The phases of hot/dense/magnetized QCD from the lattice. Gergely Endrődi

Transcription

1 The phases of hot/dense/magnetized QCD from the lattice Gergely Endrődi Goethe University of Frankfurt EMMI NQM Seminar GSI Darmstadt, 27. June 2018

2 QCD phase diagram 1 / 45

3 Outline relevance of background magnetic fields and isospin asymmetries hot/magnetized QCD including magnetic fields including very strong magnetic fields results: phase diagram hot/asymmetric QCD including the isospin asymmetry pion condensation results: phase diagram further applications conclusions 2 / 45

4 Introduction

5 Strong interactions explain 99.9% of visible matter in the Universe 3 / 45

6 Strong interactions explain 99.9% of visible matter in the Universe elementary particles: quarks and gluons 3 / 45

7 Strong interactions explain 99.9% of visible matter in the Universe elementary particles: quarks and gluons elementary fields: ψ(x) and A µ (x) QCD Lagrangian L QCD = 1 4 Tr F µν(g s, A) 2 + ψ[γ µ ( µ + ig s A µ ) + m]ψ 3 / 45

8 Strong interactions explain 99.9% of visible matter in the Universe elementary particles: quarks and gluons elementary fields: ψ(x) and A µ (x) QCD Lagrangian L QCD = 1 4 Tr F µν(g s, A) 2 + ψ[γ µ ( µ + ig s A µ ) + m]ψ g s = O(1) non-perturbative physics 3 / 45

9 Path integral and lattice field theory path integral [Feynman 48] ( Z = DA µ D ψ Dψ exp ) d 4 x L QCD (x) discretize spacetime on a lattice with spacing a [Wilson 74] Monte-Carlo algorithms to generate configurations dimensional integrals high-performance computing 4 / 45

10 Path integral and lattice field theory path integral [Feynman 48] ( Z = DA µ D ψ Dψ exp ) d 4 x L QCD (x) discretize spacetime on a lattice with spacing a [Wilson 74] Monte-Carlo algorithms to generate configurations dimensional integrals high-performance computing 4 / 45

11 QCD and external parameters running coupling g s (E) relevant parameters that control the energy scale: temperature T : excites all states baryon density n B n u + n d : excites p + and n isospin asymmetry n I n u n d : creates p + -n asymmetry, excites π + background magnetic field B: forces quarks on Landau levels (chemical potentials conjugate to densities: µ B, µ I ) 5 / 45

12 Magnetic fields: heavy-ion collisions off-central events generate magnetic fields [Kharzeev, McLerran, Warringa 07] strength: B = T B earth 5m 2 π impact: chiral magnetic effect, anisotropies, elliptic flow, hydrodynamics with B,... reviews: [Fukushima 12] [Kharzeev, Landsteiner, Schmitt, Yee 14] [Kharzeev 15] 6 / 45

13 Magnetic fields magnetars neutron stars with strong surface magnetic fields [Duncan, Thompson 92] strength on surface: B = T strength in core: B = T B earth impact: convectional processes, mass-radius relation, gravitational collapse/merger [Anderson et al 08],... 7 / 45

14 Magnetic fields: early universe I large-scale intergalactic magnetic fields 10 µg = 10 9 T I origin in the early universe I generation through a phase transition: electroweak epoch B 1019 T [Vachaspati 91, Enqvist, Olesen 93] 8 / 45

15 Isospin asymmetry: nuclei and neutron stars neutron to nucleon ratio in nuclei Z A 0.4 but: neutron skin near surface neutron to nucleon ratio in interior of neutron stars Z A / 45

16 Order parameters

17 Order parameters quark condensate ψψ (chiral symmetry breaking) ψψ = T log Z V m = 1 Tr /D + m Polyakov loop P (deconfinement) P = Tr P exp 1/T 0 A 4 (x, τ) dτ 10 / 45

18 Order parameters T c inflection point nature of phase transition singularity in slope at T c l,s 1.0 Continuum 0.8 N t 16 N t 12 N t N t T MeV Renormalized Polyakov loop 1.0 Continuum N t N t 12 N t N t T MeV [Aoki, Endrődi et al 06, Borsányi et al. 10] 11 / 45

19 Order parameters T c inflection point nature of phase transition singularity in slope at T c l,s 1.0 Continuum 0.8 N t 16 N t 12 N t N t T MeV Renormalized Polyakov loop 1.0 Continuum N t N t 12 N t N t T MeV [Aoki, Endrődi et al 06, Borsányi et al. 10] analytical crossover 11 / 45

20 Background magnetic fields

21 Impact of magnetic fields on quark condensate: primary B ψ ψ on Polyakov loop: secondary B A 4 A 4 12 / 45

22 Magnetic catalysis chiral condensate spectral density around 0 [Banks, Casher 80] ψψ tr /D 1 ρ(0) large magnetic fields reduce dimensionality and induce degeneracy B to maintain ψψ > 0 [Gusynin et al 96] B = 0 ρ(p)dp Tp 2 dp strong interaction is needed B m 2 ρ(p)dp TB dp the weakest interaction suffices 13 / 45

23 Magnetic catalysis: lattice simulations numerical simulation of the path integral Z = DU D ψ Dψ exp( S QCD ) obtain condensate from ψψ = 1 V 4 log Z m physical m π, continuum limit [Bali,Bruckmann,Endrődi,Fodor,Katz,Schäfer 12] 14 / 45

24 Magnetic catalysis zero temperature magnetic catalysis at zero temperature is a robust concept: χpt, NJL, AdS-CFT, linear σ model, lattice QCD,... [Andersen, Naylor 14] [Bali,Bruckmann,Endrődi,Fodor,Katz,Schäfer 12] 15 / 45

25 Effect of magnetic fields: nonzero temperature

26 Phase diagram: models recall catalysis argument ψψ ρ(0) model calculations at T > 0: magnetic catalysis for all T Tc (B) increases for example the PNJL model [Gatto, Ruggieri 11] eb 19 eb 15 eb 10 eb T MeV T Χ, T P MeV D ΧSR C ΧSB 1.0 TΧ 0.9 TP T B MeV eb m Π 16 / 45

27 eb/m Phase diagram: models majority of low-energy models give the same qualitative result linear sigma model + Polyakov loop [Mizher, Chernodub, Fraga 10] quark-meson model + functional renormalization group [Kamikado, Kanazawa 13] NJL model + Polyakov loop [Ferreira, Costa, Menezes, Providencia, Scoccola 13]... T (MeV) With vacuum corrections Chiral transition Deconfinement transition Expected magnetic field generated in the LHC eb(m π ) T pc (B)/T pc (0) full FRG LPA Mean Field Tc (MeV) Tc (MeV) T u c T d c T χ c T Φ c PNJL EPNJL eb (GeV 2 ) 17 / 45

28 Phase diagram: lattice simulations lattice QCD, physical m π, continuum limit [Bali,Bruckmann,Endrődi,Fodor,Katz,Krieg,Schäfer,Szabó 11, 12] surprise: magnetic catalysis turns into inverse magnetic catalysis (IMC) around T c [Bruckmann, Endrődi, Kovács 13] 18 / 45

29 Phase diagram: lattice simulations lattice QCD, physical m π, continuum limit [Bali,Bruckmann,Endrődi,Fodor,Katz,Krieg,Schäfer,Szabó 11, 12] surprise: magnetic catalysis turns into inverse magnetic catalysis (IMC) around T c [Bruckmann, Endrődi, Kovács 13] 18 / 45

30 Phase diagram: lattice simulations impact on the QCD phase diagram [Bali, Bruckmann, Endrődi, Fodor, Katz, Krieg, Schäfer, Szabó 11] 19 / 45

31 Phase diagram: lattice simulations impact on the QCD phase diagram [Bali, Bruckmann, Endrődi, Fodor, Katz, Krieg, Schäfer, Szabó 11] [Endrődi 15] 19 / 45

32 Phase diagram: comparison T Χ, T P MeV D ΧSR C ΧSB TΧ TP T B MeV T (P) c eb m Π 2 model lattice T c (B) increases decreases and T ( ψψ) c diverge converge condensate magnetic catalysis T inverse catalysis T T c 20 / 45

33 Phase diagram: comparison T Χ, T P MeV D ΧSR C ΧSB TΧ TP T B MeV T (P) c eb m Π 2 model lattice T c (B) increases decreases and T ( ψψ) c diverge converge condensate magnetic catalysis T inverse catalysis T T c models and lattice simulations are as different as can be 20 / 45

34 Inverse magnetic catalysis related to secondary effect of B on gluons encoded in effective potential for P in models tune this potential or coupling constants of models with B [Fraga et al. 13, Ferreira et al. 14, Ayala et al. 15, ] 21 / 45

35 Large B: anisotropic effective theory

36 Large B limit what happens to L QCD at eb Λ 2 QCD, T 2? first guess: asymptotic freedom says α s 0 i.e. complete decoupling of quarks and gluons but: B breaks rotational symmetry and effectively reduces the dimension of the theory for quarks gluons also inherit this spatial anisotropy, κ(b) B [Miransky, Shovkovy 2002; Endrődi ] L QCD B tr B 2 + tr B2 + [1 + κ(b)] tr E 2 + tr E2 22 / 45

37 Simulating the anisotropic effective theory pure (but anisotropic) gauge theory: can be simulated on the lattice [Endrődi ] Polyakov loop susceptibility peak height scales with V histogram shows double peak-structure at T c 23 / 45

38 Simulating the anisotropic effective theory pure (but anisotropic) gauge theory: can be simulated on the lattice [Endrődi ] Polyakov loop susceptibility peak height scales with V histogram shows double peak-structure at T c the transition is of first order 23 / 45

39 Critical point analytical crossover for 0 eb 3.25 GeV 2 first-order transition for B there must be a critical point in between [Cohen, Yamamoto 13] estimate: extrapolate width of susceptibility peak to 0 eb CP 10(2) GeV 2 24 / 45

40 Phase diagram 25 / 45

41 Isospin asymmetry

42 Isospin chemical potential quark chemical potentials (3-flavor) µ u = µ B 3 + µ I µ d = µ B 3 µ I µ s = µ B 3 µ S zero baryon number, zero strangeness, but nonzero isospin µ u = µ I µ d = µ I µ s = 0 pion chemical potential µ π = µ u µ d = 2µ I isospin density n I = n u n d 26 / 45

43 Pion condensation QCD at low energies pions chiral perturbation theory chemical potential for charged pions: µ π at zero temperature µ π < m π vacuum state µ π m π Bose-Einstein condensation [Son, Stephanov 00] T <u γ 5 d>=0 A m π < π >=0 <u γ 5 d>=0 µ I 27 / 45

44 Bose-Einstein condensate accumulation of bosonic particles in lowest energy state [Anderson et al 95 JILA-NIST/University of Colorado] velocity distribution of Ru atoms at low temperature peak at zero velocity (zero energy) 28 / 45

45 Setup on the lattice [Brandt, Endrődi, Schmalzbauer 17]

46 Symmetry breaking QCD with light quark matrix M = /D + m ud 1 + µ I γ 0 τ 3 + iλγ 5 τ 2 chiral symmetry (flavor-nontrivial) SU(2) V U(1) τ3 29 / 45

47 Symmetry breaking QCD with light quark matrix M = /D + m ud 1 + µ I γ 0 τ 3 + iλγ 5 τ 2 chiral symmetry (flavor-nontrivial) SU(2) V U(1) τ3 29 / 45

48 Symmetry breaking QCD with light quark matrix M = /D + m ud 1 + µ I γ 0 τ 3 + iλγ 5 τ 2 chiral symmetry (flavor-nontrivial) SU(2) V U(1) τ3 spontaneously broken by a pion condensate π ± = ψγ 5 τ 1,2 ψ a Goldstone mode appears 29 / 45

49 Symmetry breaking QCD with light quark matrix M = /D + m ud 1 + µ I γ 0 τ 3 + iλγ 5 τ 2 chiral symmetry (flavor-nontrivial) SU(2) V U(1) τ3 spontaneously broken by a pion condensate π ± = ψγ 5 τ 1,2 ψ a Goldstone mode appears add small explicit breaking 29 / 45

50 Symmetry breaking QCD with light quark matrix M = /D + m ud 1 + µ I γ 0 τ 3 + iλγ 5 τ 2 chiral symmetry (flavor-nontrivial) SU(2) V U(1) τ3 spontaneously broken by a pion condensate π ± = ψγ 5 τ 1,2 ψ a Goldstone mode appears add small explicit breaking extrapolate results λ 0 29 / 45

51 Simulation details nx +ny +nz +nt staggered light quark matrix with η 5 = ( 1) ( ) D(µ / M = I ) + m λη 5 λη 5 /D( µ I ) + m we have γ 5 τ 1 -hermiticity η 5 τ 1 Mτ 1 η 5 = M determinant is real and positive det M = det( /D(µ I ) + m 2 + λ 2 ) > 0 partition function via Monte-Carlo Z = DU det( /D(µ I ) + m 2 + λ 2 ) }{{} light quarks det( /D(0) + m s ) }{{} strange quark e Sg }{{} gluons 30 / 45

52 Need for improvement simulations at λ > 0 are far away from desired λ 0 limit 31 / 45

53 Need for improvement simulations at λ > 0 are far away from desired λ 0 limit improvement using the singular value representation of M 31 / 45

54 Singular value representation pion condensate Σ π = λ log det( /D(µ I ) + m 2 + λ 2 2λ ) = Tr /D(µ I ) + m 2 + λ 2 singular values spectral representation /D(µ I ) + m 2 ψ i = ξ 2 i ψ i Σ π = T V i 2λ ξ 2 i + λ 2 V dξ ρ(ξ) 2λ ξ 2 + λ 2 λ 0 πρ(0) first derived in [Kanazawa, Wettig, Yamamoto 11] 32 / 45

55 Singular value representation pion condensate Σ π = λ log det( /D(µ I ) + m 2 + λ 2 2λ ) = Tr /D(µ I ) + m 2 + λ 2 singular values spectral representation /D(µ I ) + m 2 ψ i = ξ 2 i ψ i Σ π = T V i 2λ ξ 2 i + λ 2 V dξ ρ(ξ) 2λ ξ 2 + λ 2 λ 0 πρ(0) first derived in [Kanazawa, Wettig, Yamamoto 11] compare to Banks-Casher-relation at µ I = 0 32 / 45

56 Singular value density integrated spectral density N(ξ) = ξ 0 dξ ρ(ξ ), ρ(0) = lim ξ 0 N(ξ)/ξ 33 / 45

57 Singular value density integrated spectral density N(ξ) = ξ 0 dξ ρ(ξ ), ρ(0) = lim ξ 0 N(ξ)/ξ compare ρ(0) to velocity distribution around zero 33 / 45

58 Singular value density integrated spectral density N(ξ) = ξ 0 dξ ρ(ξ ), ρ(0) = lim ξ 0 N(ξ)/ξ compare ρ(0) to velocity distribution around zero Bose-Einstein condensation! 33 / 45

59 Results: phase transition

60 Condensates pion and chiral condensate after λ 0 extrapolation read off chiral crossover T pc (µ I ) and pion condensation boundary µ I,c (T ) 34 / 45

61 Order of the transition volume scaling of order parameter shows typical sharpening collapse according to O(2) critical exponents [Ejiri et al 09] 35 / 45

62 Order of the transition volume scaling of order parameter shows typical sharpening collapse according to O(2) critical exponents [Ejiri et al 09] indications for a second order phase transition at µ I = m π /2, in the O(2) universality class 35 / 45

63 Continuum extrapolations compare (pseudo)critical temperatures for different lattice spacings a = 1/(N t T ) take continuum limit a 0 (N t ) 36 / 45

64 Phase diagram meeting point of chiral crossover and pion condensation boundary: pseudo-triple point at T pt = 151(7) MeV, µ I,pt = 70(5) MeV 37 / 45

65 Deconfinement vs chiral symmetry breaking Polyakov loop contour lines apparently insensitive to pion condensation boundary existence of a condensed but deconfined phase? 38 / 45

66 Conjectured phase diagram 39 / 45

67 Conjectured phase diagram quark-gluon plasma hadronic phase BCS phase BEC phase 39 / 45

68 Conjectured phase diagram quark-gluon plasma hadronic phase BCS phase BEC phase BCS phase expected on general grounds for high µ I [Son, Stephanov 00] 39 / 45

69 Potential applications

70 Magnetic fields in the early universe? I large-scale intergalactic magnetic fields 10 µg = 10 9 T origin in the early universe I generation through a phase transition: electroweak epoch B 1019 T 600 GeV2 /e [Vachaspati 91, Enqvist, Olesen 93] I how large is B that survives until the QCD epoch? 40 / 45

71 Magnetic fields in the early universe? I large-scale intergalactic magnetic fields 10 µg = 10 9 T origin in the early universe I generation through a phase transition: electroweak epoch B 1019 T 600 GeV2 /e [Vachaspati 91, Enqvist, Olesen 93] I how large is B that survives until the QCD epoch? 40 / 45

72 Pion condensation in the early universe? weak equilibrium u d e ν e (simplicity: one family of particles) charge neutrality n Q = 0, baryon symmetry n B = 0 but nonzero lepton number n L 0 chemical potentials µ Q, µ B and µ L can µ Q = 2µ I > m π be reached? for sufficiently large n L, yes [Abuki, Brauner, Warringa 09] 41 / 45

73 Pion condensation in compact stars? equation of state for isospin-dense system at low T, neutralized by leptons [Brandt, Endrődi, Fraga, Hippert, Schaffner-Bielich, Schmalzbauer 18] solve Tollman-Oppenheimer-Volkov equations for gravitational stability 100 black holes π + e M [M ] 10 π + µ π R [km] weak decays under investigation 42 / 45

74 Magnetic fields in heavy-ion collisions? off-central events generate magnetic fields [Kharzeev, McLerran, Warringa 07] test charge-dependence in RHIC isobar run [bnl.gov] 43 / 45

75 CME-sensitive observables for nonzero density correlation of topology and electric polarization [Bali, Bruckmann, Endrődi, Fodor, Katz, Schäfer 14] D u ( ) = d 4 x q top (x)e z (x + ) correlations affected by isospin chemical potential? 44 / 45

76 Summary phase diagram for strong background magnetic fields phase diagram for nonzero isospin-asymmetry new Banks-Casher-type relation establish pion condensation improve various observables 45 / 45