Cross-over versus first-order phase transition in holographic gravity-single-dilaton models of QCD thermodynamics
CCross-over versus first-order phase transition in holographicgravity–single-dilaton models of QCD thermodynamics
R. Yaresko, J. Knaute, B. K¨ampfer
Helmholtz-Zentrum Dresden-Rossendorf,POB 51 01 19, 01314 Dresden, Germany andTU Dresden, Institut f¨ur Theoretische Physik, 01062 Dresden, Germany
Abstract
A dilaton potential is adjusted to recently confirmed lattice QCD thermodynamics data in thetemperature range (0 . . . . . T c where T c = 155 MeV is the pseudo-critical temperature. Theemployed holographic model is based on a gravity–single-field dilaton dual. We discuss conditionsfor enforcing (for the pure gluon plasma) or avoiding (for the QCD quark-gluon plasma) a first-orderphase transition, but still keeping a softest point (minimum of sound velocity). PACS numbers: 11.25.Tq, 47.17.+e, 05.70.Ce, 12.38.Mh, 21.65.Mn a r X i v : . [ h e p - ph ] M a r . INTRODUCTION The celebrated AdS/CFT correspondence [1] has sparked a large number of dedicated in-vestigations of strongly coupled systems (cf. [2] for recent surveys). A particular field ofapplication is provided by the strong coupling nature of QCD at low momentum/energyscales. While employing 5d Einstein gravity in the dual description is strictly justified onlyin the large N c and large t’Hooft coupling limits of the boundary theory, which ensuresuppression of loop and stringy corrections to classical gravitation theory, by such meansnevertheless one could study models which are expected to exhibit a behavior resemblantof QCD. The aim is then often to understand, on a qualitative level, phenomena which arehardly accessible in the 4d quantum field theory. A prominent example is given by real-time phenomena, e.g. within QCD. Other phenomena, such as the hadron spectrum or theequation of state, are accessible by lattice QCD calculations - but here one would like tounderstand qualitatively the emerging numerical results by means of transparent models.Many facets of the QCD equation of state are fairly known by now, both for the physicalparameter section and for various limits of parameters (e.g. quark masses, dimension ofthe gauge group, flavor number, adjoint representations of quarks etc.). This statementapplies only for finite-temperature ( T ) QCD at zero baryo-chemical potential ( µ ). However,in relativistic heavy ion collisions, the bulk of excited matter has µ >
0, as inferred fromthe analysis of hadron abundancies [3]. The knowledge of the QCD equation of state is,therefore, presently incomplete (in particular beyond the range accessible by the µ/T (cid:28) T − µ phase diagram where the cross-over turns in afirst-order phase transition. Mainly based on universality arguments, a multitude of modelshave been employed to locate the critical point [4, 5], but also more directly QCD anchoredapproaches, e.g. Dyson-Schwinger equations as integral formulation of QCD, have been used[6]. Parallel to the theoretical attempts, also special experimental searches are conducted,e.g. the beam energy scan at RHIC [7].Coming back to options for modeling a phase diagram similar to QCD with the conjecturedcritical point, we mention [8], where in a holographic model, including gravity, a dilatonfield and a U(1) gauge field, the possibility of such a realisation has been demonstrated.The set-up of [8] is based on a dilation potential, which features qualitatively the equation2f state at µ = 0, supplemented by a dynamical strength function, which is adjusted tothe quark number susceptibility, again at µ = 0. While in the infinitely heavy quark mass( m q → ∞ ) limit of QCD, which becomes then a pure Yang-Mills theory, the equation ofstate is known since some time [9] and has been confirmed by high-precision lattice QCDsimulations [10], the status of QCD with physical quark masses has been settled only veryrecently. After refinements in the lattice discretization schemes and actions and continuumextrapolations the results of two independent collaborations [11, 12] became consistent.Given this new situation and having in mind e.g. an application in the spirit of [8] to the QCDphase diagram modeling, one should seek for an appropriate dilaton potential, reproducingsufficiently accurately the by now known QCD equation of state at µ = 0. This is the aimof the present note. We have hereby the attitude to take the AdS/CFT dictionary literally,i.e. translate, without corrections due to N c = 3 or finite coupling, the 5d Riemann metricinto 4d energy-momentum tensor components (or correlators) and vice versa. II. ADJUSTING A DILATON POTENTIAL At µ = 0, the equation of state, in parametric form, follows from [13] LT ( φ H ) = V ( φ H ) πV ( φ ) exp (cid:16) A ( φ ) + (cid:90) φ H φ dφ (cid:104) X + 23 X (cid:105)(cid:17) , (1) G s ( φ H ) = 14 exp (cid:16) A ( φ ) + 34 (cid:90) φ H φ dφ X (cid:17) , (2)for entropy density s and temperature T , where the scalar function X ( φ ; φ H ) [14] is deter-mined by the system (a prime means a derivative w.r.t. φ ) X (cid:48) = − (cid:16) Y − X (cid:17)(cid:16) X V (cid:48) V (cid:17) , (3) Y (cid:48) = − (cid:16) Y − X (cid:17) YX , (4)which is integrated from the horizon φ H − (cid:15) to the boundary φ with initial conditions X ( φ H − (cid:15) ) = − V (cid:48) ( φ H ) V ( φ H ) + O ( (cid:15) ) , (5) Y ( φ H − (cid:15) ) = − X ( φ H − (cid:15) ) (cid:15) + O ( (cid:15) ) , (6)and (cid:15) →
0. The quantity A ( φ ) encodes the near-boundary behavior of the model. Weassume L V ( φ ) ≈ −
12 + L M φ for φ → φ = 0 which results in A ( φ ) = log φ ∆ − , wherebywe have set L Λ = 1 [13] and, as usual, L M = ∆(∆ − < ∆ < s and T , the pressure follows as p ( φ H ) = (cid:90) φ H ∞ d ˜ φ H dT ( ˜ φ H ) d ˜ φ H s ( ˜ φ H ) , (7)where p ( ∞ ) = 0 holds if V (cid:48) /V < (cid:112) / V (cid:48) /V | φ →∞ → const, corresponding to a”good” singularity at φ = ∞ [14]. We consider only such cases.Besides of a proper adjustment of the dilaton potential V ( φ ) to the equation of state, themodel parameters G /L and L must be fitted, too. Since a direct mapping procedure ofan input equation of state to the potential is not at our disposal, we use as trial ansatz v D ( φ ) ≡ V (cid:48) D V D = − L M φ + s φ for φ ≤ φ m , (cid:16) t tanh( t φ − t ) + t (cid:17)(cid:16) − b cosh( b φ − b ) (cid:17) for φ ≥ φ m , (8)(demanding differentiability of v D at φ m fixes L M and s ) and findfit to φ m t t t t b b b G /L v s s/T LT c = 1 . v s from [11]) or LT c = 0 . s/T from [11]). This ansatzobeys the Chamblin-Reall IR behavior L V ( φ → ∞ ) ∼ e ( t + t ) φ . The approach belongs to asimilar class of holographic models as the model class B in [15]: it has no confinement in thesense of [14] for t + t < (cid:112) / ζ/s as found in the present setting. To set a scale,we determine T c in the holographic model by the inflection point of s/T as a function of T , and T c = 155 MeV [11] is used in the lattice QCD data [11]. The resulting velocity ofsound squared, v s = d log Td log s = d log Tdφ H (cid:0) d log sdφ H (cid:1) − , the scaled entropy density, s/T , the scaledpressure, p/T , and the scaled interaction measure I/T = ( sT − p ) /T are exhibited inFig. 1 together with the lattice QCD data [11, 12]. The solid blue (dotted red) curves areour optimum fits of v s ( s/T ) with the parameters of (9). Circles depict the respectivequantities at T c . One observes that the softest point, i.e. the minimum of v s as a functionof T /T c , is slightly below unity (see upper left panel in Fig. 1) and the maximum of theinteraction measure is a little bit up-shifted in comparison with the lattice QCD results (see4ower right panel in Fig. 1). One observes also some other minor imperfections of our fits,in particular at the lowest temperatures covered by the lattice QCD data, and also at largetemperatures for the interaction measure. T/T c v s T/T c s / T T/T c p / T T/T c I / T FIG. 1: Velocity of sound squared v s (upper left panel), scaled entropy density s/T (upper rightpanel), scaled pressure p/T (lower left panel), and scaled interaction measure I/T (lower rightpanel) as functions of T /T c . Solid blue curves: fit to v s data, dashed red curves: fit to s/T data,circles: position of thermodynamic quantities at T c . Lattice QCD data: black plusses from [11],green crosses from [12]; for both sets we use the pseudo-critical temperature T c = 155 MeV. In the present setting, the ratio of shear viscosity to entropy density is η/s = 1 / (4 π ) [16], asusual for the Hilbert action on the gravity side [17, 18]. The ratio of bulk to shear viscositycan be calculated via the Eling-Oz formula [18] ζη (cid:12)(cid:12)(cid:12) φ H = (cid:16) d log sdφ H (cid:17) − = (cid:16) v s d log Tdφ H (cid:17) − , (10)5r, equivalently [19], via the Gubser-Pufu-Rocha formula [17] ζη (cid:12)(cid:12)(cid:12) φ H = V (cid:48) ( φ H ) V ( φ H ) | h ( φ ) | , (11)where h ( φ ) is extracted from solving the perturbation equation h (cid:48)(cid:48) + 1 X (cid:16) Y − X (cid:17)(cid:16) X V (cid:48) V (cid:17) h (cid:48) − YX (cid:16) Y − X (cid:17)(cid:16) X V (cid:48) V (cid:17) h = 0 , (12)with initial conditions h ( φ H − (cid:15) ) = 1 and h (cid:48) ( φ H − (cid:15) ) = 0 for (cid:15) →
0. The result is
T/T c ζ / T T/T c ζ / η FIG. 2: Scaled bulk viscosity, ζ/T , (left) and ratio of bulk to shear viscosity, ζ/η , (right) as afunction of T /T c . Line codes as in Fig. 1. exhibited in Fig. 2. Remarkable is the reduction of ζ/η by 50% at T c in comparison withthe SU(3) gluon plasma (YM) considered in [20]. To understand this difference, recall theadiabatic approximation of [13]: X ( φ ) ≈ − V (cid:48) ( φ ) V ( φ ) . In this approximation, the non-local termin (11) becomes unity, h = 1 (since the coefficient of h in (12) vanishes, see also [21]),and, comparing the values of V (cid:48) /V ≈ . .
8) for the QGP (pure glue, see below) at T c we find the ratio (cid:0) ( ζ/η ) QGP / ( ζ/η ) Y M (cid:1) ( T c ) ≈
56% (cf. also [15, 22] for recent holographiccalculations of transport coefficients).On the other hand, for ζ/T , the situation is reversed: at T c , the QGP value is 50% largerin comparison to the gluon plasma case; the peak of ζ/T is located at a larger value T /T c ≈ .
3. This difference between ζ/η and ζ/T for QGP and for the gluon plasma canbe attributed to the different number of degrees of freedom as reflected by the scaled entropydensity s/T . 6 II. CROSS-OVER VS. FIRST-ORDER PHASE TRANSITION
Remarkably, the ansatz (8) for the dilaton potential is the same as used in [20] to describethe SU(3) gluon plasma, which displays a first-order phase transition. (This is actually notso surprising, as [13] has demonstrated that a two-parameter ansatz for the potential allowseither for a cross-over, or a first-order phase transition or a second-order transition, depend-ing on the choice of the parameters. Also, [15] uses a unique ansatz with two parameter setsto arrive at a first-order phase transition or a cross-over.) To elucidate the origin of such adifference we exhibit in Fig. 3 a few relevant quantities of both optimized models. φ H φ H FIG. 3: V (cid:48) /V (solid black curves), v s (dashed red curves, dot-dashed curves are for the adiabaticapproximation) and 0 . T /T c (blue dotted curves) as functions of φ H . Left panel: for the puregluon plasma (the grey band covers the unstable and metastable regions), right panel: for QCDquark-gluon plasma (fit to v s ). Vertical dotted lines bracket the fit range to the lattice data. For the pure gluon plasma (left panel) the quantity V (cid:48) /V has a first maximum of about 0 . φ H ≈ .
1. In the adiabatic approximation [13] the velocity of sound squared is v s ≈ − (cid:16) V (cid:48) V (cid:17) + . . . , (13)i.e. a local maximum (minimum) of V (cid:48) /V is related to a local minimum (maximum) of v s .If V (cid:48) /V is sufficiently large, v s can go to zero. In fact, the adiabatic approximation is quiteaccurate (compare the red dot-dashed curve (adiabatic approximation) and the dashed curve(exact result) in Fig. 3 - left panel). That implies, lifting V (cid:48) /V sufficiently causes a first-order7hase transition, here signalled by v s = 0. The entropy density s ( φ H ) is a monotonouslydropping function, as holds true for the considered examples (and is assumed to hold ingeneral in the thermodynamically stable phase, see [14]). Hence, v s = 0 corresponds to anextremum of T ( φ H ), which is a minimum (maximum) if dv s /dφ H < dv s /dφ H > V (cid:48) /V is adjusted such that v s ( φ H ) becomes negative in some φ H interval and then, forlarger φ H , rises to become positive again, the local minimum of T ( φ H ) is followed by a localmaximum. (These extrema are very shallow in the left panel of Fig. 3 and hardly visible onthe used scale.) Such a behavior of T ( φ H ) leads in turn to the usual loop structure of p ( T ),characteristic of a first-order phase transition. For the case at hand, the pressure is alwayspositive. (In contrast, the IHQCD model [14] has one global minimum of T ( φ H ) which givesrise to the high-temperature branch and the unstable section of p ( φ H ); the low-temperaturebranch is represented by the line p = 0 corresponding to the thermal gas.)Inspection of the same quantities for our fit of the QCD equation of state (see right panel inFig. 3) reveals V (cid:48) /V < . φ from 0 (UV region) upto 10 (towards the IR region), that is v s > . − T c , the ansatz (8) does not qualify tocontinue towards the deep IR region, since, e.g., v s becomes negative for φ H (cid:38) .
5, signallingthe break-down of the ansatz’s capabilities. (From the IHQCD viewpoint such a behavioris admissible: the point where v s = 0 would signal a Hawking-Page phase transition to the p = 0 phase, and desirable: the model becomes zero- T confining [14]. In contrast, our ansatz(8) is an ad hoc construction to mimic the Yang-Mills equation of state for T > . T c (up to10 T c ), corresponding to φ H (cid:46) . φ H ≈ . φ H (cid:38) .
5. Thus, it is meaningless to derive from (8) propertiesof the boundary theory in the IR region.) In contrast, for the QCD parameter adjustment(see (9)), the ansatz (8) seems to be applicable towards the deep IR region.Upon an integration of V (cid:48) /V the potentials V ( φ ) emerge, displayed in Fig. 4. In contrastto V (cid:48) /V , the potentials look quite featureless, both in the region where the softest point(minimum of v s ) appears for the QCD equation of state (depicted by the arrows) and in the8egion of the first-order phase transition for the Yang-Mills equation of state (grey polygon). φ V / ( − L ) QCD YM
FIG. 4: The dilaton potentials V ( φ ) in units of − L for the ansatz (8) with parameter sets from(9) (solid blue or red dashed curve for the fit of v s or s/T ; the arrows point to the location where v s has the minimum). The potential (8) with parameters adjusted to the Yang-Mills equation ofstate [20] is exhibited by the dotted black curve (the un/metastable region is depicted by the greypolygon). Vertical dashed (dotted) lines bracket the fit range to QCD [11] (Yang-Mills [10]) latticedata. IV. DISCUSSION AND SUMMARY
In contrast to the IHQCD model, which covers quite a lot of QCD features both for the puregluon plasma [14, 23] and for QCD in the Veneziano limit [24], at finite as well as at zerotemperature together with a direct account of the two-loop t’Hooft running coupling, weconsider here a simple holographic gravity–single-dilaton model without any explicit a prioriscale setting. All parameters are adjusted to finite-temperature lattice QCD thermodynam-ics in a selected temperature range. We formulate a simple criterion to see already at thedilaton potential (actually its scaled derivative) whether a first-order phase transition canemerge, as for the pure gluon plasma, or a cross-over is encoded, as for QCD at µ = 0. While9ur focus is clearly on features in a limited temperature range at T ≥ T c , also some sectionof the low-temperature region can be successfully accomodated in the model, leaving thedeep IR region for further studies. We also stress that we do not require a specified behaviorof the model outcome in the UV region. Note here that the influence of both asymptoticregimes on the equation of state in the considered temperature interval is fairly small: asshown in [20] the influence of the UV region on dimensionless thermodynamic quantitiesshould not exceed a few percent; for T > . T c , the deep IR region contributes to p and I as a small integration constant, while s , T , v s , and the viscosities are independent of it.Hence, although our dilaton potentials ignore QCD features at T → T → ∞ , weargue that they qualify for further investigations. For instance, supplemented by a fit of thequark number susceptibility one can repeat the analysis of [8] with an up-to-date input to aholographic study of the phase diagram. Even prior to that we note the interesting drop ofthe ratio ζ/η by 50% at T c when including quarks.Another obvious extension of our studies would be the inclusion of a field dual to the chiralcondensate (cid:104) q ¯ q (cid:105) , which is responsible not only for the breaking of conformal invariancein addition to the gluon condensate as expressed by the trace anomaly but, even moreimportantly, it is in the chiral limit an order parameter of chiral symmetry breaking inQCD. Extensive investigations in this direction, albeit for the Veneziano limit QCD, wereperformed in [24], where chiral symmetry breaking is realized by tachyon dynamics.In summary, we present an adjustment of a single-field dilaton potential to recently confirmedlattice QCD thermodynamics data in the temperature range (0 . − . T c . A criterion isdelivered for ensuring a cross-over at the softest point.Acknowledgements: The work is supported by BMBF grant 05P12CRGH1 and EuropeanNetwork HP3-PR1-TURHIC. 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