The Phases of QCD Thomas Schaefer North Carolina State University 1
Motivation Different phases of QCD occur in the universe Neutron Stars, Big Bang Exploring the phase diagram is important to understanding the phase that we happen to live in Structure of hadrons is determined by the structure of the vacuum Need to understand how vacuum can be modified QCD simplifies in extreme environments Study QCD matter in a regime where quarks and gluons are the correct degrees of freedom 2
QCD Phase Diagram Big Bang T Quark Gluon Plasma RHIC Hadron Gas CERN Supernova Nuclear Liquid Color Superconductivity Neutron Stars µ 3
Quantumchromodynamics Elementary fields: Quarks Gluons color a = 1,..., 3 (q α ) a f spin α = 1, 2 A a color a = 1,..., 8 µ spin ɛ ± µ flavor f = u, d, s, c, b, t Dynamics: non-abelian gauge theory L = q f (id/ m f )q f 1 4 Ga µνg a µν G a µν = µ A a ν ν A a µ + gf abc A b µa c ν id/ q = γ µ ( i µ + ga a µt a) q 4
Phases of Gauge Theories Coulomb Higgs Confinement V (r) e2 r V (r) e mr r V (r) kr Standard Model: U(1) SU(2) SU(3) 5
Phases of Matter phase order broken rigidity Goldstone param symmetry phenomenon boson crystal ρ k translations rigid phonon magnet M rotations magnetization magnon superfluid Φ particle number supercurrent phonon supercond. ψψ gauge symmetry supercurrent none (Higgs) 6
Gauge Symmetry Local gauge symmetry U(x) SU(3) c ψ Uψ D µ ψ UD µ ψ A µ UA µ U + iu µ U F µν UF µν U Gauge symmetries cannot be broken Gauge symmetries can be realized in different modes Coulomb Higgs confined d.o.f: 2 (massless) 3 (massive) 3 (massive) Distinction between Higgs and confinement phase not always sharp 7
Define left and right handed fields ψ L,R = 1 2 (1 ± γ 5)ψ Chiral Symmetry L R Fermionic lagrangian, M = diag(m u, m d, m s ) L = ψ L (id/ )ψ L + ψ R (id/ )ψ R L L R R + ψ L Mψ R + ψ R Mψ L L R R L M M M = 0: Chiral symmetry (L, R) SU(3) L SU(3) R ψ L Lψ L, ψ R Rψ R 8
Chiral Symmetry Breaking Chiral symmetry implies massless, degenerate fermions m (1/2)+ N = 935 MeV m (1/2) N = 1535 MeV Chiral symmetry is spontaneously broken ψ f L ψg R + ψ f L ψg R (230 MeV)3 δ fg SU(3) L SU(3) R SU(3) V Consequences: dynamical mass generation m Q = 300 MeV m q m N = 890 MeV + 45 MeV (QCD, 95%) + (Higgs, 5%) 9
Low Energy Effective Lagrangian Low energy degrees of freedom: Goldstone modes U(x) = exp(iπ a λ a /f π ) Effective lagrangian controls L = L = f 2 π 4 Tr[ µu µ U ] + (BTr[MU] + h.c.) +... Goldstone boson scattering Coupling to external currents Quark mass dependence 10
Symmetries of the QCD Vacuum: Summary Local SU(3) gauge symmetry confined: V (r) kr Chiral SU(3) L SU(3) R symmetry Axial U(1) A symmetry spontaneously broken to SU(3) V Vectorial U(1) B symmetry anomalous : µ A 0 µ = N f 16π 2 Ga µν G a µν unbroken: B = d 3 x ψ ψ conserved 11
Notes QCD with general N f, N c (with or without SUSY) There are asymptotically free theories without confinement and/or chiral symmetry breaking QCD with N f = 3 confinement implies chiral symmetry breaking symmetry breaking pattern SU(3) L SU(3) R SU(3) V unique order parameter ψψ 0 12
14 N f 12 10 8 6 4 QCD Phase Diagram: Nc and Nf QCD SUSY QCD 14 12 10 N IILM, chisb lattice, conf 2 3 4 c 8 6 4 conformal Coulomb N c 2 3 4 no chisb, confinement chiral symmetry breaking, confinement b=0 b=2 IR free conformal b =0 chiral symmetry breaking, confinement no ground state 13 weak chiral symmetry breaking
Approaching the Phase Diagram: Symmetries and Weak Coupling Arguments T critical QGP critical µ e pion gas critical µ color superconductor neutron gas 14
Approaching the Phase Diagram: Strongly Correlated Phases T critical QGP sqgp Hagedorn critical µ e pion gas critical nuclear liquid BEC BCS µ color superconductor neutron gas 15
Approaching the Phase Diagram: Experiments and Numerical Simulations T lattice RHIC s HIC s µ e neutron stars µ 16
Part I: QCD at Finite Temperature The Heavy Ion Program at RHIC T sqgp RHIC s QGP sqgp µ e µ 17
The High T Phase: Qualitative Argument High T phase: Weakly interacting gas of quarks and gluons? typical momenta p 3T Large angle scattering involves large momentum transfer effective coupling is small Small angle scattering is screened (not anti-screened!) coupling does not become large Quark Gluon Plasma 18
Lattice QCD Euclidean partition function Z = da µ dψ exp( S) = da µ det(id/ ) exp( S G ) Lattice discretization: n n+ µ U µ (n) = exp(igaa µ (n)) D µ φ 1 a [U µ(n)φ(n + µ) φ(n)] (G a µν) 2 1 a 4 Tr[U µ(n)u ν (n + µ)u µ (n + µ + ν)u ν (n + ν) 1] Monte Carlo: da µ e S {U (1) µ (n), U (2) µ (n),...} 19
Lattice Results 0.6 0.3 order parameters 0.5 0.4 0.3 0.2 0.1 m q /T = 0.08 0 5.2 5.3 5.4 m L 0.2 0.1 L m q /T = 0.08 0 0 5.2 5.3 5.4 1.0 0.5 equation of state 5 4 3 2 1 0 p/t 4 p SB /T 4 3 flavor 2+1 flavor 2 flavor pure gauge T [MeV] 100 200 300 400 500 600 20
BNL and RHIC 21
Multiplicities Star TPC 22
Multiplicities 400 Au+Au 19.6 GeV 600 Au+Au 130 GeV dn ch /dη 300 200 dn ch /dη 400 100 200 0-5 0 5 η 0-5 0 5 η 800 0-6% 6-15% Au+Au 200 GeV 25-35% 35-45% dn ch /dη 600 400 15-25% 45-55% 200 0-5 0 5 η 23 Phobos White Paper (2005)
Heavy Ion Collisions τ =const t η =const hadrons QGP pre equilibrium projectile target z Scaling (Bjorken) expansion all comoving observers are equivalent 24
Bjorken Expansion 25
Radial Flow Radial expansion leads to blue-shifted spectra 1/(2π) d 2 N / (m T dm T dy) [c 4 /GeV 2 ] 10 2 10 STAR preliminary Au+Au central s NN = 200 GeV K - π - 1/(2π) d 2 N / (m T dm T dy) [c 4 /GeV 2 ] 1 10-1 K - STAR preliminary p+p min. bias s NN = 200 GeV π - p 10-2 p 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 m T - m 0 [GeV/c 2 ] m T - m 0 [GeV/c 2 ] v T 0.6c! m T = p 2 T + m2 26
Elliptic Flow Hydrodynamic expansion converts coordinate space anisotropy to momentum space anisotropy Anisotropy Parameter v 2 0.3 0.2 0.1 0 Hydro model π K p Λ PHENIX Data π + +π - + - K +K p+p STAR Data 0 K S Λ+Λ b 0 2 4 6 Transverse Momentum p T (GeV/c) source: U. Heinz (2005) 27
Elliptic Flow II v 2 /ε 0.25 HYDRO limits 0.2 0.15 0.1 E lab /A=11.8 GeV, E877 E lab /A=40 GeV, NA49 0.05 E lab /A=158 GeV, NA49 s NN =130 GeV, STAR s NN =200 GeV, STAR Prelim. 0 0 5 10 15 20 25 30 35 (1/S) dn ch /dy source: U. Heinz (2005) 28
Viscosity Energy momentum tensor T µν T µν + η( µ u ν + ν u µ trace) perturbative QCD η = 107T 3 /(g 4 log(g 1 )) universal bound (D. Son)? η/s 1/(4π) bound saturated in strong coupling SUSY theories with gravitational dual v 2 (p T ) 0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 b 6.8 fm (16-24% Central) STAR Data Γ s /τ o = 0 Γ s /τ o = 0.1 Γ s /τ o = 0.2 0.02 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 p T (GeV) source: D. Teaney (2003) 29
Jet Quenching R AA = n AA N coll n pp 1.6 1.4 1.2 Au-Au, Cent Au-Au, Periph d-au, min-bias 1 R AA 0.8 0.6 0.4 0.2 0 0 2 4 6 8 10 p T (GeV/c) source: Phenix White Paper (2005) 30
Jet Quenching II Disappearance of away-side jet φ) dn/d( φ) dn/d( 1/N 1/NTRIGGER Trigger 0.2 0.1 d+au FTPC-Au 0-20% p+p min. bias Au+Au Central 0-1 0 1 2 3 4 φ (radians) source: Star White Paper (2005) 31
Jet Quenching: Theory 10.0 RHIC data energy loss governed by ˆq = ρ q dq 2 2 dσ dq 2 q (GeV 2 /fm) 1.0 0.1 pion gas QGP cold nuclear matter 0.01 0.1 1 10 100 larger than pqcd predicts? (GeV/fm 3 ) source: R. Baier (2004) 32
Gauge Theory at Strong Coupling: Holographic Duals The AdS/CFT duality relates large N c (Conformal) gauge theory in 4 dimensions correlation fcts of gauge invariant operators string theory on 5 dimensional Anti-de Sitter space S 5 boundary correlation fcts of AdS fields exp dx φ 0 O = Z string [φ( AdS) = φ 0 ] z x The correspondence is simplest at strong coupling g 2 N c strongly coupled gauge theory classical string theory 33
Gauge Theory at Strong Coupling: Finite Temperature Thermal (conformal) field theory AdS 5 black hole CFT temperature Hawking temperature of black hole CFT entropy Hawking-Bekenstein entropy =area of event horizon weak coupling s = π2 2 N 2 c T 3 = 3 4 s 0 s strong coupling λ = g 2 N Extended to transport properties by Policastro, Son and Starinets η = π 8 N 2 c T 3 34
Summary (Experiment) Matter equilibrates quickly and behaves collectively Little Bang, not little fizzle Initial energy density in excess of 10 GeV/fm 3 Conditions for Plasma achieved Evidence for strongly interacting Plasma ( sqgp ) Fast equilibration τ 0 1 fm Strong energy loss of leading partons 35
Part II: QCD at Finite Density Nuclear Systems T µ e nuclear liquid neutron gas µ 36
QCD at Finite Density Partition function [ Z = Tr e β(h µn)] β = 1/T N = d 3 x ψ ψ Path integral representation (euclidean) Z = da µ det(id/ + iµγ 4 )e S = da µ e iφ det(id/ + iµγ 4 ) e S Sign problem: importance sampling does not work Also: No general theorems (a la Vafa-Witten) Phase structure much richer (breaking of translational, rotational, parity, isospin,... symmetry) 37
Perfect Liquids Neutron Star (Crab) sqgp Trapped Fermi Gas 38
Universality What do these systems have in common? dilute: rρ 1/3 1 a r k 1 F strongly correlated: aρ 1/3 1 Neutron Matter a nn = 18 fm, r nn = 2.7 fm 0.001ρ 0 ρ 0.3ρ 0 Feshbach Resonance in 6 Li a = 45 a.u., k 1 F 103 a.u. a = a ( 1 + B B 0 ) 39
Universality Consider limiting case ( Bertsch problem) Universal equation of state How to find ξ? (k F a) (k F r) 0 E ( E ) A = ξ = ξ 3 ( k 2 F A 0 5 2M Numerical Simulations Experiments with trapped fermions Analytic Approaches ) 40
Neutron Matter on the Lattice P (MeV/fm 3 ) 0.14 0.12 0.1 0.08 0.06 0.04 fc f 1 f 2 b 1 b 2 s 1 s 2 FP 0.02 0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 ρ/ρ N P 0.5P 0 41
Analytic Approaches Two body correlations: Sum particle-particle ladders 1 0.75 0.5 E A = k2 F 2M pp (f 2(k F a)/(3π) -0.5 PP -> f PP ) 1 6 pp (D -> 8 ) 35π (11 2 log(2))(k F a) -0.75-1.5-1 -0.5 0 0.5 E/E 0 0.25 0-0.25 pp ladders pp (f PP ->2) 1/k F a Systematic Approach: Large d Expansion ξ = 0.5 + O(1/d) 42
Part III: QCD at Finite Density Color Superconductivity T color superconductor µ e µ 43
Very Dense Matter Consider baryon density n B 1 fm 3 quarks expected to move freely Ground state: cold quark matter (quark fermi liquid) n(p) P F only quarks with p p F scatter p F Λ QCD coupling is weak P F p No chiral symmetry breaking, confinement, or dynamically generated masses 44
Color Superconductivity Is the quark liquid stable? P F PF P F P F P P Attractive interaction leads to instability F P F F P F Dominant interaction: Uses Fermi surface coherently qq condensate, superfluidity/superconductivity, gap in fermion spectrum, transport without dissipation QCD: gluon exchange attractive in 3 channel 3 3 = 6 S + 3 A flux reduced attractive Spin-flavor-color wave function ( ) (ud du) (rb br) s = 0, I = 0, c = 3 45
Phase Diagram: First Version Universality, lattice results T 2 nd E Plasma 1st order transition ends Hadrons E Nuclear Matter 1 µ st Why? Competition between <qq> and <qq> critical endpoint (E) persists even if m 0 46
Lattice Results T/MeV 185 180 175 170 165 160 155 150 N f =2, [FP] N f =2+1, [FK] N f =2, [All] N f =4, [EL] 145 0 200 400 600 800 µ B /MeV [FK] Improved re-weighting, [FP] imaginary chemical potential [All] Taylor expansions 47
Phase Diagram: Freezeout T (MeV) 250 200 150 crossover quark gluon plasma end point 100 50 NA49 SIS, AGS RHIC hadrons 0 0 500 1000 µ B (MeV) 48
Superconductivity quark-quark scattering (µ Λ QCD ) Gap equation: double logarithmic behavior (p 0 ) = g2 18π 2 dq 0 {log ( bm p 0 q 0 ) } +... (q 0 ) q 2 0 + (q 0 ) 2 collinear log BCS log 0 = 512π 4 µg 5 exp ( ) ) π2 + 4 exp ( 3π2 8 2g 49
µ : CFL Phase Consider N f = 3 (m i = 0) F L C L C R F R q a i q b j = φ ɛ abi ɛ iji ud = us = ds <q q > <q q > L L R R rb = rg = bg Symmetry breaking pattern: Rotate left flavor Compensate by rotating color SU(3) L SU(3) R [SU(3)] C U(1) SU(3) C+F All quarks and gluons acquire a gap... have to rotate right flavor also! ψ L ψ L = ψ R ψ R 50
Phase Diagram: N f = 3 QCD T Plasma H superfluidity Hadrons (Hyper) Nuclear CFL Matter <qq> Matter 2 <(qq) > µ Quark Hadron Continuity 51
QCD quarks, gluons 2p F Effective Field Theories 2 HDET CFLChTh quasi particles, holes, gluons Goldstone bosons p = p F Quantumchromodynamics High density effective theory L = ψ(id/ + µγ 0 )ψ 1 4 Ga µνg a µν L = ψ v(iv D)ψ v 2 ψt vcψ v 1 4 Ga µνg a µν +... Chiral effective theory L = f 2 π 4 Tr [ 0 Σ 0 Σ v 2 Σ Σ ] + Tr ( N iv µ D µ N ) +... 52
Mass Terms: Match HDET to QCD L = ψ R MM 2µ ψ R + ψ L M M 2µ ψ L R R R L R + MM M + M + C µ 2 (ψ R Mλa ψ L )(ψ R Mλa ψ L ) R g 2MM L R g M L R L R L g M mass corrections to FL parameters ˆµ, v F and V ++ 0 53
Mass Terms: Match HDET to CFLχTh Kinetic term: ψ L X Lψ L + ψ R X Rψ R D 0 N = 0 N + i[γ 0, N], Γ 0 = V 0 + 1 2 ( ξxr ξ + ξ X L ξ ) 0 Σ = 0 Σ + ix L Σ iσx R vector (axial) potentials Contact term: (ψ R Mψ L)(ψ R Mψ L) L = 3 2 4π 2 { [Tr(MΣ)] 2 Tr(MΣMΣ) } meson mass terms 54
Phase Structure and Spectrum Phase structure determined by effective potential V (Σ) = f 2 π 2 Tr ( X L ΣX R Σ ) ATr(MΣ ) B 1 [ Tr(MΣ ) ] 2 +... V (Σ 0 ) min Fermion spectrum determined by L = Tr ( N iv µ D µ N ) + Tr ( N γ 5 ρ A N ) + 2 { Tr (NN) [Tr (N)] 2}, ρ V,A = 1 2 { ξ M M 2p F ξ ± ξ MM 2p F } ξ ξ = Σ 0 55
Phase Structure and Spectrum 20 10 0 CFL CFLK + 80 µ Q (MeV) -10-20 CFLπ CFLK 0 60 40-30 20-40 10-8 10-6 10-4 10-2 10 0 m s 2 /2µ (MeV) 10 20 30 20 40 60 80 100 meson condensation: CFLK m s (crit) m 1/3 u 2/3 gapless modes? (gcflk) µ s (crit) 4 3 56
Phase Diagram: m s 0 T E Plasma Hadrons E nuclear superfluid Nuclear Matter exotics CFL 2SC CFL K µ p CFLK, LOFF Phase structure at moderate µ (and m s, µ e 0) complicated and poorly understood. Systematic calculations m 2 s µ 2, m s µ 2, g 1 Use neutron stars to rule out certain phases What are the most useful observables? 57
Conclusion: The Many Phases of QCD high T QGP large N c strings large N f conformal phase QCD low density nuclear matter high energy color glas large density color superconductivity 58