Nonequilibrium quantum field theoryand the lattice

Jürgen BergesDarmstadt University of Technology

Content

I. Motivation fast thermalization in heavy ion collisionsearly universe instabilities and prethermalizationstrongly coupled quantum gases

II. Nonequilibrium dynamicstwo-particle irreducible expansionslimitations of (semi-)classical descriptions

III. Real-time quantum fields on a latticereal-time stochastic quantizationnonabelian gauge theory

I. Motivation

Facility for Antiproton and Ion Research (GSI)

Relativistic Heavy Ion Collider (BNL)

Large Hadron Collider (CERN)

Heavy ion collisions

Phasediagramm

(schematic)

QCD critical point in the universality class of the Ising model!Berges, Rajagopal; Halasz et al.; Stephanov et al. `99 ... ; Lattice-QCD: Fodor, Katz `02; ...

Far-from-equilibrium dynamics

Heavy-ion collisions (BNL,CERN,GSI) explore strong interaction matter starting from a transient nonequilibrium state

• Thermalization ?Properties of the equilibriumphase diagram of QCD ?Braun-Munzinger, Redlich, Stachel,

QGP3 (2004) 491; ...

• Theoretical justification of earlylocal thermal equilibrium? Hydrodynamics after .1 fm/c?

Kolb, Heinz, QGP3 (2004) 634; ...

Fast thermalization?

Xu, Greiner, Phys. Rev. C 71 (2005) 064901; ...

Shuryak, Zahed, Phys. Rev. C70 (2004) 021901; ...

Mrowczynski, Phys. Lett. B 314 (1993) 118 Arnold, Moore, Yaffe, Phys. Rev. Lett. 94 (2005) 072302;Rebhan, Romatschke, Strickland, Phys. Rev. Lett. (2005) 102303;Romatschke, Venugopalan, Phys. Rev. Lett. 96 (2006) 062302; ...

Berges, Borsanyi, Wetterich, Phys. Rev. Lett. 93 (2004) 142002

New properties (sQGP)?

Plasma instabilities:

Prethermalization? Different quantities effectively thermalizeon different time scales: Early equation of state → Hydrodynamics

Fast thermalization from kinetic theory?

...

Early Universe

End of Inflation reheating CMB

`entropy´ production

thermal spectrumwith fluctuations

far-from-equilibrium`initial´ state

time

Reheating• Explosive particle production from nonequilibrium instabilities

Vergleiche: Parametrische Resonanz in der klassischen Mechanik

CLASSICAL: Traschen, Brandenberger, PRD 42 (1990) 2491; Kofman, Linde, Starobinsky, PRL 73 (1994) 3195;Khlebnikov, Tkachev, PRL 77 (1996) 219; ...

QUANTUM: Berges, Serreau, PRL 91 (2003) 111601 Arrizabalaga, Smit, Tranberg, JHEP 0410 (2004) 017

quasistationaryevolution

explosive particleproduction

Parametric resonance reheating:

• Quasistationary evolution leads to extremely slow thermal equilibration→ non-thermal fixed points

• Prethermalization Berges, Borsanyi, Wetterich, Phys. Rev. Lett. 93 (2004) 142002Podolsky, Felder, Kofman, Peloso, Phys. Rev. D 73 (2006) 023501;…

SU(2)×SU(2) ‘quark-meson‘ model (2PI 1/NF to NLO):

Prerequisite for hydrodynamics!

tdamp teqtpt

tpt

Approximatively thermal equation of state after tpt ¿ trelax ¿ teq!

Ultra-cold quantum gases

B-field

Attract Repel• Tunable BEC self-interaction!Strong coupling (Feshbach resonance)

⇒ Measure BEC size, shape: ⇒ B(t) faster than atom motion:

OD

0

1

OD

0

1

OD

0

1

550 a0

3000 a0

a = 70 a0

In trap focussed burst atoms

BEC remnant

480 µmCornish et al. Phys. Rev. Lett. 85 (2000) 1795

Ultracold atomic gas dynamics of 23Na in 1D

Gasenzer, Berges, Schmidt, Seco, PRA 72 (2005) 063604

t

Method: 2PI 1/N expansionBerges, NPA 699 (2002) 847

II. Nonequilibrium quantum fields

Standard QFT techniques fail out of equilibrium

`Secularity´ `Universality´

• nonlinear dynamics necessaryfor late-time thermalization

• uniform approximations in timerequire infinite pert. orders

2-particle irreducible generating functionals

⇒ systematic 2PI loop-, coupling- or 1/N-expansions available

⇒ far-from-equilibrium dynamics as well as late-time thermalization in QFT

Berges, Cox ´01; Aarts, Berges ´01; Berges ´02; Cooper, Dawson, Mihaila ´03; Berges, Serreau ´03; Berges, Borsányi, Serreau ´03; Cassing, Greiner, Juchem ´03; Arrizabalaga, Smit, Tranberg ´04 ...

Luttinger, Ward ´60; Baym ´62; Cornwall, Jackiw, Tomboulis ´74

E.g. scalar N-component field theory to NLO in 2PI 1/N-expansion:

includesNLO 1PI !

Berges ´02 ; Aarts, Ahrensmeier, Baier, Berges, Serreau ´02

Time evolution equations

statistical propagator ∼ h{Φ,Φ}i

spectral function ∼ h[Φ,Φ]i

Nonequilibrium:

Equilibrium/Vacuum: (fluct.-diss. relation)

Nonequilibrium instability:(parametric resonance)

Nonperturbative!

III. Quantum fields on a lattice

Real time:

non-positive definite probability measure!

Euclidean stochastic quantization• Classical Hamiltonian in (d+1)-dimensional space-time

• Expectation values for quantum theory with action :

,

• Replace canonical ensemble averages by micro-canonical:

Classical dynamics in ‘fifth‘-time (t5) to compute quantum averages!

• discretization to second order in

• conjugate momenta have Gaussian distribution; randomly refreshafter every single step → Langevin dynamics

,

Parisi, Wu ’81; …

,with white noise

Real-time stochastic quantizationKlauder ’83; Parisi ’83; Hüffel, Rumpf ’84; Okano, Schülke, Zheng ’91 …

Replace embedded d-dimensional Euclidean by Minkowskian action:

with d‘Alembertian

for Euclidean stochastic quantization⇒

for real-time stochastic quantization⇒

Langevin dynamics:

i.e. , in general complex!

Simulating nonequilibrium quantum fieldsBerges, Stamatescu, Phys. Rev. Lett. 95 (2005) 202003

t at-1

t at-1

Scalar λφ4-theory:

classical starting configuration (t5 = 0), Langevinupdating takes into account quantum corrections

λ = 0

λ ≠ 0

Convergence:

t at-1

Langevin time

apparently good convergence properties

• same initial (t = 0) conditions

’null’ starting configuration (t5 = 0)

‘run-away’ trajectoriesmuch suppressed by smaller step-size

⇒

⇒

Precision tests

Berges, Borsanyi, Sexty, Stamatescu, in preparation

Anharmonic quantum oscillator:

• real-time thermal equilibrium

• comparison with solutionof Schrödinger equation weak coupling

strong coupling

0

0.05

0.1

0.15

0.2

0.25

0 0.1 0.2 0.3 0.4 0.5

<ϕ(

0)ϕ(

t)>

t

stochastic Schrödinger: (real contour)

(complex contour)

−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 0.5 1 1.5 2 2.5 3 3.5 4

<ϕ(

0)ϕ(

t)>

t

stochasticSchrödinger

• short real-time contour:

⇒ good agreement of stochasticquantization and `exact´ results

Fixed points of the Langevin flowStationary solutions at late t5 fulfill:

⇒

, , …similarly for

⇒

⇒

⇒

...

infinite set of Dyson-Schwinger equations for n-point functions!

-0.2

-0.15

-0.1

-0.05

0

0.05

0 1 2 3 4 5 6 7 8

Langevin time

t=0.375

tfinal=2

LHS (0,0)RHS (0,0)LHS (0,t)RHS (0,t)LHS (t,t)RHS (t,t)

-0.1

-0.08

-0.06

-0.04

-0.02

0

0 1 2 3 4 5 6 7 8

Langevin time

t=0.375

tfinal=1

LHS (0,0)RHS (0,0)LHS (0,t)RHS (0,t)LHS (t,t)RHS (t,t)

thermal fixed point

LHS RHS

non-unitary fixed point

Dyson-Schwinger equation:

• fulfilled by both thermal as well as non-unitary fixed point (symmetrized)

-0.2-0.1

0 0.1 0.2

0 5 10 15 20 25 30

Im G

(t,t)

contour point index

0.2

0.3

0.4

0.5

Re

G(t

,t)

tfinal=1tfinal=2

Nonabelian gauge theoryReal-time lattice action: (plaquette)

with anisotropic couplings

Langevin dynamics:

, ,

,

(not ∼ gµν for Minkowski theory!)

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1 2 3 4 5

spat

ial p

laqu

ette

ave

rage

Langevin time

Euclideancontour tilt tan(α)=2.2

tan(α)=1.1tan(α)=0.6

0

1

2

3

4

5

6

0.1 1 10

ϑ cro

ssov

er

Contour tilt: tan(α)

τ+=0.125τ+=0.25

τ+=2 , symmetric

τ+

SU(2) gauge theoryon a contour:

• thermal fixed point only approximate(intermediate Langevintimes) !

,( )Dyson-Schwinger equation for plaquette:

µ

N2(N − 1)2

βµγ iN

1N

γµ

γµ

γ

µ

γ

µ

γ+−

}

{=

−

Σ −

−

LHS RHS

2

3

4

5

6

7

8

9

0 0.5 1 1.5 2 2.5 3 3.5 4

Sch

win

ger-

Dys

on e

quat

ions

Langevin time

LHSRHS

thermal

crossover

non-unitary

Conclusions• Loop-, or 1/N-expansions of 2PI effective action suitable to resolve secularity and universality

⇒ Limited range of validity of kinetic approaches⇒ Far-from-equilibrium dynamics & thermalization in QFT

• 2PI 1/N-expansion provides quantitative description of nonperturbative dynamics as instabilities or critical phenomena ⇒ 2PI 1/N for SU(N) gauge theories?

• Nonperturbative lattice simulations of real-time quantum fields:

⇒ Stochastic quantization solves hierarchy of real-timeDyson-Schwinger equations, however, solutions not unique

⇒ Short-time evolution of scalar fields⇒ Thermal fixed point unstable for SU(2) gauge theory

Nonequilibrium Dynamics in Particle Physics and Cosmology

Jan. 14 to March 28, 2008Kavli Institute for Theoretical Physics, Santa Barbara

Organizers: J. Berges (Darmstadt), L. Kofman (CITA), L. Yaffe (U. of Washington)

Limitations of kinetic theory

• gradient expansion in

• memory loss (t0 →∞, s0 ∈ (-∞,∞) with X0 finite)

• (quasiparticle picture)

Based on Berges, Borsányi, Phys. Rev. D74 (2006) 045022

,

Lowest-order gradient expansion:

Imaginary part real partof self-energy

NLO gradient expansion:

withand Poisson brackets

Quantitative example• weak-coupling g2φ4-model, 2PI three-loop

occupationnumber

⇒

p tra

nsve

rse

plongitudinal

• characteristic anisotropy measure: (isotropy → ∆F ≡ 0)

:

tdamp

valid kinetic description

• LO/NLO results only quantitative after tdamp (memory loss)→ not suitable for studying fast thermalization (t ¿ tdamp )

tdamp

valid kinetic description

tdamp

valid kinetic description

: :

• NLO gradient corrections insignificant for ∆F (cf. isotropization)

• NLO gradient corrections significant for F (cf. thermalization)

• NLO results quantitative for t & tdamp