Ginzburg-Landau approach to QCD phase transitions Motoi Tachibana (Saga University, Japan) contents Introduction to QCD and its phase structure Chiral-super interplay in high density QCD (Comments on UcA/QCD correspondence ) Summary (1) Tetsuo Hatsuda, Naoki Yamamoto, Gordon Baym and M. T., Phys. Rev. Lett. 97 (2006) 122001. N. Yamamoto, T. Hatsuda, G. Baym and M. T., Phys. Rev. D 76 (2007) 074001. T. Hatsuda, N. Yamamoto and M. T., Phys. Rev. D 78 (2008) 011501. EMMI, Feb. 20, 2009

In this workshop, we have heard QCD, QCD phase diagram Phase transitions, phase coexistence Chiral symmetry breaking Axial anomaly Color superconductivity Critical point(s) Bose-Einstein condensation (BEC) My talk is also related to these.

Constituents

Quark Flavors Tc ~ 200 MeV μc ~ 400 MeV ms~100MeV Quark flavors Kobayashi & Maskawa (1973) Light quarks mu~3MeV md~5MeV ms~100MeV Heavy quarks mc~1.3 GeV mb~4.3 GeV mt~171 GeV Tc ~ 200 MeV μc　 ~ 400 MeV 　 ms~100MeV same order As far as we are interested in physics around several hundred MeV, we can take heavy quark mass to ∞.

Dynamics

Quark Colors and Quantum Chromo Dynamics 1966 SUc(3) YM theory as a model of strong interaction Nambu (’66) 1965-1972 Precursors of asymptotic freedom 　Vanyashin & Terenteev (’65), Khriplovich (’69), ’t Hooft (’72) 1973 Discovery of asymptotic freedom Gross & Wilczek, Politzer (’73)

Finally, I discovered QUARK! which is more like liquid than I expected. It says this is QUARK, but I cannot see its color. Maybe this should be called “QUARKs” ? “QUARK” in Germany

Features

QCD running coupling “Bergkatze” Color Confinement Asymptotic freedom scale （ＧｅV) Running coupling (αs = ｇ２/４π） αs Color Confinement　 Asymptotic freedom　 (int)/(kin) << 1 “Bergkatze”

Dynamical Breaking of Chiral Symmetry Chiral basis : QCD Lagrangian : classical QCD symmetry (massless quarks) Quantum QCD vacuum (massless quarks) Chiral condensate : spontaneous mass generation Axial anomaly : quantum violation of U(1)A

Phases of QCD H2O 　http://boojum.hut.fi/research/theory/typicalpt.html 4He

T Schematic phase diagram in QCD QGP (quark-gluon plasma) CSC (color superconductivity) QGP (quark-gluon plasma) cSB (chiral symmetry breaking)

Color superconductivity at high baryon density CFL dSC uSC 2SC flavor color “Majorana mass” major differences from the standard BCS superconductor 1. Relativistic fermi system color-magnetic int. dominant High Tc : Tc/eF ~ 0.1 Compact pair : r~ 1-10 fm Son, PRD59 (’99), Schafer & Wilczek, PRD60 (’99) Pisarski & Rischke, PRD61 (’00) 2. Color-flavor entanglement Various phases (c.f. Ice, 3He) 2SC, uSC, dSC, CFL etc

Origin of each “phase” and Extreme QCD (EQCD) strong residual int. pre-formed pairs Hatsuda & Kunihiro, 　Phys. Rev. Lett. 55 (‘85) DeTar, Phys.Rev. D32 (‘85) Asymptotic freedom + Debye screening  　deconfinement Collins & Perry, Phys.Rev.Lett. 34 (‘75) cSB CSC QGP mB T Asakawa & Yazaki, 　Nucl. Phys. A504 (’89) Critical point quark-anti-quark pairing Chiral instability Nambu & Jona-Lasinio, 　Phys.Rev. 122 (‘61) quark-quark pairing  Cooper instability Bailin & Love, Phys.Rep.107 (‘84)

QCD and high temperature superconductivity (HTS) cSB CSC QGP 1. Competition between different orders 2. Strong coupling 　 Common features in QCD, HTS, and ultracold atoms Sigrist and Ueda, (‘91) Babaev, Int. J. Mod. Phys. A16 (‘01) Kitazawa, Nemoto, Kunihiro, PTP (‘02) Abuki, Itakura & Hatsuda, PRD (’02) 　　 Chen, Stajic, Tan & Levin, Phys. Rep. (’05) Baym, Hatsuda, Tachibana & Yamamoto (’06)

Chiral-super interplay in high density QCD　　 cSB CSC QGP ?

How to study phase structure ? Ginzburg-Landau-Wilson (GLW) approach : model independent, analytic 1. Topological structure of the phase diagram 2. Order of the phase transition 3. Critical properties Lattice QCD : exact, limited to μ~0, numerically heavy Models (NJL model, strong coupling QCD etc) : qualitative, semi-analytic σ (x) : Order parameter field Same symmetry with underlying theory K = {T, m, μ, … } : External parameters Recipe Ginzburg-Landau = Saddle point approximation Wilson = Fluctuations by renormalization group method ・ Valid for continuous or weak 1st order transitions ・ Choice of σ (x) is an “art” ・ Results should be eventually checked by e.g. lattice QCD Caution

GL analysis for chiral-super interplay in QCD (Nf=3) QCD Symmetry: Chiral field: diquark field: GL potential Pisarski & Wilczek, PRD 29 (’84) ・Iida & Baym, PRD 63 (’01) ・Iida, Matsuura, Tachibana 　& Hatsuda, PRD 71 (’05) Yamamoto, Tachibana, Baym & Hatsuda, PRL 97 (’06)

Complete classification of the GL potential (m=0) Axial anomaly L L R L R R = anomaly-induced terms Yamamoto, Tachibana, Baym & Hatsuda, PRL 97 (’06), PRD 76 (’07)

Chiral-CFL interplay in three massless flavors Chiral-CFL interplay in Nf=3 Color-flavor Locking (CFL) Natural parameter relations : anomaly γ term acts as an external field for σ, washing out the 1st order transition for large γd as in magnetic system with an external field. 2 Appearance of New Critical Point

Possible phase diagram in QCD cSB CSC QGP “Anomaly driven critical point in high density QCD” Yamamoto, Tachibana, Baym & Hatsuda, PRL 97 (’06)

Comments Finding precise location of new critical point requires phenomenological models, and lattice QCD simulation. (GL approach just can tell us the topological structures) To make schematic phase diagram more realistic should include *realistic quark masses *for neutron stars, charge neutrality and beta equilibrium *Interplay with confinement (characterized by Polyakov loop) [e.g., R. Pisarski, PRD62 (2000); K. Fukushima, PLB591 (2004); C.Ratti, M. Thaler, W. Weise PRD73 (2006); C.Ratti, S. Rössner and W. Weise, PRD (2007) hep-ph/0609281 ]. * thermal gluon fluctuations * possible spatial inhomogeneities (LOFF states)

Two massless flavors tetracritical pt. bicritical point Assume 2-flavor CSC phase (2SC) then (No cubic terms; cf. three flavors) tetracritical pt. bicritical point

Instanton-induced crossover in dense QCD N. Yamamoto, JHEP0812:060(2008) Phase Diagram of “Instantons” T mB QGP CFL χSB “instanton molecule” “instanton liquid” “instanton plasma“

Comment on UcA/QCD correspondence cSB CSC QGP

GL free energy for multicomponent ultracold fermion system with superfluidity and magnetism (Demler et al., 2007) magnetization pairing gap

BEC-BCS crossover in quark matter ? Abuki, Itakura & Hatsuda, PRD65 (’02) BEC-like BCS-like μ(MeV) ξc/dq Ladder QCD at finite m ξc : coherenth length dq : interquark distance 40K : JILA group, PRL 92 (2004) 040403 6Li : Innsbruck group, PRL 92 (2004) 120401 MIT group, PRL 92 (2004) 120403 40K Cond. of Fermionic-Atom Pairs N0/N = 10% 5% 1% ξc dq tightly bound loosely bound

Bose-Fermi mixture in Ultracold Atoms and Dense QCD -- superfluid of composite-fermions -- K. Maeda, Master thesis (’09) Maeda, Baym & Hatsuda, (’09) b,f norml gas N normal gas b-BEC N-BCS weak strong T Cold Atom 40K 87Rb dense QCD

Summary and Future 1. QCD phase structure and Extreme QCD (EQCD) ・ Three major phases in QCD: ChiSB,　QGP and CSC ・ Axial anomaly (Kobayashi-Maskawa-’tHooft) plays crucial roles 2. Chiral-super interplay in dense QCD ・ Possible new critical point driven by axial-anomaly ・ Hadron-quark continuity ・ Connection to similar system in UcA ・ Instanton-induced crossover 3. Comment on UcA/QCD correspondence ・ BEC-BCS crossover in UcF <-> CSC in QCD ・ composite fermion superfluid in UcBF <-> neutron superfluid in QCD ・　UcA as a tabletop lab. for dense QCD Exp. Supercom. Theory UcA

phase diagram (without d-σ coupling) : 1st order : 2nd order μ T

phase diagram (with d-σ coupling) μ A new critical point driven by the axial anomaly

m condensates Continuity in the ground state Low m High m p(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 & Hatsuda., PR D76 (’07) ○ Continuity of vector mesons Octet vector meson <-> octet gluon 　 Tachibana,Yamamoto & Hatsuda, PRD78 (2008) Explicit realization of spectral continuity

Backup slides

Chiral Transition at Finite T　　 cSB CSC QGP ?

X ~ GLW analysis of hot QCD Symmetry: Chiral field: Pisarski & Wilczek (’84) Axial anomaly X Symmetry: ~ Chiral field: Chiral transformation: SU(Nf)LxSU(Nf)RxU(1)A SU(Nf)LxSU(Nf)R quark mass term

Some examples of GL potential ・ 2nd order phase transition Z(2) Ising model Nf=2 QCD ・ 1st order phase transition Z(3) Potts model Nf=3 QCD ・ Tri-critical behavior Meta-magnet Nf=2+1 QCD

Order of the thermal QCD transition (μ=0) Pisarski and Wilczek, PRD29 (’84) Svetitsky & Yaffe, NPB210 (’82) 1st cross over 2nd m s small large u,d Nf=0 Nf=1 Nf=2 Nf=3

Chiral phase transition (Nf=3) μ T Chiral field: ? (Note: 2nd order for Nf=2) 1st order

Color superconductivity phase transition μ T ? Diquark field: 2nd order

Natural parameter relations: μ T Chiral-super interplay Hatsuda-Tachibana- Yamamoto-Baym (‘06) ? Natural parameter relations:

phase diagram (with d-σ coupling) μ A new critical point driven by the axial anomaly

Realistic phase diagram in Nf=2+1 ? mu,d,s = 0 (3-flavor limit) T μ mu,d = 0, ms=∞ (2-flavor limit) T μ T μ 0 ≾ mu,d<ms≪∞ (realistic quark masses) High T critical point Low T critical point

高密度星に関係する様々なカラー超伝導相 （電荷中性条件とβ平衡条件が重要） 高密度星に関係する様々なカラー超伝導相 （電荷中性条件とβ平衡条件が重要） nd > nu > ns u d s CFL dSC uSC 2SC cSB CFL 2SC uSC dSC FFLO 2SC : Bailin and Love, Phys. Rep. (’84) CFL : Alford, Rajagopal and Wilczek, NPB (’99) dSC : Iida, Matsuura, Tachibana and Hatsuda, PRL (’04) uSC : Ruster, Werth, Buballa, Shovkovy and Rischke, PRD (’05) FFLO, gapless phase, CSL, K-cond. etc

Symmetry realization in hot/dense QCD（for mu,d,s=0 case） QGP : Collins & Perry, PRL 34 (1975) cSB CSC QGP χSB : Nambu, PRL 4 (1960) CSC : Alford, Rajagopal & Wilczek, NP B537 (1999)

Spectral continuity at finite μ cSB CSC QGP

QCD sum rules in the superconducting medium Vector current： Current correlation function： Operator Product Expansion (OPE) up to O(1/Q6) : 4-quark condensate Diquark condensate Chiral condensate

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

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

QCDの相構造 バリオン密度 温度

magnetically controllable ・density 1014 - 1015 cm-3 K. Maeda, Master thesis (2009) 10 12 10 9 10 7 T[K] 10 2 10 1 10 -3 10 -7 Quark-gluon plasma Superfluid neutron matter Center of sun Boiling water Freezing water Liquid nitrogen Superfluid, superconductor Superfluid of 3He Ultracold atoms Ultracold Atoms (UcA) 　　・T ~ 10-7　K ・hyperfine states magnetically controllable 　　・density 1014 - 1015 cm-3 ( cf. Air ~ 1019cm-3 )

Y. Nambu, Nobel Lecture (Dec.8, 2008), page 24/25

Strangely looking? CSC (color QGP (quark-gluon plasma) cSB superconductivity) QGP (quark-gluon plasma) cSB (chiral symmetry breaking)