1. QCD – from the vacuum to high temperature an analytical approach

2. 3 lectures Condensates and phase diagram Condensates and phase diagram Higgs picture of QCD Higgs picture of QCD Functional renormalization group Functional renormalization group

3. Condensates and Phase Diagram Action and functional integral Action and functional integral Generating functionals Generating functionals Chiral symmetry Chiral symmetry Condensates and symmetry breaking Condensates and symmetry breaking Phase transition Phase transition Has T c been measured? Has T c been measured?

4. Understanding the phase diagram

5. Cosmological phase transition… …when the universe cools below 175 MeV 10 -5 seconds after the big bang 10 -5 seconds after the big bang

6. QCD at high density Nuclear matter Heavy nuclei Neutron stars Quark stars …

7. Why analytical approach ? Complementary to numerical investigations (lattice – QCD ) (lattice – QCD ) Many questions cannot be addressed by lattice - QCD

8. QCD

9. Quantum numbers quarkquarkgluonmesonbaryon particleudg π+π+π+π+p electric charge 2/3-1/3011 isospin1/2-1/2011/2 color33811

10. Light particles Light particles mesons pseudoscalar octet: (π +,π -,π 0,K +,K -,K 0,K 0,η) pseudoscalar octet: (π +,π -,π 0,K +,K -,K 0,K 0,η) vector meson octet + … vector meson octet + …baryons octet : (p,n,Λ,Σ +,Σ -,Σ 0,Ξ 0,Ξ - ) octet : (p,n,Λ,Σ +,Σ -,Σ 0,Ξ 0,Ξ - ) decuplet + … decuplet + … -

11. quarks and gluons at high momenta Visible as jets at high virtual momenta Visible as jets at high virtual momenta Perturbation theory valid Perturbation theory valid

12. Short and long distance degrees of freedom are different ! Short distances : quarks and gluons Short distances : quarks and gluons Long distances : baryons and mesons Long distances : baryons and mesons How to make the transition? How to make the transition? confinement confinement

13. Action +

14. QCD is microscopic theory of strong interactions

15. Local gauge symmetry Local gauge symmetry

16. Running coupling : QCD effective gauge coupling depends on momentum scale μ

17. Generating functionals, Effective action

18. Partition function and functional integral Integration over all configurations of gauge field A and quark fields ψ Fermion fields are anticommuting Grassmann variables This is the difficult part ! Needs regularisation ! Lattice QCD ; gauge fixing,ghosts,Slavnov Taylor identities …

19. Correlation functions Only gauge invariant correlation functions can differ from zero. Local gauge symmetries are not spontaneously broken Local gauge symmetries are not spontaneously broken ( exact proof ) ( exact proof ) Scalar correlation functions Scalar correlation functions describes propagation of (pseudo)scalar mesons describes propagation of (pseudo)scalar mesons in suitable channel in suitable channel Vector meson correlation function Vector meson correlation function

20. Generating functionals Generating functionals Add to the action a source term Add to the action a source term Partition function becomes a functional of the sources, Z[j]. Can be evaluated for arbitrary j(x) Partition function becomes a functional of the sources, Z[j]. Can be evaluated for arbitrary j(x)

21. Generating functional and correlation functions Concentrate on scalars – other channels can be treated in complete analogy other channels can be treated in complete analogy Order parameter Order parameter Correlation function Correlation function

22. Generating functional for connected Green’s functions W[ j ]=ln Z[ j ] W[ j ]=ln Z[ j ] G c : propagator or G c : propagator or connected two point function connected two point function

23. Effective action Effective action Effective action is defined by Legendre transform Matrix of second functional derivatives Γ (2) is the inverse propagator

24. Physical sources With this convention for physical sources the quark mass term is not included in S anymore. S: action for N f massless quarks Γ has the full chiral symmetry

25. “Solution” of QCD Effective action ( for suitable fields ) contains all the relevant information of the solution of QCD Gauge singlet fields, low momenta: Order parameters, meson-( baryon- ) propagators Order parameters, meson-( baryon- ) propagators Gluon and quark fields, high momenta: Perturbative QCD Perturbative QCD Aim: Computation of effective action

26. Symmetries If functional measure preserves the symmetries of S : If functional measure preserves the symmetries of S : Γ has the same symmetries as S Γ has the same symmetries as S If not : anomaly If not : anomaly Examples for symmetries of Γ : Examples for symmetries of Γ : - gauge symmetry, - gauge symmetry, - global chiral flavor symmetry SU(N f )xSU(N f ) - global chiral flavor symmetry SU(N f )xSU(N f ) Example for anomaly: global axial U(1) Example for anomaly: global axial U(1)

27. Chiral symmetry breaking

28. Chiral symmetry For massless quarks the action of QCD is invariant under independent phase rotations acting on left handed and right handed quarks. SU(N f ) L x SU(N f ) R SU(N f ) L x SU(N f ) R

29. Quark masses Quark mass terms mix left handed and right handed quarks Quark mass terms mix left handed and right handed quarks They break the axial symmetry They break the axial symmetry Equal quark masses leave vectorlike “diagonal” SU(N) unbroken. Equal quark masses leave vectorlike “diagonal” SU(N) unbroken. Quark mass terms can be treated as sources for scalar quark-antiquark bilinears. Quark mass terms can be treated as sources for scalar quark-antiquark bilinears.

30. Effective potential Evaluate effective action for homogenous field φ for homogenous field φ

31. Spontaneous symmetry breaking SYM SYM =0 =0 SSB =φ 0 ≠ 0 =φ 0 ≠ 0

32. Pions Two-flavor QCD, σ : 2x2 matrix Effective potential

33. Spontaneous chiral symmetry breaking For sufficiently small m² and positive λ 1,2 : For sufficiently small m² and positive λ 1,2 : ( even for vanishing quark masses ) ( even for vanishing quark masses )

34. Goldstone bosons Flat directions in potential, dictated by symmetry breaking massless (pseudo)scalar fields : pions massless (pseudo)scalar fields : pions Presence of quark mass term shifts minimum and gives mass to pions

35. Nonlinear σ - model Nonlinear description (neglects the scalar excitations beyond π,η)

36. Kinetic term

37. Chiral perturbation theory

40. Meson fluctuations Insert factor unity Insert factor unity New action and functional integral involves also σ - field New action and functional integral involves also σ - field Hubbard,Stratonovich

41. σ - interactions Four fermion interaction can cancel corresponding piece induced by loops in QCD Four fermion interaction can cancel corresponding piece induced by loops in QCD Typical form Typical form Mass and kinetic term for σ Mass and kinetic term for σ Yukawa interaction between σ and quarks Yukawa interaction between σ and quarks Source now for σ Source now for σ

42. Connection with quark bilinears

43. Effective action with scalar fluctuations

44. QCD phase transition

45. Phase diagram Phase diagram R.Pisarski =φ 0 ≠ 0 =φ 0 ≠ 0 =0 =0

46. Chiral symmetry restoration at high temperature High T SYM =0 =0 Low T SSB =φ 0 ≠ 0 =φ 0 ≠ 0 at high T : less order more symmetry examples: magnets, crystals

47. QCD at high temperature Quark – gluon plasma Quark – gluon plasma Chiral symmetry restored Chiral symmetry restored Deconfinement ( no linear heavy quark potential at large distances ) Deconfinement ( no linear heavy quark potential at large distances ) Lattice simulations : both effects happen at the same temperature Lattice simulations : both effects happen at the same temperature

48. Lattice results e.g. Karsch,Laermann,Peikert Critical temperature in chiral limit : N f = 3 : T c = ( 154 ± 8 ) MeV N f = 2 : T c = ( 173 ± 8 ) MeV Chiral symmetry restoration and deconfinement at same T c

49. QCD – phase transition Quark –gluon plasma Quark –gluon plasma Gluons : 8 x 2 = 16 Gluons : 8 x 2 = 16 Quarks : 9 x 7/2 =12.5 Quarks : 9 x 7/2 =12.5 Dof : 28.5 Dof : 28.5 Chiral symmetry Chiral symmetry Hadron gas Hadron gas Light mesons : 8 Light mesons : 8 (pions : 3 ) (pions : 3 ) Dof : 8 Dof : 8 Chiral sym. broken Chiral sym. broken Large difference in number of degrees of freedom ! Strong increase of density and energy density at T c !

50. Pressure Pressure

51. Analytical description of phase transition Needs model that can account simultaneously for the correct degrees of freedom below and above the transition temperature. Needs model that can account simultaneously for the correct degrees of freedom below and above the transition temperature. Partial aspects can be described by more limited models, e.g. chiral properties at small momenta. Partial aspects can be described by more limited models, e.g. chiral properties at small momenta.

52. Quark descriptions ( NJL-model ) fail to describe the high temperature and high density phase transitions correctly High T : chiral aspects could be ok, but glue … (pion gas to quark gas ) (pion gas to quark gas ) High density transition : different Fermi surface for quarks and baryons ( T=0) quarks and baryons ( T=0) – in mean field theory factor 27 for density at given chemical potential – – in mean field theory factor 27 for density at given chemical potential – Confinement is important : baryon enhancement Berges,Jungnickel,… Berges,Jungnickel,… Chiral perturbation theory even less complete

53. Universe cools below 170 MeV… Both gluons and quarks disappear from Both gluons and quarks disappear from thermal equilibrium : mass generation thermal equilibrium : mass generation Chiral symmetry breaking Chiral symmetry breaking mass for fermions mass for fermions Gluons ? Gluons ? Analogous situation in electroweak phase transition understood by Higgs mechanism Analogous situation in electroweak phase transition understood by Higgs mechanism Higgs description of QCD vacuum ? Higgs description of QCD vacuum ?

54. Order of the phase transition is crucial ingredient for experiments ( heavy ion collisions ) and cosmological phase transition

55. Order of the phase transition

56. Second order phase transition

57. First order phase transition First order phase transition

58. Matsubara formalism Only change for T ≠ 0 : Only change for T ≠ 0 : Euclidean time on torus with circumference β =1/T Euclidean time on torus with circumference β =1/T

59. Matsubara formalism for fermions

60. Thermodynamic potentials and observables Evaluate effective potential U at minimum free energy free energy pressure pressure energy density energy density vanishing chemical potential and sources

61. Thermodynamics Effective action, when evaluated on torus for euclidean time with circumference 1/T, (and in presence of chemical potential ) contains full information about thermodynamic behavior of QCD. Effective action, when evaluated on torus for euclidean time with circumference 1/T, (and in presence of chemical potential ) contains full information about thermodynamic behavior of QCD.

62. Has T c been measured ? Observation : statistical distribution of hadron species with “chemical freeze out temperature “ T ch =176 MeV Observation : statistical distribution of hadron species with “chemical freeze out temperature “ T ch =176 MeV T ch cannot be much smaller than T c : hadronic rates for T ch cannot be much smaller than T c : hadronic rates for T< T c are too small to produce multistrange hadrons (Ω,..) T< T c are too small to produce multistrange hadrons (Ω,..) Only near T c multiparticle scattering becomes important Only near T c multiparticle scattering becomes important ( collective excitations …) – proportional to high power of density ( collective excitations …) – proportional to high power of density P.Braun-Munzinger,J.Stachel,CW T ch ≈T c

63. Heavy ion collision Heavy ion collision

64. Hadron abundancies Hadron abundancies

65. Very rapid density increase …in vicinity of critical temperature Extremely rapid increase of rate of multiparticle scattering processes ( proporional to very high power of density )

66. Energy density Lattice simulations Lattice simulations Karsch et al Karsch et al

67. Production time for Ω multi-meson scattering π+π+π+K+K -> Ω+p Ω+p strong dependence on pion density strong dependence on pion density P.Braun-Munzinger,J.Stachel,CW

68. T ch ≈ T c T ch ≈ T c

69. Phase diagram Phase diagram hadrons hadrons quarks and gluons

71. Running coupling : QED

72. Chiral perturbation theory Chiral perturbation theory

73. Matsubara formalism