can the weibel-instability explain fast thermalisation in
TRANSCRIPT
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!