Baryons with Holography

Hideo SUGANUMA ( Kyoto Univ. )Toru KOJO ( Kyoto Univ. )

Kanabu NAWA ( RCNP )in collaboration with

According to the concept of ‘holography’ ,non-perturbative aspects of QCD (meson dynamics)can be analyzed by a tree-level superstring theory.

Sakai-Sugimoto model : Holograhic QCD[T.Sakai and S.Sugimoto, Prog. Theor. Phys. 113, 843 (2005)]

INTRODUCTION

Meson spectra of (axial) vector mesons.Hidden local symmetry.Vector meson dominance.KSRF relation.GSW model.

[Bando, et al.,1985]

[Sakurai, 1969]

[Kawarabayashi, Suzuki, Riazuddin, and Fayyazuddin, 1966]

[GellMann, Sharp, and Wegner, 1962]

Many phenomenological aspects of mesons are uniquely derived, starting from one superstring model.

INTRODUCTION

Holographic description is reliable for large- limit.

Large- QCD becomes equivalent with the weak coupling system of mesons (and glueballs); baryon does not directly appear as a dynamical degrees of freedom.

How to describe a baryon in the large- holographic model ?

*Chiral soliton picture .Baryon as a topological soliton in 4dim. meson effective actioninduced by Holographic QCD (Brane-induced Skyrmion).

[‘tHooft, Nucl. Phys. B72, 461(1974); B75, 461(1974)]

According to the concept of ‘holography’ ,non-perturbative aspects of QCD (meson dynamics)can be analyzed by a tree-level superstring theory.

Sakai-Sugimoto model : Holograhic QCD[T.Sakai and S.Sugimoto, Prog. Theor. Phys. 113, 843 (2005)]

Holography

Holographic QCD (Sakai-Sugimoto model)

Baryons in holographic QCD***

CONTENTS

Gauge theory on D-branes

Superstring theory has 10 dim. space-time to avoid ‘anomaly’.

D-brane is introduced as fixed edges of open strings.(‘D’ comes from ‘Dirichlet boundary condition’)

Dp-brane = (p+1) dim. membrane

open string

‘parallel ’

‘perpendicular ’ ; scalar (NG) field.

; gauge field.

(p+1) dim. gauge theory appears‘on’ Dp-brane !!!

10dim.

Holography

10dim.

Dp-brane×Nc

1 space dim. with finite curvature.

(p+1) dim. gauge theory

([p+ ]+1) dim. super gravity1

; low energy; weak coupling

; strong couplingStrong-weak duality

(S-duality)

D-brane = black hole (black brane)[Polichinsky (1995)]

[Maldacena (1997)]

Holographic QCD

Construction of massless QCD (quarks and gluons)

on D4/D8/D8-branes.

[T.Sakai and S.Sugimoto, Prog. Theor. Phys. 113, 843 (2005)]

D4-brane×NC

D8-brane×Nf , L

4-4

4-8

4-8D8-brane×Nf , R

Index on D4 brane ( color )Index on D8 brane ( flavor )

: quark (L)

: quark (R)

: gluon

10dim.

Probe approximation with

Mass of Dp-brane is proportional to its sheets number.

D4×NcD8, D8×Nf ‘‘probe’’

SUGRA background

light probe

heavy object

gravitational background

m Nf∝

D4-brane×NC

D8×Nf

D8×Nf

10dim. space time

D8

light probe

4-4

4-8

4-8

8-8

M NC∝

[‘tHooft, Nucl. Phys. B72, 461(1974); B75, 461(1974)]

D4-brane×NC

D8×Nf

D8×Nf

4-4

4-8

4-8

Chiral symmetry breaking and color confinement

10dim. space time

； Chiral symmetry breaking .

Colored particles are located behind horizonand become invisible from outside.

quark (L)

quark (R)

gluon

8-8

D8

horizon

L

R； Color confinement .

Same critical temperature (Tc=170MeV)in lattice QCD.

Mesons appear from 8-8 strings on the residual D8-branes,without explicit baryons ; property of large-Nc QCD.

~

DBI action of probe D8-brane with D4 SUGRA background

： surface tension

9 　 4 = 5

5dim. Yang-Mills theory

D8-brane is 9dim. membrane.

： dilaton fieldD4 SUGRA backgroud

Integration out symmetric 4dims.

Unified theory of mesons with pion and infinite number of (axial) vector mesons !!!

00

1

Mode expansion

: vector

: vector

: axial-vector

mass :parity :

Z Z

pion (axial) vector mesons

: axial-vector

complete basis :

5dim. gauge field

Pion mainly coupleswith lightest vector mesons (ρ)

Vector and axial-vector mesonsalternately appear in the excitation spectra.

(pions and ρ-mesons)

4dim. meson effective action from holographic QCD

,

,

pion field :

1-form of pion field :

axial vector current :

vector current :

pion decay constant :

ρ-meson mass :

Holographic QCD hasjust two parameters.

All the couplings are uniquely determined !!!

[K. Nawa et al., Phys. Rev. D75, 086003 (2007)]

(pions and ρ-mesons)

4dim. meson effective action from holographic QCD

,

,

pion field :

1-form of pion field :

axial vector current :

vector current :

Traditional phenomenologiescan be discussed in the action.

Hidden local symmetry.

Vector meson dominance.

KSRF relation.

[Bando, et al.,1985]

[Sakurai, 1969]

[Kawarabayashi et al., 1966]

[K. Nawa et al., Phys. Rev. D75, 086003 (2007)]

hidden local symmetry* [M. Bando, T. Kugo, and K. Yamawaki, Prog. Theor. Phys. 73, 1541 (1985); Phys. Rep. 164, 217 (1988)]

Comparison with phenomenology

Hidden local symmetric terms :

Couplings remain as parameters. All the couplings are uniquely determined.

Holographic QCD is not just the low-energy effective theoryonly with the constraint of symmetries in 4 dim.

chiral perturbation theory*：

： ,,

[I. Zahed and G.E. Brown, Phys. Rep. 142, 1 (1986)]

2-derivative term

4-derivative term

Instability of Skyrme soliton.5dim. YM theoryfrom holographic QCD:

Topological picturefor baryon is also supported bysuperstring theory.

L2

L4

,,

Baryons in Holographic QCD

Large- QCD becomes equivalent with the weak coupling system of mesons (and glueballs); baryon does not directly appear as a dynamical degrees of freedom. [‘tHooft, Nucl. Phys. B72, 461(1974); B75, 461(1974)]

How to describe a baryon in large-Nc holographic model ?

Skyrme term naturally appears ;(topological picture for baryon is also supported by superstring theory).*

Baryon as a chiral soliton[K. Nawa, H. Suganuma, and T. Kojo, Phys. Rev. D75, 086003 (2007)]

Baryon as Chiral Soliton

[T.H.R. Skyrme, Proc. R. Soc. A260, 127 (1961)]

π ;ρ ;

Pauli matrix

Hedgehog soliton with B=1

Shrinkage

Pions are slightly replaced by ρ-mesons through interactionsin holographic QCD. Shrinkage of pion amplitude F(r) .

Stable hedgehog soliton solution exists !!!**

(pion) (ρ-meson)

[K. Nawa, H. Suganuma, and T. Kojo, Phys. Rev. D75, 086003 (2007)]

-meson components in the core of a Baryon (Skyrmion)

(Comparison of total enegy density and each energy density in -meson channels)

-mesons are actively affect the total mass of a baryonin the deeper interior of itby using various interaction channels.

Mass and radius of baryon

Holographic QCD has just two parameters ;

; ultraviolet cut-off scale.

; large-Nc effective theory.

Experimental inputs for pion decay constant and ρ-meson mass ;

Mass : ,

MeV MeV

Exp. ：

Radius :

Isospin rotation as semi-classical quantization.

Dynamical pion cloud.

Increase of total size !!!

(for charge radius)

Summary and outlook‘Holography’ gives the duality between (p+1) dim. gauge theory without gravity and ([p+1]+1) dim. supergravity.

S-duality gives powerful analysis for each side from the other side.S-duality makes difficult the exact proof of gauge/gravity duality.

Non-perturbative aspects of QCD :glueball mass :

baryon properties :

meson properties :

Non-perturbative aspects of gravity :critical exponent in blackhole formation :Hawking temperature of blackhole :

*

*

[E. Witten, Adv. Theor. Math. Phys. 2 (1998), 505]

[T.Sakai and S.Sugimoto, Prog. Theor. Phys. 113, 843 ; 114, 1083 (2005)]

[K. Nawa, H. Suganuma, and T. Kojo, Phys. Rev. D75, 086003 (2007)]

[L. Alvarez-Gaume et al., PLB 649, 478 (2007)]

[D.K.Hong et al. (M. Rho group), Phys. Rev. D76 : 061901 (2007)][H. Hata et al.(Particle Theory Group, Kyoto Univ.), hep-th/0701280]

[S. Mukherji et al., JHEP 0205 (2002) 026][E. Witten, Adv. Theor. Math. Phys. 2, 505 (1998)]