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QCD – from the vacuum to high temperature

QCD – from the vacuum to high temperature. an analytical approach. Functional Renormalization Group. from small to large scales. How to come from quarks and gluons to baryons and mesons ?.

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QCD – from the vacuum to high temperature

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

  2. Functional Renormalization Group from small to large scales

  3. How to come from quarks and gluons to baryons and mesons ? Find effective description where relevant degrees of freedom depend on momentum scale or resolution in space. Microscope with variable resolution: • High resolution , small piece of volume: quarks and gluons • Low resolution, large volume : hadrons

  4. Scales in strong interactions

  5. /

  6. Flow equation for average potential

  7. Simple one loop structure –nevertheless (almost) exact

  8. Infrared cutoff

  9. Wave function renormalization and anomalous dimension for Zk (φ,q2) : flow equation isexact !

  10. approximations On the exact level : New flow equation forZk (φ,q2) needed ! Often approximative form ofZk (φ,q2) is known or can be simply computed e.g.small anomalous dimension

  11. Partial differential equation for function U(k,φ) depending on two ( or more ) variables Z k= c k-η

  12. Regularisation For suitable Rk: • Momentum integral is ultraviolet and infrared finite • Numerical integration possible • Flow equation defines a regularization scheme ( ERGE –regularization )

  13. Momentum integral is dominated by q2 ~ k2 . Flow only sensitive to physics at scale k Integration by momentum shells

  14. Scalar field theory e.g. linear sigma-model for chiral symmetry breaking in QCD

  15. Scalar field theory

  16. O(N) - model • First order derivative expansion

  17. Ising model Flow of effective potential CO2 T* =304.15 K p* =73.8.bar ρ* = 0.442 g cm-2 Experiment : S.Seide …

  18. Critical behaviour

  19. Critical exponents

  20. Scaling form of evolution equation On r.h.s. : neither the scale k nor the wave function renormalization Z appear explicitly. Fixed point corresponds to second order phase transition. Tetradis …

  21. Flow equation contains correctly the non-perturbative information ! (essential scaling usually described by vortices) Essential scaling : d=2,N=2 Von Gersdorff …

  22. Kosterlitz-Thouless phase transition (d=2,N=2) Correct description of phase with Goldstone boson ( infinite correlation length ) for T<Tc

  23. Exact renormalization group equation

  24. Generating functional

  25. Effective average action Loop expansion : perturbation theory with infrared cutoff in propagator

  26. Quantum effective action

  27. Proof of exact flow equation

  28. Truncations Functional differential equation – cannot be solved exactly Approximative solution by truncation of most general form of effective action

  29. Exact flow equation for effective potential • Evaluate exact flow equation for homogeneous field φ . • R.h.s. involves exact propagator in homogeneous background field φ.

  30. Nambu Jona-Lasinio model

  31. Critical temperature , Nf = 2 Lattice simulation J.Berges,D.Jungnickel,…

  32. Chiral condensate

  33. pion mass sigma mass temperature dependent masses

  34. Criticalequationofstate

  35. Scalingformofequationof state Berges, Tetradis,…

  36. Universal critical equation of state is valid near critical temperature if the only light degrees of freedom are pions + sigma with O(4) – symmetry. Not necessarily valid in QCD, even for two flavors !

  37. Chiral quark–meson model J.Berges, D.Jungnickel…

  38. Effective low energy theory • Imagine that for scale k=700 MeV all other fields except quarks have been integrated out and the result is an essentially pointlike four quark interaction • Not obviously a valid approximation !

  39. In principle, m can be computed from four quark interaction in QCD Meggiolaro,… Connection with four quark interaction

  40. Chiral quark-meson model – three flavors - • Φ : (3,3) of SU(3)L x SU(3)R • Effective potential depends on invariants

  41. Spontaneous chiral symmetry breaking

  42. Limitations • Confinement not included • Pointlike interaction at scale kφnot a very accurate description of the physics associated with gluons • substantial errors in nonuniversal quantities ( e.g. Tc )

  43. Conclusions Non-perturbative flow equation ( ERGE ) • Useful non-perturbative method • Well tested in simple bosonic and fermionic systems • Interesting generalizations to gauge theories , gravity

  44. Flow equations for QCD Much work in progress by various groups • Gluodynamics ( no quarks ) • Quark-meson models • Description of bound states • For realistic QCD : if Higgs picture correct, this would greatly enhance the chances of quantitatively reliable solutions

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