Download presentation

Presentation is loading. Please wait.

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

2
Functional Renormalization Group from small to large scales 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. Find effective description where relevant degrees of freedom depend on momentum scale or resolution in space. Microscope with variable resolution: Microscope with variable resolution: High resolution, small piece of volume: quarks and gluons quarks and gluons Low resolution, large volume : hadrons

4
Scales in strong interactions

7
/

10
Flow equation for average potential

11
Simple one loop structure – nevertheless (almost) exact

12
Infrared cutoff

13
Wave function renormalization and anomalous dimension for Z k (φ,q 2 ) : flow equation is exact !

14
approximations On the exact level : New flow equation for Z k (φ,q 2 ) needed ! Often approximative form of Z k (φ,q 2 ) is known or can be simply computed e.g.small anomalous dimension

15
Partial differential equation for function U(k,φ) depending on two ( or more ) variables Z k Z k = c k -η

16
Regularisation For suitable R k : Momentum integral is ultraviolet and infrared finite Numerical integration possible Flow equation defines a regularization scheme ( ERGE –regularization )

17
Integration by momentum shells Momentum integral is dominated by q 2 ~ k 2. q 2 ~ k 2. Flow only sensitive to physics at scale k physics at scale k

18
Scalar field theory e.g. linear sigma-model for chiral symmetry breaking in QCD

19
Scalar field theory Scalar field theory

20
O(N) - model First order derivative expansion

21
Flow of effective potential Ising model CO 2 T * =304.15 K p * =73.8.bar ρ * = 0.442 g cm-2 Experiment : S.Seide …

22
Critical behaviour

23
Critical exponents

24
Scaling form of evolution equation Tetradis … On r.h.s. : neither the scale k nor the wave function renormalization Z appear explicitly. Fixed point corresponds to second order phase transition.

25
Essential scaling : d=2,N=2 Essential scaling : d=2,N=2 Flow equation contains correctly the non-perturbative information ! (essential scaling usually described by vortices) Von Gersdorff …

26
Kosterlitz-Thouless phase transition (d=2,N=2) Correct description of phase with Goldstone boson Goldstone boson ( infinite correlation length ) ( infinite correlation length ) for T<T c for T<T c

27
Exact renormalization group equation

28
Generating functional Generating functional

29
Loop expansion : perturbation theory with infrared cutoff in propagator Effective average action

30
Quantum effective action

31
Proof of exact flow equation

32
Truncations Truncations Functional differential equation – cannot be solved exactly cannot be solved exactly Approximative solution by truncation of most general form of effective action most general form of effective action

35
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.

36
Nambu Jona-Lasinio model

37
Critical temperature, N f = 2 J.Berges,D.Jungnickel,… Lattice simulation

38
Chiral condensate Chiral condensate

39
temperature dependent masses temperature dependent masses pion mass pion mass sigma mass sigma mass

40
Critical equation of state

41
Scaling form of equation of state Berges, Tetradis,…

42
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 !

43
Chiral quark–meson model J.Berges, D.Jungnickel…

44
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 !

45
Connection with four quark interaction Connection with four quark interaction In principle, m can be computed from four quark interaction in QCDMeggiolaro,…

46
Chiral quark-meson model – three flavors Φ : (3,3) of SU(3) L x SU(3) R Effective potential depends on invariants -

47
Spontaneous chiral symmetry breaking Spontaneous chiral symmetry breaking

48
Limitations Confinement not included Pointlike interaction at scale k φ not a very accurate description of the physics associated with gluons substantial errors in nonuniversal quantities ( e.g. T c )

49
Conclusions Non-perturbative flow equation ( ERGE ) Useful non-perturbative method Well tested in simple bosonic and fermionic systems Interesting generalizations to gauge theories, gravity

50
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

51
end

Similar presentations

© 2020 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google