS black hole background are quite similar tothose of the large black hole background of the previous section, and therefore we will bevery brief here. The results are summarized in Figures 20 and 21. For each temperaturewe again find an (cid:96) max above which the connected string solution does not exist, and aphase transition from connected to disconnected string solution occurs at (cid:96) crit < (cid:96) max as we increase (cid:96) . Once again, (cid:96) crit is found to decrease with temperature suggesting thedeconfined nature of the qq pair at higher and higher temperatures. With the chemicalpotential turned on, we find that (cid:96) crit shows no dependence on µ for temperatures near T c , and only for high temperatures T (cid:38) T c , (cid:96) crit shows a mild dependence on µ . qq pair For the thermal AdS background, F con is the only relevant solution which is independentof temperature. Subsequently, the entropy of the quark pair in the dual confined phase isalso zero, S confined = 0 (4.8)This is in sharp contrast to the specious-confined phase of the previous section, which wasshown to be dual to a small black hole phase, where the entropy was found to enhancewith temperature and to grow to a large magnitude near the transition temperature. crit T ST s L Figure 22:
Entropy of the qq pair as a func-tion of distance in the deconfined phase for varioustemperatures. Here µ = 0 and red, green and bluecurves correspond to T /T crit = 1 .
1, 1 . . TT crit T ST s L Figure 23:
Entropy of the qq pair as a functionof temperature in the deconfined phase for variousvalues of chemical potential µ . Here red, green,blue and brown curves correspond to µ = 0, 0 . . . F con for small separation length whereas for large separation lengthit is given in terms of F discon . The main results are shown in Figures 22 and 23 . Theessential features of the entropy are again in agreement with lattice QCD. In particular,it can be observed from Figure 22 that the entropy is an increasing function of separa-tion length, which saturates to a constant value at large separations. We have testedseveral other forms of the scale factor A ( z ) as well (apart from A ( z ) and A ( z )) andfound similar results for the deconfined phase. At this point we should emphasize thatclose results were already described in other holographic models too, both in top-down aswell as in bottom-up constructions. For instance, the entropy of a quark pair in N = 4supersymmetric Yang-Mills theory and in bottom-up improved holographic models wasshown to increase with inter-quark distance in [14, 47]. This result therefore seems tobe a universal feature of the dual deconfined phase in holographic theories. Moreover,our analysis further suggests that the same behaviour occurs in the presence of chemicalpotential too. It would be interesting to see whether lattice QCD could predict analogousresults in the latter case, at least in the parametric region of small chemical potentialwhich is amenable to some extent to Euclidean lattice simulations. An important thermodynamical observable that is sensitive to the phase transition andattracted a lot of interest of late, both in lattice QCD and holographic QCD, is the speedof sound C s [53, 100–104]. In conformal theories, the value of speed of sound is fixed C s = 1 / C s depends non-trivially on the temperature. Indeed, lattice QCD has predicted arapid decrease in the magnitude of C s near the deconfinement transition temperature,which approaches its conformal value from below at very high temperatures [100, 101]. The magnitude of S has been suppressed by a factor of 5 in (cid:96) > (cid:96) crit region of Figure 22 in order to makeit more readable. herefore it is of great interest to see how C s behaves in our specious-confined /deconfinedand confined/deconfined phases. TT crit C s Figure 24: C s as a function of temperature forvarious values of µ . Here red, green, blue and browncurves correspond to µ = 0 .
0, 0 .
1, 0 . . C s = 1 /
3. Using A ( z ) = A ( z ). TT crit C s Figure 25: C s as a function of temperature for var-ious values of µ . Here red, green, blue and browncurves correspond to µ = 0 .
0, 0 .
1, 0 . . C s = 1 /
3. Using A ( z ) = A ( z ).The general expression of C s in the grand canonical ensemble is given by, C s = S BH T (cid:0) ∂S BH ∂T (cid:1) µ + µ (cid:0) ∂ρ∂T (cid:1) µ . (5.1)where ρ is the charge density of the boundary field theory. It can be extracted from theasymptotic expansion of A t and can be expressed in terms of µ using the gauge/gravityduality mapping. Using eqs. (2.12), (2.13) and (2.14), we get ρ = µ (cid:82) z h dx xe cx = µce cz h − A ( z ) = A ( z ). Our numerical results for C s as afunction of temperature for various values of µ are shown in Figure 24. We find that C s sharply decreases near the specious-confined /deconfined critical temperature. In particu-lar, C s decreases near the critical temperature from the specious-confined phase side too.These results are again quite similar to unquenched lattice QCD results, further advocat-ing a close relation between specious-confined and genuine confined phases, although thatrecent lattice results rather predict a smooth minimum of the speed of sound near thetransition temperature, instead of the cusp-like variation as found in our model [105,106].Moreover, we find that the effect of µ is maximal around T crit and shows a mild depen-dence away from T crit . In particular, higher values of µ try to suppress C s near T crit .However, at extremely low and high temperatures, C s approaches a µ independent con-stant value.The results with A ( z ) = A ( z ) are shown in Figure 25. Here too C s shows a sharpdecrease near T crit . The temperature dependence of C s in the deconfined phase is quitesimilar to the previous case, however now, since there is no notion of temperature, wecannot study the temperature dependence of C s in the confined phase. Again, higher alues of µ try to suppress C s near T crit and at very high temperature C s approaches a µ independent constant value. This seems to be a universal feature of the boundary gaugetheory.It is important to investigate the high temperature behaviour of C s as it is generallybelieved that QCD becomes conformal at high temperatures. In our setup, the hightemperature limit of C s can be obtained by doing a series expansion around z h = 0. Inthis limit, we get C s = 13 + 8 z h A (cid:48) (0)15 + . . . = 13 + 8 A (cid:48) (0)15 πT + . . . (5.3)where in the last equation we have used T = 1 / ( πz h ). Since A (cid:48) (0) is negative for both A ( z ) = A ( z ) and A ( z ) = A ( z ) (and for that matter for any A ( z ) which leads to an arealaw of the Wilson loop in the boundary theory [27]), it suggests that in our models toothe speed of sound approaches its conformal value from below at large temperatures. Wealso have numerically checked for sufficiently large temperatures that C s is always lessthan or equal to 1 /
3. This can also be explicitly seen from Figures 24 and 25. Further, forvery high temperatures, the effects of chemical potential only appear at the order T − . In this paper, we used the gauge/gravity duality to investigate QCD thermodynamics inrelation to its lattice version. We considered an Einstein-Maxwell-dilaton gravity modelto study the entropy of a qq pair and the speed of sound in the boundary theory. Wefirst expressed the gravity solution in terms of a scale function A ( z ) and then consideredtwo different profiles for it, each of which led to various kinds of phase transitions inthe gravity side. For A ( z ) = A ( z ), we found a first order small/large black hole phasetransition which on the dual boundary side corresponds to a specious-confined /deconfinedphase transition whereas for A ( z ) = A ( z ) we found a Hawking/Page phase transitionwhich corresponds to the standard confined/deconfined phase transition. In both casesthe critical temperature is found to decrease with the chemical potential. We discussedthat although this specious-confined phase does not strictly correspond to the standardconfined phase of quenched QCD, it does exhibit many properties which are quite simi-lar to it and even more alike unquenched QCD, making it an interesting self-consistentholographic model in se. Importantly this specious-confined phase, as opposed to thestandard confined phase, has a notion of temperature which allows us to study thermalproperties of various observables in the confined phase. We studied the free energy andentropy of a qq pair and showed that our holographic model qualitatively describes theknown lattice QCD results. In particular, one of the main results of our holographicmodel is that it predicts a sharp rise in the entropy of a qq pair near the deconfinementtransition temperature that can be seen from the specious-confined phase side too. Wefurther provided a holographic estimate for the qq entropy when a chemical potential isswitched on, which hopefully can be compared to future lattice QCD predictions. For thestandard confined phase (dual to thermal-AdS) the entropy of the qq pair is inherentlyzero. The properties of the deconfined phase are similar for both A ( z ) and A ( z ) and toother holographic models studied in the literature. We also probed the speed of sound inour constructed specious-confined /deconfined phases and again report results which arequalitatively similar to lattice QCD. e conclude this paper by suggesting some problems which would be interesting toinvestigate in the future. The most pertinent one would be to investigate the connectionbetween entanglement entropy and the QCD phase diagram. A few works have appearedin this direction especially in the soft wall models [107–111], however not much has beendiscussed in self-consistent holographic QCD models like the one presented here. Anotherinteresting problem would be to investigate the effects of a magnetic field on the entropyof a qq pair. As it is by now well known the confinement/deconfinement transition tem-perature is quite sensitive to the value of magnetic field and one might think that it maythus also affect the qq entropy near the transition temperature. This question is importantboth from lattice QCD and from holographic point of view. In lattice QCD, it might givenon-trivial evidence in favour of the influence of a magnetic field on quarkonium suppres-sion, and in holography, it might provide another scenario where testable and qualitativepredictions from the gauge/gravity duality are achievable. However the introduction ofa magnetic field will necessarily introduce additional equations on the gravitational side,coming from off-diagonal components of the Einstein equations, for which simple closedanalytic results might no longer be obtainable, and one might have to turn to numericallyconstructed metrics or employ a perturbative expansion in some small parameter(s). Weleave these and other issues for future work. Acknowledgments
It is a pleasure to thank L. Palhares and O. Andreev for useful discussions. S. M. also likesto thank S. Sugimoto for several helpful discussions and clarifications. We are grateful toO. Kaczmarek for the permission to take the Figures 1 and 2 from [12]. We are grateful forvarious useful comments made by the anonymous referee. The work of S. M. is supportedby a PDM grant of KU Leuven.
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