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.

The approaches based on the Anti de Sitter/conformal field theory (AdS/CFT) correspondence [

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

In Appendices

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.

The AdS/CFT correspondence, first suggested by Maldacena [

The duality emerges from a careful consideration of the D-brane dynamics. A

The configurations of

Now, we take

In fact, the original system also contains closed string states. The higher-order derivative corrections for the Lagrangian (

Now we view the same system from a different angle. Since D branes are massive and carry energy and Ramond-Ramond (RR) charge,

In the flat spacetime, the circumference of the circle surrounding an origin at a distance

The two descriptions of the

To be more specific, let us start with type IIB string theory for

The geometry by the D3 branes is sketched in Figure

So far, we have considered two seemingly different descriptions of the

If

On the other hand, if

The gauge/gravity correspondence is nothing but the conjecture connecting these two descriptions of

The sketch of the AdS/CFT correspondence.

The two descriptions can be viewed as two extremes of

The operator-field correspondence between operators in the four-dimensional gauge theory and corresponding dual fields in the gravity side was given in [

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 [

To attain a realistic gravity dual description of (large

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

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:

To write down the Lagrangian of the linear sigma model, we consider (global) chiral symmetry of QCD. Since the mass of light quark

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

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

Putting things together, we could arrive at the following bulk Lagrangian with local

The finite temperature could be neatly introduced by a black hole in

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

At low energy or momentum scales roughly smaller than 1 GeV,

The gluon condensate

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

Now we consider quark-gluon-mixed condensate

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.

Glueballs are made up of gluons with no constituent quarks in them. The glueball states are in general mixed with conventional

The spectrum of glueballs is one of the earliest QCD quantities calculated based on the AdS/CFT duality. In [

Now we consider a scalar glueball (_{3} [_{5} Euclidean black hole background. The equation of motion for

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 [

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

Meson spectroscopy from the hard-wall model and from its variations: Model I [

Model I | Model II | Model A | Model B | Experiment | |
---|---|---|---|---|---|

775.8 | 832 | ||||

1348 | 1244 | 1363 | 1220 | ||

80.5 | 84.0 | ||||

334 | 330 | 329 | 353 | ||

481 | 459 | 486 | 440 | ||

139.3 | 141 | ||||

4.46 | 4.87 | 4.48 | 5.29 |

We remark that the sensitivity of calculated hadronic observables to the details of the hard wall model was studied in [

In addition to mesons, baryons were also studied in the hard wall model [

Now we collect some results from the soft wall model [

Meson spectroscopy from the modified soft wall model [

1 | 475 | 775.5 | 1185 | 1230 |

2 | 1129 | 1282 | 1591 | 1647 |

3 | 1429 | 1465 | 1900 | 1930 |

4 | 1674 | 1720 | 2101 | 2096 |

5 | 1884 | 1909 | 2279 | 2270 |

6 | 2072 | 2149 | ||

7 | 2243 | 2265 |

As long as confinement and non-Abelian chiral symmetry are concerned, the Sakai-Sugimoto model [

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 [

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

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 [

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 [

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

We finish this subsection with a summary of the discussion in [

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

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 [

QCD phase diagram.

Understanding the QCD phase structure is one of the important problems in modern theoretical physics; see [

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

The nature of the chiral transition of QCD depends on the number of quark flavors and the value of the quark mass. For pure

We first discuss the deconfinement transition. In holographic QCD, the confinement to deconfinement phase transition is described by the Hawking-Page transition [

Here we briefly summarize the Hawking-Page analysis of [

This work has been extended in various directions. The authors of [

The effect of the number of quark flavors

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

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

The finite temperature phase structure of the Sakai-Sugimoto model was analyzed in [

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

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 [

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 [

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

Physics of dense matter in Sakai-Sugimoto model has been developed with/without the source term for baryon charge [

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 “qualitative” since it is capturing only large

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 [

Deep inelastic scattering has been studied in gauge/gravity duality [

Unusual bound states of quarks are also interesting subjects to work in holographic QCD. In [

Low-energy theorems of QCD and spectral density of the Dirac operator were studied in the soft wall model [

The equation of state for a cold quark matter was calculated in the soft wall metric model with a

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

We first consider a free massive scalar field whose action is given by

Consider a massive

Now for completeness, we list the relations between the conformal dimension

scalars [

spinors [

vectors (entries 3. and 4. are for forms with Maxwell type actions.):

first-order

spin-3/2 [

massless spin-2 [

In the original AdS/CFT, the duality between type IIB superstring theory on

It was shown in [

The D7 branes are added in such a way that they extend parallel in Minkowski space and extend in spacetime as given in Table

The D3/D7-brane intersection in

0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | |
---|---|---|---|---|---|---|---|---|---|---|

D3 | ||||||||||

D7 |

The addition of D7 branes to this system as in Table

If the D7 brane is separated from the D3 branes in the 89-plane direction by distance

One of the significant features of QCD is chiral symmetry breaking by a quark condensate

In [

The open string modes with both ends on the flavour D7 branes are in the adjoint of the

In previous sections, we have focused on gauge theories and their gravity dual at zero temperature. To understand the thermal properties of gauge theories using the holography, we work with the AdS-Schwarzschild black hole which is dual to

We now introduce D7 branes in this background. It is convenient to change the variable in the metric (

The asymptotic solution at large

Two classes of regular solutions in the AdS black hole background.

We see that there can be two different classifications for the D7 brane embeddings. First, for large quark masses the D7 brane ends outside the horizon. It can be interpreted that the D7 brane tension is stronger than the attractive force of the black hole. Such a D7 brane solution is called a

The D3/D7 system is a supersymmetric configuration which gives gauge theories in the ultra-violet and only has a

The D4/D8/

0 | 1 | 2 | 3 | (4) | 5 | 6 | 7 | 8 | 9 | |
---|---|---|---|---|---|---|---|---|---|---|

D4 | ||||||||||

D8- |

In this configuration,

Sketch of the

In order to obtain a holographic dual of the large

Next, we consider the induced metric on the D8 probe brane in the D4 background with an ansatz

Now, we consider the gauge field on the probe D8 brane configuration. The gauge field on the D8 brane

By solving (

The surface gravity is the gravitational acceleration experienced by a test body (with negligible mass) close to the surface of an object. For a black hole, the surface gravity is defined as the acceleration of gravity at the horizon. The acceleration of a test body at a black hole event horizon is infinite in relativity; therefore one defines the surface gravity in a different way, corresponding to the Newtonian surface gravity in the nonrelativistic limit. Thus for a black hole, the surface gravity is defined in terms of the Killing vector which is orthogonal to the horizon and here its event horizon is a Killing horizon. For the Schwarzschild case this value is well defined. Alternatively, one can derive the same value as a period of the imaginary time in the Euclidean signature. We will discuss both points of view.

The horizon of a black hole is a null surface. It means that any vector normal to the surface is a null vector. Let us consider the Killing vector that generates time translations,

The expression (

The

Now, we consider the following metric ansatz [

The partition function for canonical ensemble is given by

The gravity action we study here is

There is a solution into AdS space which minimizes (

There is another solution,

The geometry

Actually, both

Deconfinement at high temperature can be understood by the spontaneous breaking of the center of the gauge group. The corresponding order parameter is the Polyakov loop that is defined by a Wilson loop wrapping around

The gravity dual calculation for the expectation value

For the low-temperature phase, the space is

The topology of the thermal AdS geometry and the AdS-Schwarzschild geometry. (a) thermal AdS. (b) AdS-Schwarzschild.

On the other hand, for the high-temperature phase, the relevant space is

The authors thank Jihun Kim, Yumi Ko, Ik Jae Shin, and Takuya Tsukioka for useful comments on the manuscript. They are also grateful to Sergey Afonin, Oleg Andreev, Stanley J. Brodsky, Miguel Costa, Guy F. de Teramond, Hilmar Forkel, Jian-Hua Gao, and Marco Panero for their comments on the the manuscript. Y. Kim expresses his gratitude to Hyun-Chul Kim, Kyung-il Kim, Yumi Ko, Bum-Hoon Lee, Hyun Kyu Lee, Sangmin Lee, Chanyong Park, Ik Jae Shin, Sang-Jin Sin, Takuya Tsukioka, Xiao-Hong Wu, Ulugbek Yakhshiev, Piljin Yi, and Ho-Ung Yee for collaborations in hQCD. We acknowledge the Max Planck Society(MPG), the Korea Ministry of Education, Science and Technology (MEST), Gyeongsangbuk-Do and Pohang City for the support of the Independent Junior Research Group at APCTP.