1. 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)

3. ☆ In-medium modification of  /  mesons CERES/CERN KEK-PS E325 CB/TAPS@ELSA

4. ☆ 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 CB/TAPS@ELSA

5. ☆ Brown-Rho scaling implies dropping  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.

6. 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

7. 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

8. 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)

9. ☆ 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

10. 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

11. ☆ Expansion Parameter ◎ ordinary ChPT for  chiral symmetry breaking scale ◎ ChPT with HLS ☆ Validity of the expansion ? ?

12. ? ・・・ justified in the large N c QCD This is true for any models ! This is NOT enough for a systematic expansion !!

13. ◎ e.g., in Matter Field Method may cause 1/m corrections ρ ２ gauge invariance ・・・ well-defined limit of m → 0 ρ ◎ In HLS with R ξ - like gauge fixing ? ・・・ guaranteed by the gauge invariance in the HLS

14. ☆ Expansion Parameter in the ChPT with HLS ☆ Validity of the expansion O.K. in the large N c QCD O.K. in the HLS ☆ Order Counting ・・・ same as ordinary ChPT loop expansion = low-energy expansion

15. ☆ Effect of scalar meson ? ◎ σ(600) m σ = 560 MeV < m ρ = 770 MeV (Γ σ = 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 N c QCD ・ 2-quark state → m σ = m a0 = 980 MeV > m ρ in the large N c QCD σ is not needed in the large N c QCD ?

16. ◎ No need of scalar meson in large N c QCD M.H., F.Sannino, J.Schechter, PRD69, 034005 (2004) Unitarity in  scattering is satisfied without scalar meson up untill E ≦ 4  F  for N c ≧ 6 0.5 0 real part of S-wave amplitude Nc=3 0 Nc=6Nc=7 Nc=4 Nc=5 (F  ) 2 ～ N c g 2 ～ 1/N c a = 2 (fixed)

17. 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  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 available should be detemined from QCD

18. ☆ 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 integrate out

19. ☆ 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 π ρ γ

20. ☆ Inclusion of the effect of current quark masses M.H., T.Fujimori and C.Sasaki, in preparation bare parameters ρ π ρ K + quantum corrections improved by RGEs + + ・・・ + very good agreement !

21. 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 (m q = 0)

22. ・・・ 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)

23. Quark Structure and Chiral representation coupling to currents and densities (S. Weinberg, 69’) longitudinal components

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

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

26. ☆ View of the VM in Hot Matter ◎ Assumptions ・ Relevant d.o.f until near T c -ε ・・・ only π and ρ ・ Other mesons (A 1, σ,...) ・・・ still heavy ・ Partial chiral restoration already at T c -ε

27. ☆ Application of the Wilsonian matching at T > 0 QCD quarks and gluons Bare HLS for  and  matching Λ 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 dependence signature of internal structure of hadrons (Hadrons are constructed from quarks and gluons.) (perturbative treatment : OPE)

28. ☆ Wilsonian matching at T → T c -  current correlators in the OPE ☆ Can we satisfy G V → G A in the HLS ?

29. ◎ current correlators in the bare HLS ☆ Can we satisfy G V → G A for T → T c in the HLS ? ☆ Yes ! ◎ VM Conditions in hot matter for T → T c

30. ☆ ρ pole mass for T → T c bare theory VM conditions quantum effect through RGEs fixed point of RGE hadronic thermal effects π π ρ ρ ・・・ Vector Manifestation → 0

31. ☆ Is m  (T) → 0 related to the chiral symmetry restoration ? ◎ Wilsonian matching near T c add the quantum and hadronic thermal corrections ◎ Quantum theory m ρ → 0 ・・・ signal of the chiral symmetry restoration ! G.E.Brown and M.Rho, PRL 66, 2720 (1991)

32. ◎ 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 ・・・ m ρ → 0 for T → T c ⇒ m ρ → 0 ・・・ signal of the chiral symmetry restoration !

33. ◎ Predictions of the VM in hot matter ・ Vector and axial-vector susceptibilities at T c 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 T c M.H. and C.Sasaki, Nucl. Phys. A 736, 300 (2004) ・ Pion velocity near T c determined by the intrinsic thermal effects M.H., Y.Kim, M.Rho and C.Sasaki, Nucl. Phys. A 730, 379 (2004) for T → T c ⇔ Prediction in the non-linear σ model v  (T c ) → 0 for T → T c D.T.Son and M.A.Stephanov, PRL88, 202302