QCD at finite Temperature

Transcription

1 QCD at finite Temperature II in the QGP François Gelis and CEA/Saclay

2 General outline Lecture I : Quantum field theory at finite T Lecture II : in the QGP Lecture III : Out of equilibrium systems François Gelis 2007 Lecture II / III 2nd Rio-Saclay meeting, CBPF, Rio de Janeiro, September p. 2/46

3 Lecture II : Collective effects in the QGP in the QGP François Gelis 2007 Lecture II / III 2nd Rio-Saclay meeting, CBPF, Rio de Janeiro, September p. 3/46

4 ¾ Convergence? Example: perturbative calculation of the QGP pressure : 1.4 È È ¼ Ì Ì Does not converge at all... The bare quanta of the naive perturbative expansion are quite different from the actual (dressed) quanta in the QGP François Gelis 2007 Lecture II / III 2nd Rio-Saclay meeting, CBPF, Rio de Janeiro, September p. 4/46

5 Degrees of freedom Length scales François Gelis 2007 Lecture II / III 2nd Rio-Saclay meeting, CBPF, Rio de Janeiro, September p. 5/46

6 Degrees of freedom Degrees of freedom Length scales Quarks : 2 (spin) 3 (color) = 6 (per flavor) dn q d 3 xd 3 k = 1 e ω/t + 1 (Fermi-Dirac) Gluons : 2 (spin) 8 (color) = 16 dn g d 3 xd 3 k = 1 e ω/t 1 (Bose-Einstein) Average energy per particle : E T Particle density : ρ T 3 Average distance between particles : l 1/T François Gelis 2007 Lecture II / III 2nd Rio-Saclay meeting, CBPF, Rio de Janeiro, September p. 6/46

7 Length scales Degrees of freedom Length scales 1/T : wavelength of particles in the plasma 1/gT : typical distance for collective phenomena Thermal masses of quasi-particles Screening phenomena Damping of plasma waves 1/g 2 T : distance between two small angle scatterings Color transport Photon emission 1/g 4 T : distance between two large angle scatterings Momentum, electric charge transport characteristic scale of hydrodynamic modes In the weak coupling limit (g 1), there is a clear hierarchy between these scales Distinct effective theories according to the characteristic scale of the problem under study François Gelis 2007 Lecture II / III 2nd Rio-Saclay meeting, CBPF, Rio de Janeiro, September p. 7/46

8 Vacuum fluctuations Degrees of freedom Length scales At distances scales l 1/T, medium effects are irrelevant At such scales the dynamics is simply described by QCD in the vacuum François Gelis 2007 Lecture II / III 2nd Rio-Saclay meeting, CBPF, Rio de Janeiro, September p. 8/46

9 Thermal fluctuations Degrees of freedom Length scales Distance scales 1/T l 1/gT control the bulk thermodynamic properties. The system can be studied by QCD at finite temperature The leading thermal effects can be treated by an effective theory that encompasses the main collective effects, and that has the form of a collision-less Vlasov equation François Gelis 2007 Lecture II / III 2nd Rio-Saclay meeting, CBPF, Rio de Janeiro, September p. 9/46

10 Small angle scatterings Degrees of freedom Length scales When it is necessary to follow a plasma particle over distances 1/g 2 T l, we must take into account soft (small angle) collisions with other particles of the plasma This can be done simply by adding a collision term to the previous Vlasov equation François Gelis 2007 Lecture II / III 2nd Rio-Saclay meeting, CBPF, Rio de Janeiro, September p. 10/46

11 Collision rate Degrees of freedom Length scales Collisional width (up to logs) : Γ coll = p 2 g 4 T 3 m debye d 2 p p 4 g 2 T λ 1/Γ coll is the mean free path between two small angle scatterings (θ g) Note : the mean free path between two large angle scatterings (θ 1) is 1/g 4 T François Gelis 2007 Lecture II / III 2nd Rio-Saclay meeting, CBPF, Rio de Janeiro, September p. 11/46

12 Large angle scatterings Degrees of freedom Length scales Over distance scales l 1/g 4 T, one must take into account the large angle collisions, that change significantly the direction of motion of the particle (this is necessary e.g. for calculating transport coefficients) The most efficient way to describe the system at these scales is via a Boltzmann equation for color/spin averaged particle distributions François Gelis 2007 Lecture II / III 2nd Rio-Saclay meeting, CBPF, Rio de Janeiro, September p. 12/46

13 Hydrodynamical regime Degrees of freedom Length scales The hydrodynamical regime is reached for length scales that are much larger than the mean free path : 1/g 4 T l In order to describe the system at such scales, one needs : Hydrodynamical equations (Euler, Navier-Stokes) Conservation equations for the various currents Equation of state, viscosity François Gelis 2007 Lecture II / III 2nd Rio-Saclay meeting, CBPF, Rio de Janeiro, September p. 13/46

14 Scale gt Scale g2t log(1/g) Scale g2t François Gelis 2007 Lecture II / III 2nd Rio-Saclay meeting, CBPF, Rio de Janeiro, September p. 14/46

15 Reminder : Length scales Scale gt Scale g2t log(1/g) Scale g2t 1 / gt 1 / g 2 T 1 / g 4 T François Gelis 2007 Lecture II / III 2nd Rio-Saclay meeting, CBPF, Rio de Janeiro, September p. 15/46

16 Length scales Scale gt Scale g2t log(1/g) Scale g2t A mode is perturbative if its kinetic energy is much larger than its potential energy Kinetic energy : K ( A) 2 k 2 A 2 Potential energy : U g 2 A g 2 A Thus, a mode k is perturbative if g 2 A 2 k 2 When discussing the order of magnitude of A 2, it is useful to distinguish the contribution of the various momentum scales by defining A 2 κ κ d 3 k E k n B (E k ) François Gelis 2007 Lecture II / III 2nd Rio-Saclay meeting, CBPF, Rio de Janeiro, September p. 16/46

17 Length scales Scale gt Scale g2t log(1/g) Scale g2t Hard modes : k T, A 2 T T 2. Thus, K U Soft modes : k gt, k 2 g 2 A 2 T But the contribution of soft modes to A 2 is A 2 gt gt 2, and k 2 g 2 A 2 gt The soft modes interact strongly with the hard modes, but weakly among themselves they can be described perturbatively after the hard modes have been resummed Ultrasoft modes : k g 2 T, A 2 g 2 T 2, k 2 g 2 A 2 g 2 T g 2 T The dynamics of the ultrasoft modes is completely non-perturbative, because their self-interactions are as large as their kinetic energy François Gelis 2007 Lecture II / III 2nd Rio-Saclay meeting, CBPF, Rio de Janeiro, September p. 17/46

18 Hard thermal loops Scale gt Scale g2t log(1/g) Scale g2t Braaten, Pisarski (1990), Frenkel, Taylor (1990) Obtained from the bare perturbative expansion by the resummation of Hard Thermal Loops (HTL) : L HT L (gluons) = m2 g 2 Z dωˆv 4π F v α v β µα (v D) F 2 β µ, v µ = (1, ˆv) Can be formulated as a (local) collisionless transport theory for classical particles (Blaizot, Iancu ( )) : (1) [D µ, F µν ] = m 2 g (2) [v D, W(x, ˆv)] = ˆv E(x) Z dωˆv 4π vν W(x, ˆv) W(x, ˆv) is the density of hard particles (ω T) at the location x, with a velocity in the direction ˆv (1) : Yang-Mills equation for the soft field modes (ω gt) (2) : Vlasov equation for the hard particles François Gelis 2007 Lecture II / III 2nd Rio-Saclay meeting, CBPF, Rio de Janeiro, September p. 18/46

19 Effective theory at the scale g2t ln(1/g) Scale gt Scale g2t log(1/g) Scale g2t Bödeker (1999) Soft collisions have a mean free path of λ (g 2 Tln(1/g)) 1 still perturbative after resummation of the gt modes By integrating out the modes gt, one obtains a Boltzmann-Langevin equation for the long wavelength variations of the density W(x, ˆv): [v D, W(x, ˆv)] = ˆv E(x) + ξ(x, ˆv) «Z gt +g 2 NT ln dωˆv Λ 4π I(ˆv, ˆv )W(x, ˆv ) ξ(x, ˆv) is a Gaussian noise, of correlation : «ξ(x 1, ˆv 1 ), ξ(x 2, ˆv 2 ) = 2 g2 NT 2 gt ln I(ˆv 1, ˆv 2 )δ(x 1 x 2 ) Λ I(ˆv, ˆv ) is a collision term due to the interactions with field m 2 g modes of momentum gt (small angle collisions) François Gelis 2007 Lecture II / III 2nd Rio-Saclay meeting, CBPF, Rio de Janeiro, September p. 19/46

20 Dimensional reduction Scale gt Scale g2t log(1/g) Scale g2t By summing the Matsubara modes whose frequency is non-zero (fermions, bosons for n 0), one gets a 3-dimensional Yang-Mills theory coupled to an adjoint Higgs : L E = 1 4 F 2 ij + tr[d i, A 0 ] 2 + m 2 E tra λ E 2 (tra 0 2 ) 2 + A 0 is the gluon zero mode m E, λ E are determined by matching to the underlying theory (i.e. QCD) By integrating out the massive A 0, one gets a 3-dimensional pure Yang-Mills theory : L M = 1 4 F 2 ij + its coupling g M is determined order by order from L E this Yang-Mills theory is non-perturbative, and must be simulated on a lattice (this is much simpler than simulations of 4-dim QCD) François Gelis 2007 Lecture II / III 2nd Rio-Saclay meeting, CBPF, Rio de Janeiro, September p. 20/46

21 Æ Ì Ì Dimensional reduction Scale gt Scale g2t log(1/g) Scale g2t Kajantie, Laine, Rummukainen, Schröder (2002) Calculation of the QGP pressure to g 6 ln(1/g) (4 loops) : È È ¼ Æ ÙÐÐ ¹ÐÓÓÔ Æ ¼ ½ ¾ Æ Æ ½ François Gelis 2007 Lecture II / III 2nd Rio-Saclay meeting, CBPF, Rio de Janeiro, September p. 21/46

22 Dressed propagator Quasi-particles Debye screening Landau damping in the QGP François Gelis 2007 Lecture II / III 2nd Rio-Saclay meeting, CBPF, Rio de Janeiro, September p. 22/46

23 Dressed propagator Quasi-particles Debye screening Landau damping Phenomena involving many elementary constituents Long wavelength compared to the typical distance between constituents Small frequency or energy Major collective phenomena : Quasi-particles Debye screening Landau damping Collisional width François Gelis 2007 Lecture II / III 2nd Rio-Saclay meeting, CBPF, Rio de Janeiro, September p. 23/46

24 Dressed propagator Dressed propagator Quasi-particles Debye screening Landau damping In order to assess how the medium affects the propagation of excitations, one must compute the photon (gluon) polarization tensor Π µν (x, y) J µ (x)j ν (y) The photon (or gluon for QCD) self-energy can be resummed on the propagator. Diagrammatically, this amounts to summing : The properties of the medium can be read off the analytic properties of this resummed propagator (cuts, poles,...) François Gelis 2007 Lecture II / III 2nd Rio-Saclay meeting, CBPF, Rio de Janeiro, September p. 24/46

25 Dressed propagator Dressed propagator Quasi-particles Debye screening Landau damping Reminder : the photon polarization tensor Π µν is transverse. At T = 0, this implies : Π µν (P) = g µν P «µ P ν P 2 Π(P 2 ) this is due to gauge invariance and Lorentz invariance Exercise : this property ensures that the photon remains massless at all orders of perturbation theory This formula is not valid at T > 0, because there is a preferred frame (in which the plasma velocity is zero) the tensorial decomposition of Π µν is more complicated, and the photon acquires an effective mass François Gelis 2007 Lecture II / III 2nd Rio-Saclay meeting, CBPF, Rio de Janeiro, September p. 25/46

26 Dressed propagator Dressed propagator Quasi-particles Debye screening Landau damping At finite T, the tensorial decomposition of Π µν is : Π µν (P) = P µν T (P)Π T µν (P) + P (P)Π (P) L L with the following projectors (in the plasma rest frame) P ij T (P) = gij + pi p j p 2, P 0i T (P) = 0, P 00 T (P) = 0 P ij L (P) = (p0 ) 2 p i p j p 2 P 2, P 0i L (P) = p0 p i P, P 00 2 L (P) = p2 P 2 Therefore, we have Π µ µ(p) = 2Π T (P) + Π L (P), Π 00 (P) = p2 P 2 Π L (P) This leads to the following propagator : D µν (P) = P µν T (P) 1 P 2 Π T (P) + P µν L (P) 1 P 2 Π L (P) François Gelis 2007 Lecture II / III 2nd Rio-Saclay meeting, CBPF, Rio de Janeiro, September p. 26/46

27 Dressed propagator - Exercise Dressed propagator Quasi-particles Debye screening Landau damping Check the following properties of the tensors P µν T,L : P µ T µ = 2 P µ L µ = 1 P µ T α P αν = P µν T T P µ L α P αν = P µν L L P µ T α P αν L = 0 François Gelis 2007 Lecture II / III 2nd Rio-Saclay meeting, CBPF, Rio de Janeiro, September p. 27/46

28 Dressed propagator Dressed propagator Quasi-particles Debye screening Landau damping The calculation of Π µ µ and Π 00 can be done for a discrete Matsubara frequency ω p, and one performs the analytic continuation iω p p 0 afterwards Because one is after the long distance properties of the plasma, one also makes the approximation p k (Hard Thermal Loops : Braaten, Pisarski ) For instance, the fermionic contribution to the spatial part Π ij of the polarization tensor reads : ω, p (bv k k/ k ) = g2 N f T 2 Z d 3 k (2π) 3 bvi k n F ( k) k l " δ jl bv j k bvl k ω bv k p + iǫ # Note : with the gluon loop, the only change is N f N f + 2N c François Gelis 2007 Lecture II / III 2nd Rio-Saclay meeting, CBPF, Rio de Janeiro, September p. 28/46

29 Quasi-particles Dressed propagator Quasi-particles Debye screening Landau damping The functions Π T,L (P) read : Π T (P) = e2 T 2 h p Π L (P) = e2 T 2 3 p 2 + p 0 2p 1 p2 0 p 2 ln p0 + p i p 0 p i h ih 1 p2 0 1 p 0 p 2 2p ln p0 + p p 0 p Quasi-particles correspond to poles in the propagator. Their dispersion relation is the function p 0 = ω( p) that defines the location of the pole The inverse of the imaginary part of p 0 is the lifetime of the quasi-particles (If Im(p 0 ) = 0, they are stable). In order to be able to talk about quasi-particles, one must have Im(p 0 ) Re(p 0 ) François Gelis 2007 Lecture II / III 2nd Rio-Saclay meeting, CBPF, Rio de Janeiro, September p. 29/46

30 Quasi-particles Dressed propagator Quasi-particles Debye screening Landau damping Dispersion curves of particles in the plasma : ω m g (T) (L) gluons Thermal masses due to interactions with the other particles in the plasma : m q m g gt At this order, the quasi-particles are stable p François Gelis 2007 Lecture II / III 2nd Rio-Saclay meeting, CBPF, Rio de Janeiro, September p. 30/46

31 Singularities Dressed propagator Quasi-particles Debye screening Landau damping In the complex plane of ω/ p, the dressed propagator has poles (quasi-particles) and a cut (Landau damping) : ω / p L (1) T (2) François Gelis 2007 Lecture II / III 2nd Rio-Saclay meeting, CBPF, Rio de Janeiro, September p. 31/46

32 Debye screening Dressed propagator Quasi-particles Debye screening Landau damping A test charge polarizes the particles of the plasma in its vicinity, in order to screen its charge : r V(r) = exp( - m debye r) r The Coulomb potential of the test charge decreases exponentially at large distance. The effective interaction range is : l 1/m debye 1/gT Note : static magnetic fields are not screened by this mechanism (they are screened over length-scales l mag 1/g 2 T ) François Gelis 2007 Lecture II / III 2nd Rio-Saclay meeting, CBPF, Rio de Janeiro, September p. 32/46

33 Debye screening Dressed propagator Quasi-particles Debye screening Landau damping Place a quark of mass M at rest in the plasma, at r = 0 Scatter another quark off it. The scattering amplitude reads M = ˆgu( k )γ µ u( k) ˆgu( P )γ ν u( P ) X µν Pα (Q) α= T,L Q 2 Π α (Q) If P = 0 (test charge at rest), only α = L contributes k k P P Q = k-k From (P + Q) 2 = M 2, we get a 2πδ(q 0 )/2M For the scattering off an external potential A µ, the amplitude is M = [ gu( k )γ µ u( k) ] A µ (Q) Thus, the potential created by the test charge at rest is : A µ (Q) = g u( P )γ ν u( P ) 2M 2πδ(q 0 )P µν (0, q) L q 2 + Π L (0, q) = 2πgδµ0 δ(q 0 ) q 2 + Π L (0, q) François Gelis 2007 Lecture II / III 2nd Rio-Saclay meeting, CBPF, Rio de Janeiro, September p. 33/46

34 Debye screening Dressed propagator Quasi-particles Debye screening Landau damping By a Fourier transform, we obtain the Coulomb potential : Z A 0 (t, r) = g d 3 q (2π) 3 e i q r q 2 + Π L (0, q) If we are in the vacuum, Π L = 0, and the Fourier transform gives the usual Coulomb law : Z A 0 vac(t, r) = g d 3 q e i q r (2π) 3 q 2 = g 4π r In a plasma, Π L (0, q) = g2 T 2 3 m 2. The Fourier transform D can also be done exactly Z A 0 (t, r) = g d 3 q (2π) 3 e i q r q 2 + m 2 D = g 4π r e m D r the potential is unmodified at r 1/m D, but exponentially suppressed at large distance François Gelis 2007 Lecture II / III 2nd Rio-Saclay meeting, CBPF, Rio de Janeiro, September p. 34/46

35 Debye screening Dressed propagator Quasi-particles Debye screening Landau damping It is easy to see here why the naive perturbation theory works pretty badly Suppose we want to calculate the Coulomb potential of a test charge in the QGP in perturbation theory. The term of order g 2n+1 would be : A 0 2n+1(t, r) = ( 1) n g m 2n D d 3 q (2π) 3 e i q r q 2n+2 all these corrections are very divergent in the infrared. No truncation in the series over n gives the correct long distance behavior of the potential François Gelis 2007 Lecture II / III 2nd Rio-Saclay meeting, CBPF, Rio de Janeiro, September p. 35/46

36 Landau damping Dressed propagator Quasi-particles Debye screening Landau damping The self-energies Π L,T (p 0, p) have an imaginary part when p 0 p. This implies that the propagation of space-like modes is attenuated A wave propagating through the plasma is damped because its quanta may be absorbed by particles of the plasma : The characteristic frequency of this damping is : ω c gt François Gelis 2007 Lecture II / III 2nd Rio-Saclay meeting, CBPF, Rio de Janeiro, September p. 36/46

37 Medium effects: anisotropic François Gelis 2007 Lecture II / III 2nd Rio-Saclay meeting, CBPF, Rio de Janeiro, September p. 37/46

38 Medium effects (anisotropic) Medium effects: anisotropic Most of the previous analysis can be carried through in the case of a plasma with an anisotropic distribution of particles. In particular, the formula for the polarization tensor in terms of f( k) remains valid : Π ij (ω, p) = g 2 T Z d 3 k (2π) 3 bvi k f( k) k l " δ jl bv j k bvl k ω bv k p + iǫ # Model for an anisotropic distribution : start from a generic isotropic distribution f(k 2 ) and squeeze it : p f(p 2 ) f(p 2 + ξ( n p) 2 ) p z n^ François Gelis 2007 Lecture II / III 2nd Rio-Saclay meeting, CBPF, Rio de Janeiro, September p. 38/46

39 Medium effects (anisotropic) Medium effects: anisotropic Within this model, it is easy to factorize the integration over the argument of n (i.e. p 2 + ξ( n p) 2 ) : Π ij (ω, p) = m 2 D Z d 2 bv k 4π bvi k bv l k + ξ(bv k bn)bn l (1 + ξ(bv k bn) 2 ) 2 " δ jl bv j k bvl k ω bv k p + iǫ # with m 2 D g2 2π 2 Z 0 dk k 2 df(k 2 ) dk 2 m D sets the magnitude of all the medium effects on the gauge bosons Only the remaining integral over the unit vector v k is affected by the anisotropy (ξ 0) The tensorial decomposition of Π µν is more complicated than in the isotropic case, because the vector n µ can be used in the construction of the basis François Gelis 2007 Lecture II / III 2nd Rio-Saclay meeting, CBPF, Rio de Janeiro, September p. 39/46

40 Singularities (anisotropic) In the anisotropic case, some poles of the dressed propagator have moved away from the real axis : Medium effects: anisotropic unstable modes ω / p damped modes Some poles have migrated to the upper half plane, and lead to instabilities These imaginary poles exist no matter how small the squeezing parameter ξ is (but their imaginary part goes to zero when ξ 0) François Gelis 2007 Lecture II / III 2nd Rio-Saclay meeting, CBPF, Rio de Janeiro, September p. 40/46

41 Instability Medium effects: anisotropic The poles in the upper half plane lead to the indefinite growth of some small fluctuations. Indeed, the forward propagation of a small fluctuation a(x) in the medium is given by : a(x) = Z d 3 y G R (x, y) h 0 y 0 y i a in (y 0, y) where a in (y) is the initial condition for the fluctuation, and G R the retarded propagator in the medium Consider a toy model for a propagator with such a pole : G R (k) 1 (k 0 iγ) k 2 with Γ > 0 and a plane wave as the initial condition a in (y) exp( iq y) One finds : a(x) = e i(q 0x 0 q x) e Γ(x 0 y 0 ) {z } unbounded growth of the fluctuation François Gelis 2007 Lecture II / III 2nd Rio-Saclay meeting, CBPF, Rio de Janeiro, September p. 41/46

42 Growth rate spectrum In QED and QCD, the growth rate Γ depends on the momentum q of the plane wave fluctuation : Medium effects: anisotropic Γ(q) larger ξ q q * = m D At moderate anisotropies, the most unstable mode is of the same order as the Debye mass At very large anisotropies, all modes from soft to hard are unstable François Gelis 2007 Lecture II / III 2nd Rio-Saclay meeting, CBPF, Rio de Janeiro, September p. 42/46

43 Numerical results Small anisotropy. From : Rummukainen (Trento, Jan. 2007) Medium effects: anisotropic (k*/2) 4 L asym = 2 L max = 16 energy density / m B 2 /2 E 2 /2 m 0 = 1/a m 0 = 0.77/a m 0 = 0.55/a m 0 = 0.45/a m 0 t François Gelis 2007 Lecture II / III 2nd Rio-Saclay meeting, CBPF, Rio de Janeiro, September p. 43/46

44 Numerical results Large anisotropy. From : Rummukainen (Trento, Jan. 2007) Medium effects: anisotropic L asym = 24 L max = 32 a m 0 = energy density / m (k*/2) a m 0 = B 2 /2 E 2 / t m 0 François Gelis 2007 Lecture II / III 2nd Rio-Saclay meeting, CBPF, Rio de Janeiro, September p. 44/46

45 Numerical results Isotropization. From : Rummukainen (Trento, Jan. 2007) Medium effects: anisotropic L max = 32, L asym = 28, m 0 a = 0.3, Energy (arbitrary units) Modes (0,0,k z ) Modes (k x,0,0) t m 0 François Gelis 2007 Lecture II / III 2nd Rio-Saclay meeting, CBPF, Rio de Janeiro, September p. 45/46

46 Lecture III : Out of equilibrium systems Outline of lecture III Schwinger-Keldysh formalism, Long time pathologies From fields to kinetic theory Collisionless kinetic equations Boltzmann equation Transport coefficients François Gelis 2007 Lecture II / III 2nd Rio-Saclay meeting, CBPF, Rio de Janeiro, September p. 46/46