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Nonabelian plasma instabilities
Anton Rebhan
Technical University Vienna, Austria
Nonabelian plasma instabilities – p. 1
Contents
Nonabelian plasma instabilities:important collective phenomenon in wQGP prior to thermalization/isotropization
• Why even consider wQGP (i.e. extrapolation of g ≪ 1 physics to g ∼ 1)
• Hard-Loop Effective Theory (gauge-covariant Boltzmann-Vlasov)
• Review of numerical results for nonabelian plasma instabilities
• Extension to nonstationary free-streaming expanding non-Abelian plasma
Nonabelian plasma instabilities – p. 2
wQGP or sQGP?
Pressure of pure-glue QCD: lattice (Bielefeld) vs. perturbative result to order g5 with
renormalization scale dependence for µMS = πT . . . 4πTstrictly pert. [Arnold & Zhai, PRD50(’94)7603] vs. improved/optimized [Blaizot, Iancu & AR, PRD68(’03)025011]
1.5 2 2.5 3 3.5 4 4.5 50.5
0.6
0.7
0.8
0.9
1
1.5 2 2.5 3 3.5 4 4.5 50.5
0.6
0.7
0.8
0.9
1T=T
P=P0
g5full 3-loopPMSFAC
RHICLHC
sQGP wQGP ?Nonabelian plasma instabilities – p. 3
wQGP or sQGP?
Entropy in pure-glue QCD: lattice vs. Hard-Thermal-Loop quasiparticle entropy with
Next-to-Leading Approximations of asymptotic thermal masses
suggestive of dominance of weakly interacting (hard) quasiparticles for T & 3Tc
[Blaizot, Iancu & AR, PRD63(’01)065003]
1 2 3 4 50.7
0.75
0.8
0.85
0.9
0.95
1567
1 2 3 4 50.7
0.75
0.8
0.85
0.9
0.95
1567
Pure glue QCD
NLA
S/S0
T/Tc
lattice
λ|µMS=2πT
cΛ:
0
12
1
2
Recent improvements/fits by inclusion of (sizeable) LO quasiparticle width: Peshier, CassingNonabelian plasma instabilities – p. 4
wQGP or sQGP?
No lattice EOS results for N = 4 SYM,
but essentially unique Pade approximant R[4,4] = 1+αλ1/2+βλ+γλ3/2+δλ2
1+αλ1/2+βλ+γλ3/2+δλ2
for known weak and strong coupling results
0 2 4 6 8 10 12 140.7
0.75
0.8
0.85
0.9
0.95
1
0 2 4 6 8 10 12 140.7
0.75
0.8
0.85
0.9
0.95
1
N = 4 super-Yang-MillsS/S0
λ ≡ g2N
weak-coupling to order λ3/2
strong-coupling to order λ−3/2
cΛ = 0
cΛ = 2 112
λ1
λ → ∞
NLA Pade
QCD @ 3.5Tc
QCD @ 2Tc
[J.-P. Blaizot, E. Iancu, U. Kraemmer & AR, JHEP 06(2007)035]Nonabelian plasma instabilities – p. 5
wQGP or sQGP?
Perhaps neither!
Actual QGP should be approached from both strong and weak coupling
Weak coupling approach sensible at least for observables where hard(quasi-)particles dominate
At very least: needed for comparison
Nonabelian plasma instabilities – p. 6
Scales of wQGP
• T : energy of hard particles
• gT : thermal masses, Debye screening mass,Landau damping, plasma instabilities [Mrowczynski 1988, 1993, . . . ]
• g2T : magnetic confinement, color relaxation, rate for small angle scattering
• g4T : rate for large angle scattering, η−1 T 4
Effective theory at scale gT : Hard-(Thermal-)Loop Effective Action[Frenkel, Taylor & Wong; Braaten & Pisarski 1991]
equivalent to: gauge-covariant Boltzmann-Vlasov [Mrowczynski, AR & Strickland ’04]
[Blaizot & Iancu 1993, Kelly, Liu, Lucchesi & Manuel 1994]
in particular required for:• Bottom-up thermalization [Baier, Mueller, Schiff & Son 2000]
teq ∝ g−13/5 → g−? [Arnold, Lenaghan, Moore, JHEP 08 (’03) 002]
• Shear viscosity [Arnold, Moore & Yaffe]
(η/s)−1 = g4 ln(1/g)f(ln(1/g)) +(η/s)−1anomalous (!!)
[Asakawa, Bass & Muller, PRL 96 (’06) 252301]Nonabelian plasma instabilities – p. 7
Boltzmann-Vlasov equations
With color-neutral background distribution v · ∂ f0(p,x, t) = 0, vµ = pµ/p0
gauge covariant Boltzmann-Vlasov:
v · D δfa(p,x, t) = gvµFµνa ∂(p)
ν f0(p,x, t) = −g(Ea + v ×Ba) · ∇pf0,
DµFµνa = jν
a = g
∫
d3p
(2π)3pµ
2p0δfa(p,x, t).
So far: mostly stationary f0(p) with ∂µf0 ≡ 0
• isotropic: f0(p) = f0(|p|), ∇pf0 ∝ v
v · D δfa(p,x, t) = −gEa · ∇pf0 (stable)
• anisotropic f0(p), ∇pf0 6∝ v
v · D δfa(p,x, t) = −g(Ea + v × Ba) · ∇pf0 unstable!
Nonabelian plasma instabilities – p. 8
Filamentation (Weibel) instabilities
Initially homogeneous superposition of counterstreaming particles unstableagainst filamentation [Weibel 1959]
(parallel currents of same sign attract!)
X X
unstablemodes
Non-Abelian plasmas: nonlinear dynamics even before backreaction onhomogeneous background becomes important
Nonabelian plasma instabilities – p. 9
Discretized Hard Loop Theory
Integrating out hard momentum scaleleaves dependence on velocities of hard particles.
Local set of effective field equations in terms of auxiliary fields Wµ(x,v)δfa(p, x) = −gW a
µ (x,v)∂µ(p)f0(p)
Dµ(A)F µν = jν [A]
jµ[A] = −g2
∫
d3p
(2π)3
1
2|p|pµ ∂f0(p)
∂pβW β(x;v)
[v · D(A)]Wβ(x;v) = Fβγ(A)vγ
Real-time lattice simulation in temporal gauge: Aaix
, Πaix
, Wax,v
Need: large spatial lattice with large number NW of auxiliary fields in adjointrepresentation
Nonabelian plasma instabilities – p. 10
1D+3V
Restriction to most unstable modes with momentum k ∝ ez:dimensional reduction 3D+3V → 1D+3V (homogeneity in transverse directions)
Numerical results: (SU(2), 10,000 sites, NW = 100, moderate anisotropy)
Energy densities E [AR, Romatschke & Strickland, PRL 94 (’05) 102303]
Nonabelian plasma instabilities – p. 11
3D+3V
More general initial conditions: 3D+3VExponential growth in non-Abelian regime saturates to weak linear growth
Magnetic energy density for moderate anisotropy:[Arnold, Moore & Yaffe, PRD72 (’05) 054003]
20 40 60 80 100 120m∞ t
10-3
10-2
10-1
100
101
102
mag
netic
ene
rgy
dens
ity [i
n un
its o
f m∞4 /g
2 ]
3+1 dim. non-Abelian3+1 dim. Abelian1+1 dim. non-Abelianex
pone
ntia
l
linear
20 40 60 80 100 120 140 160 180 200m∞ t
0
1
2
3
4
mag
netic
ene
rgy
dens
ity [
in u
nits
of
m∞4 /g
2 ]
3+1 dim. non-Abelian3+1 dim. Abelian1+1 dim. non-Abelian
linear
Nonabelian plasma instabilities – p. 12
3D+3V
Alternative discretization (v-grid instead of Ylm expansion), somewhat strongeranisotropy
50 60 70 80 90 100m 8 t
0
10
20
30
[Ene
rgy
Den
sity
]/(m
84 /g2 )
|HL|B
T
Bz
ET
Ez
[AR, Romatschke & Strickland, JHEP 09 (’05) 041]
Very strong anisotropy: saturation occurs only at very strong field strength[Bodeker & Rummukainen, arXiv/0705.0180]
Nonabelian plasma instabilities – p. 13
Non-Abelian Cascade
Saturation mechanism: non-Abelian interactions of unstable IR modes cascadetheir energy to more energetic modes (Kolmogorov turbulence)
[Arnold & Moore, PRD73 (’06) 025006]
Nonabelian plasma instabilities – p. 14
Unstable glasma
Original Color-Glass-Condensate calculations (numerical solution of YM field equations for
colliding lightlike color sources) boost-invariant
Small rapidity fluctuations unstable like plasma instabilities (hard gauge modes as plasma
particles) [P. Romatschke and R. Venugopalan, PRL 96, PRD 74 (2006)]
Longitudinal pressure (mainly from transverse magnetic fields) ∼ e#√
τ
O O O OO
O
O O
OO
OO
OO O O
500 1000 1500 2000 2500 3000
g2µτ
1e-12
1e-11
1e-10
1e-09
1e-08
1e-07
1e-06
1e-05
0.0001
0.001
ma
x τ
~ PL
(τ,ν)/
g4µ
3Lη
64x64, ∆=10-10
aη
1/2
32x64, ∆=10-10
aη
1/2
16x256,∆=10-10
aη
1/2
128x128,∆=10-6
aη
1/2O O
32x64, ∆=10-6
aη
1/2
16x256,∆=10-5
aη
1/2
16x256,∆=10-4
aη
1/2
3e-05
P. Romatschke and R. Venugopalan, hep-ph/0605045
Nonabelian plasma instabilities – p. 15
Hard-Expanding-Loop (HEL) formalism
Extension of Hard-Loop formalism to nonstationary free-streaming plasma[Romatschke & AR, PRL97 (’06) 252301]
xα = (τ,x⊥, η) (τ proper time, η spacetime rapidity)
v → azimuthal angle φ, momentum rapidity y
τ−1Dα(τF αβ) = jβ
jα[A] = −g2 1
2
Z ∞
0
p⊥dp⊥
(2π)2
Z 2π
0
dφ
2π
Z ∞
−∞dy pα ∂f0(p⊥, pη)
∂pβWβ(x; φ, y)
v · D Wα(τ, xi, η; φ, y) = vβFαβ with vα ≡ pα
|p⊥|= (cosh(y − η), cos φ, sin φ,
sinh(y−η)τ
).
Take f0(p, x) = fiso
(√
p2⊥
+ p2η/τ2
iso
)
= fiso
(
√
p2⊥
+ (p′zτ/τiso)2)
strongly oblate momentum space anisotropy for τ ≥ τ0 ≫ τiso
Asymptotic behavior of transversely constant modes eAi(τ, ν) =R
dηe−iνηAi(τ, η):
eAi(τ, ν) ∼ τ · 2F3
„3−
√1+4ν2
2,
3+√
1+4ν2
2; 2, 2 − iν, 2 + iν;−µτ
«
→ τ1/4 exp (2√
µτ) for ν ≫ 1 µ =π
8m2Dτiso
, mD = mD |τiso
Nonabelian plasma instabilities – p. 16
Transversely constant modes in Abelian regime
Abelian regime: W fields can be eliminated → integro-differential equations
1 5 10 20 30 50 100 200 300
0.1
1
10
1001 5 10 20 30 50 100 200 300
τ/τ0
ν=3ν=10
ν=30
|Bi (τ,
ν)/
Ei (τ 0
,ν)|
Numerical solution vs. asymptotic 2F3 behavior (thin bright lines)
≈ realistic gluon density (from CGC)) → uncomfortably late onset of instabilities!Nonabelian plasma instabilities – p. 17
Matching to CGC
Parameters from saturation scenario τ0 ≃ Q−1s :
n(τ0) = c (N2c −1)Q3
s
4π2Ncαs(Qsτ0)
with gluon liberation factor c ={
0.5 Krasnitz et al. (numerical)2 ln 2 Kovchegov (analytical estimate)
fiso = N fthermal with (transverse) temperature T = 0.47Qs [Krasnitz et al.]
pure glue → N =1
αs
c
8Nc(0.47)3ζ(3)
τ0
τiso
1
Qsτ0
→µ
Qs
=1
8m2
DπτisoQ−1s = π2
48·0.47·ζ(3)c ≈
{
0.182 (c = 0.5) (previous plot)
0.505 (c = 2 ln 2)
Qs ≃ 1 GeV (RHIC) . . . 3 GeV (LHC) ?
Nonabelian plasma instabilities – p. 18
Transversely constant modes in Abelian regime
1 10 20 30 40 50 60 70 80 90 1000.01
0.1
1
10
100
Most optimistic case: c = 2 ln 2 [Kovchegov] Abelian≈upper bound
|B(ν, τ)|, |E(ν, τ)| normalized to |E(ν, τ0)|
ν = 1, 2, 4, 8, 15, 30
τiso = 0.01τ0
τ/τ01 fm/c: (RHIC) (LHC) Nonabelian plasma instabilities – p. 19
Longitudinal free streaming expansion 1D+3V non-Abelian
Discretized non-Abelian HEL [AR, Strickland & Attems, in preparation]
0 5 10 15
0.1
1
10
100
1000 total energytransverse Etransverse Blongitudinal Elongitudinal Bhard -> soft energy transfer
τ/τ0
E/(g2/τ40 )
high initial anisotropy (and increasing), initial conditions from CGC [Fukushima, Gelis, McLerran ’06]
(but unrealistically high gluon density)
Nonabelian plasma instabilities – p. 20
Longitudinal free streaming expansion 1D+3V
Time evolution of initial noise with mode number cutoff:
Abelian spectrum Non-Abelian spectrum: cascade
1 2 5 10 20 50
1.´10-9
0.00001
0.1
1000
1.´107
f j
1 2 5 10 20 50 100 200
1.´10-9
1.´10-7
0.00001
0.001
0.1
10
j
Nonabelian plasma instabilities – p. 21
Outlook
• Detailed study of nonabelian regime of hard-expanding-loop dynamics ofplasma instabilities
• Done: Abelian (linearized) case
• Reference case for numerics going up to nonabelian regime• Growth of small fluctuations uncomfortably small for n(τ0) from CGC !
• Coming out soon: Non-Abelian 3D+1V free-streaming HEL
• In progress: Non-Abelian 3D+3V HEL
• Next step: inclusion of (some) backreaction on f0(x,p)NB: full backreaction in (stationary) CPIC simulations of Dumitru, Nara & Strickland,
“UV avalanche”, PRD75 (2007) 025016 (rapid thermalization once fields are
nonperturbatively large)
Nonabelian plasma instabilities – p. 22