Out-of-equilibrium dynamics ofcoherent non-abelian fields
J. Berges
IKP Theory Center
Technische Universität Darmstadt
Institute for Theoretical Physics
Universität Heidelberg

Nonequilibrium QCD
Relativistic heavy-ion collisions explore
strong interaction matter starting from atransient nonequilibrium state
Thermalization process?
Schematically:• Characteristic nonequilibrium time scales? Relaxation? Instabilities?

Nonequilibrium dynamics of coherent fields
Color Glass:‘color flux tubes‘

1) Consider extreme case: constant color magnetic field pointing in z-directionfrom(all other zero) exponential growth of fluctuations (Nielsen-Olesen instability)with maximum rate

Nielsen, Olesen ’78; Chang, Weiss ’79; … Iwasaki ’08; Fujii, Itakura ’08 …

Bz

2) Consider less extreme case: temporal modulations on scales from(all other zero) non-linear part offield strength tensor,

• Background-field equation:

Oscillating solution:with period

Compare e.g. scalar 4 theory:- early universe inflaton dynamics(preheating)

WMAP- non-rel. gas of ultracold atoms (Gross-Pitaevski),a (s-wave scattering length)
B. Novak, RG-conferenceNIST talks next week• Linearized fluctuation equation, SU(2):maximally amplified modes:or(px = py = 0)

growth rate

Oscillator with time-dependent frequency with ‘wrong sign‘ forapproximate solution: ()momentum in z direction

 similar to Nielsen-Olesen instability with time-averaged magnetic fieldNon-linear evolution: Classical-statisticallattice gauge theory
Wilson action:(real time!)

Plaquette variables
Here:  = 0 /  = s  = 4, temporal gauge, SU(2), no expansion
Sampling introduces classical-statistical fluctuations (‘loops‘)  non-linear evolution,accurate for sufficiently ’large fields/high occupation’ numbers:
anti-commutator

{A, A}  [A, A]

commutator ‘working horse‘ for instability dynamics
Romatschke, Venugopalan; Berges, Gelfand, Sexty, Scheffler, Schlichting;Kunihiro, Müller, Ohnishi, Schäfer, Takahashi, Yamamoto; Fukushima, Gelis; …

Nonequilibrium coherent fields on the lattice
Exponential growth of fluctuations:with Sexty, Scheffler, in preparation• good agreement of primary
growth with linear analysis!
Primary growth rates:• secondary rate ~2NO from non-linear (2PI)

cf. also Berges, Scheffler, Sexty, PRD 77 (2008) 034504

with Sexty, Scheffler, in preparation

Isotropization
3) Choose initial homogeneous fields randomly (ensemble)
with zero mean and width <Compare:original

with Sexty, Scheffler, in preparationvs.

ensemble average• very efficient isotropization for ensemble averaged homogeneous fields!

 early equation of state for hydrodynamics

Comparison to previous ensembles
Initial conditions: stochastically generated inhomogeneous fields with

and Qs    z (extreme oblate anisotropy)
Exponential growth of fluctuations:kT

Primary/secondary growth rates:

/1/4 = 1(Coulomb gauge)

Berges, Scheffler, Sexty, PRD 77 (2008) 034504 (SU(2)); + Gelfand, PLB 677 (2009) 210 (SU(3))

Coherence speed-up
Compare:spatially homogeneous fields vs. stochastic inhomogeneous fields

Inverse primary growth rates:

1/NO  0.3 – 0.6 fm/ce.g. RHIC  5-25 GeV/fm3, LHC  2 x RHIC

1/  1.0 – 1.8 fm/c

Non-linear dynamics leading to turbulence
analytical (2-loop 2PI)

• Scaling exponent close to perturbative Kolmogorov value at high p:  = 4/3
Berges, Scheffler, Sexty, PLB 681 (2009) 362; see also Fukushima, Gelis, arXiv:1106.1396 [hep-ph]• Nonperturbative infrared scaling behavior with  = 4 ( = 5) expected
Infrared “occupation number“  1/g2  strongly correlated! Universal!
Berges, Rothkopf, Schmidt ‘08; Berges, Hoffmeister ‘09; Scheppach, Berges, Gasenzer ’10;
Carrington, Rebhan ’10; Nowak, Sexty, Gasenzer ’10; …

 see also talks next weekQuantum corrections and fermions• Classical-statistical gauge field description accurate for sufficiently large field expectation values/highly occupied modes but quantum corrections at low occupied higher momenta
inclusion into simulations using inhomogeneous 2PI effective action
cf. Berges, Roth, NPB 847 (2011) 197• Fermions: n(p)  1 (Pauli principle) no classical-statistical approximation
 enhancement of quantum corrections from highly occupied bosons!
~ 1/g2gg~ O(1)Requires real-time lattice simulations with dynamical fermions!
cf. Berges, Gelfand, Pruschke, PRL 107 (2011) 061301

Conclusions & Outlook• coherent fields can lead to ultra-fast dynamics:
1/NO  0.3 – 0.6 fm/c
for typical LHC/RHIC energies• very efficient isotropization for ensemble averaged homogeneous fields!
 early equation of state for hydrodynamics• non-linear dynamics crucial for efficient development of turbulence perturbative Kolmogorov scaling exponent at high p:  = 4/3
 non-perturbative scaling exponent at low p? shown to be true for scalars
PRL 101 (2008) 041603• enhancement of quark corrections to O(1) from highly occupied bosons?
shown to be true for quark-meson model PRL 107 (2011) 061301
 real-time dynamical fermions on the lattice in 3+1 dimensions

Non-linear dynamics with expansion
see also Venugopalan,
Romatschke ’06;
Fukushima, Gelis ’11

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