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![Page 1: The Glasma: instabilities, turbulence, thermalization Raju Venugopalan Brookhaven National Laboratory Orsay Summer School, July 4, 2014.](https://reader037.fdocuments.us/reader037/viewer/2022110210/56649e6a5503460f94b6891b/html5/thumbnails/1.jpg)
The Glasma: instabilities, turbulence, thermalization
Raju VenugopalanBrookhaven National Laboratory
Orsay Summer School, July 4, 2014
![Page 2: The Glasma: instabilities, turbulence, thermalization Raju Venugopalan Brookhaven National Laboratory Orsay Summer School, July 4, 2014.](https://reader037.fdocuments.us/reader037/viewer/2022110210/56649e6a5503460f94b6891b/html5/thumbnails/2.jpg)
From the violence of a nuclear collision…to the calm of a quark-gluon fluid
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Approaches to thermalization
Two ``clean” theoretical limits:
Holographic thermalization (based on duality of strongly coupled(g2 Nc ∞; Nc ∞)
N=4 SUSY YM to classical gravity in AdS5×S5 )
Highly occupied QCD at weak coupling
( g2 0 ; g2 f ~ 1)
Our focus: non-equilibrium strongly correlated gluodynamics at weak coupling
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RG evolution for 2 nuclei
Log divergent contributionscrossing nucleus 1 or 2:
Gelis,Lappi,RV (2008)
and can be computed on the initial Cauchy surface
linear operator on initial surface
Contributions across both nuclei are finite-no log divergences => factorization
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Factorization + temporal evolution in the Glasma
Glasma factorization => universal “density matrices W” “matrix element”
NLO terms are as large as LO for S ln(1/x): small x (leading logs) and strong field (gρ) resummation Gelis,Lappi,RV (2008)
ε=20-40 GeV/fm3 for τ=0.3 fm @ RHIC
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Heavy ion phenomenology in weak couplingCollisions of lumpy gluon ``shock” waves
Leading order solution: Solution of QCD Yang-Mills eqns
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Heavy ion phenomenology in weak coupling-IILeading order Yang-Mills solutions are boost invariant:
State of the art phenomenology:
Match event by event Glasma (YM)+2+1-D viscous hydro
Energy dist. of YM fields YM+hydro
Solve viscous hydroequations with Glasma initial conditions
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Heavy ion phenomenology in weak coupling
Hydrodynamics: efficient translation of spatial anisotropy into momentum anisotropy
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Matching boost invariant Yang-Mills to hydrodynamics
Initial longitudinal pressure is negative: Goes to PL =0 from below with time evolution
Glasma energy density and pressure
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Imaging the force fields of QCD
Solns. of QCD Yang-Mills eqns. demonstrate that each of these color “flux tubes” stretching out in rapidity is of transverse size 1/QS
<< 1 fm
Multiparticle dynamics is controlled by sub-nucleon QCD scales
There are ~ πR2QS2 flux tubes – multiplicity, dn/dη ≈ πR2QS
2/αS
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Matching boost invariant Yang-Mills to hydrodynamics
Energy density and (ux,uy) from
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Matching boost invariant Yang-Mills to hydrodynamics
Matching to viscous hydro is “brutal” : assume very rapid isotropization at initial hydro time
Large systematic uncertainty: how does isotropization/thermalization occur on times < 1 fm/c ?
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Heavy ion phenomenology in weak coupling
Results from the IP-Glasma +MUSIC model:AdS/CFT bound = 0.08
Gale,Jeon,Schenke,Tribedy,Venugopalan, PRL (2013) 012302
RHIC data require lower average value of η/s relative to LHC
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Heavy ion phenomenology in weak coupling
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Heavy ion phenomenology in weak coupling
Remarkable agreement of IP-Glasma+MUSIC with data out to fairly peripheral overlap geometries…
Schenke,Venugopalan, arXiv:1405.3605
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The Glasma at NLO: plasma instabilities
At LO: boost invariant gauge fields Aclμ,a(xT,τ) ~ 1/g
NLO: Aμ,a (xT,τ,η) = Aclμ,a(xT,τ) + aμ,a(η) aμ,a(η)= O(1)
Small fluctuations grow exponentially as ~
Same order of classical field at
QSτ
increasing seed size
2500
Resum such contributions to all orders
Romatschke,VenugopalanDusling,Gelis,VenugopalanGelis, Epelbaum
Systematic Schwinger-Keldysh approach allows one to separate divergences before collision (factorization) from those after… Gelis,Lappi,RV
Jeon
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The Glasma at NLO: plasma instabilities
At LO: boost invariant gauge fields Aclμ,a(xT,τ) ~ 1/g
NLO: Aμ,a (xT,τ,η) = Aclμ,a(xT,τ) + aμ,a(η) aμ,a(η)= O(1)
Small fluctuations grow exponentially as ~
Same order of classical field at
QSτ
increasing seed size
2500
Resum such contributions to all orders
Leading quantum corrections can be expressed as average over a classical-statistical ensemble of initial conditions
Romatschke,VenugopalanDusling,Gelis,VenugopalanGelis, Epelbaum
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Initial conditions in the Glasma
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Initial conditions in the Glasma
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Early universe cosmology: turbulent thermalization
Model for early universe thermalization:
Weakly coupled scalar field theory (λΦ4) (λ=10-8 )
In homogeneous background field
+ vacuum fluctuations
Micha,Tkachev, PRD 70 (2004) 043538
Growth of instabilities via parametric resonance of classical fieldand vacuum fluctuations
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Turbulent thermalization in Cosmology
Λ ~ t-β
nhard ~ tα
Thermalization process characterized by quasi-stationary evolution with scaling exponents. Dynamic: α=-4/5, β=-1/5 ; Spectral: κ= -3/2
Micha,Tkachev, PRD 70 (2004)043538
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Turbulent thermalization in Cosmology
Free Turbulence:Quasi-stationary solution: universal, non-thermal spectral exponentsSelf-similar evolution with universal dynamical scaling exponents
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Kinetic interpretation
Self consistent solutions:
Stationary solution:
Fixed point equation + scaling solution
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Dependence on initial conditionsBerges,Boguslavskii,Schlichting,Venugopalan, arXiv:1312.5216
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Dependence on initial conditionsBerges,Boguslavskii,Schlichting,Venugopalan, arXiv:1312.5216
Non-trivial transport of quantities conserved in different inertial ranges of momenta
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Kinetic interpretation
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Back to the Glasma…
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Initial conditions in the overpopulated QGP
Choose for the initial classical-statistic ensemble of gauge fields
Stochastic randomvariables
Polarization vectors ξ expressed in terms of Hankel functions in Fock-Schwinger gauge Aτ =0
Controls “prolateness” or “oblateness” of initial momentum distribution
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Temporal evolution in the overpopulated QGP
Solve Hamilton’s equation for 3+1-D SU(2) gauge theory in Fock-Schwinger gauge
Fix residual gauge freedomimposing Coloumb gauge at each readout time
Berges,Boguslavski,Schlichting,VenugopalanarXiv: 1303.5650, 1311.3005
Largest classical-statistical numerical simulations of expanding Yang-Mills to date: 2562 × 4096 lattices
Classical-statistical computations performed at very weak coupling αS=10-5 for “classical dominance” at all times in simulation: -- corresponds to Qτ0 ≈ Ln2(1/αS) ≈ 100
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Result: pressure become increasingly anisotropic
PL/PT approaches universal τ-2/3 behavior
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Hard scales show universal scalingGauge invariant quantities:simple quasi-particle interpretation for weak fields
Kurkela,Moore
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Hard scales show universal scalingGauge invariant quantities:simple quasi-particle interpretation for weak fields
β = 0 suggests no shift of hard transverse momentum scale in time
γ = 1/3 system longitudinal hard scalered-shifts but slower than free streaming value of γ=1
Kurkela,Moore
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Kinetic theory in the overoccupied regime
For 1 < f < 1/αS a dual description is feasible either in terms of kinetic theory or classical-statistical dynamics … Mueller,Son (2002)
Jeon (2005)
Different scenarios:
Elastic multiple scattering dominates in the Glasma
Rescattering influenced by plasma (Weibel) instabilities
Transient Bose condensation+multiple scattering
BMSS: Baier,Mueller,Schiff,Son
KM: Kurkela, Moore
BGLMV: Blaizot,Gelis,Liao,McLerran,Venugopalan
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Kinetic theory in the overoccupied regime
For 1 < f < 1/αS a dual description is feasible either in terms of kinetic theory or classical-statistical dynamics … Mueller,Son (2002)
Jeon (2005)
Different scenarios:
Differences arise due to assumptions of non-perturbative behavior at p≈mDebye
Q
Q
mDebye
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Result: universal non-thermal fixed pointConjecture:
Moments of distribution extracted over range of time slices lie on universal curves
Distribution as function of pT displays 2-D thermal behavior
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Kinetic interpretation of self-similar behavior
Follow wave turbulence kinetic picture of Zakharov, as developed by Micha & Tkachev in context of inflation
Fixed point solution satisfies
μ=α-1
If we assume that small angle elastic scattering dominates
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Kinetic interpretation of self-similar behavior
For self-similar scaling solution, a) small angle elastic scattering b) energy conservation c) number conservation
Give unique results: α = -2/3, β = 0 , γ = 1/3
These are the same exponents (within errors) extracted from our numerical simulations !
The same exponents appear in the “bottom-up” thermalizationscenario of Baier, Mueller, Schiff, Son (BMSS)
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Non-thermal fixed point in overpopulated QGP
KM: Kurkela, MooreBGLMV: Blaizot,Gelis,Liao,McLerran,Venugopalan
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Quo vadis, thermal QGP?
The “bottom-up” scenario:Baier,Mueller,Schiff,Son
mD
QS
QS
Scale for scattering of produced gluons (t > 1/QS) set by
Build up pz (which fights the red shift of pz ~ 1/t) with mult. scattering
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Quo vadis, thermal QGP?
The “bottom-up” scenario:Baier,Mueller,Schiff,Son
mD
QS
QS
Scale for scattering of produced gluons (t > 1/QS) set by
Build up pz (which fights the red shift of pz ~ 1/t) with mult. scattering
Our simulations support this picture – no significant role of instabilities -- imaginary component in mD (conjectured by Arnold, Moore, Lenaghan and Yaffe to spoil the bottom up scenario)
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Quo vadis, thermal QGP?
QS
QS
mD
Occupation #
Classical statistical simulations break down in this regime
In quantum regime, thermalization proceeds through number changing inelastic processes
Thermalized soft bath of gluons for
Thermalization temperature of
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Quo vadis, thermal QGP?
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Quo vadis, thermal QGP?
Blaizot,Dominguez,Iancu,Mehtar-Tani
Recent kinetic theory computations by Kurkela and Lu suggest that thermalization can occur between 0.2 and 1 fm for QS = 2 GeV and αS between 0.3 and 0.2 respectively
Kurkela,Lu:1405.6318
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Proof in principle of universality
Universality:
Dynamics in the vicinity of the attractor is independent of the details of the theory and is governed by universal numbers that depend on a) dimensionality of the theoryb) generic features of the scattering, eg: 2 <-> 2 , 2 <-> 3, 2 <-> 1 + condensatec) Boundary conditions
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Proof in principle of universalityBerges,Boguslavski,Schlichting,Venugopalan,In preparation
Rescaled distributions of expanding scalar and expanding gauge theory lie on top of each other !
– stationary distributions with identical scaling exponents…
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Universal non-thermal attractor in QCD
“Big whorls have little whorls, which feed on their velocity, And little whorls have lesser whorls, and so on to viscosity.”
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Turbulence: a preamble
`I am an old man now, and when I die and go to heaven there are two matters on which I hope for enlightenment. One is quantum electrodynamics, and the other is the turbulent motion of fluids. And about the former I am rather optimistic.'- Horace Lamb