7. Vector Manifestation

1. 7. Vector Manifestation • M.H. and K.Yamawaki, Phys. Rev. Lett. 86, 757 (2001) • M.H. and K.Yamawaki, Phys. Rev. Lett. 87, 152001 (2001)

2. 7.1. Vector Manifestation of Chiral Symmetry Restoration

3. ☆ Vector Manifestation ・・・ Wigner realization of chiral symmetry ρ ＝ chiral partner of π c.f. conventional linear-sigma model manifestation scalar meson = chiral partner of π

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

5. Chiral Restoration vector manifestation linear sigma model

6. 7.2. Formulation of Vector Manifestation

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

8. When we approach to the critical point • from the broken phase, • Π・・・ dominated by the massless π • Π・・・ dominated by the massive ρ A • There exists a scale Λ around which • Π and Π are well described by the bare HLS. V A V ☆ Basic assumptions • The HLS can be matched with QCD around Λ.

9. ◎ current correlators in the bare HLS Note : F (0) → 0 can occur by the dynamics of the HLS π ◎ Wilsonian matching ◎ VM conditions

10. ◎ VM conditions + Wilsonian RGEs ☆ Vector Manifestation

11. 7.3. Chiral Phase Transition in large flavor QCD

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

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

14. 7.4. Vector Manifestation in large flavor QCD

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

16. cr ☆ Vector Manifestation occurs for N → N f f ρ = Chiral Partner of π

17. ☆ 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

18. cr ☆ Estimation of N f ;

19. 7.5. Nf-Dependence of the Parameters

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

21. ☆ Assumptions for HLS parameters ; • Wilsonian matching condition

23. ☆ N dependence of F (0) and m f π ρ

24. ρ meson couplings become small

25. KSRF II KSRF I ⇔ low energy theorem of HLS • Low energy theorem is satisfied by the on-shell quantities. • KSRF II is violated.

26. 7.6. Vector Dominance in Large Nf QCD

27. ☆ Vector dominance in Nf = 3 QCD ・ In N = 3 QCD ～ real world f characterized by

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