Masayasu Harada (Nagoya Univ.)

based on (mainly) M.H. and K.Yamawaki, Phys. Rev. Lett. 86, 757 (2001) M.H. and C.Sasaki, Phys. Lett. B 537, 280 (2002) M.H. and K.Yamawaki, Phys. Rept. 381, 1 (2003) M.H. and C.Sasaki, Nucl. Phys. A 736, 300 (2004) M.H., Y.Kim, M.Rho and C.Sasaki, Nucl. Phys. A 730, 379 (2004) M.H., T.Fujimori and C.Sasaki, in preparation

at International Conference on QCD and Hadronic Physics (June 18, 2005, Beijing)

☆ In-medium modification of / mesons

CERES/CERN

KEK-PS E325

CB/TAPS@ELSA

☆ Dropping mass (Brown-Rho scaling) can explain

dropping massbased on Brown-Rho scaling

R.Rapp-J.Wambach, ANP 25,1 (2000)KEK-PS E325

CB/TAPS@ELSA

☆ Brown-Rho scaling impliesdropping mass ⇔ chiral symmetry restoration

☆ Vector Manifestation

longitudinal = Chiral partner of

near chiral restoration point

M.H. and K.Yamawaki, Phys. Rev. Lett. 86, 757 (2001)

Theoretical description of dropping mass ?

Dropping mass ・・・ necessary for the VM.

Outline

1. Introduction

2. Hidden Local Symmetry Theory

3. Vector Manifestation of Chiral Symmetry

4. Formulation of the Vector Manifestation

in Hot Matter

5. 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)M.H., T.Fujimori and C.Sasaki, in preparation

based on chiral symmetry of QCD

ρ ・・・ gauge boson of the HLS

◎ Hidden Local Symmetry Theory ・・・ EFT for and 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)

☆ 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 ξ2iπ/ F πL†

R

ξ = e e → h ξ g±iπ / Fπiσ / FσL,R L,R L,R

†ξ = e e → h ξ g±iπ / Fπ±iπ / Fπiσ / Fσiσ / FσL,R L,R L,R

†

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

h ∈ ［ SU(N ) ］f V local

g ∈ ［ SU(N ) ］fL,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

based on chiral symmetry of QCD

・・・ gauge boson of the HLS

◎ Hidden Local Symmetry Theory ・・・ EFT for and 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)

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

loop expansion ⇔ derivative expansion

◎ Chiral Perturbation Theory with HLS

☆ Expansion Parameter

◎ ordinary ChPT for

chiral symmetry breaking scale

◎ ChPT with HLS

☆ Validity of the expansion

?

?

?

・・・ justified in the large Nc QCD

This is true for any models !

This is NOT enough for a systematic expansion !!

◎ e.g., in Matter Field Method

may cause 1/m correctionsρ２

gauge invariance

・・・ well-defined limit of m → 0ρ

◎ In HLS with Rξ- like gauge fixing

?

・・・ guaranteed by the gauge invariance in the HLS

☆ Expansion Parameter in the ChPT with HLS

☆ Validity of the expansion

O.K. in the large Nc QCD

O.K. in the HLS

☆ Order Counting

・・・ same as ordinary ChPT

loop expansion = low-energy expansion

☆ Effect of scalar meson ?

◎ σ(600)

mσ= 560 MeV < mρ = 770 MeV (Γσ = 370 MeV)

see e.g., M.H., F.Sannino and J.Schechter, PRD 54, 1991 (1996)

・ 4-quark state

→ σ does not exist in the large Nc QCD

・ 2-quark state

→ mσ = ma0 = 980 MeV > mρ

in the large Nc QCD

σ is not needed in the large Nc QCD ?

◎ No need of scalar meson in large Nc QCD

M.H., F.Sannino, J.Schechter, PRD69, 034005 (2004)

Unitarity in scattering is satisfied without scalar meson up untill E 4≦ F for Nc 6≧

0.5

0

real part of S-wave amplitude Nc=3

0

Nc=6Nc=7

Nc=4

Nc=5

(F)2 ～ Nc

g2 ～ 1/Nc

a = 2 (fixed)

based on chiral symmetry of QCD

ρ ・・・ gauge boson of the HLS

◎ Chiral Perturbation Theory with HLSH.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

loop expansion ⇔ derivative expansion

◎ Hidden Local Symmetry Theory ・・・ EFT for and 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)

☆ many parameters ! ・・・ not determined by the chiral symmetry

more experimental data are availableshould be detemined from QCD

☆ Wilsonian matching between EFT and QCD

QCD quarks and gluons

EFT for hadrons

Λ

high energy

low energy

Bare theory

bare parameters

Quantum effects

Quantum theory

physical quantities

M.H. and K.Yamawaki, PRD 64, 014023 (2001)

matching

~ 1 GeV

(perturbative treatment)

Both (perturbative) QCD and EFT are applicable

integrateout

☆ A typical prediction of the Wilsonian Matching

・ bare parameters

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

, ...

good agreement !

+ quantum corrections improved by RGEs

+ + ・・・π

πρ γ

☆ Inclusion of the effect of current quark massesM.H., T.Fujimori and C.Sasaki, in preparation

bare parameters

ρ

π

ρ

K

+ quantum corrections improved by RGEs

+ + ・・・

+ + ・・・

very good agreement !

M.H. and K.Yamawaki, Phys. Rev. Lett. 86, 757 (2001)M.H. and K.Yamawaki, Phys. Rept. 381, 1 (2003)

Note : work in the chiral limit (mq = 0)

・・・ Wigner realization of chiral symmetry

longitudinalρ ＝ chiral partner of π

c.f. conventional linear-sigma model manifestation

scalar meson = chiral partner of π

M.H. and K.Yamawaki, Phys. Rev. Lett. 86, 757 (2001)

Quark Structure and Chiral representation

coupling to currents and densities

(S. Weinberg, 69’)longitudinal components

mρ → 0 is necessary ・・・ support BR scaling

Chiral Restoration

linear sigma modelvector manifestation

M.H. and C.Sasaki, Phys. Lett. B 537, 280 (2002)M.H. and C.Sasaki, Nucl. Phys. A 736, 300 (2004)

☆ View of the VM in Hot Matter

◎ Assumptions

・ Relevant d.o.f until near Tc-ε ・・・ only π and ρ

・ Other mesons (A1, σ, ...) ・・・ still heavy

・ Partial chiral restoration already at Tc-ε

☆ Application of the Wilsonian matching at T > 0

QCD quarks and gluons

Bare HLS for and

matchingΛ

high energy

low energy

integrate out quarks and gluons in hot matter

・・・ Bare parameters have temperature dependences.

Wilsonian matching condition at T = 0

Extension of WM condition to T > 0

◎ Intrinsic temperature dependencesignature of internal structure of hadrons(Hadrons are constructed from quarks and gluons.)

(perturbative treatment : OPE)

☆ Wilsonian matching at T → Tc -

• current correlators in the OPE

☆ Can we satisfy GV → GA in the HLS ?

◎ current correlators in the bare HLS

☆ Can we satisfy GV → GA for T → Tc in the HLS ?

☆ Yes !

◎ VM Conditions in hot matter for T → Tc

☆ ρ pole mass for T → Tc

bare theory

VM conditions

quantum effect through RGEs

fixed point of RGE

hadronic thermal effects

π

π

ρ

ρ

・・・

Vector Manifestation

→ 0

☆ Is m(T) → 0 related to the chiral symmetry restoration ?

◎ Wilsonian matching near Tc

add the quantum and hadronic thermal corrections

◎ Quantum theory

mρ → 0 ・・・ signal of the chiral symmetry restoration !

G.E.Brown and M.Rho, PRL 66, 2720 (1991)

◎ Hidden Local Symmetry Theory ・・・ EFT for and Systematic low-energy expansion including dynamical loop expansion ⇔ derivative expansion

◎ Wilsonian matching between the HLS and QCD

Matching of axial-vector and vector current correlators → Determination of the bare parameters

+ quantum corrections improved by Wilsonian RGEs

Physical predictions ・・・ very good agreement !

◎ Vector Manifestation in hot matter ・・・ mρ → 0 for T → Tc

⇒ mρ → 0 ・・・ signal of the chiral symmetry restoration !

◎ Predictions of the VM in hot matter・ Vector and axial-vector susceptibilities at Tc

•M.H., Y.Kim, M.Rho and C.Sasaki, Nucl. Phys. A 727, 437 (2003)

・ Large violation of vector dominance of electromagnetic form factor of pion at Tc

•M.H. and C.Sasaki, Nucl. Phys. A 736, 300 (2004)

・ Pion velocity near Tc

determined by the intrinsic thermal effects

M.H., Y.Kim, M.Rho and C.Sasaki, Nucl. Phys. A 730, 379 (2004)

for T → Tc

⇔ Prediction in the non-linear σ model v(Tc) → 0 for T → Tc D.T.Son and M.A.Stephanov, PRL88, 202302