Kadanoff-Baym Approach to Thermalization of Quantum FIelds Akihiro Nishiyama Kyoto Sangyo University Feb. 9, Collaboration with Yoshitaka Hatta

Relativistic Heavy Ion Collision at RHIC and LHC \sqrt(s NN )=0.2 TeV Au+Au (RHIC) \sqrt(s NN )=2.76TeV Pb+Pb (LHC) Proper time Rapidity Formation of Quark-Gluon Plasma (QGP) Dynamics in an expanding system. Z=tZ=-t0 =0

Early Thermalization of gluons (0.6-1fm/c)! (RHIC and LHC). Comparable to formation time (0.2fm/c). Nonequilibrium Quantum Field Theoretical Approach Application of Kadanoff-Baym equation. Semi-classical Boltzmann eq.: (2-3fm/c). Classical statistical approaches: Not Bose-Einstein distribution at late time.

Introduction of time evolution equation for classical field and Kadanoff-Baym equation for quantum fluctuation in O(N) scalar model. To show particle production and equilibration in Numerical Analyses. Comparison of quantum and classical dynamics in weakly and strongly coupled regimes. Introduction of time evolution equation for classical field and Kadanoff-Baym equation for quantum fluctuation in O(N) scalar model. To show particle production and equilibration in Numerical Analyses. Comparison of quantum and classical dynamics in weakly and strongly coupled regimes. Purpose of this talk

O(N) scalar model In expanding metric Interaction term a=1,…,N Action of scalar O(N) model All order summation of coupling by 1/N expansion. g=diag(1, - 2, -1, …)

Time Evolution Equation I Or effect of fluctuations Equation of motion of classical field Damping of classical field for an expanding system a =Tr ([density matrix]…)

Kadanoff-Baym equation Quantum evolution equation of 2-point Green’s function (fluctuations). Boson statistical (distribution) and spectral functions Memory integral Self-energies: local mass shift, nonlocal real and imaginary part

Field-Particle Conversion: Particle production from classical field. Collision of particles → Bose-Einstein distribution Off-shell effect: Memory effects and finite spectral width binary collisions (2-to-2) → Rapid Change of distribution functions (Lindner and Muller 2006) + 3-to-1 → entropy production + chemical equilibrium. Merit binaryFinite decay width Demerit Numerical simulation needs much memory of computers. X (Parametric resonance) +

Initial condition: Classical field with vacuum quantum fluctuations (Color Glass Condensate ?) vacuum Initial condition We assume homogeneity in space 2+1 dimensions. Weak Strong m/ 0 =0 and 0.1

τ/τ 0 / 0 ) Case I (without collision term) Reproduction of J. Berges, K. Boguslavski, S. Schlichting, hep-ph Evolution of classical field φ~τ -1/3 Due to expansion m/σ 0 =0

Fluctuation 2 (t)~ (t)+… Parametric Resonance instability Flat exp( g 2/3 )Curved p =2 0 0 Berges 2004.

Case II (with Collision term) = + ・・ ・ + xxxxxx Source induced amplification. Normal collision term. Summation of Next-to-Leading Order of 1/N expansion. All covering of F from O(1) to O( 1/ ) in evolution. Our results (collaboration with Y. Hatta): Quantum collision term. X: Classical field Y. Hatta and A. N., Phys. Rev. D 86 (2012) Berges et. al. Classical Statistical approaches.

Classical Statistical Approximation F = Gelis et al. (2011) Berges et al. (2012) Neq~T/p-1/2 =FF-(1/4) For example FF>> (Weakly coupled): Good approximation FF~ (Strongly coupled): Bad. Difference between classical and quantum evolution. F~(1/ )n_p, ~1/ n_p~1/,at late time ~

Berges F=O( 0 ), = F=O( -1/2 ) F=O( -2/3 ) F=O( -1 ) F~exp(g (2/3) ) Linear regime: Parametric resonance xx =O( 0 ) Nonlinear regime: source induced amplifacation =O( 0 ) time Non-perturbative regime: quasistationary evolution =O( -1 )

Evolution of classical field … ( ) / xx + Quantum and Classical evolution (Weakly coupled =10 -4 ). + Parametric resonance >1 F pT=0 m/ 0 =0.1 >25 >40 Damping of classical field + pT=pi/8 pT=pi/4

Late time statistical function (Weakly coupled) pT =190 F Lack of difference indicates that the system is far from equilibrium at late time at =190. Time evolution O(1) to O(1/ ) We cannot fit the above function.

Expanding F Quantum and Classical evolution (Strongly coupled ). … xx + + Parametric resonance Quantum Classical pT=0 pT=pi/8 pT=pi/4 ( ) /

Late time distribution function (Strongly coupled) =150 Classical statistical approximation: Neq~ T/p-1/2. It is not true thermalization. This is a problem in classical evolution. np

We have proposed the Kadanoff-Baym approach to thermalization of O(N) scalar fields from initial background classical field with longitudinal expansion in 2+1 dimensions. Only 1/N covers all evolution of instability (F: O(1) to O(1/ ) by summing all order of coupling. In weakly coupled regimes, no difference between quantum and classical evolution appears. No hint of quantum equilibration. In strongly coupled regimes, the late-time Bose-Einstein distribution is realized only in quantum evolution. In classical statistical approximation, it is not true thermalization. Need for quantum evolution equation. Summary

Application to non-Abelian gauge theories in expanding system. Initial condition in an expanding system (Color Glass Condensate with vacuum fluctuations). Renormalization procedure in an expanding system. Tuning of program codes. Remaining problems

Thank you.

Berges (Flat metric)

Relativistic Heavy Ion Collison at RHIC and LHC Proper time Rapidity Nonequilibrium phase Quark-Gluon Plasma Mixed phase Hadron gas Freeze-out τ eq =0.6-1fm/c √s NN =0.2 TeV Au+Au (RHIC) √s NN =2.76TeV Pb+Pb (LHC)

Evolution Equation of Classical fields. Phi^2: 1/la to 0 F: 1 to 1/la/tau

F, Weakly coupled.

Phi (strongly coupled). Tau^(1/3)phi(tau): ExpandingPhi(t): Nonexpanding Dilite : due to small \int k F

F, strongly coupled.

Evolution of Green’s functions F Nonexpanding Expanding Px Pz )) σ0σ0 Quantum evolution

Evolution of Green’s functions F Nonexpanding Expanding Px Pz )) σ0σ0 Quantum evolution

Evolution of Green’s functions F Nonexpanding Expanding Px Pz )) σ0σ0 Quantum evolution

Evolution of Green’s functions F Nonexpanding Expanding Px Pz )) σ0σ0 Quantum evolution

Evolution of Green’s functions F Nonexpanding Expanding Px Pz )) σ0σ0 Quantum evolution

Evolution of Green’s functions F Nonexpanding Expanding Px Pz )) σ0σ0 Quantum evolution

Evolution of Green’s functions F Nonexpanding Expanding Px Pz )) σ0σ0 Quantum evolution

Evolution of Green’s functions F Nonexpanding Expanding Px Pz )) σ0σ0 Quantum evolution

Evolution of Green’s functions F Nonexpanding Expanding Px Pz )) σ0σ0 Quantum evolution

) Evolution of Green’s functions F Nonexpanding Expanding Px Pz ) σ0σ0 Px Pη Quantum evolution

Evolution of Green’s functions F Nonexpanding Expanding Px Pz )) σ0σ0 Px Pη Quantum evolution

Evolution of Green’s functions F Nonexpanding Expanding Px Pz )) σ0σ0 Px Pη Quantum evolution

Evolution of Green’s functions F Nonexpanding Expanding Px Pz )) σ0σ0 Px Pη Quantum evolution

Evolution of Green’s functions F Nonexpanding Expanding Px Pz )) σ0σ0 Px Pη Quantum evolution

Evolution of Green’s functions F Nonexpanding Expanding Px Pz )) σ0σ0 Px Pη Quantum evolution

Evolution of Green’s functions F Nonexpanding Expanding Px Pz )) σ0σ0 Px Pη Quantum evolution

Evolution of Green’s functions F Nonexpanding Expanding Px Pz )) σ0σ0 Px Pη Quantum evolution

Evolution of Green’s functions F Nonexpanding Expanding Px Pz )) σ0σ0 Px Pη Pz=Pη/τ p η 2 /τ 2 Quantum evolution

Strongly coupled.

Late-time distribution function (Quantum evolution) NonexpandingExpanding εpεp npnp t/t0=150τ/τ0=150

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) ○ (TAG) ○ ○

Numerical Simulation for KB eq. Truncation λΦ 4 O(N) SU(N) NLO of λNLO of 1/NLO of g 2 Symmetric phase 〈 Φ 〉 =0 Our Code 1+1 dim 2+1 dim 3+1 dim 2+1 dim 3+1 dim Others’ Code 1+1 dim 2+1 dim 3+1 dim 1+1 dim 3+1 dim ?