1. Kadanoff-Baym Approach to Thermalization of Quantum Fields Akihiro Nishiyama University of Tokyo Institute of Physics

2. τ<0 fm/c CGC √s NN =200 GeV Glasma 0<τ<0.6~ 1.0fm/c Figures from P. SORENSEN QGP τ>1.0fm/c g g g g g g g g g g g gg RHIC experiments Early thermalization for Partons Yang Mills eqn‘Soft’: field, Vlasov-Boltzmann eqn‘Hard’ :parton, Nonequilibrium gluodynamics Dense system (Boltzmann eq. should not be applied) No consideration of particle number changing process g→gg, g→ggg (Off-shell effect) In order to conduct proper simulation in hydrodynamics after thermalization, the thermalization time must be very fast compared with its perturbative analysis (gg ⇔ gg, gg ⇔ ggg).

3. Introduction of the kinetic entropy based on Kadanoff-Baym equation Proof of the H-theorem for scalar O(N) Φ 4 theory. Trial to prove the H-theorem for gauge theory. Numerical simulation of time evolution of the kinetic entropy. Introduction of the kinetic entropy based on Kadanoff-Baym equation Proof of the H-theorem for scalar O(N) Φ 4 theory. Trial to prove the H-theorem for gauge theory. Numerical simulation of time evolution of the kinetic entropy. Purpose of this work Rest of this talk Kadanoff-Baym equation in Closed Time Path formalism Scalar theory as a Toy model and gauge theory, Entropy in Rel. Kadanoff-Baym equation Summary and Remaining Problems

4. Time Contour Schwinger-Dyson equation in terms of statistical (distribution) and spectral functions Kadanoff-Baym equation For a free field Breit-Wigner type Boson in Closed-Time Path formalism 2. Kadanoff-Baym eqn 2 Particle-Irreducible Effective Action Γ Mean field is omitted. Green’s function

5. Merit Spectral function Time evolution of spectral function with decay width + distribution function Off-shell effect We can trace partons which are unstable by its particle number changing process in addition to collision effects. we can extract gg→g (2 to 1) and ggg→g (3 to 1) and the inverse prohibited kinematically in Boltzmann simulation. This process might contribute the early thermalization. (Why do we use KB eq, not Boltzmann eq?) Σ=Self-energies Memory integral The Kadanoff-Baym equation: Time evolution of statistical (distribution) and spectral function Self-energies: local mass shift, nonlocal real and imaginary part

6. Extension to the O(N) theory with Next Leading Order in 1/N expansion Extension to the O(N) theory with Next Leading Order in 1/N expansion 3. Scalar theory as a toy model φ 4 theory with no condensate =0 Next Leading Order of coupling φ 4 theory with no condensate =0 Next Leading Order of coupling O(λ 2 ) LO NLO Application for BEC, Cosmology (or reheating) and DCC dynamics? + ・・ ・ O(1) O(1/N) O(λ)

7. Entropy in Rel. Kadanoff-Baym equation Nonrel. case: Ivanov, Knoll and Voskresenski (2000), Kita (2006) The first order gradient expansion of the KB equation. For NLO λ 2 (Φ 4 ), In the quasiparticle limit We reproduce the entropy for the boson. [ ] Entropy flow spectral function H-theorem needs not to be based on the quasiparticle picture. For NLO of 1/N (O(N)) :velocity Offshell I. II. I and II type of entropy density are investigated in the numerical simulation. At thermal equilibrium (O(λ 2 ))

8. Sketch of H-theorem (O(N)) Φ4Φ4 Φ4Φ4 ~λ 2 ×∫(a-b)log(a/b) ≧ 0 O(N) + ・・ ・ = = ++ 1/N ++ × ∫(a-b)log(a/b) ≧ 0 × Coupling expansion 1/N expansion = Self-energy a,b: Product of 4 Real Green’s function

9. Non-Abelian Gauge Theory No classical field =0 Leading Order of coupling No classical field =0 Leading Order of coupling O(g 2 ) LO (local) LO (nonlocal) O(g 2 ) Green’s functionSelf-energy ⇔ In Temporal Axial Gauge (TAG), divide Green’s function and self-energy into transverse (T) and longitudinal part (L), take 1 st order gradient expansion, then (TTT) + (TTL) + (TLL) For three external gluons However

10. Singularity in longitudnal mode Gauge dependence of s μ singularity If the Ward identity is satisfied. That is controlled at thermal equilibrium. (Next page) In far from equilibrium ? Breakdown of gradient expansion. For LO skeleton expansion. Blaiziot, Iancu and Rebhan (1999) Independent of T

11. Controlled gauge dependence (Smit and Arrizabaraga (2002), Carrington et al (2005) ) Truncated effective action Gauge invariant Exact Expansion of coupling of self energy Under gauge transformation ⇔ SD equation with truncation at ⇔ Stationary point Higher order gauge dependence ⇒ Energy, Pressure and Entropy derived from δΓ/δT has controlled gauge dependence. Gauge invariance is reliable in the truncated order. ⇒ Gauge invariant Energy, Pressure and Entropy derived from δΓ/δT Nielsen (1975) (Thermodynamic potential) Schwinger-Dyson equation

12. 1+1 dimension Initial conditions: 1. nonequilibrium momentum distribution (tsunami configuration) 2. Uniform in space Approach to equilibrium with no expansion Numerical simulation (A. Aarts and J. Berges (2001), J. Berges (2002)) No thermalization for the Boltzmann case (Φ4, O(N) model)

13. Evolution of kinetic entropy (Φ 4 ) Number density Entropy density Ratio of two entropy λ/m 2 =41+1No thermalization for the Boltzmann eq. mX 0 ∫ p (1+n)log(1+n)-nlog n Without spectral width quasiparticle approximation Off-shell: with spectral width Green line: Red line: In the above figure

14. Evolution of kinetic entropy (O(N)) λ/m 2 =40, N=10, 1+1 dim Number density Entropy density Ratio of two entropy S(OS)/S(QP) No thermalization for the Boltzmann eq. in 1+1 dimensions. ∫ p (1+n)log(1+n)-nlog n Without spectral width quasiparticle approximation Off-shell: with spectral width Green line: Red line: In the above figure

15. N dependence of kinetic entropy Width is suppressed by factor 1/√N. Smaller N Wider width Longer tail Rapider saturation of entropy, (faster thermalization) Larger entropy N=10, 5 and 3 Spectral function Kinetic entropy S(OS)

16. 4. Summary and Remaining Problems We have introduced the kinetic entropy based on the Kadanoff-Baym equation. The kinetic entropy satisfies H-theorem for NLO of λ(Φ 4 ) and 1/N (O(N)). It might do for LO of SU(N). S(OffShell)>S(QuasiParticle). S(OS)/S(QP) is nearly constant. The wider spectral width, the larger S(OS) and the faster thermalization. Asymptotic behavior of the entropy near equilibrium in O(N) theory Thermal solution for the SD eq. for the LO of g 2 for the gauge theory (2+1 dimensions) Gauge invariance of the entropy in far from equilibrium, Infrared singularity in longitudinal part.

17. Time irreversibility Exact 2PI (no truncation) Truncation LO of Gradient expansion H-theorem λΦ 4 O(N) SU(N) NLO of λNLO of 1/NLO of g 2 Symmetric phase 〈 Φ 〉 =0 ××× △△ ? △ (TAG) ○ ○

18. Evolution of kinetic entropy (Φ 4 ) Number density Entropy density Ratio of two entropy Total number density N/V=∫ p n p λ/m 2 =41+1 ∫ p (1+n)log(1+n)-nlog n without spectral width quasiparticle approximation No thermalization for the Boltzmann eq. Off-shell: with spectral width mX 0

19. Time evolution of number density

20. For quasiparticle approximation 0⇔40⇔4 1⇔31⇔3 2⇔22⇔2 3⇔13⇔1

21. Microscopic processes

22. Spectral function

23. Coupling dependence

24. Self-energy Schwinger-Dyson eqn For perturbative Green’s functions Imaginary part contributes to the particle number changing process g ⇔ gg

25. Evolution of kinetic entropy (O(N)) λ/m 2 =40, N=10, 1+1 dim Number density pxpx Entropy density mX 0 Ratio of two entropy S(OS)/S(QP) No thermalization for the Boltzmann eq. in 1+1 dimensions. mX 0 ∫ p (1+n)log(1+n)-nlog n Without spectral width quasiparticle approximation Off-shell: with spectral width Green line: Red line: In the above figure