QCD in extreme conditions

Transcription

1 Department of Physics Trondheim, Norway Physics seminar, NIKHEF, donderdag 12 februari 2015

2 Introduction Phase diagram of QCD

3 Introduction QCD is considered to be the correct theory of the strong interactions. L QCD = i ψ f (γ µ D µ m f )ψ f 1 4 Ga µνg µν a Symmetries of L QCD : Local SU(3) c symmetry Massless quarks SU(N f ) L SU(N f ) R - otherwise SU(N f ) V Baryon number U(1) B

4 Introduction The world we live in: Hadrons are color singlets - bound states of quarks Chiral symmetry broken in the vacuum Pions are the (pseudo)-goldstone particles from chiral symmetry breaking: SU(2) L SU(2) R SU(2) V

5 Flux tubes and linear potential

6 Heavy-ion collisions RHIC, LHC, GSI... t f HHG h c MP q c QGP Pre-Equilibrium! 0 z y! x

7 Quark stars Structure of compact stars

8 Extreme QCD What happens to matter as you heat it up? T < Tconf linear potential V(r) constant potential T > Tconf r

9 MIT Bag model Hot hadron gas versus a quark-gluon plasma P HHG = 3 π2 T 4 90, P QGP = 2(Nc 2 1) π2 T N 7 cn f 8 P P QGP T c P HHG T 4 B π 2 T 4 90 B. Bag constant B 1 4 = 145 MeV gives T c = 150 MeV.

10 Order parameters and phase transitions Chiral symmetry broken by ψψ Polyakov loop [ β ] L = P exp i A 0 (x, τ)dτ 0 Φ e 2πin Nc Φ under center symmetry Z Nc, Φ = 1 N c TrL L ren (T) HISQ/tree: N τ =6 N τ =8 N τ =12 stout, cont. SU(3) SU(2) T [MeV] χ l /T2 <ψ ψ> sub N τ =8, f K scale T [MeV] Petreczky J.Phys. G39 (2012

11 Universality arguments Universality arguments give information about order of phase transition: Two massless quarks: transition is second order Three massless quarks: transition is first order T, GeV QGP restored chiral symmetry massless quarks T, GeV 2 massless quarks tricritical point Lattice hadron gas broken chiral symmetry vacuum Models nuclear matter quark matter phases 0 1 µb, GeV 0 1 µb, GeV Critical mass for the s quark(!)

12 Physical quark masses Physical quarks masses Crossover for µ B = 0 Line of first-order transition ends at critical point T, GeV QGP 0.1 crossover vacuum critical point hadron gas nuclear matter quark matter phases CFL 0 1 µb, GeV

13 Quark-gluon plasma At sufficiently high temperature, we expect weakly interacting quarks and gluons. Perturbation theory? Lattice gauge theories: Z = DAD ψdψe β 0 dτ d 3 x ψ[/d+m f γ 0 µ f ]ψ e S YM = DAe β 0 dτ d 3 xl YM det[/d + m f γ 0 µ f ] where det[/d + m f γ 0 µ f ] = det[(x + iµ f )(X + iµ f ) + m 2 ] ix = D 0 + iσ D Real if µ B = 0 or imaginary or µ u = µ d

14 Weak-coupling expansion The weak-coupling expansion of the free energy of QCD has been calculated to order α 3 s log α s a. Ideal&Gas& Perturbative free energy vs temperature for QCD with N F = 2 and N c = 3. Very poorly convergent. Generic problem in hot field theories (scalar field theory, QED). Goal is to find a gauge-invariant framework with better convergence properties that is also able to describe dynamical properties. a Arnold and Zhai, 94/95, Kastening and Zhai 95, Braaten and Nieto 96, Kajantie, Laine, Rummukainen, and Schröder 02.

15 Dimensional reduction Three momentum scales: Hard scale T, soft scale gt, and supersoft scale g 2 T. Integrate out hard scale T perturbatively to obtain an effective three-dimensional theory (Electrostatic QCD). 2 L eff = 1 2 Tr[G2 ij ] + Tr[(D ia 0 ) 2 ] + m 2 D Tr[A2 0 ] +... Parameters depend on T, µ f, g... 2 Braaten and Nieto 96, Kajantie, Laine, K. Rummukainen, and Shaposhnikov 96, Vuorinen 03.

16 Dimensional reduction Integrate out scale gt i.e. A 0 to obtain a second effective field theory (magnetostatic QCD) 3 P = P hard + P soft + P supersoft Hard scale T contributions from dimensional reduction. Soft scale gt contributions from calculations using EQCD. MQCD contributes first at order g 6. As in HTLpt, keep full g-dependence of m D to resum. 3 Braaten and Nieto (95), Kajantie Laine, Rummukainen, and Shaposhnikov (96), Vuorinen (03)

17 Screeened perturbation theory Massless scalar field theory L = 1 2 ( µφ) 2 + g2 24 φ4. -1 = -1 + Reorganization L 0 = 1 2 ( µφ) m2 φ 2, L int = 1 2 m2 φ 2 + g2 24 φ P + m 2 -g -m 2 1

18 Hard-thermal-loop perturbation theory QCD Lagrangian L QCD = 1 2 Tr[G µνg µν ] + i ψγ µ D µ ψ + L gh + L gf + L QCD For soft momenta gt, one needs effective propagators and vertices Γ 2 " Γ 3 " Γ 4 " Γ n "

19 Hard-thermal-loop perturbation theory Expansion point is gas of massive quasiparticles by adding HTL Lagrangian ( L HTL = 1 y α ) 2 (1 δ)m2 D Tr y β G µα (y D) 2 G µβ ŷ +(1 δ)im 2 q ψγ µ yµ y D ψ, ŷ L = (L QCD + L HTL ) g g δ + L HTL, HTLpt δ formal expansion parameter and δ = 1 at the end. Generates effective propagators and vertices.

20 Three-loop calculation 0 Compute all Feynman diagram through 3 loops using HTL propagators and vertices 1 Give a prescription for mass parameters m D and m q. 2

21 Thermodynamic potential Ω NNLO Ω 0 = 7 4 d F d A ln ˆΛ q ˆµ s F α s π 2 7 ˆµ ˆµ 4 + s α F s 7 π ) 1 ℵ(z) ˆm 3 D ( 90ˆm 2 qˆm D s 2F ( ˆµ 2 ) ( ˆµ 2 ) + 15 α s π ( ˆµ 2 ) ˆm D (1 12 ) ζ ( 1) + ˆµ2 472 ˆµ 2 ζ( 1) ) 36i ˆµℵ(2, z) + 6(1 + 8 ˆµ 2 )ℵ(1, z) + 3i ˆµ(1 + 4 ˆµ 2 )ℵ(0, z) 45 4ˆm D ˆm D 5 ( ˆµ 2 ) ( ˆµ 2 ) ˆm 2 q ( ˆµ ˆµ 4 ) ( ˆµ 2 ) 2 γ E ( ˆµ 2 ) ζ ( 1) ˆµ ˆµ 4 ζ( 1) ln 2 ˆm D ( ˆµ 2 ) ˆΛ q 2 ζ ( 3) ζ( 3) [ 8ℵ(3, z) + 3ℵ(3, 2z) 12 ˆµ 2 ℵ(1, 2z) + 12i ˆµ (ℵ(2, z) + ℵ(2, 2z)) i ˆµ( ˆµ 2 ) ℵ(0, z) ] 2(1 + 8 ˆµ 2 )ℵ(1, z) c Aα s 3π [ s F α s 319 π ( ˆµ 2 ) ( ˆµ 2 ) 235 2ˆm D 16 ˆµ ˆµ γ E 47 2 ln ˆΛ q 1 ℵ(z) ( ˆµ 2 ) 44 ˆm D ˆµ ˆµ i ˆµℵ(0, z) + (5 92 ˆµ 2 ) ℵ(1, z) + 144i ˆµℵ(2, z) + 52ℵ(3, z) ( ˆµ 2 ) γ E ˆµ ℵ(z) 7 ˆm D 11 ln ] ˆµ 2 NNLO(Λ g) + ΩYM. Ω 0 ˆΛ q ζ ( 1) ζ( 1) ˆm 2 q ˆm D 4 ( ˆµ 2 ) ln ˆm D ζ ( 3) ζ( 3) ˆµ 2 ln ˆΛ q 2

22 Results - pressure for µ B = 0 and µ B = 400 MeV Use one-loop running with α s (1.5 GeV) = Mass prescription: use m f = 0 and m D from EQCD 5 ˆm 2 D = { α s 3π c A + c2 A α s 12π γ E + 22 ln ˆΛ g + 1 [ 2 N f f s F ( ˆµ 2 f ) + c As F α s ( ) ) ˆµ 2 + 4ℵ(z f f ) + s2 α F s ( ) ˆµ 2 (1 2 + ℵ(z f f )) 3 s 2F α s ( ) ]} ˆµ 2 f 3π 2 π Μ B 0 MeV 1 loopαs ; MS 176 MeV 1.0 Μ B 400 MeV 12π (( ) ( ) ˆµ ˆµ 2 γ f f E 1 loopαs ; MS 176 MeV ideal NNLO HTLpt Wuppertal Budapest TMeV 4 Bazavov et al (2012) ideal TMeV NNLO HTLpt Wuppertal Budapest 5 Vuorinen Borsanyi et al 10 and 12, Bazavov et al 09.

23 Results - interaction measure and speed of sound c 2 s = P E. 3 T 4 5 Μ B 0MeV 1loopΑs ; 176 MeV MS 4 NNLO HTLpt 3 WB T MeV 3 T Μ B 400 MeV 1loopΑs ; MS 176 MeV T MeV NNLO HTLpt WB 0.35 Μ B 0MeV 1loopΑs ; MS 176 MeV 0.35 Μ B 400 MeV 1loopΑs ; MS 176 MeV c s NNLO HTLpt 0.15 Wuppetral Budapest Stefan Boltzman limit T MeV c s Stefan Boltzman limit T MeV NNLO HTLpt Wuppetral Budapest

24 Susceptibilities Pressure P T 4 = P 0 T 4 + ijk 1 i!j!k! χ ijk ( µu T ) i ( µd T ) j ( µs ) k T Quark - and baryon-number susceptibilities χ ijk... = i+j+k+... P(T, µ) µ i u µ j. d µk s µ=0 χ B n = n P. µ n B µb =0 [ χ uu 2 + χdd 2 + χss 2 + 2χud 2 + 2χus 2 + 2χds 2 χ B 2 = 1 9 χ uu 2 = χ 200, etc. ],

25 Results - susceptibilities 7 Χ 2 f Χ 2 B u NNLO HTLpt DR WB B,stout BNL BI B, N Τ 8 HISQ WB u,stout BNL BI u,hisq BNL BI u,p4 TIFR u, N Τ 8 SQ BNL BI s,hisq MILC s, N Τ 8 asqtad T MeV Χ 4 B Χ 4 B,f DR WBB,stout BNLBIB, N Τ 8HISQ T MeV NNLO HTLpt 7 Borsanyi et al 10 and 12, Bazavov et al 13.

26 What happens to baryonic matter as you squeeze it? Nucleons overlap and quarks can move over larger distances

27 Quark matter Free massless quarks P = 2N c N f Order-α correction d 3 k (2π) 3 (µ E k)θ(µ E k ) = N cn f 12π 2 µ4. δp = α s π N f (N 2 c 1) 16π 2 µ 4.

28 Quark matter Ring diagrams give rise to term α 2 s ln α s (plasmon contribution). 8 a) V + M b) Ω VV V V V c) Ω VM V V M d) Ω ring V n=2 M M M Order-α 2 s including strange-quark mass. 9 8 Freedman and McLerran (1977), and Baluni (1978) 9 Kurkela, Romatschke, and Vuorinen (2009)

29 Superconductivity Superconductivity in metals Attractive interaction instability of Fermi surface towards formation of Cooper pairs and superconductivity

30 Color superconductivity What is the stable ground state at asymptotically high density? At high density, i.e. µ Λ QCD, QCD is weakly coupled. p p k p k k = a Attractive Repulsive s

31 Color superconductivity Attractive interaction instability of Fermi surface towards formation of quark pairs and color superconductivity Nonperturbative phenomenon. Solve gap equation for : k q k = + q = + k p p k+q k

32 Color superconductivity Color superconductivity simpler than ordinary superconductivity - no need for phonons. Diquark condensate qq 0 is gauge variant and plays the role of the Higgs field in electroweak theory. CSC phases are the Higgs phases of QCD qq is related to spectrum of excitations - position of poles of propagators are gauge invariant.

33 Color superconductivity At very high density m s effectively zero and all quarks pair up in a symmetric way ψ a,α L,i ɛ αβ ψ b,β L,j = ψ Color-flavor-locked phase: a, α R,i ɛ α β ψb, β R,j ɛ ijkɛ abk U(1) B SU(3) c SU(3) L SU(3) R SU(3) c+l+r. Nine Goldstone bosons and eight massive gluons. Breaking of chiral symmetry distinguishes this phase from the QGP phase. Automatically color and electrically neutral, µ e = 0.

34 Model Lagrangian L = ψ (γ µ µ m) ψ + G ψγ µ T A ψ ψγ µ T A ψ Gap equation G d 3 p ɛ 2 p + 2 Λe 1 g 2. More accurate Λe 1 g

35 Neutrality constraints Bulk matter must be electrically neutral - otherwise large energy penalty: E coulomb n 2 Q R5. Constraint: Q e = Ω µ e = 0. For three massless quarks, this is achieved for µ u = µ d = µ s. Color neutrality constraint Q 3,8 = Ω µ 3,8 = 0.

36 Neutrality constraints Charge neutrality realized dynamically in QCD due the generation of condensates A 3, Must be imposed by hand in NJL type models due to global SU(N c ) symmetry µ ru = 1 3 µ B 2 3 µ e µ µ 8, µ bs = 1 3 µ B 2 3 µ e 2 3 µ 8,... =...

37 Pairing Pairing depends on density p F p F p F d M s 4µ u 2 M s 4µ s d u s d u s red green blue red green blue red green blue Mismatch of Fermi momenta p F = µ 2 m 2 s µ m2 s 2µ s

38 Stress on CFL phase - pairing with nonzero momentum p or kaon condensation. At medium densities, only u and d pair up: ψ a,α L,i ɛ αβ ψ b,β L,j = ψ 2SC phase: a, α R,i ɛ α β ψb, β R,j ɛ ij3ɛ ab3 SU(3) c SU(2) L SU(2) R SU(2) c SU(2) L SU(2) R No Goldstone modes and five massive gluons.

39 Other phases Pion condensation vs color superconductivity ¼¼ ¼¼ Åε ¼ Õ µi (MeV) 2G ūu, 2G dd 2G ūiγ5d ¹ ¼¼ Õ ¹¼¼ ¹¼¼ ¹ ¼¼ ¼ Ù Åε ¼¼ ¼¼ µ = 1 2 (µ u + µ d ), µ I = 1 2 (µ u µ d ), 10 Warringa, Boer, JOA. PRD (2005)

40 Compact stars Massive objects with a mass of M = 1 2M Radius R 10 km Density of a few times nuclear density Very strong density dependent magnetic fields B Gauss Rotate with frequencies of up to 700 s 1.

41 Compact stars Tolman-Oppenheimer-Volkov equations dp dr = GE(r)m(r) r 2 [ 1 2Gm(r) r r m(r) = 4π E( r) r 2 d r. 0 Bag model: [ 1 + P(r) E(r) ] 1, P = N c 12π 2 µ 4 f B E = 3P + 4B. Strange quark matter hypothesis? f ] [ 1 + 4πr 3 ] P(r) m(r)

42 At zero pressure, P = 0, 2-flavor quark matter is unstable while 3-flavor quark matter is stable. Constraint: 145 MeV < B 1 4 < 162 MeV, 2.0 B MeV MMsun B MeV B MeV R km

43 Quark stars Equation of state known only at very high density 2.4 Mass in solar mass units Λ = 2µ Λ = 3µ Radius (km) Figure: Pure quark star from 2-loop EoS with massless quarks. Fraga et al (2000).

44 Quark stars Equation of state for medium densities - model calculations L = ψ ] [iγ µ mu m N + γ 0 µ B ψ + 1 ( µ σ µ σ m 2 2 σσ 2) +... Match onto nuclear equation of state - mixed phases? 11 Neutrality is global(!) P HP (µ e µ n ) = P QP (mu e µ n ) χ = V QP V QP + V HP 11 Schertler, Leupold, and Schaffner-Bielich(1999)

45 p [MeV/fm 3 ] p [MeV/fm 3 ] NJL model and relativistic mean field: n [MeV] NJL QP MP HP M = 1:6 M M = 1:4 M 100 e [MeV] RMF M=M M max QP MP HP RMF NJL =0