Masayasu Harada (Nagoya Univ.) based on M.H. and C.Sasaki, Phys.Rev.D74:114006,2006 at Chiral 07 (Osaka, November 14, 2007) see also M.H. and K.Yamawaki, Phys. Rept. 381, 1 (2003) M.H. and C.Sasaki, Phys. Lett. B 537, 280 (2002) M.H., Y. Kim and M. Rho, Phys. Rev. D 66, (2002). M.H. and C.Sasaki, Nucl. Phys. A 736, 300 (2004)

Hadron phase Color-Superconducting phase T Quark-Gluon-Plasma phase ☆ ＱＣＤ in hot and dense matter μBμB 1. Introduction

☆ Melting of quark – anti-quark condensate 〈 q q 〉 Is there a signal ?

☆ Vector Manifestation longitudinal  = Chiral partner of  near chiral restoration point M.H. and K.Yamawaki, Phys. Rev. Lett. 86, 757 (2001) Dropping  mass ・・・ signal of the chiral restoration based on the VM. ☆ Brown-Rho scaling dropping  mass ⇔ chiral symmetry restoration G.E.Brown and M.Rho, Phys. Rev. Lett (1991) Theoretical description of dropping  mass. M.H. and C.Sasaki, Phys. Lett. B 537, 280 (2002) M.H., Y. Kim and M. Rho, Phys. Rev. D 66, (2002).

☆ Dropping  mass (Brown-Rho scaling) can explain dropping  mass based on Brown-Rho scaling R.Rapp-J.Wambach, ANP 25,1 (2000) KEK-PS E325

☆ These analyses seem to assume the vector dominance (VD). G. E. Brown and M. Rho, arXiv:nucl-th/ ; arXiv:nucl-th/ ☆ Strong violation of the VD ・・・ Prediction of the VM gives a substancial suppression ! Effect from the violation of the VD to the rate ? ☆ Recent experiments exclude dropping ρ ? NA60 Nucl.Phys.A774: ,2006. CERES : Talk given by P. Braun-Munzinger at KIAS-APCTP Workshop "Relativistic Heavy-Ion Collison : Present and Future" Heavy Ion Meeting (HIM ). dropping ρ??

Outline 1. Introduction 2. Hidden Local Symmetry and the Vector Dominance 3. Thermal Dilepton Spectra in the Vector Manifestation 4. Summary

2. Hidden Local Symmetry and the Vector Dominance M. Bando, T. Kugo, S. Uehara, K. Yamawaki and T. Yanagida, PRL (1985) M. Bando, T. Kugo and K. Yamawaki, Phys. Rept. 164, 217 (1988) 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  ◎ Hidden Local Symmetry ・・・ EFT for  and  based on chiral symmetry of QCD  = gauge boson of the HLS massive through the Higgs mechanism loop expansion ⇔ derivative expansion

☆ Hidden Local Symmetry U = e = ξ ξ 2 i π/ F π L † R F, F ・・・ Decay constants of π and σ πσ h ∈ ［ SU(N ) ］ fV local g ∈ ［ SU(N ) ］ f L,R global ・ Particles ρ μ = ρ μ a T a ・・・ HLS gauge boson π=π a T a ・・・ NG boson of ［ SU(N f ) L ×SU(N f ) R ］ global symmetry breaking σ=σ a T a ・・・ NG boson of ［ SU(N f ) V ］ local symmetry breaking ◎ 3 parameters at the leading order F  ・・・ pion decay constant g ・・・ gauge coupling of the HLS a = (F  /F  ) 2 ⇔ validity of the vector dominance m = a g F π ρ 22 2

e+e+ e-e- ☆ Vector dominance (  dominance) at T = 0 a = 2 ⇒ vector dominance a /2 1 – a /2 long standing problem not clearly explained in QCD ! ◎ HLS analysis [M.H. and K.Yamawaki, Phys. Rept. 381, 1 (2003)] ・ a = 4/3 in the large Nc limit cf: AdS/QCD anlysis by Sakai-Sugimoto, PTP143,843 (2005) ・ a = 2 including 1/Nc corrections see also AdS/QCD analysis by M.H., M.Matsuzaki and K.Yamawaki, PRD74, (2006).  dominance is accidental only for Nc = 3 (and T = 0)

☆  dominance at T > 0 ? e+e+ e-e- ◎ a = 2 kept fixed in several analyses (No T-dependence on a) a /2 1 – a /2 ◎ Parameters of hadronic Lagrangians depend on T. ・・・ Intrinsic temperature dependence signature of internal structure of hadrons (Hadrons are constructed from quarks and gluons.) ・ VM predicts a (T) → 1 when m  (T) → 0 for T → Tc Strong violation of  dominance in the VM Strong suppression of  contribution to the dilepton spectrum 0 → 1 1 → 1/2

☆ Intrinsic temperature dependence of parameters ・・・ obtained by integrating out heavier hadrons ・ Effects of heavy hadrons are negligible ? ・・・ Not True near the critical temperature e.g., Hagedon temperature based on string model large Nc QCD each contribution from hadrons is suppressed by 1/Nc phase transition is driven by infinite number of hadrons ・ Infinite number of hadrons contribute near Tc Integrating out infinite number of hadrons near Tc → a large T dependence of the parameters for effective models for light hadrons (e.g., π and ρ in the HLS) in real-life QCD

・・・ 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

m ρ → 0 is necessary ・・・ support BR scaling Chiral Restoration linear sigma model vector manifestation

◎ Intrinsic T dependence ・・・ basic ingredient for the Vector Manifestation (VM) ◎ VM predicts ; dropping  ; strong violation of the vector dominance a ☆ T-dependences of physical parameters ・・・ intrinsic T dependence + hadronic temperature effects from thermal π and ρ intrinsic T dependence for T > Tf = 0.7 Tc Tf/TcTf/Tc ρ massm ρ → 0 Tf/TcTf/Tc ρ width Γ ρ → 0

◎ Vector dominance ? direct γππ coupling ： 1 – a /2 Tf/TcTf/Tc VD is good strong violation of the VD ・ Strong violation of the VD occurs near T c due to the intrinsic effect.

M.H. and C.Sasaki, Phys.Rev.D74:114006,2006

☆ Effect of violation of the vector dominance VM (for T → Tc) a(T) → 1 when m  (T) → 0 VM with VD a(T) = 2 kept fixed when m  (T) → 0 T = 0.4 T c No much difference ! v.s.

◎ Near Tc VM VM with VD vacuum ρ T = 0.75 T c VM < vacuum ρ< VM with VD !! T = 0.8 T c vacuum ρ< VM < VM with VD T = 0.85 T c vacuum ρ ≪ VM ≪ VM with VD !! Signal of the VM Violation of VD is very important

◎ Hidden Local Symmetry Theory ・・・ EFT for  and  Systematic chiral perturbation including dynamical  ◎ Vector Manifestation in hot matter ・・・ m ρ → 0 for T → T c ⇒ m ρ → 0 ・・・ signal of the chiral symmetry restoration ! ・ strong violation of the VD ・・・ important for the dilepton rate ◎ Vector dominance in the HLS ・ a = 4/3 in the large Nc limit ・ a = 2 including 1/Nc corrections ◎ future direction ・ Effects of collisional broadening including A1, … ・・・ work in progress (M.H., C.Sasaki and W.Weise)