1. Cosmological constant Einstein (1917) Universe baryons 5 27 68 “Higgs” condensate Englert-Brout, Higgs (1964) bare quark 3 “Chiral” condensate Nambu (1960) quark 3 hadron QCD Spectral Functions and Dileptons T. Hatsuda (RIKEN) Condensates ⇔ Elementary excitations

2. Outline I.QCD symmetries II. Chiral order parameters III. In-medium hadrons IV. Summary “The Phase Diagram of Dense QCD” K. Fukushima + T.H., Rep. Prog. Phys. 74 (2011) 014001 “Hadron Properties in the Nuclear Medium” R. Hayano + T.H., Rev. Mod. Phys. 82 (2010) 2949 “QCD Constraints on Vector Mesons at finite T and Density” T.H., http://www-rnc.lbl.gov/DLS/DLS_WWW_Files/DLSWorkshop/dilepton.html (1997)

3. Make an estimate before every calculation, try a simple physical argument (symmetry! invariance! conservation!) before every derivation, guess the answer to every paradox and puzzle. John Wheeler's First Moral Principle from "Spacetime Physics (Taylor – Wheeler, 2 nd ed.)”

4. Symmetry realization in QCD vacuum Chiral basis : QCD Lagrangian : classical QCD symmetry (m=0) Quantum QCD vacuum (m=0) Chiral condensate : spontaneous mass generation Axial anomaly : quantum violation of U(1) A

5. Dim.3 chiral condensate in QCD free LQCD Banks-Casher relation (1980) 0 0 Di Vecchia-Veneziano formula (1980) Gell-Mann-Oakes-Renner (GOR) formula (1968)

6. Examples Axial rotation : Axial Charge : Order parameters : NOT unique ! ・・・ II. Chiral order parameters Order parameter : = 0 (no SSB) ≠ 0 (SSB) θ

7. σ （６００） Spectral evidence of SSB in QCD T.H. LBNL WS (1997)

8. ALEPH Collaboration, Phys. Rep. 421 (2005) 191 - fromτ-decays at LEP-1 ρ V (s)/s ρ A (s)/s [ρ V (s)-ρ A (s)] /s

9. Energy weighted “chiral” sum rules from QCD (m q =0) Dim.6 chiral condensate Koike, Lee + T.H., Nucl.Phys. B394 (1993) 221 Kapusta and Shuryak, PRD 49 (1994) 4694 Klingl, Kaiser and Weise, NPA 624 (1997) 527

10. -ρ pQCD (ω)) -ρ pQCD (ω))= C 4 -ρ pQCD (ω))= C 6 +C 6 ’ Dim.4 gluon condensate Dim.6 quark and gluon condensates Dilepton data (after Cocktail subtraction) Space-time average (by hydro or other models) Lattice QCD simulations (with gradient flow method) + Energy weighted “vector” sum rules from QCD (m q =0) III. In-medium hadrons

11. T.H. LBNL WS (1997) Slide p.31 c.f. GLS sum rule Adler sum rule Bjorken sum rule

12.  QGP Quark-Gluon Plasma Quark superfluid Hadron phase Chiral symmetry is always broken at finite density Baryon superfluid Hadron-Quark Continuity in dense QCD （ N c =3, N f =3 ）

13.  condensates Continuity in the ground state Tachibana, Yamamoto + T.H., PRD78 (’08) Vector Mesons = Gluons ?! Baryons = Quarks ?! Possible fate of hadrons at high density （ N c =3, N f =3 ） Low  High   (8) & H  ’ (8) & H NGs Vectors Fermions excitation  V (9)  gluons (8)  Baryons (8) Quarks (9) Continuity in the excited state ? Schafer & Wilczek, PRL 82 (’99)

14. Nonet vector mesons (heavy) Octet vector mesons (light) Octet gluons in CFL: m g =1.362Δ Gusynin & Shovkovy, NPA700 (2002) Malekzadeh & Rischke, PRD73 (2006) T.H., Tachibana and Yamamoto, PRD78 (2008) Spectral continuity of vector mesons

15. At high density: At intermediate density: At low density: Mass formula from Finite Energy Sum Rules

16. IV. Summary I. Chiral order parameters not unique : Dim.3 condensate, Dim.6 condensate, etc II. In-medium hadrons ・ chiral restoration can be seen in spectral degeneracy ・ moment analysis is important for model independence ・ interesting possibility of hadron-quark crossover (Vector mesons = Gluons, Baryons = Quarks)

17. Back up slides

18.  Vector current ：  Current correlation function ： QCD sum rules in the superconducting medium  Operator Product Expansion (OPE) up to O(1/Q 6 ) : 4-quark condensate Diquark condensate Chiral condensate

19. Low  High   (8) & H  ’ (8) & H NGs Vectors Fermions excitation  V (9)  gluons (8)  baryons (8) Quarks (9) Continuity in the excited state?? Schafer and Wilczek, PRL 82 (1999) Generalized Gell-Mann-Oakes-Renner relation : Yamamoto, Tachibana, Baym + T.H., PR D76 (’07) ○ Continuity of vector mesons Tachibana, Yamamoto +T.H., PRD78 (2008) ○ Explicit realization of spectral continuity Possible fate of hadrons at high density （ N c =3, N f =3 ） Vector Mesons = Gluons ?! Baryons = Quarks ?!  condensates Continuity in the ground state Yamamoto, Tachibana, Baym + T.H., PRL97(’06), PRD76 (’07)

20. Low  High   (8) & H  ’ (8) & H NGs Vectors Fermions excitation  V (9)  gluons (8)  baryons (8) Quarks (9) Continuity in the excited state?? Schafer and Wilczek, PRL 82 (1999) Hadron-quark continuity in dense QCD  condensates Continuity in the ground state Hatsuda, Tachibana, Yamamoto & Baym, PRL 97 (2006)

21. T.H. Slide p.2 (1997)

22. http://www-rnc.lbl.gov/DLS/DLS_WWW_Files/DLSWorkshop/dilepton.html I. Why SPF is important ?

23. FLAG Collaboration update( July 26, 2013) http://itpwiki.unibe.ch/flag/ Running masses: m q (Q) quark masses (from lattice QCD) [MeV] (MS-bar @ 2GeV) mumu 2.16 (9)(7) mdmd 4.68 (14)(7) msms 93.8 (2.4) Running coupling: α s (Q)=g 2 /4π PDG (2012) http://pdg.lbl.gov/ I. Status of QCD

24. Hadron masses from Lattice QCD Improved Wilson + Iwasaki gauge action a = 0.09 fm, L=2.9 fm, m π =135 MeV PACS-CS Coll., Phys. Rev. D 81, 074503 (2010) 3% accuracy ⇒ L~9.6 fm, m π =135 MeV on K-computer underway

25.  Nambu-Goldstone bosons  Other hadrons Examples: QCDSR from Commutator by Hayata, PRD88 (2013) Gell-Mann-Oakes-Renner relation (1968) QCDSR from OPE by SVD (1979) III. In-medium hadrons In-vacuum

26. In-medium hadrons Complex pole (even for the pion) One-parameter example (T≠0) : * For the pion, f(x) and g(x) can be evaluated for small x. See e.g. Jido, Kunhihiro + T.H., Phys. Lett. B670 (2008) * In general, experimental inputs are really necessary. * Sometimes, spectral function is better to be studied.

27. Dim.3 Chiral condensate in the medium Dim.3 Chiral condensate in the medium Lattice QCD, (2+1)-flavor Borsanyi et al., JHEP 1009 (2010) Finite Temperature (LQCD) Nuclear chiral perturbation Kaiser et al., PRC 77 (2008) Finite baryon density (χPT)

28. Mesic nuclei 2 Individual properties of NG and “Higgs” bosons π, K, η (NG), σ (Higgs), η’ (anomaly) σ  2γ, η  2γ, η’  2γ Dileptons 3 Individual properties of vector bosons ρ, ω, and φ Precision/systematic studies (dispersion relation, different targets, …) 1 Spectral difference between chiral partners π-σ, ρ-a 1, ω-f 1, etc Determination of D=6 chiral condensates in the vacuum? Tau-decay in nuclei ? Wish list

30. T.H. Slide p.7 (1997)

31. III. Exact sum rules in QCD medium T.H. Slide p.20 (1997)

32.  Nambu-Goldstone bosons  Other hadrons Examples: QCDSR from Commutator by Hayata, PRD88 (2013) Gell-Mann-Oakes-Renner relation (1968) QCDSR from OPE by SVD (1979) III. In-medium hadrons In-vacuum