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

• M.H. and K.Yamawaki, Phys. Rev. Lett. 87, 152001 (2001)

・・・ Wigner realization of chiral symmetry

ρ ＝ chiral partner of π

c.f. conventional linear-sigma model manifestation

scalar meson = chiral partner of π

Quark Structure and Chiral representation

coupling to currents and densities

(S. Weinberg, 69’)

Chiral Restoration

linear sigma modelvector manifestation

☆ Formulation of vector manifestation

◎ Chiral symmetry restoration is characterized by

◎ Π = Π is satisfied in OPEA V

How do we realize Π = Π in hadronic picture ?A V

☆ Basic assumptions

• When we approach to the critical point

from the broken phase,

Π ・・・ dominated by the massless π

Π ・・・ dominated by the massive ρ

A

V

• There exists a scale Λ around which

Π and Π are well described by the bare HLS. A V

• The HLS can be matched with QCD around Λ.

◎ current correlators in the bare HLS

◎ Wilsonian matching

◎ VM conditions

Note : F (0) → 0 can occur by the dynamics of the HLSπ

◎ VM conditions +

Wilsonian RGEs

☆ Vector Manifestation

(Increase number of light flavors)

(N = 3 in the real world ... u, d, s)f

Nf

Nfcrchiral broken phase symmetric phase 33/2 non-asymptotic free

N = 3f

application to

clue to ordinary QCD

in

☆ Running coupling in QCD N flavorsf

• two-loop β function ・・・

;

E

α

small N f

large Nf

α*

◎ b > 0 and c < 0 → α* (IR fixed point)

α* → small for N → largef

chiral restoration for N → Nf fcr

☆ F (0) → 0 occurs in HLS for large N ?π f

◎ VM conditions for N → Nf fcr

・・・ small N dependencef

; chiral restoration !

☆ Vector Manifestation occurs for N → Nf fcr

ρ = Chiral Partner of π

☆ VM and fixed point

◎ VM limit

(X, a, g) → (1, 1, 0) at restoration point

fixed point of RGEs

• VM is governed by fixed point

☆ Estimation of N fcr

;

☆ Simple anzatz for parameters in OPE

・・・ RGE invariant

☆ Assumptions for HLS parameters

• Wilsonian matching condition

;

☆ N dependence of F (0) and mf π ρ

ρ meson couplings become small

KSRF I ⇔ low energy theorem of HLS KSRF II

• Low energy theorem is satisfied by the on-shell quantities.

• KSRF II is violated.

☆ Vector dominance in Nf = 3 QCD

characterized by

・ In N = 3 QCD ～ real worldf

☆ Vector dominance in large flavor QCD

• VD is characterized by

Large violation of VD near restoration point