1. Naoki Yamamoto (University of Tokyo) 高密度 QCD における カイラル対称性 contents Introduction: color superconductivity The role of U(1) A anomaly and chiral symmetry breaking Partition function zeros and chiral symmetry breaking Summary & Outlook (1) T. Hatsuda, M. Tachibana, N.Y. and G. Baym, Phys. Rev. Lett. 97 (2006) 122001. (2) N.Y., JHEP 0812 (2008) 060. (3) N.Y. and T. Kanazawa, Phys. Rev. Lett. 103 (2009) 032001. KEK 理論センター研究会「原子核・ハドロン物理」 2009.8.11.

2. QCD phase diagram T  Quark-Gluon Plasma Hadrons RHIC/LHC CFL Color superconductivity quark matter Neutron star

3. Color Superconductivity  QCD at high density → asymptotic free Fermi surface  Attractive channel → Cooper instability [3] C ×[3] C =[6] C +[3] C E p μ q q 3 “diquark condensate” “Fermi sea” “Dirac sea”

4. Color-Flavor Locking (CFL) u d s r,g,br,g,bu,d,s  Pairing channel s-wave pairing, spin singlet → Dirac antisymmetric Attractive channel → color antisymmetric Pauli principle → flavor antisymmetric U(1) A anomaly → Lorentz scalar  3-flavor limit: Color-Flavor Locking (CFL) Alford-Rajagopal-Wilczek (NPB1999)  Gauge-invariant order parameter e.g.)  Symmetry breaking pattern:

5. CFL is positive parity ... due to the presence of U(1) A anomaly.  Consider the Kobayashi-Maskawa-’t Hooft (KMT) vertex with quark mass:  V KMT is minimized when and the positive parity state is energetically favored. Alford-Rajagopal-Wilczek (NPB1999) Kobayashi-Maskawa (PTP1970); ‘t Hooft (PRD1976) G G T. Schafer (PRD2002)

6. Chiral symmetry breaking in CFL  The chiral condensate:  Exactly calculated thanks to the screening of instantons at high μ: [Point] 1.Chiral symmetry is broken not only by the diquark condensate but also the chiral condensate in CFL. 2.Nonzero chiral condensate in CFL is model-independent. 3.Chiral-super interplay of the type is inevitable. Alford-Rajagopal-Wilczek (NPB1999) T. Schafer (PRD2002); NY (JHEP2008)

7. Possible phase structure I  Anomaly-induced critical point at high μ. Hatsuda-Tachibana-NY-Baym (PRL2006)  A realization of quark-hadron continuity. Schafer-Wilczek (PRL1999)  Critical point(s) of other origins. Kitazawa-Koide-Kunihiro-Nemoto (PTP2002); Zhang-Fukushima-Kunihiro (PRD2009); Zhang-Kunihiro, arXiv:0904.1062. T  Quark-Gluon Plasma Hadrons Color superconductivity

8. Possible phase structure III  Is there this possibility? [see also Hidaka-san’s talk] T  Quark-Gluon Plasma Hadrons CFL quark matter

9. Phase diagram of “instantons” (N f =3) T  “instanton liquid” “instanton molecule” “ instanton gas“  Chiral phase transition at high μ: instanton-induced crossover. 4-dim. generalization of Kosterlitz-Thouless transition. NY (JHEP2008)

10. Another viewpoint: Lee-Yang zeros  The partition function zeros in the complex plane at V<∞ reflects the information of the chiral condensate at V=∞:  Nonzero chiral condensate at V=∞ requires a cut through m=0. Halasz-Jackson-Verbaarschot (PRD97) [Lee-Yang zeros at μ=0] Leutwyler-Smilga (PRD92)

11. Predictions of Random Matrix Theory (RMT) Halasz-Jackson-Verbaarschot (PRD97); Halasz, et al. (PRD98)  RMT predictions: 1.Chiral symmetry restores at μ=μ c. 2.The cut will move away from origin as μ increases. → Is it consistent with the chiral symmetry breaking at high μ? [Random Matrix Theory → Ohtani-san’s talk]

12. Finite-volume QCD at high density  QCD in a large but finite torus:  ε-regime:  Elementary excitations in CFL; 9 quarks: mass gap~Δ due to the color superconductivity. 8 gluons: mass gap~Δ due to the Higgs mechanism. 8+1(+1) Nambu-Goldstone (NG) modes: nearly (or exactly) massless.  In ε-regime, Non-NG modes negligible since. Kinetic terms of NG modes negligible. NY-Kanazawa (PRL2009)

13. Partition functions in ε-regime  Chiral Lagrangian at high μ (flavor-symmetric): Son-Stephanov (PRD2000)  Exact partition function at high μ: a novel correspondence between hadronic phase and CFL phase related to quark-hadron continuity!  Dirac spectrum... at μ=0. at high μ. NY-Kanazawa (PRL2009)

14. Exact Lee-Yang zeros at high density  Asymptotic partition function and Lee-Yang zeros at μ=∞:  Chiral condensate vanishes at μ=∞.  However, many Lee-Yang zeros exist near origin even at high μ and the chiral condensate can be nonzero for μ<∞. NY-Kanazawa (PRL2009)

15. 1.Phases in dense QCD The U(1) A anomaly (or instanton) plays crucial role. Non-vanishing chiral condensate even at high μ. Chiral-super interplay is inevitable. Possible critical point(s) in dense QCD. 2.Partition function zeros in dense QCD Exact X-shaped cut in the complex mass plane at μ=∞. Chiral condensate can be nonzero for μ<∞. 3.Future problems Phases at lower or intermediate densities? Anomaly-induced interplay in NJL. Baym-Hatsuda-NY, in progress. Confinement-deconfinement transition? Microscopic understanding based on QCD? Summary & Outlook

16. Back up slides

17. Chiral vs. Diquark condensates E p pFpFpFpF -p F  Diquark condensate  Chiral condensate Y. Nambu (‘60)

18. Hadrons (3-flavor) SU(3) L ×SU(3) R → SU(3) L+R Chiral condensate NG bosons (π etc) Vector mesons (ρ etc) Baryons Color-flavor locking SU(3) L ×SU(3) R ×SU(3) C ×U(1) B → SU(3) L+R+C Diquark condensate NG bosons Gluons Quarks Phases Symmetry breaking Order parameter Elementary excitations quark-hadron continuity Continuity between hadronic matter and quark matter (color-flavor locking) Conjectured by Schäfer & Wilczek, PRL 1999

19. Instantons and chiral symmetry breaking Why instanton? : mechanism for chiral symm. breaking/restoration T=0T>T c “instanton liquid” (metal) “instanton molecule” (insulator) Schäfer-Shuryak, Rev. Mod. Phys. (‘97) See, e.g., Hell-Rößner-Cristoforetti-Weise, arXiv: 0810.1099 nonlocal NJL model  Origin of NJL model:  Then, χSB in dense QCD from instantons?

20.  Dense QCD : U(1) A is asymptotically restored. Low-energy dynamics in dense QCD convergent!  Low-energy effective Lagrangian of η ’ Manuel-Tytgat, PL(‘00) Son-Stephanov-Zhitnitsky, PRL(‘01) Schäfer, PRD(‘02)

21. Coulomb gas representation  : topological charge  : 4-dim Coulomb potential  Instanton density, topological susceptibility  Witten-Veneziano relation ：

22. Renormalization group analysis  Fluctuations ：  Change of potential after RG ：  RG trans. ： RG scale ： kinetic vs. potential  D ＝ 2 ： potential irrelevant → vortex molecule phase potential relevant → vortex plasma phase  D ≧ 3 ： potential relevant → plasma phase

23. Phase transition induced by instantons  Unpaired instanton plasma in dense QCD →Coexistence phase:  Actually, System parameter αTopological excitationsOrder of trans. 2D O(2) spin systemvortex2nd 3D compact QED magnetic monopolecrossover 4D dense QCD instantoncrossover D-dim sine-Gordon model ： Note: weak coupling QCD :

24. Color superconductivity at large N c  qq scattering Double-line notation ★ Diquarks are suppressed at large N c ! Deryagin-Grigoriev-Rubakov (‘92) Shuster-Son (‘00) Ohnishi-Oka-Yasui (‘07)

25. 0 ≾ m u,d <m s ≪ ∞ (realistic quark masses) Realistic QCD phase structure? 2nd critical point Critical point Asakawa & Yazaki, 89 m u,d,s = 0 (3-flavor limit)m u,d = 0, m s =∞ (2-flavor limit) ≿≿ T μ T μ T μ Hatsuda, Tachibana, Yamamoto & Baym 06

26. Possible phase structure II T  Quark-Gluon Plasma Hadrons Color superconductivity  Of course, 1st order chiral phase transition at T=0 is still possible.