UNIVERSITA DEGLI STUDI DI BARI` - Istituto Nazionale di ... Proceeding of IFAE2009, Incontri di...

UNIVERSITA DEGLI STUDI DI BARIDipartimento Interateneo di Fisica

DOTTORATO DI RICERCA IN FISICA

CICLO XXII

Settore Scientifico Disciplinare FIS/02

HADRONS IN ADS/QCD

Dottorando: Floriana Giannuzzi

Coordinatore: Ch.ma Prof.ssa Maria Teresa Muciaccia

Supervisori: Ch.mo Prof. Leonardo Angelini

Dr. Pietro Colangelo

ESAME FINALE 2010

A Beppe Nardulli

There lies before us, if we choose,

continual progress in happiness,

knowledge, and wisdom.

The Russell-Einstein Manifesto

Contents

List of published articles and conference proceedings 9

Introduction 11

1 The AdS/CFT correspondence 13

1.1 Historical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.2 Super Yang Mills theories . . . . . . . . . . . . . . . . . . . . . . . . 14

1.2.1 The conformal group . . . . . . . . . . . . . . . . . . . . . . . 17

1.3 Type IIB string theory and AdS space . . . . . . . . . . . . . . . . . 18

1.4 Steps towards the AdS/CFT correspondence . . . . . . . . . . . . . . 21

1.4.1 Large N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.4.2 D3-branes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

1.4.3 The holographic principle . . . . . . . . . . . . . . . . . . . . 26

1.5 The conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2 The AdS/QCD correspondence 31

2.1 QCD as a candidate for a holographic description . . . . . . . . . . . 33

2.2 Hard Wall model of AdS/QCD . . . . . . . . . . . . . . . . . . . . . 35

2.2.1 Vector and axial-vector mesons . . . . . . . . . . . . . . . . . 38

2.3 Soft Wall model of AdS/QCD . . . . . . . . . . . . . . . . . . . . . . 43

2.3.1 Vector mesons . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.3.2 Holographic scalar glueballs . . . . . . . . . . . . . . . . . . . 49

3 Holographic description of scalar mesons 53

3.1 Scalar mesons in the Soft Wall model . . . . . . . . . . . . . . . . . . 54

3.2 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4 Holographic approach to finite temperature QCD 65

4.1 Hawking-Page phase transition in the Soft Wall and Hard Wall model 66

4.2 Soft Wall model with AdS Black Hole metric . . . . . . . . . . . . . . 69

7

Contents

4.2.1 Scalar glueballs . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.2.2 Scalar mesons . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.3 Model with the Hawking-Page phase transition . . . . . . . . . . . . . 78


5 Hadron spectroscopy by an AdS/QCD QQ static potential 81

5.1 The Salpeter equation and the Multhopp method . . . . . . . . . . . 82

5.2 Wilson loop and VQQ . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.3 Potential model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.3.1 Meson spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . 92

5.3.2 Tetraquark spectrum . . . . . . . . . . . . . . . . . . . . . . . 94

5.3.3 Charmonium and bottomonium decays . . . . . . . . . . . . . 99

5.3.4 Doubly heavy baryons . . . . . . . . . . . . . . . . . . . . . . 102

5.4 Remarks inspired by the Heavy Quark Effective Theory . . . . . . . . 107


Conclusions 111

Bibliography 115

8

List of published articles and

conference proceedings

Articles

M. V. Carlucci, F. Giannuzzi, G. Nardulli, M. Pellicoro and S. Stramaglia, “AdS-

QCD quark-antiquark potential, meson spectrum and tetraquarks,” Eur. Phys.

J. C 57, 569 (2008) [arXiv:0711.2014 [hep-ph]].

P. Colangelo, F. De Fazio, F. Giannuzzi, F. Jugeau and S. Nicotri, “Light scalar

mesons in the soft-wall model of AdS/QCD,” Phys. Rev. D 78, 055009 (2008)

[arXiv:0807.1054 [hep-ph]].

F. Giannuzzi, “ηb and ηc radiative decays in the Salpeter model with the AdS/QCD

inspired potential,” Phys. Rev. D 78, 117501 (2008) [arXiv:0810.2736 [hep-

ph]].

F. Giannuzzi, “Doubly heavy baryons in a Salpeter model with AdS/QCD inspired

potential,” Phys. Rev. D 79, 094002 (2009) [arXiv:0902.4624 [hep-ph]].

F. Giannuzzi and M. Mannarelli, “Properties of charmonia in a hot equilibrated

medium,” Phys. Rev. D 80, 054004 (2009) [arXiv:0907.1041 [hep-ph]].

P. Colangelo, F. Giannuzzi and S. Nicotri, “Holographic Approach to Finite Tem-

perature QCD: The Case of Scalar Glueballs and Scalar Mesons,” Phys. Rev.

D 80, 094019 (2009) [arXiv:0909.1534 [hep-ph]].

Conference proceedings

F. Giannuzzi, “Meson spectrum and tetraquarks through an AdS/QCD inspired

potential,”

Proceeding of Quark Confinement and the Hadron Spectrum (Mainz, Germany,

1-6 September 2008),

published on Proceedings of Science CONFINEMENT8 135 (2008),

[arXiv:0811.3553 [hep-ph]].

P. Colangelo, F. De Fazio, F. Giannuzzi and S. Nicotri, “Aspects of new charm(onium)

spectroscopy,”

Proceeding of 2nd International Workshop on Theory, Phenomenology and

Experiments in Heavy Flavor Physics (Capri, Italy, 16-18 June 2008),

published on Nucl. Phys. Proc. Suppl. 185, 140 (2008).

F. Giannuzzi, “Doubly heavy baryons in a quark model with AdS/QCD inspired

potential,”

Proceeding of IFAE2009, Incontri di Fisica delle Alte Energie, (Bari, Italy,

15-17 April 2009),

published on Il Nuovo Cimento C 32, N. 3-4, p. 131 (2009), [arXiv:0909.2525

[hep-ph]].

M. Mannarelli and F. Giannuzzi, “Decay widths of charmonia in a hot equilibrated

medium,”

Proceeding of Three Days of Strong Interactions (Wroclaw, Poland, 9-11 July

2009),

arXiv:0910.3147 [hep-ph], to appear on Acta Physica Polonica B.

F. Giannuzzi, “Heavy hadron spectroscopy in a Salpeter model with AdS/QCD

inspired potential”,

Proceeding of The 2009 Europhysics Conference on High Energy Physics,

(Krakow, Poland, 16-22 July 2009),

published on Proceedings of Science EPS-HEP 2009 059 (2009).

Introduction

Quantum Chromodinamics (QCD) is the non-abelian gauge theory describing the

strong interactions, those responsible for binding quarks into nucleons and nucleons

into nuclei. At present, it is a sector of elementary particle physics intensely inves-

tigated both experimentally and theoretically. An analytic solution of QCD is far

from being found; indeed, the perturbative methods of quantum field theory cannot

be applied in a wide range of energies where the value of the strong coupling con-

stant is large. Some methods have been introduced to study the non-perturbative

regime of QCD, e.g.:

• Lattice QCD;

• QCD Sum Rules;

• Effective theories (e.g. Heavy Quark Effective Theory, Nambu Jona-Lasinio

model, Chiral perturbation theory, Non Relativistic QCD);

• Potential models;

• Schwinger-Dyson equations.

A recently developed approach is the so-called AdS/QCD. It has been inspired

by the AdS/CFT correspondence, a conjecture introduced by Maldacena in 1998

relating a string theory in a d + 1 dimensional anti-de Sitter space (AdS) times a

compact manifold with a super Yang Mills (SYM) theory in a d dimensional flat

space. Its most attractive property is that the strong coupling regime of one theory

corresponds to the weak one of the other theory: this would allow us to study the

non-perturbative regime of QCD through perturbative techniques applied to the

dual theory.

In this thesis, the main features of the AdS/CFT conjecture will be pointed

out and we shall see how the construction can be modified in order to describe

QCD in a four dimensional Minkowski space. In particular, we shall focus on a

11

Introduction

phenomenological model, the Soft Wall model.

In chapter 1 the path leading to the gauge/string conjecture will be reconstructed,

with a brief summary of the two theories linked by the correspondence, namely the

N = 4 SYM and type IIB string theory on AdS5 × S5, and with a discussion on

the large N limit of the field theory and D3-branes. Then, the conjecture will be

enunciated and analysed. In chapter 2 the application to QCD will be considered,

and the phenomenological bottom-up approach will be studied in details. In the

case of vector mesons it will be shown how to perform calculations in the so-called

Hard Wall and Soft Wall models; in the latter, the scalar glueball sector will be

investigated as well, as a starting point for subsequent analyses at finite temperature.

After these introductory notes, chapters 3, 4 and 5 will be devoted to the new results

obtained within the holographic framework, that this thesis aims at presenting. In

chapter 3 the scalar meson sector will be studied, obtaining some predictions about

spectrum, decay constants, gluon condensates and couplings to two pseudoscalar

mesons. In chapter 4 two proposals for studying QCD at finite temperature will

be introduced and applied to the scalar glueball and meson sectors. In chapter 5

it will be described how to obtain the quark-antiquark static potential through the

holographic approach; the result will be used in a potential model with relativistic

kinematics to compute masses of S-wave heavy hadrons, including heavy mesons

and some tetraquarks and baryons, and some radiative and leptonic decay widths

of charmonium and bottomonium states.

12

Chapter 1

The AdS/CFT correspondence

1.1 Historical notes

Quantum Chromodynamics (QCD) is the quantum field theory of the strong inter-

actions. This fundamental force is mediated by the exchange of quanta, the gluons,

between the matter degrees of freedom, the quarks. The charge carried by the glu-

ons is called colour and the non-abelian gauge group associated to it is SU(3)c [1],

under which quarks transform in the fundamental representation while gluons in the

adjoint one. Colour charged particles cannot be directly observed. The force con-

fining quarks inside the hadrons is very strong, with coupling about hundred times

larger than the electromagnetic one. However, it varies with the distance between

the quarks, becoming weak at short distances (i.e. high energies): this property is

known as asymptotic freedom and was predicted in [2] and observed in deep inelastic

scattering processes [3].

Before QCD, in the 1960’s string theory was introduced as a model to describe

the strong interactions [4]. It was able to explain the organisation of hadrons in

Regge trajectories, describing them as rotating strings. After the formulation of

QCD, string theory took a different direction, becoming a possible candidate for a

unified theory of all the forces.

Nevertheless, some string interpretation of hadron spectra was not abandoned;

for example, a meson is sometimes described as a quark and an antiquark connected

by a tube of strong interaction flux [5]. This picture establishes a link between QCD

and string theory, which becomes even more evident in the limit of large number

of colours N [6]. ’t Hooft proposed that in this limit the gauge theory may have a

description in terms of a tree level string theory; in particular, the leading Feynman

diagrams in the 1/N expansion are planar and look like the worldsheet of a string

theory. For example, a meson can be represented by two quark lines propagating in

13

Chapter 1. The AdS/CFT correspondence

time connected by a dense “sheet” of gluons, reminding the worldsheet swept out

by a string through time.

In 1997 these studies found a possible new framework in the so-called AdS/CFT

correspondence [7], a conjecture introduced by Maldacena relating a supergravity

theory in ten dimensions to a supersymmetric gauge theory in four dimensions. This

correspondence has been extended to a gauge theory as SU(3)c, as we shall discuss

in the following chapters, thus proving some link between QCD and a higher dimen-

sional theory in a curved space-time.

In this chapter, we illustrate the meaning of Maldacena duality. To this aim, before

introducing the conjecture, the two theories that are associated by the correspon-

dence are briefly described. In particular, concerning the gauge theory, some notes

about supersymmetry and the large N limit are collected; on the other hand, a brief

discussion on type IIB string theory in the supergravity limit and on the anti-de

Sitter (AdS) space is also carried out.

1.2 Super Yang Mills theories

In this section, the main properties of super Yang Mills theories, one side of the

correspondence, are reminded.

A super Yang Mills theory is a supersymmetric gauge theory, i.e. a gauge theory

in which the fermionic degrees of freedom match the bosonic ones. For each existing

boson, a fermionic partner must exist and viceversa, having the same mass and

quantum numbers, but with spins which differ by 1/2.

Supersymmetric (SUSY) models, which allow us to unify matter (fermions) and

interactions (bosons), have been introduced [8] to answer the questions and problems

left by the Standard Model (SM) (e.g. the hierarchy problem, the origin of dark

matter, the neutrino mass)[9], but they also aim at the ambitious purpose of unifying

the electroweak and strong interactions. In the Standard Model the three couplings

(electric, weak and strong) almost converge at an energy of about 1015 GeV, but

exact unification is excluded by about nine standard deviations, as shown in Fig.

1.1 (top panel). On the other hand, in SUSY theories the running of the couplings

changes at a certain energy, and they seem to converge at an energy of about 1016

GeV, as shown in Fig. 1.1 (bottom panel) [10, 11]. This higher scale of unification

gets closer to the reduced Planck scale (1018 GeV), suggesting that also gravity

could be somewhat included in the picture. Moreover, in this respect, there is also

a fundamental motivation. At odds with SM, SUSY allows interactions between

particles with even and odd integer spin, while the no-go theorem by Coleman and

14

1.2. Super Yang Mills theories

Mandula [12] proves that this cannot happen in non-supersymmetric theories. This

is crucial for unifying gravity with the other forces: in fact, gravity is mediated by

the exchange of a spin 2 boson (graviton), while the other interactions by spin 1

bosons. A theory describing all the observed phenomena has to be supersymmetric.

Figure 1.1: Running of the couplings in the SM (top panel) and in the MinimalSupersymmetric Model (bottom panel). The small figures are a blow up of thecrossing area among the couplings [10].

However, no experimental evidence of supersymmetric partners has been found

so far; the new hadron collider LHC at CERN will allow a direct search of these

states up to masses of several hundreds of GeV.

Let us analyse some basic aspects of the SUSY models. A supersymmetry trans-

formation mapping a particle in its superpartner is represented by a conserved charge

Q ([Q,H] = 0) changing the spin of the state on which it acts by one half:

Q |fermion〉 = |boson〉 Q |boson〉 = |fermion〉 ; (1.1)

therefore, the operator Q must be a spinor. The number of operators Q in the

model (N ) fixes the number of superpartners for each particle and characterises

the supersymmetric model. A particle and its superpartners belong to the same

15


supermultiplet, defined as the set of quantum states that can be transformed into

one another by one or more supersymmetry transformations.

The quantum number associated to Q is the R-charge, so N is also the number

of conserved R-charges. The supersymmetry algebra contains the algebra of the

Lorentz group and also

QMα , Q

Nβ = 2σµ

αβPµ δMN (1.2)

where Pµ is the energy-momentum vector, σ0 is the identity matrix, σi, i = 1, 2, 3,

are the Pauli matrices and M,N = 1, . . . ,N . The anticommutators (1.2) connect

internal (represented byQ) and geometric (represented by P ) symmetries [13], allow-

ing unification of gauge interactions (inner space) with gravity (space-time). Since

there is no evidence of superpartners having the same mass as elementary particles,

supersymmetry must be spontaneously broken, and the energy of the vacuum state

must be different from zero. This gives different masses to different members of the

supermultiplets.

The case we are interested in is the N = 4 SYM theory. The gauge group is

SU(N) and the coupling constant gYM. There are four supercharges Q, with four

real components each, so a supermultiplet has sixteen real components. There is

only one supermultiplet, comprising one gauge field AAµ (A = 1, ...N2 − 1), six real

scalar fields XAi and four Weyl spinors λAαa (α = 1, 2), where the indices i = 1, ...6

and a = 1, ...4 are linked to the R-symmetry of the system (SU(N )), for which AA is

a singlet, λ transforms under the fundamental representation and X in the adjoint;

the spinor and the scalar fields are in the adjoint representation of the gauge group.

The Callan-Symanzik β-function of this theory, representing the variation of the

coupling with the renormalisation scale, vanishes at all orders, so the theory is scale

invariant and the coupling does not run. For example, one can prove this property

at one loop [14]:

β(gYM)1loop = − 1

16π2

(11

3C(A)− 2

3

∑λ

C(λ)− 1

6

∑X

C(X)

), (1.3)

where∑

λ takes into account the sum over all Weyl fermions,∑

X the sum over

all the real scalars, and C are the quadratic Casimir operators, depending on the

representations of the gauge group. Since all the fields of the theory are in the

adjoint representation of the SU(N) group, C(A) = C(λ) = C(X) = C(adj) = N ,

16

1.2. Super Yang Mills theories

the one-loop β function vanishes:

11

3− 2

3× 4− 1

6× 6 = 0 . (1.4)

Therefore, N = 4 SYM theory respects a larger symmetry, the conformal sym-

metry, which also comprises scale invariance. Indeed, the theory is invariant under

the transformations of SU(2, 2) ∼ SO(4, 2), the four dimensional conformal group.

These transformations are particularly important for the correspondence, so the

next section is dedicated to them.

1.2.1 The conformal group

The conformal transformations act on the coordinates of the space-time leaving the

metric tensor gµν invariant up to a coefficient Λ(x):

gµν −−−→x→x′

g′µν(x′) = Λ(x)gµν(x) . (1.5)

The transformations satisfying (1.5), schematically reported in Table 1.1, are Poincare

transformations, dilations and special conformal transformations [15]. In particular,

invariance of a theory under dilations requires scale invariance.

Table 1.1: Conformal transformations and generators. Mµν is the angular mo-

mentum tensor.

Transformation Generator

Translation x′µ = xµ + aµ Pµ = −i ∂µ

Dilation x′µ = a xµ D = −i xµ ∂µ

Rotation x′µ = Mµν xν Lµν = i (xµ∂ν − xν∂µ)

Special conf. x′µ =xµ − bµx2

1− 2 b · x+ b2 x2Kµ = −i

(2xµx

ν∂ν − x2∂µ

)

A spinless field φ(x) transforms under a conformal transformation x→ x′ as

φ(x) → φ′(x′) =

∣∣∣∣∂x′∂x

∣∣∣∣−∆/d

φ(x) , (1.6)

where d is the dimension of the space and ∆ is called the conformal dimension of

the field. A field transforming as in (1.6) is called quasi-primary field. In a quantum

field theory with conformal invariance, the two-point correlation function of two

17


quasi-primary fields, defined as

〈φ1(x1)φ2(x2)〉 =1

Z[φ]

∫Dφφ1(x1)φ2(x2) e−S[φ] (1.7)

where Z[φ] is the partition function and S[φ] the action, is always of the form [16]:

〈φ1(x1)φ2(x2)〉 =

C12

|x1 − x2|2∆∆1 = ∆2 = ∆

0 ∆1 6= ∆2 ,

(1.8)

where ∆1 and ∆2 are the conformal dimensions of the fields φ1 and φ2, respectively.

Therefore, two quasi-primary fields are correlated if and only if they have the same

conformal dimension. The functional form of three-point correlation functions is

fixed by the symmetry, while higher order correlators are not constrained.

1.3 Type IIB string theory and AdS space

In this section we briefly review some properties of the theory at the other side of the

correspondence, type IIB string theory on AdS5×S5, AdS5 being a five dimensional

anti-de Sitter space and S5 a five dimensional sphere.

Type IIB string theory is one of the five consistent models describing supersym-

metric strings. It contains closed and open strings, and no tachyons. Open strings

are attached to D-branes, so their ends are fixed by Dirichlet conditions. In general,

one can also apply Neumann conditions to open strings, which act on the derivative

of the function describing the string, so, in this case, the ends of the string remain

free. It differs from type IIA string theory for having massless chiral fermions [17].

We focus on the supergravity (SUGRA) limit of the theory, in which the AdS/CFT

correspondence is more interesting. This limit is reached when the string coupling

gs is small and the curvature radius R of the space in which the theory lives is large

with respect to α′ = 12πσ

, where σ is the string tension, defined as the ratio between

the mass and the length of the string:gs 1

R√α′ 1 .

(1.9)

18

1.3. Type IIB string theory and AdS space

The supergravity theory involved in the AdS/CFT correspondence lives in the

ten dimensional space resulting from the product of a five dimensional anti-de Sitter

(AdS5) space with radius R and a five-sphere (S5) with equal radius R, namely

AdS5 × S5, with metric:

ds2 = ds2(AdS5(R)) + ds2(S5(R)) . (1.10)

The AdS5 space is a maximally symmetric Lorentzian manifold with constant

negative curvature. It can be represented as a five dimensional hyperboloid of radius

R, shown in Fig. 1.2:

X20 +X2

5 −4∑

i=1

X2i = R2 (1.11)

embedded in a flat six dimensional space with pseudo-Euclidean metric

ds2 = −dX20 − dX2

5 +4∑

i=1

dX2i (1.12)

with two time-like dimensions (X0 and X5). Xa are called embedding coordinates.

X4

X0

X5

Figure 1.2: Representation of the AdS5 space in the embedding coordinates. X0

and X5 are the time-like dimensions; the coordinates X1,2,3 are fixed [18].

Eq. (1.11) shows that the isometry group of the AdS5 space is SO(4, 2), which

is also the conformal group of the N = 4 SYM theory (see the previous section).

The metric of the AdS space can be expressed in the Poincare coordinates

19


(z, x, t), defined by the following relations [18]:

X0 =1

2z

(z2 +R2 + x2 − t2

)Xi =

Rxi

zi = 1, 2, 3

X4 =1

2z

(z2 −R2 + x2 − t2

)X5 =

Rt

z.

(1.13)

In particular, the Poincare coordinate z is given by:

1

z=X0 −X4

R2; (1.14)

therefore, two Poincare charts can be distinguished, one corresponding to the region

z > 0 (the half of the hyperboloid with X0 > X4), and the other one to z < 0. The

z = 0 brane is the boundary of the AdS space; it occurs at spatial infinity in the

embedding coordinates.

The presence of a boundary is a very important feature of the AdS space, crucial

for the AdS/CFT correspondence, since the conformal field theory is constructed on

this boundary.

The Poincare AdS space corresponds to one of these regions: usually, for the

AdS/CFT correspondence, the z > 0 chart is chosen. Its line element is:

ds2 =R2

z2(−dt2 + dx2 + dz2) z > 0 . (1.15)

The whole AdS space can be covered using the global coordinates (τ, ρ,Ωi),

which are defined by [19]:

X0 = R cosh ρ cos τ

Xi = R sinh ρ Ωi i = 1, ..., 4

X5 = R cosh ρ sin τ

(1.16)

with τ ∈ [0, 2π[, ρ ≥ 0, Ωi ∈ [−1, 1] satisfying∑

i Ω2i = 1. Note that the time

variable τ is compact, so we must “unwrap” it to get τ ∈ R by actually considering

the AdS covering space, that means an infinite set of copies of AdS spaces in the τ

20

1.4. Steps towards the AdS/CFT correspondence

direction. It is convenient to introduce a new variable θ related to ρ by tan θ = sinh ρ,

with 0 ≤ θ < π/2; the new line element reads:

ds2 =R2

cos2 θ

(−dτ 2 + dθ2 + sin2 θ

4∑i=1

(dΩi)2

). (1.17)

The boundary is at θ = π/2. Using these coordinates it is possible to draw the

Penrose diagram of the AdS space [20]: it is a solid cylinder, represented in Fig.

1.3, whose boundary (at θ = π/2) is S3 ×R, R corresponding to the time direction

τ . A light ray moves in this space as shown in the figure: it reaches the boundary

and than comes back, in finite time and with the same time delay, like a boomerang.

A massive particle behaves similarly, but it can never reach the boundary.

S3

Θ=0 Θ=Π2

Τ

Figure 1.3: Penrose diagram of the AdS space in global coordinates. The redand blue lines show the path of a light ray and of a massive particle, respectively.The lateral surface of the cylinder represents the boundary of the AdS space.

1.4 Steps towards the AdS/CFT correspondence

From the initial considerations, we have understood that the conjecture of Malda-

cena was the conclusion of a series of studies about possible relations between string

and gauge theory, and how they could be useful to each other.

The two fields that most inspired Maldacena were the studies on the large N limit of

gauge theories and the ones on D3-branes. The following sections will try to clarify

this point.

21


1.4.1 Large N

Consider a gauge theory, with gauge group SU(N), and only matter fields in the

adjoint representation, as it occurs, e.g., in the N = 4 SYM theory. Although

perturbation theory is controlled by powers of the coupling gYM, one must be careful

because a large N might compensate for a small gYM.

Let us provide an example, using the double-line notation for drawing Feynman

diagrams [6]. An adjoint field is represented as a direct product of a fundamental

and an anti-fundamental field, so its propagator is like that of a fundamental-anti-

fundamental pair. Consider the propagator of a “gluon”

〈0|T[AA

µ (x)ABν (y)

]|0〉 = Dµν(x− y)δAB , (1.18)

which is usually represented by a curly line joining the points x and y, as shown in

the top-left panel of Fig. 1.4. We introduce, instead of AAµ , the traceless N × N

matrices Aµ so that

A iµ j = AA

µ (TA)ij with A = 1, ...N2 − 1 and i, j = 1, ...N (1.19)

where TA are the generators of the SU(N) group; with (1.19), the propagator in

(1.18) becomes [21]:

〈0|T [A iµ j(x)A

kν p (y)]|0〉 = Dµν(x− y)

(δipδ

kj −

1

Nδijδ

kp

)(1.20)

and can be represented by two opposite lines, one going from i to p and the other

one from k to j, as shown in the bottom-left panel of Fig. 1.4. Similarly, a vertex

with three “gluons” can be represented by three pairs of straight opposite lines and

its amplitude is proportional to gYMδipδ

lmδ

kj (central panels of Fig. 1.4), while the

amplitude of a four-line vertex is proportional to g2YM (right panels of Fig. 1.4).

Since the limit N →∞ will be considered, we drop in (1.20) the term proportional

to 1/N .

The amplitude of a closed line, or loop, gets a factor N coming from the product

δji δ

ij. Therefore, for example, the loop obtained joining two “gluons”, represented by

two straight lines each, has an amplitude proportional to N2 (left panel in Fig. 1.5);

the amplitude of the loop obtained joining two three-line vertices is proportional to

g2YMN

3 = λN2, where λ = g2YMN is the ’t Hooft coupling (central panel in Fig. 1.5).

Since the lines have an orientation (in one direction for a fundamental index

and in the opposite direction for an anti-fundamental index), a generic diagram

defines an oriented surface, like the one in Fig. 1.6. If F is the number of faces,

22


1

1

1

i pj k

Figure 1.4: Top (resp. bottom)-left panel: a “gluon” propagator in standard(resp. double-line) notation. Top (resp. bottom)-central panel: a “gluon” three-vertex in standard (resp. double-line) notation. Top (resp. bottom)-right panel:a “gluon” four-vertex in standard (resp. double-line) notation.

Figure 1.5: Left panel: graph obtained by joining two “gluons”, i.e. two double-line graphs. Central panel: graph obtained by joining two “gluon” three-vertices.Right panel: graph obtained by joining four “gluon” three-vertices.

E the number of propagators (edges), V3 and V4 the number of three- and four-

vertices, respectively, with V = V3 + V4 the number of vertices, then the number

of propagators can be written as E = 12(4V4 + 3V3) and the scaling of a diagram is

given by:

gV3+2V4YM NF = (g2

YMN)E−VNF−E+V . (1.21)

χ = F − E + V is the Euler characteristic of the two dimensional surface on which

the diagram can be written without intersecting. For an oriented surface, the Euler

characteristic is χ = 2− 2g, where g is the genus of the surface, i.e. the number of

holes.

One can carry out a perturbative expansion of any diagram in the field theory

of the form [19]:∞∑

g=0

N2−2g

∞∑i=0

cg,iλi =

∞∑g=0

N2−2g fg(λ) , (1.22)

where fg is some polynomial in λ. In the ’t Hooft limit, or large N limit, charac-

terised by

N →∞ with λ fixed , (1.23)

23


Figure 1.6: A generic diagram in double-line notation.

only the diagrams with maximal χ contribute: these are the planar diagrams, i.e.

those that can be written on a sphere, which has χ = 2. The others are suppressed

by powers of 1/N2.

Moreover, in the strong coupling regime λ → ∞ diagrams with many vertices

are favoured: the vertices will fill the surface more densely, defining a worldsheet of

the kind one might expect in string theory [6].

The important point is that the expansion in (1.22) is the same as the topological

expansion in string theory with closed oriented strings, identifying 1/N as the string

coupling constant [22]. In the ’t Hooft limit, this is also equivalent to identify the

coupling constant of the string theory (gs) with the one of the SYM theory (g2YM),

since

gs = g2YM =

λ

N. (1.24)

Let us stress once again this result: the ’t Hooft limit of a SU(N) gauge theory

corresponds to the weak coupling regime of a string theory (gs → 0), with gs = g2YM.

Another important comment follows from (1.24). The limit λ→∞ corresponds

in the string theory to gsN → ∞. We anticipate that the radius R of the AdS5

space indicated in the AdS/CFT correspondence is R4 ∝ gsN .

Therefore, for a string theory living in the AdS space with radius R4 ∝ gsN , the

weak coupling regime (gs → 0) corresponds to the ’t Hooft limit of a SU(N) gauge

theory, with gs = g2YM. Then, the supergravity limit (gs → 0 and R→∞, see (1.9))

is associated to the strong coupling regime λ→∞ of the gauge theory.

This analysis was based on perturbation theory, and it is far from a rigorous

derivation of a relation between field and string theories. It is rather an indication

that a gauge theory at strong coupling could be described by a string theory at weak

coupling.

24


1.4.2 D3-branes

Here we want to emphasise some features of D3-branes, without entering in the

details of this argument, which are beyond the scope of this thesis.

A p-brane is a p-dimensional object, whose world-volume is (p+ 1)-dimensional.

We can split the coordinates of a d-dimensional space-time into (p+ 1) longitudinal

(xµ, µ = 0, 1, ..., p) and (d − p − 1) transverse (yi, i = p + 1, ..., d − 1) coordinates,

with r2 = yiyi.

Dirichlet p-branes [23], or Dp-branes, are p dimensional hyperplanes where open

strings can end; longitudinal (transverse) coordinates satisfy Neumann (Dirichlet)

boundary conditions.

Let us consider N coincident D3-branes in a ten dimensional space.

In the low-energy limit, the theory on the (four dimensional) world-volume coincides

with N = 4 supersymmetric Yang Mills theory with gauge group U(N) in four

dimensions [24].

On the other hand, D3-branes can also be seen as solutions of ten dimensional type

IIB supergravity, with metric of the form:

ds2 =

(1 +

L4

r4

)−1/2

(dxµ)2 +

(1 +

L4

r4

)1/2

(dr2 + r2dΩ5) ; (1.25)

L4 = 4πgsNα′2, where gs is the string coupling constant and α′ is proportional to

the inverse of the string tension. This metric interpolates between a throat geometry

and a ten dimensional Minkowski region. In fact, in the limit r →∞, the metric in

(1.25) reduces to the one of a ten-dimensional Minkowski space:

ds2 = dx2 + dr2 + r2 dΩ5 ; (1.26)

in the low-energy limit r → 0, it reduces to:

ds2 =r2

L2(dxµ)2 +

L2

r2dr2 + L2 dΩ5 , (1.27)

which is the metric of an AdS5 × S5 space. This becomes evident by changing the

radial coordinate via r → z = L2/r:

ds2 =L2

z2

(dx2 + dz2

)+ L2 dΩ5 , (1.28)

where L is the radius of curvature of both the sphere and the AdS space.

Matching the two low-energy descriptions of N coincident D3-branes, Maldacena

25


could identify N = 4 SYM theory on a four dimensional Minkowski space with type

IIB supergravity theory in AdS5 × S5 [7].

1.4.3 The holographic principle

Before introducing the duality conjecture, it can be helpful to define the concept of

Holography, which will be sometimes used in the text.

The holographic principle was proposed by ’t Hooft in [25], and then developed

by Susskind [26], who applied it to string theory. It states that, in a quantum

theory of gravity, the number of allowed configurations of the system in a finite

volume of space is proportional to the area of the boundary of the space, so the

total information contained in the volume of space can be stored in the boundary.

A review on the holographic principle can be found in [27].

This concept is reminiscent of a hologram, in fact holography is also an optical

technology by which a three dimensional image is stored on a two dimensional surface

via a diffraction pattern.

1.5 The conjecture

The AdS/CFT correspondence conjectured by Maldacena in 1997 [7] states that

Type IIB string theory on (AdS5 × S5)N plus some appropriate boundary

conditions (and possibly also some boundary degrees of freedom) is dual to

N = 4 d = 3 + 1 U(N) super-Yang-Mills.

Dual means that there is a one-to-one map between the physical quantities of

N = 4 SYM theory, with gauge group U(N) and coupling constant gYM, and type

IIB string theory on AdS5×S5, with gs = g2YM. AdS5×S5 is fixed by requiring that

the string theory has the same number of supersymmetries (32) as the N = 4 SYM

theory. The isometry group SO(2, 4) of the AdS5 × S5 acts on the boundary as the

conformal group of the SYM theory.

Later on, Witten proved [28] that, although the theory on the boundary has a

U(N) gauge group, with U(1) a free factor, however, actually, the physics in the

interior of the bulk is described by the SU(N) piece [29]. Therefore, it is more

correct to consider in the boundary a field theory with SU(N) gauge group, in

which N quarks can combine into a neutral object [24].

26

1.5. The conjecture

The above statement is referred to as the strong form of the conjecture, since

it is assumed to hold for any N and gs. There are indeed three versions of the

correspondence [30]:

1. Strong version: Type IIB string theory on AdS5×S5 (∀gs and ∀R2/α′) is dual

to SU(N) SYM theory (∀gYM and ∀N);

2. Mild Version: Weakly coupled type IIB strings on AdS5 × S5 (gs → 0 and

R2/α′ fixed) is dual to planar SU(N) SYM theory (N → ∞ and λ = g2YMN

fixed);

3. Weak Version: Type IIB supergravity on AdS5×S5 (gs → 0 and R2/α′ →∞)

is dual to planar SU(N) SYM theory at strong coupling (N →∞ and λ→∞).

The mostly adopted form of the conjecture is the third one (in the SUGRA

limit), where it is known how to compute amplitudes. As we have seen in section

1.4.1, the SUGRA limit in the AdS side corresponds to the strong coupling regime

and large N , or ’t Hooft, limit in the CFT one.

A comment is in order. It follows that in the region where one theory is weakly

coupled the other one is strongly coupled, so the two weakly coupled descriptions

are non-overlapping: this makes the conjecture hard to prove or disprove but it may

also happen that a calculation could be very difficult on the strongly coupled side

and very easy on the dual weakly coupled one.

In order to use the duality to perform calculations, we need a dictionary which

relates observables on each side. A precise way in which the two theories can be

mapped into each other was not given in the original paper of Maldacena, but was

later proposed independently by Gubser, Klebanov and Polyakov [31] and by Witten

[28]. Since the boundary of the AdS5 space, namely S3 × R, is equivalent to R3,1,

which is a copy of the Minkowski space, plus a point at infinity, the authors suggested

a recipe to link the gravity theory in the bulk (AdS space) to the field theory on

the boundary (Minkowski space). In this sense, the AdS/CFT correspondence can

be considered as a holographic projection of the supergravity theory in the bulk to

the field theory on the boundary, as symbolically depicted in Fig. 1.7.

They proposed that there is a one-to-one correspondence between the fields in the

bulk and the gauge invariant local operators of the field theory on the boundary;

the conformal dimensions of the operators are determined by the masses of fields.

Let us now see in detail how this works.

We adopt the metric of the AdSd+1 space in Poincare coordinates:

ds2 =R2

z2(dxµdxµ + dz2) z > 0 (1.29)

27


Figure 1.7: Sketch of the holographic description of the AdS/CFT correspon-dence.

where µ = 0, 1, ..., d− 1.

The supergravity partition function ZS[φ] is given by:

ZS[φ0] = exp(−S[φ]) (1.30)

with S the classical supergravity action. φ0(x) is the boundary value of the field

φ(x, z), defined as follows:

φ(x, z) =

∫∂AdSd+1

ddx′ K(z, x− x′) φ0(x′) (1.31)

K(z, x− x′) −−−−−→∂AdSd+1

zξ δd(x− x′) , (1.32)

where ∂AdSd+1 indicates the boundary of the AdSd+1 space; the value of ξ depends

on the field and its mass. The function K is called bulk-to-boundary propagator.

The ansatz for the relation between the two theories, written in [28] in the

Euclidean four dimensional space and here generalised to the Minkowski one, is

that the supergravity partition function, ZS[φ0], is equal to the generating functional

of the correlation function of the corresponding operators, provided the boundary

conditions (1.31) (1.32) are satisfied:⟨exp

∫∂AdSd+1

ddx φ0(x)O(x)

⟩CFT

= ZS[φ0] . (1.33)

φ0 is interpreted as the source of the operator O.

There is a relation between the mass of the field φ and the conformal dimension

of the operator O in the conformal field theory. Let us find this relation in the case

28

1.5. The conjecture

of a massive scalar field.

The action for a scalar field in an AdS space reads:

S[φ] =1

2

∫AdSd+1

dd+1x√|g|[gMN(∂Mφ)(∂Nφ) +m2

d+1φ2]

(1.34)

where g is the determinant of the metric and md+1 is the mass of the field. The

equation of motion is(1√g∂M

[√|g| gMN∂N

]−m2

d+1

)φ(x, z) = 0 . (1.35)

We substitute (1.31) into Eq. (1.35), with boundary condition (1.32) at a certain

point P on the boundary. If we choose P to be the point at z = +∞, both the equa-

tion of motion and the boundary condition become invariant under four dimensional

translations of the x coordinates, so that:

zd+1

R2∂z(z

−d+1 ∂zK)−m2d+1K = 0 . (1.36)

We may find a solution K ∝ zd+λ, provided

λ(λ+ d)−m2d+1R

2 = 0 , (1.37)

whose roots are

λ± = −d2±√d2

4+m2

d+1R2 . (1.38)

The solution that behaves as in (1.32) in P , namely that vanishes at z = 0 and goes

to infinity at z →∞, is K = zd+λ+ . To show that it has the same singularity in P

as a delta function, it helps to make an inversion by mapping P to z = 0:

xµ → xµ

z2 + x2, (1.39)

which transforms K to:

K =zd+λ+

(z2 + x2)d+λ+. (1.40)

Using the limitzd+2λ+

(z2 + x2)d+λ+−−→z→0

δd(x) , (1.41)

Eq. (1.40) becomes:

K −−→z→0

z−λ+δd(x) , (1.42)

29


as in (1.32). From (1.31) and (1.40), the field φ can be written as:

φ(x, z) = c

∫ddx′

zd+λ+

(z2 + |x− x′|2)d+λ+φ0(x

′)

= c z−λ+

∫ddx′

zd+2λ+

(z2 + |x− x′|2)d+λ+φ0(x

′) . (1.43)

We can now evaluate the action (1.34) on-shell, that is using (1.43), and, integrating

by parts and reducing to a surface term, we get

S[φ] =c (d+ λ+)

2

∫ddx ddx′

φ0(x)φ0(x′)

|x− x′|2(d+λ+). (1.44)

From the prescription of the correspondence (1.33), deriving twice both sides with

respect to the source φ0, it results that

〈O(x)O(x′)〉 =1

|x− x′|2(d+λ+), (1.45)

so, by virtue of (1.8), the conformal dimension ∆ of the operator dual to a scalar

field is

∆ = d+ λ+ =d

2+

√d2

4+m2

d+1R2 (1.46)

where md+1 is the mass of the scalar field in the AdSd+1 space. On the other hand,

the squared mass of a field dual to an operator with conformal dimension ∆, is

m2d+1R

2 = ∆(∆− d) . (1.47)

The generalisation to fields other than scalars is in Table 1.2 [32].

Table 1.2: Relations between the conformal dimension ∆ of operators and thesquared masses m2

d+1 of the dual fields [32].

Operators Relations

scalars m2d+1R

2 = ∆(∆− d)

spin 1/2, 3/2 |md+1|R = ∆− d/2

p-form m2d+1R

2 = (∆− p)(∆ + p− d)

spin 2 m2d+1R

2 = ∆(∆− d)

30

Chapter 2

The AdS/QCD correspondence

Taking obstacles into account, the

shortest line between two points

may be a crooked one.

Bertolt Brecht, Life of Galileo

We have learnt that the AdS/CFT correspondence relates a supergravity theory

in the weak coupling regime to a four dimensional SYM field theory in the strong

coupling regime. It would be useful if the four dimensional theory were QCD, since

this would allow us to explore its non-perturbative regime by studying a perturbative

dual theory. However, the field theory described by the correspondence is a super-

symmetric theory with conformal invariance, while QCD has none of these features.

The most important differences between QCD and SYM theories are summarised in

Table 2.1 [30].

Then, one can think of somehow modifying the supergravity theory so that its

holographic projection could describe QCD. This is known as the top-down approach,

in which a string model is built in such a way that, at low energies, it may describe

a gauge theory with features similar to QCD. For example, one can start from the

AdS/CFT correspondence, and modify the background of the supergravity theory

in order to break supersymmetry and conformal symmetry. Moreover, since the

original correspondence contains only D3-branes corresponding to adjoint degrees

of freedom in the SYM theory, fundamental matter with flavour may be added

by introducing different types of D-branes [33]. For more details about top-down

models, see, e.g., [34].

We will focus on a more phenomenological approach, known as the bottom-up

approach. It consists in formulating a five dimensional phenomenological model,

inspired by supergravity, reproducing the properties of QCD [35]. Fitting the model

31

Chapter 2. The AdS/QCD correspondence

Table 2.1: Comparison between some features of QCD and large N super YangMills theory [30].

QCD SYM

confining not confining

has a chiral condensate has no chiral condensate

discrete spectrum continuous spectrum

running coupling conformal

quarks adjoint matter

not supersymmetric maximally supersymmetric

Nc = 3 Nc →∞

parameters with some data from QCD, it is then possible to predict other quantities,

like hadron masses and form factors. It should be considered as a phenomenologically

driven approach, interpolating between the low-energy and high-energy limits of

QCD. It is inspired by the AdS/CFT correspondence, but its direct relationship to

it is not straightforward.

In both approaches, the “dictionary” linking the “AdS” and “CFT” sides is

assumed to hold a priori. The agreement between the results obtained by the

AdS/QCD models and experimental data is surprisingly good (typically of order

of 10%). There are some incalculable systematic errors, since one would expect the

results to suffer, e.g., from being at large N , from near-conformality, from the pres-

ence of (broken) super-partners.

It is important to note that while the large Nc limit is clearly not the same as QCD,

it has been observed in lattice simulations [36] that some results obtained in the

large Nc limit are close to the ones obtained at Nc = 3, namely the mass spectra.

There are two main bottom-up approaches, the Hard Wall model and the Soft

Wall model, characterised by different ways of breaking the conformal symmetry of

the five dimensional theory.

Before studying in detail these holographic models, let us review some properties

of QCD that they try to reproduce.

32

2.1. QCD as a candidate for a holographic description

2.1 QCD as a candidate for a holographic descrip-

tion

Here we review some main aspects of the theory of strong interactions, giving some

details about the QCD Lagrangian and the chiral symmetry. These concepts will

represent the starting point for the construction of the phenomenological models.

Lagrangian

Let us focus on the field content and the properties of the Lagrangian of the theory,

which is given by:

LQCD = ψ(i /D −m

)ψ − 1

4F 2 , (2.1)

where ψ denotes the quark fields with mass matrix m, Dµ is the covariant derivative

and F aµν = ∂µA

aν − ∂νA

aµ − g fabcAb

µAcν , in which Aa

µ is the gluon field, fabc are the

structure constants of SU(3) and g is the coupling constant of the theory.

The masses of the up and down quarks are of only a few MeV, much below the

natural scale of QCD, ΛQCD ∼ 200 MeV, that is the energy at which the running

coupling constant diverges, so they are commonly referred to as light quarks. The

mass of the strange quark is less than hundred MeV. Heavy quarks are the charm,

bottom and top quarks, the mass of which is very large with respect to ΛQCD.

If the up and down quark masses are considered equal, the isospin symmetry follows,

and, as a consequence, protons and neutrons are not distinguished; the hadronic

states are organised in isomultiplets the members of which are degenerate in mass.

Chiral symmetry

In the approximation of vanishing up and down quark masses, the QCD Lagrangian

gets a further symmetry, the chiral symmetry, according to which left-handed up and

down quarks can be interchanged with the right-handed ones, and vice-versa. The

chiral symmetry group is SU(2)L × SU(2)R; if the strange quark is also considered

massless, the symmetry becomes SU(3)L × SU(3)R.

However, in real hadrons, different parity states are not degenerate in mass. There-

fore, chiral symmetry must be spontaneously broken by the formation of quark

condensates: the pseudoscalar mesons (the pions, for SU(2)) remain massless while

the scalar mesons (i.e., σ) get a mass. In this respect, the pions and scalars represent

the Goldstone and (the analogous of the) Higgs bosons, respectively, the latter giving

mass to all light hadrons. In addition, since light quarks have a finite, although very

small, mass, chiral symmetry undergoes a slight explicit breaking, too: this causes

33


the pions to have a small mass, so they are commonly regarded as pseudo-Goldstone

bosons.

The value of the quark condensate of up and down quarks is about (−240 MeV)3;

it makes hadrons comprising only light quarks, like ρ, so heavy, although the current

quark masses are just a few MeV. In some effective models, QCD vacuum effects are

neglected, and the constituent quark mass is introduced, which is almost 300-400

MeV for the quark up and down.

In the quark model, hadrons comprising light quarks are classified in multi-

plets having definite spin, parity and charge conjugation (JPC). In each multiplet,

hadrons are distinguished according to their isospin and ipercharge. Let us consider,

e.g., mesons, which are made up of a quark and an antiquark, in a model with spon-

taneously broken SU(3)L × SU(3)R symmetry. The quark is in the fundamental

representation (3) while the antiquark in the anti-fundamental one (3). When they

are combined, the octet and singlet irreducible representations can form:

3⊗ 3 = 8⊕ 1 . (2.2)

There are four nonets, corresponding to different spin-parity values: 0− (pseu-

doscalars), 0+ (scalars), 1− (vectors), 1+ (axial-vectors).

Conformal limits

In the limit of massless quarks, QCD is supposed to have another property, namely

that there is a range of energy in which the coupling is approximately constant [37].

This possible infrared fixed point of the β function would make QCD conformal

in that region. Some recent results for αs, extracted from data taken with the

CLAS spectrometer and corresponding to the blue triangles in Fig. 2.1 [38], seem to

confirm that αs becomes constant at very small Q2 and agree with the generalised

Gerasimov-Drell-Hearn (GDH) sum rule prediction [39] (dashed line in Fig. 2.1).

Moreover, due to asymptotic freedom, the β function of QCD at high energies

is also very small, and vanishes for αs → 0, so that in the ultraviolet QCD is also

approximately conformal, if the quark masses are neglected.

These two arguments support QCD as a candidate for a description inspired by the

AdS/CFT correspondence.

In the following, the Hard Wall and Soft Wall model of holographic QCD will

be investigated. For each QCD operator, a five dimensional action can be written,

34

2.2. Hard Wall model of AdS/QCD

Q (GeV)

α s(Q

)/π

αs,g1/π world data

αs,τ/π OPAL

pQCD evol. eq.

JLab PLB 650 4 244

JLab CLAS

αs,F3/π

GDH limit

0.06

0.070.080.090.1

0.2

0.3

0.4

0.5

0.6

0.70.80.9

1

10-1

1

Figure 2.1: Some extractions of the effective strong coupling constant αs(Q)/π(see [38] and references therein). At low Q, the points marked by blue triangleshave been obtained from CLAS at Jefferson Lab; the dashed line representspredictions from [39].

containing the kinetic term of the dual field and, eventually, the interaction terms.

In particular, we shall see how the chiral symmetry breaking can be implemented

in the bottom-up approach, and compare the outcome of the two models.

2.2 Hard Wall model of AdS/QCD

In the Hard Wall model, the conformal symmetry is broken by inserting a cutoff on

the z axis at z = zm, with zm ∼ O(Λ−1QCD) [35, 40, 41], so that the AdS space is

reduced to z 6 zm. This corresponds to introducing a scale in the five dimensional

theory, and the coordinate z can be related to the inverse of the energy [35, 40]. In

such a model, the running of the QCD gauge coupling is neglected until the infrared

energy Qm ∼ 1/zm. zm is called infrared cutoff and z = zm is the infrared brane.

The first topic that has been studied within the Hard Wall model is chiral sym-

metry breaking. The conservation of the global chiral symmetry currents in the

four dimensional theory is described by a gauge invariance in the five dimensional

one. The field content of the holographic theory and the corresponding operators

in QCD are summarised in Table 2.2. In the first column, the QCD operators that

are important for describing the SU(nf )L × SU(nf )R chiral flavour symmetry are

reported, i.e. the left- and right-handed currents (first two rows), and the chiral

order parameter, in the third row. In the second column the corresponding five

35


dimensional fields are defined, whose boundary values, according to the AdS/CFT

correspondence, are the sources of the operators in the first column. The number p is

the order of the p-form, ∆ is the conformal dimension, and m25 is the five dimensional

mass, computed following the relation:

m25R

2 = (∆− p)(∆ + p− 4) . (2.3)

Table 2.2: Operators and dual fields of the model for chiral symmetry breaking.qL and qR are the left- and right-handed quark fields; T a are the generators ofthe SU(Nc) group, with a = 1, ..., N2

c − 1; α and β are flavour indices.

O(x) φ(x, z) p ∆ m25 R2

qLγµT aqL Aa

Lµ 1 3 0

qRγµT aqR Aa

Rµ 1 3 0

qαRq

βL 2Xαβ/z 0 3 -3

Notice from Table 2.2 that the fields dual to the conserved currents are massless,

as it should happen for gauge fields.

The fact that the scalar bulk field X is tachyonic does not affect the stability

of the theory, since fields with slightly negative masses are allowed, as discussed in

[42]. The metric of the AdS space can be written as:

ds2 =R2

z2(dxµdxµ + dz2) 0 < z 6 zm (2.4)

where the index µ runs from 0 to 3 and R is the radius of the AdS space. The

boundary z = 0 is a Minkowski space with metric ηµν and signature (−,+,+,+).

The five dimensional action for this model is:

S = − 1

kH

∫d5x

√|g| Tr

|DX|2 +m2

5 |X|2 +1

4g25

(F 2L + F 2

R)

(2.5)

where g is the determinant of the metric in (2.4), m25R

2 = −3 from Table 2.2 and

kH is a parameter making the action dimensionless. D is the covariant derivative,

such that

DMX = ∂MX − iAL MX + iXAR M (2.6)

where AL,R = AaL,RT

a, M = 0, 1, .., 4. The strength tensor is FMN = ∂MAN −∂NAM − i[AM , AN ].

36


We introduce the axial and vector fields by the definition:

A =AL − AR

2V =

AL + AR

2, (2.7)

in terms of which the covariant derivative takes the form:

DMX = ∂MX + i[X,VM ]− iX,AM (2.8)

and the strength tensors become:

FMNV = ∂MV N − ∂NV M − i[V M , V N ]− i[AM , AN ] ,

FMNA = ∂MAN − ∂NAM − i[V M , AN ]− i[AM , V N ] . (2.9)

In the approximation of taking only the part of the Lagrangian which is quadratic

in the fields, we get

F 2L + F 2

R ≈ 2(F 2V + F 2

A) , (2.10)

so the action in (2.5) becomes:

S = − 1

kH

∫d5x

√|g| Tr

|DX|2 +m2

5|X|2 +1

2g25

(F 2V + F 2

A)

. (2.11)

Let us study the field X(x, z). We write it as X(x, z) = X0(z)e2iπaT a

, where

X0(z) = 〈X(x, z)〉 is its vacuum expectation value and π is the pion field, in anal-

ogous way to Chiral Perturbation Theory [43]. The Euler-Lagrange equation of

motion for X0 is:

∂z

[√|g| gzz ∂zX0(z)

]+

3

R2

√|g|X0(z) = 0 ; (2.12)

X ′′0 (z)− 3

zX ′

0(z) +3

z2X0(z) = 0 , (2.13)

whose general solution is

X0(z) = Az +B z3 , (2.14)

where the coefficients A and B are fixed by the boundary conditions at z = 0 and

z = zm. In the UV limit, z = ε→ 0, we require: (2R/ε)X0(ε) = M , where M is the

quark mass matrix, getting A = M/(2R). The IR boundary condition should fix

the other coefficient, but, instead of doing so, we consider B as an input parameter,

equal to the quark condensate Σ [35]:

X0(z) =1

2RMz +

1

2RΣ z3 . (2.15)

37


Eq. (2.15) shows that, in case of massless quarks (M = 0), we are dealing with

a theory with spontaneous symmetry breaking, while for vanishing condensate and

finite quark mass, we are considering explicit symmetry breaking. On the other

hand, if X0 = 0 the theory is chirally symmetric, as one can see from the action

(2.5), where the contributions of the AL and AR fields are equal.

2.2.1 Vector and axial-vector mesons

We now analyse the vector and axial-vector fields VM(x, z) and AM(x, z). We focus

on the nf = 2 lightest flavours, so T a = σa/2 with σa the Pauli matrices.

We also fix the Vz = Az = 0 gauge and define the perpendicular and longitudinal

components of the axial field, Aµ = A⊥µ + ∂µϕ (notice that the vector field has

only a transverse component, since the longitudinal one does not appear in the La-

grangian).

The 4d Fourier transformed field is defined by Vµ(x, z) =∫

dq4

(2π)4eiq·xVµ(q, z) (analo-

gously for the axial field). From Eq. (2.11), keeping only the terms with no more

than two fields, we get the following equations of motion for the vector and axial-

vector fields:

∂z

(1

z∂zV

aµ (q, z)

)− q2

zV a

µ (q, z) = 0 (2.16)

∂z

(1

z∂zA

a⊥µ(q, z)

)− q2

zAa⊥µ(q, z)− g2

5 R2 v(z)2

z3Aa⊥µ(q, z) = 0 (2.17)

∂z

(1

z∂zϕ

a(q, z)

)+g25 R

2 v(z)2

z3(πa(q, z)− ϕa(q, z)) = 0 (2.18)

q2∂zϕa(q, z) +

g25 R

2 v(z)2

z2∂zπ

a(q, z) = 0 , (2.19)

where we have defined v(z) = 2X0(z).

In the next section, the vector field case will be discussed, as a typical example

of how the spectra and other quantities are computed within this approach.

An example: the ρ meson

We interpret the normalisable modes in the 5d theory as corresponding to hadrons in

QCD. Therefore, in the vector field case, the normalisable solution ψρ of Eq. (2.16)

with q2 = −m2ρ, for an arbitrary component of Vµ, describes the ρ meson field and

its radial excitations. For this aim, Eq. (2.16) has to be solved with conditions (we

38


drop the flavour and Lorentz indices):

ψρ(0) = 0

∫dz

zψρ(z)

2 = 1 . (2.20)

The solution to Eq. (2.16) subject to (2.20) is:

ψρ(z) = Az J1 (mρz) (2.21)

with A the normalisation factor. Then, applying the Neumann boundary condition

at the IR brane, we get:

∂zψρ(z)|z=zm= 0 ⇒ J0 (mρ zm) = 0 . (2.22)

This condition determines the eigenvalues and eigenfunctions of the vector field, each

mode corresponding to a radial excitation of the ρ mesons with mass mρn , where

n is the radial quantum number. The nodes of the function J0(x) can be observed

in Fig. 2.2. We use Eq. (2.22) to fix the parameter zm from the experimental

value of the ρ mass; the first node in Fig. 2.2 occurs at x = mρ zm ≈ 2.4, so, from

mρ = 0.776 GeV, we get

zm ≈ 3.1 GeV−1 , (2.23)

which is correctly of O(Λ−1QCD).

A property of the J0(x) function is that the large zeros xn, i.e. the nodes at large

x, are proportional to n, xn ∼ n [44]. In this case, from (2.22) we find that at

high excitation numbers n 1, m2ρ ∼ n2. This behaviour disagrees with data,

which rather indicate m2ρn∼ n (Regge behaviour). This discrepancy suggests a

modification of the holographic model, described in the next section.

0 5 10 15 20-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

x

J 0HxL

Figure 2.2: First nodes of the Bessel function J0(x).

39


In the Hard Wall model it is easy to study correlation functions of the operators

in Table 2.2. We remind that the fundamental relation of the AdS/QCD corre-

spondence associates the supergravity action to the generating functional of the

connected correlator of QCD:

S5[φ(x, ε)] = W4[φ0(x)] φ(x, ε) = φ0(x) at ε→ 0 . (2.24)

In QCD the two-point correlation function, defined as:

Π(x, x′) = i〈0|T [O(x)O(x′)]|0〉 , (2.25)

can be computed by deriving twice the generating functional W4 with respect to the

source φ0 of the operator O:

Π(x, x′) =δ2W4

δφ0(x)δφ0(x′)

∣∣∣∣φ0=0

, (2.26)

so, using (2.24), this is equivalent to derive twice the 5d action with respect to φ0,

once we have substituted the definition of φ0 from (1.31):

Π(x, x′) =δ2S5

δφ0(x)δφ0(x′)

∣∣∣∣φ0=0

. (2.27)

Before computing the correlation function for the vector operator, we write (1.31)

in the 4d Fourier space:

V aµ (q, z) = V (q, z)V a

µ0(q) (2.28)

where V (q, z) is the bulk-to-boundary propagator and V aµ0(q) the source of the vector

field. Substituting (2.28) in Eq. (2.16), we get the equation of motion for V (q, z):

∂z

(1

z∂zV (q, z)

)− q2

zV (q, z) = 0 , (2.29)

with boundary conditions

V (q, 0) = 1 V ′(q, zm) = 0 . (2.30)

The solution reads:

V (q2, z) = A(q2) z√q2 I1(

√q2 z) +B(q2) z

√q2K1(

√q2 z) (2.31)

with B(q2) = 1 and A(q2) = B(q2)K0(√

q2 zm)

I0(√

q2 zm).

40


Let us now go back to the action (2.11) and consider the term containing the

vector meson field:

SV = − 1

4g25 kH

∫d5x

√|g| (∂MV

aN − ∂NV

aM) FMN a

V

= − 1

2g25 kH

∫d5x

[∂M

(√|g| V a

ν FMν aV

)− V a

ν ∂M

(√|g| FMν a

V

)]= − 1

2g25 kH

∫d5x

[∂z

(√|g| gzzgµν V a

ν Fazµ V

)+ ∂α

(√|g| gαβgµν V a

ν Faβµ V

)]= − 1

2g25 kH

∫d5x ∂z

(√|g| gzzgµν V a

ν ∂zVaµ

)=

1

2g25 kH

∫d4x

√|g| gzzgµν V a

ν ∂zVaµ

∣∣∣∣z=0

=R

2g25 z kH

∫d4t

∫d4y d4y′ ηµν V a

µ 0(y′)V a

ν 0(y)V (t− y, z)∂zV (t− y′, z)

∣∣∣∣z=0

(2.32)

where we have used the equation of motion to drop the second term in the second

line, the second term in the third line vanishes at the boundary of the Minkowski

space and in the last line the bulk-to-boundary propagator has been introduced. This

procedure will be followed anytime we will compute correlation functions. Following

Eq. (2.27), we get:

Πa,bµν (x) =

δ2S

δV µ a0 (x) δV ν b

0 (0)

=R

g25 kH z

∫d4t ηµν δ

ab V (t, z)∂zV (t− x, z)

∣∣∣∣z=0

=R

g25 kH z

∫d4t

∫d4q

(2π)4

d4q′

(2π)4ηµν δ

ab eiq·t eiq′·(t−x)V (q, z)∂zV (q′, z)

∣∣∣∣z=0

=R

g25 kH z

∫d4q

(2π)4ηµν δ

ab eiq·x V (q, z)∂zV (q, z)

∣∣∣∣z=0

=

∫d4q

(2π)4eiq·x (qµqν − q2ηµν) ΠAdS(q2) +

∫d4q

(2π)4eiq·x qµqν Π′(q2) ,

(2.33)

so, in the Fourier space,

ΠAdS(q2) = − R

g25 kH q2 z

V (q, z)∂zV (q, z)

∣∣∣∣z=0

= − R

g25 kH

(γE +

K0(√q2 zm)

I0(√q2 zm)

+1

2log(q2ε2)− log 2

). (2.34)

41


Let us compare this result with QCD. The two-point correlation function (2.25) in

the 4d Fourier space is:

ΠABµν QCD(q2) = i

∫d4x eiq·x 〈0|T [OA

µ V (x)OBν V (0)]|0〉 (2.35)

with OAµ V (x) = q(x)γµT

Aq(x). This quantity has been computed in the framework

of QCD sum rules [45], including the leading power corrections to the perturbative

term in the limit q2 →∞:

ΠQCD(q2) = − 1

8π2

(1 +

αs

π

)log

q2

ν2+

3m2

q2+

1

q4〈mqq〉+

1

24q4〈αs

πG2〉+ ... , (2.36)

ν being a renormalisation scale. We identify ε in (2.34) with 1/ν in (2.36) [46]

(holography is also related to the renormalisation group [47], an issue we will not

discuss here). The perturbative term obtained by the AdS/QCD correspondence

has the same form as the one obtained in QCD. Comparing Eq. (2.34) with Eq.

(2.36), we can identifyR

g25 kH

=1

4π2. (2.37)

The two-point correlation function can also be used to compute the decay con-

stants of the vector mesons. We consider the Green’s function corresponding to Eq.

(2.16), which can be written as:

G(q; z, z′) =∑

ρ

ψρ(z)ψρ(z′)

q2 +m2ρ + iε

. (2.38)

Using the relation

V (q, z′) =1

z∂zG(q; z, z′)

∣∣∣∣z=ε

, (2.39)

we can write the two-point function as

Π(q2) =R

g25 kH

∑ρ

[ψ′ρ(ε)/ε]2

(q2 +m2ρ + iε)m2

ρ

. (2.40)

Identifying the meson masses as the poles of the two-point correlation function and

the squared decay constants Fρ, defined by 〈0|Jaµ |ρb〉 = Fρδ

abεµ, as the residues:

Π(q2) =∑

ρ

F 2ρ

(q2 +m2ρ + iε)m2

ρ

, (2.41)

42

2.3. Soft Wall model of AdS/QCD

with the AdS/CFT correspondence we get:

F 2ρ =

R

g25 kH

[ψ′ρ(ε)/ε]2 =

R

g25 kH

[ψ′′ρ(0)]2 . (2.42)

From Eq. (2.21), (2.37) and mρ = 776 MeV, we get F1/2ρ = 329 MeV. It can be

compared with F1/2ρ = 345 ± 8 MeV [48], experimentally obtained from the decay

ρ→ e+e−.

In Table 2.3 there is a summary of the boundary conditions needed to compute

the normalisable solution and the bulk-to-boundary propagator from an equation of

motion and the quantities that can be obtained from them in the Hard Wall model.

Table 2.3: Sketch of the properties of the normalisable function and bulk-to-boundary propagator in the Hard Wall model. In the second column we showthe boundary conditions needed to get each solution. In the third column the

quantities that can be computed are listed. λ+ = −d2

+√

d2

4+m2

5R2 (1.38).

Solution Boundary Outcomeconditions

Normalisableψ(0) = 0

eigenvalues and∫dz

z3−2p |ψ|2 = 1∂zψ|z=zm = 0

eigenfunctions

Bulk-to-boundary ψ(z) −−−→z→0

z−λ+ two-point functionpropagator ∂zψ|z=zm = 0 (masses, decay constants, condensates)

2.3 Soft Wall model of AdS/QCD

While in the Hard Wall model an infrared brane is inserted on the z axis, where the

holographic space abruptly ends, in the Soft Wall model the breaking of conformal

symmetry is achieved through a “dilaton term” in the metric, that provides a smooth

cutoff to space-time. This dilaton term is a background field and it is called so since

it is similar to the one defined in string theory, even if in our context it is not

dynamical. As we shall see, differently from the Hard Wall, the Soft Wall model is

able to predict meson masses arranged in Regge trajectories.

43


The general action has the form:

S =

∫d5x

√|g| e−φ(z) L , (2.43)

where φ(z) is the field called dilaton. The choice of the dilaton is crucial for hadron

spectra since it affects the infrared behaviour of the theory. Regge trajectories are

achieved choosing [49]

φ(z) = c2z2 , (2.44)

where c is a mass scale which breaks the conformal invariance and establishes the

energy scale in the Soft Wall model, so we expect c ∼ O(ΛQCD).

We start investigating the model for chiral symmetry breaking, already stud-

ied in the previous section within the Hard Wall model. We again introduce a

SU(nf )L × SU(nf )R gauge theory and the scalar field X, which is responsible for

chiral symmetry breaking, in the following action:

S = −1

k

∫d5x

√|g| e−φ(z) Tr

|DX|2 +m2

5|X|2 +1

4g25

(F 2L + F 2

R)

, (2.45)

where the constant k renders the action dimensionless. We use the metric:

ds2 =R2

z2

(−dt2 + dx2 + dz2

)(2.46)

and the same definitions as in (2.6). After introducing the vector and axial-vector

fields (2.7), we obtain:

S = −1

k

∫d5x

√|g| e−φ(z) Tr

|DX|2 +m2

5|X|2 +1

2g25

(F 2V + F 2

A)

. (2.47)

Then we repeat the same steps as in the Hard Wall model for computing the

expectation value of the scalar field and the eigenfunctions of the vector field. The

main differences in the procedure will be pointed out.

The equation of motion for the scalar field X0(z) reads:

X ′′0 (z)−

(3

z+ 2 c2z

)X ′

0(z) +3

z2X0(z) = 0 . (2.48)

To find the solution, it is convenient to perform a Bogoliubov transformation, which

consists in defining the function

X0(z) = ec2z2/2z3/2Y (z) (2.49)

44


in order to get a second order differential equation of Schrodinger type:

− Y ′′(z) +

(3

4z2+ 2c2 + c4z2

)Y (z) = 0 . (2.50)

Once found the solution of this equation, we transform it back obtaining:

X0(z) = Az U(1/2, 0, c2z2) +B z L(−1/2,−1, c2z2) , (2.51)

where U is the Tricomi confluent hypergeometric function and L is the generalised

Laguerre function. The coefficients A and B are fixed by the boundary conditions;

the Laguerre function contains an essential singularity as z → +∞, so it must be

discarded putting B = 0. The other coefficient is found requiring that the linear

part is proportional to the quark mass mq, as in the Hard Wall case; the result is:

X0(z) =1

2Rmq Γ(3/2) z U(1/2, 0, c2z2) . (2.52)

Therefore, in the Soft Wall the regularity condition cancels one of the two solutions

leaving only one parameter. This means that we cannot identify a term proportional

to the quark mass and another one proportional to the chiral condensate: here

the quark mass and the chiral condensate are not independent but related. This

makes the description of chiral symmetry breaking less reliable than in the Hard

Wall model, since there is no way of distinguishing a spontaneous from an explicit

symmetry breaking. In particular, at small z

X0(z) −−→z→0

1

2Rmq z −

1

4Rmq c

2 [1− 2γE − 2 log(cz)− ψ(3/2)] z3 (2.53)

with ψ the Euler function. Identifying the coefficient of z3 with the chiral condensate,

we see a proportionality relation between the quark mass and the chiral condensate

that is absent in QCD. To avoid such a drawback, some proposals have been put

forward, for example by adding potential terms to the action [49].

2.3.1 Vector mesons

The equation of motion for the vector field, obtained from the action (2.47), is:

∂z

(1

ze−c2z2

∂zVaµ

)− q2

ze−c2z2

V aµ = 0 . (2.54)

45


With the Bogoliubov transformation Vρ(z) =√z ec2z2/2Y (z) and m2

ρ = −q2, the

equation of motion for one component becomes

− Y ′′(z) +

(3

4z2+ c4z2

)Y (z) = m2

ρ Y (z) , (2.55)

whose solution is:

Y (z) = A1√z

e−c2z2/2 U(−m2

ρ

4c2, 0, c2z2) +B

1√z

e−c2z2/2 L(m2

ρ

4c2,−1, c2z2) . (2.56)

The hadronic states correspond to the normalisable solutions, which are obtained

only for a discrete set of m2ρ. We find that both the functions become normalisable

at

m2ρn

= 4c2(n+ 1) with n = 0, 1, 2, ... (2.57)

The corresponding eigenfunctions are written in terms of the Laguerre polynomials

L1n:

Yρn(z) =

√2n!

(n+ 1)!e−c2z2/2 c2 z3/2L1

n(c2z2) with n = 0, 1, 2, ... ; (2.58)

the first three are plotted in Fig. 2.3.

Figure 2.3: First three eigenfunctions Yρn(z), n = 0, 1, 2, from (2.58).

The proportionality between m2ρn

and n is the evidence of the Regge behaviour

of the ` = 0 spectrum of the ρ meson, in agreement with the experimental data, as

shown in Fig. 2.4 [50].

We can use (2.57) to fix the parameter c; for example, knowing that mρ0 = 0.776

GeV, we get

c ≈ 388 MeV ≈ O(ΛQCD) . (2.59)

46


X

X

X

X

X

X

ΡH770L

ΡH1450LΡH1570L

ΡH1700L

ΡH1900L

ΡH2150L

0 1 2 3 4 5 60

1

2

3

4

5

n

mΡ2

Figure 2.4: Linear fit of the experimental squared masses of the radial excitationsof the ρ meson as a function of the radial quantum number n. The blue pointsrepresent the experimental data while the red points, marked by “x”, the valuesfrom (2.57). The experimental and theoretical points corresponding to ρ(770)coincide, since this value has been used to fit the parameter c.

Let us compute the two-point correlation function of the vector operator. We

can follow the procedure used in the Hard Wall model. First, we find the solution

of Eq. (2.54) for the bulk-to-boundary propagator, defined in (2.28), requiring that

the solution must be regular at infinity (not containing essential singularities) and

that V (q, 0) = 1. The first condition eliminates the Laguerre solution L while the

second one fixes the coefficient of the hypergeometric function:

V (q, z) =q2

4c2Γ(

q2

4c2)U(

q2

4c2, 0, c2z2) . (2.60)

Repeating the same procedure as in the Hard Wall, we find the two-point corre-

lation function:

ΠAdS(q2) = − R

2 k g25

(γE +HN(

q2

4c2) + log(c2/ν2)

), (2.61)

with HN the Harmonic Number function. To compare this result with the one

obtained in QCD, we perform the limit q2 →∞:

ΠAdS −−−→q2→∞

− R

2 k g25

[2γE − log 4 + log(q2/ν2) +

+2 c2

q2− 4 c4

3 q4+

32 c8

15 q8− 1024 c12

63 q12+O

(1

q16

)].

(2.62)

47


We find out that the two-point correlation function of the Soft Wall model contains

not only a perturbative term of the same form as the one found in QCD (2.36), but

also all the non-perturbative corrections, called condensates, appearing in (2.36) and

introduced in the QCD sum rules approach. Comparing this expansion with (2.36),

from the perturbative term we fix kg25:

R

k g25

=1

4π2. (2.63)

Notice that this result does not depend on c, as in the Hard Wall case the perturba-

tive term in (2.34) does not depend on zm. Moreover, the fact that we find the same

result in both models proves that it does not depend on the IR regime but it is an

effect of the asymptotical UV conformal symmetry. Using (2.63) we can have some

predictions about the condensates; for example, for the dimension four condensate

this model predicts:

〈αs

πG2〉 =

4 c4

π2≈ 0.0092 GeV4 (2.64)

which is slightly smaller than the commonly used value 0.012 GeV4, whose uncer-

tainty is estimated about 30% [51].

Finally, from Yρn (2.58), the vector eigenfunctions can be computed:

Vρn(z) =

√2n!

(n+ 1)!c2 z2 L1

n(c2z2) with n = 0, 1, 2, ... , (2.65)

and, using the relation in (2.42), we get the decay constants:

F 2ρ =

R

kg25

[V ′′ρ (0)]2 =

8R c4 (n+ 1)

kg25

. (2.66)

For ρ(770), we find

F 1/2ρ ≈ 260 MeV , (2.67)

which is slightly smaller than the experimental result [48].

In Table 2.4 the properties of the normalisable solution and of the bulk-to-

boundary propagator in the Soft Wall model are summarised, in an analogous way

as in the Hard Wall model in Table 2.3.

48


Table 2.4: Sketch of the properties of the normalisable function and bulk-to-boundary propagator in the Soft Wall model. In the second column we showthe boundary conditions needed to get each solution. In the third column the

quantities that can be computed are listed. λ+ = −d2

+√

d2

4+m2

5R2 (1.38).

Solution Boundary Outcomeconditions

Normalisableψ(0) = 0

eigenvalues and∫dz

z3−2p e−φψ2 = 1regularity at z →∞

eigenfunctions

Bulk-to-boundary ψ(z) −−−→z→0

z−λ+ two-point functionpropagator regularity at z →∞ (masses, decay constants, condensates)

2.3.2 Holographic scalar glueballs

The vector sector is a typical example to understand how to compute spectra and

correlation functions within the Soft Wall model. Now we study the scalar glueballs

since this will be useful for the section dedicated to finite temperature states. Glue-

balls were the first objects to be analysed within the AdS/CFT framework by Witten

in his pioneering paper [28]. In this top-down approach, confinement is reached by

introducing a black hole in the AdS geometry.

Here glueballs will be studied with the phenomenological Soft Wall approach

[52]. Further results about glueballs in the Hard Wall model can be found, i.e.,

in [53]. The lowest dimension operator in 4d having the quantum numbers of a

JPC = 0++ glueball is O = βTr(G2), with conformal dimension ∆ = 4. β is the

Callan-Symanzik function. The corresponding 5d field is a massless scalar X(x, z),

since m25 = 0 from (2.3), as schematically reported in Table 2.5.

Table 2.5: Scalar glueball operator and corresponding field.

O(x) φ(x, z) p ∆ m25 R2

β Tr(G2(x)) X(x, z) 0 4 0

The action for this scalar field is:

S = − 1

2k′

∫d5x

√|g| e−φ(z) gMN ∂MX∂NX , (2.68)

49


from which the equation of motion can be computed:

∂M

(√|g| e−φ(z) gMN ∂NX

)= 0 . (2.69)

We reduce it to a Schrodinger like equation by performing the Bogoliubov transfor-

mation X(q, z) = ec2z2/2z3/2ψ(z), where X is the 4d Fourier transform of the field

X; we get:

− ψ′′ + V (z)ψ = −q2ψ (2.70)

with potential V (z) = 154z2 + 2c2 + c4z2. Defining m2 = −q2, the glueball spectrum

is found looking for the normalisable solutions, which are:

ψn(z) = An e−c2z2/2 z5/21F1(−n, 3, c2z2) (2.71)

where 1F1 is the Kummer confluent hypergeometric function and An a normalisation

factor. These modes correspond to the spectrum

m2n = 4c2(n+ 2) . (2.72)

Glueballs are organised in a Regge trajectory with the same slope as the one describ-

ing vector mesons (2.57), but with a different intercept. In particular, the squared

mass of the lightest glueball is twice the ρ squared mass. There are no experimental

observations of glueballs, so there are no measured data to be compared with (2.72);

however the masses predicted by this model are lighter than expectations by other

QCD approaches [54]. In fact, using c = 388 MeV, fitted in the previous section by

the ρ mass, we find m0 ≈ 1.1 GeV, against m0 ≈ 1.7 GeV estimated in [54].

Let us compute the two-point function of the glueball operator. We come back

to Eq. (2.70), which is the same equation of motion for the bulk-to-boundary prop-

agator X(q, z), and, requiring a regular behaviour at infinity and X(q, 0) = 1, we

find the solution:

X(q, z) = Γ(2 +q2

4c2)U(

q2

4c2,−1, c2z2) . (2.73)

The bulk-to-boundary propagator defines the following two-point function:

ΠAdS(q2) =R3

k′ z3e−c2z2

X(q, z)∂zX(q, z)

∣∣∣∣z→0

=R3

k′

−q

2ν2

2− 1

8q2(q2 + 4c2)(γE − 2 +HN(1 +

q2

4c2) + log(c2/ν2))

,

(2.74)

50


whose asymptotic behaviour at q2 →∞ is given by:

ΠAdS −−−→q2→∞

R3

k′

q4

8

(2− 2γE + log 4− log

q2

ν2

)+

+q2

(−ν

2

2+c2

4(1− 4γE + 2 log 4− 2 log

q2

ν2)

)+

−5 c4

6+

2 c6

3 q2− 4 c8

15 q4+O

(1

q6

). (2.75)

This result can be compared with the 1/q2 expansion in QCD. Writing β(αs) =

β1

(αs

π

)+ β2

(αs

π

)2+ ... and keeping only the first term with β1 = −11

6Nc + 1

3nf (Nc

is the number of colours and nf the number of acting flavours, so nf = 0), the

two-point function reads [55]:

ΠQCD −−−→q2→∞

C0q4

(− log

q2

ν2+ 2− 1

ε′

)+ C4〈O4〉+

C6

q2〈O6〉+

C8

q4〈O8〉+O

(1

q6

)(2.76)

where

C0 = 2

(β1

π

)2 (αs

π

)2

C4 = 4 β21

(αs

π

)(2.77)

C6 = 8 β21

(αs

π

)2

C8 = 8π

(β1

π

)2

α3s ,

while the gluon condensates are the expectation value of gauge invariant operators:

〈O4〉 = 〈αs

πG2〉

〈O6〉 = 〈gsfabcGaµνG

bνρG

cρµ〉 (2.78)

〈O8〉 = 14〈(fabcG

aµαG

bνα

)2〉 − 〈(fabcGaµνG

bαβ

)2〉 .Matching the QCD expression with the one from AdS/QCD, we find two impor-

tant differences. First, in QCD the dimension two condensate is absent, namely the

coefficient proportional to q2 in the expansion, because it is impossible to build a

gauge invariant operator of dimension two. Moreover, in ΠAdS we find a negative

value of the dimension four condensate, while the other determinations in QCD and

even the result found in the vector sector point to a positive value. These difficulties

may be overcome by modifying the boundary conditions, as discussed in [56], but

51


we shall not proceed in such a discussion.

52

Chapter 3

Holographic description of scalar

mesons

The scalar sector of hadron spectroscopy is still an open and debated field. In Fig.

3.1 the tower of the observed JPC = 0++ states is shown, comprising the isoscalars

(f0), and the isodoublet and isotriplet partners (K∗0 and a0, respectively).

f0H600L or Σ

K0*H800L

f0H980L a0H980L

f0H1370LK0*H1430La0H1450L

f0H1500L

f0H1710L

K0*H1950L

f0H2020L

f0H2100L

f0H2200L

f0H2330L

Figure 3.1: Tower of the observed scalar mesons. The data are from [50].

Scalar resonances are difficult to resolve because of their large decay widths.

From the quark model, we expect all these states to be arranged in a nonet, but

there are too many particles (counting all the different charged states) to be fitted

even in two nonets! From this problem, many hypotheses originated concerning the

53

Chapter 3. Holographic description of scalar mesons

structure of these particles: it has been proposed that some or all of these states

may not be ordinary mesons, but glueballs, tetraquarks, or molecules, whose mass

range is below 1.8 GeV.

In the next section, we shall investigate scalar meson spectroscopy by the Soft

Wall model. The lightest scalar meson is predicted to have a mass very close to

f0(980), so we shall conclude that, according to this model, this particle can be

considered as an ordinary scalar meson.

3.1 Scalar mesons in the Soft Wall model

In the holographic framework, scalar mesons have been investigated in [57, 58];

here a detailed analysis, including a discussion on the two-point function and of the

condensates, is presented.

To start the analysis, it is necessary to define the field that in the 5d space

describes these states. The dual field of the QCD operator OAS (x) = q(x)TAq(x),

A = 0, 1, ...8, having conformal dimension ∆ = 3, is a scalar field SA with m25R

2 =

−3 from (2.3), as schematically shown in Table 3.1. With nf = 3, it has singlet

S1(x, z) and octet Sa8 (x, z) components, forming the multiplet

S = SATA = S1T0 + Sa

8Ta , (3.1)

where T 0 = 1/√

2nf = 1/√

6 and T a are the generators of SU(3)f , with normalisa-

tion Tr(TATB

)= δAB/2.

Table 3.1: Scalar meson operator and its dual field.

O(x) φ(x, z) p ∆ m25 R2

q(x)TAq(x) SA(x, z) 0 3 -3

Since this field S(x, z) has the same features as the scalar field X(x, z) defined in

Table 2.2 to describe chiral symmetry breaking, we introduce it in the action (2.47)

S = −1

k

∫d5x

√|g| e−φ(z) Tr

[|DX|2 +m2

5|X|2 +1

2g25

(F 2V + F 2

A)

], (3.2)

54

3.1. Scalar mesons in the Soft Wall model

φ = c2z2, defining

X(x, z) = (X0(z) + S(x, z)) e2iπ(x,z) . (3.3)

We again consider the metric

ds2 =R2

z2

(−dt2 + dx2 + dz2

). (3.4)

Let us start from computing the scalar spectrum. To this aim, we consider the

free action for the scalar field SA(x, z):

SS = − 1

2k

∫d5x√|g| e−φ(z)

[gMN∂MS

A(x, z)∂NSA(x, z) +m2

5 SA(x, z)SA(x, z)

](3.5)

from which the equation of motion is obtained:

∂M

(√|g| gMN e−φ(z)∂NS

)+

3

R2

√|g| e−φ(z)S = 0 . (3.6)

The 4d Fourier transform is defined by S(x, z) =∫

d4q(2π)4

S(q, z), with equation of

motion:

∂z

(1

z3e−φ(z) ∂zS

)+

3

z5e−φ(z)S − q2

z3e−φ(z)S = 0 . (3.7)

The solutions of this equation can be found by performing a Bogoliubov transfor-

mation S = ec2z2/2z3/2Y , thus obtaining a second order linear differential equation

of the Schrodinger type:

− Y ′′(z) + V (z)Y (z) = −q2Y (z) (3.8)

with potential V (z) = 34z2 + 2c2 + c4z2.

We have learnt that the spectrum is determined looking for the normalisable solu-

tions: this condition can be satisfied only by a discrete set of m2 = −q2, which will

constitute the spectrum. We find the following eigenfunctions:

Sn(c z) =

√2

n+ 1c3z3L1

n(c2z2) , (3.9)

and the corresponding masses are:

m2Sn

= c2(4n+ 6) . (3.10)

Also in the case of scalar mesons, the spectrum is organised in a Regge trajectory,

55


whose slope is equal to the one obtained in the vector meson sector (2.57):

m2ρn

= c2(4n+ 4) (3.11)

and in the scalar glueball sector (2.72):

m2Gn

= c2(4n+ 8) . (3.12)

This depends on the fact that the spectrum is linked to the IR behaviour of the

theory, which is determined by the mass scale c of the dilaton field. Using c = 0.388

GeV, fitted from the ρ meson mass, the masses of scalar mesons are completely

fixed. In particular, we find for the first state m0 = 950 MeV, which is very close

to the mass of f0(980) and a0(980), identifying these states as the lightest scalar

mesons. The ratio between the predicted squared mass of the lightest scalar and the

lightest vector meson is independent from any parameter defining the model. In this

respect, we find a good agreement with experimental data, as one can appreciate

from Table 3.2.

Another consequence of having identical Regge trajectories in the various sectors is

that, as the radial number n increases, hadron masses become degenerate.

Table 3.2: Comparison between the predicted and experimental values of theratio (Rn) of the squared masses of scalar and vector mesons; n is the radialquantum number.

Quantity Theory Experiment

R0 =m2

S0m2

ρ0

=m2

f0(a0)

m2ρ0

32

Rf0 = 1.597± 0.033Ra0 = 1.612± 0.004

R1 =m2

S1m2

ρ1

54

Rf0(1505) = 1.06± 0.04 (or Rf0(1370) = 0.9± 0.2)Ra0(1450) = 1.01± 0.04

The mass spectrum can be also determined from the poles of the two-point cor-

relation function, which may also provide numerical predictions for the condensates,

as we have seen in the vector and glueball cases. We remind that the two-point

function is computed in AdS/QCD deriving twice the 5d action with respect to the

source, which is identified with the value that the field has at the boundary z = 0.

In the scalar case, we define the 4d Fourier transform of the source S0(q2) and the

56


4d Fourier transform of the bulk-to-boundary propagator S(q, z), so that

S(q, z) = S(q, z)S0(q2) . (3.13)

Substituting this latter expression in Eq. (3.7), we obtain the equation of motion

for the bulk-to-boundary propagator

S ′′ −(

3

z+ 2c2z

)S ′ +

(3

z2− q2

)S = 0 . (3.14)

The general solution is:

S(q, z) = Az U(q2

4c2+

1

2, 0, c2z2) +B z L(− q2

4c2− 1

2,−1, c2z2) , (3.15)

where A and B are coefficients to be fixed from boundary conditions. The two

conditions to get the bulk-to-boundary propagator are that the solution has to be

regular (with no essential singularities) at z → ∞ and S(q, z) −−→z→0

z/R. The first

condition leads to B = 0 and the second to A = Γ( q2

4c2+ 3

2)/R, so

S(q, z) =1

RΓ(

q2

4c2+

3

2) z U(

q2

4c2+

1

2, 0, c2z2) . (3.16)

Analogously to the vector meson case, we find:

ΠABAdS(q2) = δAB R3

k z3e−φ S(q, z) ∂zS(q, z)

∣∣∣∣z→0

=

= δABR

k

[ν2 + q2 log(c2/ν2) +

1

2q2

(−1 + 4γE + 2ψ(

q2

4c2+

3

2)

)+

+2c2(−1 + 2γE + ψ(

q2

4c2+

3

2) + log(c2/ν2)

)](3.17)

in which ν is the renormalisation scale. The two-point function is plotted in Fig.

3.2: for negative q2 it presents a discrete set of poles, corresponding to the scalar

spectrum (3.10), arising from the Euler function ψ. The residues of ΠAdS are:

F 2n =

R

kResidue[(2c2 + q2)ψ(

q2

4c2+

3

2)] =

R

k16 c4 (n+ 1) . (3.18)

We can compare ΠAdS with the QCD analogue in the limit q2 → ∞. In QCD,

the two-point correlation function is defined by:

ΠABQCD(q2) = i

∫d4x eiq·x〈0|T [OA

S (x)OBS (0)]|0〉 (3.19)

57


-3 -2 -1 0 1 2 3 4

-4

-2

0

2

4

6

8

q2 HGeV2L

PA

dSHG

eV2 L

Figure 3.2: Two-point function of the scalar meson field in (3.17), with ν=1GeV.

and in the limit q2 →∞ becomes [45]

ΠABQCD(q2) −−−→

q2→∞

δAB

2

[3

8π2

(1 +

11αs

3π

)q2 log(

q2

ν2) +

3

q2〈mq qq〉+

1

8q2〈αs

πG2〉

+mqgs

2 q4〈(q σµνλ

aq)Gaµν〉+

παs

q4〈(q σµνλ

aq)2〉

+2παs

3q4〈(q γµλ

aq)∑q=u,d

qγµλaq〉+O(1/q6)

]. (3.20)

The same expansion in the AdS/QCD result gives:

ΠABAdS(q2) −−−→

q2→∞δABR

k

[q2 log

q2

ν2+ q2

(2γE − log 4− 1

2

)+2 c2

(log

q2

ν2− log 4 + 2γE + 1

)+

2 c4

3 q2+

4 c6

3 q4+O(1/q6)

].

(3.21)

Matching the perturbative terms of the two expressions, we get:

R

k=

Nc

16π2, (3.22)

so g25 can be fixed from (2.63) at g2

5 = 3/4. Once known the coefficient k, it is

possible to evaluate the decay constants in (3.18):

F 2n =

Nc

π2c4(n+ 1) ; (3.23)

58


F0 =√

Nc

πc2 = 0.08 GeV2, to be compared with Fa0 = 〈0|O3

S|a0(980)〉 = (0.21±0.05)

GeV2 [59], and for Ff0 = 〈0|ss|f0(980)〉 = (0.18± 0.015) GeV2 [60].

For mq = 0, the four dimensional gluon condensate can also be predicted:

〈αs

πG2〉 = 0.004 GeV4 ; (3.24)

this value is smaller than the commonly used one 〈αs

πG2〉 = 0.012 GeV4. The other

differences with respect to QCD sum rules are that there is a disparity in the sign of

the dimension six operator at the order O(1/q4), if one uses in QCD the factorisation

approximation:

〈(qσµνλaq)2〉 ' −16

3〈qq〉2

(3.25)

〈(qγµλaq)2〉 ' −16

9〈qq〉2 ,

and the presence of a dimension two condensate in the correlation function of

AdS/QCD, which is absent in QCD.

Therefore, the Soft Wall model suggests the existence of such a condensate, which, in

principle, cannot be constructed in QCD. We should then modify this phenomeno-

logical model in order to make it more similar to the theory we want to describe.

For example, it is possible to change the dilaton profile, in particular in the region of

small z, where it is not constrained by the requirement of getting Regge trajectories

in the meson spectra. Another suggestion for solving this problem was proposed

in [56] in the glueball sector, in which, as already underlined in section 2.3.2, this

problem is also present. The authors suggest to keep both the solutions for the

bulk-to-boundary propagator, although one of them is not regular. By tuning the

coefficient of the non-regular solution, it is possible to obtain a vanishing dimension

two condensate and also a four dimensional gluon condensate with the right sign.

However, the issue concerning the dimension two condensate is debated and deserves

further investigations.

It is also possible to provide a spectral representation of the bulk-to-boundary

propagator. In this respect, two useful formulae are [61]:

U(a, b, x) =1

Γ(a)

∫ 1

0

dyya−1

(1− y)bexp

[− y

1− yx

](3.26)

and1

(1− y)2exp

[− y

1− yx

]=

∞∑n=0

L1n(x) yn . (3.27)

59


In particular, the solution (3.16) is proportional to:

U(a, 0, c2z2) =1

Γ(a)

∫ 1

0

dy ya−1 exp

[− y

1− yc2z2

]=

1

aΓ(a)c2z2

∫ 1

0

dy ya 1

(1− y)2exp

[− y

1− yc2z2

]=

1

Γ(a+ 1)c2z2

∞∑n=0

∫ 1

0

dy ya+n L1n(c2z2)

=1

Γ(a+ 1)c2z2

∞∑n=0

1

a+ n+ 1L1

n(c2z2) . (3.28)

So, substituting in (3.16) with a = q2

4c2+ 1

2and q2

n = −(4nc2 + 6c2), we get:

S(q, z) =1

R4c4z3

∞∑n=0

L1n(c2z2)

q2 − q2n + iε

(3.29)

and, finally, using (3.9) and (3.23),

S(q, z) =1

Rc

√8

Nc

π∞∑

n=0

Fn Sn(c2z2)

q2 − q2n + iε

. (3.30)

We can compute the coupling of a scalar meson S with two light pseudoscalar

fields P . For this aim, we need to know the equation of motion for the pseudoscalar

field. They have been obtained in section 2.2.1 for the Hard Wall model, in Eqs.

(2.18) and (2.19); here we write them for the Soft Wall model:

∂z

(1

ze−φ(z) ∂zA

a⊥µ

)− q2

ze−φ(z) Aa

⊥µ −g25 R

2 v(z)2

z3e−φ(z) Aa

⊥µ = 0 (3.31)

∂z

(1

ze−φ(z) ∂zϕ

a

)+g25 R

2 v(z)2 e−φ(z)

z3(πa − ϕa) = 0 (3.32)

q2∂zϕa +

g25 R

2 v(z)2

z2∂zπ

a = 0 , (3.33)

where v(z) = 2X0(z).

This interaction is described by the terms of the action (3.2) involving a product

of the three fields, since the three-point function will be obtained deriving the action

with respect to them. Therefore we have to go beyond the quadratic approxima-

tion. Writing the axial-vector bulk field in terms of the transverse and longitudinal

60


components AM = A⊥M +∂Mϕ, we get this only contribution, coming from Tr|DX|2:

SSPP = −4

k

∫d5x

√|g| e−φ(z)gMN v(z) Tr S (∂Mπ − ∂Mϕ)(∂Nπ − ∂Nϕ) . (3.34)

Separating the contributions of the two components of the scalar field, the singlet

and the octet, we get:

SSPP = −4

k

∫d5x

√|g| e−φ(z)gMN v(z)S1 (∂Mψ

a)(∂Nψb)

1√2nf

Tr[T aT b]

−4

k

∫d5x

√|g| e−φ(z)gMN v(z)Sa

8 (∂Mψb)(∂Nψ

c) Tr[T aT bT c]

= − 2

k√

6

∫d5x

1

z3e−φ(z) v(z)S1 η

MN(∂Mψa)(∂Nψ

b)

−1

kdabc

∫d5x

1

z3e−φ(z) v(z)Sa

8 ηMN(∂Mψ

b)(∂Nψc) , (3.35)

where ψa = ϕa − πa. In the 4d Fourier space, we can write a Fourier transformed

field as the product of the source and the bulk-to-boundary propagator, as done for

scalars in (3.13). For the longitudinal part of the axial-vector field we define:

ϕa(q, z) =1

q2A‖(q

2, z)(−iqµAa

‖ 0 µ(q)), (3.36)

while for ψa

ψa(q, z) =1

q2Ψ(q2, z)

(−iqµAa

‖ 0 µ(q)). (3.37)

From this latter formula we want to extract the contribution of the pseudo-Goldstone

boson, which can be obtained by writing:

ψa(q, z) =1

q2Ψ(0, z)

(−iqµAa

‖ 0 µ(q)), (3.38)

meaning that we are considering only the mode with q2 = 0 (which has been sepa-

rated from the others in Ψ) while the other modes are neglected by requiring q2 = 0

in Ψ. Imposing the chiral limit q2 = 0 in Eq. (3.33), we obtain the condition

∂zπa = 0, and Eq. (3.32) can be written as an equation for Ψ(0, z):

∂z

(1

ze−φ(z) ∂zΨ(0, z)

)− g2

5 R2 v(z)2 e−φ(z)

z3Ψ(0, z) = 0 , (3.39)

which coincides with the equation of motion for the bulk-to-boundary propagator

of the transverse component of the axial field at q2 = 0, which is obtained from Eq.

61


(3.31) with Aa⊥µ(0, z) = A(0, z)Aa

⊥ 0 µ(0). Identifying Ψ(0, z) with A(0, z), so that

ψaP (q, z) =

1

q2A(0, z)

(−iqµAa

‖ 0 µ(q)), (3.40)

the octet contribution to SSPP becomes

i SSPP = − i

kdabc

∫d4q1 d

4q2 d4q3

(2π)12(2π)4 δ4(q1 + q2 + q3)∫ ∞

0

dzR3

z3e−φ(z) v(z)S(q2

1, z) Sa8 0(q1)

[(∂zA(0, z))2 − q2 · q3A(0, z)2

](− i

q22

qµ2 A

b‖ 0 µ(q2)

)(− i

q23

qν3 A

c‖ 0 ν(q3)

). (3.41)

Analogously to the two-point function, the three-point function is obtained by func-

tional derivation of the action SSPP with respect to the sources of the operators

corresponding to the two pions, namely Ab‖ 0 α(p1) and Ac

‖ 0 β(p2), and to the scalar,

Sa8 0(q), with the result:

ΠabcAdSαβ(p1, p2) =

p1α p2β

p21p

22

2R3

kdabc

∫ ∞

0

dz1

z3e−φ(z) v(z)S(q2, z)[

(∂zA(0, z))2 − q2

2A(0, z)2

](3.42)

with q = −(p1 + p2).

In QCD, the three-point correlation function involving two pseudoscalars and one

scalar operator is:

ΠabcQCDαβ(p1, p2) = i2

∫d4x1 d

4x2 eip1·x1 eip2·x2〈0|T [Ob5 α(x1)Oa

S(0)Oc5 β(x2)]|0〉 ;

(3.43)

defining the scalar form factor FP :

〈P d|OaS|P e〉 = F dae

P (q2) , (3.44)

we can write:

ΠabcQCDαβ(p1, p2) = −p1α p2β

p21 p

22

f 2π F

abcP (q2) . (3.45)

Comparing (3.42) with (3.45), and using (3.30), we get the expression of the scalar

62

3.2. Concluding remarks

form factor and of the gSnPP couplings in AdS/QCD:

F abcP (q2) = − 2

k f 2π

dabc

∫ ∞

0

dzR3

z3e−φ(z) v(z)S(q2, z)

[(∂zA(0, z))2 − q2

2A(0, z)2

]= −dabc2

∞∑n=0

Fn gSnPP

q2 +m2n

(3.46)

with

gSnPP = − 2

k f2π

∫ ∞

0

dzR3

z3e−φ(z) v(z)

1

R c

√8

Nc

π Sn(c2z2)[(∂zA(0, z))2 +

m2S n

2A(0, z)2

]. (3.47)

To compute the couplings, we still need to know A(0, c2z2), the bulk-to-boundary

propagator of the transverse axial-vector field at q2 = 0, that is the solution of

Eq. (3.39). The scalar field v(z) = 2X0(z), obtained in (2.52), is proportional to

the quark mass mq, so the term proportional to v2 in (3.39) is small and can be

neglected; requiring as usual that A(0, z) −−→z→0

1 and that the solution must be

regular at z →∞, we find A(0, z) ≈ 1. Finally, for n = 0, S0(z) =√

2c3z3, and the

coupling between the lightest scalar field and two pseudoscalar mesons is given by:

gS0PP =

√Nc

4π

m2S 0

f 2π

R c2∫ ∞

0

dz e−c2z2

v(z) . (3.48)

The value we find is small, of order of 10 MeV depending on the quark mass used

as an input, while other phenomenological determinations indicate larger values,

showing that the scalar states are characterised by their large couplings to light

pseudoscalar mesons. For example, from experiments ga0ηπ = 12±6 GeV while from

a QCD estimate gf0K+K− ' 6−8 GeV [62]. This discrepancy confirms the difficulties

of the Soft Wall model in correctly describing chiral symmetry breaking. In section

2.3 we have discussed about the proportionality between the chiral condensate and

the quark mass found in the scalar field v(z) (2.52): this makes the chiral condensate

and v(z) very small, once fixed the coefficient which multiplies the solution from the

quark mass. Consequently, gSPP becomes small too.

3.2 Concluding remarks

We have analysed the Soft Wall model at zero temperature in three sectors, namely

vector mesons, scalar glueballs and scalar mesons. One relevant result is that the

63


masses of these states are organised in Regge trajectories with the same slope. Once

the slope has been fixed, for example through the mass of the ρ meson, the masses

of all the other states can be computed, including the ground states and the radial

excitations. For instance, the mass of the first scalar meson is predicted to be 950

MeV, confirming the hypothesis that this state can be identified with the particle

f0(980).

The two-point correlation functions have also been computed. They involve

three more parameters, two of them, k and k′, define the dimension of the actions

for the mesons and glueballs, respectively, and the other one, g25, represents the

contributions of the vector and axial-vector fields to the action. Fixing k and g25

form the perturbative term of the scalar and vector two-point functions, one can

then extract the corresponding decay constants and the condensates appearing in

the 1/q2 expansion in the space-like region.

Therefore, we have shown that this model has a large predictive capability, since,

from only three parameters, many quantities can be analitically computed. However,

there are also some unsolved problems, concerning the description of chiral symmetry

breaking and the matching between the condensates predicted here and in other

models. In this respect, we have proposed some possible modifications of the model,

which could help in solving there discrepancies. These topics have not been analysed

yet and are left for future studies.

64

Chapter 4

Holographic approach to finite

temperature QCD

We shall study some models explaining how QCD at finite temperature can be

holographically represented.

The AdS space has no natural temperature associated with it. One can con-

struct thermal states by imposing a periodicity in imaginary time, identifying the

temperature as the inverse of such period. Alternatively, one can also have black

hole solutions to the Einstein equations with a negative cosmological constant which

are asymptotic to AdS space. Imposing regularity of the solution, it is found that

the imaginary time must be periodic, with period depending on the position of the

horizon. Since in the imaginary time formulation of finite temperature quantum

field theory the periodicity of Euclidean time is proportional to the inverse temper-

ature, in this system there is a natural temperature associated with the position of

the horizon of the black hole, the Hawking temperature; it is defined in such a way

that the apparent conical singularity at the horizon can be removed by a change of

coordinates [63].

In 1983 Hawking and Page [64] computed the difference between the Euclidean

action of the black hole metric and that of AdS space characterised by the same

period in imaginary time. Varying the temperature, they found a change of sign of

this quantity, which suggests that a first order phase transition may occur.

The first proposal for applying these concepts to the AdS/CFT correspondence

has been suggested by Witten [65]. He introduced the two models, or two metrics,

described above. In the first one, which we call “Thermal AdS”, finite temperature

effects are introduced by considering the AdS space with compact Euclidean time

and the temperature is defined as the inverse of the period in the time direction.

In the second one, which we call “AdS Black Hole”, a black hole is inserted in the

65

Chapter 4. Holographic approach to finite temperature QCD

AdS space-time, restricting the fifth direction from the AdS boundary z = 0 to the

black hole horizon z = zh. The Hawking temperature is inversely proportional to

the distance zh of the black hole horizon to the AdS boundary: T = 1/(πzh), which

means that higher temperatures correspond to smaller systems.

Witten found that, if the space-time is non-compact, namely Rn−1 × S1, both

spaces are non-confining and no thermal phase transition occurs, since theAdS Black

Hole space is dominant for all non zero temperatures, i.e. the free energy is smaller.

If the space is compact, namely Sn−1 × S1, there is a thermal phase transition

between a confined phase, dual to the Thermal AdS geometry, and a deconfined

one, dual to the AdS Black Hole geometry. Confinement is studied by looking at

the behaviour of Wilson loops. After these considerations, he suggested that the

Hawking-Page phase transition in the bulk can be associated with the first order

deconfinement transition of the large N theory constructed on the boundary; the

Hawking temperature at which it occurs has been identified with the deconfinement

temperature.

In the next section we will see how this idea can be implemented in the bottom-up

approaches we have analysed so far.

4.1 Hawking-Page phase transition in the Soft Wall

and Hard Wall model

The model for the Hawking-Page phase transition has been exploited both in the

Hard Wall and in the Soft Wall in [66], aiming to define the dual theory of QCD at

finite temperature. Let us find out what is the deconfinement temperature predicted

by these models.

We start from the Hard Wall. The gravitational action is:

Sg = − 1

2κ2

∫d5x

√g

(R+

12

R2

), (4.1)

where κ ∼ 1/Nc is the gravitational coupling and R is the scalar curvature. There

are two relevant solutions to the equations of motion, namely Thermal AdS and

AdS black hole. The first is characterised by the line element

ds2 =R2

z2

(dτ 2 + dx2 + dz2

)(4.2)

with 0 < z 6 zm and Euclidean time 0 6 τ < β′; the periodicity of τ is not

constrained, it is identified with the inverse of the temperature. The second solution

66

4.1. Hawking-Page phase transition in the Soft Wall and Hard Wall model

is characterised by a metric with black hole:

ds2 =R2

z2

(f(z) dτ 2 + dx2 +

dz2

f(z)

)f(z) = 1− z4

z4h

(4.3)

with 0 < z < min(zm, zh) and 0 6 τ < πzh; zh is related to the temperature by

zh = 1/(πT ).

For both the solutions, the on-shell action is:

Sg =4

R2κ2

∫d5x

√g , (4.4)

which can be regularised by integrating to an ultraviolet cutoff z = ε. The two

regularised free energy densities are, respectively,

V1(ε) =4R3

κ2

∫ β′

0

dτ

∫ zm

ε

dz1

z5(4.5)

V2(ε) =4R3

κ2

∫ πzh

0

dτ

∫ min(zm,zh)

ε

dz1

z5. (4.6)

Requiring that the two geometries are locally the same at z = ε, where the metric

tends to the Minkowski one, we can fix β′ = πzh

√f(ε):

V1(ε) ∼ R3

κ2π zh

[− 1

2z4h

− 1

z4m

+1

ε4+O(ε)

](4.7)

V2(ε) ∼ R3

κ2π zh

[− 1

z4m

+1

ε4+O(ε)

]zm < zh[

− 1

z4h

+1

ε4+O(ε)

]zm > zh .

(4.8)

Then, the difference between the free energy of the AdS Black Hole and Thermal

AdS space, shown in Fig. 4.1 with respect to T (plain line), is:

∆VHW = limε→0

(V2(ε)− V1(ε)) =R3πzh

κ2

1

2z4h

zm < zh[1

z4m

− 1

2z4h

]zm > zh .

(4.9)

When the black hole horizon is beyond the IR cutoff (zm < zh), i.e. at low temper-

ature, ∆VHW > 0, so Thermal AdS is stable. Looking at the figure, we see that at

high temperatures AdS Black Hole becomes the stable phase, so at a certain point

67


a phase transition occurs. We find that ∆V = 0 for THWc = 21/4/(πzm) ≈ 122 MeV,

using zm = 3.1 GeV−1, as fitted from the ρ mass.

In the Soft Wall model, we repeat the same steps as before, starting with a

slightly different gravitational on shell action, which now contains also a dilaton:

Sg =4

R2κ2

∫d5x

√g e−φ(z) φ(z) = c2z2 . (4.10)

In the Thermal AdS metric the fifth coordinate z extends to infinity, whereas in

the AdS Black Hole one it is always bounded from the black hole horizon z = zh.

Again, the two free energy densities are:

V1(ε) =4R3

κ2

∫ β′

0

dτ

∫ ∞

ε

dz1

z5e−φ(z) (4.11)

V2(ε) =4R3

κ2

∫ πzh

0

dτ

∫ zh

ε

dz1

z5e−φ(z) , (4.12)

and, after expanding at O(ε), the difference between the free energy density in the

AdS Black Hole and Thermal AdS metric is, in the Soft Wall model:

∆VSW = limε→0

(V2(ε)− V1(ε)) =R3π

κ2 z3h

[1

2+ e−c2z2

h

(c2z2

h − 1)

+ c4 z4h Ei(−c2z2

h)

];

(4.13)

it is represented by the dashed line in Fig. 4.1.

0 50 100 150 200

-0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

T HMeVL

DV

Figure 4.1: Difference between the free energy density computed in AdS BlackHole and Thermal AdS. The plain line (resp. dashed line) refers to the resultobtained in the Hard Wall (resp. Soft Wall) model. R3/κ2 = 1 has been used.

Both in the Hard Wall and in the Soft Wall model we find a first order phase

transition, in agreement with expectations of large Nc field theories. The Thermal

AdS regime is associated with a confined phase [66], since the expectation value of

68

4.2. Soft Wall model with AdS Black Hole metric

the Polyakov loop vanishes, while the AdS Black Hole regime is associated with a

deconfined phase, since the presence of a horizon makes the expectation value of the

Polyakov loop no more vanishing [65]. The deconfinement temperature predicted

by the Soft Wall model is very close to the one found in other determinations; for

example, with lattice calculations it is predicted T ∼ 170 MeV for nf = 2, even if

the transition in such a theory with fermions is a crossover and not of first order

[67]. It is also found that in a pure gauge theory the deconfinement temperature

slightly depends on Nc, in particular it seems to decrease with Nc [68]. Instead, the

type of the transition is more sensitive to Nc: it is known to be of second order for

Nc = 2 [69] and first order in the limit Nc → ∞ [68]; at Nc = 3 the transition is

of first order [70], but some authors prefer to speak of a “nearly” second order or

“weakly” first order (see, e.g., [68, 71] and references therein).

Another feature of the model with the Hawking-Page phase transition is that the

spatial Wilson loop and other quantities, like glueball and meson spectra, do not

change with temperature in the confined phase, since Thermal AdS metric is in this

respect equivalent to AdS metric. This is in agreement with large Nc behaviour,

but may disagree with QCD [67].

Nevertheless, in the holographic approach one can also describe the finite tem-

perature gauge theory using only one of the two geometries described so far, without

considering the Hawking-Page phase transition. This method is consistent with the

model we have used at T = 0, in which the metric is not a dynamical object.

In the next sections, we shall study how scalar glueballs and mesons could behave

in a hot medium with two different models: in the first one, the finite temperature

regime is reached through a black hole geometry in the bulk, while in the second

one, it is described by a Thermal AdS or a black hole geometry, depending on which

one is the stablest [72].

4.2 Soft Wall model with AdS Black Hole metric

We first consider a model in which the finite temperature field theory in the boundary

is described by a bulk theory in the AdS space with a black hole. The metric is

given by:

ds2 =R2

z2

(f(z) dt2 − dx2 − dz2

f(z)

)0 < z < zh (4.14)

where the black hole is introduced through the function f(z) = 1 − z4

z4h

and zh

is the position of the horizon. As already stated, the temperature is defined by

69


T = 1/(π zh). In the next two sections, scalar glueballs and mesons, respectively,

will be studied in this framework.

4.2.1 Scalar glueballs

We have investigated scalar glueballs in a holographic approach at T = 0 in section

2.3.2. We introduce the scalar field X(x, z) described in Table 2.5 in the action:

S =1

2k′

∫d5x

√g e−φ(z) gMN ∂MX∂NX , (4.15)

with, as usual, φ(z) = c2z2 and metric (4.14). Notice that the sign in front of the

action has changed with respect to section 2.3.2 because we are using a different

metric, with reversed signs. The equation of motion for this field is:

e−φ(z)√g gµν ∂µ∂νX(x, z) + ∂z

(e−φ(z)√g gzz ∂zX(x, z)

)= 0 . (4.16)

The 4d Fourier transform of the field, X(q, z), is related to the bulk-to-boundary

propagator in the momentum space, X(q, z), by: X(q, z) = X(q, z) X0(q), with

X(q, 0) = 1, where X0(q), according to the AdS/CFT dictionary, acts as the source

of the four dimensional operator describing the glueball in the generating functional

of the boundary theory. Using this definition in Eq. (4.16), the equation of motion

for the bulk-to-boundary propagator can be determined:

X ′′(q, z)− 4− f(z) + 2 c2 z2 f(z)

z f(z)X ′(q, z) +

(q20

f(z)2− q2

f(z)

)X(q, z) = 0 (4.17)

where q = (q0, q). We first consider the case q = 0; calling ω = q0 and defining the

dimensionless variable u = z/zh, we can rewrite Eq. (4.17):

X ′′(ω, u)− 3 + u4 + 2 c2 z2h u

2(1− u4)

u (1− u4)X ′(ω, u) +

ω2 z2h

(1− u4)2X(ω, u) = 0 . (4.18)

At zero temperature the boundary conditions for the bulk-to-boundary propagator

requires X(q, 0) = 1 and regularity at z → ∞: the first must be satisfied also at

finite temperature, while the second one has to be substituted with a condition at

the horizon u = 1. Near the horizon, the general solution of the equation of motion

is a superposition of a in falling wave (X−) plus corrections and an out going wave

70


(X+) plus corrections:

X(ω, u)u→1∼ αX−(ω, u) + β X+(ω, u)

= α (1− u)−i√

ω2zh/4 [1 +O(1− u)] + β (1− u)i√

ω2zh/4 [1 +O(1− u)] .

(4.19)

To compute the retarded Green’s function, the bulk-to-boundary propagator should

behave near the horizon as the in falling solution, so the other one must be discarded

[73].

According to the AdS/CFT dictionary and analogously to the zero temperature

techniques, we compute the two-point Green’s function by two functional derivations

of the action with respect to the source X0. Following the method used so far, we

get the formula:

Π(ω2) =1

2k′R3 f(u)

u3z4h

e−φ(u)X(ω, u)∂uX(ω, u)

∣∣∣∣u=0

. (4.20)

Again, what is needed is the solution near the AdS boundary, which can be found

by expanding Eq. (4.18) near u ≈ 0:

X(ω, u) −−→u→0

A(ω2)

(1 +

ω2z2h

4u2 + ...

)+B(ω2)

(c4z4

h

2u4 + ...

). (4.21)

The condition X(ω, 0) = 1 fixes A(ω2) = 1, while the coefficient B(ω2) must be

fixed from the boundary condition near the black hole horizon, i.e. by selecting the

in falling solution [74]:

X(ω, u) −−→u→1

X−(ω, u) ∼ (1− u)−i√

ω2zh4 . (4.22)

The spectral function ρ(ω2) is defined as the imaginary part of the retarded

Green’s function. Substituting (4.21) in (4.20), we find that the only term that can

have a non-vanishing imaginary part is the one proportional to the coefficient B(ω2),

so the spectral function is:

ρ(ω2) =R3 c4

k=(B(ω2)) . (4.23)

The numerical procedure we have followed to determine B, and so ρ(ω2), is described

here:

1. We have solved Eq. (4.18), requiring that at small z (z = ε → 0) it behaves

71


like the first linearly independent function in (4.21):

X(ω2, ε) = 1 +ω2z2

h

4ε2 X ′(ω2, ε) =

ω2z2h

2ε . (4.24)

Let us call this solution X1(ω2, z).

2. We have again solved Eq. (4.18), requiring that at small z it behaves like the

second linearly independent function in (4.21):

X(ω2, ε) =c4z4

h

2ε4 X ′(ω2, ε) = 2c4z4

hε3 . (4.25)

Let us call this solution X2(ω2, z).

Then, the solution we are looking for is

X(ω2, u) = X1(ω2, u) +B(ω2)X2(ω

2, u) (4.26)

since this has the proper behaviour near u ∼ 0, as in (4.21) with A = 1.

Let us now fix B by requiring that at u→ 1 the solution is the in falling one.

3. Near u ∼ 1, we know that the solution is a superposition of the in falling (X−)

and out going (X+) function, so, in general, one can write

X1u→1∼ αX− + β X+ (4.27)

X2u→1∼ δ X− + γ X+ (4.28)

with α, β, δ, γ complex coefficients. Moreover, since X1 and X2 are real while

X− and X+ are complex, with X− = X+, then α = β and δ = γ and

X1 = (αR + i αI)X− + (αR − i αI)X+ (4.29)

X2 = (δR + i δI)X− + (δR − i δI)X+ . (4.30)

Manipulating the two expressions, we find:

(δR−iδI)X1−(αR−iαI)X2 = [(δR − iδI)(αR + iαI)− (αR − iαI)(δR + iδI)]X−

(4.31)

and B = − (αR−iαI)(δR−iδI)

. The coefficients are determined by fitting, around u ∼ 1,

X1 = 2< [(αR + i αI)X−] , (4.32)

X2 = 2< [(δR + i δI)X−] . (4.33)

72


The spectral function for the first two peaks, modulo a multiplicative factor, is

shown in Fig. 4.2 at several temperatures, using c = 0.388 GeV, fitted from vector

meson spectrum in (2.57). At low temperatures, the peaks are narrow, becoming

broader as the temperature increases. The position of the peaks is identified with

the mass of scalar glueballs, in particular the two peaks in the figure correspond

to the lowest lying state and the first excitation. The excited state dissolves at a

lower temperature than the ground state: this behaviour reflects the one found, i.e.,

for the charmonium states in a hot medium, since J/ψ melts at higher temperature

with respect to ψ′ [75]. Moreover, increasing the temperature, the position of the

peaks moves towards smaller values.

0.8 1.0 1.2 1.4 1.6 1.8 2.00

5

10

15

20

25

30

Ω2 HGeV2

L

ImB

T=44 MeVT=29 MeVT=25 MeVT=21 MeV

Figure 4.2: Spectral function, modulo a multiplicative factor, of the glueballsat several temperatures, in the model with black hole. Only the first two peakshave been shown.

The mass and the width can be found by fitting each peak with a Breit-Wigner

function [76]:

ρ(ω2) =amΓωb

(ω2 −m2)2 +m2Γ2(4.34)

with parameters a and b. The behaviour of the mass and the width with temperature

is shown in Fig. 4.3.

At temperatures below T ∼ 20 MeV (T ∼ 17 MeV for the excited state) the

mass and the width slightly change with respect to the values at T = 0. At zero

temperature, the glueball spectrum has been computed in section 2.3.2, finding

m2G n = 4c2(n + 2), and the spectral function is characterised by zero widths. In

this range of temperatures, the horizon of the black hole is far enough and the

eigenfunctions vanish before reaching it. This means that we can determine glueball

73


0 10 20 30 401.0

1.2

1.4

1.6

1.8

T HMeVL

m2HG

eV2 L

0 10 20 30 400.00

0.05

0.10

0.15

0.20

0.25

T HMeVL

GHG

eVL

Figure 4.3: Behaviour of the mass (left panel) and the width (right panel) ofthe scalar glueball with temperature. The plain blue (resp. dashed red) linecorresponds to the ground (resp. first excited) state.

masses looking for the eigenvalues of the equation

−H ′′(m2, u) +

(15

4u2+ 2 c2 z2

h + c4 z4h u

2

)H(m2, u) = m2 z2

hH(m2, u) (4.35)

which has been obtained performing a Bogoliubov transformation, as in section 2.3.2.

To test the method, we have computed glueball masses in the range T = 20 − 22

MeV both solving the eigenvalue problem and finding the position of the peaks

of the spectral function, obtaining the same results. For higher temperatures, at

T = 25− 30 MeV, the squared mass is reduced to about 80% of its value at T = 0.

The second peak disappears at T ∼ 29 MeV, while the first at T ∼ 44 MeV. In

this range of temperatures, the increasing of the width reflects the broadening of

the peaks in the spectral function.

This effect can be also observed from Fig. 4.4 (left panel), in which a fitted poly-

nomial has been subtracted from the first peak of the spectral function at different

temperatures. This behaviour of the scalar glueball mass and width is qualitatively

analogous to the behaviour observed in lattice studies [77, 78], but the temperature

scale is very different, as one can appreciate looking at Figs. 4.4. In [77], the authors

perform an analysis of the spectral function, finding that the peaks become broader

and lower while increasing the temperature; fitting the peaks, they have found the

masses and widths shown in Fig. 4.5. Comparing Fig. 4.5 with Fig. 4.3 and the

two figures in Fig. 4.4, it is evident that, even if the qualitative behaviour of these

quantities with respect to temperature is similar to the one we have found in the

holographic approach, the scale of temperature is much higher. In Fig. 4.5 it is

shown that the thermal width grows gradually with temperature, and there is an

abrupt rise near the critical temperature Tc. This is analogous to Fig. 4.3, apart

from the scale of temperature. Concerning the mass, it modestly decreases in Fig.

74


0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.50

10

20

30

40

50

Ω2 HGeV2

L

BWHΩ

2 L T=45 MeV

T=34 MeV

T=29 MeV

T=28 MeV

0

20

40

60

80

100

120

140

160

0 1000 2000 3000 4000 5000 6000

ρ(ω

)/G(0

) [G

eV−1

]

ω [MeV]

T=390MeVT=312MeVT=253MeVT=130MeV

Figure 4.4: Left panel: Breit-Wigner behaviour extracted from the first peak ofthe spectral function at different temperatures subtracting a fitted polynomial.Right panel: spectral function in [77].

4.5 by about 100 MeV, a result similar to our evaluation (notice that in Fig. 4.3 the

squared mass is plotted). An analysis with similar results has been carried out in a

recent paper [78].

1000

1500

100 150 200 250 300 350 400

!0 [

MeV

]

T [MeV]

0

500

1000

1500

100 150 200 250 300 350 400

! [M

eV]

T [MeV]

Figure 4.5: Lowest scalar glueball mass (left panel) and width (right panel)plotted against the temperature, obtained with lattice techniques in [77]. Thedotted line represents the critical temperature Tc = 280 MeV.

For non-vanishing values of the three-momentum, q 6= 0, the results are similar.

In Fig. 4.6 the spectral function at T = 30 MeV is shown for discrete values of q2 in

the range q2 = 0−0.8 GeV2: we can see that, increasing q2 the peaks of the spectral

function are shifted towards higher values of q20 and become broader. Moreover, the

difference q20 − q2 is not constant, reflecting the violation of Lorentz invariance in

finite temperature theory.

Before drawing our comments about this model, we can examine it again in

another sector, investigating how scalar mesons behave in a hot medium.

75


1.0

1.5

2.0

q02 HGeV2

L

0.0

0.2

0.4

0.6

0.8

q2HGeV2

L0

5

10

15

20

ImHBL

Figure 4.6: Spectral function, modulo a multiplicative factor, at T = 30 MeVfor different values of the three-momentum squared q2 in the range 0 6 q2 6 0.8GeV2.

4.2.2 Scalar mesons

The action for scalar mesons at zero temperature (3.2) has been introduced in section

3.1. In the model with metric (4.14), it becomes:

S =1

k

∫d5x

√g e−φ(z) Tr

[|DY |2 −m2

5|Y |2 −1

4g25

(F 2L + F 2

R)

], (4.36)

where the scalar bulk field Y = (Y0 + S) e2iπ, with m25R

2 = −3, comprises a back-

ground field Y0(z), the scalar field S(x, z) = SATA and the chiral field π(x, z). The

quadratic part in the field SA is:

S(2)S =

1

2 k

∫d5x e−φ(z)√g

[gMN∂MS

A(x, z) ∂NSA(x, z)−m2

5SA(x, z)SA(x, z)

],

(4.37)

so the equation of motion for the bulk-to-boundary propagator in the 4d Fourier

space S(q, z) is

S ′′(q, z)−2c2z2f(z) + 3 + z4

z4h

zf(z)S ′(q, z) +

3

z2f(z)S(q, z) +

(q20

f(z)2− q2

f(z)

)S(q, z) = 0

(4.38)

76


with q = (q0, q). For ~q = 0 and introducing the variable u = z/zh, the equation

becomes:

S ′′(q20, u)−

2c2z2hu

2(1− u4) + 3 + u4

u(1− u4)S ′(q2

0, u)+3

u2(1− u4)S(q2

0, u)+q20z

2h

(1− u4)2S(q2

0, u) = 0 .

(4.39)

As at T = 0, we require: S(q, 0) = u. Near the horizon u → 1 the general solution

behaves in the same way as in the glueball case:

S(q, z) −−→u→1

S− + S+ with S∓(ω2, u) = (1− u)∓i√

ω2z2h/4 . (4.40)

For u→ 0 the approximated solution is:

S(ω2, u) −−→u→0

uU(2c2 − ω2

4c2, 0, c2z2

hu2) +B(ω2)uL(−2c2 − ω2

4c2,−1, c2z2

hu2) (4.41)

where the coefficient B(ω2) defining the spectral function, is determined selecting

at u ∼ 1 the in falling solution S−, following the same procedure as in the glueball

case. Thus, the spectral function for scalar mesons at finite temperature, that is the

imaginary part of B(ω2), is shown in Fig. 4.7. It is very similar to the one we have

0.6 0.8 1.0 1.2 1.4 1.6 1.80

2

4

6

8

10

Ω2 HGeV2

L

ImB

T=75 MeVT=55 MeVT=32 MeVT=25 MeV

Figure 4.7: Spectral function, modulo a multiplicative factor, of scalar mesons atseveral values of the temperature. The first two peaks are shown, correspondingto the ground state and the first radial excitation.

found for glueballs, it has peaks becoming broader when the temperature increases.

The behaviour of the mass and the thermal width with respect to temperature, ob-

tained by a fit of each peak with the Breit-Wigner function (4.34), is shown in Fig.

4.8. At low temperatures, the position of the peaks coincides with the scalar squared

masses at T = 0 we have found in section 3.1, namely m2Sn

= c2(4n + 6). Then,

increasing the temperature, the scalar mass decreases, in agreement with what was

77


happening for glueballs, but, at odds with the previous case, at a certain temper-

ature, it starts growing again. The first excitation disappears from the spectrum

at T ∼ 35 MeV, while the ground state survives till T ∼ 70 MeV: this behaviour

seems universal in all sectors considered so far. These values are higher than the

dissociation temperatures of glueballs, but they are still too low in comparison with

lattice expectations.

0 10 20 30 40 50 60 700.6

0.8

1.0

1.2

1.4

T HMeVL

m2HG

eV2 L

0 20 40 600.0

0.1

0.2

0.3

0.4

0.5

T HMeVL

GHG

eVL

Figure 4.8: Mass (left panel) and width (right panel) of the scalar mesons atvarying the temperature. The plain blue (resp. dashed red) line corresponds tothe ground (resp. first excited) state.

Some concluding comments about this model, in which the black hole geometry

is used to describe a finite temperature field theory, are in order. This model is

able to reproduce some peculiar features of the masses and the widths of hadrons at

finite temperature. However, this qualitative agreement with the results found with

other models is not supported by a quantitative agreement, since the phenomena

reproduced seem to occur at much lower temperatures. The scale is determined by

the dimensionful constant c in the background dilaton field, which has been fixed

from the ρ meson mass at T = 0. There are no other scales in terms of which the

physical temperature can be expressed.

4.3 Model with the Hawking-Page phase transi-

tion

We now investigate scalar glueballs and mesons in the Soft Wall model with the

other approach, in which a Hawking-Page phase transition occurs between a phase

described by Thermal AdS metric and a phase described by AdS metric with black

hole. The first is stable up to Tc ∼ 191 MeV, so at low temperature we use the

78

4.3. Model with the Hawking-Page phase transition

Thermal AdS geometry to compute glueball and scalar meson masses, then the

AdS Black Hole geometry is adopted.

In the case of Thermal AdS, the equations of motion are the same as at T = 0,

which means that the glueball and scalar meson masses do not change in this region,

as shown in Fig. 4.9 for the first two states. On the other hand, for T > Tc, the AdS

Black Hole metric should be used, but the analyses in the previous section reveal

that at this high temperature the spectral function is flat, no peak appears both

in the glueball and in the meson sectors: dissociation has already occurred. So, at

T = Tc, when the black hole appears in the metric, the masses jump from m2 6= 0 to

m2 = 0 (Fig. 4.9) and dissociation occurs together with deconfinement, as it could

be expected in a first order phase transition.

THP

0 50 100 150 200 250 3000.0

0.5

1.0

1.5

2.0

T HMeVL

m2HG

eV2 L

THP

0 50 100 150 200 250 3000.0

0.5

1.0

1.5

2.0

T HMeVL

m2HG

eV2 L

Figure 4.9: Masses of scalar glueballs (left panel) and mesons (right panel) atvarying the temperature. Thermal AdS metric has been used for T < THP ,while AdS Black Hole one otherwise. The blue (resp. red) line corresponds tothe ground (resp. first excited) state.

This is what happens in large Nc theories, in which, in the confined phase, the

mass of the hadrons remains constant until a first order phase transition, from which

it begins changing with temperature.

This can be considered a successful application of the conjecture to large Nc

theory. Moreover the predicted deconfinement temperature is quantitatively very

close to the one expected in QCD, and this result may candidate this model to

describe also QCD at finite temperature. However the behaviour of the masses and

widths of hadrons, although in agreement with large Nc theories, seems not to reflect

expectations of other approaches to QCD.

79



We have studied two different approaches to describe QCD at finite temperature.

If we consider the model with the black hole metric and without the Hawking-

Page phase transition, we get a nice description of how hadrons may behave at finite

temperature; it is also qualitatively in agreement with other QCD models, even if

some differences are found in the scale of temperature at which such phenomena

occur.

If we consider the model with the Hawking-Page phase transition, we get a nice

description of the deconfinement transition in terms of the inversion of the hierarchy

between two different geometries of the bulk. Moreover, the critical temperature is

close to the deconfinement temperature of QCD predicted by other models. This

model can also reproduce many features of the large Nc theory in the confined and

deconfined phases, although, as we have already seen, other models of QCD, like

lattice simulations, predict less trivial results.

Further efforts are needed to construct a model having all these properties, in

particular, the behaviour of masses at varying the temperature and the presence of

a deconfinement phase transition.

80

Chapter 5

Hadron spectroscopy by an

AdS/QCD QQ static potential

So far we have developed a holographic approach to study hadron masses and decay

constants, in which hadrons are described as normalisable solutions of equations of

motion obtained from a five dimensional action.

Here we shall investigate the hadrons from another point of view, as bound states

of two constituents, as, e.g., the hydrogen atom, following the prescriptions of the

quark model. We regard them as “holographic hadrons” because the interaction

between the constituents will be described through a potential computed using the

prescriptions of the AdS/QCD correspondence.

The constituent quark model is a quantum-mechanical model in which the mass

and wave function of the hadron are determined solving a (relativistic or non-

relativistic) wave equation, which describes the instantaneous interaction among

the constituent quarks through a potential. For example, a meson is studied as a

two-body problem, i.e. considering the mutual interaction between the constituent

quark and antiquark. It is a challenge to determine the exact potential, being it a

non-perturbative quantity.

The structure of hadrons in terms of the constituent quarks is analysed in the

quark model, which classifies hadrons according to their spin and parity in JP

multiplets (see section 2.1).

In this part, we shall introduce a relativistic model for studying the spectrum of

mesons, baryons and tetraquarks; the model uses the quark-antiquark potential

computed by Andreev and Zakharov [79] following the AdS/QCD recipe. The next

section is dedicated to the description of the relativistic wave equation and a method

for solving it. Then, the computation of the AdS/QCD potential is presented. These

ingredients give rise to the potential model by which the spectra will be computed.

81

Chapter 5. Hadron spectroscopy by an AdS/QCD QQ static potential

5.1 The Salpeter equation and the Multhopp method

The eigenvalue equation that describes a two-body problem with relativistic kine-

matics is the Salpeter equation:(√m2

1 −∇2 +√m2

2 −∇2 + V (r)

)ψ(r) = M ψ(r) , (5.1)

where m1 and m2 are the masses of the constituents, V is the potential representing

their interaction, M and ψ are the mass and the wave function, respectively, of

hadron, i.e. the bound state that they form, in its rest frame. For example, for

mesons, m1 and m2 are the masses of the constituent quark and antiquark. This

equation arises in QCD from the Bethe-Salpeter equation, replacing the interaction

with an instantaneous local potential and considering a limited Fock space containing

qq pairs only.

For central potentials, the eigenfunctions can be factorised, separating the radial

part from the angular one:

ψ(r) = Y`m(r)φ`(r) , (5.2)

where Y`m are the spherical harmonics. Computing the Fourier transform of the

square root operator and switching to spherical coordinates, the Salpeter equation

(5.1) for the radial part φ`(r) can be written as:

[M − V (r)]φ`(r) =

2

π

∫ +∞

0

dr′ r′2∫ +∞

0

dk k2

(√k2 +m2

1 +√k2 +m2

2

)J`(kr) J`(kr

′)φ`(r′) .

(5.3)

where J`(x) are the spherical Bessel functions. For ` = 0, J0(x) = sin x/x and this

equation reduces to

[M − V (r)]u0(r) =

2

π

∫ ∞

0

dr′∫ ∞

0

dk

(√k2 +m2

1 +√k2 +m2

2

)sin (kr) sin (kr′)u0(r

′) , (5.4)

where u0(r) = r φ0(r). This is an integral equation and can be solved numerically;

here we will adopt the Multhopp method [80, 81], which transforms the integral equa-

tion into a set of linear equations introducing N parameters θk, called Multhopp’s

angles.

82

5.1. The Salpeter equation and the Multhopp method

Let us see in some detail how this method works. We first write the right hand

side of Eq. (5.4) in a different way. Defining

Hj =2

π

∫ ∞

0

dr′∫ ∞

0

dk√k2 +m2

j sin (kr) sin (kr′)u0(r′) , j = 1, 2 (5.5)

and substituting k = mj sinh x, we can write:

Hj =m2

j

2π

∫ ∞

0

dr′ u0(r′)Fj(r

′) (5.6)

with

Fj(r′) = 4

∫ +∞

0

dx coshx√

sinh2 x+ 1 sin(mj r sinh x) sin(mj r′ sinh x)

=

∫ +∞

0

dx (1 + cosh(2x)) [cos(mj |r − r′| sinh x)− cos(mj |r + r′| sinh x)] .

(5.7)

Introducing the modified Bessel function:

Kν =1

cos(νπ/2)

∫ +∞

0

dt cosh(νt) cos(x sinh t) , (5.8)

which satisfies the recurrence relation

z Kν−1(z)− z Kν+1(z) = −2ν Kν(z) , (5.9)

we can write (5.7) as:

Fj(r′) =

2K1(mj|r + r′|)mj|r + r′|

− 2K1(mj|r − r′|)mj|r − r′|

. (5.10)

Therefore

Hj =m2

j

π

∫ +∞

0

dr′ u0(r′)

[K1(mj|r + r′|)mj|r + r′|

− K1(mj|r − r′|)mj|r − r′|

]= − 1

πP.V.

∫ +∞

−∞dr′

u0(r′)

|r − r′|2+

+mj

π

∫ +∞

−∞dr′

u0(r′)

|r − r′|

[1

mj|r − r′|−K1(mj|r − r′|)

], (5.11)

where, at the second line we have defined u0(−r) = −u0(r). Then, we can go back

83


to the Salpeter equation, which, after these definitions, becomes:

[M − V (r)]u0(r) =

= − 2

πP.V.

∫ +∞

−∞dr′

u0(r′)

|r − r′|2+

+

m1

π

∫ +∞

−∞dr′

u0(r′)

|r − r′|

[1

m1|r − r′|−K1(m1|r − r′|)

]+m1 → m2

.

(5.12)

Once the equation has been written in this form, we can apply the Multhopp method.

First, we change the variable, defining r = − cot θ, and write the Salpeter equation

in the form:

Mψ(θ) =

∫ π

0

dθ′K(θ, θ′)ψ(θ′) , (5.13)

where ψ satisfies the boundary conditions ψ(0) = ψ(π) = 0 and can be written as a

Fourier series, that we truncate at j = N :

ψ(θ) ∼N∑

j=1

cj sin(jθ) (5.14)

with coefficients

cj =2

N + 1

N∑l=1

sin(jθl)ψ(θl) , θl =l

N + 1π l = 1, ..., N ; (5.15)

θl are called Multhopp’s angles. Substituting the truncated Fourier series in (5.13),

we get:

Mψ(θl) =N∑

m=1

2

N + 1

N∑j=1

[∫ π

0

dθ′K(θl, θ′) sin(jθ′)

]sin(jθm)ψ(θm) , (5.16)

so the original integral equation has been modified obtaining a system of linear

equations of the form:

N∑m=1

Blmψ(θm) = Mψ(θl) l = 1, ..., N (5.17)

whose solution is characterised by a mass M and a wavefunction ψ computed in N

points.

84

5.2. Wilson loop and VQQ

5.2 Wilson loop and VQQ

One of the most significant results obtained through the AdS/QCD correspondence

is an expression for the heavy quark-antiquark potential, the static energy of in-

teraction between a quark and an antiquark. This non-perturbative quantity has

been computed numerically, e.g., with lattice simulations [82]. It has also been

parametrized in different ways, namely as the Cornell potential [83]:

V (r) = −kr

+r

a2+ C , (5.18)

where the parameters k, a and C can be fixed fitting the meson spectrum. Eq. (5.18)

is an interpolating form between the two known behaviours: in the perturbative

regime, r 1, the potential behaves like the Coulomb potential, whereas in the

limit of large distance between the quark and the antiquark, it is expected to grow

linearly, providing confinement.

The potential can be evaluated using a Wilson loop, which is a non-perturbative,

gauge invariant quantity defined as

WC[A] = Pe−igHC dzµAµ(z) (5.19)

where Aµ is the gauge field and P indicates the path ordered exponential. The

potential of interaction between two static quarks is proportional to the expectation

value of the Wilson loop computed on a rectangular contour C in the Euclidean

space in the limit T →∞, where T is the length of the rectangular path in the time

direction:

limT→∞

〈WC〉 = e−T V (r) . (5.20)

The rectangular C describes a static quark-antiquark pair propagating in the Eu-

clidean time, as shown in Fig. 5.1. In particular, one is at position x = −r/2 and

the other one at x = r/2; since they are static, an approximation that can be well

verified by heavy quarks, they move along straight lines, tracing the rectangular

contour C.

In [84], Maldacena proposed to adopt as the dual of the Wilson loop in the

gravity theory the exponential:

〈WC〉 ∼ e−SNG (5.21)

where SNG is the Nambu-Goto action, i.e. the area of a string worldsheet, bounded

by the curve C at the boundary of the AdS space. This concept will be clear in a

85


T

r2

-r2

Τ

x

Figure 5.1: Rectangular path along which the Wilson loop is computed.

moment. Imagine that C is drawn in a Minkowski space, where x and τ are one

spatial and the temporal coordinate, respectively. The Minkowski space is at the

boundary z = 0 of the AdS space and the worldsheet is the surface spanned by the

string with endpoints attached to the quark and the antiquark, moving in time from

τ = 0 to τ = T , as shown in Fig. 5.2, where different surfaces correspond to different

values of the quark-antiquark distance.

-r/2

r/2

T

0

z0

martedì 13 ottobre 2009

Figure 5.2: Worldsheet spanned by a string joining a quark and an antiquark,moving in time. Different surfaces correspond to different distances between theparticles. The blue lines correspond to the rectangular contour in Fig. 5.1. Ithas been obtained solving Eq. (5.30) with c = 1.

Using the AdS/QCD correspondence we identify the left hand sides of Eqs. (5.20)

and (5.21), such that the interquark potential can be obtained as:

V (r) = limT→∞

1

TSNG . (5.22)

86


The Nambu-Goto action is:

SNG =1

2πα′

∫d2ξ√

det [gMN ∂αXM ∂βXN ] , (5.23)

where ξα, α, β = 1, 2, are the worldsheet coordinates. The Euclidean metric g is

given by

ds2 =R2

z2ec z2/2

(dτ 2 + dx2 + dz2

), (5.24)

where τ is the Euclidean time and R is the radius of the AdS space. This is another

way of implementing the Soft Wall model, called Soft Wall metric model, in which

the dilaton term is inserted in the metric and not directly in the action [79]. Choosing

ξ1 = x and ξ2 = τ , the matrix in (5.23) can be written as

gMN ∂αXM∂βX

N =R2 ec z2/2

z2

(1 + z′2 0

0 1

)(5.25)

where the prime means a derivative with respect to x, since, in the static approx-

imation, z depends only on x and is constant with respect to τ . After computing

the determinant of this matrix, the Nambu-Goto action becomes:

SNG =R2 T

2πα′

∫ r/2

−r/2

dx ec z2/2

√1 + z′2

z2. (5.26)

A first integral can be found:

H = L − z′δLδz′

=ec z2/2

z2√

1 + z′2= const . (5.27)

The integration constant is fixed requiring that the maximum value of z is reached

at x = 0: z(0) = z0, z′(0) = 0:

ec z2/2

z2√

1 + z′2=

ec z20/2

z20

, (5.28)

from which we extract:

z′2 =z40

z4ec(z2−z2

0) − 1 . (5.29)

The equation of motion for z can be computed:

z z′′ + (1 + z′2)(2− cz2) = 0 ; (5.30)

from Fig. 5.2 we can observe that the solution z(x) of this equation increases from

87


z = 0 to z = z0 when x goes from x = −r/2 to x = 0, while, from x = 0 to x = r/2,

it decreases, so in the first region we choose in (5.29) z′ > 0 and in the second z′ < 0.

The distance between the quark and the antiquark can be written as:

r =

∫ r/2

−r/2

dx =

∫ z0

0

dz

|z′|+

∫ 0

z0

dz

−|z′|= 2

∫ z0

0

dzz2

z20

(ec(z2−z2

0) − z4

z40

)−1/2

(5.31)

where, in writing the last term, we have used (5.29). Introducing the dimensionless

quantities v = z/z0 and λ = cz20 , we find:

r = 2

∫ 1

0

dv z0 v2(ec z2

0(v2−1) − v4)−1/2

= 2

√λ

c

∫ 1

0

dv v2(eλ(v2−1) − v4

)−1/2

.

(5.32)

This integral is real for λ < 2, since

eλ2(v2−1) > v4 ⇒ λ < min

(4 log v

v2 − 1

)= 2 ; (5.33)

this also means that there is an upper limit for the maximum value that the variable

z can assume:

z0 <

√2

c. (5.34)

In Fig. 5.2 three solutions of the equation (5.30) corresponding to different

values of z0 <√

2/c are plotted, each one extends also in time (remind that we

are using the static approximation, so the solution does not change with τ). At

small distances r between the quarks (at the endpoints of the string) the solutions

do not reach z0; at increasing r, the strings can approach z0 becoming flatter and

flatter at the brane z = z0. When r is large the surface spanned by the string can

be considered approximately rectangular and its area is simply proportional to the

product r ·T , so that the resulting potential in (5.22) is proportional to the distance

r, as we expect to be in the confinement phase. We can also observe from (5.34)

that, if c = 0, i.e. if the space is simply AdS, z0 is no more bounded: if the conformal

symmetry is not broken, we cannot get confinement in the theory.

The same steps can be repeated for the action SNG:

SNG reg =T g

π

√c

λ

∫ 1

ε

dv v−2 eλv2/2(1− v4 eλ(1−v2)

)−1/2

, (5.35)

where g = R2/α′ and ε has been introduced to regularise it. From (5.22) and writing

the expansion in ε:

Vreg =g

πε+ V +O(ε) , (5.36)

88


we find the quark-antiquark potential, expressed in the following parametric form:r(λ) = 2

√λ

c

∫ 1

0

dv v2(eλ(v2−1) − v4

)−1/2

V (λ) =g

π

√c

λ

(−1 +

∫ 1

0

dv v−2

[eλv2/2

(1− v4 eλ(1−v2)

)−1/2

− 1

]);

(5.37)

the functions r(λ), V (λ) and V (r) are plotted in Fig. 5.3.

0.0 0.5 1.0 1.5 2.00

1

2

3

4

5

6

Λ

rHΛL

0.0 0.5 1.0 1.5 2.0-3

-2

-1

0

1

2

3

Λ

VHΛL

0 2 4 6 8 10-3

-2

-1

0

1

2

r

VHrL

Figure 5.3: Top-left panel: quark-antiquark distance r as a function of thedimensionless parameter λ; top-right panel: interaction potential V with respectto λ; bottom panel: parametric plot of the potential with respect to the distancer. All the plots have been obtained using g = c = 1.

At small distances the potential in Fig. 5.3 behaves as 1/r; this can be confirmed

by performing the limit λ ∼ 0 (corresponding to r → 0, see Fig. 5.3):r −−→

λ→0

1

ρ

√λ

c

(1− 1

4λ(1− πρ2) +O(λ2)

)

V −−→λ→0

− g

2πρ

√c

λ

(1 +

1

4λ(1− 3πρ2) +O(λ2)

),

(5.38)

89


with ρ = Γ(1/4)2/(2π)3/2. Combining the two expressions, we get:

V (r) = g

(−k0

r+ σ0r +O(r3)

)r 1 (5.39)

where k0 = (2πρ2)−1 and σ0 = cρ2/4. At λ→ 2, corresponding to the limit of large

r (see again Fig. 5.3), we get:r −−→

λ→2−√

2

clog(2− λ) +O(1)

V −−→λ→2

−g e

2π

√c

2log(2− λ) +O(1) ,

(5.40)

so the potential in this limit has the form:

V (r) = g(σr +O(1)) r 1 , (5.41)

i.e. there is a linear relation between V and r, as it can be qualitatively observed

from Fig. 5.2.

This is the expression for the quark-antiquark potential (5.37) obtained using

the AdS/QCD duality.

5.3 Potential model

In the previous sections we have learnt how to solve a two-body problem through

the Salpeter equation and how to describe the interaction between a quark and an

antiquark through an instantaneous potential. Here this information is used to study

mesons and determine their masses. In particular, we deal with mesons comprising

at least one heavy quark, for which the approximation involved in describing the

interaction through a static potential is more reliable.

This model can be also applied to states comprising more than two constituent

particles but that can be approximately described through the interaction of two

objects. This is the case of tetraquarks, regarded as bound states of a diquark and

an antidiquark, or baryons comprising two heavy quarks and a light quark, regarded

as bound states of a diquark and a light quark. These configurations are sketched

in Fig. 5.4.

A diquark is a coloured bound state of two quarks; from the point of view of

SU(3)c colour group, a quark is in the fundamental representation, 3, and two

90

5.3. Potential model

Figure 5.4: Left panel: sketch of a tetraquark as a bound state of a diquark andan antidiquark. Right panel: sketch of a baryon as a bound state of a heavydiquark and a light quark.

interacting quarks can form the multiplets:

3⊗ 3 = 3⊕ 6 , (5.42)

the first one, 3, is attractive and corresponds to a diquark. Analogously, an antidi-

quark is in the representation 3. Therefore, a tetraquark can be described as the

singlet state arising from:

3⊗ 3 = 1⊕ 8 , (5.43)

where 3 and 3 correspond to a diquark and an antidiquark, respectively. The relation

(5.43) can be also used for baryons and mesons: in the former case, 3 stands for a

quark and 3 for a diquark while in the latter 3 is the quark and 3 the antiquark.

In other words, we assume that the quark-antiquark potential is almost the same as

the diquark-quark and diquark-antidiquark one. Under this hypothesis, the Salpeter

equation will be used to determine the spectrum of all these hadrons.

We consider in the potential a term accounting for the spin interaction. In the

one-gluon-exchange approximation and for S-wave states (` = 0), it has the form

[85]:

Vspin(r) = AQδ(r)

m1m2

S1 · S2 with δ(r) =

(σ√π

)3

e−σ2r2

, (5.44)

where S1 and S2 are the spin of the interacting particles, such that:

2 (S1 ·S2) = (S1 + S2)2−S1

2−S22 = sH(sH + 1)− s1(s1 + 1)− s2(s2 + 1) . (5.45)

The parameter AQ in (5.44) is proportional to αs, which is a running coupling

constant, so it gets two different values, one for hadrons containing at least one

bottom quark (Ab) and Ac otherwise.

Finally, we introduce a cutoff in the potential, such that at distances r 6 rm it is

constant and equal to V (rm) [81]. The cutoff eliminates the Coulombic divergence

91


occurring at r → 0 in (5.37), which, otherwise, would have produced an unphysical

logarithmic divergence in the wave function solution of the Salpeter equation. We

use [86]:

rm =

k

M(m1 = m2)

k′

M(m1 6= m2) .

(5.46)

The complete potential for describing the interaction between a quark and an

antiquark is:

V (r) =

VAdS(r) + Vspin(r) + V0 r > rm

VAdS(rm) + Vspin(rm) + V0 r 6 rm ,

(5.47)

where VAdS(r) indicates the potential in (5.37) extracted from the AdS/QCD corre-

spondence and V0 is a constant term.

Once the potential in Eq. (5.1) has been defined, we can start investigating

the spectrum of hadrons. The parameters of the model, namely the constituent

quark masses and the parameters appearing in the potential, are fixed from the

experimental values of meson masses. Then, predictions for the spectrum of some

tetraquarks [87] and baryons [88], and for some decay widths of charmonium and

bottomonium states [89] will be obtained.

5.3.1 Meson spectrum

Meson masses and wave functions are determined by solving the Salpeter equation

(5.1) (` = 0):(√−∇2 +m2

q +√−∇2 +m2

q + V

)ψM(r) = MM ψM(r) (5.48)

in which mq and mq are the masses of the constituent quark and antiquark, and

V (r) is the potential in (5.47).

In Table 5.2 the measured masses of the mesons [50] together with the values

computed in this model are reported, using the parameters in Table 5.1.

One encouraging result is that, as we see in Table 5.2, ηb(1S) is predicted to

have a mass of 9.387 GeV, in agreement with a recent measurement by the BaBar

Collaboration in the Υ(3S) radiative decay mode: Υ(3S) → γηb [90], according to

92


Table 5.1: Parameters defining the potential model, fitted from the meson spec-trum.

Constituent masses VAdS Vspin Cutoff

mq 0.302 GeV c 0.3 GeV2 σ 1.21 GeV k 1.48

ms 0.454 GeV g 2.75 Ac 7.92 k′ 2.15

mc 1.733 GeV V0 -0.49 GeV Ab 3.09

mb 5.139 GeV

Table 5.2: Mass spectra for heavy mesons; q = u, d. Units are GeV.

Flavour Level J = 0 J = 1

Particle Th. mass Exp. mass [50] Particle Th. mass Exp. mass [50]

cq 1S D 1.862 1.867 D∗ 2.027 2.0082S 3.393 2.598 2.6223S 2.837 2.987

cs 1S Ds 1.973 1.968 D∗s 2.111 2.112

2S 2.524 2.6703S 2.958 3.064

cc 1S ηc 2.990 2.980 J/ψ 3.125 3.0972S 3.591 3.637 3.655 3.6863S 3.994 4.047 4.039

bq 1S B 5.198 5.279 B∗ 5.288 5.3252S 5.757 5.8193S 6.176 6.220

sb 1S Bs 5.301 5.366 B∗s 5.364 5.412

2S 5.856 5.8963S 6.266 6.296

bc 1S Bc 6.310 6.286 B∗c 6.338

2S 6.869 6.8793S 7.221 7.228

bb 1S ηb 9.387 Υ 9.405 9.4602S 10.036 10.040 10.0233S 10.369 10.371 10.3554S 10.619 10.620 10.579

which Mηb= 9388.9+3.1

−2.3(stat)± 2.7(syst) MeV.

In order to test the limits of the model, we have also computed the masses of

states containing only light quarks (u, d, s). The results for their spin averaged

masses are shown in Table 5.3. For the lightest mesons we find a large deviation,

while such a discrepancy is somehow reduced in case of ss and for higher radial

93


excitations. This is reasonable, since a constituent quark model with instantaneous

interaction, computed in the static approximation, is not able to describe the chiral

dynamics of light states. The better accuracy of the ss system allows us to fix the

spin constant As in (5.44) from this channel, obtaining As = 11.3, which will be

used to compute the masses of diquarks. With this value, we obtain for ϕ a mass

m=1.011 GeV (the experimental value is 1.019 GeV) and for ϕ′ a mass m=1.663

GeV (the experimental value is 1.680 GeV).

Table 5.3: Mass spectra for spin averaged masses of light mesons; q = u, d.Units are GeV.

Flavour Level Th. mass Exp. mass [50]

qq 1S 0.792 0.6162S 1.386 1.424

qs 1S 0.932 0.7942S 1.501

ss 1S 0.981 0.9122S 1.571 ≈ 1.653

It might be useful at this stage to study the effect of the relativistic kinematics

on the equation of state. To this end we use the same potential, with parameters

indicated in Table 5.1, with two different equations: the Salpeter (5.1) and the

Schrodinger equation. The results of the two equations for mesons with JP = 1−

are reported in Table 5.4. The comparison between the two computed spectra and

the experimental one shows that, as expected, the results obtained by the Salpeter

equation are more accurate than the ones obtained by the Schrodinger equation. The

advantage of using the Salpeter equation is particularly significant for the charmed

states, since this equation takes into account a relevant source of corrections, i.e.

those due to the relativistic kinematics.

5.3.2 Tetraquark spectrum

We first compute diquark masses, which will be used as input to determine tetraquark

masses. We have already defined a diquark as a bound state of two quarks. Com-

paring (5.42) with (5.43) in the one-gluon-exchange approximation, it is found that

the energy of interaction between two quarks in the 3 representation is half the one

between a quark and an antiquark forming a singlet. Thus, diquark masses can be

computed solving the Salpeter equation (5.1) in which m1 and m2 are the masses

94


Table 5.4: Comparison between spectra of mesons with JP = 1− computedby the Salpeter equation in Eq. (5.1) and the Schrodinger equation. The samepotential V (r) is used in both cases.

Flavour Level Salpeter Schrodinger

cq 1S 2.027 2.1542S 2.598 2.877

cs 1S 2.111 2.1822S 2.670 2.843

cc 1S 3.125 3.1332S 3.655 3.695

bq 1S 5.288 5.4942S 5.819 6.204

sb 1S 5.364 5.5072S 5.896 6.154

bc 1S 6.338 6.5502S 6.879 6.922

bb 1S 9.405 9.7742S 10.040 10.055

of the quarks and the potential is half the one we have used for mesons, i.e. (5.47).

The values obtained for diquarks comprising at least a heavy quark are reported in

Table 5.5; QQ indicates a diquark with spin 1 while [QQ] a diquark with spin 0.

Even if the application of this model to light interacting particles produces larger

uncertainties, we have also attempted to compute masses of diquarks comprising a

strange and a light quark, in order to obtain an estimate of the masses of tetraquarks

with open charm. Notice that in Table 5.5 spin 0 states with two identical quarks

are absent, due to Fermi statistics [91].

We discuss now the possibility that a diquark and an antiquark combine to

produce a tetraquark state. We have already pointed out in section 3.1 that some

scalar mesons, such as σ(480), κ(800), f0(980), have been also interpreted as four-

quark states [92]. However here we will not investigate these states since they involve

only light quarks. We will rather analyse some recently discovered states with both

hidden and open charm, e.g. X, Y , DsJ , for a review see [91, 93].

Tetraquark masses can be computed by solving the following Salpeter equation:(√−∇2 +m2

d +√−∇2 +m2

d+ V

)ψT (r) = MT ψT (r) , (5.49)

95


Table 5.5: Diquark masses. QQ (resp. [QQ]) means a spin 1 (resp. S = 0) diquarkQQ. Units are GeV.

J = 1 J = 0

State Mass State Mass

qs 0.980 [qs] 0.979

ss 1.096

cq 2.168 [cq] 2.120

cs 2.276 [cs] 2.235

cc 3.414

bq 5.526 [bq] 5.513

bs 5.630 [bs] 5.619

bc 6.741 [bc] 6.735

bb 10.018

where md and md are the masses of the diquark and antidiquark. We have stated

above that, from the point of view of group theory, these states are identical to

mesons, so, in this respect, we could use the potential (5.47). However since di-

quarks are composite objects with a finite size, we have modified the potential (5.47)

through a convolution with the wave functions of the diquark ψd and antidiquark

ψd:

V (R) =1

N

∫dr1

∫dr2 |ψd(r1)|2|ψd(r2)|2V

(∣∣∣R + r1 − r2

∣∣∣) (5.50)

with

N =

∫dr1

∫dr2 |ψd(r1)|2|ψd(r2)|2 . (5.51)

The result is shown in Fig. 5.5. ψd,d(r) are computed from the diquark wave equa-

tion. This is only approximately correct because that equation provides the diquark

wave function in the diquark rest frame, whereas (5.49) holds in the tetraquark rest

frame. However for diquarks comprising heavy quarks (c, b) the average diquark

velocity is small (we estimate β ∼ 0.15 for diquarks with open charm and β ∼ 0.06

for diquarks with open bottom). Therefore we can neglect the distortion induced

by the Lorentz boost on the wave function.

In Tables 5.6 and 5.7 the values predicted for the four-quark states with hidden

charm and hidden bottom, respectively, are collected.

A peculiar feature of the diquark-antidiquark scheme for the X state is the

prediction of four different states with mass differences of a few MeV. Two of

96


0 1 2 3 4 5-4

-3

-2

-1

0

1

r HGeV-1L

VHrLHG

eVL

Figure 5.5: The potential between a static diquark-antidiquark pair (dashedline) and quark-antiquark one (solid line). Units are GeV (V) and GeV−1 (r).Data refer to the [cq]cq potential.

Table 5.6: Four-quark states with hidden charm interpreted as bound statescomprising a diquark (cq) and an antidiquark (cq). The model in [94] uses aquasipotential of the Schrodinger type. Ref. [92] uses a constituent quark model(† means that the experimental value is used as an input in this case). Units areGeV.

JPC Flavour content Mass (this work) Mass [94] Mass [92] Exp. State

0++ [cq][cq] 3.857 3.812 3.723

1++ ([cq]cq+ [cq]cq)/√

2 3.899 3.871 3.872† X(3872)

1+− ([cq]cq − [cq]cq)/√

2 3.899 3.871 3.754

0++ cqcq 3.729 3.852 3.832

1+− cqcq 3.833 3.890 3.882

2++ cqcq 3.988 3.968 3.952 Y (3940)

Table 5.7: Four-quark states with hidden bottom interpreted as bound statescomprising a diquark (bq) and an antidiquark (bq). The model in [94] uses aquasipotential of the Schrodinger type. Units are GeV.

JPC Flavour content Mass (this work) Mass [94]

0++ [bq][bq] 10.260 10.471

1++ ([bq]bq ± [bq]bq)/√

2 10.284 10.492

0++ bqbq 10.264 10.473

1+− bqbq 10.275 10.484

2++ bqbq 10.296 10.534

97


them are neutral: Xu = [cu][cu], Xd = [cd][cd], and two charged: X+ = [cu][cd],

X− = [cd][cu]. However, so far, only one state has been observed, namely X(3872),

discovered by the Belle Collaboration [95] in the decay mode J/ψπ+π− with mass

3872.0± 0.6 (stat)± 0.5 (syst) MeV, and confirmed by the CDF Collaboration [96],

measuring a mass of (3871.3±0.7 (stat)±0.4 (syst)) MeV. The average mass of this

state is (3871.2 ± 0.4) MeV [50] and its quantum numbers should be JPC = 1++.

The mass we find is compatible with the experimental one, in the sense that the

difference between them is of the same order as the differences in Table 5.2 between

the experimental and predicted meson masses.

The assignment C = +1 follows from the fact that the decay X → γJ/Ψ is ob-

served. Moreover, from the decay X → π+π−J/Ψ, one can notice that the part of

the 2π invariant mass spectrum that can be ascribed to a ρ0 decay is consistent with

S-wave decay of the X state. From this, it follows that X can be a 0−, 1+ or 2−

state. Finally the angular distribution in this channel is incompatible with J = 0

and therefore the only remaining possibilities are J = 1 or J = 2. If the peak in

the D0D0π0 decay channel at 2σ from the mass of X(3872) is interpreted as due

to this state, then the J = 2 should be excluded, which leaves us with J = 1 only.

In particular, the Belle Collaboration measured in this decay mode X → D0D0π0:

M=(3875.4± 0.7+1.2−2.0) MeV [97], while the BaBar Collaboration observed a peak in

the mass distribution of D0D0 at M=(3875.1+0.7−0.5± 0.5) MeV [98]. There is a debate

[99, 100] about the possibility that these events correspond to the same or different

neutral particles, namely Xu and Xd, since the mass shift between the two bumps

at 3871 and 3875 MeV is of the same order as the mass difference between the u

and d quarks.

The state 2++ of the hidden charm spectrum has been associated, in [92], to the

particle Y (3940), observed by the Belle Collaboration in the decay modeB → Kω J/Ψ

[101]. Its reported mass is M = (3943± 11± 13) MeV.

For the first radial excitations of the two X states with JPC = 1+− we find a

mass of 4.421 GeV and 4.418 GeV, respectively. In [102] the state Z(4433), recently

observed by the Belle Collaboration [103] through the decay Z(4433) → ψ(2S)π± is

interpreted as the first radial excitation of one of these states. This interpretation is

compatible with our results because of the theoretical errors of the present model.

However, the existence of this state is still debated since it has not been confirmed

by the BaBar Collaboration [100].

Let us finally comment on the possible existence of tetraquarks comprising a

heavy diquark and a light diquark. We present in Table 5.8 our predictions for

tetraquarks with open charm and strangeness and compare them with the outcomes

98


of the constituent quark model in [92], where the state 0+ is associated with the

particle Ds(2317) [104], 1+ with Ds(2457) [104] and 2+ with X(2632) [105]. Our

results are significantly different from those of [92]. Again, this might be due to

the limitations of one or both the constituent quark models. In any event, we do

not expect theoretical errors larger than a few hundred MeV for the results of the

present model in Table 5.8, so that this model does not support the interpretation

of the states Ds(2317), Ds(2457) and X(2632) as tetraquark charmed states with

open strangeness. In [106] these states are interpreted as a mixture of P -wave

quark-antiquark states and four-quark components.

Table 5.8: Comparison between the results of the present model and those ofRef. [92] for tetraquarks with open charm and strangeness. Units are GeV.

JP Flavour content Th. mass (this work) Th. mass (model [92])

0+ [cq][qs] 2.840 2.371

0+ cqqs 2.503 2.424

1+ cq[qs] 2.880 2.410

1+ cqqs 2.748 2.462

1+ [cq]qs 2.841 2.571

2+ cqqs 2.983 2.648

5.3.3 Charmonium and bottomonium decays

Within the potential model we can compute some decay widths of charmonium and

bottomonium states. At odds with the analyses in the previous sections concerning

mesons and tetraquarks, we fix the parameter k appearing in (5.46) instead of fitting

it, according to a QCD duality argument [86]: k = 4π3

. The input set of parameters

is reported in Table 5.9. The values obtained for the S-wave cc and bb spectra are

shown in Table 5.10; in Fig. 5.6 the corresponding wave functions are depicted.

The decay constants fP and fV of a pseudoscalar and a vector meson are defined

by:

〈0|Aµij|P (k)〉 = i kµQijfP

〈0|V µij |V (k, λ)〉 = ε(λ)µQij MV fV (5.52)

where k is the momentum, λ the helicity and ε the polarization vector of the meson.

Aµij is the axial current qiγ5γ

µqj, Vµij is the vector current qiγ

µqj and Qij is the meson

99


Table 5.9: Parameters defining the potential model, fitted from the meson spec-trum. The value of k is not fitted, but is fixed by a QCD duality argument.

Constituent masses VAdS Vspin cutoff

mq 0.34 GeV c 0.4 GeV2 σ 0.47 GeV k 4.2

ms 0.48 GeV g 2.50 Ac 14.56 k′ 2.1

mc 1.59 GeV V0 -0.47 GeV Ab 6.49

mb 5.02 GeV

1 2 3 4 5k HGeVL

-4

-2

0

2

4

6

uΗ

c Hn

SLHkL

1 2 3 4 5k HGeVL

-4

-2

0

2

4

6

uΗ

b H

nSLH

kL

1 2 3 4 5k HGeVL

-4

-2

0

2

4

6

u Ψ H

nSLH

kL

1 2 3 4 5k HGeVL

-4

-2

0

2

4

6

u

Y Hn

SLHkL

Figure 5.6: The momentum wave functions of the first four states of ηc(nS)(top left), ηb(nS) (top right), J/ψ(nS) (bottom left) and Υ(nS) (bottom right).The continuos line represents the 1S state, the dotted line represents the 2Sstate, the dashed line represents the 3S state, and the dot-dashed line representsthe 4S state. The wave functions are dimensionless: they are normalised as∫dk |u(k)|2 = 2M .

flavour matrix. Within the Salpeter model, they are given by [81]:

fP =√

31

2πM

∫ +∞

0

dk k u0(k)N12

[1− k2

(Ei +mi)(Ej +mj)

]fV =

√3

1

2πM

∫ +∞

0

dk k u0(k)N12

[1 +

k2

3(Ei +mi)(Ej +mj)

](5.53)

100


with

N =(Ei +mi)(Ej +mj)

EjEi

, (5.54)

where M is the mass of the meson, mi is the mass of the constituent quark i and Ei

its energy, u(k) is the meson reduced wave function in momentum space, obtained

by Fourier transforming the reduced radial wave function u(r) = r ψ(r); and k is

the momentum of the constituent quark in the rest frame of the meson.

The obtained decay constants are collected in Table 5.10; in particular, fηc turns

out to be compatible with a determination obtained by the CLEO Collaboration:

fηc = 335± 75 MeV [107].

Table 5.10: Masses of pseudoscalar and vector cc and bb states compared to theexperimental data. In the fourth column the decay constants, computed using(5.53), are reported.

Particle Th. mass (MeV) Exp. mass (MeV) [50] Decay const. (MeV)

ηc 3025.3 2980.3 ± 1.2 342

η′c 3603.5 3637.0 ± 4 266

η′′c 4039.3 195

J/ψ 3079.8 3096.916 ± 0.011 356

ψ′ 3624.3 3686.09 ± 0.04 237

ψ′′ 4057.0 4039±1 185

ηb 9433.9 9388.9 +3.1−2.3 (stat) ± 2.7 (syst) [90] 637

η′b 9996.8 430

η′′b 10347.5 367

Υ 9438.3 9460.30±0.26 686

Υ(2S) 9998.6 10023.26 ± 0.31 484

Υ(3S) 10348.8 10355.2 ±0.5 335

Υ(4S) 10622.3 10579.4 ±1.2 301

Using the computed values of fP and fV , it is possible to determine the widths Γγγ

of the radiative decays ηb,c(nS) → γγ, and Γ`+`− of the processes ψ(nS) → `+`− and

Υ(nS) → `+`−. They can be computed using the effective Lagrangians [108, 109]:

Lγγeff = −i c1(q γσγ5q)εµνρσF

µνAρ

L`¯

eff = −c2(q γµq)(`γµ¯) (5.55)

101


where

c1 =Q2 4π αem

(M2 + EbM)

c2 =Q 4π αem

M2. (5.56)

One obtains

Γγγ =4π Q4α2

emM3f 2

P

(M2 + EbM)2

Γ`+`− =4π Q2α2

emf2V

3M, (5.57)

where Q is the electric charge (in units of e) of the constituent quark and Eb =

2m−M is the binding energy.

The values obtained for the pseudoscalar mesons are shown in Table 5.11, to-

gether with recent theoretical results. The prediction for the ηc radiative decay

width is compatible, within the error, with the average evaluated in [50] from exper-

imental data: Γγγ ∼ (7.2 ± 0.7 ± 2.0) KeV. Moreover, the experimental branching

ratio for ηcc → γγ is (1.8 + 0.6− 0.5)× 10−4 [50], from which Γγγ ∼ 4.9 KeV, being

very close to our prediction. In the case of η′c, there is an experimental observation

by the Cleo Collaboration [110], measuring (1.3± 0.6) KeV; this value is smaller, or

marginally comparable, with our determination. In this respect, our result follows

most theoretical models [108, 111, 112, 113, 114], which predict higher values for

Γγγ(η′c), although in some cases within the experimental error. This might suggest

that the disagreement could be attributed to the systematics of the experimental

measurement, namely, to the assumption that ηc and η′c have the same branching

fractions to the final state KSKπ.

Concerning the bb pseudoscalar meson, the theoretical models in Table 5.11 pre-

dict, for the ηb → γγ decay width, values in the range 230-560 eV; the result obtained

in this paper points towards small values in this range.

For vector mesons, the predicted and the experimental values of the leptonic

decay widths are reported in Table 5.12. There is an overall agreement, excluding a

discrepancy in the Υ(3S) that could be attributed to a possible D-wave component

in this meson.

5.3.4 Doubly heavy baryons

We complete the analysis on the spectroscopy of heavy hadrons with doubly heavy

baryons, which are made up of two constituent heavy quarks and a light quark. We

102


Table 5.11: Decay widths Γγγ (in KeV) of pseudoscalar states in two photons.

Particle This work [109] [111] [112] [113]

ηc 4.252 7.46 7.18 7.14±0.95 5.5

η′c 3.306 4.1 1.71 4.44±0.48 1.8

η′′c 1.992 1.21

ηb 0.313 0.560 0.230 0.384± 0.047 0.350

η′b 0.151 0.269 0.070 0.191 ± 0.025 0.150

η′′b 0.092 0.208 0.040 0.100

Table 5.12: Decay widths Γ`+`− (in KeV) of vector mesons.

Particle This work Exp. [50]

J/ψ 4.080 5.55± 0.14±0.02

ψ′ 2.375 2.38 ± 0.04

ψ′′ 0.836 0.86±0.07

Υ 1.237 1.340 ± 0.018

Υ(2S) 0.581 0.612 ± 0.011

Υ(3S) 0.270 0.443± 0.008

Υ(4S) 0.212 0.272± 0.029

will treat them as bound states of two objects, a heavy diquark and a light quark.

It is assumed that the two heavy quarks are very close, in such a way that they are

seen as a whole static colour source by the third constituent light quark. Moreover

the relation (5.43) suggests us that the interaction between a quark and a diquark

inside a baryon can be studied in analogous way as the one between a quark and an

antiquark inside a meson.

Given these considerations, we can write the Salpeter equation as:(√m2

q −∇2 +√m2

d −∇2 + V (r)

)ψB(r) = MB ψB(r) , (5.58)

where mq and md are the masses of the quark and the diquark, MB and ψB(r)

are the mass and the wave function of the baryon, respectively. Analogously to

the tetraquark case, the potential V is obtained with a convolution of the quark-

103


antiquark potential in (5.47) with the wave function of the diquark:

V (R) =1

N

∫dr |ψd(r)|2V (|R + r|) , (5.59)

where N is a normalisation factor.

Using the parameters in Table 5.9 and with the same procedure described in

section 5.3.2, we compute again diquark masses, finding the values in Table 5.13.

Table 5.13: Diquark masses in GeV. QQnS (resp. [QQ]nS) means a spin 1(resp. spin 0) diquark QQ in S-wave with radial number n.

Diquark State Mass

ccnS 1S 3.2382S 3.589

[bc]nS 1S 6.5582S 6.882

bcnS 1S 6.5622S 6.883

bbnS 1S 9.8712S 10.165

Then, solving Eq. (5.58), we find the eigenvalues corresponding to the masses of

baryons, which are reported in Tables 5.14, 5.15 and 5.16 for baryons comprising two

charm, two bottom and a charm and a bottom quark, respectively. Since there are

no experimental data, apart from one state, it is interesting to compare the values

found with this potential model with the predictions of other works: ref. [115,

116, 117] describe baryons by a non-relativistic quark model based on a three-body

problem; in ref. [118, 119] potential models based on the quark-diquark hypothesis

are investigated, the first one relativistic and the second one non-relativistic; in ref.

[120] doubly heavy baryon masses are computed in the framework of QCD sum

rules; ref. [121, 122, 123] deal with quenched lattice QCD.

In Fig. 5.7 the wave functions of the first three radial excitations of Ωcc and Ξbb

are shown.

The only state observed so far is a candidate for Ξcc reported by the SELEX

Collaboration, which found a signal for the decay Ξ+cc → ΛcK

−π+ [124]. The same

Collaboration confirmed the production of Ξ+cc considering the decay mode Ξ+

cc →pD+K− [125], with measured mass of Ξcc:

MΞcc = 3518.9± 0.9 MeV . (5.60)

104


Table 5.14: Masses (GeV) of baryons composed by a diquark in the lowest massconfiguration cc1S and a light quark (q or s). In the case of ref. [121], thevalues have been obtained using β = 2.1.

Particle State JP Content This work [115] [116] [117] [118] [119] [120] [121] [122]

Ξcc 1S 12

+qcc1S 3.547 3.579 3.676 3.612 3.620 3.48 4.26 3.562 3.549

2S 4.183 3.8763S 4.640

Ξ∗cc 1S 3

2

+qcc1S 3.719 3.656 3.753 3.706 3.727 3.61 3.90 3.625 3.641

2S 4.282 4.0253S 4.719

Ωcc 1S 12

+scc1S 3.648 3.697 3.815 3.702 3.778 3.59 4.25 3.681 3.663

2S 4.268 4.1123S 4.714

Ω∗cc 1S 3

2

+scc1S 3.770 3.769 3.876 3.783 3.872 3.69 3.81 3.737 3.734

2S 4.3343S 4.766

Table 5.15: Masses (GeV) of baryons composed by a diquark bb1S and a lightquark (q or s).

Particle State JP Content This work [115] [116] [117] [118] [119] [120] [123]

Ξbb 1S 12

+qbb1S 10.185 10.189 10.340 10.197 10.202 10.09 9.78 10.127

2S 10.751 10.5863S 11.170

Ξ∗bb 1S 3

2

+qbb1S 10.216 10.218 10.367 10.236 10.237 10.13 10.35 10.151

2S 10.770 10.5013S 11.184

Ωbb 1S 12

+sbb1S 10.271 10.293 10.454 10.260 10.359 10.18 9.85 10.225

2S 10.830 10.6043S 11.240

Ω∗bb 1S 3

2

+sbb1S 10.289 10.321 10.486 10.297 10.389 10.20 10.28 10.246

2S 10.839 10.6223S 11.247

Taking into account the uncertainties in the quark masses and those related to

our description of the baryon, the mass found with this model can be considered

compatible with this experimental value.

Looking at the tables, we can notice that the only remarkable discrepancy be-

tween the values concerns radial excitations. In particular the values we have esti-

105


0 2 4 6 8 10-2

-1

0

1

2

k HGeVL

u ccsH

kL

0 2 4 6 8 10

-2

-1

0

1

2

3

k HGeVL

u bbqH

kL

Figure 5.7: Wave functions of the first three radial excitations of Ωcc (left) andΞbb (right). The continuous line represents the 1S wave function, the dottedline the 2S wave function and the dashed line the 3S wave function. The wavefunctions are dimensionless: they are normalised as

∫dk |u(k)|2 = 2M , being k

the modulus of the relative 3-momentum of the quark-diquark pair.

Table 5.16: Masses (GeV) of baryons composed by a diquark bc in the lowestmass configuration and a light quark (q or s).

Particle State JP Content This work [116] [117] [118] [119] [120]

Ξbc 1S 12

+qbc1S 6.904 7.011 6.919 6.933 6.82 6.75

2S 7.4783S 7.904

Ξ′bc 1S 1

2

+q[bc]1S 6.920 7.047 6.948 6.963 6.85 6.95

2S 7.4853S 7.908

Ξ∗bc 1S 3

2

+qbc1S 6.936 7.074 6.986 6.980 6.90 8.00

2S 7.4953S 7.917

Ωbc 1S 12

+sbc1S 6.994 7.136 6.986 7.088 6.91 7.02

2S 7.5593S 7.976

Ω′bc 1S 1

2

+s[bc]1S 7.005 7.165 7.009 7.116 6.93 7.02

2S 7.5633S 7.977

Ω∗bc 1S 3

2

+sbc1S 7.017 7.187 7.046 7.130 6.99 7.54

2S 7.5713S 7.985

mated here are systematically higher than the ones found in ref. [115]. This could

be explained by considering that in a model in which a baryon is described as a

bound state of a diquark and a quark the first excited state with ` = 0 does not

correspond to the 2S radial excitation of the whole baryon, but it is the one in which

106

5.4. Remarks inspired by the Heavy Quark Effective Theory

the diquark is in the 2S state [116]. To verify this hypothesis, we have computed the

masses of such states, i.e. using as input the masses of 2S diquarks, already reported

in Table 5.13. The results are shown in Table 5.17, together with the results of other

models, using the (2S diquark)-(quark) scheme as well. In this way, the values we

have found are compatible not only with the others in the table, but also with those

found in [115].

A comment is in order. We have not considered baryons comprising bc or [bc]

diquarks in the new excited level because the excited states of diquarks bc and

[bc] are not stable due to the emission of soft gluons [119].

Table 5.17: Masses (GeV) of the excited baryons in which the diquark is in the2S state.

Baryon JP Quark-diquark content This work [116] [118] [119, 126]

Ξcc12

+qcc2S 3.893 4.029 3.910 3.812

Ξ∗cc

32

+qcc2S 4.021 4.042 4.027 3.944

Ωcc12

+scc2S 3.992 4.180 4.075

Ω∗cc

32

+ scc2S 4.105 4.188 4.174

Ξbb12

+qbb2S 10.453 10.576 10.441 10.373

Ξ∗bb

32

+qbb2S 10.478 10.578 10.482 10.413

Ωbb12

+sbb2S 10.538 10.693 10.610

Ω∗bb

32

+sbb2S 10.556 10.721 10.645

5.4 Remarks inspired by the Heavy Quark Effec-

tive Theory

We have introduced a potential model, characterised by a relativistic kinematics

and by a potential computed through a recently developed method, the AdS/QCD

correspondence, which is the main theme of this thesis. The results we have obtained

about the spectroscopy and decays of heavy hadrons have already been compared

with experimental data, if possible, and outcomes of other models.

A final analysis of the model we have implemented may come from the Heavy

Quark Effective Theory; in particular, we focus on the baryon sector.

The mass of a baryon comprising a single heavy quark Q can be written as a

107


1/mQ expansion, being mQ the mass of the heavy quark in the following way [127]:

MM = mQ + Λ− λ1

2mQ

+ dMλ2

2mQ

+O(1/m2Q) ; (5.61)

Λ contains the contribution of the light degrees of freedom while λ1 and λ2 are

determined by the matrix elements:

λ1 = 〈B(v)|Qv(iD)2Qv|B(v)〉 (5.62)

dM λ2 = 2ZQ〈B(v)|Qv g Gµνσµν Qv|B(v)〉 , (5.63)

where v is the velocity of the quark Q in the baryon, dM = J` · JQ is the Clebsch

factor (J` is the angular momentum of the light quarks), ZQ is a renormalisation

factor with ZQ(µ = mQ) = 1.

Analogously, one can attempt to write an expansion for a S-wave baryon made

up of a heavy diquark and a light quark with respect to the inverse of the heavy

diquark mass mQQ:

MQQq = mQQ + Λ +λ1

2mQQ+ AQdB

λ2

2mQQ(5.64)

where dB = SQQ · Sq. Since in our model the coefficient AQ proportional to dB in

the spin term (5.44) takes two different values in case of baryons with a charm or a

bottom quark, we have displayed it here, too. The mass splitting between JP = 3/2+

and JP = 1/2+ baryons turns out to be, for example in case of ΞQQ:

Ξ∗QQ − ΞQQ = AQ

3λ2

4mQQ. (5.65)

From Eq. (5.65), assuming that λ2 does not depend on the diquark and that all

the dependence is in AQ, the ratio between the mass splitting of Ξbb and Ξcc and

between the difference of the mass squared is given by:

Ξ∗bb − Ξbb

Ξ∗cc − Ξcc

=Abmcc

Acmbb,

Ξ∗2bb − Ξ2

bb

Ξ∗2cc − Ξ2

cc

=Ab

Ac

. (5.66)

These relations are well verified, both for ΞQQ and for ΩQQ baryons, as one can

appreciate considering the results in Table 5.14 and 5.15. Moreover, a mass splitting

hierarchy is obtained:

(Ξ∗cc − Ξcc) > (Ω∗

cc − Ωcc) > (Ξ∗bb − Ξbb) > (Ω∗

bb − Ωbb) . (5.67)

108

5.5. Concluding remarks


We have studied a constituent quark model and computed masses of heavy mesons,

tetraquarks and baryons. The peculiar feature of this model is that we use a potential

computed in the AdS/QCD framework; in this respect, these states are considered

“holographic hadrons”. Given meson masses, predictions about possible new states

can be obtained. This information adds to other results obtained within similar

models using different potentials.

The predictions obtained within this model are collected in the tables presented

in the previous sections; the main results are summarised here:

X the model can reproduce the spectrum of mesons comprising at least one heavy

quark with a relative error (defined as the sum of the differences between the

experimental and theoretical values, divided by the experimental value) of

∼ 0.1%;

X there is also an a posteriori check for the fit, obtained comparing our predicted

outcome for the ηb mass with the subsequent measurement by the Babar Col-

laboration;

X the decay widths are in agreement with the experimental values, but for the

Υ(3S). One can explain this discrepancy by the hypothesis that in the ex-

perimental value there is also a contribution from the D-wave. Moreover,

the unknown widths are in agreement with the predictions of other models,

enforcing some of them;

X about doubly heavy baryons, the disagreement between the masses of the 2S

states, computed here and in other models dealing with a three-body problem,

can be explained by stating that the first excited level of a heavy diquark-quark

configuration is the one in which the diquark is in the 2S excited state and

the baryon in the 1S level;

X doubly heavy baryon masses verify some relations from HQET. The mass of

the recently observed Ξcc baryon is compatible with the one of the baryon

comprising a cc diquark and a light quark, predicted here;

X the model accepts the possibility that the particle X(3872) can be interpreted

as a tetraquark state, but it rejects this possibility for DsJ states.

109

Conclusions

Soon after Maldacena claimed the conjecture about a correspondence between a

supergravity and a strongly coupled gauge theory, a proposal for applying it to QCD

appeared. In this respect, researchers are trying to develop holographic methods,

looking for a new possible approach to the non-perturbative regime.

I have shown some results obtained in less than five years of studies on bottom-

up approaches, focusing, in particular, on the Soft Wall model, which has been

introduced in order to get hadrons arranged in Regge trajectories.

Our contributions deal with the scalar meson sector and the finite temperature

regime. We have computed masses, decay constants, and the two-point correlation

function of scalar mesons, gluon condensates and the coupling with two pseudoscalar

mesons. In a hot medium, we have analysed how the masses and widths of scalar

glueballs and mesons vary with the temperature up to the dissociation.

Another quantity that has been investigated in a slightly modified version of

this framework is the static energy of interaction of a quark-antiquark pair. We

have used this information in a constituent quark model to compute heavy meson

masses and some charmonium and bottomonium decay widths. We have also studied

tetraquarks and baryons with two heavy quarks, considering them as bound states of

a diquark and an antidiquark and of a quark and a diquark, respectively. Under these

assumptions, they can be treated as two-body problems, in which the interaction

between the constituents is almost analogous to the quark-antiquark one.

The successes and the difficulties of the holographic models have been underlined

throughout the thesis.

One of the main properties of the model we have studied is the possibility of

analytically computing many quantities, for instance the masses and decay constants

of scalar glueballs, and scalar and vector mesons at zero temperature, which turn

out to be organised in Regge trajectories. There is only one parameter that has

been fitted from the mass of the ρ meson. However, the description of the chiral

symmetry breaking in the Soft Wall model is not satisfactory since it predicts a

proportionality between the quark condensate and the quark mass, which is absent

111

Conclusions

in QCD. This is also the origin of a small predicted value for the coupling of a scalar

meson to two pseudoscalars. Some proposals have been suggested to overcome this

difficulty: one can somehow modify the metric or the dilaton term in the action;

another possibility is to add potential terms in the action for the scalar field.

We expect new developments and improvements for the analysis of QCD prop-

erties. Moreover, besides QCD, further applications of Maldacena conjecture have

recently been proposed, concerning condensed matter systems, for instance studies

on superconductivity and the Quantum Hall effect, so that exciting perspectives for

the holographic methods are foreseen in the near future.

112

Acknowledgements

Vorrei ringraziare il Prof. Leonardo Angelini, il Dr. Pietro Colangelo, la Dr.ssa

Fulvia De Fazio e il Dr. Stefano Nicotri per il sostegno e la loro disponibilita durante

questi tre anni di dottorato. Ringrazio anche la Dott.ssa Maria Valentina Carlucci,

il Dr. Massimo Mannarelli, il Prof. Mario Pellicoro e il Dr. Sebino Stramaglia per la

proficua collaborazione. In particolare, ringrazio tutti loro per lo spirito di gruppo

con cui si e lavorato, che ritengo uno degli aspetti piu belli di questo mestiere.

Vorrei aggiungere una dedica particolare a Fulvia e Pietro, non solo in merito

alla loro competenza e dedizione al lavoro e per avermi seguito scrupolosamente, ma

anche per non avermi mai fatto sentire il cambiamento, per avermi accolta nel loro

gruppo naturalmente, per non essermi mai sentita sola sul lavoro.

Ringrazio il Prof. Tri Nang Pham e il Dr. Eligio Lisi per aver letto con attenzione

la tesi e per gli utilissimi suggerimenti.

Ringrazio anche tutti i familiari e gli amici, che mi hanno sostenuta e arricchito

le mie giornate fuori dal lavoro, per la pazienza che a volte serve quando si ha a che

fare con un fisico...

Un ringraziamento speciale va a Stefano, che e riuscito a dimostrarmi cosa conta

davvero, mentre prima lo potevo solo immaginare.

Vorrei chiudere questo lavoro nella stessa maniera in cui ho cominciato, cioe

ricordando Beppe Nardulli. Finora non l’ho mai fatto, ne in un articolo ne in un

seminario, perche mi sembrava riduttivo, pur comprendendo l’importanza che per

me hanno queste cose...., ma la ritenevo una responsabilita troppo grande. In questo

caso, non ho saputo resistere e voglio dedicare la tesi di dottorato alla persona con

cui ho cominciato questa esperienza nella ricerca. Io ci ho messo tutto l’impegno

possibile nel scriverla....

113

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