Chiral symmetry breaking in dense QCD Naoki Yamamoto (University of Tokyo) contents Introduction: QCD critical point at high T Chiral-super interplay QCD phase structure from instantons QCD phase structure at large Nc Summary & Outlook (1) T. Hatsuda, M. Tachibana, G. Baym & N.Y., Phys. Rev. Lett. 97 (2006) 122001. (2) N.Y., JHEP 0812 (2008) 060. 駒場原子核理論セミナー April 15, 2009

? QCD phase diagram T mB Quark-Gluon Plasma Color superconductivity ..But, 2-flavor NJL rather than QCD Early universe T Quark-Gluon Plasma ? RHIC/LHC Color superconductivity Hadrons Neutron star & quark star mB

QCD critical point at high T

QCD critical point? First predicted by 2-flavor NJL model Asakawa-Yazaki, ‘89 Confirmed by other models, e.g., random matrix model Halasz et al. ‘98 Lattice results: still controversial de Forcrand-Philipsen ‘06, ‘08 But models have many ambiguities! e.g.) NJL-type Lagrangian: Parameters (to be fitted with pion mass/decay const.): Λ, G, m → Calculate phase diagram numerically. Thermodynamic potential:

QCD (tri)critical point (Nf=2) Potential at lowest order (m=0): T μ : 1st order : 2nd order c.f.) Coefficient in NJL: N.Y. et al., ‘07

No critical point in massless 3-flavor limit Chiral field: Pisarski-Wilczek (‘84) U(1)A anomaly μ T 1st order

QCD critical point in 2+1 flavor 0 = mu,d,s < 0 = mu,d≪ms < μ T T μ T μ 0<mu,d<ms As ms increases, Note) CP in 2-flavor limit is also model-dependent.

Some comments Unknown medium effects on model parameters easily smear out CP! QCD critical point at high T from 2+1 flavor PNJL model with gD~c0 K. Fukushima, PRD (‘08), N. Bratovich, T. Hell, S. Rößner + W. Weise (’08) c.f.) 4-fermi interaction etc. also has medium effects 3-flavor random matrix model with axial anomaly? Sano-Fujii-Ohtani, (‘09)

Location of QCD critical point? Taken from hep-lat/0701002, M. Stephanov

Chiral-super interplay

Chiral vs. Diquark condensates Chiral condensate Diquark condensate E p pF -pF Y. Nambu (‘60)

Chiral-super interplay in models Phase diagram in 2-flavor NJL model Berges-Rajagopal, ‘99 Examples of phase diagrams in 2-flavor random matrix model Vanderheyden-Jackson, ‘00

Notes Many ambiguities in NJL: With vector interaction → coexistence phase appears Kitazawa et al, ‘02 Possible higher interactions Kashiwa et al. ‘07 Medium effects on interactions (remember 3-flavor PNJL) Chen et al. ’09 Favor-dependence, quark masses, ... However, their topological structures look similar, why? → Because all models have QCD symmetries!

Ginzburg-Landau approach (Nf=2) GL potential: Most general phase diagram Hatsuda-Tachibana-Yamamoto-Baym (‘06) T μ Precise medium effects on GL coefficients needed

Anomaly-induced interplay (Nf=3) Hatsuda-Tachibana-Yamamoto-Baym (‘06) T μ : 1st order : 2nd order Non-vanishing chiral condensate at high μ due to U(1)A anomaly The possible 2nd critical point at high μ Anomaly-induced interplay in NJL Yamamoto-Hatsuda-Baym in progress

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

QCD phase structure from instantons

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

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

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

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

Phase transition induced by instantons D-dim sine-Gordon model： 　　　　　System 　　　　　　parameter α Topological excitations Order of trans. 2D O(2) spin system vortex 2nd 3D compact QED magnetic monopole crossover 4D dense QCD instanton crossover Note: weak coupling QCD: Unpaired instanton plasma in dense QCD →Coexistence phase: 　　　 Actually,

Phase diagram of “instantons” (Nf=3) mB QGP CFL χSB “instanton molecule” “instanton liquid” “instanton gas“ Chiral phase transition at high μ: instanton-induced crossover. 4-dim. generalization of Kosterlitz-Thouless transition. N. Yamamoto, JHEP 0812:060 (2008)

QCD phase structure at large Nc

QCD phase diagram at large Nc Gluodynamics (~Nc2) dominates independent of μB (~Nc). McLerran-Pisarski, NPA (‘07) see also, Horigome-Tanii, JHEP (‘07)

CSC at large Nc? ★ Diquarks are suppressed at large Nc! qq scattering Double-line notation qq scattering Deryagin-Grigoriev-Rubakov (‘92) Shuster-Son (‘00) Ohnishi-Oka-Yasui (‘07) ★ Diquarks are suppressed at large Nc!

Color Superconductivity Conjectured Phase Diagram for Nc = 3 RHIC LHC SPS FAIR AGS Confined N ~0(1) Not Chiral Baryons N ~ NcNf Chiral Debye Screened Baryons Number N ~ Nc 2 Color Superconductivity Liquid Gas Transition Critical Point Quark Gluon Plasma Quarkyonic Matter Confined Matter T From McLerran at QM2009 Not correct for 3-flavor limit: deconfinement earlier than χSR. Note that large Nc leads to No color superconductivity Weak axial anomaly indep. of μ A dynamical question: subtleness of quark masses. (flavor-dep.) A puzzle: how χSB occurs after χSR?

Summary & Outlook QCD phase structure Consensus is highly model-dependent. The QCD critical point at high T? Possible 2nd critical point at high μ. 2. Instanton plasma from low μ to high μ Instantons play crucial roles everywhere. Non-vanishing chiral condensate even at high μ. Future problems Quarkyonic vs. CSC? QCD phase structure from QCD itself? AdS/CFT application?

Finite-volume QCD at high μ N. Yamamoto, T. Kanazawa, arXiv:0902.4533. microscopic regime: Exact analytical results; Partition function (zero topological sector): a novel correspondence! Spectral sum rules: Dirac spectra at high μ are governed by the CSC gap Δ. Lee-Yang zeros: conventional random matrix model fails to reproduce CSC. Application to dense 2-color QCD is also possible. T. Kanazawa, T. Wettig, N. Yamamoto, to appear soon. at μ=0. at high μ.

Hadron-quark continuity Continuity between hadronic matter and quark matter (Color superconductivity) 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 superconductivity 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 Conjectured by Schäfer & Wilczek, PRL 1999

Back up slides

Order of the thermal transition Z(3) GL theory O(4) GL theory SUL(3)xSUR(3) GL theory

Color Superconductivity QCD at high density asymptotic freedom Attractive channel [3]C×[3]C=[3]C+[6]C Fermi surface q 3 Cooper instability dL,R :diquark E p pF -pF 3-flavor case u,d,s r,g,b u d s Color-Flavor Locking (CFL) phase Alford-Rajagopal-Wilczek (‘99)

Color superconductivity phase transition Diquark field: Iida-Baym (‘00) μ T 2nd order