Quantum Chaos in the Quantum Fluid

Quantum chaos in the quantum ﬂuid

Raju Venugopalan Brookhaven Naonal Laboratory

Joe Fest, McGill, Montreal, June 12-14, 2012 Smoking Joe and Holy Indian

Lone ranger: I think we are surrounded, Kemo sabe Tonto: What do you mean “we” $@!# Smoking Joe and Holy Indian

Lone ranger: I think we are surrounded, Kemo sabe Tonto: Indeed, Lone Ranger – no worries – they are all here to praise you on your 60th ! Talk

 Movaon: the unreasonable eﬀecveness of hydrodynamics in heavy ion collisions to paraphrase E. P. Wigner

An ab inio weak coupling approach:

 Classical coherence of wee partons in nuclear wavefuncons

 Quantum ﬂuctuaons: Factorizaon, Evoluon, Decoherence

 Isotropizaon, Bose-Einstein Condensaon, Thermalizaon ?

4 Ab inio approach to heavy ion collisions RV, ICHEP review talk, 1012.4699

Color Glass Inial Glasma sQGP - Hadron Condensates Singularity perfect ﬂuid Gas t τ0 ?  Compute properes of relevant degrees of freedom of wave fns. in a systemac framework (as opposed to a “model”)?

 How is maer formed ? What are its non-equilibrium properes & lifeme? Can one “prove” thermalizaon or is the system “parally” thermal ?

 When is hydrodynamics applicable? How much jet quenching occurs in the Glasma? Are there novel topological eﬀects (sphaleron transions?) Ab inio approach to heavy ion collisions RV, ICHEP review talk, 1012.4699

 How is maer formed ? What are its non-equilibrium properes & lifeme? Can one “prove” thermalizaon or is the system “parally” thermal ?

 When is hydrodynamics applicable? How much jet quenching occurs in the Glasma? Are there novel topological eﬀects (sphaleron transions?) Gluon Saturaon in large nuclei: classical coherence from quantum ﬂuctuaons

Wee parton ﬂuctuaons me dilated on strong interacon me scales

The gluon density saturates at

a maximal value of ~ 1/αS  gluon saturaon

Large occupaon # => classical color ﬁelds

|P>_pert |P>_classical 2 1/QS Quantum decoherence from classical coherence

Color Glass Initial sQGP - Hadron Glasma Condensates Singularity perfect fluid Gas t

Glasma (\Glahs-maa\): Noun: non-equilibrium maer between CGC and QGP

Computaonal framework Gelis,RV NPA (2006)

Schwinger-Keldysh: for strong me dependent sources (ρ ~ 1/g) , inial value problem for inclusive quanes

For eg., Schwinger mechanism for pair producon, Hawking radiaon, … Forming a Glasma in the lile Bang Lumpy classical conﬁguraons

Soluons of Yang-Mills equaons produce (nearly) boost invariant gluon field configuraons: “Glasma flux tubes”

Lumpy gluon ﬁelds are color screened in

transverse plane over distances ~ 1/QS - Negave Binomial mulplicity distribuon.

“Glasma ﬂux tubes” have non-trivial longitudinal color E & B ﬁelds at early mes --generate Chern-Simons topological charge

See Bjoern Schenke’s talk from yesterday Their quantum descendents

Color Glass Initial Glasma sQGP - Hadron Condensates Singularity perfect fluid Gas t

Two kinds of important quantum ﬂuctuaons: a) Before the collision: pη=0 modes factorized into the wavefuncons - responsible for energy/rapidity evoluon of wavefuncons a) Aer the collision pη ≠ 0; hold the key to early me dynamics - responsible for decoherence, isotropizaon, thermalizaon Quantum ﬂuctuaons in classical backgrounds: I Gelis,Lappi,RV: 0804.2630, 0807.1306,0810.4829

Suppressed Factorized into energy evoluon of wavefuncons

JIMWLK factorizaon: pη=0 (small x !) modes that are coherent with the nuclei can be factorized for inclusive observables

W’s are universal “funconal density matrices” describing distribuon of large x color sources ρ1 and ρ2 of incoming nuclei; can be extracted from DIS or hadronic collisions Inial condions for quantum evoluon

For large nuclei, general consideraons about the color structure of higher dimensional representaons of color charge density ρa probed give as an inial condion for evoluon (MV model) a a 2 ρ (x )ρ (x ) Wx [ρ]=exp d x ⊥ ⊥ 0 − ⊥ 2 µ2 µ2 = Color charge squared per unit area ~ A1/3

Other (sub-leading in A) contribuons to these inial condions

Jeon, RV Dumitru,Jalilian-Marian,Petreska  These inial condions give eikonalized gluon distribuons at leading log in x and Q2 Higher order correcons can be modeled by introducing x and b dependence (Basis of IP-sat model discussed in Bjoern’s talk) Soluon of B-JIMWLK hierarchy

∂WY [ρ] = [ρ] WY [ρ] ∂Y H a a ρ α = 2 ∇⊥ B-JIMWLK hierarchy of correlators contains all leading info on charge, rapidity and transverse posions of n-gluon correlaons in incoming wavefuncons

 This is now solved ! Correlator of Light-like Wilson lines Tr(V(0,0)V^dagger (x,y))

Rummukainen,Weigert (2003) Dumitru,Jalilian-Marian,Lappi,Schenke,RV, PLB706 (2011)219 Quantum ﬂuctuaons in classical backgrounds: II

η Romatschke,Venugopalan p ≠ 0 (generated aer collision) modes grow exponenally Fukushima,Gelis,McLerran n g exp QSτ Quant. ﬂuct. grow exponenally aer collision

As large as classical increasing seed size ﬁeld at 1/Qs !

2500 T µν (x) T µν (x)

Parametrically suppressed Exponenate and resum these The ﬁrst fermi: a master formula

Also correlators of Tμν  From soluons of B-JIMWLK

µν T LLx+Linst. = [Dρ1][Dρ2] WY Y[ρ1] WY +Y[ρ2] beam− beam [da(u)] F [a] T µν [A (ρ , ρ )+a] × init LO cl 1 2

 Gauge invariant Gaussian spectrum  3+1-D soluons of of quantum ﬂuctuaons Yang-Mills equaons

 Expression computed recently-numerical evaluaon in progress

Dusling,Gelis,RV  This is what needs to be matched to viscous hydrodynamics, event-by-event

 All modeling of inial condions for heavy ion collisions includes various degrees of over simpliﬁcaon relave to this “master” formula Big Bang Little Bang

Hot Era WMAP data (3x105 years) QGP Inﬂation

CGC/ Glasma

Plot by T. Hatsuda 17 Glasma spectrum of inial quantum ﬂuctuaons

Path integral over small ﬂuctuaons equivalent to

1 dν iνη A(x , τ, η)=Acl.(x , τ)+ dµk cνk e χk(x ) Hiν (λkτ)+c.c ⊥ ⊥ 2 2π ⊥ Gaussian random variables

Berry conjecture: High lying quantum eigenstates of classically chaoc systems, linear superposions of Gaussian random variables

Yang-Mills is a classically chaoc theory B. Muller et al.

Srednicki: Systems that sasfy Berry’s conjecture exhibit “eigenstate thermalizaon” Also, Jarzynski, Rigol, … Hydrodynamics from quantum ﬂuctuaons

Dusling,Epelbaum,Gelis,RV (2011) scalar Φ4 theory:

Energy density and pressure without averaging over ﬂuctuaons

Energy density and pressure aer averaging over ﬂuctuaons

Converges to single valued relaon “EOS” Hydrodynamics from quantum ﬂuctuaons Dusling,Epelbaum,Gelis,RV (2011)

Anatomy of phase decoherence:

ΔΘ = Δω t

Tperiod = 2π / Δω

 Tperiod ≅ 18.2 / g ΔΦmax

Different field amplitudes from different inializaons of the classical field T µ = dφ dφρ˙ (φ, φ˙) T µ(φ, φ˙) dE dθ ρ˜ (E,θ) T µ(E,θ) µ t µ ≡ t µ ≈ µ t dE ρ˜t(E) dθ Tµ (E,θ) = 0 →∞ μ Because Tμ for scalar theory is 2π t+T a total derivave dθ T µ(E,θ)= dτ T µ(φ(τ), φ˙(τ)) = 0 µ T µ and φ is periodic t Quasi-parcle descripon? Epelbaum,Gelis (2011)

Spectral funcon ρ (ω,k)

Plasmon mass

ω k Energy density of free quasi-parcles

 At early mes, no quasi-parcle descripon Energy density on the lace

 May have quasi-parcle descripon at late mes. Eﬀecve kinec “Boltzmann” descripon in terms of interacng quasi-parcles at late mes ? Quasi-parcle occupaon number 4 10 4 10 Gelis,Epelbaum (2011) t = 0 3 20 3 10 10 40 60 2 10 80 100 1 120 2 10 10 140 0 10

-1 1 10 k

f 10

-2 10 0 0.5 1 1.5 2 2.5 0 10

-1 Inial 10 mode distribuon

-2 10 0 0.5 1 1.5 2 2.5 3 3.5 k

4 t = 0 150 2000 10

50 300 5000 T/(tk-!)-1/2

System becomes over occupied relave to a thermal distribuon… Proof of concept: isotropizaon of longitudinally expanding ﬁelds in scalar Φ4

101 2PT + PL Dusling,Epelbaum,Gelis,RV: on arXiv, Friday ε

PT 100 PL

10-1

10-2

10-3 0 50 100 150 200 250 300 τ

QCD – similar framework – more challenging computaonally and conceptually Dusling, Gelis, RV: arXiv 1106.3927

Numerical development underway – results hopefully very soon Bose-Einstein Condensaon in HI Collisions ? Blaizot,Gelis,Liao,McLerran,RV: arXiv:1107.5295v2

Cold rubidium atoms in a magnec trap Gell-Mann’s Totalitarian Principle of Quantum Mechanics: Everything that is not forbidden is Compulsory

 Possible phenomenological consequences… Mickey Chiu et al., 1202.3679

Bose-Einstein Condensaon and Thermalizaon

Blaizot,Gelis,Liao,McLerran,RV: arXiv:1107.5295v2

Assumpon: Evoluon of “classical” ﬁelds in the Glasma can be matched to a quasi-parcle transport descripon See also, Mueller, Son (200)2 Jeon (2005)

All esmates are “parametric”: αS << 1

3 4 System is over-occupied: n ≈ QS /αs ; ε = QS /αS -3/4 1/4  n ε ≈ 1/αS >> 1

In a thermal system, n ε-3/4 = 1

If a system is over-occupied near equilibrium and elasc scaering dominates, it can generate a Bose-Einstein condensate Known in context of inﬂaon: Khlebnikov, Tkachev (1996) Berges et al. (2011) Bose-Einstein Condensaon and Thermalizaon

neq = feq(p); εeq = ωp feq(p) p p 1 feq(p)= In a many-body system, gluons develop a mass β(ωp µ) 1 1/2 e − − ωp=0 = m ≈ αS T

If over-occupaon persists for μ = m, system develops a condensate

3 1 feq(p)=ncδ (p)+ β(ω m) 1 e p− −

3 Qs 1/4 nc = 1 αS As αS  0, most parcles go into the condensate αS − It however carries a small fracon 1/4 4 4 ε = mn α T << T of the energy density… c c ≈ S Transport in the Glasma df p ∂ f z ∂ f = C[f] dt ≡ τ − τ pz “Landau” equaon for small angle 2  2 scaering: df Λ2 Λ df α S ∂ p2 + S f(p)(1 + f(p)) dt |coll ∼ p2 p dp Λ S

This is sasﬁed by a distribuon where

1 1 ΛS f ; p<ΛS ; ΛS

ΛS and Λ are dynamical scales determined by the transport equaon Transport in the Glasma

1/αS At τ ~ 1/QS

p ΛS=Λ=QS

1/α At τ > 1/Q S S f(p)

p ΛS Λ

When ΛS = αS Λ, the system thermalizes; 1/2 one gets the ordering of scales: Λ =T, m = Λ ΛS = α T, Λ S = αST Thermalizaon: from Glasma to Plasma

3 Fixed box: Energy conservaon gives Λ ΛS = constant 2 From moments of transport eqn., , τcoll = Λ / ΛS ~ t t 3/7 t 1/7 From these two condions, Λ Q 0 Λ Q S ∼ s t ∼ s t 0 1 1 7/4 Thermalizaon me: t therm. ∼ Q α S S

Also, Kurkela, Moore (2011)

3 3 Entropy density s = Λ increases and saturates at ttherm as T

3 2 Nquark ~ Λ = Ngluon (Λ ΛS/αS) at ttherm when ΛS = αS Λ Thermalizaon: from Glasma to Plasma

Expanding box : maer is strongly self interacng for ﬁxed anisotropy 1+δ t0 ε (t) ε(t ) 0 < δ ≤ 1/3 g ∼ 0 t (4+δ)/7 (1+2δ)/7 t0 t0 ΛS QS Λ Q ∼ t ∼ S t 7/(3 δ) 1 τ0 − Thermalizaon me ttherm = QS τ

For δ= -1, recover ﬁxed box results…

A condensate can sll form in the expanding case for δ > 1/5 What about plasma instabilies ? Kurkela, Moore(2011) Schlichtling,Mclerran, RV, in preparaon

Relevant case for us: in general need careful numerical study to gauge impact on scaling soluons Summary

 Presented ab inio picture of mul-parcle producon and thermalizaon in heavy ion collisions

 Thermalizaon is a subtle business even in weak coupling

 Hydrodynamics may be unreasonably eﬀecve because it requires rapid decoherence of classical ﬁelds and strong self-interacons, not thermalizaon

 Excing possibility of a transient Bose-Einstein Condensate Happy 60th, Joe !!

• Look forward to the 70th, 80th, … THE END P An Analogy with the Early Universe Mishra et al; Mocsy- Sorensen The Universe HIC distributions and kinetic correlations of freeze-out produced particles hadronization lumpy initial energy density

QGP phase quark and gluon degrees of freedom

Credit: NASA

Δρ/√ρref

Δφ WMAP HIC-ALICE Role of inelasc processes ? Wong (2004) Mueller,Shoshi,Wong (2006)

Power counng for n  m processes contribuons to the collision integral n+m-2 Verces contribute αS

n+m-2 Factor of (ΛS/αS) from distribuon funcons

n+m-4 Screened infrared singularity: (1/Λ ΛS) Remaining phase space integrals Λn+m-5

2 Net result is τinelas ~ Λ / ΛS = τelas At most parametrically of the same order as elasc scaering. So a transient Bose-Einstein condensate can form.

Dusling,Epelbaum,Gelis,RV, in progress Numerical simulaons will be decisive Blaizot, Liao, McLerran CGC based models and bulk distribuons Kowalski, Motyka, Wa e+p constrained ﬁts give good descripon of hadron data Tribedy, RV: 1112.2445

p+p d+Au

Also:

Collimated long range Rapidity correlaons “the ridge” A+A Di-hadron d+A correlaons Status of CGC art Rummukainen,Weigert (2003) Dumitru,Jalilian-Marian,Lappi,Schenke,RV: arXiv:1108.1764  Numerical soluon of B-JIMWLK hierarchy :

Mul-parton correlaons in nuclear wavefuncons: Rapidity, number and impact parameter ﬂuctuaons of glue

 Recent progress in NLO computaons: F2, FL, single inclusive hadron producon,… Balitsky,Chirilli; Kovchegov; Chirilli,Yuan,Xiao; Beuf  Phenomenological “Global” analysis of e+p, e+A, p+p, d+A

A+A data in CGC models works well Levin, Rezaiean; Tribedy, RV 1112.2445