On the Baryonic Density and Susceptibilities in a Holographic Model of QCD

In this paper, we calculate analytically the baryonic density and susceptibilities, which are sensitive probes to the fermionic degrees of freedom, in a holographic model of QCD both in its hot QGP phase and in its cold dense phase. Interesting patterns due to strong coupling dynamics will be shown and valuable lessons for QCD will be discussed.


Introduction
Historically string theory was developed during the attempt to build dual models for the strong interaction describing various hadrons and their scattering amplitudes [1]. After the advent of Quantum Chromodynamics (QCD) as the fundamental gauge theory of strong interaction, string theory departed into a quite different route with many fascinating discoveries. More recently, a new twist has been added to the story between string theory and QCD: the AdS/CFT correspondence [2,3,4], or more generally the gauge/gravity duality, has provided a way to study the strong coupling dynamics of non-Abelian gauge theories (like QCD) via its string theory dual in the much more tractable supergravity regime. A lot of interesting results have been achieved along this direction with applications in various aspects, for reviews see e.g. [5,6,7,8,9,10,11,12].
While a specific gravity dual for real QCD hasn't been found yet, many important insights into the nonperturbative aspects of QCD have been obtained via gauge/gravity duality. Let's just mention two remarkable examples of such kind. Study of the so-called quark-gluon plasma (QGP), the deconfined phase of QCD at temperature higher than the deconfinement transition T > T c ≈ 196MeV and low baryonic density, has been very important but proved to be rather difficult. Two powerful tools for such study include the lattice QCD and the heavy ion collisions experiments (e.g. at RHIC facility): interestingly, the former approach found that the the QGP thermodynamics (e.g. the energy density) deviates from Stefan-Boltzmann limit constantly by about 20% in the range 1.5−4T c [13], while the latter approach found that the QGP produced at RHIC (corresponding to 1 − 2T c ) has an extremely small shear viscosity to entropy density ratio η/s of the order 0.1 [14]. Both discoveries challenged naive expectation of a weakly coupled QGP right above T c and led to the paradigm shift to a strongly coupled QGP (sQGP) [14,15,16]. It turns out that both results can be much better understood in light of the strong coupling results from gauge/gravity duality: the thermodynamics of CFT plasma at infinitely strong coupling has been calculated via its dual black hole to be exactly 3 4 of the Stefan-Boltzmann limit [17], while a large class of strong coupling gauge theories with gravity dual have been shown to have their η s = 1 4π ≈ 0.08. These successes have inspired a flush of activity to extract useful information for QCD and QGP via gauge/gravity duality, see e.g. reviews in [10,11,15].
In this paper, we focus on the fermionic sector i.e. the degrees of freedom carrying baryonic charges in QCD thermodynamics, and use the gauge/gravity duality to qualitatively obtain some of their nonperturbative features. There are very rich dynamics associated with the quarks in QCD, for example the spontaneous chiral symmetry breaking in QCD vacuum and its restoration at high temperature/density. There are also interesting phase structures in the low T and high baryonic density region of QCD phase diagram where the fermionic sector becomes dominant. Color superconductivity [19] was found to occur at very high density with the phase presumably a color-flavor-locking one [20]. At moderately high density there are interesting phenomena like e.g. the interplay between the color superconductivity and chiral restoration and the pairing with mismatched Fermi surfaces [21] [22]. More recently based on wisdom from large N c argument, it has been proposed that there could be a new phase in the cold dense region, named "quarkyonic" phase [23] [24], which is confined and yet has thermodynamics scaling as N c like a system made of quarks. There are also many recent results on the baryonic density and susceptibilities from lattice QCD which show nonperturbative patterns [26,27,28,29,30]. It is thus of great interest to study these aspects of QCD with the handy tool of gauge/gravity duality. To this end, we use the Sakai-Sugimoto(SS) model [31], which have reproduced (at lease qualitatively) an impressive number of QCD results, for example: the baryon property, form factor [32], and the nuclear force [33], etc. The SS model has been studied at finite temperature [34] and at finite baryon density in [35,36,37]. We will particularly calculate the baryonic density and susceptibilities, which are sensitive probes to the fermionic degrees of freedom, from the Sakai-Sugimoto model both in the hot QGP phase and in the cold dense phase with the motivation to understand their patterns under strong coupling and find valuable lessons for QCD.
The paper is organized as follows. We give in Section.II a brief review of baryonic density and susceptibilities in QCD and in Section.III a brief review of pertinent Sakai-Sugimito results obtained before. The baryonic density and susceptibilities in Sakai-Sugimoto model will be analytically calculated for the hot QGP phase in Section.IV and for the cold dense phase in Section.V. Finally in Section.VI we summarize the results and discuss relevant lessons for QCD.

Definition and Examples
We start with a Taylor expansion of pressure P (T, µ) with respect to chemical potential µ (with the convention that quark carries unit baryonic charge) at fixed T : with the (dimensionless) baryonic susceptibilities d n (T ) defined as Note for the above d n the odd-n ones vanish by symmetry. Furthermore we see all non-zero d n except n = 0 represent certain contribution from the baryonic degrees of freedom, and importantly all non-baryonic degrees of freedom (e.g. the gluonic sector in QCD) do not directly contribute to them. So these derivatives probe the properties of the effective fermions in the system directly and sensitively. This can be seen also from the e.g. the baryonic density as given by To give an idea and to provide a benchmark of the baryonic density and susceptibilities, we explicitly evaluate these for a free gas of particles with mass M and baryonic charge B. The density is given by e √ (βM )+x 2 +(βBµ) + 1 (4) with β = 1/T and N i denoting the number of internal degrees of freedom (e.g. color,flavor,spin, etc). The susceptibilities d n can be obtained by subsequent differentiation with respect to µ. We are particularly interested in two limiting cases: the non-relativistic(NR) limit with βM → ∞ and the ultra-relativistic(UR) limit with βM → 0. In these limits we can obtain concrete results below. (i) NR limit: the baryonic density and susceptibilities in this limit are: We observe two important points at this limit: (a) all susceptibilities d n are positive and have the same dependence on T up to a constant coefficient ; (b) the ratio between successive susceptibilities is directly related to the baryonic charge carried by the degree of freedom, i.e. d n+2 /d n = B 2 , independent of M, T and n; (c) for multi-component non-interacting gas of species M i , B i , all d n are simply a sum over species with the same formulae above and they remain all positive, but one then expect the ratios d n+2 /d n =< B 2 i > to be an abundance-averaged baryonic charge which now depends on B i and M i , T, n as well.
(ii) UR limit: the baryonic density and susceptibilities in this limit are: Again for multi-component non-interacting gas of species M i , B i , the d n are simply a sum over species with the same formulae above. For such a Stefan-Boltzman gas of quarks (with B q = 1 convention) with spin N s , flavor N f and color N c , the susceptibilities per degrees of freedom (D.o.F) are simply with all higher ones vanishing. In both limits of the free gas example above, we find all non-vanishing susceptibilities to be positive and proportional to B n which implies the contribution of fermions with large B becomes larger and larger with increasing order n.
We end this part by emphasizing again that the density and susceptibilities are direct and sensitive probes to the fermions with baryonic charges in the system. In particular their deviation from the free patterns encodes important information about the dynamics and it is certainly of great interest to know the behavior of baryonic density and susceptibilities under strong coupling.   [27]. The dotted green bands represent the Stefan-Boltzmann limit.

The Susceptibilities in Lattice QCD
While the susceptibilities contain very useful information about the fermionic degrees of fredom, it is generally hard to be calculated in a strongly coupled theory like QCD. Nevertheless such susceptibilities of QCD can be studied in its lattice formulation. In particular, calculating the d n (defined at zero µ) offers an important method (via Taylor expansion) to explore the QCD phase diagram at finite density which was traditionally unaccessible for lattice QCD due to the well-known "sign problem". These susceptibilities are also directly related to baryonic charge fluctuations in the thermal QCD matter created in heavy ion collisions which can be experimentally measured [25]. The first lattice results for 2-flavor QCD [26] with a relatively large pion mass by Karsch et al from a few years ago showed highly nontrivial patterns near T c , with strong deviation from the free case in the region T ∼ 1 − 2T c . More recent 3-flavor results with a more realistic pion mass by the same group [27] still preserved the nontrivial patterns close to T c though with deviation from the free case limited to be below ∼ 1.4T c. See also other lattice works in e.g. [28][29] [30] which all show similar behavior of the susceptibilities.
We now make a detailed discussion on these lattice results. Let's start with the T < T c part, i.e. the confined hadronic phase: in this phase we actually see from Fig.1 the same signs and similar T-dependence of d 2,4,6 (including exponential-like growth close to T c ) hinting at the NR limit of a free gas as in Eq. (6). And indeed a simple hadronic resonance gas model in-cluding various baryons from the Particle Data Book can nicely fit the lattice data for T < T c , see e.g. Fig.1 in [38]. In contrast, the data for T > T c do not resemble any free gas (NR or UR) model at all. A few nontrivial features are readily visible from Fig.1, especially for 1 − 1.4T c : d 2,4,6 have rather distinctive patterns, with d 2 positive and mildly growing, d 4 positive and rapidly dropping, while d 6 negative and dropping even more abruptly. The susceptibilities results clearly disfavor any weakly interacting quasiparticle models for the deconfined quark-gluon plasma (QGP) phase right above T c , instead they indicate non-perturbative effect from strong coupling. This observation turns out to be consistent with the conclusion from other independent approaches that the QGP in the same 1 − 2T c -region is rather strongly coupled, now called sQGP. In particular the experimental study of the QGP phase by heavy ion collisions at RHIC suggests that it behaves as a very good liquid [14,15,16,10]. Lattice works on e.g. thermodynamics [39] and transport properties [40], heavy quark potentials [41] and charmonium above T c [42] also point to a similar conclusion. A microscopic explanation of such strong coupling results may rely on understanding of the specific mechanism of how the confinement occurs toward T c from above: it has been suggested under the generic spirit of electric-magnetic duality that the QCD plasma close to T c is actually a magnetic one, made of light and abundant monopoles which become Bose-condensed at T c and enforce confinement [43,44,45,46].
Returning to the non-trivial susceptibilities in 1 − 1.4T c , a theoretical understanding is still lacking. One particular suggestion involves possible bound states of quarks and gluons even in the plasma phase due to the presence of still strong coupling in 1 − 2T c [47,48,49], including both conventional colorless states (remnants of mesons and baryons) and colored states like diquarks, q-g states and even polymer-like chains. Those bound states carrying baryonic charges can contribute substantially to the susceptibilities, as first pointed out by Liao and Shuryak in [38], and especially become more and more important in higher orders. To see this, one just notes that diquarks have baryonic charge B = 2 and baryons B = 3, and that the susceptibilities go like d n ∼ B n as evident from Eq. (6,8): this means even the density of the bound states may be small compared to quarks but they still can dominate e.g. d 4 , d 6 and even higher order ones. It was shown in [38] that the bound states contribution, mainly from baryons, could be dominant for the peak and wiggle structures seen in the lattice data. This was confirmed by an even better agreement with data in later studies using the so-called PNJL model [50] in which due to Polyakov line suppression of colored states below and around T c , precisely the 3-quark colorless combinations (e.g. baryons) dominate the nontrivial susceptibilities around T c . Despite these progresses, it is still unclear how the strong coupling dynamics affects the susceptibilities from individual quarks -this is what we attempt to address in the present paper.
To conclude this Section, we have seen that the baryonic density and susceptibilities are sensitive probes to the degrees of freedom carrying baryonic charges, and can exhibit rather nontrivial structures in strong coupling regime as shown by lattice QCD data. It is natural, then, to study these properties for strongly coupled gauge theories in general and for useful models of QCD in particular. To this end, the Sakai-Sugimoto model [31] is a good choice: as a holographic model of QCD, it is calculable in strong coupling regime by virtue of the gauge/string duality and has been shown to have many realistic features of QCD. In the rest of the paper, we will calculate the baryonic density and susceptibilities in the Sakai-Sugimoto model and discuss the lessons for QCD.

Brief Review of Sakai-Sugimoto Model with Chemical Potential
In this section we summarize the Sakai-Sugimoto (SS) model for notation and completeness. For a thorough presentation we refer [31] for zero temperature and [34] for finite temperature.
The SS model, in brief, is defined by the dynamics of N f D8-D8 branes in the background field (the metric, the dilaton, and the Ramond-Ramond field) generated by N c D4-branes. In order not to disturb the background we require N f ≪ N c , which is called the "probe" limit and corresponds to the quenched approximation. The low energy dynamics of D8-D8 branes are governed by the Dirac-Born-Infeld (DBI) action and the Chern-Simons (CS) action: where φ and C 3 are the dilaton and the Ramond-Ramond field. The metric generated by D4 brane is encoded in the induced metric g M N on the D8-D8 branes. s is the tension of the D8-brane. In the following subsections we will specify the ingredient fields of the DBI and CS action: the induced metric, dilaton, RR field, and the gauge field.

Ingredient fields
The induced metric on the D8 branes from the D4 branes background metric can be written as [31,34] where 1 .

(Deconfined phase)
The embedding information is encoded only in γ and thereby g U U . For definition of the warping factor f (U) in both confined and deconfined phases, see the appendix. The dilaton and RR field are given as where g S is the string coupling constant, Ω 4 = 8π 2 /3 is the volume of the unit S 4 and ǫ 4 is the corresponding volume form.
For the gauge field we only consider the time component of the U(1) part of the U(N f ) gauge field, which is normalized as 1 In the appendix, the background metric and coordinate notation convention are shown.
where we also assumed theÂ 0 is only a function of U. This choice of the bulk gauge field is a standard holographic way to introduce the chemical potential in the boundary field theory.Â 0 (∞) of the classical solution is identified with the chemical potential.

The DBI action
the DBI action (9) reads where N s = 2 and reflects the two contributions from D8 and D8 branes and Ω 4 results from the trivial angle integration of S 4 . The symbol ′ denotes the differentiation with respect to U. Note that g 00 g U U is the only place where the confined and deconfined phases are distinguished.
In terms of the dimensionless variables defined as [36] the DBI action reads β is the inverse temperature, V 3 is the volume of X space. For simplicity in this paper we only consider the antipodal configuration, in which the D8 branes and D8 branes are maximally separated at the boundary. This implies that, in the confined phase, the D8-D8 branes are connected at U = U KK , and, in the deconfined phase, D8 and D8 branes are parallel to each other extending to the black hole horizon. Since it also implies that x 4 is constant (x ′ 4 = 0) the DBI action is reduced to The antipodal configuration simplifies the problem since we only need to solve the one variable (â 0 ) equation instead of the coupled equations of two variables (â 0 , x 4 ). It also simplifies the phase diagram of holographic QCD since there is no "cusp" configuration studied in [36]. Thus the deconfined phase at T > T c and any µ corresponds to the hot QGP phase, 2 and the confined phase at T < T c and sufficiently large µ > µ c corresponds to a cold dense phase made of baryonic matter while at small µ < µ c the phase is the trivial vacuum configuration with zero baryonic density. The onset chemical potential, µ c , is defined in (26). The antipodal configuration will allow us to obtain analytic results for both the hot QGP phase and the cold dense phase, which shall remain qualitatively the same for large non-maximal separations of D8-D8 branes.

Grand potential
The grand potential is identified with the on-shell DBI action. For convenience we will compute the rescaled grand potential defined as 3 where t is the dimensionless temparature and µ is the dimensionless chemical potential defined as

QGP phase
Let us first consider the deconfined phase, Sinceâ 0 is a cyclic coordinate, its conjugate momentum is conserved, which is defined up to a normalization as 2 In general there could also be other phases corresponding to connected configurations in deconfined phase as shown in [36]. 3 The grand potential in SS model has been computed in several papers [35,36,37] with various notational conventions. In this paper we follow the convention in [36]. andâ 0 is easily solved. Especially where a(u T ) = 0 for the regularity at the horizon. With this solution the grand potential reads where d(µ) is the function of µ via (22).

Cold dense phase
In the confined phase we can do the same analysis with However in this case we need to consider the explicit source term. Contrary to the deconfined phase, D8 and D8 branes are connected at u = u KK . For a nontrivialâ 0 there must be a singularity at u = u KK and to take this account we consider the D4 branes wrapping S 4 as the baryon source. The source of a uniform distribution of D4 branes (∼ d) can be explicitly introduced by the CS action. Referring to [36] for more detail, we simply quote the final result for the grand potential and chemical potential where µ c ≡ 1 3 u KK is the onset chemical potential of the cold dense phase.

Baryonic Density and Susceptibilities in the QGP Phase
At T > T c the Saika-Sugimoto model has a deconfined, chirally symmetric QGP phase for any µ. We make use of the analytic results for QGP phase to analyze its barynoic degrees of freedom. Using Eq. (23,22) and subtracting out the vacuum part in the grand potential, we explicitly write out the following equations as our starting point at given T, µ: In principle we need to solve d(T, µ) from the constraint and then obtain the full P [T, µ] to calculate density and susceptibilities. However, we can make use of the chain rule to do the calculation without solving the constraint. In other words, we (temporarily) consider µ as being determined by T, d: And we first evaluate the derivatives over d: The density is then given by n B = ∂P/∂d ∂µ/∂d = d, as it should. To determine its ultimate dependence on T, µ we will have to solve the constraint.
We now calculate the susceptibilities d n as defined in Eq. (2). To do that we introduce P n (T, d) ≡ ∂P n ∂µ n with d n = t n−4 · P n (d → 0) (noting that d → 0 is equivalent to µ → 0). P n can be evaluated order by order, using P n+1 = ∂Pn/∂d ∂µ/∂d . The final results to the order n = 10 are given below: , χ 6 = − 3 8 · 5 · 31 2 5 · 13 2 · 23 , χ 8 = 3 9 · 5 2 · 7 · 3011 2 7 · 11 · 13 3 · 23 , χ 10 = − 3 1 4 · 5 2 · 7 · 24546787 2 9 · 11 · 13 4 · 23 2 · 43 (32) These results show very distinctive patterns: (a) first of all we see simple power dependence on temperature but different orders depend on T very differently; (b) particularly, only d 2 has positive power of T -dependence, i.e. growing with T , while all higher order susceptibilities have negative power thus vanish in the T → ∞ limit; (c) furthermore, we notice there is an alternating sign pattern for n > 2, i.e. d 4,8 are positive while d 6,10 are negative; (d) finally we notice the ratios between successive susceptibilities are d n+2 /d n ∼ t −2 . As is evident from the above results, even in the QGP phase with quarks as the basic baryonic charge carriers, the strong interaction modifies the behavior significantly, resulting in non-perturbative patterns of these susceptibilities. At this point, it would be interesting to see the actual dependence of the susceptibilities on physical parameters by recovering all the dimensions (and neglected constants) of involved quantities, i.e. P, µ, T . This can be done via the following: We also re-scale temperature by ]. Eventually we arrive at the results below: with ξ n = χ n ·3 2n−8 /(π ·2 n−3 ) (note the sign patterns of ξ n follow from χ n , i.e. ξ 2,4,8,... > 0 while ξ 6,10,... < 0). First of all we notice the degrees of freedom counting reflects quark-like dependence, as baryonic or any other non-single combinations of quarks would result in different dependence on both N c and N f . On the other hand we see nontrivial power dependence on the coupling: again d 2 ∼ λT grows both with λ andT but all higher susceptibilities will be suppressed at very strong coupling. Finally let's discuss the asymptotic behavior of baryonic density at very large and very small density/chemical potential, i.e. µ → ∞ and µ → 0. In these limits we can solve the constraint equation to obtain the density. (i) Dense Limit in QGP Phase: In the dense limit µ → ∞ and thus d → ∞, we solve the constraint equation Eq.(28) to leading order and obtain the pressure and baryonic density to be (after recovering physical dimensions): We can also calculate the energy per particlē with E S.B. = 3µ/4 representing the energy per particle for a free gas of massless fermions at zero temperature and finite µ. Amusingly, the strong coupling result misses the non-interacting limit by only 1/21 ∼ 5% despite the strong coupling dynamics.
(ii) Dilute Limit in QGP Phase: In the dilute limit µ → 0 and thus d → 0, we note that 2 F 1 can then calculate the baryonic density to leading order This result has linear dependence on µ, in common with the leading order at small µ of free fermion gas at UR limit. The strong coupling density here however depends on coupling λ and also has T 3 dependence, differing from the free case which has no coupling and has T 2 dependence.

Baryonic Density and Susceptibilities in the Cold Dense Phase
We now turn to the cold dense phase at T = 0. To study this phase, we focus on the situation with maximal D8-D8 separation which allows tractable analytic formulae and much simplifies the calculation. For non-maximal separation, the physics at T = 0 remains qualitatively the same while the calculation is much more involved.
We start with Eq. (25,26) and further rewrite the pressure and chemical potential as a function of d into the following forms by re-scaling the quantities with u KK : In the aboved 2 ≡ d 2 /u 5 KK andμ c = µ c /u KK = 1/3. Also note in the pressure we have subtracted out the vacuum part, e.g. the (−1) within the bracket. These can be easily solved numerically: for each givenμ we can first solved from Eq.(39) and then obtainP from Eq. (38), and the results are shown as solid blue lines in Fig.2.
A few comments are in order. First we notice that forμ <μ c there is no solution ford as is evident from Eq. (39), so the solution is the trivial U-shape and the system is in vacuum phase without any dependence onμ. Whenμ >μ c baryons start to emerge and a new phase with nonzero baryonic density takes over the vacuum one. Furthermore both the pressure and the baryonic density are continuous atμ c transition while the first derivative of d versus µ (i.e. the lowest susceptibility atμ c ) is not, which implies a second order phase transition. In physical unit, one has µ c = λM KK 27π = 2 27 × λ × T c : with λ ∼ 20 − 50, the relation reasonably agrees with current rough estimate of µ c /T c in QCD.
We now make an expansion of the pressure and density in the cold dense phase close toμ c . This can be done by systematically analyze the expansion ofd → 0 order by order in the integrands of Eqs. (38,39). The result for the pressure is: 3rd(orange), 4th(magenta), and 5th(purple) order respectively. Interestingly we found that the expansion till the 2nd order, i.e. the series with (μ −μ c ) 2 and (μ −μ c ) 4 terms only, agrees remarkably well with the full result up to very largeμ, indicating some fine cancellations among all higher order terms. The expansion for the density is given by: We also note that these coefficients are related to those in the pressure expansion by f 2n−1 = (2n)·c 2n order by order as required byd = ∂μP . In Fig.2(right) we compare the full numerical result of the density (solid blue line) with its Taylor series results truncated at the 1st(red), 2nd(green), 3rd(orange), 4th(magenta), and 5th(purple) order respectively. Not surprisingly we again found that the expansion till the 2nd order, i.e. the series with (μ −μ c ) and (μ −μ c ) 3 terms only, agrees remarkably well with the full result up to very largeμ. Finally we study another quantity of interest, i.e. the energy per particlē E which at T = 0 is given byĒ = (μd −P )/d ≡ g(μ)μ. The function g(μ) is plotted in Fig.3 as solid blue line. The dashed green line indicates the free case g = 3/4. We see that close to the transition point the energy per particle deviates much from the free value and drops very sharply, while at much larger density it curves back and approaches the free value from below very slowly. We notice that the curve crosses the free line at aboutμ ≈ 2μ c : it may imply that below this density the system has repulsive interaction and above it the system has attractive interaction. In either regions, the deviation ofĒ from free case is rather modest.
We also present the results with physical scales recovered as in Eq.(33) and where we have used µ c = u KK 3 = λM KK 27π .

Summary and Discussions
In summary, we have calculated the baryonic density and susceptibilities in a holographic model of QCD. The results both for the hot QGP phase in Eq. (34) and for the cold dense phase in Eq. (44) show interesting patterns due to strong coupling dynamics. In the following we discuss relevant lessons for QCD that can be learned from the results.
We first discuss the results for hot QGP phase close to T c . To give an idea of the numbers, we re-write the results in Eq.(34) as: We first note that a few qualitative features agree well with the lattice QCD data shown in Fig.1: the sign pattern of d 2,4,6 is the same, d 2 approximately exhibits linear growth with T close to T c while d 4,6 shows quick decrease with T , and also d 6 vanishes much more abruptly than d 4 . If one takes a λ ∼ 10 − 20, 4 our d 2 values are very close to the lattice results, but the d 4 and d 6 values are too small. The comparison seems to indicate that dynamics from very strong coupling of quarks only tends to suppress higher order susceptibilities and may not be adequate to account for the d 4,6 close to T c seen in lattice QCD: this is yet another indication that those higher order susceptibilities may be attributed to multi-quark correlations like baryons which manifest themselves more and more in higher order susceptibilities as we already pointed out in Section.II. Another interesting observation is that for both phases we have noticed that in the dense regime the energy per particleĒ is very close to the Stefan-Boltzmann limit, i.e.Ē E SB (the so-called Bertsch number) is close to 1 despite the strong coupling. We remind that the result of entropy in D3-D7 system (roughly involving adjoint gauge sector but without fundamental fermions) deviates from the Stefan-Boltzmann limit by 25%. This comparison indicates that the strong coupling dynamics may somehow modify the thermodynamics of fundamental fermions much more mildly than of the gauge fields. Interestingly lattice QCD seems to show similar situation: while d 2 approaches the Stefan-Boltzmann limit already around 1.4T c , the pure gauge lattice results show no tendency toward the same limit even at 4T c .
Finally we point out that the cold dense phase of the Sakai-Sugimoto model studied above has its many features similar to the quarkyonic matter in large N c QCD [23] [24]. Both are in the confined regime but have their thermodynamics scaling as N c N f like a quark system. The transition from the vacuum phase to the dense phase for both happens at about the baryon mass threshold as a second order one, after which the baryonic density starts growing. While calculation is hard for the quarkynoic matter in QCD, one may study its holographic dual quantitatively and obtain useful results as ours in Section.V. It will be very interesting to further investigate via the holographic method the many interesting features of the quarkyonic matter, e.g. the chiral symmetry and excitations near or deep into the Fermi surface, etc, which were previously qualitatively studied. where X = X 1,2,3 , while U(≥ U KK ) and Ω 4 are the radial coordinate and four angle variables in the X 5,6,7,8,9 direction. R is given by where g s and l s are the string coupling and length respectively. Since X 4 is compacitified to a circle, to avoid a conical singularity at U = U KK , the period of δX 4 is set to The field theory is defined by Kaluza-Klein mass (M KK ) and the four-dimensional coupling constant at the compactification scale, g Y M , where g 5 (= (2π) 2 g s l s ) is the five dimensional coupling constant obtained from D4 brane DBI action. From (47) and (49) At finite temperature, there are two possibilities [52]. One is to follow the standard prescription of the finite temperature field theory. The geometry is the same as the zero temperature apart from the fact that the time direction is Euclidean (X 0 → X E 0 = iX 0 ) and compactified with a circumference β = 1/T : This corresponds to the confined phase which is thermodynamically preferred (i.e. with the smallest action) in the low temperature regime. The other possible geometry contains the black hole, which is another saddle point of the Euclidean path integral over supergravity configuration. The pertinent background is This corresponds to deconfined phase which is thermodynamically preferred (i.e. with the smallest action) at high temperature. To avoid a conical singularity at U = U T the period of δX E 0 of the compactified t E direction is set to which is identified with the inverse temperature. The confinement/deconfinement phase transition, i.e. the switching from one background geometry to the other, occurs when δX 4 = δX E 0 i.e. at the critical temperature T c ,