Quantum chaos in the quantum fluid Raju Venugopalan Brookhaven Na<onal 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 Mo<va<on: the unreasonable effec<veness of hydrodynamics in heavy ion collisions to paraphrase E. P. Wigner An ab ini<o weak coupling approach: Classical coherence of wee partons in nuclear wavefunc<ons Quantum fluctua<ons: Factoriza<on, Evolu<on, 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 systema<c 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 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 systema<c 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?) 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 occupa<on # => 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 Computa<onal framework Gelis,RV NPA (2006) Schwinger-Keldysh: for strong me dependent sources (ρ ~ 1/g) , ini&al value problem for inclusive quan&&es For eg., Schwinger mechanism for pair produc<on, Hawking radia<on, … Forming a Glasma in the lile Bang Lumpy classical configuraons Solu<ons of Yang-Mills equa<ons 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 fluctua<ons: a) Before the collision: pη=0 modes factorized into the wavefunc<ons - responsible for energy/rapidity evolu&on of wavefuncons a) Aer the collision pη ≠ 0; hold the key to early <me dynamics - responsible for decoherence, isotropiza&on, thermalizaon Quantum fluctua<ons in classical backgrounds: I Gelis,Lappi,RV: 0804.2630, 0807.1306,0810.4829 Suppressed Factorized into energy evolu<on of wavefunc<ons JIMWLK factoriza<on: 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 considera<ons about the color structure of higher dimensional representaons of color charge density ρa probed give as an ini<al condi<on for evolu<on (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) contribu<ons to these ini<al condi<ons Jeon, RV Dumitru,Jalilian-Marian,Petreska These inial condions give eikonalized gluon distribuons at leading log in x and Q2 Higher order correc<ons can be modeled by introducing x and b dependence (Basis of IP-sat model discussed in Bjoern’s talk) Solu<on 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 wavefunc<ons 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 fluctua<ons in classical backgrounds: II η Romatschke,Venugopalan p ≠ 0 (generated axer collision) modes grow exponen<ally 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 solu<ons 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 solu<ons of of quantum fluctuaons Yang-Mills equa<ons 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 ini<al condi<ons 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 fluctua<ons 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 chao<c 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 rela<on “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-par<cle descrip<on? Epelbaum,Gelis (2011) Spectral func<on ρ (ω,k) Plasmon mass ω k Energy density of free quasi-par<cles At early mes, no quasi-parcle descripon Energy density on the lace May have quasi-par<cle descrip<on at late <mes. Effec<ve kine<c “Boltzmann” descrip<on in terms of interacng quasi-parcles at late mes ? Quasi-par<cle occupa<on 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 k 10 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 rela<ve to a thermal distribu<on… 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 computa<onally 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 Assump<on: Evolu<on of “classical” fields in the Glasma can be matched to a quasi-parcle transport descripon See also, Mueller, Son (200)2 Jeon (2005) All es<mates 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 infla<on: 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 frac<on 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 <p<Λ 0;Λ <p ∼ αS ∼ αS p ∼ 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 condi<ons, Λ Q 0 Λ Q S ∼ s t ∼ s t 0 1 1 7/4 Thermalizaon me: t therm.