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 𝑁𝑐 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 D4/D8/D8 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 AdS5×𝑆5 and the strongly coupled 𝒩=4 supersymmetric Yang-Mills theory on the four-dimensional boundary of AdS5. 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. D𝑝 Brane Dynamics

The duality emerges from a careful consideration of the D-brane dynamics. A D𝑝 brane sweeps out (𝑝+1) 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 (𝑝+1) directions of the brane, see Figure 1(a), and a D𝑝 brane can be seen as a point along its transverse directions. The dynamics of the D𝑝 brane is described by the Dirac-Born-Infeld (DBI) action [5] and Chern-Simons term:𝑆D𝑝=βˆ’π‘‡π‘ξ€œπ‘‘π‘+1π‘₯π‘’βˆ’πœ™ξ”ξ€·π‘ƒ[𝑔]βˆ’detπ‘Žπ‘+2πœ‹π›Όξ…žπΉπ‘Žπ‘ξ€Έ+𝑆𝐢𝑆(2.1) with a dilaton π‘’βˆ’πœ™. Here π‘”π‘Žπ‘ is the induced metric on D𝑝. 𝑃 denotes the pullback and πΉπ‘Žπ‘ is the world-volume field strength. 𝑇𝑝 is the tension of the brane which has the following form:𝑇𝑝=1(2πœ‹)𝑝𝑔𝑠𝑙𝑠𝑝+1=1(2πœ‹)π‘π‘”π‘ π›Όξ…ž(𝑝+1)/2,(2.2) and it is the mass per unit spatial volume. Here 𝑔𝑠 is the string coupling and 𝑙𝑠 is the string length. π›Όξ…ž is the Regge slope parameter and related to the string length scale as 𝑙𝑠=βˆšπ›Όξ…ž. 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 π‘šπ‘ =π‘™π‘ βˆ’1=π›Όξ…žβˆ’1/2. Thus, at low energies 𝐸β‰ͺπ‘šπ‘ , the higher-order corrections come in powers of 𝛼′𝐸2 from integrating out the massive string modes. If there are 𝑁𝑐 stack of multiple D branes, the open strings between different branes give a non-Abelian π‘ˆ(𝑁𝑐) 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.

Now, we take 𝑝=3 and consider 𝑁𝑐 D3 brane stacks in type IIB theory. The low-energy effective action of this configuration gives a non-Abelian gauge theory with π‘ˆ(𝑁𝑐) gauge group. In addition, this gauge group can be factorized into π‘ˆ(𝑁𝑐)=π‘†π‘ˆ(𝑁𝑐)Γ—π‘ˆ(1) and the π‘ˆ(1) part, which describes the center of mass motion of the D3 branes, can be decoupled by the global translational invariance. The remaining subgroup π‘†π‘ˆ(𝑁𝑐) describes the dynamics of branes from each other. Therefore we see that in the low-energy limit, the massless open string modes on 𝑁𝑐 stacks of D3-branes constitute 𝒩=4π‘†π‘ˆ(𝑁𝑐) Yang-Mills theory [6] with 16 supercharges in (3+1) spacetime. From (2.1) for 𝑝=3, we obtain the effective Lagrangian at low energies up to the two-derivative order:1β„’=βˆ’4πœ‹π‘”π‘ ξ‚€1Tr4πΉπœ‡πœˆπΉπœ‡πœˆ+12π·πœ‡πœ™π‘–π·πœ‡πœ™π‘–βˆ’14ξ€Ίπœ™π‘–,πœ™π‘—ξ€»2+𝑖2Ξ¨πΌΞ“πœ‡π·πœ‡Ξ¨πΌβˆ’π‘–2Ξ¨πΌΞ“π‘–ξ€Ίπœ™π‘–,Ψ𝐼(2.3) with a gauge field π΄πœ‡, six scalar fields πœ™π‘–, and four Weyl fermions Ψ𝐼.

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 𝛼′𝐸2 from the massive modes and powers of the string coupling 𝑔𝑠𝑁𝑐 for loop corrections. It is known that the string coupling constant 𝑔𝑠 is related by the 10-dimensional gravity constant as 𝐺(10)βˆΌπ‘”2𝑠𝑙8𝑠 and thus the dimensionless string coupling is of order 𝐺(10)𝐸8, which is negligible in the low-energy limit. Therefore, at low energies closed strings are decoupled from open strings and the physics on the 𝑁𝑐 D3 branes is described by the massless 𝒩=4 super Yang-Mills theory with gauge group π‘†π‘ˆ(𝑁𝑐).

2.2. AdS5×𝑆5 Geometry

Now we view the same system from a different angle. Since D branes are massive and carry energy and Ramond-Ramond (RR) charge, 𝑁𝑐 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 brane𝑇3=1(2πœ‹)3𝑔𝑠𝑙4𝑠(2.4) is finite.

In the flat spacetime, the circumference of the circle surrounding an origin at a distance π‘Ÿ is 2πœ‹π‘Ÿ, 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 𝑅, an asymptotical infinite cylinder structure, or AdS5×𝑆5; see Figure 2(b). The 𝑁𝑐 D3 brane stack is located at the infinite end of the throat and this infinite end is called the β€œhorizon”. In the near horizon geometry, a D3 brane is surrounded by a five-dimensional sphere 𝑆5.

To be more specific, let us start with type IIB string theory for 𝑝=3. 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 𝑑𝐴4 and it is self-dual βˆ—πΉ5=𝐹5. The low-energy effective action is1𝑆=(2πœ‹)7𝑙8π‘ ξ€œπ‘‘10π‘₯βˆšξ‚€π‘’βˆ’π‘”βˆ’2πœ™ξ€·π‘…+4(βˆ‡πœ‘)2ξ€Έβˆ’2𝐹5!25.(2.5) We assume that the metric is spherically symmetric in seven dimensions with the RR source at the origin; then the 𝑁𝑐 parameter appears in terms of the five-form field RR-field strength on the five-sphere asξ€œπ‘†5βˆ—πΉ5=𝑁𝑐,(2.6) where 𝑆5 is the five-sphere surrounding the source for a four-form field 𝐢4. Now by using the Euclidean symmetry we get the curved metric solution [7–9] for the D3 brane:𝑑𝑠2=𝑓(π‘Ÿ)βˆ’1/2πœ‚πœ‡πœˆπ‘‘π‘₯πœ‡π‘‘π‘₯𝜈+𝑓(π‘Ÿ)1/2ξ€·π‘‘π‘Ÿ2+π‘Ÿ2𝑑Ω25ξ€Έ,(2.7) where𝑅𝑓(π‘Ÿ)=1+4π‘Ÿ4(2.8) with the radius of the horizon 𝑅:𝑅2=√4πœ‹π‘”π‘ π‘π‘βˆšπ›Όβ€²=4πœ‹π‘”π‘ π‘π‘π‘™2𝑠.(2.9)𝑑Ω5 is the five-sphere metric. For π‘Ÿβ‰«π‘… we have 𝑓(π‘Ÿ)≃1 and the spacetime becomes flat with a small correction 𝑅4/π‘Ÿ4=4πœ‹π‘”π‘ π‘π‘π‘™4𝑠/π‘Ÿ4. This factor can be interpreted as a gravitational potential since 𝐺(10)βˆΌπ‘”2𝑠𝑙8𝑠 and 𝑀D3βˆΌπ‘π‘π‘‡3βˆΌπ‘π‘/𝑔𝑠𝑙4𝑠 and thus 𝑅4/π‘Ÿ4βˆΌπΊπ‘€D3/π‘Ÿ4. In the near horizon limit, this gravitational effect becomes strong and the metric changes into𝑑𝑠2=π‘Ÿ2𝑅2πœ‚πœ‡πœˆπ‘‘π‘₯πœ‡π‘‘π‘₯𝜈+𝑅2π‘Ÿ2ξ€·π‘‘π‘Ÿ2+π‘Ÿ2𝑑Ω25ξ€Έ.(2.10) This is AdS5×𝑆5.

The geometry by the D3 branes is sketched in Figure 2(b). Far away from the D3 brane stacks, the spacetime is flat (9+1)-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 AdS5×𝑆5 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 AdS5×𝑆5 geometry.

2.3. The Gauge/Gravity Duality

So far, we have considered two seemingly different descriptions of the 𝑁𝑐 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 𝑔𝑠 as in (2.4). So the strength of the gravity effect due to 𝑁𝑐 stacks of D3 branes depends on the parameter 𝑔𝑠𝑁𝑐.

If 𝑔𝑠𝑁𝑐β‰ͺ1, from (2.9) we see that 𝑅β‰ͺ𝑙𝑠 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 𝑔𝑠 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 𝒩=4 super Yang-Mills theory with π‘†π‘ˆ(𝑁𝑐) gauge group.

On the other hand, if 𝑔𝑠𝑁𝑐≫1, 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 𝑔𝑠𝑁𝑐 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, AdS5×𝑆5.

The gauge/gravity correspondence is nothing but the conjecture connecting these two descriptions of 𝑁𝑐 D3 branes in the low-energy limit. It is a duality between the 𝒩=4 super-Yang-Mills theory with gauge group π‘†π‘ˆ(𝑁𝑐) and the type IIB closed string theory in AdS5×𝑆5; see Figure 3. The relation between the Yang-Mills coupling 𝑔YM and the string coupling strength 𝑔𝑠 is given by𝑔2YM=4πœ‹π‘”π‘ ,𝑅𝑙𝑠4=4πœ‹π‘”π‘ π‘π‘.(2.11) Then, the β€˜t Hooft coupling πœ†=𝑔2YM𝑁𝑐 can be expressed in terms of the string length scale:π‘…πœ†=𝑙4𝑠4.(2.12) Therefore, the dependence of 𝑔𝑠𝑁𝑐 becomes the question of whether the β€˜t Hooft coupling is large or small, or the gauge theory is strongly or weakly coupled.

The two descriptions can be viewed as two extremes of π‘Ÿ. For the sake of convenience, we use the coordinate 𝑧=𝑅2/π‘Ÿ. Then, the AdS5×𝑆5 metric (2.10) becomes𝑑𝑠2=𝑅2𝑧2ξ€·πœ‚πœ‡πœˆπ‘‘π‘₯πœ‡π‘‘π‘₯𝜈+𝑑𝑧2ξ€Έ+𝑅2𝑑Ω25,(2.13) which shows the conformal equivalence between AdS5 and flat spacetime more clearly. In (2.13), each 𝑧-slice of AdS5 is isometric to four-dimensional Minkowski spacetime. In this coordinate, 𝑧=0 is the boundary of AdS5, where Yang-Mills theory lives, with identifying π‘₯πœ‡ as the coordinates of the gauge theory. If π‘§β†’βˆž, the determinant of the metric goes to zero and it is the PoincarΓ© horizon. Here the factor 𝑅2/𝑧2 also has some relation with the energy scales. If the gauge theory side has a certain energy scale 𝐸, the corresponding energy in the gravity side is (𝑧/𝑅)𝐸. In other words, a gauge theory object with an energy scale 𝐸 is involved with a bulk side one localized in the z-direction at π‘§βˆΌ1/𝐸 [1, 10, 11]. Therefore, the UV or high-energy limit corresponds to 𝑧→0 (or π‘Ÿβ†’βˆž) and the IR or low-energy limit corresponds to π‘§β†’βˆž (or π‘Ÿβ†’0).

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,ξ‚¬π‘‡π‘’βˆ«π‘‘4π‘₯πœ™0(π‘₯)π’ͺ(π‘₯)ξ‚­CFT=𝑍sugra,(2.14) where πœ™0(π‘₯)=πœ™(π‘₯,π‘’β†’βˆž) and the string theory partition function 𝑍sugra at the boundary specified by πœ™0 has the form𝑍sugra=π‘’βˆ’π‘†sugra(πœ™(π‘₯,𝑒))|π‘’β†’βˆž.(2.15) 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 𝒩=1 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 𝑁𝑐 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 𝑁𝑐 QCD together with π‘ˆ(1)A chiral symmetry breaking. They showed that the chiral condensate scales as 1/π‘šπ‘ž for large π‘šπ‘ž. 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 𝑇fund could happen in deconfined phase, 𝑇>𝑇deconf, where 𝑇deconf=𝑀KK/(2πœ‹). Since the value of 𝑀KK is around 1 GeV, we can estimate 𝑇deconf∼160MeV. In this case for 𝑇deconf<𝑇<𝑇fund there exist free unbound quarks and meson bound sates of heavy quarks and above 𝑇fund 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 𝑇fund could be that of strange quarks rather than charmonium or bottomonium [20].

To attain a realistic gravity dual description of (large 𝑁𝑐) 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 𝑁𝑐. Based on a D4/D8/D8 model, Sakai and Sugimoto studied hadron phenomenology in the chiral limit π‘šπ‘ž=0, 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 β€œLight-Front Holography” which maps amplitudes in extra dimension to a Lorentz invariant impact separation variable 𝜁 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: βƒ—πœ‹βˆΌπ‘žβƒ—πœπ›Ύ5π‘ž and πœŽβˆΌπ‘žπ‘ž, where βƒ—πœ 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 π‘žπΏπ›Ύπœ‡π‘‘π‘Žπ‘žπΏβŸ·π΄π‘ŽπΏπœ‡(π‘₯,𝑧),π‘žπ‘…π›Ύπœ‡π‘‘π‘Žπ‘žπ‘…βŸ·π΄π‘Žπ‘…πœ‡(π‘₯,𝑧),π‘žπ›Όπ‘…π‘žπ›½πΏβŸ·ξ‚€2𝑧𝑋𝛼𝛽(π‘₯,𝑧).(3.1) 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 Ξ” and spin 𝑝 of the dual 4D operator in AdS𝑑+1. For instance, consider a bulk field 𝑋(π‘₯,𝑧) dual to π‘ž(π‘₯)π‘ž(π‘₯). The bulk mass of 𝑋(π‘₯,𝑧) is given by π‘š2𝑋=(Ξ”βˆ’π‘)(Ξ”+π‘βˆ’π‘‘) with Ξ”=3, 𝑝=0, and 𝑑=4, and so π‘š2𝑋=βˆ’3. 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 ∼10 MeV is negligible compared to the QCD scale Ξ›QCD∼200 MeV, we may consider the exact chiral symmetry of QCD and treat quark mass effect in a perturbative way. Under the axial transformation, π‘žβ†’π‘’βˆ’π‘–π›Ύ5βƒ—βƒ—πœβ‹…πœƒ/2π‘ž, the pion-like and sigma-like states transform as βƒ—βƒ—πœ‹β†’βƒ—πœ‹+πœƒπœŽ and βƒ—πœŽβ†’πœŽβˆ’πœƒβ‹…βƒ—πœ‹. From this, we can obtain terms that respect chiral symmetry such as βƒ—πœ‹2+𝜎2. 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 ((βƒ—πœ‹2+𝜎2)βˆ’π‘2)2 that leads to spontaneous chiral symmetry breaking due to a nonzero vacuum expectation value of the scalar field 𝜎, ⟨𝜎⟩=𝑐. In this case the explicit chiral symmetry due to the small quark mass could be mimicked by adding a term βˆ’πœ–πœŽ to the potential which induces a finite mass of the pion, π‘š2πœ‹βˆΌπœ–/𝑐. In a holographic QCD model, the chiral symmetry breaking is encoded in the vacuum expectation value of a bulk scalar field dual to π‘žπ‘ž. For instance, in the hard wall model [23, 24], it is given by βŸ¨π‘‹βŸ©=π‘šπ‘žπ‘§+πœπ‘§3, where π‘šπ‘ž and 𝜁 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 𝑧=π‘§π‘š such that the radial direction 𝑧 of dual gravity runs from zero to π‘§π‘š. Since the radial direction corresponds to an energy scale of a boundary gauge theory, 1/π‘§π‘š maps to Ξ›QCD.

Putting things together, we could arrive at the following bulk Lagrangian with local π‘†π‘ˆ(2)LΓ—π‘†π‘ˆ(2)R, the hard wall model [23, 24]:𝑆HW=ξ€œπ‘‘4π‘₯ξ€œβˆšπ‘‘π‘§ξƒ¬βˆ’1𝑔Tr4𝑔25𝐹2𝐿+𝐹2𝑅+||||𝐷𝑋2||𝑋||+32ξƒ­,(3.2) where π·πœ‡π‘‹=πœ•πœ‡π‘‹βˆ’π‘–π΄πΏπœ‡π‘‹+π‘–π‘‹π΄π‘…πœ‡ and 𝐴𝐿,𝑅=π΄π‘ŽπΏ,π‘…π‘‘π‘Ž with Tr(π‘‘π‘Žπ‘‘π‘)=(1/2)π›Ώπ‘Žπ‘. The bulk scalar field is defined by 𝑋=𝑋0𝑒2π‘–πœ‹π‘Žπ‘‘π‘Ž, where 𝑋0β‰‘βŸ¨π‘‹βŸ©. Here 𝑔5 is the five-dimensional gauge coupling, 𝑔25=12πœ‹2/𝑁𝑐. The background is given by𝑑𝑠2=1𝑧2𝑑𝑑2βˆ’π‘‘ξ‹π‘₯2βˆ’π‘‘π‘§2ξ€Έ,0β‰€π‘§β‰€π‘§π‘š.(3.3) 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 π‘’βˆ’Ξ¦ with Ξ¦(𝑧)=𝑧2 in the bulk action, the soft wall model. The form Ξ¦(𝑧)=𝑧2 in the AdS would ensure the Regge-like behavior of the mass spectrum π‘š2π‘›βˆΌπ‘›. The action is given by𝑆SW=ξ€œπ‘‘4π‘₯ξ€œπ‘‘π‘§π‘’βˆ’Ξ¦βˆšξƒ¬βˆ’1𝑔Tr4𝑔25𝐹2𝐿+𝐹2𝑅+||||𝐷𝑋2||𝑋||+32ξƒ­.(3.4) Here we briefly show how to obtain the 4D vector meson mass in the soft wall model. The vector field is defined by 𝑉=𝐴𝐿+𝐴𝑅. With the Kaluza-Klein decomposition π‘‰π‘Žπœ‡(π‘₯,𝑧)=𝑔5βˆ‘π‘›π‘£π‘›(𝑧)πœŒπ‘Žπœ‡(π‘₯), we obtain πœ•π‘§ξ€·π‘’βˆ’π΅πœ•π‘§π‘£π‘›ξ€Έ+π‘š2π‘›π‘’βˆ’π΅π‘£π‘›=0,(3.5) where 𝐡=Ξ¦(𝑧)βˆ’π΄(𝑧)=𝑧2+log𝑧 in the AdS geometry (3.3). With 𝑣𝑛=π‘’π΅πœ“π‘›, we transform the equation of motion into the form of a SchrΓΆdinger equation: πœ“π‘›ξ…žξ…žβˆ’π‘‰(𝑧)πœ“π‘›=βˆ’π‘š2π‘›πœ“π‘›,(3.6) where 𝑉(𝑧)=𝑧2+3/(4𝑧2). Here π‘šπ‘› is the mass of the vector resonances, and 𝜌 meson corresponds to 𝑛=0. The solution is well known in quantum mechanics and the eigenvalue π‘š2𝑛 is given by [33] π‘š2𝑛=4𝑐(𝑛+1),(3.7) where 𝑐 is introduced to restore the energy dimension.

The finite temperature could be neatly introduced by a black hole in AdS𝑑+1, where 𝑑 is the dimension of the boundary gauge theory. The background is given by𝑑𝑠2=1𝑧2𝑓(𝑧)𝑑𝑑2βˆ’π‘‘ξ‹π‘₯2βˆ’π‘‘π‘§2𝑓(𝑧),(3.8) where 𝑓(𝑧)=1βˆ’π‘§π‘‘/π‘§π‘‘β„Ž. The temperature of the boundary gauge theory is identified with the Hawking temperature of the black hole 𝑇=𝑑/(4πœ‹π‘§β„Ž). 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:β„’πœ‡=πœ‡π‘žπ‘žβ€ π‘ž.(3.9) Then, we introduce a bulk U(1) gauge field π΄πœ‡ which is dual to π‘žπ›Ύπœ‡π‘ž. According to the dictionary, 𝐴0(𝑧→0)βˆΌπ‘1π‘§π‘‘βˆ’Ξ”βˆ’π‘+𝑐2π‘§Ξ”βˆ’π‘, we have 𝐴0(𝑧→0)βˆΌπœ‡π‘ž. In the hard wall model, the solution of the bulk U(1) vector field is given by𝐴𝑑(𝑧)=πœ‡+πœŒπ‘§2,(3.10) where πœ‡ and 𝜌 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, π‘Ÿ>1 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 βŸ¨πΊπ‘Žπœ‡πœˆπΊπ‘Žπœ‡πœˆβŸ© 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 π‘₯ξ…žπœ‡=(1+π›Ώπœ†)π‘₯πœ‡,π΄ξ…žπœ‡=(1βˆ’π›Ώπœ†)π΄πœ‡,π‘žξ…ž=ξ‚€31βˆ’2ξ‚π›Ώπœ†π‘ž,(4.1) the trace of the energy momentum tensor reads schematically πœ•πœ‡π½πœ‡D=π‘‡πœ‡πœ‡π›Όβ‰ƒβˆ’βŸ¨π‘ πœ‹πΊπ‘Žπœ‡πœˆπΊπ‘Žπœ‡πœˆβŸ©.(4.2) Here π½πœ‡D is the dilatation current, 𝛼𝑠 is the gauge coupling, and π‘‡πœ‡πœˆ is the energy-momentum tensor of QCD. Due to Lorentz invariance, we can write βŸ¨π‘‡πœ‡πœˆβŸ©=βˆ’πœ–vacπœ‚πœ‡πœˆ, where πœ–vac 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 𝑇𝑐 regardless of the number of quark flavors. The change in the gluon condensate could lead to a dropping of the heavy quarkonium mass around 𝑇𝑐 [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 Tr(πΊπœ‡πœˆπΊπœ‡πœˆ). The 5D gravity action with the dilaton is given by 1𝑆=𝛾2πœ…2ξ€œπ‘‘5π‘₯βˆšπ‘”ξ‚Έβ„›+12𝑅2βˆ’12πœ•π‘€πœ™πœ•π‘€πœ™ξ‚Ή,(4.3) where 𝛾=+1 for Minkowski metric, and 𝛾=βˆ’1 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: 𝑑𝑠2=𝑅𝑧2ξ‚€βˆš1βˆ’π‘2𝑧8πœ‚πœ‡πœˆπ‘‘π‘₯πœ‡π‘‘π‘₯𝜈+𝑑𝑧2,(4.4) and the corresponding dilaton profile is given by πœ™ξ‚™(𝑧)=32ξ‚΅log1+𝑐𝑧41βˆ’π‘π‘§4ξ‚Ά+πœ™0,(4.5) where πœ™0 is a constant. At 𝑧=1/𝑐1/4 there exists a naked singularity that might be resolved in a full string theory consideration. Near the boundary 𝑧→0, πœ™(𝑧)βˆΌπ‘π‘§4.(4.6) Therefore, 𝑐 is nothing but the gluon condensate up to a constant. Unfortunately, however, 𝑐 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 𝐺2β‰‘βŸ¨(𝛼𝑠/πœ‹)πΊπ‘Žπœ‡πœˆπΊπ‘Žπœ‡πœˆβŸ© [42]. The value is determined to be 𝐺2=0.010Β±0.0023 GeV at zero temperature [42]. A phenomenological estimation of the gluon condensate in QCD sum rules gives ⟨(𝛼𝑠/πœ‹)πΊπ‘Žπœ‡πœˆπΊπ‘Žπœ‡πœˆβŸ©β‰ƒ0.012 GeV4 [36].

Now we consider quark-gluon-mixed condensate βŸ¨π‘žπœŽπœ‡πœˆπΊπœ‡πœˆπ‘žβŸ©, 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 βŸ¨π‘žπœŽπœ‡πœˆπΊπœ‡πœˆπ‘žβŸ©=π‘š20βŸ¨π‘žπ‘žβŸ©.(4.7) In [43], an extended hard wall model is proposed to calculate the value of π‘š20. The bulk action of the extended model is given byξ€œπ‘‘π‘†=5π‘₯βˆšξƒ¬||||𝑔Tr𝐷𝑋2||𝑋||+32βˆ’14𝑔25𝐹2𝐿+𝐹2𝑅+||||𝐷Φ2βˆ’5Ξ¦2ξƒ­,(4.8) where Ξ¦ 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:1βŸ¨π‘‹(π‘₯,𝑧)⟩=2ξ€·ξπ‘šπ‘§+πœŽπ‘§3ξ€Έ,1⟨Φ(π‘₯,𝑧)⟩=6𝑐1π‘§βˆ’1+πœŽπ‘€π‘§5ξ€Έ,(4.9) where 𝑐1 is the source term for the mixed condensate and πœŽπ‘€ represents the mixed condensate πœŽπ‘€=βŸ¨π‘žπ‘…πœŽπœ‡πœˆπΊπœ‡πœˆπ‘žπΏβŸ©. Taking 𝑐1=0, a source-free condition to study only spontaneous symmetry breaking, we determine the value of the mixed condensate or π‘š20 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 π‘š20 in [43] is 0.72 GeV2. A new method to estimate the value of π‘š20 in is suggested in [44], where a nonperturbative gauge invariant correlator (the nonlocal condensate) is calculated in dual gravity description to obtain π‘š20. With inputs from the slopes for the Regge trajectory of vector mesons and the linear term of the Cornell potential, they obtained π‘š20=0.70 GeV2, which is comparable to that from the QCD sum rules, 0.8 GeV2 [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 π‘žπ‘ž 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 (0++) on 𝐑3×𝐒1 as an example [50]. When the radius of the circle 𝐒1 is very small 𝑅→0, 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 πœ™, which is dual to tr 𝐹2 in the AdS5 Euclidean black hole background. The equation of motion for πœ™ is given by πœ•πœ‡ξ‚€βˆšπ‘”πœ•πœˆπœ™π‘”πœ‡πœˆξ‚=0,(5.1) and the metric is 𝑑𝑠2=ξ‚΅πœŒ2βˆ’π‘4𝜌2ξ‚Άβˆ’1π‘‘πœŒ2+ξ‚΅πœŒ2βˆ’π‘4𝜌2ξ‚Άπ‘‘πœ2+𝜌2+𝑑⃗π‘₯2+𝑑Ω25,(5.2) where 𝜏 is for the compactified imaginary time direction. For simplicity, we assume that πœ™ is independent of 𝜏 [12, 50] and seek a solution of the form πœ™(𝜌,π‘₯)=𝑓(𝜌)π‘’π‘˜β‹…π‘₯, where βƒ—π‘˜ is the momentum in 𝐑3. Then the equation of motion for 𝑓(𝜌) reads πœŒβˆ’1π‘‘ξ‚΅ξ€·πœŒπ‘‘πœŒ4βˆ’π‘4ξ€ΈπœŒπ‘‘π‘“ξ‚Άπ‘‘πœŒ+π‘š2=0,(5.3) where π‘š2 is the three-dimensional glueball mass, π‘š2=βˆ’π‘˜2 [12, 50]. By solving this eigenvalue equation with suitable boundary conditions, regularity at the horizon (𝜌=𝑏), and normalizability π‘“βˆΌπœŒβˆ’4 at the boundary (πœŒβ†’βˆž), 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 πœ™ dual to the scalar gluon operator Tr(πΉπœ‡πœˆπΉπœ‡πœˆ) to write down the bulk action as [54] ξ€œπ‘‘π‘†βˆΌ5βˆšπ‘”π‘’βˆ’Ξ¦π‘”π‘€π‘ξ€·πœ•π‘€πœ™πœ•ξ€Έξ€·π‘πœ™ξ€Έ,(5.4) where Ξ¦=𝑧2 as in the soft wall model. The equation of motion for πœ™(π‘ž,𝑧) can be transformed to a one-dimensional SchrΓΆdinger form: πœ“ξ…žξ…žβˆ’π‘‰(𝑧)πœ“=π‘ž2πœ“,(5.5) where πœ“=π‘’βˆ’(Ξ¦+3ln𝑧)/2πœ™ with π‘ž2=π‘š2 [54]. The glueball mass spectrum is then given as the eigenvalue of the SchrΓΆdinger type equation with regular eigenfunction at 𝑧=0 and 𝑧=∞: π‘š2𝑛=4(𝑛+2)̃𝑐,(5.6) where 𝑛 is an integer, 𝑛=0,1,2,…. ̃𝑐 is introduced to make the exponent Ξ¦ dimensionless, Ξ¦=̃𝑐𝑧2, and it will be fit to hadronic data. Since the vector meson mass in the soft wall model is π‘š2𝑛=4(𝑛+1)̃𝑐, we calculate the ratio of the lightest (𝑛=0) scalar glueball mass π‘š2𝐺0 to the 𝜌 meson mass to obtain π‘š2𝐺0/π‘š2𝜌=2 [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 𝜌(πœ”,βƒ—π‘ž)=βˆ’2Im𝐺𝑅(πœ”,βƒ—π‘ž). 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:𝑑𝑠2=πœ‹2π‘§π‘šξ€Ίξ€·sinπœ‹π‘§/2π‘§π‘šξ€·ξ€Έξ€»π‘‘π‘‘2βˆ’π‘‘π‘₯𝑖𝑑π‘₯π‘–βˆ’π‘‘π‘§2ξ€Έ,0β‰€π‘§β‰€π‘§π‘š,(5.7) and it is stated that the correction from the deformation is less than 10%. 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 reads𝑑𝑠2=1𝑧2π‘’βˆ’2𝐡(𝑧)𝑑𝑑2βˆ’π‘‘π‘₯𝑖𝑑π‘₯π‘–βˆ’π‘‘π‘§2ξ€Έ,0β‰€π‘§β‰€π‘§π‘š,(5.8) where 𝐡(𝑧)=(π‘š2π‘ž/24)𝑧2+(π‘šπ‘žπœŽ/16)𝑧4+(𝜎2/24)𝑧6. 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 π‘žπ‘ž operator. It turns out that predicted hadronic observables are not sensitive to varying scaling dimension of π‘žπ‘ž 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 π‘§π‘š 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 Ξ¦βˆΌπ‘§2 is introduced phenomenologically to explain π‘š2π‘›βˆΌπ‘›. 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 𝑋 dual to π‘žπ‘ž 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 𝑋0 such that it satisfies constraints from the AdS/CFT at UV and from phenomenology at IR: 𝑋0βˆΌπ‘šπ‘žπ‘§+πœŽπ‘§3 as 𝑧→0 and 𝑋0βˆΌπ‘§ as π‘§β†’βˆž. The constraint at IR is due to the observation [72] that chiral symmetry is not restored in the highly excited mesons. Note that 𝑋0βˆΌπ‘§ keeps the mass difference between vector and axial-vector mesons constant as π‘§β†’βˆž. With the parameterized 𝑋0, they obtained a dilaton factor Ξ¦(𝑧) [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 D4/D8/D8 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 𝑇𝑐. 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 π‘§π»π‘š for heavy quarkonia in the hard wall model which is different from the IR cutoff for light mesons, 1/π‘§πΏπ‘šβˆΌ300 MeV. Note that in the hard wall model there is a one-to-one correspondence between the IR cutoff and the vector meson mass 1/π‘§π‘šβˆΌπ‘šπ‘‰. In [87], the lowest vector 𝑐𝑐 (𝐽/πœ“) mass ∼3GeV is used as an input to fix the IR cutoff for the charmonium, 1/π‘§π»π‘šβ‰ƒ1.32GeV. With this, the mass of the second resonance is predicted to be ∼7.2GeV, which is quite different from the experiment π‘šξ…žπœ“βˆΌ3.7GeV. 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] π‘š2𝑛=4(𝑛+1)𝑐.(5.9) For charmonium system, again the lowest mode (𝐽/πœ“) is used to fix 𝑐, βˆšπ‘β‰ƒ1.55GeV. Then the mass of the second resonance πœ“β€² is π‘šπœ“ξ…žβ‰ƒ4.38GeV, which is 20% away from the experimental value of 3.686GeV [87]. Additionally, the mass of heavy quarkonium such as 𝐽/πœ“ at finite temperature is calculated to predict that the mass decreases suddenly at 𝑇𝑐 and above 𝑇𝑐 it increases with temperature. Furthermore, the dissociation temperature is determined to be around 494MeV 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 𝜌(πœ”,βƒ—π‘ž)=βˆ’2Im𝐺𝑅(πœ”,βƒ—π‘ž). 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 (π‘πœŒ and 𝑐𝐽/πœ“) are introduced. It is observed in [88] that a peak in the spectral function melts with increasing temperature and eventually is flattened at 𝑇≃1.2𝑇𝑐. 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 𝐽/πœ“ 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, 𝐽/πœ“ and πœ“ξ…ž [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 π‘‡β‰ˆ540 MeV, or 2.8𝑇𝑐 [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 Ξ›QCD∼200 MeV, the nonrelativistic SchrΓΆdinger equation could be a useful tool to study heavy quark bound states: ξ‚΅βˆ’β–Ώ22π‘šπ‘Ÿξ‚Ά+𝑉(π‘Ÿ)Ξ¨(π‘Ÿ)=𝐸Ψ(π‘Ÿ),(5.10) where π‘šπ‘Ÿ is the reduced mass, π‘šπ‘Ÿ=π‘šπ‘„/2. A tricky point with potential models for quarkonia is which potential is to be used in the SchrΓΆdinger 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 SchrΓΆdinger equation with holographic potentials [99]. They used two ansΓ€tze of the potential model: the F-ansatz (U-ansatz) which identifies the potential in the SchrΓΆdinger equation with the free energy (the internal energy), respectively. With the F-ansatz, 𝐽/πœ“ does not survive above 𝑇𝑐, while the dissociation temperature of Ξ₯ is (1.3βˆ’2.1)𝑇𝑐. For the U-ansatz, 𝐽/πœ“ dissolves into open charm quarks around (1.2βˆ’1.7)𝑇𝑐 and Ξ₯ dissociates at about (2.5βˆ’4.2)𝑇𝑐.

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 𝑠𝑠 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 πœ‹Β±+π‘’βˆ’β†’πœ‹Β±+π‘’βˆ’ 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 πΉπœ‹(π‘ž2), where π‘ž2 is given by the energy and momentum of the photon π‘ž2=πœ”2βˆ’ξ‹π‘ž2. If the pion is a structureless point particle, we have πΉπœ‹=1. The pion electromagnetic form factor is expressed by, with the use of Lorentz invariance, charge conjugation, and electromagnetic gauge invariance:𝑝1+𝑝2ξ€Έπœ‡πΉπœ‹ξ€·π‘ž2ξ€Έ=ξ«πœ‹ξ€·π‘2ξ€Έ||π½πœ‡||πœ‹ξ€·π‘1,(5.11) where π‘ž2=(𝑝2βˆ’π‘1)2 and π½πœ‡ is the electromagnetic current, π½πœ‡=βˆ‘π‘“π‘’π‘“π‘žπ‘“π›Ύπœ‡π‘žπ‘“. The pion charge radius is determined by βŸ¨π‘Ÿ2πœ‹βŸ©=6πœ•πΉπœ‹ξ€·π‘ž2ξ€Έπœ•π‘ž2|π‘ž2=0.(5.12) In a vector meson dominance model, where the photon interacts with the pion only via vector mesons, especially 𝜌 meson, the pion form factor is given by πΉπœ‹ξ€·π‘ž2ξ€Έ=π‘š2πœŒπ‘š2πœŒβˆ’π‘ž2βˆ’π‘–π‘šπœŒΞ“πœŒξ€·π‘ž2ξ€Έ.(5.13) Then we obtain the pion charge radius βˆšβŸ¨π‘Ÿ2πœ‹βˆšβŸ©=6/π‘šπœŒβ‰ƒ0.63 fm. The experimental value is βˆšβŸ¨π‘Ÿ2πœ‹βŸ©=0.672 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: Ξ“πœ‡π›Όπ›½ξ€·π‘1,𝑝2ξ€Έξ€œξ€œ=βˆ’π‘‘π‘₯𝑑𝑦𝑒(βˆ’π‘–π‘1π‘₯+𝑖𝑝2𝑦)0|||𝑇𝐽†𝛼5(π‘₯)π½πœ‡(0)𝐽𝛽5(|||0𝑦).(5.14) 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: 0||𝐽𝛽5||𝑝2𝑝2||π½πœ‡||𝑝1𝑝1||𝐽†𝛼5||0≃𝑓2πœ‹πΉπœ‹ξ€·π‘ž2𝑝𝛼1𝑝𝛽2ξ€·π‘πœ‡1+π‘πœ‡2ξ€Έ.(5.15) Note that ⟨0|𝐽𝛼5|π‘βŸ©=π‘–π‘“πœ‹π‘π›Ό, where |π‘βŸ© 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 𝜌-meson was evaluated to be βŸ¨π‘Ÿ2𝜌⟩=0.53 fm2. The number from the soft wall model is βŸ¨π‘Ÿ2𝜌⟩=0.655 fm2 [109]. The approach based on the Dyson-Schwinger equations predicted βŸ¨π‘Ÿ2𝜌⟩=0.37 fm2 [110] and βŸ¨π‘Ÿ2𝜌⟩=0.54 fm2 [111]. The quark mass (or pion mass) dependence of the charge radius of the 𝜌-meson was calculated in lattice QCD: for instance, with π‘šπœ‹β‰ƒ300 MeV, βŸ¨π‘Ÿ2𝜌⟩=0.55 fm2 [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 βˆšβŸ¨π‘Ÿ2πœ‹βŸ©=0.58 fm [113] and in [114] βˆšβŸ¨π‘Ÿ2πœ‹βŸ©=0.500 fm, βˆšβŸ¨π‘Ÿ2πœ‹βŸ©=0.576 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 ξ«πœ‹π‘ξ€·π‘ξ…žξ€Έ||Ξ˜πœ‡πœˆ||πœ‹(0)π‘Ž1(𝑝)⟩=2π›Ώπ‘Žπ‘π‘”ξ€Ίξ€·πœ‡πœˆπ‘ž2βˆ’π‘žπœ‡π‘žπœˆξ€ΈΞ˜1ξ€·π‘ž2ξ€Έ+4π‘ƒπœ‡π‘ƒπœˆΞ˜2ξ€·π‘ž2,ξ€Έξ€»(5.16) where Ξ˜πœ‡πœˆ is the energy momentum tensor, π‘ž=π‘ξ…žβˆ’π‘, and 𝑃=(π‘ξ…ž+𝑝)/2. 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].

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 βŸ¨πΏβŸ©β‰ˆlimπ‘Ÿβ†’βˆžπ‘’βˆ’π›½π‘‰(π‘Ÿ),(6.1) where 𝑉(π‘Ÿ) is the potential between a static quark-antiquark pair at a distance r, and π›½βˆΌ1/𝑇. 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 π‘†π‘ˆ(3) 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 by𝑆grav1=βˆ’2πœ…2ξ€œπ‘‘5π‘₯βˆšπ‘”ξ‚΅π‘…+12𝐿2ξ‚Ά,(6.2) where πœ…2=8πœ‹πΊ5 and 𝐿 is the length scale of the AdS5, there are two solutions for the equations of motion derived from the gravitational action. The one is the sliced thermal AdS (tAdS): 𝑑𝑠2=𝐿2𝑧2ξ€·π‘‘πœ2+𝑑𝑧2+𝑑π‘₯23ξ€Έ,(6.3) where the radial coordinate runs from the boundary of tAdS space 𝑧=0 to the cut-off π‘§π‘š. Here 𝜏 is for the compactified Euclidean time-direction with periodicity π›½ξ…ž. The other solution is the AdS black hole (AdSBH) with the horizon π‘§β„Ž:𝑑𝑠2=𝐿2𝑧2𝑓(𝑧)π‘‘πœ2+𝑑𝑧2𝑓(𝑧)+𝑑π‘₯23ξ‚Ά,(6.4) where 𝑓(𝑧)=1βˆ’(𝑧/π‘§β„Ž)4. The Hawking temperature of the black hole solution is 𝑇=1/(πœ‹π‘§β„Ž), which is given by regularizing the metric near the horizon. At the boundary 𝑧=πœ– the periodicity of the time-direction in both backgrounds is the same and so the time periodicity of the tAdS is given by 𝛽=πœ‹π‘§β„Žβˆšπ‘“(πœ–).(6.5) Now we calculate the action density 𝑉, which is defined by the action divided by the common volume factor of 𝑅3. The regularized action density of the tAdS is given by𝑉1(πœ–)=4𝐿3πœ…2ξ€œπ›½β€²0ξ€œπ‘‘πœπ‘§IRπœ–π‘‘π‘§π‘§5,(6.6) and that of the AdSBH is given by𝑉2(πœ–)=4𝐿3πœ…2ξ€œπœ‹π‘§β„Ž0ξ€œπ‘‘πœπ‘§πœ–π‘‘π‘§π‘§5,(6.7) where 𝑧=min(π‘§π‘š,π‘§β„Ž). Then, the difference of the regularized actions is given byΔ𝑉𝑔=limπœ–β†’0𝑉2(πœ–)βˆ’π‘‰1ξ€»=⎧βŽͺβŽͺ⎨βŽͺβŽͺ⎩𝐿(πœ–)3πœ‹π‘§β„Žπœ…212𝑧4β„Ž,π‘§π‘š<π‘§β„Ž,𝐿3πœ‹π‘§β„Žπœ…21𝑧4π‘šβˆ’12𝑧4β„Žξƒͺ,π‘§π‘š>π‘§β„Ž.(6.8) When Δ𝑉𝑔 is positive (negative), tAdS (AdSBH) is stable. Thus, at Δ𝑉𝑔=0 there exists a Hawking-Page transition. In the first case π‘§π‘š<π‘§β„Ž, there is no Hawking-Page transition and the tAdS is always stable. In the second case π‘§π‘š>π‘§β„Ž, the Hawking-Page transition occurs at𝑇𝑐=21/4ξ€·πœ‹π‘§π‘šξ€Έ,(6.9) and at low temperature 𝑇<𝑇𝑐 (at high temperature 𝑇>𝑇𝑐) the thermal AdS (the AdS black hole) geometry becomes a dominant background. When we fix the IR cutoff by the 𝜌 meson mass, we obtain 1/π‘§π‘š=323 MeV and 𝑇𝑐=122 MeV. In the soft wall model, 𝑇𝑐=191MeV [59].

This work has been extended in various directions. The authors of [133] revisited the thermodynamics of the hard wall and soft wall model. They used holographic renormalization to compute the finite actions of the relevant supergravity backgrounds and verify the presence of a Hawking-page type phase transition. They also showed that the entropy, in the gauge theory side, jumps from 𝑁0 to 𝑁2 at the transition point [133]. In [134], the extension was done by studying the thermodynamics of AdS black holes with spherical or negative constant curvature horizon, dual to a non-supersymmetric Yang-Mills theory on a sphere or hyperboloid respectively. They also studied charged AdS black holes [135] in the grand canonical ensemble, corresponding to a Yang-Mills theory at finite chemical potential, and found that there is always a gap for the infrared cutoff due to the existence of a minimal horizon for the charged AdS black holes with any horizon topology [134]. With an assumption that the gluon condensate melts out at finite temperature, a Hawking-Page type transition between the dilaton AdS geometry in (4.4) and the usual AdS black hole has studied in [136].

The effect of the number of quark flavors 𝑁𝑓 and baryon number density on the critical temperature was investigated by considering a bulk meson action together with the gravity action in [137]. It is shown that the critical temperature decreases with increasing 𝑁𝑓. As the number density was raised, the critical temperature begins to drop, but it saturates to a constant value even at very large density. This is mostly due to the absence of the back-reaction from number density [137]. The back-reaction due to the number density has included in [138–140]. In [141], deconfinement transition of AdS/QCD with π’ͺ(π›Όξ…ž3) corrections was investigated. In [142], thermodynamics of the asymptotically-logarithmically-AdS black-hole solutions of 5D dilaton gravity with a monotonic dilaton potential are analyzed in great detail, where it is shown that in a special case, where the asymptotic geometry in the string frame reduces to flat space with a linear dilaton, the phase transition could be second order. The renormalized Polyakov loop in the deconfined phase of a pure π‘†π‘ˆ(3) gauge theory was computed in [143] based on a soft wall metric model. The result obtained in this work is in good agreement with the one from lattice QCD simulations.

Due to this Hawking-Page transition, we are not to use the black hole in the confined phase, and so we are not to obtain the temperature dependence of any hadronic observables. This is consistent with large 𝑁𝑐 QCD at leading order. For instance, it was shown in [144, 145] that the Wilson loops, both time-like and space-like, and the chiral condensate are independent of the temperature in confining phase to leading order in 1/𝑁𝑐. This means that the chiral and deconfinement transitions are first order. The deconfinement and chiral phase transitions of an π‘†π‘ˆ(𝑁) gauge theory at large 𝑁𝑐 were also discussed in [146]. However, in reality we observe temperature dependence of hadronic quantities, and therefore we have to include large 𝑁𝑐 corrections in holographic QCD in a consistent way. A quick fix-up for this might be to use the temperature dependent chiral condensate as an input in a holographic QCD model and study how this temperature dependence conveys into other hadronic quantities [147].

6.2. Chiral Transition

Now we turn to the chiral transition of QCD based on the chiral condensate. In the hard wall model, the chiral symmetry is broken, in a sense, by the IR boundary condition. In case we have a well-defined IR boundary condition at the wall 𝑧=π‘§π‘š, we could calculate the value of chiral condensate by solving the equation of motion for the bulk scalar 𝑋. In the case of the AdS black hole we could have a well defined IR boundary condition at the black hole horizon, which allows us to calculate the chiral condensate. For instance, in [148], it is shown that with the AdS black hole background the chiral condensate together with the current quark mass is zero in both the hard wall and soft wall models. This is easy to see from the solution of 𝑋0 in the AdS black hole background [148, 149]: 𝑋0ξƒ©π‘š(𝑧)=π‘§π‘ž2𝐹114,14,12,𝑧4𝑧4β„Žξƒͺ+πœŽπ‘žπ‘§22𝐹134,34,32,𝑧4𝑧4β„Žξƒͺξƒͺ.(6.10) At 𝑧=π‘§β„Ž, both terms in 𝑋0(𝑧) diverge logarithmically, which requires to set both of them zero: π‘šπ‘ž=0, 𝜎=0. This is different from real QCD, where current quark mass can be nonzero in the regime 𝑇>𝑇𝑐.

The finite temperature phase structure of the Sakai-Sugimoto model was analyzed in [150] to explore deconfinement and chiral symmetry restoration. Depending on a value of the model parameter, it is predicted that deconfinement and chiral symmetry restoration happens at the same temperature or the presence of a deconfined phase with broken chiral symmetry [150]. Phase structure of a stringy D3/D7 model has extensively studied in [151–153].

6.3. Equation of State and Susceptibility

Apart from the chiral condensate, various thermodynamic quantities could serve as an indicator for a transition from hadron to quark-gluon phase. Energy density, entropy, pressure, and susceptibilities are such examples. We first consider energy density and pressure. Schematically, based on the ideal gas picture we discuss how the energy density and pressure tell hadronic matter to quark-gluon plasma. At low temperature thermodynamics of hadron gas will be dominated by pions which are almost massless, while in QGP quarks and gluons are the relevant degrees of freedom. Energy density and pressure of massless pions are πœ‹πœ–=𝛾2𝑇304πœ‹,𝑝=𝛾2𝑇904,(6.11) where the number of degrees of freedom 𝛾 is three. In the QGP, they are given by πœ‹πœ–=𝛾2𝑇304πœ‹+𝐡,𝑝=𝛾2𝑇904βˆ’π΅,(6.12) where 𝛾=37, and 𝐡 is the bag constant. Apart from the bag constant, the degeneracy factor 𝛾 changes from 3 to 37, and therefore we can expect that the energy density and pressure will increase rapidly at the transition point. Since the dual of the boundary energy-momentum tensor π‘‡πœ‡πœˆ is the metric, we can obtain the energy density and pressure of a boundary gauge theory from the near-boundary behavior of the gravity solution. To demonstrate how-to, we follow [154, 155]. We first rewrite the gravity solution in the Fefferman-Graham coordinate [156]: 𝑑𝑠2=1𝑧2ξ€·π‘”πœ‡πœˆπ‘‘π‘₯πœ‡π‘‘π‘₯πœˆβˆ’π‘‘π‘§2ξ€Έ.(6.13) Next, we expand the metric π‘”πœ‡πœˆ at the boundary 𝑧→0: π‘”πœ‡πœˆ=𝑔(0)πœ‡πœˆ+𝑧2𝑔(2)πœ‡πœˆ+𝑔(4)πœ‡πœˆ+β‹―.(6.14) Now we consider flat 4D metric such that 𝑔(0)πœ‡πœˆ=πœ‚πœ‡πœˆ. Then 𝑔(2)πœ‡πœˆ=0 and the vacuum expectation value of the energy momentum tensor is given by βŸ¨π‘‡πœ‡πœˆβŸ©=const⋅𝑔(4)πœ‡πœˆ.(6.15) For example, we consider an AdS black hole in the Fefferman-Graham coordinate: 𝑑𝑠2=1𝑧21βˆ’π‘§4/𝑧4β„Žξ€Έξ€·1+𝑧4/𝑧4β„Žξ€Έπ‘‘π‘‘2βˆ’ξƒ©π‘§1+4𝑧4β„Žξƒͺ𝑑π‘₯2βˆ’π‘‘π‘§2ξƒͺ.(6.16) Here the temperature is defined by βˆšπ‘‡=2/(πœ‹π‘§β„Ž). Then, we read off βŸ¨π‘‡πœ‡πœˆξƒ©3βŸ©β‰ˆdiag𝑧4β„Ž,1𝑧4β„Ž,1𝑧4β„Ž,1𝑧4β„Žξƒͺ,(6.17) which satisfies πœ–=3𝑝. There have been many works on the equations of state for a holographic matter at finite temperature [142, 157–160]. In [161], the energy density, pressure, and entropy of a deconfined pure Yang-Mills matter were evaluated in the improved holographic QCD model [162, 163]. The energy density and pressure vanish at low temperature, and at the critical temperature, π‘‡π‘βˆΌ235 MeV, they jump up to a finite value, showing the first-order phase transition. It is interesting to note that in [164] some high-precision lattice QCD simulations were performed with increasing 𝑁𝑐 at finite temperature, and the results were compared with those from holographic QCD studies.

Various susceptibilities are also useful quantities to characterize phases of QCD. For instance, the quark number susceptibility has been calculated in holographic QCD in a series of works [148, 165]. The quark number susceptibility was originally proposed as a probe of the QCD chiral phase transition at zero chemical potential [127, 131]:πœ’π‘ž=πœ•π‘›π‘žπœ•πœ‡π‘ž.(6.18) In terms of the retarded Green function πΊπ‘…πœ‡πœˆ(πœ”,π‘˜), the quark number susceptibility can be written as [166] πœ’π‘ž(𝑇,πœ‡)=limπ‘˜β†’0𝐺Re𝑅𝑑𝑑(πœ”=0,π‘˜).(6.19) In [165], it is claimed that quark number susceptibility will show a sudden jump at 𝑇𝑐 in high-density regime, and so QCD phase transition in low-temperature and high-density regime will be always first order. Thermodynamics of a charged dilatonic black hole, which is asymptotically RN-AdS black hole in the UV and AdS2×𝐑3 in the IR, including the quark number susceptibility were extensively studied in [167]. The critical end point of the QCD phase diagram was studied in [168] by considering the critical exponents of the specific heat, number density, quark number susceptibility, and the relation between the number density and chemical potential at finite chemical potential and temperature. It is shown that the critical end point is located at 𝑇=143 MeV and πœ‡=783 MeV in the QCD phase diagram [168].

6.4. Dense Baryonic Matter

Understanding the properties of dense QCD is of key importance for laboratory physics such as heavy ion collision and for our understanding of the physics of stable/unstable nuclei and of various astrophysical objects such as neutron stars.

To expose an essential physics of dense nuclear matter, we take the Walecka model [169, 170], which describes nuclear matter properties rather well, as an example. The simplest version of the model contains the nucleon πœ“, omega meson πœ”, and an isospin singlet, Lorentz scalar meson 𝜎 whose minimal Lagrangian is β„’=πœ“ξ€·π‘–/πœ•+π‘”πœŽπœŽβˆ’π‘”πœ”β•±πœ”ξ€Έ1πœ“+2ξ€·πœ•πœ‡πœŽπœ•πœ‡πœŽβˆ’π‘š2𝜎𝜎2ξ€Έβˆ’14πΉπœ‡πœˆπΉπœ‡πœˆ+12π‘š2πœ”πœ”πœ‡πœ”πœ‡.(6.20) Within the mean field approximation, the properties of nuclear matter are mostly determined by the scalar mean field 𝜎=(π‘”πœŽ/π‘š2𝜎)𝑛𝑠 and the mean field of the time component of the πœ” field πœ”0=(π‘”πœ”/π‘š2πœ”)𝑛, where 𝑛 is the baryon number density and 𝑛𝑠 is the scalar density. For instance, the pressure of the nuclear matter described by the Walecka is1𝑃=4πœ‹2ξ‚Έ23πΈβˆ—πΉπ‘3πΉβˆ’π‘šβˆ—π‘2πΈβˆ—πΉπ‘πΉ+π‘šβˆ—π‘4𝐸lnβˆ—πΉ+π‘πΉπ‘šβˆ—π‘+1ξ‚Άξ‚Ή2𝑔2πœ”π‘š2πœ”π‘›2βˆ’12𝑔2πœŽπ‘š2πœŽπ‘›2𝑠,(6.21) where πΈβˆ—πΉ=𝑝2𝐹+π‘šβˆ—π‘2,π‘šβˆ—π‘=π‘šπ‘βˆ’π‘”2πœŽπ‘š2πœŽπ‘›π‘ .(6.22) Further, many successful predictions based on the Walecka model and its generalized versions, Quantum Hadrodynamics, require large scalar and vector fields in nuclei. This implies that to gain a successful description of nuclear matter or nuclei, having both scalar and vector mean fields in the model seems crucial. The importance of the interplay between the scalar and vector fields can be also seen in the static nonrelativistic potential between two nucleons. The nucleon-nucleon potential from single 𝜎-exchange and single πœ”-exchange is given by 𝑔𝑉(π‘Ÿ)=2πœ”14πœ‹π‘Ÿπ‘’βˆ’π‘šπœ”π‘Ÿβˆ’π‘”2𝜎14πœ‹π‘Ÿπ‘’βˆ’π‘šπœŽπ‘Ÿ.(6.23) Note that single 𝜎-exchange can be replaced by two pion exchange. If π‘”πœ”>π‘”πœŽ and π‘šπœ”>π‘šπœŽ, then the potential in (6.23) captures some essential features of the two nucleon potential to form stable nuclear matter: repulsive at short distance and attraction at intermediate and long distance. We remark here that the scalar field in the Walecka model may not be the scalar associated with a linear realization of usual chiral symmetry breaking in QCD; see, for instance, [171].

The hard wall model or soft wall model in its original form does not do much in dense matter. This is primarily due to its simple structure and chiral symmetry. Suppose that we turn on the time component of a U(1) bulk vector field dual to a boundary number operator, 𝑉𝑑(𝑧)=πœ‡+πœŒπ‘§2. To incorporate this U(1) bulk field into the hard wall model, we consider U(2) chiral symmetry. The covariant derivative with U(1) vector and axial-vector is given by π·πœ‡π‘‹=πœ•πœ‡π‘‹βˆ’π‘–π΄πΏπœ‡π‘‹+π‘–π‘‹π΄π‘…πœ‡ and it becomes π·πœ‡π‘‹=πœ•πœ‡π‘‹βˆ’π‘–π‘‹(π΄πΏπœ‡βˆ’π΄π‘…πœ‡). Therefore, the U(1) bulk field π‘‰πœ‡=π΄πΏπœ‡+π΄π‘…πœ‡ does not couple to the scalar 𝑋, meaning that the physical properties of 𝑋 are not affected by the chemical potential or number density. Note, however, that the vacuum energy of the hard wall or soft wall model should depend on the chemical potential and number density by the AdS/CFT. One simple way to study the physics of dense matter in the hard or soft wall model is to work with higher-dimensional terms in the action. For instance, the role of dimension six terms in the hard wall model was studied in free space [172, 173]. If we turn on the number density through the U(1) bulk field, we have a term like 𝑋20𝐹2𝑉, where 𝐹𝑉 is the field strength of the bulk U(1) gauge field [174]. Then we may see interplay between number density and chiral condensate encoded in 𝑋0. In [175], based on the hard wall model with the Chern-Simons term it is shown that there exists a Chern-Simons coupling between vector and axial-vector mesons at finite baryon density. This mixes transverse 𝜌 and π‘Ž1 mesons and leads to the condensation of the vector and axial-vector mesons. The role of the scalar density or the scalar field in the hard wall model was explored in [176]. In [139], a back-reaction due to the density is studied in the hard wall model.

Physics of dense matter in Sakai-Sugimoto model has been developed with/without the source term for baryon charge [177–181]. For instance, in [180] localized and smeared source terms are introduced and a Fermi sea has been observed, though there are no explicit fermionic modes in the model. A deficit with the Sakai-Sugimoto model for nuclear matter might be the absence of the scalar field which is quite important together with U(1) vector field. The phase structure of the D3/D7 model at finite density is studied in [182, 183]. The nucleon-nucleon potential is playing very important role in understanding the properties of nuclear matter. For example, one of the conventional methods to study nuclear matter is to work with the independent-pair approximation, Brueckner’s theory, where two-nucleon potentials are essential inputs. Holographic nuclear forces were studied in [184–187].

7. Closing Remarks

The holographic QCD model has proven to be a successful and promising analytic tool to study nonperturbative nature of low energy QCD. However, its success should always come with β€œqualitative” since it is capturing only large 𝑁𝑐 leading physics. To have any transitions from β€œqualitative” to β€œquantitative”, we have to invent a way to calculate subleading corrections in a consistent manner. A bit biased, but the most serious defect of the approach based on the gauge/gravity duality might be that it offers inherently macroscopic descriptions of a physical system. For instance, we may understand the QCD confinemnt/deconfinement transition through the Hawking-Page transition, qualitatively. Even though we accept generously the word β€œqualitatively”, we are not to be satisfied completely since we do not know how gluons and quarks bound together to form a color singlet hadron or how hadrons dissolve themselves into quark and gluon degrees of freedom. In this sense, the holographic QCD cannot be stand-alone. Therefore, the holographic QCD should go together with conventional QCD-based models or theories to guide them qualitatively and to gain microscopic pictures revealed by the conventional approaches.

Finally, we collect some interesting works done in bottom-up models that are not yet properly discussed in this review. Due to our limited knowledge, we could not list all of the interesting works and most results from top-down models will not be quoted. To excuse this defect we refer to recent review articles on holographic QCD [28, 188–197].

Deep inelastic scattering has been studied in gauge/gravity duality [198–209]. Light and heavy mesons were studied in the soft-wall holographic approach [210].

Unusual bound states of quarks are also interesting subjects to work in holographic QCD. In [211], the multiquark potential was calculated and tetra-quarks were discussed in AdS/QCD. Based on holographic quark-antiquark potential in the static limit, the masses of the states X(3872) or Y(3940) were predicted and also tetra-quark masses with open charm and strangeness were computed in [212]. A hybrid exotic meson, πœ‹1(1400), was discussed in [213]. The spectrum of baryons with two heavy quarks was predicted in [214].

Low-energy theorems of QCD and spectral density of the Dirac operator were studied in the soft wall model [215]. A holographic model of hadronization was suggested in [216].

The equation of state for a cold quark matter was calculated in the soft wall metric model with a U(1) gauge field. The result is in agreement with phenomenology [217].


A. Bulk Mass and the Conformal Dimension of Boundary Operator

In this appendix, we summarize the relation between the conformal dimension of a boundary operator and the bulk mass of dual bulk field. We work in the Euclidean version of AdS𝑑+1:𝑑𝑠2=1ξ€·π‘₯0ξ€Έ2π‘‘ξ“πœ‡=0(𝑑π‘₯πœ‡)2.(A.1)

A.1. Massive Scalar Case

We first consider a free massive scalar field whose action is given by 1𝑆=2ξ€œπ‘‘π‘‘+1π‘₯βˆšπ‘”ξ€·πœ•πœ‡πœ™πœ•πœ‡πœ™+π‘š2πœ™2ξ€Έ.(A.2) Let the propagator of πœ™ be 𝐾(π‘₯0,βƒ—π‘₯;βƒ—π‘₯β€²). To solve πœ™ in terms of its boundary function πœ™0, we look for a propagator of πœ™, a solution 𝐾(π‘₯0,βƒ—π‘₯;βƒ—π‘₯β€²) of the Laplace equation on 𝐡𝑑+1 whose boundary value is a delta function at a point 𝑃 on the boundary. We take 𝑃 to be the point at π‘₯0β†’βˆž. The boundary conditions and metric are invariant under translations of the π‘₯𝑖, then we can consider 𝐾 as a function of only π‘₯0, and thus 𝐾(π‘₯0,βƒ—π‘₯;𝑃)=𝐾(π‘₯0). Then, the equation of motion is ξ‚€βˆ’ξ€·π‘₯0𝑑+1𝑑𝑑π‘₯0ξ€·π‘₯0ξ€Έβˆ’π‘‘+1𝑑𝑑π‘₯0+π‘š2𝐾π‘₯0ξ€Έ=0,(A.3) where we used 1βˆšπ‘”πœ•πœ‡βˆšπ‘”πœ•πœ‡=ξ€·π‘₯0𝑑+1𝑑𝑑π‘₯0ξ€·π‘₯0ξ€Έβˆ’π‘‘+1𝑑𝑑π‘₯0.(A.4) We analyze the equation of motion near the boundary, π‘₯0β†’0, and take 𝐾(π‘₯0)∝(π‘₯0)πœ†+𝑑. From the equation of motion, we haveβˆ’(πœ†+𝑑)πœ†+π‘š2=0,(A.5) where πœ† is the larger root πœ†=πœ†+. The conformal dimension Ξ” of the boundary operator is related to the mass m on AdS𝑑+1 space by Ξ”=𝑑+πœ†+. Thus, we obtain (Ξ”βˆ’π‘‘)Ξ”=π‘š2,(A.6) or 1Ξ”=2ξ‚€βˆšπ‘‘+𝑑2+4π‘š2.(A.7)

A.2. Massive 𝑝-Form Field Case

Consider a massive 𝑝-form potential [218]: 1π’œ=π’œπ‘!πœ‡1β‹―πœ‡π‘π‘‘π‘₯πœ‡1⋯𝑑π‘₯πœ‡π‘.(A.8) The free action of π’œ is1𝑆=2ξ€œAdS𝑑+1ξ€·β„±βˆ§βˆ—β„±+π‘š2π’œβˆ§βˆ—π’œξ€Έ,(A.9) where β„±=π‘‘π’œ is the field strength 𝑝+1 form. The variation of this action is ξ€œπ›Ώπ‘†=AdS𝑑+1ξ€·βˆ’(βˆ’1)π‘π›Ώπ’œβˆ§π‘‘βˆ—β„±+π‘š2π›Ώπ’œβˆ§βˆ—π’œξ€Έ,(A.10) and then the classical equation of motion for π’œ from (A.9) is(βˆ’1)π‘π‘‘βˆ—π‘‘π’œβˆ’π‘š2βˆ—π’œ=0.(A.11) In addition, π’œ satisfies π‘‘βˆ—π’œ=0. By using the metric (A.1), the equation of motion (A.10) can be written asπ‘₯0ξ€Έ2πœ•2πœ‡βˆ’(𝑑+1βˆ’2𝑝)π‘₯0πœ•0+𝑑+1βˆ’2π‘βˆ’π‘š2ξ€Έξ‚„π’œ0𝑖2…𝑖𝑝π‘₯=0,(A.12)0ξ€Έ2πœ•2πœ‡βˆ’(π‘‘βˆ’1βˆ’2𝑝)π‘₯0πœ•0βˆ’π‘š2ξ‚„π’œπ‘–1⋯𝑖𝑝=2π‘₯0ξ‚€πœ•π‘–1πœ”0𝑖2⋯𝑖𝑝+(βˆ’1)π‘βˆ’1πœ•π‘–2πœ”0𝑖3⋯𝑖𝑝𝑖1+β‹―.(A.13) Now from the vielbein π‘’πœ‡π‘Ž=π‘₯0π›Ώπœ‡π‘Ž, we introduce fields with flat indices:𝐴0𝑖2⋯𝑖𝑝=ξ€·π‘₯0ξ€Έπ‘βˆ’1π’œ0𝑖2⋯𝑖𝑝,𝐴𝑖1⋯𝑖𝑝=ξ€·π‘₯0ξ€Έπ‘π’œπ‘–1⋯𝑖𝑝.(A.14) Then the equations of motion (A.12) of 𝐴0𝑖2⋯𝑖𝑝 becomeπ‘₯0ξ€Έ2πœ•2πœ‡βˆ’(π‘‘βˆ’1)π‘₯0πœ•0βˆ’ξ€·π‘š2+𝑝2ξ€Έξ‚„π΄βˆ’π‘π‘‘0𝑖2⋯𝑖𝑝=0.(A.15) We consider𝐴0𝑖2β‹―π‘–π‘βˆΌξ€·π‘₯0ξ€Έβˆ’πœ†(A.16) as π‘₯0β†’0. Then substituting this in (A.15) gives π‘₯0=0ξ€Έ2πœ•20βˆ’(π‘‘βˆ’1)π‘₯0πœ•0βˆ’ξ€·π‘š2+𝑝2ξ€Έξ‚„ξ€·π‘₯βˆ’π‘π‘‘0ξ€Έβˆ’πœ†=π‘₯0ξ€Έ2πœ•0ξ‚€ξ€·π‘₯βˆ’πœ†0ξ€Έβˆ’πœ†βˆ’1ξ‚βˆ’(π‘‘βˆ’1)π‘₯0ξ‚€ξ€·π‘₯βˆ’πœ†0ξ€Έβˆ’πœ†βˆ’1ξ‚βˆ’ξ€·π‘š2+𝑝2π‘₯βˆ’π‘π‘‘ξ€Έξ€·0ξ€Έβˆ’πœ†ξ‚„=ξ€Ίξ€·π‘šπœ†(πœ†+1)+πœ†(π‘‘βˆ’1)βˆ’2+𝑝2π‘₯βˆ’π‘π‘‘ξ€Έξ€»ξ€·0ξ€Έβˆ’πœ†=ξ€Ίξ€·π‘šπœ†(πœ†+𝑑)βˆ’2+𝑝2π‘₯βˆ’π‘π‘‘ξ€Έξ€»ξ€·0ξ€Έβˆ’πœ†,(A.17) and therefore we obtain the relationπœ†(πœ†+𝑑)=π‘š2+𝑝2βˆ’π‘π‘‘.(A.18) With Ξ”=𝑑+πœ†, we have (Ξ”βˆ’π‘‘)Ξ”=π‘š2+𝑝2βˆ’π‘π‘‘βŸΉ(Ξ”βˆ’π‘)𝑝+(Ξ”βˆ’π‘)(Ξ”βˆ’π‘‘)=π‘š2,(A.19) and we finally arrive at (Ξ”βˆ’π‘‘+𝑝)(Ξ”βˆ’π‘)=π‘š2,(A.20) or 1Ξ”=2𝑑+(π‘‘βˆ’2𝑝)2+4π‘š2ξ‚Ά.(A.21)

A.3. General Cases

Now for completeness, we list the relations between the conformal dimension Ξ” and the mass for the various bulk fields in AdS𝑑+1:(1)scalars [3]: Ξ”Β±βˆš=(1/2)(𝑑±𝑑2+4π‘š2),(2)spinors [219]: Ξ”=(1/2)(𝑑+2|π‘š|),(3)vectors (entries 3. and 4. are for forms with Maxwell type actions.): Ξ”Β±βˆš=(1/2)(𝑑±(π‘‘βˆ’2)2+4π‘š2),(4)𝑝-forms [218]: Ξ”Β±βˆš=(1/2)(𝑑±(π‘‘βˆ’2𝑝)2+4π‘š2),(5)first-order (𝑑/2)-forms (𝑑 even) (see [220] for 𝑑=4 case.): Ξ”=(1/2)(𝑑+2|π‘š|),(6)spin-3/2 [221, 222]: Ξ”=(1/2)(𝑑+2|π‘š|),(7)massless spin-2 [223]: Ξ”=𝑑.

B. D3/D7 Model and π‘ˆ(1) Axial Symmetry

In the original AdS/CFT, the duality between type IIB superstring theory on AdS5×𝑆5 and 𝒩=4 super-Yang-Mills theory with gauge group π‘†π‘ˆ(𝑁𝑐) can be embodied by the low-energy dynamics of a stack of 𝑁𝑐 D3 branes in Minkowski space. All matter fields in the gauge theory produced by the D3 branes are in the adjoint representation of the gauge group. To introduce the quark degrees of freedom in the fundamental representation, we introduce some other branes in this supersymmetry theory on top of the D3 branes.

B.1. Adding Flavour

It was shown in [17] that by introducing 𝑁𝑓 D7 branes into AdS5×𝑆5, 𝑁𝑓 dynamical quarks can be added to the gauge theory, breaking the supersymmetry to 𝒩=2. The simplest way to treat D3/D7 system is to work in the limit where the D7 is a probe brane, which means that only a small number of D7 branes are added, while the number of D3 branes 𝑁𝑐 goes to infinity. In this limit 𝑁𝑓β‰ͺ𝑁𝑐 we may neglect the back-reaction of the D7 branes on AdS5×𝑆5 geometry. In field theory side, this corresponds to ignoring the quark loops, quenching the gauge theory.

The D7 branes are added in such a way that they extend parallel in Minkowski space and extend in spacetime as given in Table 3. The massless modes of open strings that both end on the 𝑁𝑐 D3 branes give rise to 𝒩=4 degrees of freedom of supergravity on AdS5×𝑆5 consisting of the π‘†π‘ˆ(𝑁𝑐) vector bosons, four fermions, and six scalars. In the limit of large 𝑁𝑐 at fixed but large β€˜t Hooft coupling πœ†=𝑔2YM𝑁𝑐=𝑔𝑠𝑁𝑐≫1, the D3 branes can be replace with near horizon geometry that is given by𝑑𝑠2=π‘Ÿ2𝑅2ξ€·βˆ’π‘‘π‘‘2+𝑑π‘₯21+𝑑π‘₯22+𝑑π‘₯23ξ€Έ+𝑅2π‘Ÿ2𝑑𝑦2=π‘Ÿ2𝑅2ξ€·βˆ’π‘‘π‘‘2+𝑑π‘₯21+𝑑π‘₯22+𝑑π‘₯23ξ€Έ+𝑅2π‘Ÿ2ξ€·π‘‘πœŒ2+𝜌2𝑑Ω23+𝑑𝑦25+𝑑𝑦26ξ€Έ,(B.1) where ⃗𝑦=(𝑦1,…,𝑦6) parameterize the 456789 space and π‘Ÿ2≑𝑦2. 𝑅 is the radius of curvature 𝑅2=√4πœ‹π‘”π‘ π‘π‘π›Όβ€² and 𝑑Ω23 is the three-sphere metric. The dynamics of the probe D7 brane is described by the combined DBI and Chern-Simons actions [5, 224]:𝑆D7=βˆ’π‘‡7ξ€œπ‘‘8π‘₯𝑃[𝑔]βˆ’detπ‘Žπ‘+2πœ‹π›Όβ€²πΉπ‘Žπ‘ξ€Έ+ξ€·2πœ‹π›Όξ…žξ€Έ22𝑇7ξ€œπ‘ƒξ€ΊπΆ(4)ξ€»βˆ§πΉβˆ§πΉ,(B.2) where 𝑔 is the bulk metric (B.1) and 𝐢(4) is the four-form potential. 𝑇7=1/((2πœ‹)7𝑔𝑠𝛼′4) is the D7 brane tension and 𝑃 denotes the pullback. πΉπ‘Žπ‘ is the world-volume field strength.

The addition of D7 branes to this system as in Table 3 breaks the supersymmetry to 𝒩=2. The lightest modes of the 3-7 and 7-3 open strings correspond to the quark supermultiplets in the field theory. If the D7 brane and the D3 brane overlap, then 𝑆𝑂(6) symmetry is broken into 𝑆𝑂(4)×𝑆𝑂(2)βˆΌπ‘†π‘‚(2)𝑅×𝑆𝑂(2)πΏΓ—π‘ˆ(1)𝑅 in the transverse directions to D3 and so preserves 1/4 of the supersymmetry. The 𝑆𝑂(4) rotates in 4567, while the 𝑆𝑂(2) group acts on 89 in 3. The induced metric on D7 takes the form, in general, as𝑑𝑠2D7=π‘Ÿ2𝑅2πœ‚πœ‡πœˆπ‘‘π‘₯πœ‡π‘‘π‘₯𝜈+𝑅2π‘Ÿ2ξ€·ξ€·1+𝑦5ξ…ž2+𝑦6ξ…ž2ξ€Έπ‘‘πœŒ2+𝜌2𝑑Ω23ξ€Έ,(B.3) where π‘¦ξ…ž5=𝑑𝑦5/π‘‘πœŒ and π‘¦ξ…ž6=𝑑𝑦6/π‘‘πœŒ. When the D7 brane and the D3 brane overlap, the embedding is 𝑦5=0,𝑦6=0,(B.4) and the induced metric on the D7 brane is replaced by𝑑𝑠2D7=𝜌2𝑅2πœ‚πœ‡πœˆπ‘‘π‘₯πœ‡π‘‘π‘₯𝜈+𝑅2𝜌2ξ€·π‘‘πœŒ2+𝜌2𝑑Ω23ξ€Έ.(B.5) The D7 brane fills AdS5 and is wrapping a three sphere of 𝑆5. In this case the quarks are massless and the R-symmetry of the theory is π‘†π‘ˆ(2)π‘…Γ—π‘ˆ(1)𝑅 and we have an extra π‘ˆ(1)𝑅 chiral symmetry.

If the D7 brane is separated from the D3 branes in the 89-plane direction by distance 𝐿, then the minimum length string has nonzero energy and the quark gains a finite mass, π‘šπ‘ž=𝐿/2πœ‹π›Όβ€². It is known that the R-symmetry is then only π‘†π‘ˆ(2)𝑅 and separation of D7 and D3 breaks the 𝑆𝑂(2)βˆΌπ‘ˆ(1)𝑅 that acts on the 89-plane. In this case, we can set for the embedding as 𝑦5=0,𝑦6=𝑦6(𝜌).(B.6) Then, the action for a static D7 embedding (with πΉπ‘Žπ‘ zero on its world volume) becomes 𝑆D7=βˆ’π‘‡7ξ€œπ‘‘8π‘₯𝑃[𝑔]βˆ’detπ‘Žπ‘ξ€Έ=βˆ’π‘‡7ξ€œπ‘‘8π‘₯βˆšβˆ’detπ‘”π‘Žπ‘ξ”1+π‘”π‘Žπ‘πœ•π‘Žπ‘¦π‘–πœ•π‘π‘¦π‘—π‘”π‘–π‘—=βˆ’π‘‡7ξ€œπ‘‘8π‘₯πœ–3𝜌3ξ”ξ€·πœ•1+πœŒπ‘¦5ξ€Έ2+ξ€·πœ•πœŒπ‘¦6ξ€Έ2,(B.7) where 𝑖,𝑗=5,6 and πœ–3 is the determinant from the three sphere. The ground state configuration of the D7 brane is given by the equation of motion with 𝑦5=0:π‘‘βŽ‘βŽ’βŽ’βŽ’βŽ£πœŒπ‘‘πœŒ3πœ•πœŒπ‘¦6ξ”ξ€·πœ•1+πœŒπ‘¦6ξ€Έ2⎀βŽ₯βŽ₯βŽ₯⎦=0.(B.8) The solution of this equation has an asymptotic behavior at UV (πœŒβ†’βˆž) as𝑦6π‘β‰ƒπ‘š+𝜌2+β‹―.(B.9) Now we can identify [198] that π‘š corresponds to the quark mass and 𝑐 is for the quark condensate βŸ¨πœ“πœ“βŸ© in agreement with the AdS/CFT dictionary.

B.2. Chiral Symmetry Breaking

One of the significant features of QCD is chiral symmetry breaking by a quark condensate πœ“πœ“. The π‘ˆ(1) symmetry under which πœ“ and πœ“ transform as πœ“β†’π‘’βˆ’π‘–π›Όπœ“ and πœ“β†’π‘’π‘–π›Όπœ“ in the gauge theory corresponds to a π‘ˆ(1) isometry in the 𝑦5𝑦6 plane transverse to the D7 brane. This π‘ˆ(1) symmetry can be explicitly broken by a nonvanishing quark mass due to the separation of the D7 brane from the stack of D3 branes in the 𝑦5+𝑖𝑦6 direction. Assume that the embedding as 𝑦5=0 and 𝑦6βˆΌπ‘/𝜌2 and then by a small rotation π‘’βˆ’π‘–πœ– on 𝑦5+𝑖𝑦6 generates π‘¦ξ…ž5β‰ƒπœ–π‘/𝜌2 and π‘¦ξ…ž6≃𝑦6 up to the π’ͺ(πœ–2) order.

In [18] the embedding of a D7 probe brane is embodied in the Constable-Myers background and the regular solution 𝑦6βˆΌπ‘š+𝑐/𝜌2 of the embedding 𝑦5=0, 𝑦6=𝑦6(𝜌) shows the behavior 𝑐≠0 as π‘šβ†’0 which corresponds to the spontaneous chiral symmetry breaking by a quark condensate. In [225], the chiral symmetry breaking comes from a cosmological constant with a constant dilaton configuration which is dual to the 𝒩=4 gauge theory in a four-dimensional AdS space.

B.3. Meson Mass Spectrum

The open string modes with both ends on the flavour D7 branes are in the adjoint of the π‘ˆ(𝑁𝑓) flavour symmetry of the quarks and hence can be interpreted as the mesonic degrees of freedom. As an example, we discuss the fluctuation modes for the scalar fields (with spin 0) following the argument of [226]. The directions transverse to the D7 branes are chosen to be 𝑦5 and 𝑦6 and the embedding is𝑦5=0+πœ’,𝑦6=𝐿+πœ‘,(B.10) where 𝛿𝑦5=πœ’ and 𝛿𝑦6=πœ‘ are the scalar fluctuations of the transverse direction. To calculate the spectra of the world-volume fields it is sufficient to work to quadratic order. For the scalars, we can write the relevant Lagrangian density asβ„’D7=βˆ’π‘‡7√[𝑔]βˆ’detπ‘ƒπ‘Žπ‘=βˆ’π‘‡7βˆšβˆ’detπ‘”π‘Žπ‘ξ”1+π‘”π‘Žπ‘ξ€·πœ•π‘Žπœ’πœ•π‘πœ’π‘”55+πœ•π‘Žπœ‘πœ•π‘πœ‘π‘”66ξ€Έ=βˆ’π‘‡7βˆšβˆ’detπ‘”π‘Žπ‘ξ‚™1+π‘”π‘Žπ‘π‘…2π‘Ÿ2ξ€·πœ•π‘Žπœ’πœ•π‘πœ’+πœ•π‘Žπœ‘πœ•π‘πœ‘ξ€Έβ‰ƒβˆ’π‘‡7βˆšβˆ’detπ‘”π‘Žπ‘ξ‚΅11+2𝑅2π‘Ÿ2π‘”π‘Žπ‘ξ€·πœ•π‘Žπœ’πœ•π‘πœ’+πœ•π‘Žπœ‘πœ•π‘πœ‘ξ€Έξ‚Ά,(B.11) where 𝑃[𝑔]π‘Žπ‘ is the induced metric on the D7 world-volume. In spherical coordinates with π‘Ÿ2=𝜌2+𝐿2, this can be written asβ„’D7β‰ƒβˆ’π‘‡7𝜌3πœ–3ξ‚΅11+2𝑅2𝜌2+𝐿2π‘”π‘Žπ‘ξ€·πœ•π‘Žπœ’πœ•π‘πœ’+πœ•π‘Žπœ‘πœ•π‘πœ‘ξ€Έξ‚Ά,(B.12) where πœ–3 is the determinant of the metric on the three sphere. Then the equations of motion becomeπœ•π‘Žξ‚΅πœŒ3πœ–3𝜌2+𝐿2π‘”π‘Žπ‘πœ•π‘Ξ¦ξ‚Ά=0,(B.13) where Ξ¦ is used to denote the real fluctuation either πœ’ or πœ‘. Evaluating a bit more, we have𝑅4ξ€·πœŒ2+𝐿2ξ€Έ2πœ•πœ‡πœ•πœ‡1Ξ¦+𝜌3πœ•πœŒξ€·πœŒ3πœ•πœŒΞ¦ξ€Έ+1𝜌2βˆ‡π‘–βˆ‡π‘–Ξ¦=0,(B.14) where βˆ‡π‘– is the covariant derivative on the three-sphere. We apply the separation of variables to write the modes asΞ¦=πœ™(𝜌)π‘’π‘–π‘˜β‹…π‘₯𝒴ℓ𝑆3ξ€Έ,(B.15) where 𝒴ℓ(𝑆3) are the scalar spherical harmonics on 𝑆3, which transform in the (β„“/2,β„“/2) representation of 𝑆𝑂(4) and satisfyβˆ‡π‘–βˆ‡π‘–π’΄β„“=βˆ’β„“(β„“+2)𝒴ℓ.(B.16) The meson mass is defined by𝑀2β‰‘βˆ’π‘˜2.(B.17) Now we define 𝜚=𝜌/𝐿 and 𝑀2=βˆ’π‘˜2𝑅4/𝐿2, and then the equation for πœ™(𝜌) isπœ•2𝜚3πœ™+πœšπœ•πœšξƒ©πœ™+𝑀2ξ€·1+𝜚2ξ€Έ2βˆ’β„“(β„“+2)𝜚2ξƒͺπœ™=0.(B.18) This equation was solved in [226] in terms of the hypergeometric function. To solve the equation, we first setπœ™(𝜚)=1+πœšβ„“ξ€·1+𝜚2ξ€Έβˆ’π›Όπ‘ƒ(𝜚),(B.19) where2𝛼=βˆ’1+1+𝑀2β‰₯0.(B.20) With a new variable 𝑦=βˆ’πœš2, (B.18) becomes𝑦(1βˆ’π‘¦)π‘ƒξ…žξ…ž[]𝑃(𝑦)+π‘βˆ’(π‘Ž+𝑏+1)π‘¦ξ…ž(𝑦)βˆ’π‘Žπ‘π‘ƒ(𝑦)=0,(B.21) where π‘Ž=βˆ’π›Ό, 𝑏=βˆ’π›Ό+β„“+1, and 𝑐=β„“+2. The general solution is taken by 𝛼β‰₯0, and by noting that the scalar fluctuations are real for βˆ’βˆž<𝑦≀0, one finds, up to a normalization constant, the solution of πœ™:πœ™πœŒ(𝜌)=β„“ξ€·πœŒ2+𝐿2ξ€Έπ›ΌπΉξ‚΅πœŒβˆ’π›Ό,βˆ’π›Ό+β„“+1;β„“+2;βˆ’2𝐿2ξ‚Ά.(B.22) Imposing the normalizability at πœŒβ†’βˆž, we obtainβˆ’π›Ό+β„“+1=βˆ’π‘›,𝑛=0,1,2,….(B.23) The solution is thenπœ™πœŒ(𝜌)=β„“ξ€·πœŒ2+𝐿2𝑛+β„“+1πΉξ‚΅πœŒβˆ’(𝑛+β„“+1),βˆ’π‘›;β„“+2;βˆ’2𝐿2ξ‚Ά,(B.24) and from the condition (B.23) we get𝑀2=4(𝑛+β„“+1)(𝑛+β„“+2).(B.25) Then by the definition of meson mass (B.17), we derive the four-dimensional mass spectrum of the scalar meson:𝑀𝑠(𝑛