holographic models models of the of the scalar scalar sector sector of QCD

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The AdS AdS/QCD /QCD correspondence correspondence and the and the holographic

holographic models models of the of the scalar scalar sector sector of QCD

of QCD of QCD of QCD

Frédéric Jugeau

TPCSF, IHEP, Beijing

NTU 10/06/2011


AdS/CFT correspondence provides a new way to address Physics at strong coupling

weakly-coupled AAnti dde SSitter Supergravity / strongly-coupled (super)CConformal FField TTheory

SUSY : no gluino neither squark Conformal :


• AdS AdS/CFT CFT correspondence

(Maldacena , Witten, Gubser, Klebanov, & Polyakov 1998)

•• Holographic Models of QCD or AdS/QCD correspondence

What is the gravity theory dual to QCD ?

(assumed to exist) - dimensionful parameters (m , L )

- renormalization (mass scalem)

(Witten 1998, Polchinski & Strassler 2002, Pomarol et al. , Erlich et al. 2005)


- renormalization (mass scalem)

 Hadronic spectrum

• 0 scalar (& 1 vector) glueballs

• chiral dynamics of QCD (indepindep.. explicit & implicit cSB descriptions?)

• 0 scalar mesons++ a (980), f (980), a (1450)0 0 0

++ --

Towards a weakly-coupled gravity dual description of the non-perturbative physics of strong interactions

 AdS/QCD Soft-Wall model (negative dilaton, higher-dim. potential terms, dynamical AdS/QCD)


• Wilson loop and Heavy quarkonium QQ potential :


Holographic principle principle and and AdS AdS/CFT, /CFT, AdS AdS/QCD applications /QCD applications

Maldacena (hep-th/9803002) ; Rey & Yee (hep-th/9803001) ; Sonnenschein et al. (hep-th/ 9803137)

• Heavy-light mesons :

Erdmenger et al. (hep-th/0605241) ; Herzog et al. (hep-th/0802.2956)

Chiral symmetry breaking mechanism & light mesons :

Evans et al. (hep-th/0306018) ; Erlich et al. (hep-ph/0501128) ; Da Rold & Pomarol (hep-ph/0510268)

• Spectroscopy and Form Factors :

Csáki et al. (hep-th/9806021) ; Constable & Myers (hep-th/9905081) ; Boschi-Filho & Braga (hep-th/0207071) Katz et al. (hep-ph/0510388) ; Kwee & Lebed (hep-ph/0708.4054) ; Grigoryan & Radyushkin (hep-ph/0703069)

Andreev & Zakharov (hep-ph/0604204) ; F. Jugeau (hep-ph/0812.4903)

• Baryons :

Hong et al. (hep-ph/0609270) ; Sakai & Sugimoto (hep-th/0701280); Pomarol & Wulzer (hep-ph/0904.2272)

• Deep Inelastic Scattering :

Braga et al. (hep-th /0807.1917)

• Quark-gluon plasma :

Son et al. (hep-th/0405231) ; Kiritsis et al. (hep-th/0812.0792)

• Condensed matter systems (quantum Hall effect, superconductor, superfluidity) :

Herzog, Kovtun & Son (hep-th/0809.4870) ; Hartnoll, Herzog & Horowitz (hep-th/0810.1563)

• Warped extra dimension Electroweak Physics

Gherghetta et al. (hep-ph/0808.3977)

• Astrophysics : Holographic Dark Energy

Li (hep-th/0403127)


x = (x , z)

h = diag (-1,+1,+1,+1)


Maldacena’s conjecture (1998) or conjecture (1998) or AdS AdS/CFT correspondence /CFT correspondence

IIB (oriented closed) superstring theory in N = 4 supersymmetric YM theory SU(N) in the boundary space (z 0)

compact manifold Anti de Sitter space

Holographic spacetime / bulk â

(no physical extra dimension)



M m


holographic coordinatecoordinate

(dual to an energy scale)


AdS radius Rradius R

• : solution of empty space Einstein equation

• Isometry group : SO(2,4)SO(2,4)

cosmological constant : < 0

conformal structure of flat boundary space : SO(2,4)SO(2,4) scalar curvature :


N = 4 SUSY : conformal theory (b = 0 at 3-loop level) pseudo-euclidean :


(preserves , like for )

• isometry group SO(6)SO(6)

(usually not considered for AdS/QCD)

N = 4 SUSY : global SU(4)SU(4) R-symmetry Supercharge algebra




Supergravity limit of type - IIB weakly

weakly--coupledcoupled superstring theory in 10d warped spacetime

‘t Hooft

‘t Hooft limit of N = 4 superconformal strongly

strongly--coupledcoupled Yang-Mills SU(N) in 4d Minkowski spacetime




Ncoincident D33-branes

free closed string N

Ncoincident D33-branes free closed string

SU(NN) gauge theory on the (33+1)-dim.


Parameter correspondence (closed) string

coupling constant

Gauge group of (rank+1) = N YM coupling

Regge slope


(string length

l l l l )

coupling constant

(‘t Hooft coupling )

• ‘t Hooft limit


(large Nlarge N at l fixed) : (1) (2)

of (rank+1) = N

<< 1


Tree--levellevel perturbative string theory :

• StrongStrong coupling constant l >> 1



Small scalar curvature : (2)

(string point-like particle) Supergravity



Symmetry correspondence

Gauge symmetry

4d boundary operator Global symmetry chiral

local, gauge invariant, scaling dim. D

Operator/field correspondence (Witten, Gubser, Klebanov, Polyakov 1998)

5d bulk field massive, p-form

holographic coordinate

z IR



CFT operators

AdS/CFT provides 2 languages for deriving correlation functions (2-,3-,4-points) bulk-to-boundary propagator K(x,x’) 4d boundary



x x’



Freezing behaviour behaviour of QCD effective charges of QCD effective charges at at low low Q Q

freezing behaviour of the coupling : conformal window for QCD

(Deur, Burkert, Chen & Korsch, Phys. Lett. B665:349-351, 2008)

(also Lattice QCD, Schwinger-Dyson eqs.

but also experimental evidence) 2 2

CFT applicable to study CFT applicable to study the properties of hadrons

at first approximation (AdS/QCD)

partons hadrons broken conformal behaviour transition region


Scale invariance breaking and

Scale invariance breaking and AdS AdS/QCD /QCD

dilatation invariance

different values of z : different scales at which hadrons are observed - UV regime : boundary space (z 0)

- IR regime : max. separation of quarks inside hadrons (~ x ) 2 max. value of z (like a spacetime coordinate)

scaling dim. :

canonical dim.

anomalous dim. (AdS/QCD : g = 0) dilatation charge :

- IR regime : max. separation of quarks inside hadrons (~ x ) max. value of z

• Hard wall approx.Hard wall approx. (Polchinski & Strassler 2002) :

•• Soft wall approx.Soft wall approx. (Karch et al. 2006) : background dilaton field (SW ) Kaluza-Klein mass spectrum (~ QM well potential) :

Linear Regge trajectories :

(Gherghetta et al. 2008 : dynamical justification but with an extra tachyon scalar field)

( , ) breakbreak conformal inv. of CFT : introduction of QCD scale QCD scale ΛΛQCDQCD

Caveat : strong l >> 1 at any length scales (no asymptotic freedom of QCD ?)


- AdS/CFT : String-like theories →→ QCD-like gauge theories (upup--downdown approach) - AdS/QCD : QCD properties

→ →

5d weakly-coupled dual theory (bottombottom--upup approach)

QCD operators

Holographic models of the scalar sector of QCD Holographic models of the scalar sector of QCD

• Scalar (& vector) glueballs : bound-states of gluons (well defined in large N limit)

• Scalar mesons: a (980, 1450), f (980, 1370, 1505), (0 0 s(600) ?)…

• chiral dynamics of QCD (a few operators)

Gravity dual theory in the 5d bulk left- and right handed currents : &

vector r, axial a1,



scalar glueball

scalar a0, f0, (s?)

scalar meson operator :


scalar glueball operator :

vector glueball operator : (D=3,p=1)


(D=4,p=0) (D=7,p=1) chiral order parameter :


vector r, axial a1, pseudoscalar modes

vector glueball

chiral symmetry breaking function v(z), chiral pion p

zm hard

hard wallwall


soft wallwall


Soft Wall Model of

Soft Wall Model of QCD QCD

(Karch et al.)

linear eqs. of motion :

• axial-vector :

• vector :

transverse : a1 mesons

longitudinal : pseudoscalar modes

• vector :

• pseudoscalar :

• chiral symmetry breaking function :

• scalar :

(SW )≤


n-point correlation functions in terms of bulk-to-boundary propagators

• 2-point correlation function :

cSB function scalar meson bulk field

SPP couplings quadratic eff. action : spectroscopy

QCD Soft Wall Model for

QCD Soft Wall Model for scalar scalar mesons mesons

(F.J. et al.)

scalar bulk field :

- QCD : - AdS :


Decay constants (residues) :

 current-vacuum matrix elt. (0.21≤0.05 GeV ) 4 :

Masses (simple poles of the y digamma function) :

 Ratio (1.612≤0.004):

 First radial excitation state (1.01≤0.04):

• Large q limit of the 2-point correlation function : pert. contr. + power corrections2

 4-dim. gluon condensate (0,012 GeV4 ) :

 6-dim. condensates (QCD µ - <qq> 2 ) : 6-dim. positive positive condensates


 First radial excitation state :

 becomes const. as n increases


 3-point correlator scalar form factor SPP couplings :

• 3-point correlation functions :

 effective interaction action :

longitudinal component longitudinal component of the axial

of the axial--vectorvector bulkbulk fieldfield chiral

chiral bulkbulk fieldfield scalar

scalar bulkbulk fieldfield


cSB SB functionfunction


massless pion decay constant scalarscalar holoholo. . wavewave functionfunction axialaxial--vectorvector bb--toto--b b propprop. . atat q = 0q = 022 : vanishes in the large limit chiral symmetry breaking function

boundary conditions ( finite when )

quark condensate light quark mass No consistent No consistent implicitimplicit ccSB description SB description in the Soft Wall

in the Soft Wall model SWmodel SW++


QCD Soft Wall Model for the

QCD Soft Wall Model for the scalar scalar & & vector vector glueballs glueballs

(F.J. et al.)

Spectroscopy :

• scalar gluebal :

• vector glueball :


QCDSR Lattice QCDLattice QCD

1.089 GeV



< 1

Narison (hep-ph/9612457) Paver

1.5 (0.2) AdS



(hep-lat/0508002) Morningstar


1.475(30)(65) 1.730(50)(80)

Hang, Zhang (hep-ph/9801214)

1.580(150) F.J. et al.

3.240(330)(150) 3.850(50)(190)


(hep-lat/0508002) Morningstar

(hep-lat/9901004) 1.334 GeV


Modification of the background :

modification of the dilatondilaton

metric function dilaton

• UV conformal behaviour :

• IR behaviour : linear Regge behaving mass spectrum modification of the geometry

(l: perturbative parameter)

( )

UVsubleading IRsubleading

λ < 0 Increasingmass splittings

Maximun effect : warped geometry Mass splitting • dilaton :

• geometry :



Investigating AdS AdS/QCD /QCD through through the the scalar scalar glueball glueball correlator correlator

(F.J. et al.)

A finite action requires B=0 unless (defines SW )L

: no 2d condensate <A²>

(absent in QCD)

OPE (absent in QCD)


 4d gluon condensate :

 low-energy theorem :

@ 0.007 GeV (a4 s=1.5, h @1.2) 0

If B=0 (Forkel et al.) : < 0 and

In any case, @ 0.0004 GeV (4 0.045 GeV4) and (>0 in QCD)


Soft Wall model

Soft Wall model with with a a negative negative dilaton dilaton F F(z) = (z) = --c²z² c²z² (SW ) (SW )

-- (Brodsky et al., Fen Zuo, Nicotri)

• ccSB SB functionfunction :

2 IR regular indep. solutions : explicit & explicit & implicitimplicit ccSB descriptionsSB descriptions

• vectorvector mesonmeson mass mass spectrumspectrum unchanged under :

massless pion-like vector mode ! but the on-shell action divergent in IR


massless pion-like vector mode !

• scalarscalar sectorsector : decreasing w.r.t. SW unchanged

4d gluon condensate

6d condensates negativenegative(at odds with SW ) : isoscalar s(600) meson

as a qq meson or tetraquark?




( )




Dynamical AdS AdS/QCD /QCD

(Shock et al., Forkel et al., de Paula et al.)

gravity-dilaton-tachyon action (Gherhgetta et al.):

AdS spacetime,5 at the price of

family of metric :

dilaton profile and potential only in terms of warp factor A(z) dilaton profile and potential only in terms of warp factor A(z)

• ¥ 0 : conformal AdS dominates in UV limit5

• ¥ 1 : area law of Wilson loop

• =2 : linear Regge trajectories

 dilaton background field only

 warpedwarped AdS geometry in IR ,



AdS/QCD model /QCD model with with zz--dependent dependent 5d masses 5d masses

(Vega et al.)

running qq operator : anomalous dimension - D=3+d z-dep. mass of dual scalar field X

cSB function EOM : v(z) ñ m (z)X

a EOM : Regge-like mass spectrum O(z²)1 = constant (UV)constant (UV), linear, quadratic (IR)

explicit and implicit cSB descriptions ï 2 possibles values of for each W

W=0.1 W=0.5 W=2


Conclusion Conclusion

AdS/CFT provides a new way to address Physics at strong coupling

AdS/QCD : identify the main properties of the dual theory of QCD

 chiral dynamics of QCD : no explicit & implicit cSB descriptions in naive SW ( cSB function )

 scalar glueball and meson phenomenology (masses, decay constants, condensates) : surprisingly close pheno. results regarding the simplicity of the holographic models

Higher-dimensional gravity dual theory of QCD predictions at low energy ! SW not solution of gravity-dilaton action EOM : dilaton ï non-AdS geometry

• higher-dim. scalar potential terms

• z-dep. 5d scalar bulk mass (anomalous dim.) gluon condensates ( > 4d)

s(600) ?

running of quark mass & condensate (too drastic modifications of AdS/CFT to AdS/QCD?)

but at odds with QCD

If we impose


Backup Slides




cSB SB breaking breaking in in improved improved SW SW models models (I) (I)

(Gherghetta et al.)

AdS spacetime,5 quartic term in the scalar potential

cSB function EOM : v(z) ñ F(z)

: no-cS restoration

A,B,C(m ,q S, N ,c k, l)


: linear Regge traject.

: UV conformal : no-cS restoration

, , , ,



cSB SB breaking breaking in in improved improved SW SW models models (II) (II)

(Peng Zhang)

AdS spacetime5 cubic term : Regge traject. for both mesons and nucleons

cSB function EOM : v(z) ñ F(z)

: asymp. linear nucleon spectra

A,B,C(m ,q s, N ,l,g)c


: linear Regge traject.

: UV conformal

( - 7,2% error)

, , ,

• scalar meson potential O(z²) : linear Regge trajectory

• cubic coupling l < 0 : increasing mass spectrum of the scalar mesons

• but m(f0(118)) < m(p) at odds with QCD



Lattice QCD, QCD, theoretical theoretical calculations calculations and and phenomenological phenomenological models models

Indications of the behaviour of as


Wilson loop v.e.v. (Maldacena 1998) :

holographic description of pert. QCD (consistency)

• AdS/CFT :

coulomb-like conformal behaviour at all length scales non-perturbative : non-polynomial

Description of the running of the QCD

Description of the running of the QCD coupling coupling constant ? constant ?

Stronger versions of AdS/CFT : finite l perturbative expansion

(F.J. hep-ph/0812.4903)

linear confinement at large distances when explodes

• AdS/QCD:

at short-distances, we want i.e. QCD running coupling

QCD b-function / metric function ? (Kiritsis et al. )




The large N consistent

The large N consistent behaviour behaviour of the QCD Hard Wall model of the QCD Hard Wall model The

The holographic holographic implicit implicit c cSB SB mechanism mechanism

(F.J. hep-ph/0902.3864)

• normalizable modes : ~ Large-N behaviour :

• decay constants :


UV scalar correlator behaviour in the Soft Wall modelSoft Wall model :

• mass spectrum : ~


• b-to-b propagator : - timelike

• VPP coupling constant :

• form factors : ,

- spacelike





The holographic implicit cSB mechanism :

• cSB function :

pseudoscalar mode eq. of motion

Gell-Mann-Oakes-Renner relation



IR limit




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