M.H. and K.Yamawaki, Phys. Rev. Lett. 86, 757 (2001) M.H. and K.Yamawaki, Phys. Rev. Lett. 87, (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 model vector manifestation

☆ Formulation of vector manifestation ◎ Chiral symmetry restoration is characterized by ◎ Π = Π is satisfied in OPE AV How do we realize Π = Π in hadronic picture ? AV

☆ 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 HLS π

◎ VM conditions + Wilsonian RGEs ☆ Vector Manifestation

(Increase number of light flavors) (N = 3 in the real world... u, d, s) f N f N f cr chiral broken phase symmetric phase 33/2 non-asymptotic free N = 3 f application to clue to ordinary QCD in

☆ Running coupling in QCD N flavors f two-loop β function ・・・ ; E α small N f large N f α * ◎ b > 0 and c < 0 → α* (IR fixed point) α* → small for N → large f chiral restoration for N → N ff cr

☆ F (0) → 0 occurs in HLS for large N ? πf ◎ VM conditions for N → N f f cr ・・・ small N dependence f ; chiral restoration !

☆ Vector Manifestation occurs for N → N ff cr 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 f cr ;

☆ Simple anzatz for parameters in OPE ・・・ RGE invariant

☆ Assumptions for HLS parameters Wilsonian matching condition ;

☆ N dependence of F (0) and m fπρ

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 N f = 3 QCD characterized by ・ In N = 3 QCD ～ real world f

☆ Vector dominance in large flavor QCD VD is characterized by Large violation of VD near restoration point