Can the Weibel-Instability explain fast thermalisation in ...

Can the Weibel-Instability explain fast thermalisation in heavy ion collisions?

Dénes Sexty

TU Darmstadt

Collaborators: Jürgen Berges, Sebastian Scheffler

1. Weibel Instabilities2. Classical approximation on the lattice3. Results Growth rates Prolate vs oblate initial conditions Comparison of SU(2) with SU(3)4. Power law solutions5. Summary

April 3, 2009, Budapest

The Thermalization Puzzle

Applicability of hydrodynamic models for predicting the collective flow in heavy in collisions suggests local thermal equilibrium for times:

eq~1−2 fm/c

from Boltzmann eq.: Xu, Greinger, Phys. Rev. C (2005) 064901;...

Prethermalisation is enough for hydrodynamics?Berges, Borsányi, Wetterich, PRL 93 (2004) 142002

Way out?

Plasma Instabilites is faster than scattering processes

Mrowczynski, Phys. Rev. C49 (1994) 2919Romatschke, Strickland PRD68 (2003) 036004Arnold, Leneghan, Moore JHEP08 (2003) 002Rebhan, Romatschke, Strickland PRL94 (2005) 102303Arnold, Moore, Yaffe, PRL94 (2005) 072302Romatschke, Venugopalan, PRL96 (2006) 062302Dumitru, Nara, Strickland, PRD75 (2007) 025016

from hydrodynamics: Kolb, Heinz(2004);Luzum, Romatschke (2008)

Dufaux et al, JCAP 0607 (2006) 006, ...in cosmological context: (Borsányi, Patkos, Sexty PRD68 (2003) 063512)

Non-equilibrium QFT

Need to develop and test different aprroximation schemes

2PI studies

classical simulations

HTL approximation

stochastic quantisation

Boltzmann equation

transport coefficients in eq. + hydro

All methods have different range of applicability need more than one of them to describe a heavy-ion collision

We want to describe heavy ion collisions. What can we do?

Plasma Instability (aka. Weibel Instability)

anisotropic particle distribution

seed of magnetic waves

The current is bent to amplify magnetic field (filamentation)

Exponential growth of certain modes

(Weibel, PRL 2(1959) 83, Mrowczynski, PRC 49 (1994) 2191)

Phenomena also exsits in pure gauge theories (without charged fermions)

Studies using hard loop approximation

Separation of scales● anisotropic hard modes “particles”● exponentially growing soft modes

Problems: have to deal with backreaction in experiments, there's no scale separation

Advantages: clear picture can use (or verify) HTL results

Romatschke, Strickland PRD68 (2003) 036004Arnold, Leneghan, Moore JHEP08 (2003) 002Rebhan, Romatschke, Strickland PRL94 (2005) 102303Arnold, Moore, Yaffe, PRL94 (2005) 072302Romatschke, Venugopalan, PRL96 (2006) 062302Dumitru, Nara, Strickland, PRD75 (2007) 025016

In this study: No scale separation, all modes discretised on the lattice

Not enough computers range of modes is smaller

Methods of investigations

High occupation numbers for gauge fields Classical approximation

Check for scalars Classical , 2PI

Pauli principle: fermions cannot be higly occupied pure gauge

Symmetry group: SU(2) and SU(3) no qualitative difference

Non equilibrium: Initial value problem Anisotropic initial conditions well estabilished methods for classical fields

Short time scales expansion neglected

Classical SU(2) and SU(3) gauge theory on lattice with anisotropic initial conditions

Arrizabalaga, Smit, Tranberg JHEP0410 (2004) 017Aarts, Berges PRL88 (2002) 041603Berges, Gasenzer cond-mat/0703163

No artificial separation of scales: all modes live on the same lattice

Berges, Scheffler, Sexty PRD (2008)

Lattice formulations of classical EOM for gauge fields

Lagrange formulation:

Hamiltonian formulation

Space time lattice with temporal and spatial links

DSDUx

=0EOM is calculated from the action:

Electric fields live in group space (represented by temporal plaquettes)

Space discretisation first: H Ea ,U=

12E

a2∑spatial plaq.

Tr U

Ea t t /2=E

a t− t /2−DaH Ea ,U

Ut t =expi t aEaUt

Electric fields live in Lie algebra space

Then Hamiltonian EOM is discretised in time

Needed: matrix exponentialization

Tr U a=ba U=?Needed: inversion of Tr U a

Lagrangian Lattice implementation

Wilson action: S=s∑1

2Tr1TrUspatial−t∑

12Tr1

TrUtemporal

Link variables: Ux=eigAax aa

Equations of Motion

DSDUix

=0Gauss Constraint

DSDU0x

=0

parellel transporter from toxa x

U x=U xU xaU −1xaU

−1xplaquette variable

t=2Tr1

g02 s=

2Tr1gs

2=

as

at

Temporal axial gauge

g0=gs=1 ≈10 N s3=64 3⋯128 3

U 0=1 A0a=0

initially G.C.=0 later also fulfilled

Simulation at

Initial Conditions

anisotropic distribution: lots of particles in the transverse planefew particles in the longitudinal planes

Gaussian initial configuration with:

⟨∣A iak ,t=0∣2⟩=Cexp −kx

2ky2

22 −kz

2

2z2 ≫z

is given by fixing energy densityC

zero initial momenta Gauss constraint is trivially fulfilled

To avoid numerical problems, a small plateau is addedmimicing the quantum n=1/2

⟨∣A iak ,t=0∣2⟩=MAX Cexp −kx

2ky2

22 −kz

2

2z2 ,Amp

∣k∣ Amp≈10−12

Results

t=log10

∑i ,b ,ppx

2py2∣ Ai

b t ,p∣2

∑i ,b,ppz

2∣ A ibt ,p∣2Anisotropy parameter:

primary

seco

ndar

y

Converting to physical units

=LAT

g2 a−4=400 MeV 4⋯700 MeV 4 “pessimistic” or “optimistic” choice

Using the optimistic value of =30 GeV/fm3 , g=1

Energy density needed for

1/=0.1fm/c

=300 TeV/fm3

fourth root and square root: mild dependence on , g

and

Growth rates

1 / sec≈0.3 fm/c1 / sec≈0.8 fm/c

Timescales from secondary rates:

optimisticallypessimistically

=30 Gev/fm3

=1Gev/fm3

Prolate vs Oblate

Pancakes collide + free streaming leads to oblate distribution: transverse plane highly occupied longitudinal modes are empty

What happens if there's no free streaming regime? instability starts before oblate distribution reached prolate distribution: longitudinal highly occupied transversal plane anoccupied

Growth rates 50% faster, expansion helps isotropisation

Or is it Nielsen-Olesen instability?

Fujii, Itakura, Iwazaki (2009)

Homogeneous, constant magnetic field Some modes acquire “Zeeman energy”

∝e t pz=g B−pz

2

±2 g B

SU(2) pure gauge theory recast as theory of

“electromagnetic field” “charged vector fields”

A3

=A1i A

2

2= pz

2g B2 N1±2gB

Nonhomogenic field: instability present for

Rinhom1/g BNonzero growth at pz=0

Pressure modes

Non equilibrium px=−Tzzx spacetime dependent pressure

Fourier transformation: gauge invariant quantityTzzp

Bottom-up isotropisation

Isotropisation time for lower momentum modes is shorter for 0pz/1/41

Inverse rates for isotropisation:

1-2 fm/c for 0pz700 MeV optimistically, using

2.3 – 4.6 fm/c for 0pz300 MeV pestimistically, using

=30 GeV /fm3

=1GeV /fm3

Diagrams

are the secondaries loop induced?

Measure 2 point functions on the lattice

Insert into conributions to selfenergy (here sunset/2p.f. is plotted)

More complex than scalars, hereinifintely many diagrams contribute

scalar study in: Berges, Serreau PRL91 (2003) 11601

Secondaries are caused by fluctuations of the already excited modes

Structure of EOM for F:∂t

2k 2F k t=k t

where has contributions from all modesk

Comparison of SU(2) and SU(3) results

Berges, Gelfand, Scheffler, Sexty PLB (2009)

Tr U a U=?Solving the EOM:

ba=−12

Tr Ua U=1−ba bai baaSU(2) is simple:

SU(3) : Newton method is used for numerical solution

Growth rates

SU(2) and SU(3) qualitatively similar

SU(3) primary growth rates are 25% lower

primaries and secondaries are still present

secondaries start later

Initial growth of the longitudinal modes caused by tadpole

Other diagrams suppressed by theanisotrophy ratio mT=

Energy density is kept fixed

mT2∝

NN 2−1

mT , SU 3=34

mT , SU 2

Explanation of slower growth

Non-equilibrium fixedpoint?

Slow evolution after the instability

A power law region emerges at small momenta

Fitted power law behavior at t=1000

Time dependence of fitted exponent

Analytics of non thermal fixedpoints

No one diagram dominates at g^4 order

Different diagrams give different exponents via Saharov transformations

No easy explanation as in case of scalars

Conclusions

Instabilities are present in our model of QGP (scale separation and hard particles are not necessary)

Bottom up isotropisation, but rates are a bit too small

Using SU(3) or prolate initial condition, no qualitative change observed

Power laws solution seen numerically analytically more challenging

Wilson loopsTransverse plane: Longitudinal plane:

early:

late:

Wilson loops

Spatial Wilson loops show are law: W CX,Y~e−X Y

spatial string tension:

LAT=0.05 =0.5

Nonperturbative dynamics!