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The Phases of QCD

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The Phases of QCD The Phases of QCD Thomas Schaefer North Carolina State University 1 Motivation Di®erent 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 modi¯ed QCD simpli¯es 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 Plan 1. QCD and Symmetries 2. The High Temperature Phase: Theory 3. Exploring QCD at High Temperature: Experiment 4. QCD at Low Density: Nuclear Matter 5. QCD at High Density: Quark Matter 6. Matter at Finite Density: From the Lab to the Stars 4 Quantumchromodynamics Elementary ¯elds: Quarks Gluons color a = 1; : : : ; 3 a 8 a color a = 1; : : : ; 8 (q®) spin ® = 1; 2 A 8 f > ¹ § < < spin ²¹ flavor f = u; d; s; c; b; t > : :> Dynamics: non-abelian gauge theory 1 L = q¹ (iD= ¡ m )q ¡ Ga Ga f f f 4 ¹º ¹º a a a abc b c G¹º = @¹Aº ¡ @º A¹ + gf A¹Aº ¹ a a iD= q = γ i@¹ + gA¹t q ¡ ¢ 5 \Seeing" Quarks and Gluons L LLLL 3 LLLLL333 e+ q e+ q g e− q e− q 6 Running Coupling Constant 7 Asymptotic Freedom Study modi¯cation of classical ¯eld by quantum fluctuations 1 1 k2 A = Acl + ±A F 2 ! + c log F 2 ¹ ¹ ¹ g2 µg2 µ¹2 ¶¶ ¤ ¤ £ ¡ £ ¢¡ ¥ ¤ © © ¦¨§ ¦¨§ dielectric ² > 1 paramagnetic ¹ > 1 dielectric ² > 1 ¹² = 1 ) ² < 1 g3 1 2 ¯(g) = 2 ¡ 4 Nc + Nf (4¼) nh3 i 3 o 8 About Units QCD Lite¤ is a parameter free theory The lagrangian has a coupling constant, g, but no scale. After renormalization g becomes scale dependent g is traded for a scale parameter ¤ ¤ is the only scale, the QCD \standard kilogram" ¡1 ¤QCD ' 200 MeV ' 1 fm ¤ QCD Lite is QCD in the limit mq ! 0, mQ ! 1 9 V ( Coulomb r ) Standa » e r 2 Phases rd Mo ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ del: ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ V ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ of ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ( ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ Higgs r ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ U Gauge ) ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ » (1) 10 ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ e ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ £ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ r mr ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ S ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ Theo ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ U ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ (2) Con¯nement ries £ S V U ( (3) r ) » k r Phases of Matter phase order broken rigidity Goldstone param symmetry phenomenon boson crystal ½k translations rigid phonon magnet M~ rotations magnetization magnon superfluid h©i particle number supercurrent phonon supercond. hÃÃi gauge symmetry supercurrent none (Higgs) 11 Gauge Symmetry Local gauge symmetry U(x) 2 SU(3)c à ! Uà D¹Ã ! UD¹Ã y y y A¹ ! UA¹U + iU@¹U F¹º ! UF¹º U Gauge \symmetries" cannot be broken Gauge \symmetries" can be realized in di®erent modes Coulomb Higgs con¯ned d.o.f: 2 (massless) 3 (massive) 3 (massive) Distinction between Higgs and con¯nement phase not always sharp 12 Chiral Symmetry De¯ne left and right handed ¯elds L 1 à = (1 § γ )à R L;R 2 5 Fermionic lagrangian, M = diag(mu; md; ms) ¹ ¹ L = ÃL(iD= )ÃL + ÃR(iD= )ÃR L L R R M M ¹ ¹ + ÃLMÃR + ÃRMÃL L R R L M = 0: Chiral symmetry (L; R) 2 SU(3)L £ SU(3)R ÃL ! LÃL; ÃR ! RÃR 13 Chiral Symmetry Breaking Chiral symmetry implies massless, degenerate fermions (1=2)+ ¤ ¡ mN = 935 MeV mN (1=2) = 1535 MeV Chiral symmetry is spontaneously broken ¹f g ¹f g 3 fg hÃLÃR + ÃLÃRi ' ¡(230 MeV) ± SU(3)L £ SU(3)R ! SU(3)V (G ! H) Consequences: dynamical mass generation mQ = 300 MeV À mq mN = 890 MeV + 45 MeV (QCD; 95%) + (Higgs; 5%) 14 a Goldstone Bosons: Consider broken generator Q5 a a a a a a [H; Q5] = 0 Q5j0i = j¼ i Hj¼ i = HQ5j0i = Q5Hj0i = 0 Low energy e®ective theory for the Goldstone modes Step 1: Parameterize G=H =pseudoscalar GB's y U(x) : U ! LUR (L; R) 2 SU(3)L £ SU(3)R Vacuum U fg = ±fg. Massless fluctuations (G=H) a a a § 0 § 0 0 U(x) = exp(iÁ ¸ =f¼) Á = (¼ ; ¼ ; K ; K ; K¹ ; ´) Step 2: Write most general G invariant e®ective lagrangian f 2 L = ¼ Tr[@ U@¹U y] + : : : 4 ¹ Non-linear sigma model 15 Expand lagrangian (SU(2) sector) 1 1 @4 L = (@ Áa)2 + (Áa@ Áa)2 ¡ (Áa)2(@ Áb)2 + O 2 ¹ 6f 2 ¹ ¹ µf 4 ¶ ¼ £ ¤ ¼ Step 3: Low energy expansion (power counting) 2 2 4 4 T¼¼ » k1;2=f¼ + k1;2=f¼ + : : : Relation to f¼: Couple weak gauge ¯elds § ¨ π W @¹U ! (@¹ + igW¹ ¿ )U e f § ¹ ¨ π ν L = gf¼W¹ @ ¼ 16 Quark Masses y Non-zero quark masses: L = ùLMÃR + ùRM ÃR M ! LMRy spurion ¯eld M Chiral lagrangian at leading order in M L = BTr[MU] + h:c: Mass matrix M = diag(mu; mdms). Minimize e®ective potential Uvac = 1; Evac = ¡BTr[M] hùÃi = ¡B Expand around Uvac: pion mass 2 2 ¹ m¼f¼ = (mu + md)hÃÃi Chiral expansion m n 4 @U m¼ L = f¼ ¤Â = 4¼f¼ µ ¤Â ¶ µ ¤Â ¶ 17 Symmetries of the QCD Vacuum: Summary Local SU(3) gauge symmetry con¯ned: V (r) » kr Chiral SU(3)L £ SU(3)R symmetry spontaneously broken to SU(3)V Axial U(1)A symmetry N anomalous : @ A0 = f Ga G~a ¹ ¹ 16¼2 ¹º ¹º Vectorial U(1)B symmetry unbroken: B = d3x Ãyà conserved R 18 Notes QCD with general Nf ; Nc (with or without SUSY) ¯nd theories without con¯nement and/or chiral symmetry breaking QCD with Nf = Nc = 3 con¯nement implies chiral symmetry breaking ¤ symmetry breaking pattern SU(3)L £ SU(3)R ! SU(3)V unique order parameter hùÃi 6= 0 19 N 10 12 14 4 6 8 f QCD 2 chiral symmetry breaking, confinement 'weak' chiral symmetry breaking b=0 conformal b=2 Phase QCD 3 b'=0 N Diagram: c lattice, conf IILM, chisb 4 20 10 12 14 4 6 8 ¡ ¡ ¡ 2 ¡ ¡ ¡ ¡ ¡ chiral symmetry breaking, confinement no chisb, confinement IR free ¡ ¡ ¡ ¡ N ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ c ¡ ¡ ¡ ¡ SUSY QCD ¡ ¡ ¡ ¡ and ¡ ¡ ¡ ¡ conformal 3 ¡ ¡ ¡ ¡ Coulomb no ground state ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ N ¡ ¡ ¡ ¡ N ¡ ¡ ¡ ¡ c f ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ 4 ¡ ¡ ¡ ¡ Approaching the Phase Diagram: Symmetries and Weak Coupling Arguments T QGP critical critical critical color pion superconductor µ gas e µ neutron gas 21 Approaching the Phase Diagram: Strongly Correlated Phases T QGP critical sQGP Hagedorn critical critical color BEC− superconductor pion nuclear liquid BCS µ gas e µ neutron gas 22 Approaching the Phase Diagram: Experiments and Numerical Simulations T RHIC's lattice HIC's neutron stars µ e µ 23.
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