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

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Functional Renormalization Group from small to large scales from small to large scales

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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

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Scales in strong interactions

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Flow equation for average potential

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Simple one loop structure – nevertheless (almost) exact

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Infrared cutoff

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

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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

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

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

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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

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

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Scalar field theory Scalar field theory

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O(N) - model First order derivative expansion

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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 …

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Critical behaviour

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Critical exponents

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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.

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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 …

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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

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Exact renormalization group equation

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Generating functional Generating functional

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

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Quantum effective action

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Proof of exact flow equation

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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

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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.

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Nambu Jona-Lasinio model

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

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Chiral condensate Chiral condensate

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

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Critical equation of state

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Scaling form of equation of state Berges, Tetradis,…

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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 !

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

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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 !

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

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

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Spontaneous chiral symmetry breaking Spontaneous chiral symmetry breaking

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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 )

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

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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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