1. THERMODYNAMICS OF THE HIGH TEMPERATURE QUARK-GLUON PLASMA Jean-Paul Blaizot, CNRS and ECT* Komaba - Tokyo November 25, 2005

2. Plan of the lecture Introduction. QCD and its phase diagram. Perturbation theory and its difficulties. Role of fluctuations in weak coupling expansions.. Effective theories. Dimensional reduction. Hard thermal loops. Resummations. Entropy and the quasiparticle picture. Large Nf as a test of weak coupling expansions Conclusions.

4. QCD and its phase-diagram Control parameters

5. T BB Hadronic matter Quark-Gluon Plasma Nuclei Colour superconductor The QCD phase diagram

10. Perturbation theory at high temperature

12. The pressure to order

14. Perturbation theory up to order Lattice:

15. The problem is not limited to QCD : similar difficulties are met in scalar theories

17. Origin of the difficulty: weak coupling but many active degrees of freedom

18. Role of fluctuations in perturbation theory Thermal fluctuations Compare ‘kinetic energy’ with ‘interaction energy’ Magnitude of fluctuations depends on typical wavenumber

19. Hard modes (‘plasma particles’) Soft modes (collective modes) Special scale Perturbation theory OK But expansion in g The dynamics of soft modes is non perturbatively renormalized by their coupling to hard modes

20. Scales, fluctuations and degrees of freedom in the quark-gluon plasma

21. Classical field approximation Dimensional reduction

22. DIMENSIONAL REDUCTION (scalar field) Effective action for the « zero mode » where Hard: Soft:

23. DIMENSIONAL REDUCTION (QCD) Integration over the hard modes In leading order

24. Integration over the soft modes Non perturbative contribution

26. Collective phenomena Hard thermal loops Effective theories

29. Collective modes and thermodynamics - the static mode of dimensional reduction may not be the most relevant degree of freedom - it is possible to include dynamical information about the physical modes of the plasma in the calculation of the thermodynamics For the calculation of the thermodynamical functions - the bad behaviour of perturbative expansion is not directly attributed to it (e.g. scalar case)

33. Screened perturbation theory

34. +Petitgirard, PRD66,0851016(2002); hep-ph/0302069

35. (From J.-P. B., E. Iancu and A. Rebhan, hep-ph/0303045)

37. Skeleton expansion

38. Thermodynamic potential In terms of dressed propagators: : sum of all two-particle-irreducible (2PI) skeleton diagrams

39. Thermodynamic potential Contributions to (2PI diagrams) (Ghost-free diagrams) G. Baym, PR127,1391(1962) J. M. Luttinger and J. C. Ward, PR 118, 1417 (1960) J. M. Cornwall, R. Jackiw, and E. Tomboulis, PR D10, 2428 (1974) full propagators From Andreas Ipp, ECT*

40. Thermodynamic potential Contributions to (2PI diagrams) (Ghost-free diagrams) 2-loop approximation From Andreas Ipp, ECT*

41. Stationarity property Entropy and number densities: At 2-loop level further simplifications :

42. Entropy is simple!

43. Entropy and number density Simple formulae at 2-loop level: UV finite! can use HTL J.-P. B, E. Iancu, A. Rebhan: Phys.Rev.D63:065003,2001 Note:  functional dropped out.

45. HTL approximation with In 2PI entropy formula: Correctly reproduces g 2 term, but gives only part of the g 3 term HTL works well for soft scales ~gT not so good for hard scales, except at small virtuality

46. HTL-NLA: correct g 3 Andreas Ipp, ECT* In order to obtain also the correct g 3 term, we have to include corrections to the asymptotic thermal masses These are weighted averages, and tuned to reproduce also the correct g 3 term in the pressure.

47. Two successive approximations

48. State of the art Compare Lattice – 2PI J.-P. B., E. Iancu, A. Rebhan: Phys.Rev.D63:065003,2001 F. Karsch, Nucl.Phys.A698:199-208,2002; G. Boyd et al., Nucl. Phys. B469, 419 (1996). from J.-P. B., E. Iancu, A. Rebhan: Nucl.Phys.A698:404-407,2002 pure-glue SU(3) Yang-Mills theory

49. Large N f

50. Thermodynamic potential Large N f limit Contributions to (2PI diagrams) (Ghost-free diagrams) 2-loop approximation Next-to-leading order leading order gets exact at

51. Thermal pressure at large N f Leading order Next-to-leading order +++…= Ring diagram resummation

52. Thermal pressure at large N f Leading order Next-to-leading order Note: Formula is independend of 

53. Landau pole Finite Large : QCD is asymptotically free Large N f not asymptotically free anymore!

54. Landau pole Result exact to LO in 1/N f expansion Landau pole Necessity of regularization  large momentum cutoff Cutoff must be Euclidean invariant (prevent potential fake logarithmic divergencies) Choose cutoff with

55. Moore, JHEP 0210 A.Ipp, Moore, Rebhan, JHEP 0301 Pressure at Large N f

56. Φ-derivable approximations in the Large N f limit

57. 2PI formula contains Σ Separating off ideal gas contribution: To test the 2PI formula in the large Nf limit, we have to evaluate  (k) on the light-cone. Andreas Ipp, ECT* 2PI formula itself is UV safe, but one has to be careful about  (k): Cutoff necessary there.

58. Fermionic self- energy Numerically demanding Supercomputer BEN@ECT*

59. Non-trivial test successful Fermionic entropy contributionS f calculated by integration over fermion self-energy Σ agrees with previous calculation to 2-3 digits. (3 days for 25 nodes of ECT* TeraFlop cluster) J.-P. B., A.Ipp, A. Rebhan, U. Reinosa, hep-ph/0509052

60. Fermionic self-energy Possibility to test NLA approximation, and go beyond  NLO J.-P. B., A.Ipp, A. Rebhan, U. Reinosa, hep-ph/0509052

61. Entropy in large N f Surprisingly good agreement! J.-P. B., A.Ipp, A. Rebhan, U. Reinosa, hep-ph/0509052

62. Outlook Full QCD NLO calculation Replace average by k-dependent Hope for further improvement! Test 2PI at finite chemical potential

63. Summary Accuracy of perturbation theory depends on momentum scale of relevant fluctuations; effective theories can be used efficiently While strict perturbation theory is meaningless, weak coupling methods provide an accurate description of the QGP for Simple physical picture emerges for the quark-gluon plasma at high temperature in terms of quasiparticles What happens to these quasiparticles when one approaches the phase transition? Weak coupling techniques, with resummation, are remarkably succesfull at large Nf, even in regimes where the coupling is not small