Abstract

The purpose of this review is to provide basic ingredients of holographic QCD to nonexperts in string theory and to summarize its interesting achievements in nuclear and hadron physics. We focus on results from a less stringy bottom-up approach and review a stringy top-down model with some calculational details.

1. Introduction

The approaches based on the Anti de Sitter/conformal field theory (AdS/CFT) correspondence [1–3] find many interesting possibilities to explore strongly interacting systems. The discovery of D-branes in string theory [4] was a crucial ingredient to put the correspondence on a firm footing. Typical examples of the strongly interacting systems are dense baryonic matter, stable/unstable nuclei, strongly interacting quark gluon plasma, and condensed matter systems. The morale is to introduce an additional space, which roughly corresponds to the energy scale of 4D boundary field theory, and try to construct a 5D holographic dual model that captures certain nonperturbative aspects of strongly coupled field theory, which are highly nontrivial to analyze in conventional quantum field theory based on perturbative techniques. There are in general two different routes to modeling holographic dual of quantum chromodynamics (QCD). One way is a top-down approach based on stringy D-brane configurations. The other way is so-called a bottom-up approach to a holographic, in which a 5D holographic dual is constructed from QCD. Despite the fact that this bottom-up approach is somewhat ad hoc, it reflects some important features of the gauge/gravity duality and is rather successful in describing properties of hadrons. However, we should keep in mind that a usual simple, tree-level analysis in the holographic dual model, both top-down and bottom-up, is capturing the leading contributions, and we are bound to suffer from subleading corrections.

The goal of this review is twofold. First, we will assemble results mostly from simple bottom-up models in nuclear and hadron physics. Surely we cannot have them all here. We will devote to selected physical quantities discussed in the bottom-up model. The selection of the topics is based on authors’ personal bias. Second, we present some basic materials that might be useful to understand some aspects of AdS/CFT and D-brane models. We will focus on the role of the AdS/CFT in low-energy QCD. Although the correspondence between QCD and gravity theory is not known, we can obtain much insights on QCD by the gauge/gravity duality.

We organize this review as follows. Section 2 reviews the gauge/gravity. Section 3 briefly discusses developments of holographic QCD and demonstrates how to build up a bottom-up model using the AdS/CFT dictionary. After discussing the gauge/gravity duality and modeling in the bottom-up approach, we proceed with selected physical quantities. In each section, we show results mostly from the bottom-up approach and list some from the top-down model. Section 4 deals with vacuum condensates of QCD in holographic QCD. We will mainly discuss the gluon condensate and the quark-gluon mixed condensate. Section 5 collects some results on hadron spectroscopy and form factors from the bottom-up model. Contents are glueballs, light mesons, heavy quarkonium, and hadron form-factors. Section 6 is about QCD at finite temperature and density. We consider QCD phase transition and dense matter. Section 7 is devoted to some general remarks on holographic QCD and to list a few topics that are not discussed properly in this article. Due to our limited knowledge, we are not able to cover all interesting works done in holographic QCD. To compensate this defect partially, we will list some recent review articles on holographic QCD.

In Appendices A–F, we look back on some basic materials that might be useful for nonexperts in string theory to work in holographic QCD. In Appendix A: we review the relation between the bulk mass and boundary operator dimension. In Appendix B: we present a D3/D7 model and axial U(1) symmetry in the model. In Appendix C: we discuss non-Abelian chiral symmetry based on model. In Appendices D and E: we describe how to calculate the Hawking temperature of an AdS black hole. In Appendix F: we encapsulate the Hawking-Page transition and sketch how to calculate Polyakov loop expectation value in thermal AdS and AdS black hole.

We close this section with a cautionary remark.Though it is tempting to argue that holographic QCD is dual to real QCD, what we mean by QCD here might be mostly QCD-like or a cousin of QCD.

2. Introduction to the AdS/CFT Correspondence

The AdS/CFT correspondence, first suggested by Maldacena [1], is a duality between gravity theory in anti de Sitter space (AdS) background and conformal field theory (CFT). The original conjecture states that there is a correspondence between a weakly coupled gravity theory (type IIB string theory) on and the strongly coupled supersymmetric Yang-Mills theory on the four-dimensional boundary of . The strings reside in a higher-dimensional curved spacetime and there exists some well-defined mapping between the objects in the gravity side and the dual objects in the four-dimensional gauge theory. Thus, the conjecture allows the use of non-perturbative methods for strongly coupled theory through its gravity dual.

2.1. Brane Dynamics

The duality emerges from a careful consideration of the D-brane dynamics. A brane sweeps out world-volume in spacetime. Introducing D branes gives open string modes whose endpoints lie on the D branes and the open string spectrum consists of a finite number of massless modes and also an infinite tower of massive modes. The open string end points can move only in the parallel directions of the brane, see Figure 1(a), and a brane can be seen as a point along its transverse directions. The dynamics of the brane is described by the Dirac-Born-Infeld (DBI) action [5] and Chern-Simons term: with a dilaton . Here is the induced metric on . denotes the pullback and is the world-volume field strength. is the tension of the brane which has the following form: and it is the mass per unit spatial volume. Here is the string coupling and is the string length. is the Regge slope parameter and related to the string length scale as . In general, states in the closed string spectrum contain a finite number of massless modes and an infinite tower of massive modes with masses of order . Thus, at low energies , the higher-order corrections come in powers of from integrating out the massive string modes. If there are stack of multiple D branes, the open strings between different branes give a non-Abelian gauge group; see Figure 1(b). In the low-energy limit, we can integrate out the massive modes to obtain non-Abelian gauge theory of the massless fields.

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(b)

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Figure 1 

The configurations of stack of D3 branes in 10d spacetime. (a) D3 branes sweep (3+1) dimensions in (9+1) space time. (b) = 3 stack of D3 branes and all the possible classes of open strings.

Now, we take and consider D3 brane stacks in type IIB theory. The low-energy effective action of this configuration gives a non-Abelian gauge theory with gauge group. In addition, this gauge group can be factorized into and the part, which describes the center of mass motion of the D3 branes, can be decoupled by the global translational invariance. The remaining subgroup describes the dynamics of branes from each other. Therefore we see that in the low-energy limit, the massless open string modes on stacks of D3-branes constitute Yang-Mills theory [6] with 16 supercharges in spacetime. From (2.1) for , we obtain the effective Lagrangian at low energies up to the two-derivative order: with a gauge field , six scalar fields , and four Weyl fermions .

In fact, the original system also contains closed string states. The higher-order derivative corrections for the Lagrangian (2.3) come both in powers of from the massive modes and powers of the string coupling for loop corrections. It is known that the string coupling constant is related by the 10-dimensional gravity constant as and thus the dimensionless string coupling is of order , which is negligible in the low-energy limit. Therefore, at low energies closed strings are decoupled from open strings and the physics on the D3 branes is described by the massless super Yang-Mills theory with gauge group .

2.2. Geometry

Now we view the same system from a different angle. Since D branes are massive and carry energy and Ramond-Ramond (RR) charge, D3 branes deform the spacetime around them to make a curved geometry. Note that the total mass of a D3 brane is infinite because it occupies the infinite world-volume of its transverse directions, but the tension, or the mass per unit three-volume of the D3 brane is finite.

In the flat spacetime, the circumference of the circle surrounding an origin at a distance is , and it simply shrinks to zero if one approaches the origin. But if there is a stack of D3 branes, it deforms the spacetime and makes throat geometry along its transverse directions. Thus, near the D3 branes, the radius of a circle around the stack approaches a constant , an asymptotical infinite cylinder structure, or ; see Figure 2(b). The D3 brane stack is located at the infinite end of the throat and this infinite end is called the “horizon”. In the near horizon geometry, a D3 brane is surrounded by a five-dimensional sphere .

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Figure 2 

The two descriptions of the D3 configuration. (a) Flat spacetime for . D3 branes deform the spacetime.

To be more specific, let us start with type IIB string theory for . We find a black hole type solution which is carrying charges with respect to the RR four-form potential. The theory has magnetically charged D3 branes, which are electrically charged under the potential and it is self-dual . The low-energy effective action is We assume that the metric is spherically symmetric in seven dimensions with the RR source at the origin; then the parameter appears in terms of the five-form field RR-field strength on the five-sphere as where is the five-sphere surrounding the source for a four-form field . Now by using the Euclidean symmetry we get the curved metric solution [7–9] for the D3 brane: where with the radius of the horizon : is the five-sphere metric. For we have and the spacetime becomes flat with a small correction . This factor can be interpreted as a gravitational potential since and and thus . In the near horizon limit, this gravitational effect becomes strong and the metric changes into This is .

The geometry by the D3 branes is sketched in Figure 2(b). Far away from the D3 brane stacks, the spacetime is flat -dimensional Minkowski spacetime and the only modes which survive in the low-energy limit are the massless-closed string (graviton) multiplets, and they decouple from each other due to weak interactions. On the other hand, close to the D3 branes, the geometry takes the form and the whole tower of massive modes exists there. This is because the excitations seen from an observer at infinity are close to the horizon and a closed string mode in a throat should go over a gravitational potential to meet the asymptotic flat region. Therefore, as we focus on the lower-energy limit, the excitation modes should be originated deeper in the throat, and then they decouple from the ones in the flat region. Thus in the low-energy limit the interacting sector lives in geometry.

2.3. The Gauge/Gravity Duality

So far, we have considered two seemingly different descriptions of the D3 brane configuration. As we mentioned, each of the D3 branes carries the gravitational degrees of freedom in terms of its tension, or the string coupling as in (2.4). So the strength of the gravity effect due to stacks of D3 branes depends on the parameter .

If , from (2.9) we see that and therefore the throat geometry effect is less than string length scale. Thus the spacetime is nearly flat and the fluctuations of the D3 branes are described by open string states. In this regime the string coupling is small and the closed strings are decoupled from the open strings. Here the closed string description is inapplicable since one needs to know about the geometry below the string length scale. If we take the low-energy limit, the effective theory, which describes the open string modes, is super Yang-Mills theory with gauge group.

On the other hand, if , then the back-reaction of the branes on the background becomes important and spacetime will be curved. In this limit the closed string description reduces to classical gravity which is supergravity theory in the near horizon geometry. Here the open string description is not feasible because is related with the loop corrections and one has to deal with the strongly coupled open strings. Again, if we take the low-energy limit, the interaction is described by the type IIB string theory in the near-horizon geometry, .

The gauge/gravity correspondence is nothing but the conjecture connecting these two descriptions of D3 branes in the low-energy limit. It is a duality between the super-Yang-Mills theory with gauge group and the type IIB closed string theory in ; see Figure 3. The relation between the Yang-Mills coupling and the string coupling strength is given by Then, the ‘t Hooft coupling can be expressed in terms of the string length scale: Therefore, the dependence of becomes the question of whether the ‘t Hooft coupling is large or small, or the gauge theory is strongly or weakly coupled.

Figure 3 

The sketch of the AdS/CFT correspondence.

The two descriptions can be viewed as two extremes of . For the sake of convenience, we use the coordinate . Then, the metric (2.10) becomes which shows the conformal equivalence between and flat spacetime more clearly. In (2.13), each -slice of is isometric to four-dimensional Minkowski spacetime. In this coordinate, is the boundary of , where Yang-Mills theory lives, with identifying as the coordinates of the gauge theory. If , the determinant of the metric goes to zero and it is the Poincaré horizon. Here the factor also has some relation with the energy scales. If the gauge theory side has a certain energy scale , the corresponding energy in the gravity side is . In other words, a gauge theory object with an energy scale is involved with a bulk side one localized in the z-direction at [1, 10, 11]. Therefore, the UV or high-energy limit corresponds to (or ) and the IR or low-energy limit corresponds to (or ).

The operator-field correspondence between operators in the four-dimensional gauge theory and corresponding dual fields in the gravity side was given in [2, 3]. Then the AdS/CFT correspondence can be stated as follows, where and the string theory partition function at the boundary specified by has the form The relation (2.14) implies that the generating functional of gauge-invariant operators in CFT can be matched with the generating functional for tree diagrams in supergravity.

3. Holographic QCD

Ever since the advent of the AdS/CFT correspondence, there have been many efforts, based on the correspondence, to study nonperturbative physics of strongly coupled gauge theories in general and QCD in particular.

Witten proposed [12] that we can extend the correspondence to non-supersymmetric theories by considering the AdS black hole and showed that this supergravity treatment qualitatively well describes strong coupled QCD (or QCD-like) at finite temperature: for instance, the area law behavior of Wilson loops, confinement/deconfinement transition of pure gauge theory through the Hawking-Page transition, and the mass gap for glueball states. In [13] symmetry breaking by expectation values of scalar fields were analyzed in the context of the AdS/CFT correspondence, which is essential to encode the spontaneous breaking of chiral symmetry in a holographic QCD model. Regular supergravity backgrounds with less supersymmetries corresponding to dual confining super-Yang-Mills theories were proposed in [14, 15]. It has been shown by Polchinski and Strassler [16] that the scaling of high-energy QCD scattering amplitudes can be obtained from a gravity dual description in a sliced AdS geometry whose IR cutoff is determined by the mass of the lightest glueball. Important progress towards flavor physics of QCD has been made by adding flavor degrees of freedom in the fundamental representation of a gauge group to the gravity dual description [17]. Chiral symmetry breaking and meson spectra were studied in a nonsupersymmetric gravity model dual to large nonsupersymmetric gauge theories [18], where flavor quarks are introduced by a D7-brane probe on deformed AdS backgrounds. Using a D4/D6 brane configuration, the authors of [19] explored the meson phenomenology of large QCD together with chiral symmetry breaking. They showed that the chiral condensate scales as for large . A remarkable observation made in [19] is that in addition to the confinement/deconfinement phase transition the model exhibits a possibility that another transition set by could happen in deconfined phase, , where . Since the value of is around 1 GeV, we can estimate MeV. In this case for there exist free unbound quarks and meson bound sates of heavy quarks and above the meson states dissociate into free quarks, which in some sense mimics the dissociation of heavy quarkonium in quark-gluon plasma (QGP). However, we should note that meson bound states in Dp/Dq systems are deeply bound, while the heavy quarkonia in QCD are shallow bound states. In this sense the bound state that disappears above could be that of strange quarks rather than charmonium or bottomonium [20].

To attain a realistic gravity dual description of (large ) QCD, non-Abelian chiral symmetry is an essential ingredient together with confinement. Holographic QCD models, which are equipped with the correct structure for the problem, namely, chiral symmetry and confinement, have been suggested in top-down and bottom-up approaches. They are found to be rather successful for various hadronic observables and for certain processes dominated by large . Based on a model, Sakai and Sugimoto studied hadron phenomenology in the chiral limit , and the chiral symmetry breaking geometrically [21, 22]. More phenomenological holographic QCD models were proposed [23–25]. In [23, 24], chiral symmetry breaking is realized by a nonzero chiral condensate whose value is fitted to meson data from experiments. Hadronic spectra and light-front wave functions were studied in [26] based on the “Light-Front Holography” which maps amplitudes in extra dimension to a Lorentz invariant impact separation variable in Minkowski space at fixed light-front time. Light-Front Holography has led to many successful applications in hadron physics including light-quark hadron spectra, meson and baryon form factors, the nonperturbative QCD coupling, and light-front wave-functions; see [27–29] for a review on this topic. In [30], a relation between a bottom-up holographic QCD model and QCD sum rules was analyzed.

Now, we demonstrate how to construct a bottom-up holographic QCD model by looking at a low-energy QCD. For illustration purposes, we compare our approach with the (gauged) linear sigma model. The D3/D7 model is summarized in Appendix B with some calculational details. For a review of the linear sigma model, we refer to [31]. Some material in this section is taken from [32]. Suppose that we are interested in two-flavor QCD at low energy, roughly below 1 GeV. In this regime usually we resort to the effective models or theories of QCD for analytic studies since the QCD lagrangian does not help much.

To construct the holographic QCD model dual to two flavor low-energy QCD with chiral symmetry, we first choose relevant fields. To do this, we consider composites of quark fields that have the same quantum numbers with the hadrons of interest. For instance, in the linear sigma model we introduce pion-like and sigma-like fields: and , where is the Pauli matrix for isospin. In the AdS/CFT dictionary, this procedure may be dubbed operator/field correspondence: one-to-one mapping between gauge-invariant local operators in gauge theory and bulk fields in gravity sides. Then we introduce An interesting point here is that the 5D mass of the bulk field is not a free parameter of the model. This bulk mass is determined by the dimension and spin of the dual 4D operator in . For instance, consider a bulk field dual to . The bulk mass of is given by with , , and , and so . For more details, see Appendix A.

To write down the Lagrangian of the linear sigma model, we consider (global) chiral symmetry of QCD. Since the mass of light quark  MeV is negligible compared to the QCD scale  MeV, we may consider the exact chiral symmetry of QCD and treat quark mass effect in a perturbative way. Under the axial transformation, , the pion-like and sigma-like states transform as and . From this, we can obtain terms that respect chiral symmetry such as . Similarly we ask the holographic QCD model to respect chiral symmetry of QCD. In AdS/CFT, however, a global symmetry in gauge theory corresponds local symmetry in the bulk, and therefore the corresponding holographic QCD model should posses local chiral symmetry. This way vector and axial-vector fields naturally fit into chiral Lagrangian in the bulk as the gauge boson of the local chiral symmetry.

We keep the chiral symmetry in the Lagrangian since it will be spontaneously broken. Then we should ask how to realize the spontaneous chiral symmetry breaking. In the linear sigma model, we have a potential term like that leads to spontaneous chiral symmetry breaking due to a nonzero vacuum expectation value of the scalar field , . In this case the explicit chiral symmetry due to the small quark mass could be mimicked by adding a term to the potential which induces a finite mass of the pion, . In a holographic QCD model, the chiral symmetry breaking is encoded in the vacuum expectation value of a bulk scalar field dual to . For instance, in the hard wall model [23, 24], it is given by , where and are proportional to the quark mass and the chiral condensate in QCD. In the D3/D7 model, chiral symmetry breaking can be realized by the embedding solution as shown in Appendix B.

The last step to get to the gravity dual to two flavor low-energy QCD is to ensure the confinement to have discrete spectra for hadrons. The simplest way to realize it might be to truncate the extra dimension at such that the radial direction of dual gravity runs from zero to . Since the radial direction corresponds to an energy scale of a boundary gauge theory, maps to .

Putting things together, we could arrive at the following bulk Lagrangian with local , the hard wall model [23, 24]: where and with . The bulk scalar field is defined by , where . Here is the five-dimensional gauge coupling, . The background is given by Instead of the sharp IR cutoff in the hard wall mode, we may introduce a bulk potential that plays a role of a smooth cutoff. In [33], this smooth cutoff is introduced by a factor with in the bulk action, the soft wall model. The form in the AdS would ensure the Regge-like behavior of the mass spectrum . The action is given by Here we briefly show how to obtain the 4D vector meson mass in the soft wall model. The vector field is defined by . With the Kaluza-Klein decomposition , we obtain where in the AdS geometry (3.3). With , we transform the equation of motion into the form of a Schrödinger equation: where . Here is the mass of the vector resonances, and meson corresponds to . The solution is well known in quantum mechanics and the eigenvalue is given by [33] where is introduced to restore the energy dimension.

The finite temperature could be neatly introduced by a black hole in , where is the dimension of the boundary gauge theory. The background is given by where . The temperature of the boundary gauge theory is identified with the Hawking temperature of the black hole . In Appendices D and E, we try to explain in a comprehensive manner how to calculate the Hawking temperature of a black hole.

Now we move on to dense matter. According to the AdS/CFT dictionary, a chemical potential in boundary gauge theory is encoded in the boundary value of the time component of the bulk U(1) gauge field. To be more specific on this, we first consider the chemical potential term in gauge theory: Then, we introduce a bulk U(1) gauge field which is dual to . According to the dictionary, , we have . In the hard wall model, the solution of the bulk U(1) vector field is given by where and are related to quark chemical potential and quark (or baryon) number density in boundary gauge theory. It is interesting to notice that in chiral perturbation theory, a chemical potential is introduced as the time component of a gauge field by promoting the global chiral symmetry to a local gauge one [34, 35].

4. Vacuum Structures

At low energy or momentum scales roughly smaller than 1 GeV,  fm, QCD exhibits confinement and a nontrivial vacuum structure with condensates of quarks and gluons. In this section, we discuss the gluon condensate and quark-gluon mixed condensate.

The gluon condensate was first introduced, at zero temperature, in [36] as a measure for nonperturbative physics in QCD. The gluon condensate characterizes the scale symmetry breaking of massless QCD at quantum level. Under the infinitesimal scale transformation the trace of the energy momentum tensor reads schematically Here is the dilatation current, is the gauge coupling, and is the energy-momentum tensor of QCD. Due to Lorentz invariance, we can write , where is the energy of the QCD vacuum. Therefore, the value of the gluon condensate sets the scale of the QCD vacuum energy. In addition, the gluon condensate is important in the QCD sum rule analysis since it enters in the operator product expansion (OPE) of the hadronic correlators [36]. At high temperature, the gluon condensate is useful to study the nonperturbative nature of the QGP. For instance, lattice QCD results on the gluon condensate at finite temperature [37, 38] indicate that the value of the gluon condensate shows a drastic change around regardless of the number of quark flavors. The change in the gluon condensate could lead to a dropping of the heavy quarkonium mass around [39].

In holographic QCD, the gluon condensate figures in a dilaton profile according to the AdS/CFT since the dilaton is dual to the scalar gluon operator . The 5D gravity action with the dilaton is given by where for Minkowski metric, and for Euclidean signature. We work with Minkowski metric for most cases in this paper. The solution of this system is discovered in [40, 41] by solving the coupled dilaton equation of motion and the Einstein equation: and the corresponding dilaton profile is given by where is a constant. At there exists a naked singularity that might be resolved in a full string theory consideration. Near the boundary , Therefore, is nothing but the gluon condensate up to a constant. Unfortunately, however, is an integration constant of the coupled dilaton equation of motion and the Einstein equation, and therefore, it will be determined by matching with physical observables. In [41], the value of the gluon condensate is estimated by the glueball mass. An interesting idea based on the circular Wilson loop calculation in gravity side is proposed to calculate the value of the gluon condensate [42]. The value is determined to be  GeV at zero temperature [42]. A phenomenological estimation of the gluon condensate in QCD sum rules gives   [36].

Now we consider quark-gluon-mixed condensate , which can be regarded as an additional order parameter for the spontaneous chiral symmetry breaking since the quark chirality flips via the quark-gluon operator. Thus, it is naturally expressed in terms of the quark condensate as In [43], an extended hard wall model is proposed to calculate the value of . The bulk action of the extended model is given by where is a bulk scalar field dual to the 4D operator on the left-hand-side of (4.7). Then the chiral condensate and the mixed condensate are encoded in the vacuum expectation value of the two scalar fields: where is the source term for the mixed condensate and represents the mixed condensate . Taking , a source-free condition to study only spontaneous symmetry breaking, we determine the value of the mixed condensate or by considering various hadronic observables. In this sense the mixed condensate is not calculated but fitted to experimental data like the chiral condensate in the hard wall model. The favored value of the in [43] is 0.72 . A new method to estimate the value of in is suggested in [44], where a nonperturbative gauge invariant correlator (the nonlocal condensate) is calculated in dual gravity description to obtain . With inputs from the slopes for the Regge trajectory of vector mesons and the linear term of the Cornell potential, they obtained  , which is comparable to that from the QCD sum rules, 0.8 Ge [45].

5. Spectroscopy and Form Factors

Any newly proposed models or theories in physics are bound to confront experimental data, for instance, hadron masses, decay constants, and form factors. In this section, we consider the spectroscopy of the glueball, light meson, heavy quarkonium, and hadron form factors in hard wall model, soft wall model, and their variants.

5.1. Glueballs

Glueballs are made up of gluons with no constituent quarks in them. The glueball states are in general mixed with conventional states; so in experiments we may observe these mixed states only. Their existence was expected from the early days of QCD [46, 47]. For theoretical and experimental status of glueballs, we refer to [48, 49].

The spectrum of glueballs is one of the earliest QCD quantities calculated based on the AdS/CFT duality. In [12], Witten confirmed the existence of the mass gap in the dilaton equation of motion on a black hole background, implying a discrete glueball spectrum with a finite gap. Extensive studies on the glueball spectrum were done in [50, 51] and also comparisons between the supergravity results and lattice gauge theory results were made.

Now we consider a scalar glueball () on as an example [50]. When the radius of the circle is very small , only the gauge degrees of freedom remain and the gauge theory is effectively the same as pure QCD₃ [12, 50]. Using the operator/field correspondence, we first find operators that have the quantum numbers with glueball states of interest and then introduce a corresponding bulk field to obtain the glueball masses. In this case we are to solve an equation of motion for a bulk scalar field , which is dual to tr  in the AdS₅ Euclidean black hole background. The equation of motion for is given by and the metric is where is for the compactified imaginary time direction. For simplicity, we assume that is independent of [12, 50] and seek a solution of the form , where is the momentum in . Then the equation of motion for reads where is the three-dimensional glueball mass, [12, 50]. By solving this eigenvalue equation with suitable boundary conditions, regularity at the horizon (), and normalizability at the boundary (), we can obtain discrete eigenvalues, the three-dimensional glueball masses. In the context of a sliced AdS background of the Polchinski and Strassler set up [16], which is dual to confining gauge theory, the mass ratios of glueballs are studied in [52, 53].

More realistic or phenomenology-oriented approaches follow the earlier developments. In the soft wall model the mass spectra of scalar and vector glueballs and their dependence on the bulk geometry and the shape of the soft wall are studied in [54]. The exact glueball correlators are calculated in [55], where the decay constants as well as the mass spectrum of the glueball are also obtained in both hard wall and soft wall models. Here we briefly summarize the scalar glueball properties in the soft wall model [54, 55]. Following a standard path to construct a bottom-up mode, we introduce a massless bulk scalar field dual to the scalar gluon operator to write down the bulk action as [54] where as in the soft wall model. The equation of motion for can be transformed to a one-dimensional Schrödinger form: where with [54]. The glueball mass spectrum is then given as the eigenvalue of the Schrödinger type equation with regular eigenfunction at and : where is an integer, . is introduced to make the exponent dimensionless, , and it will be fit to hadronic data. Since the vector meson mass in the soft wall model is , we calculate the ratio of the lightest scalar glueball mass to the meson mass to obtain [54]. The properties of the glueball at finite temperature are studied in the hard wall model [56] and also in the soft wall model [56, 57] by calculating the spectral function of the glueball in the AdS black hole backgroudn. The spectral function is related to various Green functions, and it can be defined by the two-point retarded Green function as . The retarded function can be computed in the real-time AdS/CFT, following the prescription proposed in [58]. Both studies using the soft wall model predicted that the dissociation temperature of scalar glueballs is far below the deconfinement (Hawking-Page) transition temperature of the soft wall model. See Section 6.1 and Appendix F for more on the Hawking-Page transition. Note that below the Hawking-Page transition temperature, the AdS black hole is unstable. In [56], the melting temperature of the scalar glueball from the spectral functions is about 40–60 MeV, while the deconfinement temperature of the soft wall model is about 190 MeV [59]. This implies that we have to build a more refined holographic QCD model to have a realistic melting temperature [56, 57].

5.2. Light Mesons

There have been an armful of works in holographic QCD that studied light meson spectroscopy. Here we will try to summarize results from the hard wall model, soft wall model, and their variants.

In Table 1, we list some hadronic observables from hard wall models to see if the results are stable against some deformation of the model. In the table, * means input data and the model with no * is a fit to all seven observables: Model A and Model B from the hard wall model [23], Model I from a hard wall model in a deformed AdS geometry [60], and Model II from a hard wall model with the quark-gluon-mixed condensate [43]. In [24], the following deformed AdS background is considered: and it is stated that the correction from the deformation is less than . The backreaction on the AdS metric due to quark mass and chiral condensate is investigated in [60]. One of the deformed backgrounds obtained in [60] phenomenologically reads where . In Table 1, we quote some results from this deformed background. Dynamical (back-reacted) holographic QCD model with area-law confinement and linear Regge trajectories was developed in [61].

We remark that the sensitivity of calculated hadronic observables to the details of the hard wall model was studied in [63] by varying the infrared boundary conditions, the 5D gauge coupling, and scaling dimension of operator. It turns out that predicted hadronic observables are not sensitive to varying scaling dimension of operator, while they are rather sensitive to the IR boundary conditions and the 5D gauge coupling [63].

In addition to mesons, baryons were also studied in the hard wall model [64–67]. It is pointed out in [65, 66] that one has to use the same IR cutoff of the hard wall model for both meson and baryon sectors.

Now we collect some results from the soft wall model [33]. There were two nontrivial issues to be resolved in the original soft wall model. Firstly, so called, the dilaton factor is introduced phenomenologically to explain . The dilaton factor is supposed to be a solution of gravity-dilaton equations of motion. Secondly, the chiral symmetry breaking in the model is a bit different from QCD since the chiral condensate is proportional to the quark mass in the soft wall model. In QCD, in the chiral limit, where the quark mass is zero, the chiral condensate is finite that characterizes spontaneous chiral symmetry breaking. Several attempts have made to improve these aspects and to fit experimental values better [68–71]. In [69], a quartic term in the potential for the bulk scalar dual to is introduced to the soft wall model to incorporate chiral symmetry breaking with independent sources for spontaneous and explicit breaking; thereby the chiral condensate remains finite in the chiral limit. Then, the authors of [69] parameterized the vev of the bulk scalar such that it satisfies constraints from the AdS/CFT at UV and from phenomenology at IR: as and as . The constraint at IR is due to the observation [72] that chiral symmetry is not restored in the highly excited mesons. Note that keeps the mass difference between vector and axial-vector mesons constant as . With the parameterized , they obtained a dilaton factor [69]. We list some of results of [69] in Table 2. An extended soft wall model with a finite UV cutoff was discussed in [73, 74]. In [75], the authors studied a dominant tetra-quark component of the lightest scalar mesons in the soft wall model, where a rather generic lower bound on the tetra-quark mass was derived.

As long as confinement and non-Abelian chiral symmetry are concerned, the Sakai-Sugimoto model [21, 22] based on a brane configuration (see Appendix C) is the only available stringy model. In this model, properties of light mesons and baryons have been greatly studied [21, 22, 77–84].

In a simple bottom-up model with the Chern-Simons term, it was also shown that baryons arise as stable solitons which are the 5D analogs of 4D skyrmions and the properties of the baryons are studied [85].

5.3. Heavy Quarkonium

The properties of heavy quark system both at zero and at finite temperature have been the subject of intense investigation for many years. This is so because, at zero temperature, the charmonium spectrum reflects detailed information about confinement and interquark potentials in QCD. At finite temperature, due to the small interaction cross-section of the charmonium in hadronic matter, the charmonium spectrum is expected to carry information about the early hot and dense stages of relativistic heavy ion collisions. In addition, the charmonium states may remain bound even above the critical temperature . This suggests that analyzing the charmonium data from heavy ion collision inevitably requires more detailed information about the properties of charmonium states in QGP. Therefore, it is very important to develop a consistent nonperturbative QCD picture for the heavy quark system both below and above the phase transition temperature. For a recent review on heavy quarkonium see, for example, [86].

Now we start with the hard wall model to discuss the heavy quarkonium in a bottom-up approach. A simple way to deal with the heavy quarkonium in the hard wall model was proposed in [87]. Since the typical energy scales involved for light mesons and heavy quarkonia are quite different, we may introduce an IR cutoffs for heavy quarkonia in the hard wall model which is different from the IR cutoff for light mesons,  MeV. Note that in the hard wall model there is a one-to-one correspondence between the IR cutoff and the vector meson mass . In [87], the lowest vector () mass is used as an input to fix the IR cutoff for the charmonium, . With this, the mass of the second resonance is predicted to be , which is quite different from the experiment . This is in a sense generic limitation of the hard wall model whose predicted higher resonances are quite different from experiments. Moreover, having two different IR cutoffs in the hard wall model may cause a problem when we treat light quark and heavy quark systems at the same time. In the soft wall model, the mass spectrum of the vector meson is given by [33] For charmonium system, again the lowest mode () is used to fix , . Then the mass of the second resonance is , which is away from the experimental value of [87]. Additionally, the mass of heavy quarkonium such as at finite temperature is calculated to predict that the mass decreases suddenly at and above it increases with temperature. Furthermore, the dissociation temperature is determined to be around in the soft wall model [87].

To compare heavy quarkonium properties obtained in a holographic QCD study with lattice QCD, the finite-temperature spectral function in the vector channel within the soft wall model was explored in [88]. The spectral function is related to the two-point retarded Green function by . The retarded function can be computed following the prescription [58]. Thermal spectral functions in a stringy set-up, D3/D7 model, were extensively studied in [89]. To deal with the heavy quarkonium in the soft wall model, two different scales ( and ) are introduced. It is observed in [88] that a peak in the spectral function melts with increasing temperature and eventually is flattened at . It is also shown numerically that the mass shift squared is approximately proportional to the width broadening [88]. Another interesting finding in [88] is that the spectral peak diminishes at high momentum, which could be interpreted as the suppression under the hot wind [90, 91]. A generalized soft wall mode of charmonium is constructed by considering not only the masses but also the decay constants of the charmonium, and [92]. They calculated the spectral function as well as the position of the complex singularities (quasinormal frequencies) of the retarded correlator of the charm current at finite temperatures. A predicted dissociation temperature is  MeV, or [92].

Alternatively, heavy quarkonium properties can be studied in terms of holographic heavy-quark potentials. Since the mass of heavy quarks is much larger than the QCD scale parameter  MeV, the nonrelativistic Schrödinger equation could be a useful tool to study heavy quark bound states: where is the reduced mass, . A tricky point with potential models for quarkonia is which potential is to be used in the Schrödinger equation: the free energy or the internal energy. In the context of the AdS/CFT, there have been a lot of works on holographic heavy quark potentials [93–103]. Hou and Ren calculated the dissociation temperature of heavy quarkonia by solving the Schrödinger equation with holographic potentials [99]. They used two ansätze of the potential model: the F-ansatz (U-ansatz) which identifies the potential in the Schrödinger equation with the free energy (the internal energy), respectively. With the F-ansatz, does not survive above , while the dissociation temperature of is . For the U-ansatz, dissolves into open charm quarks around and dissociates at about .

We finish this subsection with a summary of the discussion in [20] on the usefulness of Dq/Dp systems in studying heavy quark bound states. A Dq/Dp system may be good for bound states at high temperature since the mesons in the Dq/Dp system are deeply bounded, while heavy quarkonia are shallow bound states. However, there exist certain properties of heavy quarkonia in the quark-gluon plasma that could be understood in the D4/D6 model such as dissociation temperature.

5.4. Form Factors

Form factors are a source of information about the internal structure of hadrons such as the distribution of charge. We take the pion electromagnetic form factor as an example. Consider a pion-electron scattering process through photon exchange. The cross section of this process measured in experiments is different from that of Mott scattering which is for the Coulomb scattering of an electron with a point charge. This deviation is parameterized into the pion form factor , where is given by the energy and momentum of the photon . If the pion is a structureless point particle, we have . The pion electromagnetic form factor is expressed by, with the use of Lorentz invariance, charge conjugation, and electromagnetic gauge invariance: where and is the electromagnetic current, . The pion charge radius is determined by In a vector meson dominance model, where the photon interacts with the pion only via vector mesons, especially meson, the pion form factor is given by Then we obtain the pion charge radius fm. The experimental value is  fm [104]. To evaluate the form factor, we consider the three-point correlation function of two axial vector currents which contains nonzero projection onto a one pion state and the external electromagnetic current: Alternatively, we can consider two pseudoscalar currents instead of the axial vector currents. The three-point correlation function can be decomposed into several independent Lorentz structures. Among them we pick up the Lorentz structure corresponding to the pion form factor: Note that , where is a one pion state. For more details on the form factor, we refer to [105–107].

In a holographic QCD approach, we can easily evaluate the three-point correlation function of two axial vector currents (or two pseudoscalar currents) and the external electromagnetic current. In [108], the form factors of vector mesons were calculated in the hard wall model and the electric charge radius of the -meson was evaluated to be  . The number from the soft wall model is   [109]. The approach based on the Dyson-Schwinger equations predicted   [110] and   [111]. The quark mass (or pion mass) dependence of the charge radius of the -meson was calculated in lattice QCD: for instance, with  MeV,   [112]. The pion form factor was studied in the hard wall model [113] and in a model that interpolates between the hard wall and soft wall models [114]. The results obtained are fm [113] and in [114]  fm,  fm, depending on their parameter choice. The gravitational form factors of mesons were calculated in the hard wall model [115, 116]. The gravitational form factor of the pion is defined by where is the energy momentum tensor, , and . There are also interesting works that studied various form factors in holographic QCD [117–120]. Form factors of vector and axial-vector mesons were calculated in the Sakai-Sugimoto model (Figure 4) [121].

Figure 4 

QCD phase diagram.

6. Phases of QCD

Understanding the QCD phase structure is one of the important problems in modern theoretical physics; see [122–126] for some recent reviews. However, a quantitative calculation of the phase diagram from the first principle is extraordinarily difficult.

Basic order parameters for the QCD phase transitions are the Polyakov loop which characterizes the deconfinement transition in the limit of infinitely large quark mass and the chiral condensate for chiral symmetry in the limit of zero quark mass. The expectation value of the Polyakov loop is loosely given by where is the potential between a static quark-antiquark pair at a distance r, and . The expectation value of the Polyakov loop is zero in confined phase, and it is finite in deconfined phase, while the chiral condensate, which is the simplest order parameter for the chiral symmetry, is nonzero with broken chiral symmetry, vanishing with a restored chiral symmetry. Apart from these order parameters, there are thermodynamic quantities that are relevant to study the QCD phase transition. The equation of state is one of them. The energy density, for instance, has been found to rise rapidly at some critical temperature. This is usually interpreted as deconfinement: liberation of many new degrees of freedom. The fluctuations of conserved charges such as baryon number or electric charge [127–130] are also an important signal of the quark-hadron phase transition. The quark (or baryon) number susceptibility, which measures the response of QCD to a change of the quark chemical potential, is one of such fluctuations [127, 131].

The nature of the chiral transition of QCD depends on the number of quark flavors and the value of the quark mass. For pure gauge theory with no quarks, it is first order. In the case of two massless and one massive quarks, the transition is the second-order at zero or small quark chemical potentials, and it becomes the first order as we increase the chemical potential. The point where the second order transition becomes the first order is called tricritical point. With physical quark masses of up, down, and strange, the second order at zero or low chemical potential becomes the crossover, and the tricritical point turns into the critical end point.

6.1. Confinement/Deconfinement Transition

We first discuss the deconfinement transition. In holographic QCD, the confinement to deconfinement phase transition is described by the Hawking-Page transition [132], a phase transition between the Schwarzschild-AdS black hole and thermal AdS backgrounds. This identification was made in [12]. One simple reasoning for this identification is from the observation that the Polyakov expectation value is zero on the thermal AdS geometry, while it is finite on the AdS black hole. See Appendix F for some more description of the Hawking-Page transition and the Polyakov expectation in thermal AdS and AdS black hole. In low-temperature confined phase, thermal AdS, which is nothing but the AdS metric in Euclidean space, dominates the partition function, while at high temperature, AdS-black hole geometry does. This was first discovered in the finite volume boundary case in [12]. In the bottom-up model, it is shown that the same phenomena happen also for infinite boundary volume if there is a finite scale associated with the fifth direction [59].

Here we briefly summarize the Hawking-Page analysis of [59] done in the hard wall model. In the Euclidean gravitational action given by where and is the length scale of the , there are two solutions for the equations of motion derived from the gravitational action. The one is the sliced thermal AdS (tAdS): where the radial coordinate runs from the boundary of tAdS space to the cut-off . Here is for the compactified Euclidean time-direction with periodicity . The other solution is the AdS black hole (AdSBH) with the horizon : where . The Hawking temperature of the black hole solution is , which is given by regularizing the metric near the horizon. At the boundary the periodicity of the time-direction in both backgrounds is the same and so the time periodicity of the tAdS is given by Now we calculate the action density , which is defined by the action divided by the common volume factor of . The regularized action density of the tAdS is given by and that of the AdSBH is given by where . Then, the difference of the regularized actions is given by When is positive (negative), tAdS (AdSBH) is stable. Thus, at there exists a Hawking-Page transition. In the first case , there is no Hawking-Page transition and the tAdS is always stable. In the second case , the Hawking-Page transition occurs at and at low temperature (at high temperature ) the thermal AdS (the AdS black hole) geometry becomes a dominant background. When we fix the IR cutoff by the meson mass, we obtain  MeV and  MeV. In the soft wall model, MeV [59].