Integrating out Holographic QCD Models to Hidden Local Symmetry

Masayasu Harada (Nagoya University)

Dense strange nuclei and compressed baryonic matter @ Yukawa Institute, Kyoto, Japan (April 21, 2011)

MH, S.Matsuzaki and K.Yamawaki, Phys. Rev. D 74, 076004 (2006) MH, S.Matsuzaki and K.Yamawaki, Phys. Rev. D82, 076010 (2010) MH and M. Rho, arXiv:1102.5489

ＱＣＤ (Strong Coupling Gauge Theory)

Hadron Phenomena

Q C D

Low Energy hadron Phenomena

Lattice ＱＣＤ Effective models

☆ Holographic QCD Models Effective models of QCD

・ Large Nc limit

QCD ⇒ weakly interacting theory of mesons

Baryons are given as solitons.

○ infinite number of mesons

・ large λ = Nc g2 (’t Hooft coupling) limit

Correspondence in real-life QCD ?

Contribution from infinite tower can be included

In holographic models

Example :

Short distance behavior of N-N potential

∝ 1/r2 (with infinite tower contribution summed up) Hashimoto-Sakai-Sugimoto, PTP122, 427 (2009)

☆ Predictions of hQCD models

○ momentum independent quantities

e.g., mass, coupling, …

It is not difficult to compare model predictions

with experiments.

○ Momentum dependent quantities

e.g., form factors, scattering cross sections, …

It seems difficult to compare model predictions

with experiments,

since it is difficult to add up contributions from

infinite number of mesons.

Integrating out heavy modes

☆ Proposal

hQCD models

HLS model Most general effective model

for p and r

Low Energy hadron Phenomena

This may give an interpretation of hQCD results

in terms of lowest lying mesons (r and p).

This may give a clue to understand

what we can learn from hQCD on real life QCD ?

Outline

1.Introduction

2.Hidden Local Symmetry

3.A Method for Integrating Out

4.Form Factors in Sakai-Sugimoto Model

5.Application to Nucleon Form Factors

6.Summary

M. Bando, T. Kugo, S. Uehara, K. Yamawaki and T. Yanagida, PRL 54 1215 (1985)

M. Bando, T. Kugo and K. Yamawaki, Phys. Rept. 164, 217 (1988)

M.H. and K.Yamawaki, Phys. Rept. 381, 1 (2003)

based on chiral symmetry of QCD

ρ ・・・ gauge boson of the HLS

◎ Hidden Local Symmetry Theory ・・・ EFT for r and p M. Bando, T. Kugo, S. Uehara, K. Yamawaki and T. Yanagida, PRL 54 1215 (1985)

M. Bando, T. Kugo and K. Yamawaki, Phys. Rept. 164, 217 (1988)

M.H. and K.Yamawaki, Physics Reports 381, 1 (2003

☆ Chiral Lagrangian

Non-Linear Realization of Chiral Symmetry

SU(N ) ×SU(N ) → SU(N ) f f f L R V

◎ Basic Quantity

U = e → g U g R 2iπ T /F a

a π

L

† ; g ∈ SU(N ) L,R f L,R

◎ Lagrangian

L = tr［∇ U ∇ U ］ F π

2

4 μ

μ †

∇ U ≡∂ U － i L U + i U R μ μ μ μ

L , R ; gauge fields of SU(N ) μ μ f L,R

☆ Hidden Local Symmetry

［SU(N ) ×SU(N ) ］ ×［SU(N ) ］ → ［SU(N ) ］f f fL R Vglobal local Vf global

［SU(N ) ×SU(N ) ］ ×［SU(N ) ］ → ［SU(N ) ］f f fL R Vglobal local Vf global

U = e = ξ ξ 2iπ/ F π L

†

R

ξ = e e → h ξ g±iπ / Fπiσ / Fσ

L,R L,R L,R†

ξ = e e → h ξ g±iπ / Fπ±iπ / Fπiσ / Fσiσ / Fσ

L,R L,R L,R†

F , F ・・・ Decay constants of π and σ π σ

h ∈ ［SU(N ) ］ f V local

g ∈ ［SU(N ) ］ f L,R L,R global

・ Particles

ρμ = ρμa T a ・・・ HLS gauge boson

π=πaTa ・・・ NG boson of ［SU(Nf)L×SU(Nf)R］global symmetry breaking

σ=σaTa ・・・ NG boson of ［SU(Nf)V］local symmetry breaking

◎ 3 parameters at the leading order

Fp ・・・ pion decay constant

g ・・・ gauge coupling of the HLS

a = (Fs/Fp)2 ⇔ validity of the r meson dominance

m = ag F π ρ 2 2 2

• Maurer-Cartan 1-forms

• Lagrangian

based on chiral symmetry of QCD

ρ ・・・ gauge boson of the HLS

◎ Chiral Perturbation Theory with HLS H.Georgi, PRL 63, 1917 (1989); NPB 331, 311 (1990):

M.H. and K.Yamawaki, PLB297, 151 (1992)

M.Tanabashi, PLB 316, 534 (1993):

M.H. and K.Yamawaki, Physics Reports 381, 1 (2003)

Systematic low-energy expansion including dynamical r

loop expansion ⇔ derivative expansion

◎ Hidden Local Symmetry Theory ・・・ EFT for r and p M. Bando, T. Kugo, S. Uehara, K. Yamawaki and T. Yanagida, PRL 54 1215 (1985)

M. Bando, T. Kugo and K. Yamawaki, Phys. Rept. 164, 217 (1988)

Leading order Lagrangian is counted as O(p2),

Next order terms are of O(p4).

◎ Typical examplesof O(p4) terms

M.H., S.Matsuzaki and K.Yamawaki, Phys. Rev. D 74, 076004 (2006) M.H., S.Matsuzaki and K.Yamawaki, Phys. Rev. D 82, 076010 (2010)

☆ hQCD models which include 5-dimensional

gauge field at intermediate step

In the following, I will use Sakai-Sugimoto model

Kinetic term in 4-Dim. Mass term in 4-Dim

5-D gauge transformation

Residual gauge symmetry = Hidden Local Symmetry

Gauge fixing :

◎ Mode expansion of the gauge field

Eigenvalue equation

Axial-vector mesons Vector mesons

SS model

input

・・・

・・・

Pions as NG bosons

☆ m(r) ～ E ≪ m(a1)

・ Integrate out vector and axial-vector mesons

other than r meson.

Solve the equations of motions

with kinetic terms neglected.

HLS with a particular choice of parameters

Note that z1～8 are all determined by the model,

which include effects of heavy mesons.

≠ truncation of heavy mesons

M.H., S.Matsuzaki and K.Yamawaki, Phys. Rev. D 82, 076010 (2010)

p EM form factor

•In Sakai-Sugimoto model infinite tower of r mesons contributes.

k=1 : r meson k=2 : r’ meson k=3 : r” meson …

= 1.31 + (-0.35) + (0.05) + (-0.01) + …

r’ r r’’ r’’’

・ r meson dominance ⇒ ;

• In the Hidden Local Symmetry p EM form factor is parameterized as

・ Reduction of Sakai-Sugimoto model ⇒

◎ Alternative way to relate hQCD to HLS k=1 : r meson k=2 : r’ meson k=3 : r” meson …

mr ～ Q2 ≪ mr’

Sum Rules

p EM form factor

rmeson dominance c2/dof = 226/53=4.3

;

SS model : c2/dof = 147/53=2.8 best fit in the HLS : c2/dof=81/51=1.6

Exp data : NA7], NPB277, 168 (1996) J-lab F(pi), PRL86, 1713(2001) J-lab F(pi), PRC75, 055205 (2007) J-lab F(pi)-2, PRL97, 192001 (2006)

Infinite tower works well as the r meson dominance !

MH, S.Matsuzaki, K.Yamawaki, PRD82, 076010 (2010) cf : MH, K.Yamawaki, Phys.Rept 381, 1 (2003)

wp transition form factor MH, S.Matsuzaki, K.Yamawaki, arXiv:1007.4715 cf : MH, K.Yamawaki, Phys.Rept 381, 1 (2003)

• best fit in the HLS c2/dof=24/30=0.8

• Sakai-Sugimoto model : c2/dof=45/31=1.5

・r meson dominance c2/dof=124/31=4.0

Violation of r/w meson dominance may indicate existence of the contributions from the higher resonances.

p g g* form factor

g

g*

g

g*

mr = mw is used.

SS model : c2/dof = 63/5 = 13

rmeson dominance c2/dof = 4.8/5=1.0

best fit in the HLS c2/dof=3.0/4 = 0.7

w → p+p-p0 decay

A B

A/B ≪ 1 ⇒ r meson dominance is well satisfied in SS model

MH and M. Rho,. arXiv:1010.1971 [hep-ph]

☆ Hong-Rho-Yee-Yi hQCD Model

5-D effective model

including 5-D baryon field + 5-D gauge field

⇒ 4-D effective model with baryon (nucleon)

and an infinite tower of vector and axial-vector mesons

Nucleon form factos

PRD76, 061901 (2007)

JHEP 0709, 063 (2007)

Example 3: Proton EM form factor M.H. and M.Rho, arXiv:1102.5489 [hep-ph]

rmeson dominance : c2/dof=187

• best fit in the HLS : c2/dof=1.5

a = 4.55 ; z = 0.55

Violation of the r meson dominance (well-known) can explained by the existence of the infinite tower

•Hong-Rho-Yee-Yi model : c2/dof=20.2

a = 3.01 ; z = -0.042

◎ We relate holographic QCD models to the HLS model by integrating out heavier mesons in hQCD models. ・ Showed that the infinite tower of vector mesons can contribute even to pion EM form factor → can fit the data well as the r meson dominance ・Violation of r meson dominance in wp transition form factor can be explained by the existence of infinite tower ・Violation of r/w meson dominance in the proton form factor is well explained by the existence of infinite tower

6. Summary