1. Introduction

At low energy, only hadrons can be observed. Due to the large coupling of the strong interaction on large distance scale, the genuine constituents of the nuclear matter are confined within the baryons and mesons. They can be explored only with a high energy probe, for example, in the deep inelastic scattering (DIS) experiments. When the energy scale involved is sufficiently large, roughly few hundred MeVs, the interaction among quarks and gluons become perturbatively weak, the phenomenon known as the asymptotic freedom. The quarks and gluons subsequently become “deconfined” from the confinement of the strong interaction.

For effectively free quarks and gluons, perturbative treatment of the Quantum Chromodynamics (QCD) has been proven very successful in making verifiable quantitative and reliable predictions. The QCD background calculations of the scattering of quarks and gluons at the Tevatron give accurate and vital results, which are crucial in providing the benchmark for the search of New Physics beyond the Standard Model. Nevertheless, unexpected ridge-like signals related to the strong interaction are already observed from the collision of proton and proton at the Large Hadron Collider (LHC) [1]. Further investigations are required in order to determine whether this ridge structure could be explained by the perturbative QCD or if it is nonperturbative in nature.

A general picture of the deconfinement process of the quarks and gluons within hadrons is currently incomplete at the most. Naively, from argument of the RGE (Renormalization Group Equation) running of the beta function, effectively free quarks and gluons are expected to appear at high energies and/or temperatures. Transition from non-perturbative phase of nuclear matter to the perturbative regime, where the perturbative QCD is reliable, is explored most successfully in the lattice approach. Lattice studies of the QCD predicts the deconfinement temperature around 175 MeV [2]. Nuclear matter at such temperature would undergo a phase transition into a deconfined phase called the quark-gluon plasma (QGP). Most bound states of light quarks would melt down at this temperature leaving free quarks and gluons in the plasma. Remarkably, the mesonic states of heavy quarks (e.g., charmonium) in the nuclear matter at such high temperature tend to persist melting at least until [3–5] due to the remaining screened Coulomb-type binding potential between quark and antiquark. Multiquark states such as baryons can also exist in the QGP up to certain temperatures provided that the baryonic charge density is sufficiently large.

In the confined phase, only colour singlet states can exist as free particle due to the confinement. Above the deconfinement, quarks and gluons with colour charges can propagate with more freedom in the plasma. It is therefore possible that the coloured multiquark states such as diquarks could also exist in the deconfined nuclear medium. Similar to the mesonic states of the heavy quarks, these multiquarks could persist melting up to relatively high temperature above the deconfinement. We can expect the multiquarks to be abundant in the nuclear matter when the density is large up to temperature well above the deconfinement temperature. Consequently, it is interesting to investigate the physical properties of the multiquarks as well as their thermodynamical phase diagram in details. Unfortunately, perturbative QCD based on quarks and gluons is not reliable during the deconfinement phase transition. Lattice QCD is applicable only when the baryon density is small.

An alternative approach to study the strongly coupled gauge theory is the holographic model based on the AdS/CFT correspondence [6, 7]. A string theory in the curved background generated by D-branes source is conjectured to be dual to the gauge theory on the branes. The duality suggests a correspondence between the strongly coupled gauge theory on the branes and the weakly coupled string theory in the bulk. Extension of the duality to the finite temperature gauge theory can be done by adding a black hole horizon to the near-horizon limit of the background spacetime [8]. Baryons and multiquarks can be holographically constructed using the baryon vertex and strings [9–12].

In this paper, we will review the physics of the holographic multiquarks in the quark-gluon plasma using mainly the Sakai-Sugimoto (SS) model [13, 14]. The SS model and the holographic setup of the multiquarks is discussed in Section 2. Section 3 describes the thermodynamical properties and the phase diagram of the multiquark nuclear phase. Magnetic properties of the multiquark phase and the corresponding phase diagram are discussed in Sections 4 and 5, respectively. Section 6 concludes our paper. To present the main results of this paper, a summary table of the deconfined nuclear phase in the SS model is given in Table 1.

Table 1: Summary table of the phases in the deconfined Sakai-Sugimoto model, represents the external magnetic field.

2. Multiquark States and the Holographic Models

In addition to baryons and mesons, the possibility of multiquark states were recognized by Gell-Mann since the proposal of the quark model. -multiquark () such as the tetraquark and dibaryon were proposed since 1977 by Jaffe [15–17] using the MIT bag model. There are theoretical models of colour-singlet multiquarks using interactions of various origins, for example, chromomagnetism, flux tube confinement, and hadronic molecules. Despite the theoretical possibilities, conclusive discovery of the multiquarks has yet to be confirmed experimentally (see [18] and references therein).

Series of experimental results from RHIC suggests that the produced QGP is strongly coupled (sQGP) [19–22]. The fact that the QGP is strongly coupled near-and-above the deconfinement temperature suggests the possibility of the existence of exotic bound states with colour degrees of freedom in the deconfined QGP. Recall that an interaction between two heavy quarks in the confined phase at can be described empirically by the screened Cornell potential where is the Debye screening mass depending on and is the effective coupling. The first part represents the (colour-screened) confining force due to QCD string with the effective string tension ; it is around as suggested by the lattice studies. The second part represents the effective (colour-screened) Coulomb potential due to transverse string oscillation. By definition, the effective string tension vanishes at . As a result, only the screened Coulomb part contributes to the interaction between quarks but within the range of screening length . Yet, as suggested by [23], a short string-like configuration of colour fields at low becomes longer strings at-and-near which contribute to the binding between quarks and gluons. Therefore, the bound states of gluons and quarks can exist in both colour-singlet and colour-nonsinglet forms in the sQGP.

The studies of the multibody bound states in the sQGP were initiated by Shuryak and colleagues [23–25]. Based on the studies in [23], three proposed multibody bound states: (i) diquark or “polymer-chain" (); (ii) baryons (); (iii) closed (3-)chains of gluons () seem to exist only for (1–1.5). Importantly, the existence of these bound states could affect the thermodynamical and hydrodynamical properties of the sQGP.

The holographic models of colour-singlet baryon was originally investigated by Witten et al. [9, 10]. In the AdS₅ background, a D5-brane wrapping the subspace with strings attached is proposed to be a dual description of a baryon. A holographic dual of a -quark () with colour degrees of freedom is discussed in [11] (see also [26]) for the supersymmetric background. There is a number of interesting articles investigating various possibilities of the multiquarks in both confined and deconfined medium, some of them consider deformed baryon vertex [12, 27–33]. Notably, [12] uses a simplified configuration with only one point-like baryon vertex to describe a variety of classes of the multiquarks with and without the colour degrees of freedom. We will focus our attention to such multiquark model in this paper.

In recent years, the AdS/CFT correspondence has attracted interests in its applicability to the phenomenological studies of non-perturbative QCD. However, this correspondence cannot provide the gravity dual of the large QCD. As its name suggests, the AdS/CFT have the gauge theory side, which is conformal, differing from the confining behaviour of the real-world QCD. There has been many attempts to engineer the holographic model whose the confining feature is taken into account [8, 34–36].

One natural way is to consider a stack of D4-branes, in Type IIA string theory, whose the world volume possesses one compact spatial direction [8]. In the near-horizon metric of a near-extremal D4-brane, the compactified spatial circle shrinks to zero size at some finite value of the radial direction representing a smooth cutoff of the spacetime. This feature can provide us with the confining spacetime background in which the potential between a holographic quark-antiquark bound state is mainly contributed by the tension of string lying along the “hard-wall.” Consequently, the potential is linearly proportional to the separation between two ends of the string resulting in the confinement of quarks and antiquarks in the dual gauge theory.

At finite temperature, the time coordinate becomes Wick-rotated, and the asymptotic circumference of the time circle equals to the inverse of the temperature, . Consequently, the confining spacetime background at finite temperature has two compact directions. The metric of the geometry then can be written as where is the Euclidean time with temperature dependent period , is the compact spatial circle with period , and . Notice that equals zero for but equals one as approaches infinity. This factor renders the subspace a cigar-like shape, while the subspace has a cylindrical shape. However, there is an alternative supergravity solution whose the time and the compact spatial coordinates exchange the role. That is, is the compact spatial coordinate with fixed circumference, is the Euclidean time with period , and . In other words, there are two geometries which can be the supergravity solution. The comparison of the free energy between these two competing geometries tells us about the deconfinement phase transition in the gauge theory side. It is important to emphasize that the asymptotic circumference of the time circle can be variable depending on the temperature, namely, , while the -circle has a fixed circumference. As a result, the phase transition occurs once the asymptotic circumferences of the two circles become the same in both geometries such that they have the same value of free energy. This gives rise to the deconfinement transition line in the phase diagram of the holographic nuclear matter [8, 37]. For a concise review, see [38].

2.1. The Sakai-Sugimoto Model

More realistic holographic dual of the large QCD is the Sakai-Sugimoto (SS) model [13, 14]. The brane construction of the SS model is a stack of D4 branes intersecting with D8- and anti-D8- branes, where such that the presence of the probe branes D8/anti-D8-branes does not affect the D4 background. This is called the probe limit, corresponding to the quenched approximation in the lattice QCD.

Stack of D8 and branes are introduced as the flavour branes. They are located at separation distance along the compactified direction at the boundary . Open-string excitation with one end on the flavour branes behave like a chiral “quark." In the setup where D8 and are parallel in the () projection, each chiral excitation on each stack of branes transform independently, therefore the theory has a chiral symmetry. For the setup where D8 and connect, forming a U-shape or a V-shape configuration in the () projection, chiral symmetry is broken.

To obtain a SUSY broken QCD at low energy, the boundary conditions of the superpartners in the direction are chosen so that the zeroth modes vanish (Scherk-Schwarz mechanism). For energies below the first KK modes, the gauge theory therefore contains only gluons and chiral quarks. If the number of the stack of D4-branes source is chosen to be 3, this low-energy gauge theory will look exactly like QCD. The brane configuration of the SS model is shown in Table 2.

Table 2: Brane configuration of the Sakai-Sugimoto model.

Note that the “x” sign signifies that the coordinate is occupied by an infinite extending direction of the D-brane world volume and the “o” sign means that the coordinate is occupied by a compact direction of the D-brane world volume. This holographic model is a QCD-like theory in many aspects. (i) It is nonsupersymmetric resulting from the antiperiodicity for superpartners around the circle. (ii) It has the confining behaviour and the deconfinement phase transition in the same way as mentioned above. In the confined phase, the coordinate is the cigar-like compact direction and (the Euclidean time) is the cylindrical compact direction. In the deconfined phase, the two coordinates exchange their roles. To summarize, the coordinates and in (2.2) can be specified in the confined and deconfined phase as shown in Table 3.

Table 3: Geometric assignment of the compactified bulk coordinates in the Sakai-Sugimoto model.

(iii) It has dynamical quarks resulting from the presence of the flavour branes. (iv) The phases of chiral symmetry breaking and chiral symmetric quark-gluon plasma (S-QGP) can be realized. There exist two configurations of the flavour D8- and anti-D8-branes, both satisfy the equation of motion. One is the connected configuration of the D8- and anti-D8-branes representing the chiral symmetry breaking phase. Another is the parallel configuration of the D8- and anti-D8-branes lying along the radial direction of the bulk spacetime representing the chiral symmetric phase. Note that when the separation between the D8- and anti-D8-branes ; the radius of the circle, while when [37].

Since the SS model is the holographic model which gives exactly the particle content of the QCD at low energy, we will consider the holographic multiquarks in the deconfined SS model. The idea is to construct a gravity dual of the 5-dimensional gauge theory with chiral fermions which gives approximately the 4-dimensional QCD at low energy. The inevitable supersymmetry of the dual gauge theory in the string construction is broken at the position of the flavour branes used to introduce the chiral fermions. To construct the SS model, stack of D4-branes is used as the source to generate a curved background of the type IIA string theory. After taking the near-horizon limit and adding a black hole horizon, we arrive at the following background metric: where , . Note that the compact coordinate ( transverse to the probe D8-branes), with arbitrary periodicity , never shrinks to zero. The volume of the unit four-sphere is denoted by and the corresponding volume 4-form by . is the 4-form field strength, is the string length, and is the string coupling. The dilaton in this background has -dependence, and its value changes along the radial direction . This is a crucial difference in comparison to the AdS-Schwarzschild metric case where dilaton contribution is constant.

In the Sakai-Sugimoto model of D4-D8-branes construction, the D4-brane wrapping the is used as the baryon vertex. Remarkably, it was found that the baryon can also be realized as an instanton in the bulk of D4-brane-induced background spacetime, corresponding to baryon in the Skyrme model on the gauge theory side. This instanton can be described in terms of the Chern-Simons action in the bulk. Therefore, these two pictures of baryon are equivalent.

A D-brane wrapping an internal subspace accommodate a which will couple to certain -form field of the string background and becomes charged under the . To cancel this charge for the entire background, a number of strings emerging from the vertex is required. While a string emerging from the vertex contributes a negative charge, a string entering the vertex in the opposite orientation contributes a positive unit of charge. Therefore, as long as the number of strings emerging from the vertex subtracting the number of strings entering the vertex is , the configuration is allowed to exist since the total charge of the background is still zero.

Based on the charge cancelation at the vertex, three classes of exotic multiquarks are proposed in [12]. Namely, they are -baryons, -baryon, and -mesonance (strongly coupled bunch of mesons), corresponding to diquark, some exotic baryons such as pentaquark and a bunch of mesons, respectively. We parameterize as the number of hanging strings which extends from the vertex to the boundary and as the number of radial strings extending from the vertex to the horizon. Figure 1 shows 3 classes of the holographic multiquarks. Their conditions are summarized as the following.

Figure 1: An illustration of the holographic multiquark states (a) -baryon with and , (b) -baryon with and , and (c) -mesonance with and .

For -baryon,

For ()-baryon,

For -mesonance, Note that the values of and can be as large as .

2.2. Force Balance Condition

Using the field background shown in the last section, the total action of these exotic multiquark states can be generally written as The DBI action of D4-brane and the Nambu-Goto action of hanging strings and radial strings can be written as respectively. Note that is the total time over which we evaluate the action and is the position where the D4-brane vertex is located.

Now let us write the force condition. As will be seen later, this is the equilibrium condition for the existence of the multiquark states. Assume the vertex to be a point at the cusp position that does not receive any distortion from the attached strings. The distortion of the baryon vertex due to the attached strings is discussed in details in [39, 40]. Because of the spherical symmetry of the configuration in the (, , ) subspace, the action is sensitive to only the variation in the holographic direction . The variation of the action gives the volume term as well as the surface term. The equation of motion is obtained by requiring that the volume term and surface term vanishes separately. The volume term gives the Euler-Lagrange equation which determines the shape of the hanging strings. On the other hand, the surface term provides the equilibrium condition of the configuration at the tip under the variation in the direction, that is, the force balance condition at the cusp. It can be written as [12] where Obviously, is always less than one, thus we obtain the equilibrium condition Together with (2.5), (2.6), and (2.7), we obtain the lower bound of the hanging string parameter for each multiquark configuration as

for -baryon,

for ()-baryon,

for -mesonance, Note that at and it is an increasing function of the temperature.

2.3. Binding Energy and Screening Length

Theoretically, all of these bound states are allowed to exist. But a question arises which multiquark state is more stable than another. This can be addressed by considering the binding energies of each class of the multiquarks. Naturally, the binding energy of each of these holographic bound states is the total energy of the configuration subtracted by the energy of the free quarks. Similar to the calculation of Wilson loop in [41], the binding energy in the large limit could be estimated to be the total classical action divided by .

The solution or the shape of the hanging strings can be obtained by using the Nambu-Goto action from (2.10), the regulated energy of the hanging strings (subtracted by energy of the free quarks) is From the equilibrium condition corresponding to the surface term, that is, (2.11) and (2.12), we obtain where Consider in (2.10), the Lagrangian, does not explicitly depend on , such that we can define the constant of motion This leads to Eliminating using (2.18) and (2.22), we have the relation Using the above equation, we obtain the size of the radius of the multiquark state as and together with (2.17), we also have Therefore, the total energy of the vertex D-brane and the radial strings are where represents the terms within the brace of (2.25).

By numerical calculations, we compare the (the energy per degrees of freedom) versus (the size of radius of the bound states) of the 3 classes of the multiquark as in Figure 2. The deeper the binding energy is, the harder the multiquark will melt in the thermal bath. From Figure 2, the colour singlet -baryon has deeper binding well than the -baryon and -baryon. As expected, the -multiquark is bound more tightly as gets larger. For -baryon, the bound state is less tightly bound, as increases. Similarly, a -mesonance has the binding energy less than mesonic states. It becomes closer to mesons as grows.

Figure 2: Comparison of the potential per between -baryon, -baryon, and -baryon for , and at temperature .

The screening length can also be numerically calculated. It is defined to be the value at which the binding energies become zero from negative values at small distances. The numerical results, as shown in Figure 3, indicate that the multiquark states of all classes have smaller screening lengths as the temperature increases, with approximately for a fixed , , and . Furthermore, is larger as and increases for -baryon and -mesonance, respectively, while it is smaller as increases for -baryon. Interestingly, the saturation of -mesonance’s screening length occurs as , where approaches the screening length of a meson .

Figure 3: (a) the screening lengths of -baryons with respect to , (b) the screening lengths of ()-baryons with respect to , and (c) the screening lengths of -mesonance with respect to for 0.15–0.35.

3. Thermodynamic Properties of Holographic Multiquark

In the non-antipodal SS model, the holographic plasma can have two distinctive phase transitions; a deconfinement and the chiral symmetry restoration [37]. The deconfinement could occur at lower temperature than the chiral symmetry restoration. For the temperature in between the two transitions, quarks and gluons are deconfined from the confining flux tube but still interact strongly among each other through the remaining screened Coulomb-type potential. Therefore, it is possible to have the multiquark phase in the temperature range between that of the deconfinement and the chiral phase transition. This is consistent, at least in a qualitative way, with the studies of the multibody bound states in the sQCD in the framework of the real QCD [23] mentioned in the previous section.

To actually understand the physics of deconfined QGP, it is thus crucial to investigate the thermodynamical properties of the holographic multiquark phase. In order to extract the thermodynamic potential from the gravity dual model, the path integral approach in quantum gravity [42] has been used. In this technique, the time direction is circled with period in the same manner as the thermal circle in the finite temperature quantum field theory. As discussed in [43] based on the early works [44, 45], the grand canonical potential, or the Gibbs free energy, has the leading contribution from the classical Euclidean action of the bulk theory in the grand canonical ensemble, that is, . Similarly, the Helmholtz free energy has the leading contribution from the Legendre transform with respect to the baryonic charge of the classical Euclidean action, that is, in the canonical ensemble. If we are interested in the situation of nonfixed baryon number density but fixed chemical potential, the relevant thermodynamic potential is the grand canonical potential.

The deconfinement phase transition can be realized as the Hawking-Page transition due to the competition between the action of the background geometry corresponding to the confined phase and the action of the background corresponding to the deconfined phase [8]. Since the coloured multiquark matter can exist only in the deconfined phase (however, the colour-singlet multiquarks such as a baryon can also exist in the confined phase), its grand canonical potential in the Sakai-Sugimoto model is times the combination of the classical action of the deconfining spacetime geometry and the configuration of flavour sector, which includes D8--branes, the probe D4-brane vertex, and the radial strings. Note that the part of hanging strings, extending from the baryon vertex to the flavour branes, is neglected, and we assume that there is no distortion of the vertex due to the connecting strings (such distortion is discussed in [39, 40]). As a result, the baryon vertex is embedded into the flavour branes and becomes an instanton on them.

Intriguingly, whereas the deconfining spacetime geometry action (scales as ) dominates the action of the fundamental matter sector (scales as ), the dominating part can be ignored in the consideration of the holographic phase transition in the deconfined phase. Above the deconfinement, the multiquarks phase competes with the vacuum phase and the chiral-symmetric quark-gluon plasma. In this section, we will explore the phase diagram of the deconfined nuclear matter especially the region of the parameter space where the multiquark phase is dominant. Then we will study the thermodynamics of the multiquark nuclear matter in the dilute and dense limits.

3.1. Phase Diagram

In order to determine a phase diagram, we need to find which phase of nuclear matter is thermodynamically preferred to others in a particular region of the parameter space. For the grand canonical ensemble at a fixed , the thermodynamically preferred phase is the configuration with the grand canonical potential smaller than that of all other phases.

We will first determine the brane configuration in the presence of the external sources by minimizing the classical action. The position of the tip of the D8- will be determined from the equilibrium condition at the tip. The resulting brane configuration corresponds to the multiquark nuclear phase. On the other hand, the vacuum phase corresponds to the configuration with zero sources and density, and the S-QGP phase is dual to the brane configuration with parallel branes without a tip. The projecion of the brane configuration for each deconfined phase is schematically shown in Figure 4. Then we will define the normalized grand canonical potential using the action of the D8-branes. The action of each brane configuration is divergent from the limit . By subtracting with the action of the vacuum configuration, we can regulate the grand canonical potential of each configuration. By comparing the grand canonical potential of each phase, we finally draw a phase diagram in parameter space.

Figure 4: Configurations of S-QGP (a) vacuum (b), and multiquark nuclear phase (c) in projection.

Start with the DBI action of the D8-branes where is the induced metric of the D8-world volume and the field strength tensor of the gauge group living in the flavour branes is While the full D8-brane world-volume gauge fields is we turn on only the time component of the diagonal subgroup part, in order to introduce the finite chemical potential, or equivalently finite baryon density [43, 46]. From the deconfining spacetime metric, (2.3), the DBI action of the D8-branes becomes where the factor is defined to be as the result of integrating out all world-volume coordinates except the holographic direction . And are defined as

The action does not depend on , hence we can define a constant which can be interpreted as the electric displacement field along the holographic direction. Similarly for the variation of action with respect to , we can define another constant of motion, says , so that we can rearrange to obtain from which at large where is the separation between D8 and branes at defined by The parameter can be thought of as the curvature of the D8--branes around the cusp. It becomes zero when the flavour embedding is in the parallel configuration representing the chiral-symmetric QGP. According to [47], this means that it can be used as an order parameter of the nuclear matter/S-QGP phase transition.

We will set to allow the possibility of the chiral symmetry restoration as separate phase transition from the deconfinement. Apply the equilibrium condition at the cusp (see the appendix), we obtain in terms of , where Note that the formula of is derived from the variation of the total action, in which the D8-branes action has been transformed to possess the dynamical variable rather than , This is reminiscent of the way we obtain the equilibrium condition of the multiquark vertex, (2.11), minimizing the surface terms with respect to .

It is important to emphasize that the parameter is the number of radial strings in the unit of . Due to the zero length of the hanging strings, we cannot distinguish the different classes of multiquarks proposed in Section 2 for a particular value of , but some possibilities, -mesonance for example, can be ruled out by examining their thermodynamic stability. This will be shown in this subsection.

Before going further to calculate the classical action, let us comment about the electric displacement . It has been shown in [48] that it is related to the baryon number density. The baryon number density corresponds to the number density of instantons, , on the D8-branes. It also contributes to the Chern-Simons (CS) action of the flavour branes [13].

Beginning with the D8-brane CS term [49] It is convenient to rescale the RR (Ramond-Ramond) field following the Appendix of [13] such that where the last expression is obtained through the integration by part. is the RR 4-form field strength and is the CS 5-form: satisfying . Using the fact that integrating the flux over the in the D4-branes background gives and the relevant term is only the first term in the CS 5-form, (3.18), once turning on only the time-component of the diagonal field, we obtain Assuming a uniform distribution of D4-branes in at , we have [50] where is defined to be the (dimensionless) number density of instantons, or the wrapped D4-branes, at . From the viewpoint that the low-energy effective theory on the D8-brane includes the Skyrme model [13], it is natural to interpret as the baryon number density.

Using (3.6), (3.20), and (3.21), we obtain From both the DBI and CS parts of D8-branes action, the equation of motion with respect to the gauge field gives [48] Note that this reflects the one-dimensional electrostatic effect in which the point electric charges are put at , generating constant electric field in the holographic direction.

The normalized grand canonical potential from the holographic model can be defined using the D-brane action as Since the D-brane action diverges from the limit of the integration, the grand canonical potential needs to be regulated by subtracting with the grand canonical potential of the vacuum phase at the same temperature.

Apart from the grand canonical potential, the chemical potential also needs to be holographically identified in the dual bulk theory. To this end, the time component of the gauge field is taken into account. From the field/operator matching scheme, a bulk field evaluated at , that is, the boundary of the spacetime background plays a role as the source of the dual operator in the generating function of correlation functions in quantum field theory. In other words, this nonnormalizable mode of the bulk field is dual to the coefficient of the field operator. Since the chemical potential is the coefficient of the charge density operator term, it can be holographically identified as . By rescaling for convenience, we can write the dimensionless chemical potential as Similarly, the baryon number density in our normalization is given by even though the true baryon number density is defined in (3.23). Consequently, can then be used to denote the baryon number density.

Since the free energy in the canonical ensemble is the combination of the on-shell Legendre-transformed D8-brane action and the source term, it is convenient to obtain through where the free energy is holographically defined as the Legendre-transformed D8-brane action plus the source terms The Legendre-transformed action is given by The chemical potential can then be written as The second, third, and fourth terms drop out. It is clear from (3.15) corresponding to the equilibrium at the cusp that the third and fourth terms vanish. For the second term, it is because is constant as can be seen from (3.30) that depends only on . Integrating over the remaining gives , which is zero, due to the scale fixing condition . Hence we obtain

Now, it is ready to express the grand canonical potential for the multiquark (baryon corresponds to ) phase. Using (3.24), (3.4), (3.8), (3.9), we obtain the formulae of the grand canonical potential for the multiquark matter. The chemical potential can be calculated from (3.32) by eliminating via (3.8) and substituting (3.9). The grand canonical potential and the baryon chemical potential of the phases can be expressed as the following:

Nuclear (Including Exotics) Phase
Recall that depends on , , and according to (3.12) and (3.14). The last two terms in (3.34) come from the derivative of the source-term action with respect to .
There are at least other two phases that compete with the multiquark phase: the vacuum phase and the chiral-symmetric QGP phase. From the above formula of and , we can obtain the grand canonical potential and the chemical potential of the vacuum simply by (i) setting , (ii) dropping the source terms in (3.34), (iii) changing the lower bound of integration from to , and (iv) replacing by the constant of motion in the vacuum configuration, from , . Thus we obtain

Vacuum Phase,
Similarly, the grand canonical potential and the chemical potential of the S-QGP phase can be obtained by setting , reflecting its parallel configuration, and turning off the source terms. That is, setting in (3.33) and (3.34), changing lower bound of integration to and dropping the source terms in (3.34) give

S-QGP Phase,

The phase transition in the parameter space is obtained by comparing the grand canonical potential between two phases at a particular and . Let us say the transition between phase 1 with the grand canonical potential and phase 2 with . Phase 1 is thermodynamically preferred once and vice versa. There is a first order phase transition when . This kind of phase transition can be seen in the transition between the vacuum and S-QGP phases and the transition between the -QGP and the nuclear matter. However, the phase transition between the vacuum and nuclear matter phases is of the second-order as seen from (3.34). The density is continuous near , which is , and behaves as . Note that this reflects the absence of the interactions between the multiquarks and baryons. As a result, the critical chemical potential defined to be the value, at which has a discontinuity, is given by . By numerical calculations, the phase transition lines can be obtained as shown in Figure 5. The phase diagram between the chiral-broken vacuum and the chiral symmetric QGP phases was first obtained in [51]. The phase diagram of all 3 deconfined phases including the baryonic nuclear phase (without the multiquarks) is originally discussed in [48].
This phase diagram also shows the presence of the multiquark phase which can be mixed in the region of normal baryon phase (with ), say for , and C for . The multiquark matter with is less stable than the normal baryon due to the larger value of the grand canonical potential. Above , it can be shown that the multiquark phase is unstable to density fluctuations, that is, , in some regions of high and certain range of . For approximately , the multiquark phase is unstable thermodynamically to density fluctuations for most of the temperatures.
If the multiquark matter can exist in the quark-gluon plasma, it should mix with the normal baryon states in thermal equilibrium with the populations following the Boltzmann factor where is the binding energies for the states. It is interesting to explore more about the population of these multiquark states in the quark-gluon plasma potentially produced in the heavy-ion collision experiments such as the RHIC and the LHC. Existence of these multiquark states contribute significantly to the hydrodynamical and thermodynamical properties of the deconfined plasma.

Figure 5: The phase diagram of deconfined nuclear matters in the Sakai-Sugimoto model. Multiquark phase is shown as the region on the lower right corner where it is divided into 3 parts according to the value of the colour strings . , , represent the region where multiquark phase with (-baryon), 0.1, 0.3 is the most thermodynamically preferred.

3.2. Thermodynamic Relations

In the grand canonical ensemble, the grand canonical potential is the function of the dynamical variables: the volume , the temperature , and the chemical potential . Its differential is where the coefficients , , and are the pressure, entropy, and the total number of particles, respectively. It is better to understand the system of QGP in terms of volume density of extensive parameters. Let us define the volume density of , , and to be , , and , respectively. Hence, the pressure is

From (3.26) and (3.41), we use the chain rule to obtain so that where we have assumed that the regulated pressure is zero when there is no nuclear matter, that is, .

While the equations of motion cannot be obtained analytically, we can find them in the limit of very small and very large density. Using (3.34) and (3.43), we take and use the binomial expansion, then (see [52] for details) where Note that we have used the fact that of (3.14) becomes , where is with replaced by . Similarly, is defined to be with . On the other hand, for the limit of large in (3.34), the pressure from (3.43) becomes [12]

Numerically, the relations between the pressure and the density of the multiquark matter for different values of are plotted in Figure 6. This is consistent with the results of analytic calculations that for small and for large . Since the relations are not sensitive to the change of , therefore we present only the plots at . The transition from small to large is apparent at . We can also see the dependence of pressure on from the plots. The pressure of the multiquarks with larger is smaller for small . On the other hand, the pressure of the multiquarks with smaller is merely slightly larger in the large limit. Actually, the pressure is nearly insensitive to the changing of for as is implied from (3.46).

Figure 6: Pressure versus density of the multiquark phase in logarithmic scale at , zoomed in around the transition region.

From the differential of the free energy, the entropy density can be written as where is the free energy density which relates to the grand potential density as . Using (3.41), the entropy density becomes Since the pressure and the contribution of D8-branes to the baryon chemical potential are insensitive to the changing of the temperature, the entropy density is dominated by the derivative of with respect to , That is,