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