Heavy Ion Collisions: Achievements and Challenges
HHeavy Ion Collisions:Achievements and Challenges
Edward Shuryak
Department of Physics and Astronomy, Stony Brook University,Stony Brook, New York 11794-3800, USA (Dated: October 29, 2015)A decade ago brief summary of the field could be formulated as a discovery of strongly-coupledQuark-Gluon-Plasma, sQGP, making a very good liquid with surprisingly small viscosity. Since2010 we have LHC program, which added a lot to our understanding, and now there seems to be aneed to consolidate what we learned and formulate a list of issues to be studied next. Hydrodynam-ical perturbations, leading to higher harmonics of angular correlations, are identified as long-livedsound waves. Recently studied reactions involving sounds include phonon decays into two (“loopviscosity”), phonon+magnetic field into photons/dileptons (sono-magneto-luminescence), and twophonons into a gravity wave, a penetrating probe of the Big Bang. The mainstream issues in thefield now include a quest to study transition between pp, pA and heavy ion AA collisions, with anaim to locate ”the smallest drops” of the sQGP displaying collective/hydrodynamics behavior. Theissues related to out-of-equilibrium stage of the collisions, and mechanisms of the equilibration, inweak and strong coupling, are also hotly debated. I. A WIDER PICTURE
Before we immense ourselves in recent discoveries andpuzzles of heavy ion physics itself, one perhaps shouldlook in a wider field of ”strong interaction physics”. Itsdefinitions have many different meanings, ranging from anarrow one, (i) hadronic and nuclear physics phenomenabased on its fundamental theory, QCD; to much widerone (ii) aiming at understanding of all gauge theories,including QCD-like and supersymmetric ones, and evenincluding dualities of those to string/gravity theories. Wewill keep mostly to the former end, with only some sec-tions venturing to the latter definition.QCD-like theories at finite temperature and densityhas a number of regimes and phases. Due to asymp-totic freedom, the simplest of them to understand is thehigh T (or µ or both) limit, known as weakly coupled Quark-Gluon Plasma (wQGP). Well developed tools ofthe manybody theory – perturbative diagrams and theirre-summations – were used to show its main features, theelectric screening and formation of finite-mass quasipar-ticles. This is where the name “plasma” came from [1].For early reviews on high-T QCD see [2, 3].Prior to RHIC era, it was generally assumed thatwQGP regime extends down to the phase transition re-gion,
T > T c . Lattice-based thermodynamic quantitiesshowed relatively small ( 20% or less) deviation fromthose for non-interacting quarks and gluons. Many if notmost theorist were skeptical, thinking that RHIC pro-gram aimed at “production of new form of matter” wouldbasically fail. Perturbative equilibration estimates indi-cated unrealistically long time. Models predicted “fire-work of minijets” rather than collective effects.Yet strong hydrodynamical flows were observed atRHIC from the first months, supplemented by strong jetquenching. All of that was later confirmed at the LHC,which provided even much more accurate data on manydetails. For example, as we will discuss below, there are unmistakable signs that the excitation modes of the mat-ter produces is basically the sound modes. The system-atics of those lead to viscosity value which is remarkablysmall, more than an order of magnitude smaller thanthat predicted originally by the perturbative theory. So,the window in which matter produced at RHIC/LHC is, T c < T < T c , was renamed into a “strongly coupledQGP” , sQGP for short.So, which physical phenomena take place there andwhy? What exactly happens as T decreases from thewQCD to sQGP domain? The growing coupling inducestwo different kinds of phenomena.(i) One preserves the basic picture of quark and gluonquasiparticles and plasma-related phenomena, with grad-ual change of parameters consistent with smooth logarith-mic running g ∼ /log ( T /
Λ). While in the wQCD do-main the electric screening (Debye) mass are small com-pared to their energy, M E /T = O ( g ) (cid:28)
1, in sQGP do-main one learned from lattice simulations that those arenot small, M E /T = 2 − > exponentially , leading tomuch stronger power dependences on the temperature exp (cid:18) − constg ( T ) (cid:19) ∼ ( Λ T ) power (1)Those come from gauge field configurations – semiclas-sical solitons – which are qualitatively distinct fromperturbative quarks and gluons. One such example isthe (Euclidean 4-dimensional) topological solitons called instantons . Their physics is tunneling through certainbarriers into “valleys” of topologically distinct gauge a r X i v : . [ h e p - ph ] O c t field configurations. Perturbative theory , an expansionaround zero fields, knows little about existence of suchvalleys. (Except in a very subtle way, via divergences ofits series.There is recent progress in this direction, knownas a resurgence theory , which however still focuses onquantum mechanical and low-dimensional scalar theoriesrather than gauge theories. See for recent update in thetalks at [7] .)Discussion of the instanton theory will take us too farfrom our main direction, so I only mention its two ma-jor breakthroughs. In 1980-1990’s it has been shownhow instantons break the chiral symmetries , the U A (1)explicitly and SU ( N f ) spontaneously, via collectiviza-tion of fermionic zero modes, for a review see [5]. Re-cently it morphed into plasma made of instanton con-stituents, Lee-Li-Kraan-van Baal (LLKvB) instanton-dyons , or instanton-monopoles. It has been shown thatthose, if dense enough, can naturally enforce both con-finement and chiral symmetry breaking. For recent dis-cussion see talks at [6].Another – still very puzzling – set of effects, predictedby the Operator Product Expansion (OPE), are relatedwith “higher twist effects” and VEVs of various operatorsappearing in the QCD sum rule framework. The so called renormalons are supposed to induce effects with variousinteger powers of the scale , e.g. T . In the gauge theorywe still do not know which solitons, if any, are related tothem, but in simpler theories there was recent progress:see again [7].An extremely important notion of the theory is thatof electric-magnetic duality . Asymptotically free gaugetheories, QCD-like or supersymmetric ones, have weak“electric” coupling in the UV, but in the IR they gener-ically flow to strong coupling. “Magnetic” objects, suchas color-magnetic monopoles, are expected to becomelighter and more numerous along this process. The keyobservation I tirelessly emphasize is the good old Diraccondition . If one wants simultaneously describe objectswith electric and magnetic charges ( e and g in his oldnotations/normalization) singularities of the gauge po-tential of the form of the Dirac strings are unavoidable.The requirement that should be pure gauge artifacts andthus invisible leads to 2 eg (cid:126) c = 1 (2)(or in general an integer in the r.h.s.). Thus e and g must run in unison, in the opposite directions. Thus oneexpects in the infrared limit of gauge theories to findthe so called dual description based on magnetic degreesof freedom, which are light and weakly coupled. Thequestion is whether one can construct a Lagrangian ofthis dual theory, and also if/when one can/needs to useit.(One great example of the kind is provided by cel-ebrated N =2 supersymmetric Seiberg-Witten theory:near certain special points of the Higgs VEV themonopoles are nearly massless and weakly interacting: (ii) the ratio of magnetic-to-electric coupling g/e . Themain issues discussed are how the transport properties(in particular the shear viscosity ) of the plasma dependon them. More specifically, the issue is whether admix-ture of weaker-coupled MQPs increases or decreases it. T e=g linem − dominatede − confined m − confinede − dominatede − dominatedm strongly correlatedm − dominatede stronglycorrelated µ H CSQGP
FIG. 1. (color online) A schematic phase diagram on a(“compactified”) plane of temperature and baryonic chemicalpotential T − µ . The (blue) shaded region shows “magneti-cally dominated” region g < e , which includes the e-confinedhadronic phase as well as “postconfined” part of the QGPdomain. Light region includes “electrically dominated” partof QGP and also color superconductivity (CS) region, whichhas e-charged diquark condensates and therefore obviouslym-confined. The dashed line called “e=g line” is the line ofelectric-magnetic equilibrium. The solid lines indicate truephase transitions, while the dash-dotted line is a deconfine-ment cross-over line. A. Strongly coupled Quark-Gluon plasma in heavyion collisions
A realization [3,4] that QGP at RHIC is not a weaklycoupled gas but rather a strongly coupled liquid has ledto a paradigm shift in the field. It was extensively de-bated at the “discovery” BNL workshop in 2004 [5] (atwhich the abbreviation sQGP was established) and mul-tiple other meetings since.Collective flows, related with explosive behavior of hotmatter, were observed at RHIC and studied in detail: theconclusion is that they are reproduced by the ideal hydro-dynamics remarkably well. Indeed, although these flowsaffect different secondaries differently, yet their spectraare in quantitative agreement with the data for all ofthem, from π to Ω − . At non-zero impact parameterthe original excited system is deformed in the transverseplane, creating the so called elliptic flow described by v ( s, p t , M i , y, b, A ) = < cos (2 φ ) > (3)where φ is the azimuthal angle and the others standfor the collision energy, transverse momentum, particlemass, rapidity, centrality and system size. Hydrodynam-ics explains all of those dependence, for about 99% of theparticles .Naturally, theorists want to understand the nature ofthis behavior by looking at other fields of physics whichhave prior experiences with liquid-like plasmas. One ofthem is related with the so called AdS/CFT correspon-dence between strongly coupled N =4 supersymmetricYang-Mills theory (a relative of QCD) to weakly coupledstring theory in Anti-de-Sitter space (AdS) in classicalSUGRA regime. We will not discuss it in this work: fora recent brief summary of the results and references seee.g. [6].Zahed and one of us [4] argued that marginally boundstates create resonances which can strongly enhancetransport cross section. Similar phenomenon does hap-pen for ultracold trapped atoms, due to Feshbach-typeresonances at which the binary scattering length a → ∞ ,which was indeed shown to lead to a near-perfect liquid.van Hees, Greco and Rapp [7] studied ¯ qc resonances, andfound enhancement of charm stopping.Combining lattice data on quasiparticle masses and in-terparticle potentials, one finds a lot of quark and gluonbound states [8,9] which contribute to thermodynami-cal quantities and help explain the “pressure puzzle” [8],an apparent contradiction between heavy quasiparticlesnear T c and rather large pressure. The magnetic sec-tor discussed in this paper provides another contribution,that of MQPs (monopoles and dyons), which will help toresolve the pressure puzzle.A very interesting issue is related with counting of thebound states of all quasiparticles. Here the central notionis that of curves of marginal stability (CMS), which arenot thermodynamic singularities but lines indicating asignificant change of physics where a switch from onelanguage to another (like E ⇀↽ M ) is appropriate or evenmandatory.Let us mention one example related with quite interest-ing “metamorphosis” discussed in literature, in the con-text of N =2 SUSY theories. The CMS in question isrelated with the following reaction gluon ↔ monopole + dyon (4)in which the r.h.s. system is magnetically bound pair(obviously with zero total magnetic charge). The curveitself is defined by the equality of thresholds, M ( gluon ) = M ( dyon ) + M ( monopole ) (5) The remaining ∼
1% resigning at larger transverse mo-menta p t > GeV are influenced by hard processes and jets. And prevention of the double counting. FIG. 1: A schematic phase diagram on a (compactified) planeof temperature and baryonic chemical potential T − µ , from[8]. The (blue) shaded region shows magneti- cally dominatedregion g < e , which includes the e-confined hadronic phase aswell as postconfined part of the QGP domain. Light regionincludes electrically dominated part of QGP and also colorsuperconductivity (CS) region, which has e-charged diquarkcondensates and therefore obviously m-confined. The dashedline called e=g line is the line of electric-magnetic equilibrium.The solid lines indicate true phase transitions, while the dash-dotted line is a deconfinement cross-over line. the magnetic theory is then scalar QED. Its beta func-tion, as expected, has the opposite sign to that of theelectric theory. Even greater example is N =4 theory: itis electric-magnetic selfdual, so the beta function is equalto minus itself and thus is identically zero.)Returning to QCD, one can summarize the picture ofthe so called “magnetic scenario” by a schematic plotshown in Fig. 1, from [8]. At the top – the high T domain – and at the right – the high density domain– one finds weakly coupled or “electrically dominated”regimes, or wQGP. On the contrary, near the origin ofthe plot, in vacuum, the electric fields are, subdominantand confined into the flux tubes. The vacuum is filledby the magnetically charged condensate, known as “dualsuperconductor”. The region in between (relevant formatter produced at RHIC/LHC) is close to the “equi-librium line”, marked by e = g on the plot. (People forwhom couplings are too abstract, can for example defineit by an equality of the electric and magnetic screeningmasses.) In this region both electric and magnetic cou-pling are equal and thus α electric = α magnetic = 1: soneither the electric nor magnetic formulations of the the-ory are simple.Do we have any evidences for a presence or impor-tance for heavy ion physics of “magnetic” objects? Hereare some arguments for that based on lattice studies andphenomenology, more or less in historical order:(i) In the RHIC/LHC region T c < T < T c the VEVof the Polyakov line < P > is substantially different from1. Hidaka and Pisarski [9] argued that < P > must beincorporated into density of thermal quarks and gluons,and thus suppress their contributions. They called suchmatter “semi-QGP” emphasizing that say half of theirdegrees of freedom are actually operational.And yet, the lattice data insist that the thermal energydensity remains close to the T trend nearly all the wayto T c . “Magnetic scenario” [8] may explain this puzzleby introducing extra contributions of the magnetic quasi-particles. (They are not subject to < P > suppressionbecause they lack the electric charge.)(ii) Lattice monopoles so far defined are gauge depen-dent. And yet, they were found to behave as physicalparticles. Their motion definitely shows Bose-Einsteincondensation at T < T c [10]. Their spatial correlationfunctions are very much plasma-like. Even more strikingis the observation [11] revealing magnetic coupling which grows with T . Moreover, it is indeed an inverse of theasymptotic freedom curve.(iii) Plasmas with electric and magnetic charges showunusual transport properties [8]: Lorenz force enhancescollision rate and reduce viscosity. Quantum gluon-monopole scattering leads to large transport cross section[12], providing small viscosity in the range close to thatobserved at RHIC/LHC.(iv) high density of (non-condensed) monopoles near T c affect electric flux tubed, and perhaps explain [122]lattice observations of high tension in the potential en-ergy (not free energy) of the heavy-quark potentials . Wewill discuss those in section X.(v) Last but not least, high density of monopoles near T c seem to be directly relevant for jet quenching. Thisissue- a very important part of heavy ion physics - wewill discuss in section XI.(Completing this brief introduction to monopoles, it isimpossible not to mention the remaining unresolved is-sues. Theories with adjoint scalar fields – such as e.g.celebrated N =2 Seiberg-Witten theory – naturally haveparticle-like monopole solutions. Yet in QCD-like theo-ries without scalars it has to be dynamically generated,not yet explained. So far lattice definitions of monopolesare gauge-dependent. A good news is that most if notall of monopole physics can be taken care of via theinstanton-dyons we mentioned above: in this case therole of the adjoint “Higgs” is played by the time compo-nent of the gauge potential A . It is real on the Euclideanlattice, but would get i in Minkowski continuation: so itcannot be directly a “particle” in ordinary sense. )Another famous duality is AdS/CFT gauge-string du-ality [14], basically used at large number of colors asa gauge-gravity duality. It directly relates equilibratedQGP at strong coupling to a certain black hole solutionsin 5 dimensions, with the plasma temperature being itsHawking temperature, and the QGP entropy being itsBekenstein entropy. In section VII we will discuss re-cent studies of out-of-equilibrium settings, in which black hole is dynamically generated. Holographic models ofthe AdS/QCD types also lead to new views on the QCDstrings, Reggeons and Pomerons: see section VIII C.This brief introduction shows that heavy ion physicsdid acquired a lot of outreach into many other areas ofphysics. II. THE MAIN ISSUES IN QCD AND HEAVYION PHYSICS
Like any other rapidly developing field, heavy ionphysics has has “growth problems”. Quark Matter con-ferences (and their proceedings) provide regular snap-shots of the field’s development. Yet those provide poorservice for young people or outsiders who want to un-derstand what is going on. The talk time is alwaysseverely limited, so naturally only the latest results arereported. They are kaleidoscopic set of compressed an-swers to questions which were never formulated, as thereis no time for that. Discussion session are becoming ex-tremely short (or non-existing), and never recorded. Fur-thermore, large conferences have multiple parallel ses-sions, and it is hard to follow more than one subfield.Traditionally the “second pillar” of any field consistsof books, reviews, lecture notes. Those formulate thebasic questions and the the lessons learned. They alsoprovide a list of “challenges”, the questions which needto be answered next. With ongoing RHIC program andthe now maturing LHC heavy ion program, it is perhapstime for a review. Looking through the talks at the latestQuark Matter conferences, I feel a relative weakness ofthe theory, compared to overwhelming amount of datafrom RHIC and LHC. and decided it is time to do it.This review certainly is not complete, many importantdirections/achievements are not covered. Jet quenchingis nearly missing. Fluctuations in the gauge topology, re-lated to chiral magnetic effect, are not discussed: it needsto be developed further before some summary emerges.It is a subjective overview of the field, focusing on bigpicture, questions and trends rather than specific results.(On a personal note: the first review I wrote [2], onfinite temperature QCD proclaiming a search for QGP asa new goal for community, was written due to an advicegiven to me by Evgeny Feinberg. Let me also mentionhere Gerry Brown: long before I met him I benefitedfrom his vision, in founding Physics Reports devoted tosuch reviews of subfields. The instanton review in RMPof 1996 [5] is in fact my most cited paper. In 2004, whenthe sQGP paradigm of “a perfect liquid” has been born,many people wrote summary papers. My most extensiveversion was the second part of my book [4].)Let me start with few “super-questions” (and com-ments on them), which are common to the whole stronginteraction physics, extending well beyond the bound-aries of the heavy ion field.I.
Can one locate the “soft-to-hard” boundary ,in whatever observables under consideration, where thetransition from weak to strong coupling regimes takeplace?II.
Can one locate the ‘micro-to- macro” bound-ary , where the transition from large mean-free-path(ballistic) to small one (hydrodynamic) regimes happen?In particular, where in the observable discussed one findsa transition from effects due to single-parton distribu-tions to those due to collective explosion?III.
Can we experimentally identify signals ofthe QCD phase transition , in particularly locate the ”softest” and the QCD critical point?
Brief comments on them are:(Ia) Since 1970’s the “golden” test of pQCD at largemomenta transfer was deep inelastic scattering (DIS), aswell as hard exclusive processes, e.g. the pion and nu-cleon formfactors. Unlike jets in DIS, the latter havecolorless final state, they can in principle be accuratelymeasured, and thus promised to test the pQCD predic-tions due to the specific lowest order diagrams. And yet,to the highest Q ∼ GeV measured so far, neither ofthem had been found to reach quantitative agreementwith the pQCD predictions, because even at such Q thenon-perturbative effects still dominate.Closer to our field is the “mini-jet” issue. While theidentified jets have rather large momenta, say p ⊥ > GeV or so, it is widely assumed that the parton de-scription is good down to much smaller momenta. Howmuch smaller? Following DGLAP evolution toward small Q all the way to ∼ GeV one eventually reach a neg-ative gluon density. Many other arguments tell us thatat this scale, 1 GeV , pQCD cannot be used. At whichscale Q min one has to stop is defined by the “highertwist effects”, not yet studied quantitatively. Thus weare still left with uncomfortably large gap, between 1and 20 GeV .(Ib) The elementary process fundamental for our fieldis the pp scattering. Its total cross section and elasticamplitude is described by the so called Pomeron phe-nomenology. Elastic amplitude is function of the momen-tum transfer t = − q and its Bessel-Fourier transform isthe so called profile function F ( b ) depending on the im-pact parameter b . Small b is understood via perturbativeBFKL Pomeron, while large b via some string-exchangemodels. In this case the experimental data actually do in-dicate sharp transition between these regimes. Recentlyan attempt to understand both regimes in a single modelhas been quite successful, in the so called AdS/QCDframework. Furthermore, it has been suggested that thecritical b is related to critical temperature T c of the phasetransition in the gauge theory: we will discuss this in sec-tion VIII C.Ic. Proceeding from elastic to inelastic collisions, un-der which conditions we should describe the initial snap-shots of hadrons and nuclei in terms of perturbative par-tons (quarks and gluons) or non-perturbative effectiveobjects (strings,constituent quarks, etc)? As we will dis-cuss, hydro sound modes survive till freezeout, and thus this make these initial snapshots visible to the detectors.Therefore, one can now estimate at least the number ofsuch “objects”.(IIa) Furthermore, now we have a variety of cases –AA, pA and pp collisions with widely variable multiplic-ities. All of them show collective phenomena – radial,elliptic and even triangular flows -at high enough mul-tiplicity. Pushing such observables down in multiplic-ity is a current frontier. One would like to find someregime changes there, experimentally and theoretically.No sharp changes are so far detected. People apply mi-croscopic theory at one end of the pA and pp collisionsand macroscopic theory (thermo and hydrodynamics) atanother, checking to what extent one can justify them,but this process is far from converged so far.(IIb) Where exactly is the boundary between the microand macro theories? Textbook answer is that one cancompare the micro or “mean free path” scale l to the sizeof the system L (cid:29) l (3)and if the l/L ratio is small one can use the macroscopictheories. Given a very small phenomenological viscos-ity η/s ∼ . III. SOUNDS ON TOP OF THE “LITTLEBANG”A. Introductory comments on hydrodynamics
Since the start of the RHIC era in 2000, it has becomesoon apparent that the data on particle spectra (the ra-dial flow) and the elliptic flow confirm nicely predictionsof hydrodynamics, supplemented by hadronic cascade atfreezeout [16–18]. All relevant dependences – as a func-tion of p ⊥ , centrality, particle mass, rapidity and col-lision energy – were checked and found to be in goodagreement. Since the famous 2004 RBRC workshop inBrookhaven, with theory and experiment summaries col-lected in Nucl.Phys.A750 , the statements that QGP ”isa near-perfect liquid” which does flow hydrodynamicallyhas been endlessly repeated. QGP is recognized to bein the strongly coupled regime, now called sQGP, andhundreds of theoretical papers are written, developingdynamics at strong coupling. Several second-order hy-dro+cascade models had been developed in the last de-cay, which do an excellent job in describing the data. Theinterested readers should look at recent reviews such as[34]: we will take this to be a “well established domain”.(Of course, the data are constantly getting more impres-sive with time: let me mention e.g. the elliptic flow v ofthe deuterons recently measured by ALICE and shownat QM2015.)The interest has now been shifted to understandingthe limits of the hydro description. Of course, the exactboundary depends on which version of it is implied. Inthe simplest case, when one thinks of ideal hydro plus vis-cous corrections , this happens when the viscosity timesthe velocity gradients is as large as the main terms. Now,there can be two different reasons why this correctiongets large: (i) either the fluid is not good, viscosity-to-entropy ration η/s is not small, or (ii) the gradients aretoo large. Studies of the former have recently been doneusing anisotropic hydro and exact solution of Boltzmannequation in Gubser setting, as well as by the so calledanisotropic hydrodynamics : we will discuss those in sec-tion IV C. The latter situations can be approached viathe so called higher order hydrodynamics . Gradient re-summation was in particularly attempted by Lublinskyand myself: we will discuss those in section VII E. In allof these cases, an improved hydrodynamics is supposedto shift its initiation time somewhat earlier, or promiseto treat somewhat smaller systems. Yet, while the “out-of-equilibrium” initial stage get reduced, of course it cannever be eliminated. The distinction between the “ini-tial” and “equilibrated” stages is a matter of definition:but physical outputs – e.g. the total amount of entropyproduced – should ideally be independent of that. Un-fortunately, in practice we are still far from this idealscenario: studies of entropy generation at an initial stageis still in its infancy.Few other issues remains open till now, related withthe boundary of hydrodynamical description.One is the boundary at high p t . When the first RHICdata came, it has been gratifying to see that agreementwith hydro-based spectra went down more than order ofmagnitude and reached p t ∼ GeV . The it has beenextended much further, till p ⊥ ∼ GeV . This regionincludes more than 99.9% of all secondaries! Indeed, onlyfew – out of thousands – particles in a heavy ion collisiondo come from hard tails of the spectra.As the regime changes at p ⊥ ∼ GeV , one needs tounderstand why. One important point is that particlesat high p t come from an edge of the fireball, at whichthe magnitude of the hydro flow is maximal. Using thesaddle point method for Cooper-Frye integral [19] onecan see that the region from which such particles comeshrinks, as p t grows. (We will return to this point in con-nection to high multiplicity pp collisions and HBT radii,see sectionV E.) Eventually this region shrinks to a sin-gle hydro cell (defined by the mean free path), and thenhydro description needs to be modified. Teaney [20] in-troduced the viscosity effect in the cell, and noticed thatdue to extra derivative these effects should be enhancedat the edge. They also should have extra power of p t , heargued, and the deviation should be downwards, as in-deed is observed in harmonic flows. Still, physics in thewindow 4 < p ⊥ < GeV , between the hydro-dominatedand clearly jet-dominated regions is not yet understood.The second issue is the relative role of sQGP andhadronic stages in the expansion and flows. At AGS/SPS energy domain one clearly had to understand both quan-titatively: otherwise the results got wrong. For example,the paper [59] was all about the sequential freezeout ofdifferent kinds of secondaries, and how it was seen in ra-dial flow. It has been fairly important for radial flowat RHIC: yet elliptic and other harmonics of flow wereformed earlier and the hadronic stage were less impor-tant.The last 4 years had provided further confirmation ofthe hydro paradigm at LHC. One of the benefits is that,even for radial flow, the main acceleration is done in thesQGP phase, to such an extent that one can neglect thesequential freezeout and related complications. Let mejust provide one example of how well it works, from AL-ICE.Out of all individual properties of the secondaryhadron, only one – its mass – is important for hydromodels. All one needs to know is that putting an objectof mass m into a flow with velocity v , the momentumwill be mv . Nice demonstration that it is indeed so canbe made by direct comparison of the spectra for a pair ofhadrons with the same mass, with otherwise completelydifferent quantum numbers, cross sections etc. The clas-sic example is p and φ : those can hardly be more dif-ferent, the former is non-strange baryon, the latter is astrange-antistrange meson, etc. And yet, as e.g. ALICEmeasurements shows, their spectra are– with impressiveprecision – identical, up to p T ∼ GeV /c . The very factthat p/π, φ/π ratios can increase by two orders of mag-nitude and reach O(1) at such p t is by itself a dramatichydro effect, never seen in generic (non-high-multiplicity) pp collisions.I think it puts to rest many non-flow explanations ofthe particle spectra in AA collisions.Among those I count a widely popularized idea of theso called “quark scaling” of the elliptic and other flows,that suggests that if a hadron is made of n quarks then itsflow harmonics should be proportional to n , v ( h ) m ∼ nv ( q ) m ,with some the “quark” v ( q ) m . The logic behind this idea[151] comes from the expression (82) of that paper whichassumes that the quark distribution function in the phasestale is “factorized” w ( R ; p ) = w i ( R ; p )[1 + 2 v a ( p ⊥ ) cos (2 φ )] . (4)Note that the bracket does not depend on the location R . So, the authors assume v to originate from someanisotropic momentum distribution in every local hydrocell . This assumption is in direct contradiction to theflow picture, which instead postulate that each cell hasan equilibrium (=isotropic) distribution of particles [162]Anisotropic flows are due to different number of cells moving in different directions. This n -parton structurecannot possibly matter: a hadron can be made of 2, zeroor 22 partons: all you need to know is its mass. P ( v / 〈 v 〉 ) , P ( ε / 〈 ε 〉 ) v / 〈 v 〉 , ε / 〈ε 〉 p T > 0.5 GeV| η | < 2.565-70% ε IP-Glasma v IP-Glasma + MUSIC v ATLAS 0.01 0.1 1 10 100 0 0.5 1 1.5 2 2.5 3 P ( v / 〈 v 〉 ) , P ( ε / 〈 ε 〉 ) v / 〈 v 〉 , ε / 〈ε 〉 ε IP-Glasma v IP-Glasma + MUSIC v ATLAS 0.01 0.1 1 10 100 0 0.5 1 1.5 2 2.5 3 P ( v / 〈 v 〉 ) , P ( ε / 〈 ε 〉 ) v / 〈 v 〉 , ε / 〈ε 〉 ε IP-Glasma v IP-Glasma + MUSIC v ATLAS
FIG. 2. (Color online) Data points correspond to the event-by-event distribution of v , v , and v in the respective max-imal peripheral bin measured by the ATLAS collaboration[32]. These are compared to the distributions of initial ec-centricities in the IP-Glasma model and the distributions of v n from fluid dynamic evolution with IP-Glasma initial con-ditions. energy density. This radius by definition depends on thechoice of ε min . This choice however only affects the over-all normalization of r max ; it does not affect the depen-dence of r max on the number of charged particles N ch [34]. There is also some uncertainty in the radii comingfrom the choice of the infrared scale m that regulates thelong distance tail of the gluon distribution (see [3, 4, 28]).It can be mostly compensated for by adjusting a normal-ization constant K .In Fig. 3 we show the result for r max in p+p, p+Pb,and Pb+Pb collisions and compare to R inv from theEdgeworth fit to the two-pion correlation function mea-sured by the ALICE collaboration [27]. We adjust K to match to the p+p results. We determine centralityclasses in the model and assign the N ch value quoted byALICE [27] for each centrality class.Because the emission of pions occurs throughout the R i n v , K r m a x [f m ] 〈 N ch 〉 ε min = Λ QCD4 m = 0.2 GeVK = 1.25
ALICE Pb+Pb 2.76 TeV
ALICE p+Pb 5.02 TeV
ALICE p+p 7 TeV
Pb+Pb 2.76 TeV p+Pb 5.02 TeV p+p 7 TeV
FIG. 3. (Color online) R inv measured by the ALICE col-laboration [27] compared to Kr max determined using the IP-Glasma model and fluid dynamic expansion. The lower endof the band indicates the size of the initial state, the upperend the maximal value of r max during the hydrodynamic evo-lution. evolution, R inv lies somewhere between the initial radiusand the maximal radius reached during evolution. Weindicate the range of radii between these two extrema bya band in Fig. 3. We find that our estimate of the systemsize is compatible with the experimental HBT measure-ment for all systems simultaneously. The Pb+Pb resultclearly favors the presence of hydrodynamic expansion.For events with the same multiplicity (for exampleat ⟨ N ch ⟩ / ≈ m = 0 . m = 0 . v and v as a function of N offlinetrk , measuredby the CMS collaboration. Fig. 4 shows the calculated v in peripheral Pb+Pb col-lisions and central p+Pb collisions with the same N offlinetrk in comparison to experimental data by the CMS collab-oration [35]. While the Pb+Pb result reproduces the To obtain N offlinetrk we determine the centrality class in the IP-Glasma simulations and match to the N offlinetrk quoted for thatcentrality class by the CMS collaboration in [35]. N offlinetrk ≈ FIG. 2: (a) From [34]: average deformations < (cid:15) n > =
Now, assuming that the average pattern of the fireballexplosion has been well established, we are going to per-turb it. The fluctuations and and their correlations isthus the next topic of our discussion.There are two schools doing this. One uses the so calledevent-by-event hydro, with ensemble of initial conditionsdefined by some model. Yet I will argue that practi-cally all one learned from such expensive studies can alsobe understood from a much simpler approach, in whichone adds small and elementary perturbations on top ofsmooth average fireball, and solve for the complete spec-trum of excitations. Using an analogy, instead of beat-ing the drums strongly, with both hands and all fingers,one may first touch it gently with a drumstick, at dif-ferent locations, recording the spectra and intensities ofthe sounds produced. Eventually, one may understandnot only its spectrum, but also all the relevant modes ofexcitations. The one can work out linear and nonlinearsuperpositions of those elementary modes.( The reader should understand that by no means thelatter approach undermines good work which the latterdoes. People developing stable hydro codes and ensem- bles of initial conditions, averaging hydro results overthousands of configurations with complicated shapes. Allthat works spectacularly well: the v n moments of theflow perturbations in azimuthal angle φ as a functionof transverse momentum, particle type and centrality v n ( p t , m, N p ) are all reproduced. Hardly any further con-vincing of that is now needed: multiple versions of theplots proving that are shown at all conferences and per-haps seen by a reader: so I will not discuss this and takeit for granted as well.)Now is the time of synthesis of all that information. Inorder to summarize what we have learned from fluctua-tion/correlation studies one needs to go back to the dataand to the results of hydro calculation and separate theiressence from unimportant complications. Some simplepocket formula fits to analytic expressions, revealing thesystematics of physical effects in question, can be veryuseful at this stage, and we will discuss them below.But before we dip into details, let me formulate themain answers, using my drum analogy. First of all, wedid learned that perturbations on top of the little bangbasically behave as sounds propagating on the drum. Themain phenomenon is the viscous damping, and the valueof sQGP viscosity is the main physics output of thesestudies. Different time evolution of different modes isthe next issue: unlike the usual drum, the little bang isnot static but exploding: therefore oscillating behavior issuperimposed with time dependence of the amplitudes.Different excitations are excited if the drum is struck atdifferent places: similarly here, and we only start under-standing the excitation modes themselves.Let me continue where these calculations typicallystart: from angular deformations of the initial state. InFig.2(a) one finds the dependence of the mean harmonics(eccentricities) (cid:15) n = < cos ( nφ ) > (5)where n is integer and φ is the azimuthal angle. Theangular brackets mean average over events, usually forcertain centrality bin (indicated in the upper left cor-ner as a fraction of the total cross section, which scale as bdb ). The one 0-0.2% is called the ultra-central collisions, b ≈
0, and 50 −
60% are some peripheral collisions. Thefirst obvious comment to this plot is that n = 2 , all harmonics for the central bins, arebasically independent on n and centrality. What thattells us is that statistically independent “elementary per-turbations” have small angular size δφ (cid:28) π , so onebasically deals with an angular Fourier transform of thedelta function.The next observation is that the deformations getssmaller toward the central collisions, to just few percentlevel. This is also natural: larger fireballs have more par-ticles and thus they fluctuate less, in relative terms. In FIG. 3. (a) v , vs. N part for p T = 1 − v n /ε n ) vs. 1 / ¯ R for the data shown in (a): (c - e) centrality dependenceof the ε n /ε ratios extracted from fits to ( v n ( p T ) /v ( p T )) n ≥ with Eq. 6; ε n /ε ratios for the MC-Glauber [33, 37] and MC-KLN[34] models are also shown: (f) extracted values of β vs. centrality: (g) extracted values of α vs. centrality (see text).FIG. 4. (a) ln( v n /ε n ) vs. n from viscous hydrodynamical calculations for three values of specific shear viscosity as indicated.(b) ln( v n /ε n ) vs. n for Pb+Pb data. The p T -integrated v n results in (a) and (b) are for 0.1% central Pb+Pb collisions at √ s NN = 2 .
76 TeV [38]; the curves are linear fits. (c) β vs. 4 πη/s extracted from the curves shown in (a) and (b). within errors, the full data set for v n ( p T , cent) can be un- derstood in terms of the eccentricity moments coupled to a single (average) value for α and β (respectively). This observation is compatible with recent viscous hydrody- namical calculations which have been successful in repro- ducing v n ( p T , cent) measurements with a single δf (˜ p T ) ansatz and an average value of η/s [26, 27]. Therefore, these values of α and β should provide an important set of constraints for detailed model calculations. To demonstrate their utility, we have used the results from recent viscous hydrodynamical calculations [38] to calibrate β and make an estimate of η/s . This is illus- trated in Fig. 4. The p T -integrated v n results from vis- cous hydrodynamical calculations for three separate η/s values, for 0.1% central Pb+Pb collisions are shown in Fig. 4(a). They indicate the expected linear dependence
FIG. 3: (a) Atlas data, from Ref. [35], for v , v vs. N part :(b) ln ( v n /(cid:15) n ) vs. 1 /R for the same data fact one expect < (cid:15) n > ∼ √ N cells (6)where N cells = A cell /A fireball is the number of statisti-cally independent cells in the transverse plane. (Whatexactly is a cell is not simple and we will discuss it be-low in the section on the initial state. Depending on themodel it changes from a fraction to few f m .)Models of the initial state give not only the averagedeformation but also their distributions and correlations.Going a bit ahead of myself, let me note that, remarkably,the experimentally observed distributions over eccentric-ities of particle momenta distributions P ( v n ) appearsto directly reflect distributions of angular anisotropies P ( (cid:15) n ) at the initial time, see e.g. (cid:15) , v distributions inFig.2(b). (Thus, apparently no noise is generated by thehydro evolution, from the initial state (cid:15) n to the final state v n : why is it so we will discuss below.) C. Acoustic systematics: the viscous damping
There is a qualitative difference between the radialflow and higher angular harmonics. While the formermonotonously grows with time, driven by the outwardpressure gradient with a fixed sign, the latter are basi-cally sounds, or a (damped) oscillators. Therefore thesignal observed should, on general grounds, be the prod-uct of the two factors: (i) the amplitude reduction factordue to viscous damping and (ii) the phase factor contain-ing the oscillation at the freezeout. (We will discuss theeffects of the phase in the next section.)Let us start with the “acoustic systematics” which in-cludes only the viscous damping factor. It provides good qualitative account of the data and hydro calculationsinto a simple expression, reproducing dependence on theviscosity value η , the size of the system R and the har-monic number n in question. Let us motivate it as fol-lows. We had already mentioned “naive” macro and mi-cro scales (3): now we define it a bit better by insertingthe famous viscosity-to-entropy ratio η/s = lT , lL = ηs LT (7)This “true micro-to-macro ratio”, corresponding to themean free path in kinetic theory, defines the minimal sizeof a hydro cell.One effect of viscosity on sounds is the damping of theiramplitude. The so called “acoustic damping” formula,suggested by Staig and myself [47] , is given by v n (cid:15) n ∼ exp (cid:20) − Cn (cid:16) ηs (cid:17) (cid:18) T R (cid:19)(cid:21) (8)where C is some constant. The number n appearssquared because the damping includes square of the gra-dient, or momentum of the wave. So, we have the follow-ing predictions: (i) the viscous damping is exponential in n ; (ii) the exponent contains the product of two smallfactors, η/s and 1 /T R , (iii) the exponent contains 1 /R which should be understood as the largest gradient inthe system, often modelled as 1 /R = 1 /R x + 1 /R y .Extensive comparison of this expression with the AAdata, from central to peripheral, has been done in Ref.[35] from which we borrow Fig.3 and Fig.4. The Fig.3 (a)shows the well known centrality dependence of the ellipticand triangular flows. v is small for central collisionsdue to smallness of (cid:15) , and also small at very peripheralbin because viscosity is large at small systems. Fig.3 FIG. 3. (a) v , vs. N part for p T = 1 − v n /ε n ) vs. 1 / ¯ R for the data shown in (a): (c - e) centrality dependenceof the ε n /ε ratios extracted from fits to ( v n ( p T ) /v ( p T )) n ≥ with Eq. 6; ε n /ε ratios for the MC-Glauber [33, 37] and MC-KLN[34] models are also shown: (f) extracted values of β vs. centrality: (g) extracted values of α vs. centrality (see text).FIG. 4. (a) ln( v n /ε n ) vs. n from viscous hydrodynamical calculations for three values of specific shear viscosity as indicated.(b) ln( v n /ε n ) vs. n for Pb+Pb data. The p T -integrated v n results in (a) and (b) are for 0.1% central Pb+Pb collisions at √ s NN = 2 .
76 TeV [38]; the curves are linear fits. (c) β vs. 4 πη/s extracted from the curves shown in (a) and (b). within errors, the full data set for v n ( p T , cent) can be un- derstood in terms of the eccentricity moments coupled to a single (average) value for α and β (respectively). This observation is compatible with recent viscous hydrody- namical calculations which have been successful in repro- ducing v n ( p T , cent) measurements with a single δf (˜ p T ) ansatz and an average value of η/s [26, 27]. Therefore, these values of α and β should provide an important set of constraints for detailed model calculations. To demonstrate their utility, we have used the results from recent viscous hydrodynamical calculations [38] to calibrate β and make an estimate of η/s . This is illus- trated in Fig. 4. The p T -integrated v n results from vis- cous hydrodynamical calculations for three separate η/s values, for 0.1% central Pb+Pb collisions are shown in Fig. 4(a). They indicate the expected linear dependence
FIG. 4: (a) Atlas data, from Ref. [35], ln ( v n /(cid:15) n ) vs. n from viscous hydrodynamical calculations for three values ofspecific shear viscosity as indicated. (b) ln ( v n /(cid:15) n ) vs. n forPb+Pb data. The p ⊥ -integrated v n results in (a) and (b) arefrom ATLAS 0.1% central Pb+Pb collisions at sNN = 2.76TeV; the curves are linear fits. (c) exponent vs. viscosity-to-entropy ratio 4 π/s for curves shown in (a) and (b). (b) shows the ln ( v n /(cid:15) n ), which according to the formulais the exponent. As a function of the inverse system’ssize 1 /R both elliptic and triangular flows show perfectlylinear behavior. Further issues – the n dependence aswell as linear dependences of the log ( v m /(cid:15) m ) on viscosityvalue – are also very well reproduced, see Fig.4. Note thatthis expression works all the way to rather peripheral AAcollisions with R ∼ f m and multiplicities comparableto those in the highest pA binds. It also seem to worktill the largest n so far measured.So, the acoustic damping provides correct systematicsof the harmonic strength. This increases our confidencethat – in spite of somewhat different geometry – the per-turbations observed are actually just a form of a soundwaves.Since we will be interested not only in large AA sys-tems but also in new – pA and pp – much smaller fireballs,one may use the systematics to compare it with the newdata. Or, using it, one can estimate how many flow har-monics can be observed in these cases. For central PbPbat LHC collisions with1 T R = O (1 /
10) (9)its product of η/s is O (10 − ). So one can immediately seefrom this expression why harmonics up to n = O (10) canbe observed. Proceeding to smaller systems by keeping asimilar initial temperature T i ∼ M eV ∼ / (0 . f m )but a smaller size R , results in a macro-to-micro param-eter that is no longer small, or 1 /T R ∼ . ,
1, respec-tively. For a usual liquid/gas, with η/s >
1, there wouldnot be any small parameter left and one would have toconclude that hydrodynamics is inapplicable for such asmall system. However, since the quark-gluon plasma isan exceptionally good liquid with a very small η/s , onecan still observe harmonics up to m = O ( √ ∼
3. Andindeed, v , v have been observed in the first round ofmeasurements (for latest data see Fig.17). D. The Green function: waves from a pointperturbation
The problem appears very complicated: events havemultiple shapes, described by multidimensional proba-bility distributions P ( (cid:15) , (cid:15) ... ). Except that it is not. Allthose shapes are however mostly a statistical noise. Thereason is as follows: rows of nucleons sitting at differentlocations in the transverse plane cannot possibly knowabout each other fluctuations at the collision moment, sothey must be statistically independent. An “elementaryexcitation” is just one delta-function in the transverseplane (in reality, of the size of a nucleon) , on top of asmooth average fireball. In other words, the first thingto do is to calculate the Green function of the linearizedhydro equations.A particular model of the initial state expressing lo-cality and statistical independence of “bumps” has been FIG. 5: The perturbation is shown by small blue circle atpoint O: its time evolution to points x and y is described bythe Green function of linearized hydrodynamics shown by twolines. Perturbed region – shown by grey circle – is inside thesound horizon. The sound wave effect is maximal at the in-tersection points of this area with the fireball boundary: ∆ φ angle is the value at which the peak in two-body correlationfunction is to be found. Shifting the location of the pertur-bation, from (a) to (b), result in a rather small shift in ∆ φ . formulated in [24]: the correlator of fluctuations is givenby the Poisson local expression < δn ( x ) δn ( y ) > = ¯ n ( x ) δ ( x − y ) (10)where ¯ n ( x ) is the average matter distribution. The im-mediate consequence of this model is that, for the centralcollisions on which we now for simplicity focus, (cid:15) m are thesame for all m < m max = O (10) (till the bump size getsresolved).In order to calculate perturbation at later time oneneeds to calculate the Green functions, from the originallocation O to observation points x and y as shown inFig.5(a). That has been first done in [36], analyticallyfor Gubser flow. One finds that the main contributionscome from two points in Fig.5, where the “sound circle”intersects the fireball boundary. In a single-body angulardistributions those two points correspond to two excessesof particles at the corresponding two directions. Let uscall the angle between them ∆ φ : at Fig.5(a) it is about120 o or 2 rad. This is because the freesout time timesspeed of sound R h = c s τ freezeout happens to be close tothe radius of the fireball.The correlation function calculated in [36], is shown inFig. 6(a). One its feature is a peak at zero δφ = 0: it isgenerated if both observed particles come from the sameintersection point. If two particles come from differentpoints, one finds two peaks, at ∆ φ = ± φ changes toward itsmaximal value, π radian, or 1/2 of the circle.)This calculation has been presented at the first dayof Annecy Quark Matter bef ore the experimental data.The ATLAS correlation function (for the “super-centralbin”, with the fraction of the total cross section 0-1%) isshown in Fig. 6(b). The agreement of the shape is notperfect – because a model is with conformal QGP anda bit different shape – but all predicted elements of itsshape are indeed observed.While there is no need to use Fourier harmonics – Iinsist that the correlation function itself teaches us morethan harmonics, separately studied – one can certainlydo so. Note that for ultra-central collisions we now dis-cuss the largest harmonics is v (the blue curve in Fig.6(b)) and not v (the green one). Since starting defor-mations (cid:15) n are basically the same for all n , the differencemust come from hydro, and it does: we just explained itabove, using notion of sound horizon. Alternative expla-nation can be done in terms of the freezeout phases ofthe harmonics φ nfreezout .A related arguments are quite famous in the theoryof Big Bang perturbations. All harmonics get excited atthe same time by Big Bang – hydro velocities at timezero are assumed to be zero for all harmonics – and arefrozen at the same time. The acquired phases dependon n – harmonics with a larger n oscillate more rapidly.Binary correlator is proportional to cos ( φ nfreezout ) andharmonics with the optimal phases close to π/ π/ n = 3 , v ) one can still seeclear deviation from smooth damping curve ∼ exp ( − n ∗ const ). Note that the v is higher than in all hydro out-put shown, while v is lower. Let me conclude this discus-sion with a note that while certain oscillatory deviationsfrom “acoustic systematics” are there in the experimen-tal data, their origin and even qualitative reproductionby the hydro models remains uncertain.Let me conclude this section with brief discussion ofthe following issue. The “flow harmonics”, solutions oflinearized equations on top of the average smooth hydrosolution should possess a complete spectrum of pertur-bations, whose time evolutions are different from eachother. The perturbations of course depend not only onthe angle φ , as e imφ , but of course on other coordinates r, η as well.For Gubser setting (see appendix) one can use comov-ing coordinates ρ, θ, φ, η and dependence on all coordi-nates separate nicely, with known analytic expressions forharmonics and their time evolution. The flow in trans-verse plane is conveniently combined into a single angularmomentum harmonics Y l ( θ, φ ), combining azimuthal an- (cid:45) ! " " -4 -2 0 2 4 ) " , ! " C ( < 3 GeV bT ,p aT ATLAS
Preliminary a) -1 b µ Ldt = 8 $ !" ) ! " C ( |<5 bT , p aT ATLAS
Preliminary b) n , n v -5 -4 -3 -2 c) ATLAS
Preliminary |<5 bT , p aT -1 b µ Ldt = 8 $ n v -2 -1 d) ATLAS
Preliminary |<5 bT , p aT -1 b µ Ldt = 8 $ n -6 % n -3 % n , n v < 3 GeV bT ,p aT v v v v v v ATLAS
Preliminary -1 b µ Ldt = 8 $ e) n v v v v v v < 3 GeV bT ,p aT ATLAS
Preliminary -1 b µ Ldt = 8 $ f) Figure 2: The steps involved in the extraction of the v n for 2-3 GeV fixed- p T correlation: a) the two-dimensional correlation function (shown for | ∆ η | < .
75 to reduce the fluctuations near the edge), b)the one-dimensional ∆ φ correlation function for 2 < | ∆ η | < ffi cient v n , n vs n ,and d) v n vs n . The bottom two panels show the full dependence of v n , n and v n on ∆ η . The v is notshown since it breaks the factorization from v n , n to v n of Eq. 13. The shaded bands in c)-f) indicate thesystematic uncertainties. The range 2 < p aT , p bT < FIG. 6: (a) Calculated two-pion distribution as a function ofazimuthal angle difference ∆ φ , for viscosity-to-entropy ratios η/s = 0 . gle and r . Another simple eigenfunctions are plane waveswith some momenta k in the rapidity direction η .Can one define similar set of independent harmonicsfor a generic non-Gubser setting? And, even more im-portantly, can those be observed? A nice step in thisdirection has been recently made by Mazeliauskas andTeaney [50]. Using experimental data (and of course hy-dro calculations) they define such “subheading harmon-ics”. Instead of describing how they do it, here is just onepicture from that work, for triangular flow, indicating adifference between the leading and subleading flows: un-like the former, the latter gets a sign change along theradial direction.0 Planck collaboration:CMBpowerspectra&likelihood D [ µ K ] lensed CMB30 to 35370100143143x217217353x143 Figure11.
Planck power spectra and data selection. The coloured tick marks indicate the ` -range of the four cross-spectra includedin CamSpec (andcomputedwiththesamemask,seeTable4).Althoughnotused,the70GHzand143x353GHzspectrademonstratethe consistency of the data. The dashed line indicates the best-fit
Planck spectrum.
Table 4.
Overview of of cross-spectra, multipole ranges andmasks used in the
Planck high- ` likelihood. Reduced s withrespect to the best-fit minimal ⇤ CDM model are given in thefourth column, and the corresponding probability-to-exceed inthe fifth column.
Spectrum Multipolerange Mask ⇤ CDM /⌫ dof PTE100 ⇥
100 . . . . . . 50–1200 CL49 1.01 0.40143 ⇥
143 . . . . . . 50–2000 CL31 0.96 0.84143 ⇥
217 . . . . . . 500–2500 CL31 1.04 0.10217 ⇥
217 . . . . . . 500–2500 CL31 0.96 0.90Combined . . . . . . 50–2500 CL31 /
49 1.04 0.08 quencycombinationareshowninFig.11,andcomparedtospec-tra derived from the 70 GHz and 353 GHz
Planck maps.We use the likelihood to estimate six ⇤ CDM cosmolo-gical parameters, together with a set of 14 nuisance paramet-ers (11 foreground parameters, two relative calibration para- meters, and one beam error parameter , described in Sect. 3.Tables5 and 6 summarize these parameters and the associatedpriors . Apart from the beam eigenmode amplitude and calibra-tion factors, we adopt uniform priors. To map out the corres-ponding posterior distributions we use the methods describedin Planck Collaboration XVI (2013), and the resulting marginaldistributions are shown in Fig. 12. Note that on the parameters A tSZ , A kSZ and A CIB143 weareusinglargerpriorrangesascomparedto Planck Collaboration XVI (2013).Figure12showsthestrongconstrainingpowerofthe
Planck data, but also highlights some of the deficiencies of a ‘
Planck -alone’ analysis. The thermal SZ amplitude provides a good ex-ample; the distribution is broad, and the ‘best fit’ value is ex- The calibration parameters c and c are relative to the 143 ⇥ ⇥ We use the approximation ✓ MC to the acoustic scale ✓ ? (the ra-tio of the comoving size of the horizon at the time of recombination, r S , to the angular diameter distance at which we observe the fluctu-ations, D A ) which was introduced by Hu&Sugiyama (1996). ✓ MC iscommonly used, e.g., in CosmoMC , to speed up calculations; see alsoKosowskyetal.(2002)forfurtherdetails.12
FIG. 7: Power spectrum of cosmic microwave backgroundradiation measured by Planck collaboration [93].FIG. 8: (a) The lines are hydro calculations of the correlationfunction harmonics, v m , based on a Green function from apoint source [36] for four values of viscosity 4 πη/s =0,1,1.68,2(top to bottom at the right). The closed circles are the At-las data for the ultra-central bin. (b) v n { } plotted vs n .Blue closed circles are calculation of via viscous even-by-eventhydrodynamics [25], “IP Glasma+Music”, with η/s = 0 . E. Non-linear effects for harmonics
We argued above that one can view perturbations assuperposition of uncorrelated local (delta-function-like)sources. We then further argued that only those closeto the fireball surface are visible. How many of themare there? For central collisions the circumference of thefireball 2 πR A ≈ f m . The correlation length is perhapsthe typical impact parameter in NN collisions, which atLHC energies is b ∼ (cid:112) σ/π ∼ . f m . Their ratio suggestexistence of about N sources ≈ πR A b ≈
25 (11)cells on the fireball surface which fluctuate indepen-dently, up or down from the mean value. Glauber initialconditions indeed have about a dozen sound-producing“bumps”.The absolute scale of modulation in the experimentalcorrelators are of the order of a percent in magnitude,see e.g. Fig.6(b). Of course, it comes from incoherentsum over the single bumps, so one should divide by theirnumber and get the individual correlation from a bumpto be of O (10 − ) in magnitude. It however is quadraticin the wave amplitude: taking the square root of it weget sound amplitudes to be of the order of 1 /
30 or so.Summarizing: the picture looks like that from abouta dozen stones thrown into the pond. While the soundcircle visibly interfere, they do not really interact witheach other, since the amplitude is too small.Yet the conclusion that since individual sounds areweak, the linear theory is completely correct, is prema-ture. Even when the amplitude of the sound is small,they can produce large effects at the large p t end. In-deed at p t ∼ GeV the elliptic flow reach values whichmakes angular distribution 100% asymmetric. This hap-pens because a weak sound may sit “on the shoulders ofthe giants”, the explosions themselves.Similarly, the non-linear effects of flows at comparablelarge p t are non-negligible and were in fact observed. Forexample v is getting a contribution proportional to (cid:15) , v from (cid:15) , etc. Detailed studies of such effects one can1find in [31], including an interesting case of v comingfrom (cid:15) ∗ (cid:15) .These nonlinear effects come not from nonlinear termsin the hydro equation, but from an expansion of theCooper-Fry exponent exp ( p µ u µ /T ) in powers of the per-turbation. Obviously they become more important athigh p t . Furthermore, one can see that linear termsshould be linear in p t , the non-linear we mentioned shouldbe quadratic ∼ p t , etc, see more in [31]. F. Event-by-event v n fluctuations/correlations At the beginning of this section we had already empha-sized that the main source of the v n fluctuations is thatof the original perturbations (cid:15) n themselves, see e.g. Fig.2(b). Now we return to the question: Why does the ra-tio v n /(cid:15) n , evaluated by hydro, has such a small spread? .While the practitioners of the event-by-event hydrody-namics use huge variety of the initial configurations, itturns out that just one number – (cid:15) n – insufficient to char-acterize them. Even adopting the simplest model advo-cated above – that the perturbations come from point-like sources – it is nontrivial that their event-by-eventfluctuations of strength and locations do not create anyspread in the ratio v n /(cid:15) n . (Or, using my drum analogy,why does it sound the same, if hit in different locations?)Trying to understand this, let us come back to Fig.5.The source located at the fireball edge, fig.(a), leads tocorrelations which are at ∆ φ ≈ rad , as we emphasizedabove. Projected on harmonics, it will excite m = 3mostly, as 2 rad is about 1/3 of 2 π . When the sourcemoves inward, fig.(b), the perturbations move to theopposite points of the fireball ∆ φ = π , and the lead-ing excitation becomes elliptic m = 2. Estimates showthat in the latter case the correlation weakens. The ob-served shape of the correlator, for ultra-central collisionsdoes have a minimum at ∆ φ = π , supports that. Asthe source moves further toward the fireball center (notshown in Fig.5), the whole fireball becomes included in-side the sound horizon. The angular correlation dilutes toall angles and its contributions to harmonics gets negligi-ble. In summary: the sources located near the boundaryof the fireball are mostly responsible for the harmonicswe see. All of them thus generate the same angular shapeof the correlator. What remains is the overall strength,a single parameter captured in the (cid:15) n magnitude.Harmonic correlations is a rapidly developing field.Those can be divided to correlations sensitive to rela-tive phases of the harmonics and those which are not.An example of the later is SC ( m, n ) = < v m v n > − < v m >< v n > (12)on which Alice provided good data at QM 2015 for SC (4 ,
2) and SC (3 , SC (4 , > SC (3 , <
0, which is qualitatively re-produced in hydro. Now, in order to explain these data,one needs first to have the initial state model which is the sounds plot log(M)near ! Tc R fireball . GeV GeV freezeoutsQGP
GLASMA ⌧ B l + l ⌧ i ⌧ f l o g ( p r o p e r t i m e ) k FIG. 10: The log-log plane proper time τ – sound momentum k . The solid curve indicates the amplitude damping by afactor e : only small- k sounds thus survive till freeze out. Theshadowed region on the right corresponds to that in whichsonomagnetoluminiscence effect may produce extra dileptons. good enough to predict the relevant (cid:15) n and their respec-tive correlations. Suppose we do so using the Bhalerao-Ollitrault model [24] and the relation (10): the resultsdepend on integrals like (cid:82) d rr P ¯ n ( r ) with large powers P = 6 ,
8. What this tells us is that the correlation isstrongly localized at the very surface of the system, andthus subject to significant uncertainties.
G. The map of the sounds
We will argue below that the number of harmon-ics needed to describe the initial state is rather large,counted in hundreds. In Fig.10 we show a map of those,in terms of momentum (rather than angular momentum).The curved line – corresponding to “acoustic systemat-ics” discussed above – show their lifetime. This curvecrosses the freezeout time: smaller k waves can be ob-served at freezeout. Larger k cannot: they are weakeneddue to viscous damping. (A suggestion to detect thosevia the MSL process will be discussed in section IX B.)Fluctuation-dissipation theorem tells us that while theinitial perturbations are damped, new ones should beproduced instead. There should be some noise producedat the hydro stage. For a paper exploring hydro withnoise see Young et al [46]. Unfortunately, the authorshave not yet identified observables which sensitive onlyto the late − time perturbations. I suggest those shouldbe azimuth+rapidity correlations, exploring the fact thatearly time perturbations must be rapidity-independentThis idea came from the paper [48] in which a veryspecific late-time fluctuations were considered, namelysounds from collapsing QGP clusters inside the hadronicphase. Those collapse must happen, since this phase isunstable in the bulk, once T < T c : its physics is simi-2lar to the celebrated Rayleigh bubble collapse. Anotherdiscussed – and as yet unobserved – are sounds fromjets depositing its energy into the ambient matter andthen propagated via hydro perturbations, known as Machcone. Extensive recent discussion of how one can observethose can be found in Ref. [49]. H. Sounds in the loops
The hydrodynamical longitudinal pressure waves – thesounds – are the best quasiparticles we have. They areGoldstone modes, related with spontaneous breaking ofthe translation invariance by matter, and thus their in-teraction fall into certain patter familiar from physics ofthe pions. For large wavelengths they have long lifetime,exceeding the freezeout time. Therefore, in both the Lit-tle and the Big Bangs, one can observe “frozen” tracesof the initial perturbations, provide one looks at largeenough wavelengths.So, the set of sounds have their own lifetime scales,and one may wander about an ensemble of sounds, theirinteractions etc. One way to introduce those is to add tohydro equations a Langevin-type noise term, with someGaussian distribution, and then rewrite the theory withcertain artificial fields into a QFT-like form. Progressinto this direction has been recently summarized by Kov-tun in a nice review [96]. Discussion of formal issues can-not be made in this review, however, and thus I illustratethe physics involved by one example, also due to Kovtunand collaborators [97]. Let me remind that matter vis-cosity can be defined via certain limit of the stress tensorcorrelator, known as the Kubo formula. Kovtun et alcalculated a “loop corrections” to this correlator inducedby the equilibrium sounds. Technically the calculation isdone as follows: in the < T µν T µ (cid:48) ν (cid:48) > correlator one sub-stitute hydrodynamical expression for stress tensor con-taining sound perturbation velocities and make it into aloop diagram with the “sound propagators”∆ mn = (cid:90) d xe − ip α x α < u m ( x ) u n (0) > (13)for two pairs of the velocities. (We use latin indices in-dicating that they are only space-like here. For shearviscosity those used are m = x, n = y ). Skipping thederivation I jump to the answer obtained from this calcu-lation, which can be put into the form of loop correctionto the viscosity δη loop = 17120 π p max T ( (cid:15) + p ) η (14)which is UV divergent and thus includes p max , the largestmomentum for sound which still makes sense. What isimportant here is that the zeroth-order viscosity entersinto the denominator. This should not be surprising: avery good liquid with small η support very long-livedsounds, which can transfer momentum far, which means t ( T c-1 )00.0250.050.075 U Interm., (cid:100) =0.4
Weak, (cid:100) =0.1
Weak, (cid:100) =0.2
Weak, (cid:100) =0.4
Weak, (cid:100) =0.6 U f U (cid:113) t ( T c-1 )020040060080010001200 (cid:108) g w R * - ( (cid:161) + p ) - U f- ( G T c - ) Interm., (cid:100) =0.4
Weak, (cid:100) =0.1
Weak, (cid:100) =0.2
Weak, (cid:100) =0.4
Weak, (cid:100) =0.6
Sources+of+gravita'onal+waves+ • Look+at++ – rms+fluid+velocity+ – equivalent+field+quan'ty+ • Gravita'onal+wave+energy+density+rises+linearly+a`er+transi'on+ • Source:+fluid+
Gravita'onal+waves+...+Mark+Hindmarsh+ ¯ U f ¯ U = p h ( r ) i Evolu'on+of+power+spectra+ • Velocity+power+spectrum:+ – Peaked+at+wavenumber+corresponding+to+mean+bubble+separa'on++ – Mostly+compressional+(grey)+ – Small+rota'onal+component+(black)+ • GGwave+power+spectrum+ – Peaked+at+wavenumber+corresponding+to+mean+bubble+separa'on++
Gravita'onal+waves+...+Mark+Hindmarsh+ k ( T c )1e-111e-101e-091e-081e-071e-061e-050.00010.001 d V / d l n k k ( T c )1e-091e-081e-071e-061e-050.00010.0010.010.1110 d (cid:108) G W / d l n k ( G T c ) k -1 FIG. 11: (From [76]) (a) The density of generated gravitywaves ρ GW versus time t , in units of inverse T c . The densitycontinue to grow linearly even at the longest times, well afterthe phase transition itself is over. (b) The power spectrum ofthe velocity squared versus the wave momentum k . The greyupper curves are for sounds, from down up as time progresses.The black curves downwards are for rotational excitations. they produce large contribution to the effective viscos-ity! (Like ballistically moving phonons in liquid heliumor neutrinos in supernova, the most penetrating modesalways dominate the transport.)Completing this section we would like to remind thereader about existence of other hydro modes, the rota-tional ones. It is those ones which are e.g. responsiblefor the atmospheric turbulence. Unlike sounds, which arealways damped, rotational modes can get excited undercertain conditions. Very little theory work has been donein this direction so far. Floerchinger and Wiedemann [94]analyzed rotational modes on top of the Bjorken flow,and formulated conditions for one mode to get unstable.Csernai et al [95] had also found an unstable hydro mode,for non-central collisions with rotation. While this effectdoes not develop into a full-scale turbulence in heavy ioncollisions, due to limited time it exists, it does contributeto an overall fireball rotation and thus it can perhaps beobserved.3 I. Cosmological gravitational waves from sounds ofthe QCD phase transition
We think that our Universe has been “boiling” at itsearly stages at least three times: (i) at the initial equili-bration, when entropy was produced, at (ii) electroweakand (iii) QCD phase transitions. On general grounds,these should have produced certain out-of-equilibrium ef-fects. It remains a great challenge to us, to observe theirconsequences experimentally, or at least evaluate theirmagnitude theoretically. The first question is then, whatobservable may have a chance to see through the subse-quent evolution, back to the early moments of the BigBang.From the onset of the QGP physics in heavy ion colli-sions a specially important role has been attributed to the“penetrating probes”, production of photons/dileptons[84]. In this section (only) we will jump from our cus-tomary Little Bang to the discussion of the Big Bang. Soit is quite logical to start with a question whether it alsopossesses some kind of a “penetrating probe”. The an-swer is quite clear: while the electromagnetic and weakinteractions are in this case not weak enough, the gravi-tational one is. Thus the only “penetrating probe” of theBig Bang are the gravity waves (GW).30 years ago Witten [75] had discussed the cosmo-logical QCD phase transition, assuming it to be of thefirst order: he pointed out bubble production and coales-cence, producing inhomogeneities in energy distributionand mentioned production of the gravity waves. Jumpingmany years to recent time, we mention that Hindmarsh etal [76] recently found the hydrodynamical sound waves tobe the dominant source of the GW, while doing numericalsimulations of (variant of) the electroweak (EW) phasetransition in the first order transition setting. Since thiswork had triggered our interest to the subject, we startillustrating its main findings shown in Fig.11. The upperfigure shows time evolution of the GW energy, in differentsimulations. In all cases the evolution is simply linear:it means that the GW generation rate remains constant long after the phase transition itself is over. Their furtherstudies had shown that GW originates from the long-wavelength sound waves, not from the rotational modes,which are several orders of magnitude down, see lowerFig.11.The sound velocity squared is proportional to the en-ergy of the sound waves, which one can also view as ∼ ω k n k k , where ω k = c s k is the sound frequency and n k is the density of the momentum distribution, to bediscussed below. So, a flat spectrum seen in lower Fig.11corresponds to n k ∼ /k . The spectra go 2 orders ofmagnitude from the UV scale T down, and evolutiontime about 3 orders of magnitude. The question is whathappens near the IR end of the dynamical range, manyorders of magnitude away of what was simulated in thatwork.Kalaydzyan and myself [77] discussed sound-based GWproduction and argued that generation of the cosmolog- ical GW can be divided into four distinct stages, eachwith its own physics and scales. We will list them start-ing from the UV end of the spectrum k ∼ T and endingat the IR end of the spectrum k ∼ /t life cutoff by theUniverse lifetime at the era :(i) the production of the sounds(ii) the inverse cascade” of the acoustic turbulence, mov-ing the sound from UV to IR(iii) the final transition from sounds to GW.The stage (i) remains highly nontrivial, associated withthe dynamical details of the QCD (or electroweak EW)phase transitions. The stage (ii), on the other hand, isin fact amenable to perturbative studies of the acous-tic cascade, which is governed by Boltzmann equation.It has been already rather well studied in literature onturbulence, in which power attractor solutions has beenidentified. Application of this theory allows to see howsmall-amplitude sounds can be amplified, as one goes tosmaller k . The stage (iii) can be treated via standardloop diagram for sound+sound → GW transition,Before we come to it, let us briefly remind the numbersrelated to the QCD and electroweak transition. Step oneis to evaluate redshifts of the transitions, which can bedone by comparing the transition temperatures T QCD =170 MeV and T EW ∼
100 GeV with the temperature ofthe cosmic microwave background T CMB = 2 .
73 K. Thisleads to z QCD = 7 . × , z EW ∼ × . (15)At the radiation-dominated era – to which both QCDand electroweak ones belong – the solution to Friedmannequation leads to well known relation between the timeand the temperature [163] t = (cid:18) π N DOF ( t ) (cid:19) / M P T (16)where M P is the Planck mass and N DOF ( t ) is the ef-fective number of bosonic degrees of freedom (see de-tails in, e.g., PDG, Big Bang cosmology). Plugging inthe corresponding T one finds the the time of the QCDphase transition to be t QCD = 4 × − s , and electroweak t EW ∼ − s . Multiplying those times by the respec-tive redshift factors, one finds that the t QCD scale to-day corresponds to about 3 × s ∼ year, and the elec-troweak to 5 × s ∼ day.GW from the electroweak era are expected to besearched for by future GW observatories in space, suchas eLISA. The observational tools for GW at the periodscale of years are based on the long-term monitoring ofthe millisecond pulsar phases, with subsequent correla-tion between all of them. The basic idea is that whenGW is falling on Earth and, say, stretches distances ina certain direction, then in the orthogonal direction oneexpects distances to be reduced. The binary correlationfunction for the pulsar time delay is an expected func-tion of the angle θ between them on the sky. There areexisting collaborations – North American Nanohertz Ob-servatory for Gravitational Radiation, European Pulsar4Timing Array (EPTA), and Parkes Pulsar Timing Ar-ray – which actively pursue both the search for new mil-lisecond pulsars and collecting the timing data for someknown pulsars. It is believed that about 200 known mil-lisecond pulsars constitute only about 1 percent of thetotal number of them in our Galaxy. The current boundon the GW fraction of the energy density of the Universeis approximatelyΩ GW ( f ∼ − Hz) h < − . (17)Rapid progress in the field, including better pulsar timingand formation of a global collaborations of observers, isexpected to improve the sensitivity of the method , per-haps making it possible in a few year time scale to detectGW radiation, either from the QCD Big Bang GW ra-diation we discuss, or that from colliding supermassiveblack holes.The temperature T provides the micro (UV) scaleof the problem: here the phase transition provides thesound source. The cosmological horizon is the IR cut-off on the gravitational radiation wavelength: here twosounds generate the GW. In between UV and IR scalesthere is the “dynamical range”, of about 18 orders ofmagnitude! The challenge is to understand if and howthe inverse acoustic cascade can be developed there, andwhat n k dependence is generated. It turns out that theanswer crucially depend on the sign of the third deriva-tive of the sound dispersion curve ω = c s k + Ak + ... (18)which remains unknown, both for QGP and electroweakplasma.If A > direct – that is toward large k orUV – cascade, not the one we are interested in.However when the dispersive correction coefficient A < ↔ ↔ inverse cascade, with a particle flow directed to IR. In this case ofthe weak turbulence, the index of the density momentumdistribution n k ∼ k − s is known to be s weak = 10 / k leads to violation of weak turbulence appli-cability condition and the regime is known as “strong tur-bulence”. We follow the philosophy of Berges et al renor-malization of similar cascades in the case of relativis-tic scalar with quartic interaction. Unfortunately soundwaves have triple vertex are they are Goldstone particles,so the problem is much more involved. The scatteringis dominated by t -channel diagrams with small denomi-nators, producing small angle scattering with large crosssections, like it is the case for gluons. We do not yet have a complete solution of this problem, only an estimate ofthe renormalized index which we think is increasing to s strong ∼
5. Given huge – 18 decades for QCD – dy-namical range of the problem at hand this will imply asignificant increase of n k at IR.Another key result of our paper [77] is the calculationof the transition rate of the sound to gravity waves. Wefound it to be rather simple process, an on shell collisionof the two sound waves. The calculation of the rate [77]is straightforward, following from the evaluation of the“sound loop” diagram (already discussed in the previoussection, but in different kinematics). Since two soundscollide, it depends quadratically on n k : so the GW pro-duction can be hugely amplified by the inverse acousticcascade. IV. THE PRE-EQUILIBRIUM STATE, GLOBALOBSERVABLES AND FLUCTUATIONSA. Perturbative versus non-perturbativeparadigms
The requirements to any theory of the early stage canbe formulated as follows: It should be able to(i) specify certain the wave function of the colliding par-ticle, in a wide rapidity range;(ii) explain what happens at the collisions and just afterit;(iii) explain how it evolves into the final observedhadronic state.It is perhaps fair to say, that approaches based on theweak coupling (pQCD) has been able to explain (i) and(ii) but not (iii). Strong coupling ones (AdS/CFT) cando (iii) but not (i,ii).More specifically, perturbative (pQCD) regime is nat-ural for hard processes, for which the QCD running cou-pling is weak. Already in 1970’s the pQCD developed factorization framework , which divided production am-plitudes into “past”,“during” and “after” parts. The“past” and “after” parts are treated empirically, by struc-ture (or distribution) and fragmentation functions. The“during”, near-instantaneous, part is described by ex-plicit partonic process under consideration. (These threestages is the simplest example of three goals formulatedabove.) The strength of this approach is based on theseparation of hard and soft scales, by some normaliza-tion scale µ , on which the final answer should not de-pend. Dependence of PDFs and fragmentation functionson µ is described by renormalization group tool, the socalled DGLAP evolution: it let us tune them to the par-ticular kinematics at hand. It works very well for veryhard processes: high energy physicists at LHC looking atthe electroweak scale and beyond, Q >
GeV , do notneed to know anything else. People who want to studymini-jets have to worry to “higher twist” corrections toGLAP, not under control yet.Yet pQCD approach has serious weaknesses as well.5The PDFs describe only the average nucleon (or nu-cleus). As soon as a particle is touched – e.g. theimpact parameter (multiplicity bin) is selected – factor-ization theorems are no longer applicable. The absenceof good practical models describing partonic state withfluctuations remains a problem: e.g. for understand-ing pp collisions with multiplicity several times the av-erage one. As we will discuss below in detail, pQCD canhardly be used for assessing the transverse plane distri-butions/correlations of partons.For “baseline” processes – minimally biased pp, pA collisions, with rather low multiplicity – pQCD pro-cesses can be supplemented by string fragmentation,leading to rather successful “event generators”, descen-dent of the so called Lund model. In the approximationof independent string fragmentation their relative posi-tions/correlations/interactions are unimportant. How-ever, experiments which trigger higher multiplicity binsand look for subtle correlations have found phenomenaclearly beyond the reach of such simple models: we willdiscuss those in detail.At the opposite case of high multiplicity collisions –central pA, AA – the theory of the initial state is in firstapproximation classical, and general initial conditions forhydrodynamics is in the zeroth approximation given justby nuclear shape and N N cross section. The main pa-rameter hydro needs is the total entropy generated: thisis taken from parton density times some empirical coef-ficient, entropy/parton, not yet explained. In the firstapproximation – including fluctuations in the positionsof nucleons in versions of the Glauber or eikonal mod-els – one also finds simple and reasonable predictions for (cid:15) n and their correlations/fluctuations. The next approx-imations, involving fluctuating nucleons in terms of theirparton substructure, have been proposed, but their rele-vance is disputed.Partonic description of the initial state of the colli-sion at asymptotically high parton density evolved intothe so called Color-Glass-Condensate (CGC)- GLASMAparadigm originated from McLerran-Venugopalan model[78]. Let me briefly remind the main points. Since thenumber of colored objects is large, charge fluctuationswill eventually also become large producing strong gaugefields known as CGC. If gluonic fields become so strongthat the occupation numbers reach O (1 /α s ), the deriva-tive and the commutator terms are comparable and onecan use classical Yang-Mills nonlinear equations for itsdescription. GLASMA is a state made of such randomclassical fields, starting from CGC at the collision timeand then evolving as the system expands, till the oc-cupation numbers reduce to O (1). Self-consistency ofthe model is provided by the fact that 2-d parton den-sity define the characteristic patron saturation momen-tum n ∼ Q s , which is treated as a scale in the evolutionequations. At early time the charges at each “glasmacell”, of area ∼ /Q s , start separate longitudinally, pro-ducing longitudinal electric and magnetic fields. Cellsare statistically independent and fluctuate with their own Poisson-like distributions. The explicit modelling of re-sulting field, from cells in the transverse plane, is nowknown as an impact − parameter (IP) glasma models.High-multiplicity initial state then evolve into sQGP:we know that because it must be complemented by hy-dro evolution, to describe RHIC/LHC observations. Wewill go over observables below in more detail, let me illus-trate it here by one particular observable, the elliptic flow v . Suppose there is no sQGP stage: patrons – gluonsand quarks – simply “get real” after collision, more orless like the Weitzsacker-Williams photons do in QED,fragment into mini-jets and fly to the detector. Hardpartons at large momentum scale Q s of GLASMA be-longing to individual cells cannot possibly know aboutother cells and nuclear geometry: those would be mainlyproduced isotropically in the transverse plane. If theyre-interact later, the probability of that is higher wherethere is more matter – contributing negative correctionto v . Much softer patrons, with momenta Q ∼ /R ,will know about the “overlap almond” shape of the initialstate: their distribution will be anisotropic, perhaps evenwith v of the order of several percents, as observed formomentum-integrated data. Thus the prediction wouldbe of v decreasing with p t , to negative values. Needlessto say, hydro-based theory and experiment had shown v ( p t ) to be instead growing up to p ⊥ ∼ GeV , produc-ing anisotropies as large as O (1).Strong-coupling models of the initial stage and equili-brated matter fall into two categories. One is based onclassical strongly-coupled plasmas developed in [28] andbased on the notion that the ratioΓ = V interaction T > −
10 one deals with strongly correlated liquids .Screening in this regime was studied in [29], and viscosityand diffusion constant in [8] in a version with electric andmagnetic charges. Apart of jet quenching applications,it has not yet been used for heavy ion physics, and thuswe will not discuss it here.The second – much wider known – strong couplingframework is based on holography and AdS/CFT corre-spondence: we will also discuss recent progress in detail.It describes rapid equilibration by black hole formationin the dual theory space: we will discuss it in significantdetail below.
B. Centrality, E ⊥ and fluctuations Before diving into theory developments, let me brieflyremind some basic facts about the global observables andtheir fluctuations. One of the first practical question forAA collisions is determination of centrality classes, re-lated to observables like the number of participant nu-6
M. J. Tannenbaum 10 ! Identical shape of distributions indicates a nuclear-geometrical effect ! The geometry is the number of constituent quark participants/nucleon participant ! Eremin&Voloshin, PRC 67, 064905(2003) ; De&Bhattacharyya PRC 71; Nouicer EPJC 49, 281 (2007) ! New RHIC data for Au+Au at √ s NN =0.0077 TeV show the same evolution with centrality ! Quark Matter 5/19/2014 ! Remember, constituent quarks also gave universal scaling for v /n q vs KE T /n q ! qp N0 200 400 600 800 1000 [ G e V ] η / d T d E
200 GeV Au+Au130 GeV Au+Au62.4 GeV Au+Au dE T /d η is “strictly proportional” to Nqp ! M. J. Tannenbaum 15 ! A fit of dE T /d η =a × N qp +b at each √ s NN gives b=0 in all 3 cases which establishes the linearity of dE T /d η with N qp ! Quark Matter 5/19/2014 ! FIG. 12: Distributions over participant nucleons (a) andparticipant quarks (b), from [15]. cleons N p , correlated to total multiplicity N or trans-verse energy E ⊥ . The N p is defined by forward-backwardcalorimeters, while the others by the central detectors.Correlation plots between those and precise cuts defin-ing the centrality classes are the basic technical issues ofthe field.Historically, the ratio of the E ⊥ rapidity distributionsfor AA and pp collisions were fitted by a parameterization dE AA ⊥ dη / dE pp ⊥ dη = (1 − x ) N p xN coll (21)with a parameter x interpreted as an admixture of the“binary collisions” N coll to the main “soft” term, pro-portional to the number of participants. It was a nicefit: yet we now know that a “hard” components in par-ticle spectra - is 3-4 orders of magnitude down from the“soft” exponent (thermal-hydro contribution). So, theinterpretation of x becomes questionable.One possibility can be that “hard” interpretation justmentioned remains correct at early time, yet with sub-sequent equilibration and disappearance from spectra.Since hydro preserves entropy, one may argue that it willstill show up in the total multiplicity. It is however moredifficult to imagine preservation of E ⊥ at the hydro stage.But this is not the end of the story. The multiplicityand E ⊥ distributions were measured and connected to the geometric distribution over the r.h.s. of (21). Thesestudies bring along a notion of some intermediate “clus-ters” (or ancestors as ALICE papers call it) which gener-ate the secondaries in independent random processes oftheir decay.Tannenbaum [15] provided another, and very simple,interpretation to these distributions. A nucleon is repre-sented by 3 constituent quarks which interact separately– the additive quark model of 1960’s. Defining the num-ber of “quark participants” N qp he showed that – withina 1% accuracy (!) – it is proportional to the r.h.s. of(21). Thus, the E ⊥ is perfectly linear in N qp , see fig29(b). If so, each participant quark is connected by theQCD string to the other one, and those strings are the“clusters” or ancestors for final secondaries. (We will re-turn to “wounded quarks” concept in the discussion ofthe Pomeron in section VIII C.)On the other hand, the additive quark model does notagree with pQCD paradigm, which views color dipoles– not charges – as having separate cross sections (orcouplings to the Pomeron). The CGC/GLASMA the-ory adds another suggestion: it is its GLASMA cellsthat are expected to be statistically independent “clus-ters”, producing entropy/secondaries. (The McLerran-Venugopalan ”flux tubes” differ from the QCD strings:unlike those they exist in dense deconfined phase, are notquantized and they do not have universal tension).A view that the CGC-glasma picture is a high den-sity regime, while simpler Lund-type models with QCDstrings (and perhaps constituent quarks) are in low den-sity regime must be correct. The problem is we do notsee transition which will help us to tell at which densitywe need to switch between those two pictures.Let us seek further guidance from phenomenology.Note that the number of “clusters” N which fluctuateindependently define the width of observed distributions,being O ( N − / ). So, we take these three models: (i) theusual Glauber in which N is the number of participantnucleons N p ; (ii) its Tannenbaum’s modification with thenumber of participant constituent quarks N pq ; (iii) andthe CGC-glasma, and calculate the fluctuations.In the last case N GLASMA ∼ ( πR ) Q s (22)For central AA the area is geometrical Area=100 f m and N p ∼ , N pq ∼ , N GLASMA ∼ (23)For central pA the area is given by the NN cross section σ ∼ mb = 10 f m . So one gets very different numberof “clusters” N p ∼ , N pq ∼ , N GLASMA ∼ (24)Therefore these models predict vastly different fluctua-tions.There are two ways to measure fluctuations. The firstis based on multiplicity fluctuations . Going to the tail of7it, beyond the most central collisions, in AA,pA and pp we find some tail usually fitted by the negative binomialor similar distribution with two parameters, or convolu-tion of two random processes with different parameters.Its second moment should tell us how many “progenitors”(clusters, clans, ancestors) the system goes through.The second one is to look at angular deformations (cid:15) n . If created by statistically independent small-size ob-jects, they are all very similar and again parametrically O ( N − / ) [164]. We return to them in section VI C.For E t distributions, one finds that participant quarkmodel describes AuAu and dAu data extremely well,while in pp it clearly underpredicts the tail of the dis-tribution. Even 6 participant quarks – the maximal ofthe model – is not enough. Recalling our density esti-mates above, one may think that the highest multiplic-ity pp is the first case when “soft” models become in-sufficient. The models which have pQCD gluons in thewave function are doing better on the “tails”. Popu-lar quantitative model with pQCD component is Pythia(pQCD+strings): the version used by CMS describes themultiplicity tail of pp reasonably well, while it underpre-dicts a bit the tail in pA . (see e.g. CMS pages withpublic info). C. Anisotropy of the excited matter and theboundaries of hydrodynamics
Partonic initial state has small p t and large longitudi-nal momenta: so initial out-of-equilibrium stage of thecollision is highly anisotropic. However, after collisions,patrons are naturally separated in time according to dif-ferent rapidities, and create “floating matter” in whichcells have a spread of longitudinal momenta smaller than the transverse one, reversing the sign of momentumanisotropy. At later hydrodynamical stages the viscosityeffects produce local anisotropy of the particle distribu-tions, which is however small due to small viscosity ofthe fluid.What happens in between is still a matter of debates.Weak coupling approaching –partonic cascades – predictanisotropy to be rising to quite large values, while thestrong coupling (holographic) approaches lead to rapidconvergence to small values, consistent with hydrody-namics. (For more detailed discussion see e.g. [51]).The issue of anisotropy has two practical aspects. Theexperimental one – to which we return in section IV C– is a question how one can experimentally monitor theanisotropy of matter, at various stages of the evolution.The theoretical question is whether one can extend thehydrodynamical description for strongly anisotropic mat-ter. Recently there were significant development alongthe line of the so called anisotropic hydrodynamics , oraHydro. The idea [51, 52] is to introduce the asymmetryparameter into the particle distribution, and then deriveseparate equation of motion for it from Boltzmann equa-tion. More recently, solutions of various versions of hy- drodynamics were compared to the exact solution of theBoltzmann equation itself, derived for Gubser geometri-cal setting in [53]. This paper contains many instructiveplots, from which I selected the normalized temperatureshown in Fig.13 and the sheer stress Π ξξ shown in Fig.14. In both cases the pairs of points correspond to smalland very high viscosity values, separated by two ordersof magnitude and roughly representatives of the stronglyand weakly coupled regimes.Gubser’s variable ρ is the “time” coordinate. At allfour plots one can see that all curves coincide in the in-terval − < ρ <
2, but deviate from each other bothat large negative values, corresponding to the very earlystage, and for large positive ones, corresponding to verylate times. In fact all practical applications of hydrody-namics were indeed made inside this interval of ρ , withother regions being “before formation” and “after freeze-out”.Solutions for two – hugely different - viscosities showa very similar trends. Israel-Stuart hydro seems to fol-low Boltzmann in the most accurate way. Even the freestreaming regime is not very far from all hydros and ex-act Boltzmann: this would be surprising for the readerif we would not already know that the radial flow – un-like higher harmonics – can indeed be “faked”. If theseauthors would calculate the elliptic and higher flows, theresults would be quite different. It is not done yet, butone expects that for 4 πη/s = 100 those would be com-pletely obliterated.The plots for the shear stress show different behaviorsfor small and large viscosity, but again all curves coincideinside the “hydro window” of − < ρ <
2. Even goingwell outside that domain, we never see discrepancies be-tween them by more than say 20%.The overall conclusion one can draw from all of thoseimpressive works is quite simple: all versions of hydroused in practice are very accurate for realistic viscosities4 πη/s ≈ V. THE SMALLEST DROPS OF QGP
We have described above some successes of hydrody-namics for description of the flow angular harmonics,showing that those are basically sound waves generatedby the initial state perturbations. We also emphasizeda significant gap which still exists between approachesbased on weak and strong couplings, in respect to equi-libration time and matter viscosity. Needless to say, thekey to all those issues should be found in experimenta-tions with systems smaller than central AA collisions.They should eventually clarify the limits of hydrodynam-ics and reveal what exactly happen in this hotly disputed“the first 1 fm/c” of the collisions.Let us start this with another look at the flow harmon-ics. What is the spatial scale corresponding to the highest n of the v n observed? A successful description of the n -th8 FIG. 13: From [53]: the normalized temperature for 4 πη/s =1 , harmonics along the fireball surf ace implies that hydrostill works at a wavelength scale 2 πR/n : taken the nu-clear radius R ∼ f m and the largest harmonic studiedin hydro n = 6 one concludes that this scale is still fewfm. So, it is still large enough, and it is impossible to tellthe difference between the initial states of the Glaubermodel (operating with nucleons) from those generatedby parton or glasma-based models (operating on quark-gluon level) . And indeed, we don’t see harmonics withlarger n simply because of current statistical limitationsof the data sample. Higher harmonics suffer strongerviscous damping, during the long time to freezeout. Inshort, non-observation of v n , n > are unrelated to thelimits of hydrodynamics.(Except in the region of high p ⊥ ∼ GeV in whichindeed the relevant region becomes too thin to betreated macroscopically. Unfortunately, there is still lit-tle progress in understanding v n at and above such mo-menta.)In principle, one can study AA collisions for smallerand smaller systems, looking for the lighter nuclei whichstill see flow harmonics. Note that in such case the timeto freeze out scales with the radius, so angular correla-tions stay about at the same angles and n . In our pocketformula for viscous damping, it changes T R downward, allowing to understand the sound damping phenomenafrom another angle.However, it is not the way of the actual development,as unexpected discoveries of flows in very small systems– pp and pA collisions, with high multiplicity trigger. Itis those which we discuss in this chapter. As we will see,there are similarities but also important differences of thetwo cases.Before we go into details, let us try to see how largethose systems really are. At freezeout the size can bedirectly measured, using femtoscopy method. (Brief his-tory: so called Hanbury-Brown-Twiss (HBT) radii. Thisinterferometry method came from radio astronomy. Theinfluence of Bose symmetrization of the wave functionof the observed mesons in particle physics was first em-phasized by Goldhaber et al [106] and applied to proton-antiproton annihilation. Its use for the determinationof the size/duration of the particle production processeshad been proposed by Kopylov and Podgoretsky [107]and myself [108]. With the advent of heavy ion collisionsthis “femtoscopy” technique had grew into a large in-dustry. Early applications for RHIC heavy ion collisionswere in certain tension with the hydrodynamical models,although this issue was later resolved [109].)The corresponding data are shown in Fig.15, whichcombines the traditional 2-pion with more novel 3-pioncorrelation functions of identical pions. An overallgrowth of the freezeout size with multiplicity, roughlyas < N ch > / , is expected already from the simplestpicture, in which the freezeout density is some univer- FIG. 14: From [53]: shear stress for 4 πη/s = 1 , J.F. Grosse-Oetringhaus / Nuclear Physics A 931 (2014) 22–31
Fig. 6.
Left panel:
Proton to φ ratio as a function of p T for different Pb–Pb centrality classes [47].
Right panel:
Femto-scopic radii extracted from two- and three-pion cumulants together with the associated λ parameters [50]. shape is driven by radial flow. Combining this finding with that for the v suggests that the mass (and not the number of constituent quarks) drives v and spectra in central Pb–Pb collisions for p T < GeV /c . It is interesting to note that also in p–Pb collisions the shape of the p T spectra of φ and p become more similar for high-multiplicity events [3]. Identified-particle spectra
The
ALICE
Collaboration has presented yields and spectra for particle species ( π , K ± , K ∗ , K , p, φ , Λ , Ξ , Ω , d, He, Λ H) in up to collision systems (and, for pp collisions, different center of mass energies). In particular the measurement of the p T and centrality dependence of the d and the nuclei ( He, Λ H) spectra should be pointed out [25]. It is interesting to note that the yields of d, He and Λ H are correctly calculated in equilibrium thermal models. Furthermore, the yields of multi-strange baryons have been measured as a function of event multiplicity showing a smooth evolution from pp over p–Pb to Pb–Pb collisions for the yield ratios to π or p [2]. The large amount of data allows a stringent comparison to thermal models which describe particle production on a statistical basis [49]. Source sizes
For the first time, femtoscopic radii were extracted with three-pion cumulants [16,50].
This approach reduces non-femtoscopic effects contributing to the extracted radii significantly. Fig. 6
FIG. 15: Alice data on the femtoscopy radii (From [26]) (up-per part) and “coherence parameter” (lower part) as a func-tion of multiplicity, for pp, pP b, P bP b collisions. sal constant. For AA collisions this simple idea roughlyworks: 3 orders of magnitude of the growth in multiplic-ity correspond to one order of magnitude growth of thesize.Yet the pp, pA data apparently fall on a different line,with significantly smaller radii, even if compared to theperipheral AA collisions at the same multiplicity. Whydo those systems get frozen at higher density, than thoseproduced in AA? To understand that one should recallthe freezeout condition : “the collision rate becomes com-parable to the expansion rate” < nσv > = τ − coll ( n ) ∼ τ − expansion = dn ( τ ) /dτn ( τ ) (25)Higher density means larger l.h.s., and thus we need alarger r.h.s.. So, more “explosive” systems, with largerexpansion rate, freezeout earlier. We will indeed arguebelow that pp, pA high-multiplicity systems are in factmore “explosive”: it is seen from radial flow effects onspectra as well as HBT radii.But how those systems become “more explosive” inthe first place? Where is the room for that, people usu-ally ask, given that even the final size of these objectsis smaller than in peripheral AA, which has a rather weak radial flow. Well, the only space left is at the be-ginning: those systems must start accelerating earlier ,from even smaller size. Only then they would be able toget enough acceleration, and eventually strong collectiveflow, by their freezeout. A. Collectivity in small systems
Let us briefly recall the time sequence of the mainevents. The first discovery – in the very first LHC run– was due to CMS collaboration [101] which found a“ridge” correlation in high multiplicity pp events, en-hanced by a special trigger. That was required because,unfortunately, the effect was first seen only in events witha probability P ∼ − [165].Switching to most central pA CMS [102] and other col-laborators had observed a similar ridge there, now withmuch higher – few percent – probability. By subtractinghigh multiplicity and low multiplicity correlators CMSand ALICE groups soon had concluded, that “ridge” isaccompanied by the “anti-ridge” in other hemisphere,and thus it is basically a familiar elliptic flow v .PHENIX collaboration at RHIC also found a ridge-likecorrection in central dAu collisions, Furthermore, their v is larger than in pP b at LHC, by about factor 2.This was soon attributed to different initial conditions,for d and p beams, since the former have “double explo-sion” of two nucleons. Quantitative prediction came frompioneering hydrodynamical studies of such collisions byBozek [42], and then many others. Hydro predicts theeffect correctly: and that was the first indication for thecollectivity of the phenomenon. Another contribution ofPHENIX was the observation that dA HBT radii displaythe famous decreasing trend with p t well known for AA collisions, which is another – and very direct – evidencefor presence of the collective flow.Truly amazing set of data came from CMS [27]. Their v measurements from 4,6 and even 8 particles are shownin Fig.16. Previous data for AA collisions had shownperfect agreement between those, and new data for pA are in this respect the same. This establishes collectivity of the flow in pA, “beyond the reasonable doubt”.Taken collectivity for granted, one can further ask ifthe v observed is caused by the pre-collision correlationsor by the after-collision collective flows. A very nice con-trol experiment testing this is provided by dA and He A collisions. Two nucleons in d are in average far from eachother and 2 MeV binding is so small that one surely canignore their initial state correlations. So, whatever is the“initial shape” effect in pp , in dA it should be reduced by1 / √ He A by 1 / √
3, if the same logicholds.Hydrodynamical predictions are opposite: double (ortriple) initial explosions lead to a common fireball, withthe anisotropies larger than in pA . Data from RHICby PHENIX and STAR on dAu, He Au do indeed show0 Raphael Granier de Cassagnac Quark Matter 2014, Darmstadt
Multiparticle correlations • v stays large when calculated with multi-particles – v (4)=v (6)=v (8)=v (LYZ) within 10% – True collectivity in pPb collisions! Talk by Wang PAS-HIN-14-006
PbPb pPb (event multiplicity) v FIG. 16: CMS data [27] for v calculated using 2,4,6, 8particle correlations, as well as Lee-Yang zeroes (basically allparticles). Good agreement between those manifest collectiv-ity of the phenomenon.FIG. 17: v n for n − , , , p t in GeV, for high multi-plicity bin indicated on the figure. The points are from Atlas,lines from CMS (presentation at QM2015). such an increase of the v , v , relative to pAu , again inquantitative agreement with hydro [43, 44]. It looks likethis issue is now settled.Recently Atlas was able to perform the first measure-ments of higher harmonics v n , n = 4 , pP b ,see Fig. 17. Except at very high p t , those two harmonicsseem to be comparable in magnitude: it is the first con-tradiction to “viscous damping” systematics seen so far.(No idea so far why does it happen.) B. Pedagogical digression: scale invariance ofsQGP and small systems
Acceptance of hydrodynamical treatment of “smallsystem explosions” need to pass a psychological barrier:people repeatedly ask if it is indeed possible to treat asystem of less than 1 fm in size as macroscopic. (Andindeed, just 15 years ago the same question was askedabout systems of 6 fm size.) So, let us take a step backfrom the data and consider the issue of scales.If one takes smaller and smaller cells of ordinary fluid –
RTR
A DCB ETc ex.0ex.1
FIG. 18: (color online) Temperature T versus the fireball size R plane. Solid blue line is the adiabatic S = const , approx-imately T R = const for sQGP. Example 0 in the text cor-responds to reducing R , moving left A → B . Example 1 ismoving up the adiabatic A → C . Example 2 corresponds toadiabatic expansion, such as A → E , C → E . If in reality C corresponds to pA , the freezeout occurs at the earlier point D . such as water or air – eventually one reaches the atomicscale, beyond which water or air, as such, do not ex-ist: just individual molecules. QGP is not like that: itis made of essentially massless quarks and gluons whichhave no scale of their own. The relevant scale is givenby only one parameter T – thus QGP is approximatelyscale invariant. (The second scale Λ QCD only enters vialogarithmic running of the coupling, which is relativelyslow and can in some approximation be ignored.)As lattice simulation show, above the phase transition
T > T c QGP thermodynamics soon becomes scale invari-ant (cid:15)/T , p/T ≈ const ( T ). The comparison of LHC toRHIC data further suggests that it is similarly true forviscosity as well η/s ≈ const ( T ) (although with less ac-curacy so far). Thus, QGP does not have a scale of itsown, and thus would show exactly the same behavior ifconditions related by the scale transformation R A /R C = ξ, T A /T C = ξ − (26)are compared.Consider a thought experiment 1 , in which we com-pare two systems on the same adiabatic A and C . Forscale invariant sQGP the points A, C are related by thisscale transformation mentioned above, and have the sameentropy. Since the scale transformation is approximatesymmetry, we expect the same output. A smaller-but-hotter plasma ball C will explode in exactly the sameway as its larger-but-cooler version A .Let us now proceed to the thought experiment 2 , whichis the same as above but in QCD, with a running cou-pling. In the sQGP regime it leads to (very small, as1lattice tells us ) running of s/T , some (unknown) run-ning of η/T , etc. The most dramatic effect is howevernot the running coupling per se , but the lack of supersym-metry, which allows for the chiral/deconfinement phasetransition, out of the sQGP phase at T = T c to hadronicphase. The end of the sQGP explosion D thus has an absolute scale , not subject to scale transformation!So let us consider two systems A , C of the same to-tal entropy/multiplicity, initiated in sQGP with condi-tions related by scale transformation and left them ex-plode. The sQGP evolution would be related by nearlythe same set of intermediate states (modulo running cou-pling) till T ≈ T c , after which they go into the “mixed”and hadronic stages, which are not even close to be scaleinvariant! Thus the result of the explosions are not thesame. In fact the smaller/hotter system will have an ad-vantage over the larger/cooler one, since it has larger ratio between the initial and final scales T i /T f .(In the language of holographic models the scale is in-terpreted as the 5-th coordinate x , and evolution is de-picted as gravitational falling of particles,strings, fireballsetc toward the AdS center. The ratio of the scales is thedistance along the 5-th coordinate: thus in this languagetwo systems fall similarly in the same gravity, but smallersystem starts “higher” and thus got larger velocity at thesame “ground level” given by T c .)The hydro expansion does not need to stop at the phaseboundary D . In fact large systems, as obtained in centralAA collisions are known to freezeout at T f < T c , downto 100 MeV range (and indicated in the sketch by thepoint E . However small systems, obtained in peripheralAA or central pA seem to freezeout at D , as we will showat the end of the paper.Short summary of these thought experiments: not onlyone expects hydro in the smaller/hotter system to bethere, it should be very similar to that in larger/coolersystem, due to approximate scale invariance of sQGP. C. Preliminary comparison of the peripheral AA,central pA and high multiplicity pp Now is the time to go from thought experiment withsome ideal systems to the real ones. We will do it in twosteps, first starting in this section with “naive” estimatesfor three cases at hand, based on standard assumptionsabout the collision dynamics, and then returning to morerealistic studies of the last two cases in the next subsec-tions.We aim at initial transverse radii and density, thus weuse the initial size of the nuclei rather than that of thefireball at freezeout, after hydro expansion. The multi-plicity is the final one, but due to (approximate) entropyconservation during the hydro stage we think of it as aproxy for the entropy at early time as well. (Entropy gen-erated by viscosity during expansion is relatively smalland can be corrected for, if needed.)(i) Our most studied case, the central AuAu or PbPb, 〈 p T 〉 [ G e V / c ] 〈 N tracks 〉 pp pPbpp pPbpp pPb π ± K ± p, − pCMS ALICE PbPbALICE PbPbALICE PbPb Y i e l d r a t i o s 〈 N tracks 〉 pp pPbpp pPb(K + +K − )/( π + + π − )(p+ − p)/( π + + π − )CMS ALICE PbPbALICE PbPb Figure 9: Average transverse momentum of identified charged hadrons (pions, kaons, protons;left panel) and ratios of particle yields (right panel) in the range | y | < | h | < p s NN = h p T i and yield ratios werecomputed assuming a Tsallis-Pareto distribution in the unmeasured range. Error bars indicatethe uncorrelated combined uncertainties, while boxes show the uncorrelated systematic uncer-tainties. For h p T i the fully correlated normalization uncertainty (not shown) is 1.0%. In bothplots, lines are drawn to guide the eye (gray solid – pp 0.9 TeV, gray dotted – pp 2.76 TeV, blackdash-dotted – pp 7 TeV, colored solid – pPb 5.02 TeV). The ranges of h p T i , K/ p and p/ p valuesmeasured by ALICE in various centrality PbPb collisions (see text) at p s NN = by ALICE in PbPb collisions at p s NN = N tracks than is shown in the plot. Although PbPb data are not available at p s NN = p s NN = h p T i values willincrease by about 5% when going from p s NN = T ′ [ G e V / c ] m [GeV/c ] 8325884109135160185210235pPb, √ s NN = 5.02 TeV, L = 1 µ b -1 CMS T ′ [ G e V / c ] m [GeV/c ] 〈 N tracks 〉 = 8AMPTEPOS LHCHijing 2.1pPb, √ s NN = 5.02 TeVCMS 〈 N tracks 〉 = 235AMPTEPOS LHCHijing 2.1 Figure 10: Inverse slope parameters T from fits of pion, kaon, and proton spectra (both charges)with a form proportional to p T exp ( m T / T ) . Results for a selection of multiplicity classes,with different N tracks as indicated, are plotted for pPb data (left) and for MC event generatorsA MPT , E
POS L HC , and HIJING (right). The curves are drawn to guide the eye.For low track multiplicity ( N tracks . N tracks &
50) the h p T i is lower for pPb than in pp. The first ob-servation can be explained since low-multiplicity events are peripheral pPb collisions in whichonly a few proton-nucleon collisions are present. Events with more particles are indicativeof collisions in which the projectile proton strikes the thick disk of the lead nucleus. Inter-estingly, the pPb curves (Fig. 9, left panel) can be reasonably approximated by taking the ppvalues and multiplying their N tracks coordinate by a factor of 1.8, for all particle types. In otherwords, a pPb collision with a given N tracks is similar to a pp collision with 0.55 ⇥ N tracks forproduced charged particles in the | h | < h p T i than seen in central PbPb collisions. While in the PbPb case eventhe most central collisions possibly contain a mix of soft (lower- h p T i ) and hard (higher- h p T i )nucleon-nucleon interactions, for pp or pPb collisions the most violent interaction or sequenceof interactions are selected.The transverse momentum spectra could also be successfully fitted with a functional form pro-portional to p T exp ( m T / T ) , where T is called the inverse slope parameter, motivated by thesuccess of Boltzmann-type distributions in nucleus-nucleus collisions [29]. In the case of pi-ons, the fitted range was restricted to m T > c in order to exclude the region whereresonance decays would significantly contribute to the measured spectra. The inverse slopeparameter as a function of hadron mass is shown in Fig. 10, for a selection of event classes,both for pPb data and for MC event generators (A MPT , E
POS L HC , and HIJING ). While the data
FIG. 19: (color online) (From [41].) (a) Average transversemomentum of identified charged hadrons (pions, kaons, pro-tons; left panel) and ratios of particle yields (right panel)in the range | y | < | η | < .
4, for pp collisions (open symbols)at several energies, and for pPb collisions (filled symbols) at √ s NN = 5 . T eV. (b) The slopes of the m ⊥ distribution T (cid:48) (in GeV) as a function of the particle mass. The numbers onthe right of the lines give the track multiplicity. is the obvious benchmark. With the total multiplicityabout N AA ≈ and transverse area of nuclei πR A ≈ f m one gets the density per area n AA = NπR A ∼ f m − (27)which can be transformed into entropy if needed, in astandard way.(ii) Central pA (up to few percent of the total crosssection) has CMS track multiplicity of about 100. Ac-counting for unobserved range of p t , y and neutrals in-2creases it by about factor 3, so N centralpA ∼ b in pp collisions, or π < b > = σ pp ≈ f m . The density perarea is then n centralpA = N centralpA σ pp ∼ f m − (28)or 1/3 of that in central AA. Using the power of LHCluminosity CMS can reach – as a fluctuation with theprobability 10 − — another increase of the multiplicity,by about factor 2.5 or so, reaching the density N maxpA /σ pp in AA. Another approach used is a comparison of central pA with peripheral AA of the same multiplicity, or moreor less same number of participants. Similar matter den-sity is obtained.(iii) Now we move to the last (and most controver-sial) case, of the high multiplicity pp collisions. (Need-less to say the density is very low for min.bias events.)“High multiplicity” at which CMS famously discoveredthe “ridge” starts from about N maxpp > ∗ what is the area? Unlike inthe case of central pA, we don’t utilize standard Glauberand full cross section (maximal impact parameters): weaddress now a fluctuation which has small probability.In fact, nobody knows the answer to that. Based on theprofile of pp elastic scattering (to be discussed in sectionVIII C) I think it should correspond to impact parameter b in the black disc regime. If so πb b.d. ∼ / f m , whichleads to density per area n maxpp ≈ N maxpp πb b.d. ∼ f m − (29)Other evidences about glue distribution in a protoncomes from HERA diffractive production, especially of γ → J/ψ : they also suggest a r.m.s. radius of only0 . f m , less than a half of electromagnetic radius.Let us summarize those (naive) estimates: in terms ofthe initial entropy density one expects the following orderof the densities per area involved dN pAmaximal dA ⊥ ∼ dN AAperipheral dA ⊥ (cid:28) dN AAcentral dA ⊥ (cid:28) dN ppmaximal dA ⊥ (30)One may expect that the radial flow follows the samepattern: yet the data show it is not the case. D. The “radial flow puzzle” for central pA
The simplest consequence of the radial flow is increas-ing mean transverse momentum. CMS data on those, asa function of multiplicity, are shown in Fig.19(a). While pp and pA data are shown by points, the AA ones (fromALICE) are shown by shaded areas: the central ones cor-respond to its upper edge. While one invent many mech-anisms of the mean p t growth – e.g. rescattering, larger saturation momentum Q s at higher multiplicity, etc –those explanations generally do to explain why heavierparticles – protons – get this effect stronger than the pi-ons.True experimental signatures of the radial flow arebased on the observation that collective flow affect spec-tra of secondaries of different mass differently. While(near) massless pions retain exponential spectra, withblue-shifted slope, massive particles have spectra of amodified shape. Eventually, for very heavy particles (e.g. d or other nuclei) their thermal motion gets negligible andtheir momenta become just mv where v is the velocity ofthe flow. Its distribution has a characteristic peak at thefireball’s edge.More specifically, a measure proposed in [99] is to lookat the so called “violation of the m ⊥ scaling”. The m ⊥ slopes T (cid:48) are defined by the exponential form (above cer-tain p t ) dNdydp ⊥ = dNdydm ⊥ ∼ exp ( − m ⊥ T (cid:48) ) (31)and they are the best indicators of the radial flow. A sam-ple of such slopes for pA collisions, from CMS, is shownin Fig.19 (similar data from ALICE but for smaller mul-tiplicities are also available, see Fig.58). The multiplicitybins (marked by 8 and 32 at the bottom-right) show thesame T (cid:48) for all secondaries: this is the m ⊥ scaling indicat-ing that the flow is absent. This behavior is natural forindependent string fragmentation, rescattering or glasmamodels.Flow manifests itself differently. For pions T (cid:48) is simplythe freezeout temperature, blue-shifted by the exponentof the transverse flow rapidity T (cid:48) = T f e κ (32)For more massive particles – kaons, protons, lambdas,deuterons etc – the slopes are mass-dependent . As seenfrom Fig.19(b), they are growing approximately linearlywith the mass, and the effect gets more pronounced withmultiplicity. This is the signature of the collective flow.Furthermore, the central pA bin has slopes exceeding those in central PbPb collisions at LHC, the previousrecord-holding! (Predicted to happen before experiment:see version v1 of [90].)This gives rise to what we call the radial flow puzzle .Indeed, naive estimates of densities in the previous sub-section may suggest that explosion in highest multiplicity pA case should still be weaker than in AA. Indeed, boththe system is smaller and the initial entropy density seemto be smaller as well. Yet the data show the opposite: theobserved radial flow strength follows a different pattern y AA,central ⊥ < y pA,central ⊥ < y pp,highest ⊥ (33)Hydrodynamics is basically a bridge, between the ini-tial and the final properties of the system. For the radialflow dependence on the size of the system it is convenient3 FIG. 20: (color online) The freezeout surface in universal di-mensionless time t and radial distance r coordinates. (Blue)thick solid line in the middle corresponds to central AA(PbPb) collisions, (red) thick solid line on the top to thehighest multiplicity pp . Two (black) thin ones correspondto central p Pb case, before and after collapse compression,marked pA i , pA f respectively. The arrow connecting themindicates the effect of multi string collapse. to follow Ref. [90] based on Gubser’s flow , see section B. One single analytic solution describes all cases consid-ered: we will proceed from the dimensional variables ¯ τ , ¯ r with the barto dimensionless variables t = q ¯ τ , r = q ¯ r (34)using rescaling by a single parameter q with dimensionof the inverse length. In such variables there is a singlesolution of ideal relativistic hydrodynamics, which for thetransverse velocity (B3 ) and the energy density (B4)dependence on time and space is known. The secondequation fixes the shape of the freezeout surface, usuallyan isotherm T = T f .Before turning to actual plot of such surfaces, let usrecall our thought experiment 1 in the subsection V B:two collisions which are conformal copies of each otherwould look the same in dimensionless variables. And in-deed, the blue line marked AA in Fig.20 corresponds notonly to central PbPb collisions at LHC, but actually toany AA collisions. (Its parameters, for the record, are q = 1 / . f m, ˆ (cid:15) = 2531 , T f = 120 M eV , a benchmarkwith “realistic” radial flow.) Two black lines are for thepPb case: they both have T f = 170 M eV and the samemultiplicity but different scale parameters: q = 1 / . f m for the lower dotted line but twice smaller spacial scale q = 1 / . f m for the upper thin solid line. As an arrowindicates, in order to explain the data one has to starthydro not from the “naive” initial size, the former line,but from the “compressed” size, according to “spaghetticollapse” scenario we will discuss in section V B. If this is done, the freezeout surface “jumps over” our AA bench-mark blue line, and its radial flow gets stronger. Themaximal transverse velocities on these curves (locatednear the turn of the freezeout surface downward) are v pAu ⊥ = 0 . < v AA ⊥ = 0 . < v pAu,f ⊥ = 0 .
84 (35)The upper red line is our guess for the maximal multi-plicity pp collisions, assuming its q = 1 / . f m : it haseven stronger radial flow, with maximal v pp ⊥ ≈ .
93. So,paradoxically, small systems are in fact larger than AAin appropriate dimensionless variables, and that is whytheir radial flow is better developed.In summary: the observed pattern of radial flow mag-nitude can only be explained if the initial size of the pA system is significantly reduced, compared to the naiveestimates in the preceding section. E. Radial flow in high multiplicity pp According to our preliminary discussion of the densi-ties per area, in this case it is much higher than in AAcollisions. The initial state must be in a GLASMA state,if there is one. Yet we do not have theoretical guidanceabout the size. ( It is not at all surprising: those are fluc-tuations with small probability ∼ − , and understandtheir precise dynamics is difficult.) Lacking good theoryguidance, one may invert our logical path, and proceed asfollows: (i) extract the magnitude of the flows – radial, v , v – from the data, at freezeout. Then (ii) “solvedhydro backwards”, deriving the initial conditions neededto generate such a flow.One phenomenological input can be the mean p t andspectra of the identified particles in high multiplicity pp :some of those we have already shown in Fig.19.Another approach is to use is the femtoscopy method.It allows to detect the magnitude and even deformationsof the flow. Makhlin and Sinyukov [110] made the im-portant observation that HBT radii decrease with the in-crease of the (total) transverse momentum (cid:126)k + (cid:126)k = (cid:126)k t of the pair. Modification of their argument for our pur-poses is explained in a sketch shown in Fig.21. At small k t the detector sees hadrons emitted from the whole fire-ball, but the larger is k t , the brighter becomes its small(shaded) part in which the radial flow is (i) maximaland (ii) has the same direction as (cid:126)k t . This follows frommaximization of the Doppler-shifted thermal spectrum ∼ exp ( p µ u µ /T freezeout ). One way to put is is to notethat effective T in it is increased by the gamma factor ofthe flow.Hirono and myself [113] had calculated such effect andcompare the results with the ALICE HBT data [112]shown in Fig.22. We deduced the magnitude of the flowin high multiplicity pp collisions, directly visible in thosedata. The effect is best seen in the so called “out”-directed radius R out (the top plot). While low multiplic-ity data (connected by the blue dashed line) are basically4 FIG. 21: Sketch of the radial flow (arrows directed radiallyfrom the fireball center) explaining how it influences the HBTradii. At small k t the whole fireball (the large circle) con-tributes, but at larger k t one sees only the part of the fireballwhich is co-moving in the same direction as the observed pair.This region – shown by shaded ellipse – has a smaller radiiand anisotropic shape, even for central collisions. | < 0.16 GeV/c side,long |q ALICE pp @ 7 TeV 23-29 ch N (0.3, 0.4) T k| < 0.16 GeV/c out,long |q + π + π + π + π fit | < 0.16 GeV/c out,side |q out,side,long q11.511.511.5 ) l ong C ( q ) s i de C ( q ) ou t C ( q FIG. 10. Projections of the 3D Cartesian representations of the cor-relation functions for events with 23 ≤ N ch ≤
29 and pairs with0 . < k T < . c . To project onto one q -component, the oth-ers are integrated over the range 0 − .
16 GeV/ c . Dashed lines showanalogous projections of the Gaussian fit. in Section IV C. In Fig. 10 the same correlation is shown asprojections of the 3D Cartesian representation. The other q components are integrated over the range of 0 − .
16 GeV/ c .The fit, shown as lines, is similarly projected. In this plotthe fit does not describe the shape of the correlation perfectly;however, the width is reasonably reproduced. IV. FIT RESULTSA. Results of the 3D Gaussian fits
We fitted all 72 correlation functions (4+8 multiplicityranges for two energies times 6 k T ranges) with Eq. (7). Weshow the resulting femtoscopic radii in Fig. 11 as a function of k T . The strength of the correlation λ is relatively independentof k T , is 0.55 for the lowest multiplicity, decreases monoton-ically with multiplicity and reaches the value of 0.42 for thehighest multiplicity range. The radii shown in the Fig. 11 arethe main results of this work. Let us now discuss many aspectsof the data visible in this figure.Firstly, the comparison between the radii for two ener-gies, in the same multiplicity/ k T ranges reveals that they areuniversally similar, at all multiplicities, all k T ’s and all di-rections. This confirms what we have already seen directlyin the measured correlation functions. The comparison to √ s =
200 GeV pp collisions at RHIC is complicated by thefact that these data are not available in multiplicity ranges.The multiplicity reach at RHIC corresponds to a combination ALICE pp @ 7 TeV 1-11 ch N 12-16 ch N 17-22 ch N 23-29 ch N 30-36 ch N 37-44 ch N 45-57 ch N 58-149 ch Na)
ALICE pp @ 0.9 TeV 1-11 ch N 12-16 ch N 17-23 ch N 24-80 ch Nb)
STAR pp @ 200 GeVc) d) T k0.51121212 s i de G / R ou t G R ( f m / c ) l ong G R ( f m / c ) s i de G R ( f m / c ) ou t G R FIG. 11. Parameters of the 3D Gaussian fits to the complete set ofthe correlation functions in 8 ranges in multiplicity and 6 in k T for pp collisions at √ s = k T for pp collisions at √ s = . k T bin shouldbe at the same value of k T , but we shifted them to improve visibility.Open black squares show values for pp collisions at √ s =
200 GeVfrom STAR [10]. Lines connecting the points for lowest and highestmultiplicity range were added to highlight the trends. of the first three multiplicity ranges in our study. No strongchange is seen between the RHIC and LHC energies. It showsthat the space-time characteristics of the soft particle produc-tion in pp collisions are only weakly dependent on collisionenergy in the range between 0 . k T ranges. Obviously the √ s = k T integrated) correlation function for the twoenergies is different.Secondly, we analyze the slope of the k T dependence. R Glong falls with k T at all multiplicities and both energies. R Gout and R Gside show an interesting behavior – at low multiplicity the k T dependence is flat for R Gside and for R Gout it rises at low k T andthen falls again. For higher multiplicities both transverse radiidevelop a negative slope as multiplicity increases. At high FIG. 22: HBT radii versus the pair transverse momentum k T , for various multiplicities of the pp collisions, from ALICE[112]. independent on the pair momentum, at high multiplic-ity (stars and red dashed line) they are decreasing, bya rather large factor. Another consequence of the flowis anisotropy of radii. In the bottom plot the ratio of two radii are shown: at small multiplicity it is always1 – that is the source is isotropic – but at high multi-plicity the source becomes anisotropic, the radii in twodirections are quite different, with their ratio droppingto about 1 /
3, at the largest k t . It means, only 1/3 ofthe fireball emits pairs of such momenta, a direct conse-quence of the flow.In Fig.23 we show a series of calculations in which theinitial QGP stage of the collision is modelled by numeri-cal hydro solution close to Gubser analytic solution withvariable parameter q . (The late stages need to deviatefrom Gubser since near T c the EOS is very different fromconformal (cid:15) = 3 p assumed in Gubser’s derivation).Let us summarize what we learned in this subsectionso far. Unlike central pA , the highest multiplicity pp events are significantly denser/hotter than central AA.Very strong radial flow, seen in spectra of identified par-ticles and HBT radii, require very small, sub femtometer,initial size of the system. In spite of high cost associatedwith those events, their studies are of utmost importancebecause here we produce the most extreme state of mat-ter ever created in the lab. F. Can flows in small systems be “fake”?
The question and subsequent development is due toRomatschke [121], who proposed a particular model ofwhat I call “the fake flow”. In this scenario quarks andgluons at the QGP phase have no interactions and freestream from the point of the initial scattering. Onlyat some hadronization surface the system switches tohadronic cascade.In Fig.25 one see a comparison of the radial flow pro-files of the two cases, with and without interaction atthe QGP phase. One can see that crudely the profilesare in fact very similar, becoming linear Hubble-like as R o [f m ] Kt [GeV] q=0.5 q=0.7 q=0.9 q=1.1 q=1.3 q=1.5 ALICE data
FIG. 23: From [113]: HBT radii compared to ALICE data(closed circles), for solutions starting with different initial sizeof the fireball, indicated by Gubser scale parameter q (whichis inversely proportional to the size). FIG. 24: The integrated v { } for PbPb and pPb vs. mul-tiplicity from [23]. Left: Original values. Right: The fluctu-ation dependent elliptic flow, with the geometrical part sub-tracted. This geometrical part was calculated using the Pho-bos Glauber Model and is not a fit.FIG. 25: Comparison of the flow profile, for hydrodynamicsand free streaming , from [121]. time goes on. (In fact free streaming generates even abit stronger flow, because free streaming uncouples fromlongitudinal direction, and equilibrated hydro mediumdoes not.) Comparing particle spectra and HBT radiiRomatschke shows that this “fake” radial flow is indeed indistinguishable from the hydro one.What about flow harmonics? The results for PbPbcollisions are shown in Fig.26. As one can see, with-out hydro those (crudely speaking) disappear. It is notsurprising since if there is no hydrodynamics then thereshould be no sound waves, and initial bumps are simplydissolved without trace.(In fact it is an interesting question how any v n canbe generated in the free streaming. The initial momen-tum distribution of partons is isotropic, and so it mustbe related to the interaction after hadronization. Ro-matschke found that indeed before hadronization theyare absent. However this happens because two compo-nent of T µν , the flow ∼ u µ u ν part and the dissipativeΠ µν part, have nonzero values but cancel each other insum. After hadronization the hadronic interaction killsthe second component Π µν → p ⊥ ; and(ii) they do not show strong decrease with the number ∼ exp ( − n ) induced by the viscosity during the timebefore hadronizationSo, we now see that, unlike the radial flow, higher har-monics in large (PbPb) systems cannot be faked. Whatabout smaller systems? Romatschke gives the results for pP b at LHC and dAu and He Au for RHIC energies. Weshow the first case in Fig.27 , the rest can be looked atthe original paper. Again the free streaming model seemsto be failing for v , is somewhat marginally possible for d N / ( π d Y p T dp T ) [ G e V - ] p T [GeV]Particle Spectra: Hydro versus Non-Interacting Gasx 0.1x 0.01Pb+Pb √ s=2760 GeV30-40% ( π + + π - )/2 Free-Streaming( π + + π - )/2 Hydro η /s=0.08(K + +K - )/2 Free-Streaming(K + +K - )/2 Hydro η /s=0.08(p+pbar)/2 Free-Streaming(p+pbar)/2, Hydro η /s=0.08 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0 0.5 1 1.5 2 2.5 v p T [GeV]Elliptic Flow: Hydro versus Non-Interacting GasPb+Pb, √ s=2760 GeV,30-40%( π + + π - )/2 Free-Streaming( π + + π - )/2 Hydro η /s=0.08(K + +K - )/2 Free-Streaming(K + +K - )/2 Hydro η /s=0.08(p+pbar)/2 Free-Streaming(p+pbar)/2, Hydro η /s=0.08 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0 0.5 1 1.5 2 2.5 v p T [GeV]Triangular Flow: Hydro versus Non-Interacting GasPb+Pb, √ s=2760 GeV,30-40%( π + + π - )/2 Free-Streaming( π + + π - )/2 Hydro η /s=0.08(K + +K - )/2 Free-Streaming(K + +K - )/2 Hydro η /s=0.08(p+pbar)/2 Free-Streaming(p+pbar)/2, Hydro η /s=0.08 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0 0.5 1 1.5 2 2.5 v p T [GeV]Quadrupole Flow: Hydro versus Non-Interacting GasPb+Pb, √ s=2760 GeV,30-40%( π + + π - )/2 Free-Streaming( π + + π - )/2 Hydro η /s=0.08(K + +K - )/2 Free-Streaming(K + +K - )/2 Hydro η /s=0.08(p+pbar)/2 Free-Streaming(p+pbar)/2, Hydro η /s=0.08 FIG. 6. Simulations of granular Pb+Pb collisions at √ s = 2 .
76 TeV. Shown are final particle spectra and anisotropic flowcoefficients v n ( p T ) for identified particles for free-streaming evolution (no-interaction) and almost ideal hydrodynamics ( η/s =0 . through fitting a Gaussian form to the function S , S ( K , r ) ∝ e − x R − y R − z R (15)defining the femtoscopic radii R out , R side , R long . The results for these extracted radii for pions are shown in Fig. 7for d+Au , He+Au , p+Pb and Pb+Pb collisions, comparing hydrodynamic and non-interacting evolution. Fromthis figure, one can observe a striking similarity for all the extracted radii between strongly interacting evolution(hydrodynamics) and non-interacting evolution (free streaming) for all simulated systems, small and large. Similarlyto what was found for the case of radial flow, the femtoscopic radii are essentially insensitive to the details of thesystem evolution, as long as energy and momentum are conserved.In essence, this disqualifies the use of pion femtoscopic measurements as serving as evidence for a hydrodynamicphase during the system evolution.
IV. SUMMARY AND CONCLUSIONS
In this work flow signatures arising from two very different dynamics in the hot QCD phase following relativisticion collisions very studied. In the first case, the hot phase dynamics was assumed to be described by non-interactingparticles. In the second case, the hot phase dynamics was assumed to be described by extremely strongly interactingmodes leading to almost ideal hydrodynamics. In both cases, the exact same initial conditions were implementedand the dynamics was required to correspond to the same equation of state. Also, in both cases the resultingenergy-momentum tensor information was recorded on the same space-time grid and then passed on a hadron cascade“afterburner” using the same switching procedure. d N / ( π d Y p T dp T ) [ G e V - ] p T [GeV]Particle Spectra: Hydro versus Non-Interacting Gasx 0.1x 0.01Pb+Pb √ s=2760 GeV30-40% ( π + + π - )/2 Free-Streaming( π + + π - )/2 Hydro η /s=0.08(K + +K - )/2 Free-Streaming(K + +K - )/2 Hydro η /s=0.08(p+pbar)/2 Free-Streaming(p+pbar)/2, Hydro η /s=0.08 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0 0.5 1 1.5 2 2.5 v p T [GeV]Elliptic Flow: Hydro versus Non-Interacting GasPb+Pb, √ s=2760 GeV,30-40%( π + + π - )/2 Free-Streaming( π + + π - )/2 Hydro η /s=0.08(K + +K - )/2 Free-Streaming(K + +K - )/2 Hydro η /s=0.08(p+pbar)/2 Free-Streaming(p+pbar)/2, Hydro η /s=0.08 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0 0.5 1 1.5 2 2.5 v p T [GeV]Triangular Flow: Hydro versus Non-Interacting GasPb+Pb, √ s=2760 GeV,30-40%( π + + π - )/2 Free-Streaming( π + + π - )/2 Hydro η /s=0.08(K + +K - )/2 Free-Streaming(K + +K - )/2 Hydro η /s=0.08(p+pbar)/2 Free-Streaming(p+pbar)/2, Hydro η /s=0.08 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0 0.5 1 1.5 2 2.5 v p T [GeV]Quadrupole Flow: Hydro versus Non-Interacting GasPb+Pb, √ s=2760 GeV,30-40%( π + + π - )/2 Free-Streaming( π + + π - )/2 Hydro η /s=0.08(K + +K - )/2 Free-Streaming(K + +K - )/2 Hydro η /s=0.08(p+pbar)/2 Free-Streaming(p+pbar)/2, Hydro η /s=0.08 FIG. 6. Simulations of granular Pb+Pb collisions at √ s = 2 .
76 TeV. Shown are final particle spectra and anisotropic flowcoefficients v n ( p T ) for identified particles for free-streaming evolution (no-interaction) and almost ideal hydrodynamics ( η/s =0 . through fitting a Gaussian form to the function S , S ( K , r ) ∝ e − x R − y R − z R (15)defining the femtoscopic radii R out , R side , R long . The results for these extracted radii for pions are shown in Fig. 7for d+Au , He+Au , p+Pb and Pb+Pb collisions, comparing hydrodynamic and non-interacting evolution. Fromthis figure, one can observe a striking similarity for all the extracted radii between strongly interacting evolution(hydrodynamics) and non-interacting evolution (free streaming) for all simulated systems, small and large. Similarlyto what was found for the case of radial flow, the femtoscopic radii are essentially insensitive to the details of thesystem evolution, as long as energy and momentum are conserved.In essence, this disqualifies the use of pion femtoscopic measurements as serving as evidence for a hydrodynamicphase during the system evolution.
IV. SUMMARY AND CONCLUSIONS
In this work flow signatures arising from two very different dynamics in the hot QCD phase following relativisticion collisions very studied. In the first case, the hot phase dynamics was assumed to be described by non-interactingparticles. In the second case, the hot phase dynamics was assumed to be described by extremely strongly interactingmodes leading to almost ideal hydrodynamics. In both cases, the exact same initial conditions were implementedand the dynamics was required to correspond to the same equation of state. Also, in both cases the resultingenergy-momentum tensor information was recorded on the same space-time grid and then passed on a hadron cascade“afterburner” using the same switching procedure. d N / ( π d Y p T dp T ) [ G e V - ] p T [GeV]Particle Spectra: Hydro versus Non-Interacting Gasx 0.1x 0.01Pb+Pb √ s=2760 GeV30-40% ( π + + π - )/2 Free-Streaming( π + + π - )/2 Hydro η /s=0.08(K + +K - )/2 Free-Streaming(K + +K - )/2 Hydro η /s=0.08(p+pbar)/2 Free-Streaming(p+pbar)/2, Hydro η /s=0.08 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0 0.5 1 1.5 2 2.5 v p T [GeV]Elliptic Flow: Hydro versus Non-Interacting GasPb+Pb, √ s=2760 GeV,30-40%( π + + π - )/2 Free-Streaming( π + + π - )/2 Hydro η /s=0.08(K + +K - )/2 Free-Streaming(K + +K - )/2 Hydro η /s=0.08(p+pbar)/2 Free-Streaming(p+pbar)/2, Hydro η /s=0.08 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0 0.5 1 1.5 2 2.5 v p T [GeV]Triangular Flow: Hydro versus Non-Interacting GasPb+Pb, √ s=2760 GeV,30-40%( π + + π - )/2 Free-Streaming( π + + π - )/2 Hydro η /s=0.08(K + +K - )/2 Free-Streaming(K + +K - )/2 Hydro η /s=0.08(p+pbar)/2 Free-Streaming(p+pbar)/2, Hydro η /s=0.08 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0 0.5 1 1.5 2 2.5 v p T [GeV]Quadrupole Flow: Hydro versus Non-Interacting GasPb+Pb, √ s=2760 GeV,30-40%( π + + π - )/2 Free-Streaming( π + + π - )/2 Hydro η /s=0.08(K + +K - )/2 Free-Streaming(K + +K - )/2 Hydro η /s=0.08(p+pbar)/2 Free-Streaming(p+pbar)/2, Hydro η /s=0.08 FIG. 6. Simulations of granular Pb+Pb collisions at √ s = 2 .
76 TeV. Shown are final particle spectra and anisotropic flowcoefficients v n ( p T ) for identified particles for free-streaming evolution (no-interaction) and almost ideal hydrodynamics ( η/s =0 . through fitting a Gaussian form to the function S , S ( K , r ) ∝ e − x R − y R − z R (15)defining the femtoscopic radii R out , R side , R long . The results for these extracted radii for pions are shown in Fig. 7for d+Au , He+Au , p+Pb and Pb+Pb collisions, comparing hydrodynamic and non-interacting evolution. Fromthis figure, one can observe a striking similarity for all the extracted radii between strongly interacting evolution(hydrodynamics) and non-interacting evolution (free streaming) for all simulated systems, small and large. Similarlyto what was found for the case of radial flow, the femtoscopic radii are essentially insensitive to the details of thesystem evolution, as long as energy and momentum are conserved.In essence, this disqualifies the use of pion femtoscopic measurements as serving as evidence for a hydrodynamicphase during the system evolution.
IV. SUMMARY AND CONCLUSIONS
In this work flow signatures arising from two very different dynamics in the hot QCD phase following relativisticion collisions very studied. In the first case, the hot phase dynamics was assumed to be described by non-interactingparticles. In the second case, the hot phase dynamics was assumed to be described by extremely strongly interactingmodes leading to almost ideal hydrodynamics. In both cases, the exact same initial conditions were implementedand the dynamics was required to correspond to the same equation of state. Also, in both cases the resultingenergy-momentum tensor information was recorded on the same space-time grid and then passed on a hadron cascade“afterburner” using the same switching procedure.
FIG. 26: Comparison of the flow harmonic, for hydrodynam-ics and free streaming for PbPb collisions, from [121]. v , except for protons. The v are comparable but bothtoo weak to be observed.So, flow harmonics seem not to be “faked” even forsmaller systems. Yet, taking into account remaining un-certainties of the initial stage models and thus (cid:15) n values,this conclusion is not as robust as for the AA. Perhapssome scenarios intermediate between equilibrated hydroand free streaming, with larger viscosity, may still bepossible in this cases. G. Shape fluctuations: central pA vs peripheral AA Scaling relation between central pA and peripheral AAhas been proposed and tested by Basar and Teaney [30]. d N / ( π d Y p T dp T ) [ G e V - ] p T [GeV]Particle Spectra: Hydro versus Non-Interacting Gasx 0.1x 0.01p+Pb √ s=5 TeV ( π + + π - )/2 Free-Streaming( π + + π - )/2 Hydro η /s=0.08(K + +K - )/2 Free-Streaming(K + +K - )/2 Hydro η /s=0.08(p+pbar)/2 Free-Streaming(p+pbar)/2, Hydro η /s=0.08 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0 0.5 1 1.5 2 2.5 v p T [GeV]Elliptic Flow: Hydro versus Non-Interacting Gasp+Pb, √ s=5 TeV( π + + π - )/2 Free-Streaming( π + + π - )/2 Hydro η /s=0.08(K + +K - )/2 Free-Streaming(K + +K - )/2 Hydro η /s=0.08(p+pbar)/2 Free-Streaming(p+pbar)/2, Hydro η /s=0.08 0 0.02 0.04 0.06 0.08 0.1 0 0.5 1 1.5 2 2.5 v p T [GeV]Triangular Flow: Hydro versus Non-Interacting Gasp+Pb, √ s=5 TeV( π + + π - )/2 Free-Streaming( π + + π - )/2 Hydro η /s=0.08(K + +K - )/2 Free-Streaming(K + +K - )/2 Hydro η /s=0.08(p+pbar)/2 Free-Streaming(p+pbar)/2, Hydro η /s=0.08 0 0.02 0.04 0.06 0.08 0.1 0 0.5 1 1.5 2 2.5 v p T [GeV]Quadrupole Flow: Hydro versus Non-Interacting Gasp+Pb, √ s=5 TeV( π + + π - )/2 Free-Streaming( π + + π - )/2 Hydro η /s=0.08(K + +K - )/2 Free-Streaming(K + +K - )/2 Hydro η /s=0.08(p+pbar)/2 Free-Streaming(p+pbar)/2, Hydro η /s=0.08 FIG. 3. Simulations of granular p+Pb collisions at √ s = 5 TeV. Shown are final particle spectra and anisotropic flow coefficients v n ( p T ) for identified particles for free-streaming evolution (no-interaction) and almost ideal hydrodynamics ( η/s = 0 . (see section II A for a detailed discussion). For each of these events, the subsequent dynamics is simulated using eithera hydrodynamic evolution or a free-streaming evolution, followed by the same hadron cascade for the low temperaturephase. Unlike the simplified case discussed in the section II E, the granular nature of each individual event gives riseto all anisotropic flow harmonics v n with n ≥
1, not just the elliptic flow v .The results for p+Pb collisions at √ s = 5 .
02 TeV are shown in Fig. 3. Considering the identified particle spectra, onefinds that the additional radial flow generated in the free-streaming dynamics compared to hydrodynamics is almostnegligible, and the resulting spectra are essentially indistinguishable. One reason for this may be the comparativelyshorter evolution time spent in the hot phase
T > T SW for p+Pb collisions compared to the case of smooth nucleus-nucleus collisions considered in Sec.II E.The comparison between free-streaming dynamics and hydrodynamics for the elliptic flow coefficient v are con-sistent with the findings for smooth nucleus-nucleus collisions considered above: the coupled free-streaming andhadron gas dynamics gives rise to a non-negligible amount of v , but it is considerably less than the v generated inhydrodynamics.Considering the higher flow harmonics v , v , the comparison between free-streaming and hydrodynamics revealsthat it becomes more difficult to distinguish between the two scenarios in terms of flow magnitude. For instance, the v found for free-streaming plus hadron cascade dynamics is very similar in magnitude to that for hydrodynamicsplus hadron cascade. Maybe even more interesting, the v amplitude for the free-streaming plus cascade simulationin p+Pb collisions turns out to be larger than the corresponding result from hydrodynamics with η/s = 0 .
08 (seeFig. 3).Overall one finds that free-streaming dynamics followed by hadron cascade dynamics generates approximately thesame magnitude of anisotropic flow for v , v and v , e.g. independent from the order of the harmonic. This is clearlyvery different from hydrodynamics, where successively higher orders are more strongly suppressed.The results obtained for the p+Pb collisions at √ s = 5 .
02 TeV should be compared to the results for d+Au and He+Au collisions at √ s = 200 GeV energies shown in Figs. 4,5. Overall, the same trends that were identified in d N / ( π d Y p T dp T ) [ G e V - ] p T [GeV]Particle Spectra: Hydro versus Non-Interacting Gasx 0.1x 0.01p+Pb √ s=5 TeV ( π + + π - )/2 Free-Streaming( π + + π - )/2 Hydro η /s=0.08(K + +K - )/2 Free-Streaming(K + +K - )/2 Hydro η /s=0.08(p+pbar)/2 Free-Streaming(p+pbar)/2, Hydro η /s=0.08 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0 0.5 1 1.5 2 2.5 v p T [GeV]Elliptic Flow: Hydro versus Non-Interacting Gasp+Pb, √ s=5 TeV( π + + π - )/2 Free-Streaming( π + + π - )/2 Hydro η /s=0.08(K + +K - )/2 Free-Streaming(K + +K - )/2 Hydro η /s=0.08(p+pbar)/2 Free-Streaming(p+pbar)/2, Hydro η /s=0.08 0 0.02 0.04 0.06 0.08 0.1 0 0.5 1 1.5 2 2.5 v p T [GeV]Triangular Flow: Hydro versus Non-Interacting Gasp+Pb, √ s=5 TeV( π + + π - )/2 Free-Streaming( π + + π - )/2 Hydro η /s=0.08(K + +K - )/2 Free-Streaming(K + +K - )/2 Hydro η /s=0.08(p+pbar)/2 Free-Streaming(p+pbar)/2, Hydro η /s=0.08 0 0.02 0.04 0.06 0.08 0.1 0 0.5 1 1.5 2 2.5 v p T [GeV]Quadrupole Flow: Hydro versus Non-Interacting Gasp+Pb, √ s=5 TeV( π + + π - )/2 Free-Streaming( π + + π - )/2 Hydro η /s=0.08(K + +K - )/2 Free-Streaming(K + +K - )/2 Hydro η /s=0.08(p+pbar)/2 Free-Streaming(p+pbar)/2, Hydro η /s=0.08 FIG. 3. Simulations of granular p+Pb collisions at √ s = 5 TeV. Shown are final particle spectra and anisotropic flow coefficients v n ( p T ) for identified particles for free-streaming evolution (no-interaction) and almost ideal hydrodynamics ( η/s = 0 . (see section II A for a detailed discussion). For each of these events, the subsequent dynamics is simulated using eithera hydrodynamic evolution or a free-streaming evolution, followed by the same hadron cascade for the low temperaturephase. Unlike the simplified case discussed in the section II E, the granular nature of each individual event gives riseto all anisotropic flow harmonics v n with n ≥
1, not just the elliptic flow v .The results for p+Pb collisions at √ s = 5 .
02 TeV are shown in Fig. 3. Considering the identified particle spectra, onefinds that the additional radial flow generated in the free-streaming dynamics compared to hydrodynamics is almostnegligible, and the resulting spectra are essentially indistinguishable. One reason for this may be the comparativelyshorter evolution time spent in the hot phase
T > T SW for p+Pb collisions compared to the case of smooth nucleus-nucleus collisions considered in Sec.II E.The comparison between free-streaming dynamics and hydrodynamics for the elliptic flow coefficient v are con-sistent with the findings for smooth nucleus-nucleus collisions considered above: the coupled free-streaming andhadron gas dynamics gives rise to a non-negligible amount of v , but it is considerably less than the v generated inhydrodynamics.Considering the higher flow harmonics v , v , the comparison between free-streaming and hydrodynamics revealsthat it becomes more difficult to distinguish between the two scenarios in terms of flow magnitude. For instance, the v found for free-streaming plus hadron cascade dynamics is very similar in magnitude to that for hydrodynamicsplus hadron cascade. Maybe even more interesting, the v amplitude for the free-streaming plus cascade simulationin p+Pb collisions turns out to be larger than the corresponding result from hydrodynamics with η/s = 0 .
08 (seeFig. 3).Overall one finds that free-streaming dynamics followed by hadron cascade dynamics generates approximately thesame magnitude of anisotropic flow for v , v and v , e.g. independent from the order of the harmonic. This is clearlyvery different from hydrodynamics, where successively higher orders are more strongly suppressed.The results obtained for the p+Pb collisions at √ s = 5 .
02 TeV should be compared to the results for d+Au and He+Au collisions at √ s = 200 GeV energies shown in Figs. 4,5. Overall, the same trends that were identified in d N / ( π d Y p T dp T ) [ G e V - ] p T [GeV]Particle Spectra: Hydro versus Non-Interacting Gasx 0.1x 0.01p+Pb √ s=5 TeV ( π + + π - )/2 Free-Streaming( π + + π - )/2 Hydro η /s=0.08(K + +K - )/2 Free-Streaming(K + +K - )/2 Hydro η /s=0.08(p+pbar)/2 Free-Streaming(p+pbar)/2, Hydro η /s=0.08 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0 0.5 1 1.5 2 2.5 v p T [GeV]Elliptic Flow: Hydro versus Non-Interacting Gasp+Pb, √ s=5 TeV( π + + π - )/2 Free-Streaming( π + + π - )/2 Hydro η /s=0.08(K + +K - )/2 Free-Streaming(K + +K - )/2 Hydro η /s=0.08(p+pbar)/2 Free-Streaming(p+pbar)/2, Hydro η /s=0.08 0 0.02 0.04 0.06 0.08 0.1 0 0.5 1 1.5 2 2.5 v p T [GeV]Triangular Flow: Hydro versus Non-Interacting Gasp+Pb, √ s=5 TeV( π + + π - )/2 Free-Streaming( π + + π - )/2 Hydro η /s=0.08(K + +K - )/2 Free-Streaming(K + +K - )/2 Hydro η /s=0.08(p+pbar)/2 Free-Streaming(p+pbar)/2, Hydro η /s=0.08 0 0.02 0.04 0.06 0.08 0.1 0 0.5 1 1.5 2 2.5 v p T [GeV]Quadrupole Flow: Hydro versus Non-Interacting Gasp+Pb, √ s=5 TeV( π + + π - )/2 Free-Streaming( π + + π - )/2 Hydro η /s=0.08(K + +K - )/2 Free-Streaming(K + +K - )/2 Hydro η /s=0.08(p+pbar)/2 Free-Streaming(p+pbar)/2, Hydro η /s=0.08 FIG. 3. Simulations of granular p+Pb collisions at √ s = 5 TeV. Shown are final particle spectra and anisotropic flow coefficients v n ( p T ) for identified particles for free-streaming evolution (no-interaction) and almost ideal hydrodynamics ( η/s = 0 . (see section II A for a detailed discussion). For each of these events, the subsequent dynamics is simulated using eithera hydrodynamic evolution or a free-streaming evolution, followed by the same hadron cascade for the low temperaturephase. Unlike the simplified case discussed in the section II E, the granular nature of each individual event gives riseto all anisotropic flow harmonics v n with n ≥
1, not just the elliptic flow v .The results for p+Pb collisions at √ s = 5 .
02 TeV are shown in Fig. 3. Considering the identified particle spectra, onefinds that the additional radial flow generated in the free-streaming dynamics compared to hydrodynamics is almostnegligible, and the resulting spectra are essentially indistinguishable. One reason for this may be the comparativelyshorter evolution time spent in the hot phase
T > T SW for p+Pb collisions compared to the case of smooth nucleus-nucleus collisions considered in Sec.II E.The comparison between free-streaming dynamics and hydrodynamics for the elliptic flow coefficient v are con-sistent with the findings for smooth nucleus-nucleus collisions considered above: the coupled free-streaming andhadron gas dynamics gives rise to a non-negligible amount of v , but it is considerably less than the v generated inhydrodynamics.Considering the higher flow harmonics v , v , the comparison between free-streaming and hydrodynamics revealsthat it becomes more difficult to distinguish between the two scenarios in terms of flow magnitude. For instance, the v found for free-streaming plus hadron cascade dynamics is very similar in magnitude to that for hydrodynamicsplus hadron cascade. Maybe even more interesting, the v amplitude for the free-streaming plus cascade simulationin p+Pb collisions turns out to be larger than the corresponding result from hydrodynamics with η/s = 0 .
08 (seeFig. 3).Overall one finds that free-streaming dynamics followed by hadron cascade dynamics generates approximately thesame magnitude of anisotropic flow for v , v and v , e.g. independent from the order of the harmonic. This is clearlyvery different from hydrodynamics, where successively higher orders are more strongly suppressed.The results obtained for the p+Pb collisions at √ s = 5 .
02 TeV should be compared to the results for d+Au and He+Au collisions at √ s = 200 GeV energies shown in Figs. 4,5. Overall, the same trends that were identified in FIG. 27: Comparison of the flow harmonic, for hydrodynam-ics and free streaming in the pPb central bin , from [121].
Step one of their paper has been prompted by the fact(noticed in the CMS paper already): at the same mul-tiplicity, v in central pA and peripheral AA are basi-cally the same. Some people called for new paradigmsbased on this fact: but in fact it is hardly surprising:equal multiplicity means equal number of participant nu-cleons, and thus equal fluctuations of the shape. AfterBasar and Teaney removed the geometrical contributionto v in peripheral AA, they found that the remaining –fluctuation-driven – part of elliptic flow is also the samein both cases, see Fig.24.Their second proposal is that the p t dependence of (thefluctuating part) of the v n has an universal shape, andAA and pA data are only different by a scale of mean p t v pAn ( p t ) = v pAn ( p t κ ) (36)where the scaling factor is defined as κ = < p T > pP b < p T > P bP b ≈ .
25 (37)and is due to difference in the radial flow. This relationalso works well.two possible effects, as the multiplicity grows:(i) an increases the initial temperature T i . Since the finalone is fixed by hadronization near the phase transition T f ≈ T c , the contrast between them gets larger and hydroflow increases;(ii) the initial size of the fireball decreases , increasingthe initial temperature T i even further. VI. EQUILIBRATION IN QCD-BASED MODELSA. CGC and turbulent GLASMA
The idea of continuity, from a state before collisionto early time after it, is most directly realized in the socalled CGC-GLASMA approach. Technically it is basedon McLerran-Venugopalan argument [78] that high den-sity of partons leads to large color charge fluctuations,which should create strong color fields. If fields are strongenough, then classical Yang-Mills equations are sufficient,and those can be solved numerically. It is important thatat this stage the fields get strong, the occupancy of glu-ons n g ∼ /α s (cid:29) n g ∼ Q s ( x ), separating it from the dilute or par-tonic phase. It is expected to grow with collision energy(smaller x ) and higher atomic number A .7A good news from the plot is that at the highest LHCenergies and A Q s ∼ GeV , perhaps in the perturba-tive domain. I however would not accept that the “con-fining regime” at the bottom of the figure is delegatedto very small Q < . GeV . In fact the boundary ofpQCD is a factor 30 or more higher: there are no gluonswith virtuality below 1 GeV! Nambu-Jona-Lasinio noted(already in 1961!) that some strong non-perturbativeforces turns on at Q < GeV , creating chiral symmetrybreaking. By now we have plenty of evidences from thelattice simulations that all correlation functions of thegauge fields do not behave perturbatively at distanceslarger than 0.2 fm or so. Heavy quark potential is alsonor Coulombic but string-like, and the string width isonly .2 fm or so.Theoretical study of parton equilibration in weak cou-pling domain has a long history. The so called “bottomup” approach by Baier, Mueller, Schiff and Son [79] wasbased on soft gluons radiated by scattered hard partons.The name means that thermal occupation starts fromthe IR end. (Note that it is opposite to the “top-down”equilibration in holographic models we will discuss be-low.) The main prediction of that model was the equili-bration time and the initial temperature scaling with thecoupling τ eq ∼ / ( α / s Q s ) , T i ∼ α / s Q s (38)Some details were changed as later one tried to incor-porate Weibel, Nielsen-Olesen and any other instabili-ties which may occur into the model. The validity do-main of this theory is restricted by its core assumptionof small angle scattering of the gluons, justified by largeimpact parameters of the order of inverse (perturbative)Debye mass. Perturbative means M D ≈ gT (cid:28) T orsmall g (cid:28)
1. If so, g = 1 , α s = 1 / π and one findsvery long equilibration time τ eq Q s ∼
700 exceeding du-
FIG. 28: The CGC phase diagram: the saturation momen-tum Q s as a function of fraction of momentum x and theatomic number A , from [82]. ration of the collisions. (Not surprisingly, the pre-RHICperturbative predictions have been that no QGP will beproduced, just some “fireworks of mini-jets”, with smallrescattering corrections.) One may take “more realistic” α s = 1 / , / α s but evenits log as a large parameter log (1 /α s ) (cid:29)
1, allowing thetotal GLASMA evolution scale τ GLASMA ∼ log (1 /α s ) Q s (39)to be considered large.Significant progress in this directions has been incorpo-ration of certain ideas from general theory of turbulentcascades. Not going into its long history, let me justmention Kolmogorov-Zakharov stationary power-like so-lutions for a vast number of systems with nonlinear in-teraction of waves for Boltzmann equation, and time-dependent self-similar solutions. (For sound cascades inearly universe some of that we already discussed in sec-tion III I.) J.Berges and collaborators developed it forscalar and gauge fields, pointing out different regimesfor UV and IR-directed cascades, and identifying suchregimes in impressive numerical simulations.New paradigm, resulting from this body of work, isthat the pre-equilibrated plasma may be captured by a nontrivial turbulent attractor – certain self-similar powersolution, before progressing toward thermal equilibrium.let me comment on one aspect of the result, on the mo-mentum anisotropy p l /p ⊥ . Epelbaum and Gelis [80] per-formed a next-order GLASMA simulation, with g = 0 . p l /p ⊥ keep approximately constant value duringthe whole time of the simulation τ Q s = 10 −
40. Berges,Schenke, Schlichting, Venugopalan [81] found that, at g = 0 .
3, the longitudinal pressure p l /(cid:15) remains close tozero at similar times. These results are shown in Fig.29.Recently Kurkela and Zhu [83] performed cascade sim-ulations at larger couplings. In Fig.30 from this paperone can see two sets of trajectories, shown by solid anddashed lines, starting from two different initial distribu-tions. At zero coupling (upper left curve) the longitudinalmomenta of particles gets very small compared to trans-verse one, and anisotropy steadily increases. This is ascaling-like classical regime with a nontrivial fixed point.However all other paths stay more or less at the sameinitial anisotropy, and then rapidly turn downward, to lo-cally isotropic distributions (marked by diagonal crossesat the bottom).It is a pity there are no simulations done with λ be-tween 0 and 0.5: perhaps at some critical coupling abifurcation of the trajectories happens, separating those8 N UMERICAL RESULTS [TE,G
ELIS (2013)] ↵ s = - ( g = . ) -101/31/2+10.1 1.0 10.01Q s (cid:1)(cid:1) [fm/c]0.01 0.1 10.0 20.0 30.0 40.02 3 4 P T / (cid:2) P L / (cid:2) LO -0.0500.050.1 0 0.5 1 1.5 2 2.5 3 g τ P L / ( g µ ) , g τ P T / ( g µ ) g µ τ τ P L τ P T T HOMAS E PELBAUM
EarlyIsotropizationoftheQuarkGluonPlasma
FIG. 29: (a) Upper green curve is p t /(cid:15) , lower red is p l /(cid:15) , as afunction of time in units of saturation scale, τ Q s , at g = 0 . p t /(cid:15) , lower red is p l /(cid:15) , at g = 0 . who proceed toward the new and the equilibrium fixedpoints. Yet the issue is rather academic, since the realis-tic relevant coupling value α s = g / π ≈ . λ = α s N c π ≈
10, whichis the largest value shown in this figure (bottom right).The corresponding curve rapidly approaches the equilib-rium point, with coordinates (1,1) at this figure.
B. From Glasma to hydro
Summarizing the previous subsection in few words,one may say that weak coupling cascades predict highlyanisotropic pressure p l (cid:28) p t . The question is whether itpersists for the time of few fm/c, or is rapidly reduced tothe values prescribed by viscous hydrodynamics. The lat-ter scenario is predicted by strong coupling approaches,to be discussed later. We do not know when exactly atransition between these regimes happen. This questionis central for our field, and one should not jump to con-clusions without serious evidences,one way or the other.(Example of a premature statement:“No need forstrong coupling to get hydrodynamization” is conclusionof Epelbaum’s QM14 talk. Hydrodynamics is not justsome instantaneous value of the stress tensor, but theequations to follow. The value of viscosity (mean freepath) is central to their validity: and simulations doneat g = 0 . , α s = 0 .
02 correspond to viscosity hundredstimes larger than the actual values of sQGP extracted λ f>/
110100100010000 A n i s o t r opy : P T / P L λ =0 λ =1.0 λ =5.0 λ =10~x -1/2 ξ =10 ξ =4 λ =0.5 FIG. 1: Trajectories of runs with different initial conditions ξ = 4 (Solid lines) and ξ = 10 (dahsed lines) and varyingcoupling λ in a plane of mean occupancy (weighted by theenergy of particles) and anisotropy. The λ = 0 line corre-sponds to classical field approximation. The violet dots referto the times in Fig. 2. The simulations at finite coupling reachthermal equilibrium located at points indicated by the blackcrosses. grid points in angular direction and at least 100 in the p -direction. We have varied the number of grid pointsto verify that the results are insensitive to the number ofgrid points. To significantly reduce variance of the MonteCarlo estimation, we use importance sampling dq/q forthe q integral in Eq. (3). Our algorithm, the discrete- p method of [13], conserves energy exactly and has exactlycorrect particle number violation for 1 ↔ RESULTS
We shall now apply EKT to simulate the prethermalevolution of the expanding fireball created in a heavy-ion collision. In saturation framework, the initial condi-tion is typically described in terms of “gluon liberationcoefficient” c and mean transverse momentum ⟨ p T ⟩ /Q s [26, 27]. The gluon liberation coefficient is proportionalto the total gluon multiplicity per unit rapidity dN init.g d x ⊥ dy = c d A Q s πλ = 2 d A τ ! d p (2 π ) f (11)after the classical fields have decohered and can be de-scribed in terms of quasi-particles. Q s is the (adjointrepresentation) saturation scale. Lappi [28] finds inJIMWLK evolved MV model values relevant for heavy-ion collisions relevant for LHC of roughly ⟨ p T ⟩ ≈ . Q s and c ≈ .
25 extracted at time Q s τ = 12 from a 2Dclassical Yang-Mills simulation. By construction the dis-tribution then has ⟨ p z ⟩ = 0. But it is has been noted [25]that certain plasma instabilities will broaden the distri-bution in p z in a time scale Qτ ∼ / log ( λ − ). There-fore, as a rough estimate of the initial condition we in-stead take somewhat arbitrarily our initial condition at the time Qτ = 1 to be f ( p z , p t ) = 2 λ Af ( p z ξ/ ⟨ p T ⟩ , p ⊥ / ⟨ p T ⟩ ) , (12) f (ˆ p z , ˆ p ⊥ ) = 1 " ˆ p ⊥ + ˆ p z e − p ⊥ +ˆ p z ) / , (13)choosing A such that comoving energy density τϵ = ⟨ p T ⟩ dN/d x dy is fixed. We then vary ξ = 4 ,
10 to quan-tify our ignorance of the initial nonperturbative dynam-ics.Figure 1 displays a set of trajectories from simula-tions with varying λ and ξ = 4 ,
10 on a plane of meanoccupancy (weighted by the energy of particles) andanisotropy measured by the ratio of the transverse andlongitudinal pressures P T /P L . The line with λ = 0 cor-responds to the classical field limit λ → λf ,which is obtained in EKT by including only the highestpower of f ’s in Eqs.(2,9). Thermalization is an inherentlyquantum phenomenon and classical field theory can notthermalize due to the Rayleigh-Jeans catastrophe. In-deed, instead of thermalizing, classical field theory flowsto a stationary scaling solution. By performing classi-cal Yang-Mills simulations Berges et. al have establishedthat the scaling solution can described by a scaling formof the distribution function [4], f ( p z , p ⊥ , τ ) = ( Q s τ ) − / f S (( Q s τ ) / p z , p ⊥ ) , (14)where f S is approximately constant as a function oftime. In Fig. 1 this scaling is indicated by the dashedline P T /P L ∝ ( ⟨ pλf ⟩ / ⟨ p ⟩ ) − / . This behavior is demon-strated in more detail in Figure 2 where we plot a sectionof rescaled distribution function f S measured at varioustimes as a function for ˜ p z ≡ ( Q s τ ) / p z at fixed p ⊥ fol-lowing Berges et al.. Our results corroborate that sucha scaling solution exists at late times within the classicalapproximation and we observe that the scaling regime isreached after a time Q s τ ∼ λ = 1 , . f ∼
1, thereis a qualitative change in the dynamics of the system asBose enhancement is lost. This has the effect that an-siotropy freezes but the system still continues to get moredilute. Only in the last stage which is characterized bya radiational break up of the particles at the scale Q s ,does the trajectory turn back and reach thermal equi-librium, denoted by the black crosses in the Figure 1.For larger values of coupling λ = 5 . ,
10, these featuresbecome however less pronounced and the system takesmore straight trajectory towards equilibrium.
FIG. 30: From [83]: Trajectories of the systems on theoccupancy-anisotropy plane for various settings. The param-eter λ near curves show the corresponding ’t Hooft couplingconstant λ = g N c . All solid line originate from one initialdistribution characterized by the anisotropy parameter ξ = 4,the dashed lines originate from a different point with ξ = 10.. from data. Anyone who doubt that should just try toderive harmonic flows from such cascades, without hy-dro.)Surprisingly, there are little discussion of how to testfor persistence of such anisotropy in experiment, so let mestart with some comments on that. First, one can indeedmeasure an early-time anisotropy, via dilepton polariza-tion, see section IX C. Second, one can study the effectof the longitudinal pressure on the rapidity distribution.Historically, the original papers of Landau focused on thelongitudinal expansion, driven by gradient of p l . HoweverLandau’s initial condition – the instantaneous stopping –looks rather unrealistic today. Since all hydro modelingstarts after certain time – when partons already passedeach other – it needs certain initial distribution. Its widthin rapidity, appended by the calculated longitudinal hy-dro effect, was compared with the data. Uncertainties,in particular related with removal of the fragmentationregions near beam rarities, makes an accuracy of p l ex-traction poor.Since 1990’s hydro effects focused on the transversemotion, the radial and elliptic flows. Since sQGP is be-lieved to be near-conformal, the trace of stress tensor re-mains zero even out of equilibrium: so even for vanishing p l = 0 one finds p t = (cid:15)/ (cid:15)/ τ ∼ f m or so, thisdifference would already be seen in the magnitude of theelliptic flow. This issue deserves however dedicated stud-ies.Let us look what practitioners actually do while com-bining glasma and hydrodynamics. Fig.31 illustrate whatis done in the so called impact parameter (IP) glasma ap-proach. An important feature of glasma is independentfluctuations of color in different cells, which seeds theharmonic flows. At certain proper time – 0.2 fm/c in9 FIG. 31: From [23]: Transverse energy profile from the IPglasma model for a semiperipheral (b = 8 fm) Au+Au collisionat s = 200 A GeV, at times τ = 0.01, 0.2, and 5.2 fm/c. From τ = 0.01 fm/c to 0.2 fm/c the fireball evolves out of equilibriumaccording to the Glasma model. this example – glasma evolution is stopped and then theenergy momentum tensor is matched to ideal fluid. Fortechnical reasons the value of the viscous tensor is put tozero.How important is the time 0.2 fm/c selected? Notefirst that the second picture is hardly different from thefirst, except the overall scale of the energy density is re-duced. Indeed, even moving with a speed of light onecan only shift by 0.2 fm, a very small distance relative tothe nuclear size. Glasma is diluted by purely longitudinalstretching: thus transition time can be shifted up. Manyother practitioner start hydro at 0.6 fm/c or so.By starting hydro right from the second pictureSchenke and collaborators implicitly assume that hydrocells can indeed be as small as 0.2 fm, and that their codecan cope with huge gradients between the cells. v inp+Pb collisions underestimates the data by a factor ofapproximately 3.5. We have checked that even in theideal case ( η/s = 0) the data is still underestimated byapproximately a factor of 2. We also varied the freeze-out temperature and switching time τ , but no choiceof parameters could achieve much better agreement withthe experimental data. For v , shown in Fig. 5, we finda similar result: Pb+Pb data are well described, whilep+Pb data are underestimated for N offlinetrk >
60. Idealfluid dynamics (not shown) increases the v significantlyby nearly a factor of 4. Its N offlinetrk dependence is ratherflat, slightly decreasing with increasing N offlinetrk , oppositeto the trend seen in the experimental data. 〈 v 〉 / N trkoffline η /s=0.18 CMS Pb+Pb v CMS p+Pb v Pb+Pb v p+Pb v FIG. 4. (Color online) Multiplicity dependence of the root-mean-square elliptic flow coefficient v in Pb+Pb (open sym-bols) and p+Pb collisions (filled symbols) from the IP-Glasma+ music model (connected triangles) compared to ex-perimental data by the CMS collaboration [35]. 〈 v 〉 / N trkoffline η /s=0.18 CMS Pb+Pb v CMS p+Pb v Pb+Pb v p+Pb v FIG. 5. (Color online) Multiplicity dependence of the root-mean-square triangular flow coefficient v in Pb+Pb (opensymbols) and p+Pb collisions (filled symbols) from the IP-Glasma+ music model (connected triangles) compared to ex-perimental data by the CMS collaboration [35]. The primary reason for the small v n in p+Pb collisionsis that the initial shape of the system closely follows theshape of the proton (see [34]), which is spherical in ourmodel. The subnucleonic fluctuations included generatenon-zero values of the v n , but they do not fully accountfor the larger experimentally observed values. As notedabove, modifications of the (fluctuating) proton shapeare necessary to account for the larger observed v and v in p+Pb collisions. If the hydrodynamic paradigmis valid, the results of the high-multiplicity p+Pb andp+p collisions could then in principle be used to extractdetailed information on the spatial gluon distribution inthe proton.There are hydrodynamical models that describe as-pects of the p+Pb data. These models should also de-scribe key features of Pb+Pb collisions where hydrody-namics is more robust. A model where the spatial geom-etry of p+Pb collisions is different from ours is that of[13–17], where the interaction region is determined fromthe geometric positions of participant nucleons. How-ever, as noted, this model falls into the class of modelsthat are claimed [26] not to be able to reproduce thedata on event-by-event v n distributions in A+A colli-sions. Whether this particular model can do so needsto be examined. We also note that the v centrality de-pendence in the model differs from the CMS data forp+Pb collisions [16].Another model which claims large v and v in p+Pbcollisions determines the system size from the position of“cut pomerons” and strings [18, 36]. The multiplicitydependence of the v n in this model has not yet beenshown. The v n distributions in A+A collisions shouldalso provide a stringent test of this model.In addition to the important quantitative tests im-posed on different hydrodynamical models by the exper-imental data, there are conceptual issues that arise dueto the possible breakdown of the hydrodynamic paradigmwhen extended to very small systems. As shown in re-cent quantitative studies, viscous corrections can be verysignificant in p+Pb collisions but play a much smallerrole in Pb+Pb collisions [34, 37]. In particular, an anal-ysis of Knudsen numbers reached during the evolution inA+A and p+A collisions finds that viscous hydrodynam-ics breaks down for η/s ≥ .
08 in p+A collisions [37].An alternative to the hydrodynamic picture and itssensitivity to the proton shape is provided within theGlasma framework itself by initial state correlations ofgluons that show a distinct elliptic modulation in relativeazimuthal angle [21–25]. If these are not overwhelmedin p+Pb collisions by final state effects, as they are inA+A collisions, they can contribute significantly to theobserved v , and possibly v . The initial state correla-tions are those of gluons and do not address features ofthe data such as the mass ordering in particle spectra.While natural in hydrodynamical models, mass orderingmay also emerge due to universal hadronization effects, FIG. 32: (Color online) Multiplicity dependence of the root-mean-square elliptic flow coefficient v2 in Pb+Pb (opensymbols) and p+Pb collisions (filled symbols) from the IP-Glasma+music model (connected triangles) [23] compared toexperimental data by the CMS collaboration. (Note that the bottom figure, at time 5.2 fm/c, looksvery different. The original bumps disappeared and in-stead a new one is formed. The reason is sound refuses tostand still and is moving with a speed of sound. At inter-sections of “sound circles” from the primary bumps ran-dom enhancements of the density are observed. Yet sincethe bumps are statistically unclorrelated, those shouldget averaged out, at least in 2-particle correlations, andonly correlations from the same circle will stay.)How many harmonics are needed to describe pictureslike that shown in Fig.31 ? Taking 0.2 fm as a resolu-tion and 4 fm as a size, one finds that the upper picturerequires about 20*20=400 pixels to be represented by cer-tain stress tensor components. At the freezeout there areonly several angular harmonics observed, so 99% of theinformation inclosed in those pictures does not survivetill the freezeout. In the hydro simulation just describedthose disappear predictably, via viscous damping. It ispossible that this systematics will fail at shorter wave-length: so it is worth trying to measure higher harmon-ics which perhaps deviate from it. Other ways to observedensity waves can perhaps be invented: one of such ispotential observation of those in the dilepton via the socalled MSL process.
C. The initial state and angular correlations
Importance of the initial state information can bedemonstrated further using “small systems”, such as cen-tral pA collisions. When a nucleon is going along the di-ameter of heavy nucleus the mean number of participantnucleons is < N p > = n σ NN R A (40)0so for pPb at LHC one gets < N p > ≈
16. The questionhowever remains, where exactly the deposited energy is?In Fig.34 we sketched two opposite models of the initialstate. In (a) we show each of the N p participants repre-sented by N g gluons (ignoring sea quarks and antiquarks)from their PDFs each, so the total number of partons N p N g . We assume all gluons uncorrelated in the spotof the size of pp cross section. In (b) we had shown analternative picture, the stringy Pomeron, in which thereare no gluons but 2 N p QCD strings instead. Since thoseare “cold” (unexcited) we show those by straight lines.Note that the picture a bit exaggerates the ratio of twoparameters involved. Those are (i) the mean impact pa-rameter between the nucleons b ∼ (cid:112) σ NN /π ≈ . f m and (ii) the size of the quark-diquark dipole d ∼ . f m ,so b (cid:29) d .Let us estimate the deformation of the initial state ineach case. Since it is central collision, there is no meangeometrical effects and all deformations comes from fluc-tuations. As discussed above, for all n one expects thesame magnitude (cid:15) n ∼ √ N (41)where N = N p N g for (a) and only N = N p for (b).Evaluating N g from PDF’s at LHC energy includes in-tegration from x min ∼ − to 1: one gets roughly theratio (cid:15) ( b ) n (cid:15) ( a ) n ∼ (cid:112) N g ∼ v , v flows in (very peripheral) AA andcentral pA. They found that the IP-glasma model theydeveloped does a very good job for the former and un-derpredicts them in the latter case, see Fig. 32.As we already discussed above, in the peripheral AA (cid:15) is large, O(1), in any model, and in order to get the right Figure 7 presents the charged particle pseudorapidity density for p + Pb collisions at p s NN = .
02 TeVin the pseudorapidity interval | ⌘ | < . η -3 -2 -1 0 1 2 3 η / d c h d N ATLAS
Preliminary -1 b µ = 1 int p+Pb L = 5.02 TeV NN s = -0.465 cm y Figure 7: dN ch / d ⌘ measured in di ↵ erent centrality intervals. Statistical uncertainties, shown with verticalbars are typically smaller than the marker size, colour band shows the systematic uncertainty of theresults.(centrality interval 60-90%) dN ch / d ⌘ has what appears to be a double-peak structure, similar to that seenin proton-proton collisions [35, 49]. In more central collisions, the shape of dN ch / d ⌘ becomes progres-sively more asymmetric, with more particles produced in the Pb-going direction than in the proton-goingdirection.To investigate further the centrality evolution, the distributions in the various centrality intervals aredivided by the distribution in the 60-90% centrality interval. The ratios are shown in Fig. 8. The doublepeak structure seen in the distributions in Fig. 7 disappears in the ratios. The ratios are observed to grownearly linearly with pseudorapidity, with a slope that increases from peripheral to central collisions. In the0-1% centrality interval, the ratio increases by almost a factor of two over the measured ⌘ -range. Theseratios are fit with a second-order polynomial function, and the fit results are summarized in Table 2.Figure 9 shows the dN ch / d ⌘ divided by the number of participant pairs ( h N part i /
2) as a functionof h N part i for three di ↵ erent implementations of the Glauber model; standard Glauber (top panel) andGlauber-Gribov model with ⌦ = .
55 and 1.01 in the middle and lower panels respectively. Since thecharged particle yields have significant pseudorapidity dependence, the dN ch / d ⌘/ ( h N part i /
2) is presentedin five ⌘ intervals including the full pseudorapidity interval, . < ⌘ < . dN ch / d ⌘/ ( h N part i /
2) values from the standard Glauber model are approximately constant up to h N part i ⇡
10 and then increase for larger h N part i . This trend is absent in the Glauber-Gribov model with ⌦ = .
55, which shows a relatively constant behaviour for the integrated yield divided by the number ofparticipant pairs. Finally, the dN ch / d ⌘/ ( h N part i /
2) values from the Glauber-Gribov model with ⌦ = . FIG. 33: Rapidity distribution in pPb collisions for differentcentrality classes, from [62] . FIG. 34: Sketch of the initial state in central pA collisions.The plot (a) corresponds to IP-glasma model, with coloredcircles representing multiple gluons. Fig.(b) is for N p = 16Pomerons, each represented by a pair of cold strings. Theopen circles are quarks and filled blue circles are diquarks. v one has to have correct viscosity – which apparentlythese authors have. The central pA is indeed the testcase: we argued above that the density is not yet largeenough to apply the IP-glasma model, and that stringyPomerons should be more applicable one. If so, using (42)we should increase the v by a factor of 4, which brings itto an agreement with the CMS measurements. We thusconclude that the stringy model Fig.34(b) is preferableover the picture (a), the uncorrelated gluons.(The actual IP glasma model is not the uncorrelatedgluons: it should be uncorrelated glasma cells of 1 /Q s size: so our model (a) is just a straw man holding placebefore more consistent calculation is done.)Above we simplistically assumed a Glauber picture inwhich each wounded nucleon (or a participant) interactwith the projectile proton by the usual single Pomeron.This means one color exchange with each participant,who therefore is connected to the projectile by (at least) two strings.If this picture be correct and the strings were simplyindependently fragmenting, the rapidity distributions ofsecondaries would be flat (rapidity independent) for allcentrality classes. This is not the case in reality, as isseen in ATLAS data shown in Fig.33. As one can see,the peripheral bins have flat rapidity distribution: thiscorresponds to few strings extending from one fragmen-tation region to the other one without a change. Yetcentral bins have rather asymmetric distributions, withlarger multiplicity at the Pb side.In the Pomeron language it is explained by the so called“fan diagrams” in which one Pomeron can split into two.The “triple Pomeron vertex” is however small and we donot have developed “Pomeron cascades”. The multiplic-ity difference between the r.h.s. an the l.h.s. of the plot isnot too dramatic, certainly not factor N p ∼
20 as the oldwounded nucleon model would suggest. For example, forthe most commonly used centrality bin 1-5% the rapiditydensity dn ch /dη changes from about 35 to 55, across therapidity interval shown in this figure. If on the P b endthere are say N s > N p ≈
40 strings, then on the p endthere are not one or few, but still 20 strings or so. Since1the area on the l.h.s. is reduced by an order of magnitudeor so, and the number only by factor 2, it is by far moredense system than the r.h.s.!If one thinks of that, one may conclude that flows anddevelopment of collectivity should strongly depend onrapidity. Yet, at least in some crude sense, this is notthe case: for example, the famous v “ridge” is ratherflat in rapidity. So, the issue needs more attention.Finally, let me mention high multiplicity pp . We donot yet know (cid:15) n in this case. Experiment should do 4,6particle correlators and separate non-collective 2-particlecorrelations from collective ones. Hydro practitionersstill have to do v n /(cid:15) n and establish its viscosity depen-dence and accuracy. Theoretical predictions for pp coverthe whole range: from elongated transverse string [63]predicting large (cid:15) ∼ (cid:15) instead. D. Multi-string state: spaghetti
A version of the initial state theory, alternative toglasma picture, is old Lund model, used in event gen-erators like PYTHIA. It is suppose to be applicable forlower matter density, remaining in the confined phase.Multiple color charges, moving relativistically from eachother after collision, are in such case connected by multi-ple QCD strings. As those are rapidly stretched longitu-dinally, strings become nearly parallel to each other, andwe will call this state “spaghetti” for short.Transition from such a spaghetti state to GLASMAcan be relatively smooth, since in both pictures the colorfields have similar longitudinal structure. Florkowski ata talk at QM14 [55] discussed dynamics of spaghetti-likeGLASMA state. Interesting oscillations of such systemis predicted, see Fig.35.Transition between two picture is naturally expectedwhen the diluteness of the QCD strings become of the
FIG. 35: Oscillation of the energy density in simulations start-ing from “glasma”-like initial conditions. k = 5 is the numberof fluxes through the flux tubes, from [55] . FIG. 36: (Color online). (a) Static quark-antiquark pair areindicated by shaded circles: those are connected by the fluxtube (QCD string). At distance r from the tube one place theoperator ¯ qq , whose average is measured, with and without thestatic quark-antiquark pair. (b) Normalized chiral condensateas a function of the coordinate r transverse to the QCD string(in lattice units). Points are from the lattice calculation [39].The curve is the expression (44) with C = 0 . s string =0 .
176 fm. order 1, so they can no longer be separated N string Area ∼ πr string ∼ f m − (43)where in the numerical value we use the field radius inthe string r s ≈ . f m ∼ /GeV .The interaction between the QCD strings in a - - - - - FIG. 37: (Color online) Instantaneous collective potential inunits 2 g N σ T for an AA configuration with b = 11 fm, g N σ T =0 . N s = 50 at the moment of time τ = 1 fm /c . Whiteregions correspond to the chirally restored phase. σ . Indeed, the theoretical expression (cid:104) ¯ qq ( r ⊥ ) W (cid:105)(cid:104) W (cid:105)(cid:104) ¯ qq (cid:105) = 1 − CK ( m σ ˜ r ⊥ ) , (44)(where K is the modified Bessel function and the “reg-ularized” transverse distance ˜ r ⊥ is˜ r ⊥ = (cid:113) r ⊥ + s string , (45)which smoothens the 2D Coulomb singularity ∼ ln( r ⊥ )at small r ⊥ ) described lattice data well, see Fig. 36. Thesigma mass here was taken to be m σ = 600 MeV as aninput, and not fitted/modified). According to our fit,the “intrinsic” width is s string (cid:39) .
176 fm, confirmed byother lattice results.Since the strings in a spaghetti are almost parallel toeach other, the problem is reduced to the set of point par-ticles on a plane with the 2D Yukawa interaction. Fromthe fit (44) one can see [73], that the main parameter ofthe string-string interaction (in string tension units) isnumerically small, g N σ T = (cid:104) σ (cid:105) C σ T (cid:28) , (46)typically in the range 10 − − − . So, it is correctlyneglected in the situations, for which the Lund modelhas been originally invented – when only O (1) stringsare created.The interaction starts playing a role when this small-ness can be compensated by a large number of strings.As seen from Fig. 36, a magnitude of the quark conden-sate σ = |(cid:104) ¯ qq (cid:105)| at the string position is suppressed byabout 20% of its vacuum value. So, in a “spaghetti”state one should think of the quark condensate suppres-sion of about 0.2 times the diluteness, which is still lessthan 1.On the other hand, about 5 overlapping strings wouldbe enough to eliminate the condensate and restore thechiral symmetry. If N s >
30 strings implode into anarea several times smaller than σ in (which is the caseas we will argue below), then the chiral condensate willbe eliminated inside a larger region of 1 fm in radius, orabout 3 fm in area. This is nothing but a hot QGPfireball.As discussed above, the strings can be viewed as a 2Dgas of particles (in transverse plane) with unit masses atpositions (cid:126)r i . The forces between them are given by thederivative of the energy (44), and so¨ (cid:126)r i = (cid:126)f ij = (cid:126)r ij ˜ r ij ( g N σ T ) m σ K ( m σ ˜ r ij ) (47)with (cid:126)r ij = (cid:126)r j − (cid:126)r i and “regularized” ˜ r (45).In our simulations we used a classical molecular dy-namics code. The evolution consists of two qualitatively What physics is captured by the AdS/CFT description?Consider the excitations of an infinite uniform static plasma system I Small disturbances of the uniform static plasma ⌘ smallperturbations of the black hole metric ( ⌘ quasinormal modes(QNM)) g D ↵ = g D , black hole ↵ + g D ↵ ( z ) e i ! t + ikx I This gets translated on the gauge theory plasma side to T µ⌫ = T static µ⌫ + t µ⌫ e i ! t + ikx I Dispersion relation fixed by linearized Einstein’s equations. from Kovtun,Starinets hep-th/0506184
Vary k ! Determine ! i ( k ) I Lowest mode: hydrodynamic sound mode...
FIG. 38: A set of frequency modes, on ω complex plane, from[66]. Dots are for a particular wave vector k , arrows indicatethe direction of motion as k increases. distinct parts: (i) early implosion, which converts poten-tial energy into the kinetic one, which has its peak whenfraction of the particles “gravitationally collapse” intoa tight cluster; and (ii) subsequent approach to a “mini-galaxy” in virtual quasi-equilibrium. Only the first one isphysical, as the imploded spaghetti has density sufficientfor production of QGP fireball, and after that explodeshydrodynamically. The whole scenario thus resemblesthe supernovae: implosion leads to more violent explo-sion later.In Fig. 37 we show an example of the instantaneouscollective potential produced by the strings in the trans-verse plane. The white regions correspond to the valuesof potential smaller than − · g N σ T (fm − ) ≈ −
400 MeV,i.e. the chiral symmetry can be completely restored inthose regions. Large gradient of this potential at its edgecan cause quark pair production, similar to Schwingerprocess in electric field: one particle may flow outwardand one falls into the well. Such phenomenon is a QCDanalog to Hawking radiation at the black hole horizon.
VII. HOLOGRAPHIC EQUILIBRATIONA. Near equilibrium
Since this development is with heavy ion communityfor over a decade, no general introduction is perhapsneeded. And relating it to string theory background fromwhich AdS/CFT correspondence came will take us toofar from the topics. I will only comment on a couple ofissues to be relevant to discussion later in this chapter.The equilibrium setting includes the so called “AdS-black hole” metric, with its horizon located at the 5-thcoordinate z h = 1 /πT , so the gauge theory – located atthe z = 0 boundary feels the Hawking radiation tem-perature T . (Hawking radiation in asymptotically flatbackground leads to evaporation of BH. Indeed, one BHcannot heat up an infinite Universe. But AdS asymp-totics means basically a finite box, and in this case BHcan get into an equilibrium state with the Universe.)Various gravity waves propagating in this backgroundhave dispersion relations ω ( (cid:126)k ) with calculable real and3imaginary parts, see an example in Fig.38. (Suchquasinormal[166] modes are known for various examplesof BHs for a long time, these particular ones were calcu-lated by Kovtun and Starinets [66] ). In this channel thelowest eigenvalue, shown by larger dots, is close to theorigin and describes the sound mode. For reference letme mention several known terms at small k (from [54]) ω πT = ± ˜ k √ (cid:20) (cid:18) − ln 23 (cid:19) ˜ k − .
088 ˜ k (cid:21) − i ˜ k (cid:20) − −
212 ˜ k − .
15 ˜ k (cid:21) , (48)where ˜ k ≡ ( k/ πT ). First, note that at small k the imagi-nary viscous term is Imω ∼ k : we already used that factin discussion of the “acoustic damping” phenomenology.Second, the dispersive correction to the speed of soundhas positive coefficient: thus sound wave can decay intotwo: we already discussed that in discussion of the cos-mological acoustic turbulence. Third: note that higherorder corrections to viscosity are both negative. This isin contrast to some popular second order ad hoc schemessuch as Israel-Stuart.All “non-hydrodynamical modes” have Im ( ω ) / (2 πT ) = O (1), indicating that during thetime of the order of z h ∼ / (2 πT ) they all “fall into theblack hole”. (It is amusing to note that a mystifyingprocess of QGP equilibration is, in AdS/CFT setting,reduced to the problem number one in physics history,the Galilean falling stone in gravity field. ) The essenceof the AdS explanation of the rapid equilibration is thussimple: any initial objects gets irrelevant, all of thoseare absorbed by a black hole. The only memory whichremains is their total mass, which the BH transformsinto corresponding amount of Bekenstein entropy. (Plusconserved charges, if they are there.) B. Out of equilibrium 1: the shocks
Shocks are the classic objects used in studies of out-of-equilibrium phenomena. They traditionally are dividedinto weak and strong shocks. In the former case thereis small difference between matter before and after theshock: so those can be treated hydrodynamically, e.g.using the Navier- Stokes (NS) approximation. Strongshocks have finite jumps in matter properties. Theirprofiles have large gradients: so one needs some morepowerful means to solve the problem, not relying on hy-drodynamics, which is just an expansion in gradients.The reason we put this example as number one isbecause it is the only one which can be considered in stationary approximation . Indeed, in the frame whichmoves with the velocity of the shock, its profile is time-independent .Strong shocks in AdS/CFT setting were discussed inmy paper [128]. A particular example worked out in de-tail starts with NS profile shown in Fig.39. From left to (11)(11) > > > > > > (3)(3)(15)(15) x1 K K EE0:=simplify(eval(subs( f11(x1,r)=0.,f12(x1,r)=0.,f22(x1,r)=0.,EE))):ES0:=simplify(eval(subs( f11(x1,r)=0.,f12(x1,r)=0.,f22(x1,r)=0.,ES)));
ES0 := 1 r K h x1 dd x1 h x1 r cosh h x1 K r sinh h x1 dd x1 h x1 dd x1 h x1 K
64. dd x1 h x1 cosh h x1 dd x1 h x1 C h x1 r dd x1 h x1 sinh h x1 h x1 dd x1 h x1 K r dd x1 h x1 K r dd x1 h x1 FIG. 2: (color online) An example of a solution to the NShydrodynamics, the rapidity (black dashed line) and pressure(blue solid line. Pressure is in units of its initial value: thusthe curve starts near 1 on the left.
IV. STRONG SHOCKS AND THE RESUMEDHYDRODYNAMICS
Let us now proceed to shocks with the rapidity jump Y = O (1), for which there is no apparent small parameterand thus the NS equation cannot be expanded. Sincepressure appears only linearly, one can easily manipulate01 and 11 equations into one single di↵erential equationfor rapidity (the second equality) p = (4 / ⌘c ( dy/dx ) + p i (4 s i + 1)4 s + 1= ⌘s ( dy/dx ) + 4 p i s i c i sc (4.1)where we use the short-hand notations c = cosh ( y ( x )) , s = sinh ( y ( x ))All quantities with the index i = initial are the cor-responding values before the shock, at x = 1 . Thesolution to the NS equation cannot be obtained analyti-cally, so we use a numerical solver. A particular exampleis shown in Fig.2, for the rapidity and the pressure. Itsparameter were selected at random: the rapidity jumpsup by a factor of 2 and the pressure down by about factor3. Now, how reliable is the NS solution, for such a strongshock? The implicit assumption is that all nonlocal(higher gradient) terms are small. We need to know(i) the individual coe cients – higher viscosities – ofthose terms;(ii) the values of the higher gradients;and (iii) the combined e↵ect of their sum , or convergenceof the series. For usual fluids such as water to air, the gradients areusually so small that we do not need higher viscosities. Inpractice their empirical values are not even known, andso it would be hard to even estimate the magnitude ofthose terms. Yet for sQGP we have the AdS/CFT cor-respondence, which provides in principle a complete setof such higher viscosities, and many of those has alreadybeen evaluated in literature. The summary of these re-sults and their resummation was the subject of the “im-proved hydrodynamics” by Lublinsky and myself [12], LSfor short.The linearized correlators in AdS/CFT can in principleprovide the values of all kinetic coe cients, as the coef-ficient of certain powers of ! and k (the frequency andthe wave vector). Since there are several kinematicallydi↵erent channels excited by stress tensor, these calcula-tions also provide su cient number of crosschecks, Forexample, ensuring that the first viscosity always is (1.1)anywhere viscosity appears. The “improved hydrody-namics” tries to combine all known coe cient into some“resumed” model functions. The main result was the 3-term Model 1 which is based on PADE approximation,which may be called LS1. It reproduces exactly eightfirst coe cients and overall behavior of the correlatorsquite well. Furthermore, we found that the second andthe third poles largely cancel each other, and for an es-timate of the e↵ect of higher gradients we will use oursimpler model LS2 in which the e↵ective viscosity is ⌘ LS = ⌘ ⌘ , k / (2 ⇡T ) i!⌘ , / (2 ⇡T ) (4.2)written in units in which all eta coe cients are dimen-sionless numbers. Furthermore, as there is no time de-pendence in stationary problems we now discuss, we willonly need one simple coe cient ⌘ , =
12 (4.3)Due to its sign, the resumed factor reduces the e↵ectof the NS term as k grows. This is indeed what wasknown for the lowest quasi normal modes (sounds): itsimaginary part grows as k till some value, and then itstops growing.As the rest of this paper below will be discussing theAdS/CFT setting, it is convenient to us the holographicthermal horizon as the natural unit of length. Thus thehorizon location in the holographic coordinate is z h = 1 /⇡T = 1 (4.4)Using it and the holographic value of the first viscosity(1.1) one finds conveniently that the combinations whichoften appears is greatly simplified, for example the ratioin the r.h.s. of NS equation is ⌘p = ⌘s T = 1 ⇡T = 1 (4.5)Of course, our problem has with two temperatures andpressures, p i and p f at both x ! ±1 : but as the final FIG. 39: An example of a strong shock in QGP, according tothe Navier-Stokes hydro. Pressure shown by (blue) solid lineis in units of its initial value: thus the curve starts near 1 onthe left side. The (black) dashed line is the rapidity. From[128]. right the pressure decreases, and rapidity increases: theenergy and momentum fluxes are tuned to be the same.One may think of it as a process in which higher-densityQGP on the left, moving more slowly, “burns” into alower density QGP floating out with higher rapidity.In the AdS/CFT setting one however can solve theproblem from first principles, by solving the Einsteinequations. Of course the setting has an extra holographicdimension, and so all the functions depend not on one buton two variables: the longitudinal coordinate x and z .So I plugged in the NS solution, and worked out correc-tions to it (by a somewhat novel variational approach).Not going to the details of that, let me jump directlyto the surprising conclusion: those corrections happento be small, at a scale of few percent, even without anyapparent small parameter in the problem.Another tool used to correct the NS solution was theso called “re-summed hydrodynamics” by Lublinsky andmyself: it also lead to corrections at the percent level.(Unfortunately, the accuracy I had on AdS/CFT solutionwas insufficient to tell whether both agree or not).The lesson: all higher order gradient corrections to theNS solution have strong tendency to cancel each other. C. Out of equilibrium 2: the falling shell
This setting has been proposed by Lin and myself [129]and it is in a way orthogonal to the previous one: there istime dependence but no dependence on space point. Themotion occurs only along the holographic 5-th direction.4The physical meaning of this motion is as follows. Firstof all, recall that the 5-th coordinate z = 1 /r correspondsto a “scale”, with small values near the boundary (large r ) corresponding to the UV end of the scales, and large z , small r corresponding to the IR or small momentaend. Since everything happens much quicker in UV ascompared to IR, the equilibration process should natu-rally proceed from UV to IR, also known as “top-down”equilibration.One can imagine that this process can in some sense bereduced to a thin “equilibration shock wave”, propagat-ing in the z direction. The key idea of the paper [129] wasthat this shock can be thought of as a certain materialobjects – a shell or an elastic membrane – falling underits own weight . (See the sketch in Fig.40(a)). If this isthe case, the total energy of the membrane is conserved (potential energy goes into kinetic). The consequences ofthat are very important: while the shell is falling towardthe AdS center, the metric – both above and below themembrane – is actually time independent, as it can onlydepend on its total energy. So there is no need to solvethe Einstein equations. (Recall the Newton’s proof thata massive sphere has the outside field the same as a pointmass, and that there is no gravity inside the sphere. Thisis also true if the sphere is falling. It also remains truein GR.) In the case of an extreme black hole at the AdScenter (the blue dot in Fig.40(a)) the solution consistsof (i) thermal Schwarzschield-AdS metric above the shelland (ii) “empty vacuum” or the AdS solution below it.The only equation of motion which needs to be derivedand solved are those of the shell itself, r ( t ) = 1 /z ( t ). It isnot so trivial to derive it, since the coordinates below andand above the shell are discontinuous. Fortunately, a thinshell collapse has already bend solved in GR literature:the key to it is the so called “Israel junction condition”.The shell velocity (in time of the distant observer t ) isgiven by the following expression dzdt = ˙ z ˙ t = f (cid:113) ( κ p ) + ( κ p ) (1 − f ) − f κ p + κ p (1 − f ) (49)where κ p is the 5-dim gravity constant and shell elasticconstant and f = 1 − z /z h is the “horizon function” inthermal AdS. The qualitative behavior of the solution isas follows: The shell start falling with zero velocity fromcertain hight get accelerated, to near the speed of light,and then is “braking” toward the horizon position z h ,which the shell asymptotically approaches with velocityzero.After solution is found, one can calculate what differentobservers at the boundary – that is, in the gauge theory– will see. In particular, one may ask if/how such anobserver can tell a static black hole (the thermal statewith stationary horizon) from that with a falling shell?A “one-point observer” O Fig.40(b) would simply seestress tensor perturbation induced a gravitational prop-agator indicated by the red dashed line. Since the met-ric above the shell is thermal-AdS, he will find time-
FIG. 40: (a) A sketch of a falling shell geometry. Its radius r ( t ) = 1 /z ( t ) which is used in the text. (b) Single-point ob-server O and the two-point observers O , O independent temperature, pressure and energy densitycorresponding to static final equilibrium. Yet more so-phisticated “two-point observers” O , O can measurecertain correlation functions of the stress tensors. Theywill see effects from gravitons flying along the line shownby the solid line above the shell, that is in the thermalmetrics, as well as from gravitons flying over the pathshown by the dashed line which penetrate below the shell:those would notice deviations from equilibrium. Solv-ing for various two-point functions in the backgroundwith falling shell/membrane we found such deviations.They happen to be oscillating in frequency around ther-mal ones. This observation is explained [129] by certain“echo” times due to a signal reflected from the shell. So,here is the lesson: one can in principle experimentallyobserve an echo, from the 5-th dimension!For further discussion of the scenarios of top-downequilibration, with infalling scalar fields etc – the readeris referred to [130] and subsequent literature. D. Out of equilibrium 3: anisotropic plasma
Our next example, due to Chesler and Yaffe, is a set-ting in which one starts with some anisotropic but ho-mogeneous metric, and then follow its relaxation to equi-librium. It is nicely summarized by Chesler [67], so Iwill be brief. The metric is of the form of diagonal, timedependent but space-independent components, and Ein-stein equation is solved numerically. Rapid relaxation toAdS-Schwarzschield-like solution is observed. The typi-cal behavior found is displayed in Fig.41. The character-istic relaxation time is shorter than 1 /T , even when thesystem is initially quite far from equilibrium.A number of initial states can be compared, with thecondition that all of the evolve to the same equilibriumenergy density (or horizon, or T). While at early time the5 FIG. 41: Pressures anisotropy as a function of time, from [67].FIG. 42: (left) From Heller et al [57]: The temperature evolu-tion combination dlog ( w ) /dlogτ for different initial conditions(black thin curves) converging into a universal function of w = T τ , compared to hydro. (right) The pressure anisotropyfor one of the evolutions compared to 1-st (NS), 2-nd and 3-edorder hydrodynamics. momentum asymmetry can be very large – say, an orderof magnitude – it becomes very small exponentially intime. Any deviations from equilibrium get strongly red-shifted as they approach the horizon.If a more detailed analysis is made, one can locatefew specific quasi normal modes. There are no hydromodes since the setting is homogeneous, and the lowestis λ = (2 .
74 + i . πT . So, the basic lesson is that inthe strong coupling regime the isotropization time is asshort as τ isotropization ∼ . T (50) E. Out of equilibrium 4: rapidity independentcollisions
Various “bulk” objects may fall into the AdS center. Inan early paper Sin, Zahed and myself [114] argued that all“debris” created in high energy collisions form bulk blackhole which then falls toward the AdS center. Its hologramis the exploding/cooling fireball seen by observer on theboundary. The specific solution discussed in that paperwas spherically symmetric (and thus more appropriatefor cosmology than for heavy ion applications).Subsequent series of papers by Janik and collaborators(see e.g. [56] and references therein) had developed ra-pidity independent “falling” horizons, corresponding toBjorken hydro solution. In this case one has homoge-neous x-independent horizon falling as a function of time z h ( τ ). At late time it converges into hydro and z h ( τ ) = 1 /πT ( τ ) ∼ τ / (51)The variable w = T τ has the meaning of the macro-to-micro scale ratio: at late time it grows indicating that thesystem becomes more macroscopic and hydro more accu-rate. The question is when hydro description becomesvalid and with what accuracy .Fig.42 from Heller et al [57] displays time evolutionsof many initial states, all approaching the same hydro-dynamical solution. Fig.42(left) shows that this happensvia convergence to certain universal function of the vari-able w = τ T defined by dwd ln τ = F ( w ) , (52)Existence of such universality is the essence of the “re-summed hydro” proposed by Lublinsky and myself [54].Depending on accuracy required, one may assign specificvalue of w at which “hydrodynamics starts”. Its valueis in the range w i = 0 . .. − .
6. The plot on the rightdemonstrates that at such time the anisotropy is stilllarge and viscosity is important.The lesson from this work can be better explained bycomparing its result to ”naive expectations” , that hy-dro starts when macro and micro times are the same, w = τ T >
1, and that the accuracy of hydro should bebad O(1), say 100%. Calculations show instead that at twice smaller time w ∼ / pp : dN P bP b dy ( y = 0 , s ) ∼ s . dN pp dy ( y = 0 , s ) ∼ s . (53)From the RHIC energy ( E = 0 . T eV ) to the LHC, thedouble ratio dNdη | P bP b,LHC / dNdη | pp,LHCdNdη | AuAu,RHIC / dNdη | pp,RHIC = 1 . . (54)shows a noticeable change with the energy, which callsfor an explanation.Lublinsky and myself [72] proposed a simple form forthe function F ( w ) and calculate the entropy produced,from the time w i on. It turns out to be about 30%. Fur-thermore, we get the following expression for the con-tribution to this double ratio ≈ η ( LHC ) − ¯ η ( RHIC )]2 w i +3¯ η ( RHIC ) and show, that the observed growth can be naturallyexplained by the viscosity growth, from RHIC to LHC,predicted by a number of phenomenological models. VIII. COLLISIONS IN STRONG COUPLINGA. Trapped surfaces and the entropy production
The simplest geometry to consider is the wall-on-wallcollisions, in which there is no dependence on two trans-verse coordinates, and only the remaining three – time,longitudinal (rapidity), and the holographic direction –remain at play. Needless to say, it is a very formidableproblem, solved by Chasler and Yaffe via clever “nesting”of Einstein equations. The reader can find explanationsin a summary [67].Collisions of finite size objects are even more difficultto solve, but those historically brought into discussionvery important issues of trapped surface formation andthe entropy production . It was pioneered by Gubser, Pufuand Yarom [133] who considered head-on (zero impactparameter) collisions of point black holes. The setting isshown in Fig.43(a).Trapped surface is a technical substitute to the hori-zon – for this review it is not too technical to discuss thedifference – and its appearance in the collision basicallymeans that there exists a black hole inside it. Classically,all information trapped inside it cannot be observed fromoutside, and lost information is entropy . (Needless tosay, for known static black hole solutions this area doesgive the black hole Bekenstein entropy.) So, locating thissurface allows one to limit the produced entropy from be-low , by simply calculating its area . The reason why thisentropy estimate is from below is because the trappedsurface area is calculated at the early time t = 0 of thecollision, not at its end. No particle can get out fromtrapped surface, but some can get into it during the sys-tem’s evolution, increasing the black hole mass and thusits entropy. H z=L CS S H R xx z FIG. 43: (a) From [133]:A projection of the marginallytrapped surface onto a fixed time slice of the AdS geometry.(b) The area of the trapped surface versus impact parameter,with the comparison of the numerical studies [134] shown bypoints and analytic curves from [135]. The vertical line showslocation of the critical impact parameter b c beyond whichthere is no trapped surface. The setting of Gubser et al have the parameter L ,the distance separating colliding b.h. from the bound-ary: we will discuss its physical meaning below. Naively,central collisions have only axial O(2) symmetry in thetransverse plane x ⊥ , but using global AdS coordinatesthese authors found higher O(3) symmetry of the prob-lem, which becomes black hole at the collision momentat its center, thus in certain new coordinate q = (cid:126)x ⊥ + ( z − L ) zL (55)the 3-d trapped surface C at the collision moment be-comes a 3-sphere, with some radius q c . One can find q c and determine its relation to CM collision energy andBekenstein entropy. For large q c these expressions arejust E ≈ L q c G , S ≈ πL q c G , (56)7from which, eliminating q c , the main conclusion follows:the entropy grows with the collision energy as S ∼ E / L / (57)Note that this power is in general ( d − / ( d −
2) directlyrelates to the d =5-dimensional gravity, and is differentfrom the 1950’s prediction of Fermi/Landau who pre-dicted E / as well as from the data, which indicate thepower of about 0 .
30 (53).Let me now return to the meaning of the parameter L .Gubser et al relate the “depth” of the colliding objectswith the nuclear size, L ∼ /R A which cannot depend onthe energy. Lin and myself [134] argued that L shouldrather be identified with the inverse “saturation scale”,the typical parton’s momenta in the wave function of thecolliding objects L ≈ Q s ( E ) ∼ E − α (58)It becomes especially clear if one would like to turn backto wall-on-wall collisions, in which the nuclear size R A goes to infinity while Q s remains fixed. If so, L has acompletely different scale, it is not of the O (10 f m ) scalebut rather O (0 . f m ). Furthermore, it is expected to decrease with the energy L ∼ /Q s ( E ) with certain em-pirical index α ≈ /
4. Including this dependence into(57) one gets S ∼ E (2 / − (5 / α ∼ s . (59)now in reasonable agreement with the observed multi-plicity.The generalization of this theory to non-central col-lisions, by Lin and myself [134], have technical detailsI would omit here and proceed directly to the resultsshown in Fig.43(b). The figure shows dependence of thetrapped surface area on the impact parameter is actuallyfrom the paper by Gubser, Pufu and Yarom [135], com-paring our results (points) with analytic series of curvesobtained by these authors.From gravity point of view the qualitative trend shownis clear: two colliding objects may merge into a commonblack hole only provided that the impact parameter isless than some critical value b c ( E ), depending on the col-lision energy. Indeed, with b rising, the trapped energydecreases while the total angular momentum increases:so at some point Kerr parameter exceeds 1 and thus noblack hole can be formed. Interestingly, the calculationhad shown that it happens as a jump – the first ordertransition in impact parameter. Just a bit below thisimpact parameter reasonable trapped surface and blackhole exist and nothing obvious indicates that at large b none is formed.(Note that conclusion about the first order transition isconsequence of the large N c approximation and classicalgravity: those perhaps may be smoothened at finite N c .Furthermore, the problem of trapped surface in quantumgravity is way too complicated, not studied so far.) z (a)(b)(c) x x , FIG. 44: (a) An early time snapshot of a pair of strings cre-ated after one color exchange. The coordinates are explainedon the right: the colliding objects move with a speed of lightaway from each other and strings are stretched. (b) Latertime snapshot, in QdS/QCD background. Strings reachedthe “levitation surface” shown by a rectangular shape andstart oscillate around it; (c) In case of high density of manystrings they can be approximated by a continuous membrane.
This transition should be interpreted as follows: at b < b c a creation of QGP fireball takes place, whilethe system is in hadronic phase for peripheral b > b c collisions. Still, from a physics point of view a jumpin entropy as a function of impact parameter is sur-prising. Interesting, that the experimental multiplicity-per-participant plots do show change between non-QGPsmall systems and QGP-based not-too-peripheral AA collisions. It would be interesting to compare it with re-cent information on small systems, undergoing transitionto explosive regime.In the same paper [134] we have pointed out that thesimplest geometry of the the trapped surface would bethat for a wall-wall collision, in which there is no depen-dence on transverse coordinates x , x . Thus a sphere be-comes just two points in z , above and below the collidingbulk objects. We elaborated on this in [136], consideringcollision of two infinite walls made of material with dif-ferent “saturation scales” (e.g. made of lead and cotton)and studied conditions for trapped surface formation. B. From holographic to QCD strings
AdS/CFT is a duality with a string theory, so funda-mental strings are naturally present in the bulk. In factalready the first calculation, made in Maldacena’s orig-inal paper, the “modified Coulomb law”, was based on8the evaluation of the shape and total energy of a “pend-ing string”, sourced by “quarks” on the boundary. (Thisexample is so classic, that must be known to most read-ers.)Lin and myself in two papers [131, 132] extended itto non-static strings. We first solved for a falling stringwith ends moving away from each other with velocities ± v , and the second calculating its hologram (stress tensordistribution) at the boundary. This study can be thoughtof as a ”strongly coupled version of the e + e − annihilationinto two quarks”. The hologram showed a near-sphericalexplosion: the early indication that at strong couplingsetting there are no jets .These works used the setting associated with conformalgauge theory: in AdS string falling continues forever.This is of course not what we observe in the real world,in which there is confinement and there are jets. Modernstrong coupling models moved into what is collectivelyknown as AdS/QCD , for review see e.g. [137, 138].In contrast with the original AdS/CFT, the backgroundmetric is not conformally invariant and incorporates both conf inement in the IR and the asymptotic freedom inUV. These models use additional scalar (“dilaton”) field,with certain potential, complementing a gravity back-ground.In such settings bulk strings (and other objects) can“levitate”, at some position z ∗ , at which the downwardgravity force is compensated by the uplifting dilationgradient. The hologram of such levitating string at theboundary is the QCD string : its tension, width and stresstensor distribution inside are calculable. The potentialbetween point charges is given by a pending string: inthe AdS/QCD background it changes from Coulombic atsmall r to linear at large r . Furthermore, allowing funda-mental fermions in the bulk – via certain brane construc-tion –and including their back reaction in a consistentmanner one can get quite nice description of the so calledVeneziano-QCD ( N c , N f → ∞ , x = N f /N c = f ixed )[139].Since in the UV these models also possess a weak cou-pling regime, one can also model perturbative glasma,by putting certain density of color “sources” at the twoplanes, departing from each other. In such setting therewould be smooth transition between two alternatives de-scription of the initial state we discussed above – from theperturbative glasma made of longitudinal field cells to a“spaghetti” made of the QCD strings. When time τ issmall, Fig.44(a), strings are in the UV domain (at smalldistance z ∼ /Q s from the boundary) their hologramis Coulombic or glasma-like. When strings fall furtherand reach the “levitation point” z ∗ they start oscillatingaround it [140], Fig.44(b). This is very similar to oscil-lations discussed by Florkowski (see sect.VI D) withoutAdS/QCD.At this point one may wander about AdS/QCD pre-dictions for string-string interaction. This subject is nowunder investigation in [140]. We already discussed thisissue in the QCD context above, and concluded that its long-range attraction is dominated by the σ meson ex-changes (just like between nucleons, in nuclear forces): sowe need to see if it is indeed the case in the AdS/QCD,and whether one can learn something new using it. Thisis indeed the case, and in order to explain why we needto briefly remind the reader some basics on how hadronicspectroscopy comes out in this setting.AdS/QCD has very few fields in the bulk – gravity,dilation and (quark-related) “tachion”. Each of them (ortheir mixtures) produce towers of 4-dimensional hadronicstates via the quantization of their dynamics in the 5-th coordinate. One may say that hadrons are stand-ing waves in it, and their masses are just quantized 5-thmomentum. So, a new perspective on hadrons one getsfrom this approach is that one can calculate not only themasses but also the wave functions in a scale space.So the issue studied in detail in [140] is the mechanismof hadronic flavorless scalars, which includes the σ andothers. Without any changes in the setting of AdS/QCDwe found very good description of (quite involved) mixingpattern of the scalars, which puzzled spectroscopists fordecades. Since strings are gluonic objects and σ interactstrongly with quarks, understanding of such mixing iscrucial for obtaining realistic string-string forces.Note that there is so far no temperature or entropy inthe problem: the dynamics is given by classical mechanicsof strings moving in certain backgrounds.That is because we so far considered a single (or few)strings. If there are very many charge exchanges and thenumber (or density) of strings is high enough, Fig.44(c),one should perhaps include the back reaction of theirgravity and dilaton field, or even include mutual attrac-tion of strings. Such AdS/QCD version of multi-stringdynamics from [92] of what we discussed in section VI Dfollowing [91].Not going into detail, let me comment on just one issue.Unlike GR, in which there is only gravity and thus uni-versal horizon definition, in holographic models physicsis more complex and interesting. In particularly, gluonsare strings vibrations. String have their 2D worldvol-umes which can be curved and have black hole of theirown. What it means is the wave on the string can havemodes propagating left and right, but also can have re-gions where both waves move right. So gluons can fallinto those, but cannot escape. Such feature can be in-duced, in particularly, by collective gravity/dilaton of thestrings themselves. Its meaning is a QGP fireball, andits experimental signature is hydro explosion. C. Holographic Pomeron
Pomeron description of hadronic cross sections andelastic amplitudes originates from phenomenology, start-ing from 1960’s. Veneziano amplitudes derived in a “res-onance bootstrap” ideology were the original motivationfor existence of QCD strings.The Pomeron is an effective object corresponding to9 glueball reggeons (9)(9)(11)(11)(10)(10) > > > > (5)(5) > > > > > > MM K K the difference and intersepts is then a d a2 d K $ M0pp ^2 M2pp ^2 ; a K a2 ; a := 1.08 a2 := K aprime d M2pp ^2 ; aprime2 d M2pp ^2 ; aprime := 0.2033531639 aprime2 := 0.3244997296the slope at MM=0 (called prime) is propotional to size squared, so the size ratio is(the larger slope of the daughter means the object has smaller size!) sqrt aprime2aprime ; 1.263227882 t ( GeV ) J FIG. 45: Glueball masses calculated on the lattice (dia-monds) organized in Regge trajectories (lines). (From [144]). the “leading” Regge trajectory α ( t ) which dominates thehigh energy asymptotics of the hadron-hadron cross sec-tions. Fig.45 is a recent version of the Regge plot (angu-lar momentum J versus the mass squared m for glue-balls. The Pomeron corresponds to scattering and thushas small non-physical mass t < J slightly above 1: the trajectory α ( t ) has of course physi-cal states as well. It enters the elastic cross section in aform dσdt ≈ (cid:18) ss (cid:19) α ( t ) − ≈ e ln( s )[( α (0) − α (cid:48) (0) t ] (60)Dualism of the Pomeron is seen from the very differentroles which two main parameter play, the intercept α (0)and the slope α (cid:48) ( t = 0). The former is a dimensionlessindex, describing the power with which the total crosssection rises. Perturbative description of the Pomeron iswell developed: the so called BFKL [141] re-summationof the gluon ladders provides perturbative O ( α s ) valuefor it. The Pomeron slope α (cid:48) ( t = 0) has dimension[ mass − ] and is nonperturbative for sure, related to thestring tension. (By the way, the “string scale” in the fun-damental string theory still is traditionally called α (cid:48) , theonly piece of QCD phenomenology left in it, for historicreasons.) The meson/baryon Reggion slope is related toa single open string and their α (cid:48) = 1 / (2 πσ ), glueballssuch as Pomeron are closed strings should naively havethe double tension or half the slope: phenomenologicallyit is closer to 1/3. It perhaps indicates changes due to two string interaction .Another way to see dual nature of the Pomeron phe-nomenology is when one considers the so called collisionprofile as a function of the impact parameter b . At small b people think of color dipoles and gluon exchanges. Atlarge impact parameter b = 1 − FIG. 46: (Color online) (a) Dipole-dipole scattering configu-ration in Euclidean space. The dipoles have size a and are b apart. The dipoles are tilted by ± θ/ x x L plane. (b) A sketch illustrat-ing Pomeron exchange for baryon-baryon scattering: only onepair of quarks become “wounded quarks”. of AdS/QCD. As the main representative of the relatedwork let me mention two works of Zahed and collabo-rators [142] and [143]. Strings live in the 5-dimensionalspace, which is weakly coupled in UV and strongly cou-pled in IR. Furthermore, a semiclassical (instanton-like)derivation of the Pomeron amplitude is given in terms ofclosed string production – similar to Schwinger pair pro-duction in an electric field. The string worldvolume hasthe shape depicted in Fig.46(a): it is a “tube” connect-ing two flat strips, the world volume of propagating colordipoles. In Fig.46(b) we sketch a Pomeron in a collisionof two nucleons, now consisting to 3 quarks and 3 string,joined by a string junction: in this case a Pomeron tube“punctures” one of the three surfaces. This produces one“wounded quark”, as we discussed in connection to Tan-nenbaum’s description of fluctuations.Direct semiclassical derivation of the scattering am-plitude was obtained in [143] using minimization of theNambu-Goto action (the tube’s area). Fig.(a) indicateEuclidean setting in which difference in rapidity is rep-resented by twisted angle θ between the direction of twodipoles. Fig.(b) illustrates a baryon-baryon scattering, inwhich the Pomeron tube can be connected to any of the0available dipoles, explaining the concept of the “woundedquarks” we mentioned in section IV B. More than onePomeron means more tubes, connecting perhaps otherquarks.The classical action of this configuration provides the α (cid:48) term, while the intercept deviation from 1 is due tothe next order (one-loop) corrections due to string vibra-tions. It gives the amplitude squared: if cut in half by theunitarity cut the “tube” provides two strings of certainshape, which should be taken as the shape which “jumpsfrom under the barrier” and should be used as the initialconditions to real-time evolution.Not going into details, we just note that the amplitudehas alternative derivation from string diffusion equation,and that 5-d space is actually important numerically.Scattering pp data as well as deep-inelastic ep (DESYdata) are well reproduced by this model, see [143].It was further argued by Zahed and myself [144] thatbecause the “tube” has a periodic variable, resemblingthe Matsubara time, its fluctuations take the thermalform. Appearance of an effective temperature and en-tropy was new to the Pomeron problem: but once recog-nized the analogy to thermal strings can be exploited. Wein particularly argued that between two known regimesof the Pomeron – (cold) string exchange at large impactparameter b and perturbative gluon exchange at small b – there should be a third distinct regime in which astring is highly excited due to the Hagedorn phenomenon.This is what we know to happen at finite temperaturegluodynamics, which possesses a“mixed” phase betweenthe confined (hadronic) and the deconfined (partonic)phases. As it is known from decades of theoretical andnumerical (lattice) research, this is described most nat-urally by the near-critical strings , namely the strings inthe Hagedorn regime.In Fig.47(a) one finds the so called profile of the elas-tic scattering, the Bessel transform of the imaginary partof the elastic amplitude. Basic prediction of the modelis shown by the dashed line, which has the “first ordertransition”. By circles we show more realistic predictionof the model coming from averaging of that expressionover the dipole sizes: it should be compared to empiricalfit (solid line) to the LHC elastic scattering data. Thedifference is relatively small, although it becomes notice-able in scattering with momentum transfer Q beyond theminimum, see Fig.47(b). Apparently the predicted “tran-sition edge” in b is a bit sharper than in the experimentaldata.Concluding this section: decades-old Pomeron am-plitude can be derived in a “stringy” manner, in theAdS/QCD setting. Surprisingly, this approach relatesthe Pomeron to the thermal theory of excited strings,with an effective temperature proportional to the inverseimpact parameter. Rapid “phase transition” in b , fromblack to gray, corresponds to the deconfinement transi-tion, with an intermediate regime dominated by highlyexcited strings (“string-balls”). So far only for the tun-neling (Euclidean) stage of the system is considered, pre- profile2 F(b) ( GeV ) > > > > > > > > > > > > (12)(12)(4)(4)(11)(11) all d all C evalf GN u , ln 2. , 0.5 $ stepu : od : all ;1.000375273the norm is quite OK nowassignment of b,u1,u2 is a bug, I need to change it all to some local names numax:=20: stepu:=1./numax; nbmax:=1000: stepb:=0.03: for nb from 1 to nbmax do bb:=(nb-0.5)*stepb: all := 0: for n1 from 1 to numax do for n2 from 1 to numax do uu1 := stepu*(n1-.5);uu2 := stepu*(n2-.5); all := all+evalf(K(xif(bb,uu1,uu2,1.34))*GN(uu1, ln(2.), 3.)*GN(uu2, ln(2.), 3.)*stepu^2): od: od: p[nb]:=[bb,all]; od: p[10]; stepu := 0.050000000000.285, 1.000045077 pl1 d plot seq p ll $ ll = 10..100 , style = point , symbol = circle , symbolsize = 15, color = black ; display pr0 , pl1 ; pl1 := PLOT ... b > > (4)(4)(11)(11) > > -3 -2 -1 t = q ( GeV ) FIG. 47: (a) The solid line is the empirical LHC dataparametrization. The dashed line is the shape correspondingto the “excited string” approximation for fixed sizes of thedipoles, while the circles correspond to the profile averagedover the fluctuating dipoles. (b) The corresponding elasticamplitude (the absolute value squared of the profile Besseltransform) as a function of the momentum transfer. Modelprediction agrees with parameterization well at small t , up tothe dip. dicting elastic amplitude. The remaining challenge is todescribe the subsequent evolution of the system and toextend this theory to inelastic collisions. D. Collisions at ultrahigh energies
Discussions about Pomeron regimes ultimately driveus to the old question: what happens at the ultrahighenergies, well above the LHC reach?The highest observed energies, by Pierre Auger Obser-vatory and similar cosmic ray detectors, reach E lab (cid:46) E max ∼ eV . (61)limited by the so-called Greisen-Zatsepin-Kuzmin (GZK)bound at which interaction with cosmic microwave pho-tons become important.1For future comparison with the LHC observation itis convenient to convert the laboratory energy into theenergy in the center of mass frame and use a standardMandelstam invariant, assuming it is a pp collision, √ s max = (2 E max m p ) / ≈
450 TeV . (62)While significantly higher than current LHC pp energy √ s LHC = 8 TeV, the jump to it from LHC is com-parable to that from Tevatron √ s = 1 TeV or RHIC √ s RHIC = 0 . s -dependence of many observables, the extrapolation toLHC worked relatively well, and further extrapolationmay seem to be a rather straightforward task. And yet,smooth extrapolations using standard event generatorsplus, of course, the cascade codes do not reproduce cor-rectly the experimental data of the Pierre Auger collab-oration (e.g., the so called muon size of the showers).This makes the issue subject to speculations. As anexample of an exotic model let me mention the paper [64]which suggest that somehow freezeout is entirely differentand the pion production becomes suppressed. Accordingto their simulations, a model in which mostly nucleonsare produced explains the Pierre Auger data better.More modest (but still significant) change betweenLHC and ultrahigh energies, has been proposed by Kalay-dzhyan and myself [65]. Our main statement is that atthe ultra-high energies √ s max observed in cosmic rays,the “explosive regime” even in pA collision is expectedto change from a very improbable P ∼ − fluctuationto the mean behavior, with P = O (1). The reason for itis simply an increase in mean particle (entropy) densitywith energy √ s . Extrapolations suggest an increase of dN/dy by about factor 3. In Fig.48 we show the free outsurfaces and hydro velocities at them, corresponding toour calculations. By “compression” we mean the phe-nomenon of “spaghetti collapse” discussed above. Themean transverse momenta of secondaries is shown in theTable.Another generic reason for a change is that both pri-mary collisions and subsequent cascade of ultra high en-ergy cosmic rays all happen in the Earth atmosphere, sothe targets are not protons but light ( N or O ) nuclei. Fur-thermore, the projectiles themselves are also most likelyto be not protons but some nuclei. It is either also somelight nuclei or some mixture including heavier ones, be-lieved to be up to F e .Taking into account large pp cross section at ultra highenergies, ∼
150 mb, one finds that its typical impact pa-rameters b ≈ R O ≈ pO collisions most of its 16 nucleons would becomecollision “participants”. For light-light AA collisions like OO the number of participants changes from 32 (central)to zero. Accidentally, the average number of participantsis comparable to the average number of participant nu-cleons (cid:104) N p (cid:105) ≈
16 in central pP b collisions at the LHC.The question of principle still remains: and what (a)(b)
FIG. 48: (color online) (a) The freezeout surfaces in the (¯ τ , ¯ r )plane (no rescaling) and (d) the distribution of the transverseflow velocity on those surfaces. In both plots the green solidcurve at the top is our “benchmark”, the central P bP b colli-sions at LHC. Black solid line is for light-light nuclei collisions,black dashed (coincident with green by chance) are light-lightcollisions with the size compression. Similarly, red solid andred dashed are heavy-light collisions without and with the sizecompression, respectively.particles FeO FeO comp. pO pO comp. PbPb π ± K ± p, ¯ p p T [GeV/c] for pions, kaons and protons ob-tained from the particle spectra. By “comp” we mean com-pressed initial state, as explained in the text. should happen when the collision energy goes to infin-ity? Qian and Zahed [154] had proposed an argumentwhich would stop multiplicity growth. The argument isthat when the produced string becomes so long and heavythat it basically collapses due to its self-interaction intoa black hole, the entropy produced would be limited by2the Bekenstein bound. In their opinion, not only the par-ton density growth, but also growth of the cross section,should stop at certain collision energy. IX. ELECTROMAGNETIC PROBESA. Brief summary
Let us on the onset remind standard terminology to beused below. The sources of the dileptons are split intothe following categories:(i) instantaneous ¯ qq annihilation, known as the Drell-Yanpartonic process;(ii) the pre-equilibrium stage, after the nuclei pass eachother;(iii) the sQGP stage, in which matter is usually assumedto be equilibrated.(iv) hadronic stage in which rates are calculated via cer-tain hadronic models;(v) “cocktail” contribution of leptonic decays of secon-daries occurring af ter freezeout.The corresponding windows in the dilepton mass are: M > GeV for (i), 1 < M < GeV for (iii) and
M < GeV for (iv). So the early stage dileptons mostlyfall into the 3-4 GeV window. While it contains also
J/ψ, ψ (cid:48) , ψ ” states, one should rely on their subtraction,which depends on mass resolution of the detector. Thiscoincidence adds difficulty to the measurements.Status of the dilepton/photon measurements, in brief,is as follows.At SPS: NA60 has been very successful, in particularly:(a) Large enhancement at small masses
M < m ρ anddeformation of the spectral density of the electromagneticcurrent is documented;(b) using p t slope as a function of M one sees that M < GeV are produced when flow is developed,(c) while intermediate mass dileptons (IMD) 1
GeV Thermal rate in QGP & HG, with hydro (viscous/non viscous) or blastwave evolution Microscopic transport (PHSD) New early contributions Non-equilibrium effects (glasma, etc.) Enhanced thermal emission in large B-fields Modified formation time and initial conditions New effects at phase boundary Extended emissionEmission at hadronization Direct Photon Puzzle Example: viscous hydro + thermal emission Large yield and v n challenge understanding of sources, emission rates and space-time evolution *list not complete FIG. 49: Illustrations to “direct photon puzzle” (fromA.Drees, PHENIX presentation at QM2015).The yield (a), v (b) and v of the direct photons. Points aredata and curves are from theory based on hydrodynamicalmodel, the reference indicated on the figure. Both (c,d) together are known as a “direct photon puz-zle”, illustrated by the latest data in Fig. 49. My com-ment is that this field still uses a paradigm set up fromits very beginning [2], using basically perturbative ratesor those evaluated from hadronic models. Now we knowthat such approach fails elsewhere, in viscosity and jetquenching, and it is perhaps time to approach the issuein more flexible phenomenological way. If the photon pro-duction rates at the late (near T c ) stage of the collisionsis increased, both (c) and (d) problems will disappear.Like for jet quenching parameter ˆ q I would argue thatmay be due to additional scattering on monopoles, notpresent yet in any models. B. New sources of photons/dileptons: multi-gluonor phonon+magnetic field Theoretically, the production of photons and dilep-tons is tied to the presence of quarks, and is thus sen-sitive to the issue of quark chemical equilibration and itstiming. The initial stages of the high energy collisionsare believed to be dominated by gluons. Old perturba-tive arguments [86] show that chemical equilibration viaquark-antiquark pair production is relatively slow andshould be delayed relative to thermal equilibration of theglue. This idea led to a “hot glue” scenario in which thequark/antiquark density at early stage is suppressed bypowers of quark fugacity ξ q < 1. The basic process of thedilepton production q + ¯ q → γ ∗ → l + + l − (63)3is expected to be suppressed quadratically, ∼ ξ q .This scenario has been challenged: higher order pro-cesses with virtual quark loops can produce electromag-netic effects as well, even without on-shell quarks. First,contrary to general expectations, the quark loop effectin GLASMA has been suggested to be significant [85],enhanced due to multigluon -to virtual quark loop -todilepton processes like ggg → (quark loop) → γ ∗ → l + + l − . (64)To see how much correction to production rates theseprocesses contribute one needs more explicit calculationof the absolute rates of such processes, which are stilllacking.An explicit calculation [87] of the rate of such type hasbeen made, the two gluon to two photon rate gg → γγ ,in which one of the photons is the ambient magnetic fieldand the gluons are combined into a colorless matter stresstensor, and one of the photons is rotated into the initialstate T µµ + (cid:126)B → (quark loop) → γ ∗ (= dileptons ) (65)The terminology introduced in this paper is as follows:The process in which glue appears as “average” mat-ter stress tensor < T µν > , producing photons ( real orvirtual) due to time-dependent magnetic field, is called Magneto-Thermo-Luminescence , MTL for short. By “av-erage” we mean that the value of the stress tensor isaveraged over the fireball and is nearly constant, withnegligible momentum harmonics p ∼ /R .Individual events, however, are known to also possessfluctuations of the matter stress tensor δT µν , with com-plicated spatial distribution and thus non-negligible mo-menta. Although these fluctuations include both lon-gitudinal and transverse modes, in a somewhat a looseway we will refer to all of them as “sounds”. Wewill thus call the interaction of the ambient electromag-netic field and the fluctuations of the matter stress ten-sor that produces photons and dileptons Magneto-Sono-Luminescence , MSL.The paper is rather technical, and it is probably notvery useful to present here specific lengthy formulae, solet us only make some general comments on its potentialimportance. If observed, the MSL process tests both theamplitudes of the short-wavelength sounds and also har-monics of the magnetic field. We already discuss manyuses of sounds above: let us here only comment on themagnetic field. It is easy to evaluate its early values,resulting from Maxwell equations and the currents dueto the spectators in peripheral collisions. Since sQGPis believed to be a good conductor, these fields are ex-pected to create currents capturing a fraction of the fieldinside the plasma [88], perhaps lasting for many fm/c.Magnetic field evolution is important to know for otherapplications as well, e.g. chiral magnetic effects and thelike. So far, luminosity and acceptance limitations had leadexperiments to focus on most luminous central collisions.Now it is perhaps time to look at dileptons in semi-peripheral collisions as well. C. Dilepton polarization and the (early time)pressure anisotropy It is well known that when spin-1/2 particles (such asquarks) annihilate and produce lepton pairs, the crosssection is not isotropic but has the following form dσd Ω k ∼ (1 + a · cos θ k ) (66)where the subscript correspond to a momentum k of,say,the positively charged lepton and θ k is its direction rel-ative to the beam. The anisotropy parameter a in theDrell-Yan region – stage (i) in the terminology introducedat the beginning of this section– is produced by annihi-lation of the quark and antiquark partons, are collinearto the beams, and therefore a = 1.In my note [89] it is suggested that parameter a can beused to control anisotropy of the early stage of the colli-sion. In particular, if it is anisotropic so that longitudinalpressure is small relative to transverse, p l < p t , the anni-hilating quarks should mostly move transversely to thebeam, which leads to negative a < 0. (Such regime is ex-pected because such anisotropic parton distribution withsmall differences in longitudinal momenta is produced bya “self-sorting” process, in which partons with differentrapidities get spatially separated after the collision.)For illustration I used a simple one-parameter angulardistribution of quarks over their momenta p in a form W ∼ exp [ − αcos θ p ] (67)and calculate a ( α ) resulting from it. It does show that a may reach negative values as low as -0.2 at stage (ii),before it vanishes, when equilibration is over, at stages(ii-iv). X. CHARM AND CHARMONIAA. Charmonium theory: an overview Quarkonia have always been among the first classicQGP signature, although rather confusing one. Theviews on what QGP actually does for charmonia yieldhad changed in a complicated path, which proceeded viaa couple of turns of the famous logical spiral. Thus re-views which follow the historical development usually arerather confusing.Instead, ignoring the historical order, we will proceedpedagogically, from simpler to more complicated settings.We will proceed though three fundamentally different set-tings:4(i) time independent equilibrium state of charmonia;(ii) equilibration processes and rates;(iii) heavy ion collisions.But before we discuss those, let us start with even sim-pler forth setting, (0) the case of static heavy quark. Theso called static potentials which has been measured on thelattice, at zero and nonzero temperatures . A sample of2-flavor QCD results from Bielefeld group [118] are shownin Fig.50. The linear potential at zero T is indicated atall plots, for comparison. Static potential gives the freeenergy, related to entropy and internal energy V ( T, r ) by V ( T, r ) = F ( T, r ) − T ∂F∂T = F ( T, r ) + T S ( T, r ) (68)So, determining the entropy term one can subtract T S ( T, r ) term and get the internal energy . As one cansee from Fig.50. they are rather different. The force –potential gradient - has a maximum just below T c , and U itself reaches 4 GeV in magnitude. What are theirphysical meaning and which one, if any, should one useas the potential in the charmonium problem?Liao and myself [122] had proposed a model describingthese lattice findings, based on monopole-generated fluxtubes. Its detailed discussion is out of context here: letme just mention how an entropy associated with staticquarks can be understood. The key idea is the levelcrossing phenomena, occurring while the separation be-tween charges is changed. Suppose a pair of static charges(held by external hands) are slowly moved apart in ther-mal medium at certain speed v = ˙ L . For each fixed L , there are multiple configurations of the medium pop-ulated thermally. When L is changed, the energies ofthese configurations are crossing each other, and at eachlevel crossing there is certain probability to change popu-lation of the states, depending on the speed of separation v = dL/dt . If the motion is adiabatically slow, then allthe level crossing processes happen with probability 1: inthermodynamical context this leads to maintained equi-librium and maximal entropy/heat generation. If how-ever the pair is separated fast, then the level crossinghappens probabilistically, and certain entropy is devel-oped. The medium is no longer in equilibrium with thepair. The amount of entropy generated is less than in theadiabatic case. In the extreme case one may expect thatthe pair, if moving on a time scale much much shorterthan the medium relaxation time scale, decouples fromthe media and produce negligible entropy. It is plausi-ble, therefore, to identify the adiabatic limit as probingthe free energy F ( T, L ) measured on the lattice with thepresence of static Q ¯ Q pair. The internal energy V ( T, L ),on the other hand, is different from F ( T, L ) by subtract-ing the entropy term and thus can be probed in the ex-tremely fast limit in which possible transitions amongmultiple states via level crossing do not occur and no en-tropy is generated. We emphasize that such phenomenonin thermal medium is a direct analogue of what exists inpure quantum mechanical context. Perhaps the oldestexample is the so called Landau-Zener phenomenon of electron dynamics during the vibrational motion of twonuclei in a diatomic molecule. Specific electron quantumstates are defined at fixed L (the separation between thetwo nuclei) with energies E n ( L ), and certain levels crosseach other as the value of L changes. The issue is theprobability of the transition during such crossing of twolevels. Consider two levels with their energies given ap-proximately by E ( L ) = σ L + C and E ( L ) = σ L + C near the crossing point. When the two nuclei approachthe crossing point adiabatically slowly v = dL/dt → dL/dt is finite, then the transitionto both levels at crossing point may happen, with certainprobability. This is how a pure state becomes a mixture,described by the density matrix, and entropy is produced.(More quantitatively, Landau and Zener solved theproblem and showed that the probability to remain in theoriginal state (i.e. no transition) is exponentially smallat small velocity vP = exp [ − πH v | σ − σ | ] (69)where H is the off-diagonal matrix element of a two-level model Hamiltonian describing the transition be-tween the two levels. In the opposite limit of rapid cross-ing, the system remains in the original state, and no en-tropy is produced again. So it has maximum at somespeed. )For more discussion of the role of the “entropic force”in charm motion, see paper by Kharzeev [119].Now we return from static quarks to physical charmo-nium, and discuss cases (i-iii) subsequently.(i) Suppose one put a J/ψ in matter at some tempera-ture T . However small is its value, transitions from J/ψ to its excited states will happen. Eventually those willcome to ¯ DD threshold and propagate further. Since anysubsystem tends to equilibrium, at given T , and ¯ DD caneventually occupy an infinite volume, they will win overthe bound states. So, given enough time, a given initial J/ψ will dissolve always, at any T .On the other hand, random thermal transitions shouldalso proceed in the opposite direction as well: ¯ DD canproduce charmonia and those can get de-excited backto J/ψ . Thus recombination of charmonia states shouldalso happen. Given infinite time, and equilibrium densityof J/ψ will be reached.Pure thermal occupation rate for J/ψ ∼ exp ( − M J/ψ /T ) is tiny at all temperatures of inter-est. But heavy quarks c, b are produced in hardprocesses, not in thermal reactions. Since heavy ioncollision time is small compared to weak decays of those,their number is conserved. This leads to the concept ofcharm chemical potential µ c . It is not coupled to charmquantum number, is the same for c and ¯ c , and thus theoverall charm neutrality is still intact. Now one mayask what is the density of certain quarkonia states in i ( T, µ c ) in equilibrium with a particular µ c .5 Static quark anti-quark free and internal energy in 2-flavor QCD Olaf Kaczmarek -50005001000 0 0.5 1 1.5 2 2.5 3r [fm]F [MeV] 0.76T c c c c c c c c c c c c r [fm] J/ ψψ ’ χ c r med r D Figure 1: (left) The colour singlet quark anti-quark free energies, F ( r , T ) , at several temperatures as func-tion of distance in physical units. Shown are results from lattice studies of 2-flavour QCD (from [1]). Thesolid line represents the T = V ( r ) . The dashed error band corresponds to the stringbreaking energy at zero temperature, V ( r breaking ) ≃ − r breaking ≃ . − . r D ≡ / m D ( N f =0: open squares, N f =2 filled squares), and the scale r med ( N f =0: open circles, N f =2:filled circles, N f =3: crosses) defined in (2.1) as function of T / T c . The horizontal lines give the mean squaredcharge radii of some charmonium states, J / ψ , χ c and ψ ′ (see also [3, 4]) and the band at the left frame showsthe distance at which string breaking is expected in 2-flavor QCD at T = m π / m ρ ≃ . 1. Introduction A simple Ansatz to study the possible existence of bound states above the critical temperatureis to use effective temperature dependent potentials that model the medium modifications of stronginteractions in a quark gluon plasma. To what extend a suitable effective potential at finite tem-perature can be defined by quark antiquark free or internal energies and furthermore how realisticsuch (simple) descriptions of bound states in a deconfined medium are is still an open question.By comparing the screening radii obtained from lattice results on singlet free energies in 2-flavourQCD to the mean squared charge radii we obtain first estimates on the temperatures where char-monium bound states may be influenced by medium effects. In more realistic potential modelcalculations effective temperature dependent potentials that model medium effects are used in theSchrödinger equation. We present the heavy quark free energies and their contributions, i.e. en-tropy and internal energy, and discuss the different results obtained using those contributions inpotential models. 2. Screening radii and medium modifications In Fig. 1 (left) we show results for the heavy quark anti-quark free energies in 2-flavour QCD[1]. While in the limit of short distances F ( r , T ) shows no or only little medium effects, i.e. F ( r → ) ≃ V ( r ) , at large distances the free energies approach temperature dependent constant values, F ∞ ( T ) ≡ F ( r → ∞ , T ) . To characterise distances at which medium effects become important weintroduce a screening radius, r med , defined by the distance at which the value of the zero temperature PoS(LAT2005)192 192 / 2 Static quark anti-quark free and internal energy in 2-flavor QCD Olaf Kaczmarek [MeV] r [fm] 0.88T c c c [MeV] r [fm] 1.09T c c c c c c c Figure 4: The colour singlet quark anti-quark internal energies, U ( r , T ) , at several temperatures below (left)and above (right) the phase transition obtained in 2-flavour lattice QCD. In (left) we also show as horizontallines the asymptotic values which are approached at large distances and indicate the flattening of U ( r , T ) .The solid lines represent the T = V ( r ) [1, 7]. 5. Bound states in potential models Various potential model calculations in terms of solving the Schrödinger equation using eitherfree energies [8], internal energies [9] or a linear combination of both [10] were recently performedleading to different results for the temperature dependence of binding energies of heavy quarkbound states in the quark gluon plasma. Some quarkonium dissociation temperatures obtained byassuming vanishing binding energy are summarised in Tab. 1. Although the results differ, withthe smallest dissociation temperatures obtained using F ( r , T ) and the highest using U ( r , T ) , theyindicate that at least J / ψ may survive the deconfinement transition as a bound state, while thesituation for the higher states is still not obvious.state J / ψ χ c ψ ′ ϒ χ b ϒ ′ χ ′ b ϒ ′′ E is [ GeV ] T d / T c T d / T c ∼ . ∼ . 05 unbound ∼ . ∼ . ∼ . 18 - - T d / T c > ∼ ∼ . ∼ . Table 1: Estimated dissociation temperatures T d in units of T c obtained from potential models using freeenergies [8] (green), a linear combination of F and U [10] (blue) and internal energies [9] (red) as effective T -dependent potentials. 6. Conclusions We have compared the screening radii of heavy quark anti-quark pairs in the quark gluonplasma phase to the (zero temperature) mean squared charge radii of charmonium states and foundindications that the J / ψ may survive the phase transition as a bound state, while χ c and ψ ′ areexpected to show significant thermal modifications at temperatures close to the transition.Beyond this simple approximation of the medium modifications of charmonium bound states above PoS(LAT2005)192 192 / 5 Static quark anti-quark free and internal energy in 2-flavor QCD Olaf Kaczmarek [MeV] r [fm] 0.88T c c c [MeV] r [fm] 1.09T c c c c c c c Figure 4: The colour singlet quark anti-quark internal energies, U ( r , T ) , at several temperatures below (left)and above (right) the phase transition obtained in 2-flavour lattice QCD. In (left) we also show as horizontallines the asymptotic values which are approached at large distances and indicate the flattening of U ( r , T ) .The solid lines represent the T = V ( r ) [1, 7]. 5. Bound states in potential models Various potential model calculations in terms of solving the Schrödinger equation using eitherfree energies [8], internal energies [9] or a linear combination of both [10] were recently performedleading to different results for the temperature dependence of binding energies of heavy quarkbound states in the quark gluon plasma. Some quarkonium dissociation temperatures obtained byassuming vanishing binding energy are summarised in Tab. 1. Although the results differ, withthe smallest dissociation temperatures obtained using F ( r , T ) and the highest using U ( r , T ) , theyindicate that at least J / ψ may survive the deconfinement transition as a bound state, while thesituation for the higher states is still not obvious.state J / ψ χ c ψ ′ ϒ χ b ϒ ′ χ ′ b ϒ ′′ E is [ GeV ] T d / T c T d / T c ∼ . ∼ . 05 unbound ∼ . ∼ . ∼ . 18 - - T d / T c > ∼ ∼ . ∼ . Table 1: Estimated dissociation temperatures T d in units of T c obtained from potential models using freeenergies [8] (green), a linear combination of F and U [10] (blue) and internal energies [9] (red) as effective T -dependent potentials. 6. Conclusions We have compared the screening radii of heavy quark anti-quark pairs in the quark gluonplasma phase to the (zero temperature) mean squared charge radii of charmonium states and foundindications that the J / ψ may survive the phase transition as a bound state, while χ c and ψ ′ areexpected to show significant thermal modifications at temperatures close to the transition.Beyond this simple approximation of the medium modifications of charmonium bound states above PoS(LAT2005)192 192 / 5 FIG. 50: from [118]. Free energy singlet potentials F ( T, r )(top plot) and the potential energy U ( T, r ) below and above T c . While at low T this is perfectly good question, but itbecomes difficult as T increases: the notion of a stategets ambiguous. Proper field theory object to study isa correlation function of local gauge invariant operators,such as charmed scalar/vector currents K ( x, y ) = < ¯ c Γ c ( x )¯ c Γ c ( y ) > (70)where the average is over the heat bath and x or time x = t can be Minkowskian or Euclidean. In both casesit is related with the same spectral density ˜ K ( ω, k ) char-acterizing all physical excited states with given energyand momentum. At low T one can see individual statesas certain peaks at the lines ω = ω i ( k ) in the spectraldensity, but with increasing T they merge into a smoothcontinuum.Euclidean correlation functions had been numericallycalculated on the lattice: for a recent review see [117]. Unfortunately the problem of spectral density reconstruc-tion from those is mathematically badly defined. In prac-tice, highly accurate (and very expensive) Euclidean cor-relation functions are converted into a relatively poorlydefined spectral density. Even when the individual statesare seen, as some peaks in spectral density, there ishardly any accuracy to define their widths. Above cer-tain T all peaks corresponding to charmonium statesmerge into one “near-threshold bump”, the imprint ofthe Sommerfeld-Gamow enhancement due to an attrac-tive potential ∼ e − V/T > 1. .(ii) A more detailed – time dependent – set of questionscan be asked about transition rates in equilibrium matter.Those has been addressed (at least) at three levels, (a)real time QFT; (b) quantum mechanical; (c) classicaldiffusion.Real time QFT, also known as Schwinger-Keldysh for-malism, can follow a system from some initial to somefinal state using the full Hamiltonian < i | P exp ( − (cid:90) fi dtH ) | f > (71)which is viewed as a sum of that for the subsystem inquestion H and matter perturbation V . Diagrammaticexpansion, including two-time contours as well as Mat-subara portion of an Euclidean time for thermal mediaare widely used in conduced matter problems, but theyare not much used so far in the problem we discuss.If H corresponds to non relativistic quantum mechan-ical description of quarkonium, we will call it quantummechanical approach. One can evaluate matrix elementsof V over various quarkonia states. Already my first 1978paper on QGP signals [84] had charmonia (called psionsin its title). The first considered in it was J/ψ excitation to unbound states of ¯ cc due to photoeffect-like reactionsof one gluon absorption J/ψ + g → ¯ cc . For heavy quarko-nia the diagonal part of the real and imaginary part of theperturbation V can be considered as a modified potential:for recent review see [116]. More generally, one can definetransition rates between states and write a rate equation .The fundamental question here is of course whether the“matter perturbation” V is small or not. (We will ar-gue below that at very low and very high T perturbativeapproach may work, but at least for charmonium in thenear- T c matter the answer to this question is negative).Suppose the “perturbation” V is not small comparablewith the interparticle interaction: then quantum quarko-nium states are no longer special and one can as well usefor H just free particles. Using mass as large parameter,one can argue [145, 146] that even in strong coupling set-ting the heavy quark motion should be described by clas-sical stochastic equations, the Langevin or Fokker-Planktype. Let me only mention two crucial consequences ofthe argument. First, motion is diffusive, with x ∼ √ t asit happens in random uncorrelated directions. Second,each step in space is very small. Suppose a perturbationdelivers a kick of the order T to a heavy quark of mass M (cid:29) T . Its velocity is changed little, by ∆ v ∼ T /M t ∼ /T the shift incoordinate is small ∆ x ∼ /M .Suppose a quark needs to diffuse a distance largeenough so that the gradient of the potential no longerpulls it toward the antiquark. From internal potentialsdisplayed above one can see that the distance it needsto go is about 1.5 fm, or ∼ x jumps it can make.However, since it is moving diffusively, to get that farthe quark would need ∼ jumps, which can well belarger than time available. Quantitative study of classi-cal diffusion to charmonium made by Young and myself[120] confirmed that to climb out of the attractive poten-tial in multiple small steps is hard. Contrary to commonprejudice, using the realistic charm diffusion constant wefound that the survival probability of J/ψ is not smallbut is of the order of 1/2 or so.(iii) Finally, in order to model the fate of heavy quarks/quarkonia in heavy ion collisions, one need to follow themall the way, from initial hard collisions to the freezeout.In classical diffusion approach one starts with pair dis-tribution in the phase space, as defined by the partonmodel, and at the end project the resulting distributionto the charmonia states using their Wigner functions.If the reader is insufficiently puzzled by all that, let mefinish this section presenting two opposite answers to thequestion: What is the effect of the QGP production onthe charmonium survival? In 1986 Matsui and Satz [115]famously argued that QGP, via the Debye screening ofthe color potential, kills charmonium states sequentially ,from excited states down to the ground one. In 2008Young and myself [120] argued that strongly coupledQGP helps to preserve charmonium states, since smalldiffusion constant prevents Q ¯ Q to move away from eachother. There are perhaps no identifiable charmoniumstates during the process: but so what? Close pairs atthe end get projected back to the bound states. B. Charmonium composition Let me start with the most radial model of all, thatof statistical hadronization of charmonium at chemicalfreeze out [37]. It assumes that charmonium states, likeall light hadrons, are produced in thermal equilibriumstate at this time. If true, all questions discussed inthe previous subsection are completely irrelevant, sinceall charmonium history during the intermediate stages issimply forgotten.The data at RHIC and LHC show that this is onlypartially true, and so we witness two component of thecharmonium population, the “survived” one and the “re-combination” contribution. Observations of the recombi-nation component is one of the most important recent re-sults from heavy ion experiments. Instead of showing theactual data, let me ask the reader to look at the originalsources, such as Andronic’s summary talk at QM2014,which does it very well.Let me proceed to next order questions, related in par- ticular with relative population of charmonium states. Ifall of them come the statistical hadronization at chemicalfreezeout, the consequence should be N ψ (cid:48) N J/ψ = exp (cid:18) − M ψ s − M J/ψ T ch (cid:19) (72)and similar formulae for other states. However if wehave two out-of-equilibrium components, with differenthistory, the answer should be different. The “survival”component, with its flow of probability from small tolarge r , should be richer in lower states. The “recombi-nation” component flows the opposite way, and it shouldhave more higher states instead. In general, two compo-nents have different centrality and p t dependences, andin principle can be separated.Experimental data are rather incomplete: there is how-ever puzzling double ratios found by CMS [38] which hintthat the situation can be rather interesting. C. New types of “quarkonia states” in a stronglycoupled medium Like in other parts of heavy ion theory, there is aweakly coupled and strongly coupled points of view onthis question. Which one to chose is now parametric: ifthe quark mass is very large M (cid:29) T , quarkonia are non-relativistic and perturbative. In zeroth approximationone thus starts from the unmodified vacuum states, witheffects such as excitations and Debye screening includedsubsequently. Those can be described by corrections toreal part of the potential δReV and appearance of itsimaginary part ImV , see e.g. [116] for nice summary ofthis theory.The opposite picture of very strong coupling has beendiscussed in [120]. If ImV is large, using the initial vac-uum states make no sense. Indeed, there are no 2-particlebound states in a liquid, just certain correlations betweenthe charges. Classical theory has two input parame-ters. The first is the diffusion constant D of the charmquark, and the second is the “interaction strength”, usu-ally characterized by the “plasma parameter”Γ = mean potential energyT (73)in which the ambient temperature T stands for typicalkinetic energy. Classical strongly coupled systems haveΓ > cc pair, locally produced, re-mains stuck at the same place during the sQGP periodof time. At the end matter returns to hadronic phaseand the ground state J/ψ is recreated, simply because ithas the largest wave function at zero. If so, there shouldbe no J/ψ suppression at all! sQGP helps preserve thecharmonia.7More realistic simulations with finite diffusion constantcan be set using Langevin or Fokker-Planck (FP) equa-tions. One setting starts with close ¯ cc pair, which makes positive flow toward large relative distance r → ∞ . Theother – recombination – setting starts with originally un-related ¯ cc at large r , with negative diffusion current to-ward small r . Both are followed for some time of sQGPera, and at the hadronization the obtained distributionsin the phase space are projected back to the vacuumquarkonia states, using their Wigner functions.Which picture is closer to the truth, weak or strongcoupling, depends on parameters such as the quark mass:perhaps bottonium is closer to the former regime whilecharm perhaps closer to the second one.Not going into details here, let me just say that boththe diffusion coefficient for a charm quark in QGP andthe potential are reasonably well defined[167] and thesimulations were done in [120] and in subsequent paperfor the recombination: the difference is the direction ofthe diffusion, equations are the same.The point I try to make does not depend on the detailsof those papers but concerns their qualitative observa-tions: both the inward/outward diffusions are very slow,and the reason for that is quite interesting. The spatialdistribution rapidly reaches certain shape which persistswith only slow growth of its tail. The example is shownin Fig. 51. In this case the attractive Coulomb-like po-tential has been complemented by a repulsive quantumeffective potential ∼ (cid:126) /mr which generates the hole inthe distribution at small r and prevents classical fallingof the charge on the center.We called solutions with a nearly-permanent shape andsmall flux “quasi-equilibrium” solutions. This concept is– to my knowledge – not yet been noticed in this par-ticular field, but it deserved to be. Let me show howquasi-equilibrium solutions works for the quark diffusionproblem, using the FP equation ∂P∂t = ∂∂(cid:126)r D ( ∂P∂(cid:126)r + βP ∂V∂(cid:126)r ) (74)where P ( t, (cid:126)r ) is the distribution over the ¯ cc separation (cid:126)r at time t , D is the diffusion constant, β = 1 /T and V ( r )is the interquark potential. First of all, when P ∼ exp ( − βV ) (75)two terms in the r.h.s. bracket cancels: this is stationar-ity of the true equilibrium. The bracket in the r.h.s. of(74) – the particle flux – is then zero.Note however what happens when this bracket isnonzero but constant(r) : the divergence of the flux stillmakes the r.h.s. to vanish. Then the l.h.s. is also zero:these solutions are stationary. Such solutions, while sta-tionary, still possess a constant flow of particles. Thedirection depends on the sign of the constant, it can befrom small to large r as in charmonium suppression prob-lem, or from large to small r for recombination. FIG. 2: Probability of a ¯ cc pair going into the J/ψ -state vs.time, for very early time. quark pair’s separation increases. Once the time reachesabout 1 fm/c, the probability for a pair to go into a J/ψ -state is about where we started, and we quit look-ing at this approach after this much time because themean transverse momentum for a quark is the thermalaverage.After this first 1 fm/c of the QGP phase, the¯ cc distribution has thermalized in momentum space andthe evolution in position space (diffusion) needs to beexamined. The root mean square distance for diffusivemotion is given by the standard expression ! x " = 6 D c τ (20)where τ is the proper time and the interaction betweenthe quarks has been neglected. The “correlation volume”in which one finds a quark after time τ is V corr = 4 π D c τ ) / (21)and one may estimate for the probability of the ¯ cc pairto be measured in the J/ψ -state as P ( τ ) ∼ R J/ψ / (6 D c τ ) / (22)So neglecting the pair’s interaction leads to a small prob-ability that J/ψ -states will survive by the hadronizationtime at the RHIC ( τ ∼ 10 fm /c ), even for small values ofthe diffusion coefficient.To get an idea for how this simple result is changed bythe inclusion of an interaction between the constituentquarks in a given ¯ cc -pair, let us examine the Fokker-Planck equation for the ¯ cc distribution in relative posi-tion: ∂P∂t = D ∂∂ r f ∂∂ r ( P/f ) (23) − − − FIG. 3: (Color online.) Numerical solution of theone-dimensional Fokker-Planck equation for an interacting¯ cc pair. The relaxation of the initial narrow Gaussian distri-bution is shown by curves (black, red,brown,green,blue, or topto bottom at r=0) corresponding to times t = 0 , , , 10 fm,respectively. where f ( r ) ∝ exp ( − V eff ( r ) /T ) is the equilibrium dis-tribution in the magnitude of relative position r . Bysubstituting the potential shown above at T = 1 . T c and D c × (2 πT ) = 1 into the Fokker-Planck equation(for demonstration in a single spatial dimension only) wesolve it numerically and find how the relaxation processproceeds. A sample of such calculations is shown in Fig.II. It displays two important features of the relaxationprocess:(i) during a quite short time T, f ,where the effective potential is most attractive;(ii) the second stage displays a slow “leakage”, duringwhich the maximum is decreasing while the tail of thedistribution at large distances grows. It is slow becausethe right-hand side of the Fokker-Planck equation is closeto zero, as the distribution is nearly f . The interactiondrastically changes the evolution of the ¯ cc distributionin position space, and this will be demonstrated again inthe full numerical simulation of the next section. FIG. 51: from [120]: one-dimensional Fokker-Planck equa-tion for an interacting c?c pair. The relaxation of the ini-tial narrow Gaussian distri- bution is shown by curves (black,red,brown,green,blue, or top to bottom at r=0) correspondingto times t = 0, 1, 5, 10 fm, respectively. Experimental indications It is possible to compare the experimental D-meson R AA with the theoretical expectation in the case of kinetic equilibrium Spectrum in pp given by POWHEG+PYTHIA setupFinal spectrum in AA given by hydro + Cooper-Frye p T (GeV/c) R AA D therm (T F0 =155 MeV), σ QQbar with shadSTAR data (stat.err. only) 0-10%Au-Au coll. @ 200 GeV0-10% centr. class (b=3.27 fm) p T (GeV/c) R AA D therm (T F0 =155 MeV), σ QQbar with shadALICE data (stat.err. only) 0-20%Pb-Pb coll. @ 2.76 TeV0-10% centr.class (b=3.32 fm) Evidence of peak from radial flow at RHIC, while more data at low- p T (waiting for ALICE ITS upgrade, S. Siddhanta talk) necessary at LHC; inany case charm at least partially out of kinetic equilibrium Andrea Beraudo Dynamics of heavy flavor quarks in high energy nuclear collisions From quarks to hadrons: effect on R AA and v R AA and a sizable v one wouldlike to interpret as a signal of charm radial flow and thermalization p T (GeV/c) R AA D therm (T F0 =155 MeV), σ QQbar with shadSTAR data (stat.err. only) 0-10%Au-Au coll. @ 200 GeV0-10% centr. class (b=3.27 fm) p T (GeV/c) v D thermal spectrum (T FO =155 MeV)ALICE data D aver. (stat. err. only) 30-50%Pb-Pb coll. @ 2.76 TeV30-40% centr.class (b=9.25 fm) Andrea Beraudo Dynamics of heavy flavor quarks in high energy nuclear collisions FIG. 52: STAR data on RAA of D mesons, and ALICE v2 ofD, both from [150] D. Do charmed mesons and charmonia flow? Let me start with an observation that D mesons havea maximum in R AA shown in Fig. 52(a) resembling thatfor protons: and we know the latter is of hydrodynamicalorigin. However the peak is at p t = 1 . GeV , lower than8for the nucleon, while the mass of D is higher. Directcomparison to hydro at Cooper Fry (the line) does notdo such a good job as it does for light flavor hadrons. Theobserved elliptic flow shown in Fig.52(b) is nonzero butsmaller that hydro predicts (curve). So, (i) either charmis heavy enough to be out of equilibrium with matter, or(ii) it is in kinetic equilibrium with it but spectra werenot correctly calculated. (The meaning of the secondoption will be soon clarified.)LHC data on charmonium dependence on p ⊥ is tran-sitionally done in the form of R J/ψAA ( p ⊥ ), normalizing ita la parton model. A comparison of PHENIX,RHIC andALICE,LHC data, shown in Fig.53(a) (from [149]) dis-play drastic change. There are much more charmonia atLHC than at RHIC at small p t : those presumed to bedue to the “recombined” ones. The observed v of J/ψ Fig.53(b) is non-zero but smaller than hydro predicted.The recombined component is produced from matterand thus it is maximal at small p ⊥ , while the primor-dial are mostly fast (relativistic) P/M > 1. It is pre-sumed that recombined component flows with matter.The question is whether the primordial component alsodo so, or “slip” relative to the medium: the data seem tofavor the latter option. J/ ψ vs. p T - data [email protected] midrapidity forward rapidity ) c (GeV/ T p AA R - e + e →ψ Inclusive J/ |<0.8, centrality 0-40% y =2.76 TeV, | NN s ALICE Preliminary, Pb-Pb, |<0.35, centrality 0-40% y =0.2 TeV, | NN s PHENIX, Au-Au, |<1, centrality 0-60% y =0.2 TeV, | NN s STAR, Au-Au, - e + e →ψ Inclusive J/ |<0.8, centrality 0-40% y =2.76 TeV, | NN s ALICE Preliminary, Pb-Pb, |<0.35, centrality 0-40% y =0.2 TeV, | NN s PHENIX, Au-Au, |<1, centrality 0-60% y =0.2 TeV, | NN s STAR, Au-Au, ) c (GeV/ T p AA R = 0.2 TeV NN s = 2.76 TeV and Au-Au NN s Pb-Pb ± 20% global syst. = <4, centrality 0% y , 2.5< - µ + µ → ψ ALICE J/ 10% ± 20% global syst. = |<2.2, centrality 0% y , 1.2<| - µ + µ → ψ PHENIX J/ further support of (dominance of) a new production mechanism [(re)generationin QGP or at chemical freeze-out] ALICE, arXiv:1311.0214 (& prelim., Book, HF 4) J/ ψ flow [email protected] ALICE, PRL 111 (2013) 162301 CMS (Moon, HF 4) ) c (GeV/ T p v -0.100.10.20.3 < 4.0 y = 2.76 TeV), centrality 20%-60%, 2.5 < NN sALICE (Pb-Pb Y. Liu et al., b thermalized Y. Liu et al., b not thermalized X. Zhao et al., b thermalized ± global syst. = further support of production in QGP or at chemical freeze-out at the LHC (requiring thermalization of c, ¯ c and generically leading to flow) Recall: non-zero v was measured at SPS (“leakage effect”)...the RHIC case (STAR, v ∼ ) remains open ...upcoming data will settle it FIG. 53: Comparison between ALICE and Phenix charmo-nium R AA (a) and charmonium elliptic flow (b), both from[150]. E. Inhomogeneous distributed charm He, Fries and Rapp [74] had demonstrated that if oneartificially increases the drag coefficient, one reproducesthe p t distribution obtained from hydrodynamical flowof charm. In other words, they nicely checked that athigh enough scattering cross section charm is kineticallyequilibrated with matter. In this calculation the originalparticles were homogeneously distributed over the fire-ball at some early time, which is indeed needed to getconsistency between the two calculations.Yet what would happen if the drag/cross section bestill large, but the initial charm be distributed over thefireball in a non − unif orm way. Introducing charm fu-gacity f = exp ( µ c /T ) in a standard manner, one shouldthink of it as some function of time and space point, de-pending on the production mechanism. Charm recombi-nation into charmonium states is calculated at the chem-ical freezeout surface, and it should be proportional to anaverage of over it < f ( t, (cid:126)r ) > , not square of the average < f > as it is done now. In short, I wander what oneshould do for thermal recombination when the charm fu-gacity is not homogeneously distributed over the system,as is usually assumed.This lead to nice hydro exercise I will briefly tell about.Let me start with the longitudinal hydro equation writtenas entropy conservation[ s ( t, x ) u µ ] ; µ = 0 (76)with the semicolon meaning the covariant derivative. Thehomogeneous distribution means that n/s = const ( t, x ),and thus any solution for the entropy generates solutionfor n ( τ, r ) as well. If one would like to put an arbitrarydistribution of charm (or any other conserved quantity)density at some initial time, in order to find its distri-bution later one has to solve its conservation equation[ n ( τ, r ) u µ ] ; µ = 0 at all times. I started with Gubser flow,in coordinates proper time transverse distance τ, r , forwhich s ( τ, r ) is known and put n ( τ, r ) = f ( τ, r ) ∗ s ( τ, r ).The resulting “fugacity equation” reads2 rτ ∂ r f + (1 + r + τ ) ∂ τ f = 0 (77)It has a solution with an arbitrary function φ of the fol-lowing argument f = φ ( 2 r − r + τ ) (78)The function φ can be fixed from the initial conditions at τ = 0. Using it, one finds the corresponding distributionof our quantum number (charm) at the freezeout surfaceas well.The problem was solved, but the meaning of this an-swer looked mysterious. It was further clarified when Iused the co-moving coordinates (see appendix) ρ, θ . Allone needs to know is that ρ has meaning of time and θ ofspace, and the particle conservation in a comoving frame9reads simply as ∂n∂ρ = 0 (79)meaning that any function of θ independent on coordi-nate ρ solves the problem. And constant theta impliesconstancy of the combination I obtained in the originalcoordinates. The lesson I got from this is that even innumerical solutions of the hydro equations one perhapsshould try to find the co-moving coordinates. Thosewould lead to very natural distributions of external ob-jects like charm.I did checked that for recombinant J/ψ the effect of < f > (cid:54) = < f > is actually small and can perhaps beignored. I however further found that the inhomogeneityeffect suppresses v of charm, especially at higher p t , be-cause the fireball edge is a bit less populated by charmthan its middle. My calculation gives v inhomogeneous v homogeneous | p ⊥ =3 GeV = 0 . 65 (80)which brings hydro prediction closer to experimental oneson v J/ψ . Perhaps recombined charmonia do flow with thematter, after all. XI. JET QUENCHING Let me mention on the onset that hard QCD is ex-pected to be perturbative, and this is not my field. Ithus would not go into gluon radiation, in vacuum andin matter, in any detail. This small section contains onlysome remarks on recent theory developments. A. “Quasi-equilibrium” in jets I think it has been an important development relatingjet-in-matter physics with a more general turbulence the-ory. In it I will follow ref.[123] by Blaizot et al. For a largeenough medium, successive gluon emissions can be con-sidered as independent: multiple emissions can be treatedas probabilistic branching processes, with the BDMPSZspectrum playing the role of the elementary branchingrate. The inclusive gluon distribution function dNdlog ( x ) d (cid:126)k = D ( x, (cid:126)k, t )(2 π ) (81)satisfy certain diffusion-brunching equation. Integratingover transverse momentum one gets the so called zerothmoment D ( x, t ) = (cid:82) k D ( x, k, t ) on which we focus forsimplicity. This satisfies the equation t ∗ ∂D ( x, t ) ∂t = (cid:90) dzK ( z ) (cid:20)(cid:114) zx D ( xz , t ) − z √ x D ( x, t ) (cid:21) (82) with the gain and the loss terms. The details such asthe shape of the kernel K and time parameter t ∗ can befound in ref.[123]. The central feature I want to focus onis the analytic solution D ( x, t ) = ( t/t ∗ ) √ x (1 − x ) / exp ( − πt t ∗ (1 − x ) ) (83)Essential singularity at x = 1 is expected: it is known asthe Sudakov suppression factor. The remarkable news isthat apart of the exponent, the shape of the x-dependenceremains the same at all times, only the normalizationchanges.Let us see how it works in the most important small x << x − / dependence, the gain and loss terms simply cancel. Thisis the quasi-equilibrium solution under consideration. Asa result, jets in matter are expected to approach someuniversal shape, not determined by the particular initialconditions, but by the quasiequilibrium solution to whichit gets attracted as the process proceeds.It is essentially the same phenomenon as we have seenin Fig.51 for diffusing charmonium: the shape itself isdictated by the balance of the gain and loss. Both are“quasi-equilibrium” attractor solutions: their main fea-ture is constant flux of certain quantity, from one end ofthe spectrum to the other. (The flux in decaying charmo-nium comes from small to large distance between quarks,in the jet case it comes from large to small x .) Onceagain, the constancy of the flux in such solutions is thekey idea going back to Kolmogorov’s theory of hydrody-namical turbulence, which I find quite remarkable. AsEinstein once observed (approximate quote from mem-ory): “... the number of good ideas in physics is so small,that they keep being repeated again and again in variouscontext”. B. Is jet quenching dominated by the near- T c matter? Let me start with Fig. 54, emphasizing some fea-tures of the LHC (ALICE) data. The first one showsthe suppression factor R AA compared with theory basedon pQCD expressions. The first obvious feature I wantto point out is the pronounced dip in the data, clearlyseparating two distinct regimes. Although the magni-tude is centrality dependent, its location at p dipt ∼ GeV changes a little. (It also about the same as at RHIC,where the dip is minor.). The region above the dip p t > p dipt is well described by jet quenching models, whilebelow the dip it is perhaps some tail of hydro-related phe-nomena, yet to be understood.Now we proceed to angular distribution of jet quench-ing, fig.(b) on which v ( large p t ) is plotted. Already atthe very beginning of RHIC era I noticed [124] a prob-lem: direction-dependent in or out-of reaction plane by0as R in/outAA = R AA (1 ± v ) (84)apparently is incompatibly large to most theories of jetquenching. As one can see from the second Fig. 54 ,it is still very much true at LHC. Theory curves whichdescribe so well R AA , all the way down to the dip p dipt , lead to v (at the dip) much smaller than needed!What is new here are LHC measurements at much higher p t > GeV , demonstrating v values which are consis-tent with the pQCD models. Yet it is still true that in therange 6 < p t < GeV something significant is missing.(Steeply falling another contribution Fig. 54 (a) belowthe dip makes it unlikely to be important.)A solution has been proposed by Liao and myself [125]:we found that one can get near the observed v valueif the jet quenching is strongly enhanced at the near- T c matter. We motivated it by presence of magneticmonopole in the near- T c region.A model based on this idea can be reconciled withRHIC and LHC data. Let me jump to the recent paper[148] in which the corresponding model is well developed.In Fig.54 one finds fits of the R A A and v of the jets formodels with and without magnetic component. In Fig.55we show the ˆ q ( T ) of the model for a jet with 20 GeV. XII. NEAR THE PHASE BOUNDARY:FLUCTUATIONS AND THE FREEZEOUTSA. Chemical freezeouts Statistical equilibration is a famous success story, andit hardly needs to be emphasized here. Let me just showALICE data with thermal fits in Fig.56, presented atCPOD 2014 by A. Kalweit.Note that even light nuclei – d, t, He and their an-tiparticles are also included. One may wander how d ,with its 2 MeV binding energy, can be found in an envi-ronment with ambient temperature T ∼ M eV . It isliterally finding a snowflake jumping out of a hot oven.Shouldn’t d be instantly destroyed in it? The answer tothis and similar questions is well known: thermodynam-ics does not care about d lifetime. As it is destroyed withsome rate, perhaps large, in equilibrium it is recreated bythe inverse process with the same rate, so that its averagepopulation is conserved. What thermodynamics gives usis that average.Note that one deviation is K ∗ : the model predictsmore than observed. This is expected: it is short livedresonance which decays when the density is still non-negligible, the products can be re-scattered and their in-variant mass moved out of the peak. Corrections to thatcan be made using any cascade codes.Another deviation is for p + ¯ p . Some argued thatone should take into account possible annihilation onthe way out, and again use available practical solution FIG. 54: (color online) From [148]: jet suppression and ellipticparameter v , data versus models.FIG. 55: (color online) the NTcE model with enhanced near- T c quenching FIG. 56: Alice data on particle yields compared to thermalmodel [37]. The main fit parameters are indicated in theupper plot. like UrQMD or similar code to calculate it. Here how-ever comes an objection: all of them include annihilation p + ¯ p → nπ, n ∼ not the inverse reaction. Thisis an old story [153]: contrary to popular beliefs the in-verse reactions are not suppressed, in fact at equilibriumtheir rate are exactly the same as the direct one. Usinga cascade code which does not respect the detailed bal-ance – and thus the very concept of thermal equilibrium– to calculate corrections to thermal model is just logicalnonsense.I would not show the freezeout points on the phase di-agram, which has been done many times. Let me justremind that those seem to be remarkably close to thephase boundary, defined on the lattice. Why should thisbe the case? An answer suggested by Braun-Munzinger,Stachel and Wetterich [69] is also related with the multi-particle reaction rates. Since those depend on very highpower of the temperature, they all should decoupled veryclose to critical line | T ch − T c | T c (cid:28) T , in mag-nitude.New development is usage of event-by-event fluctua-tions. As it has been first pointed out in my paper [70],those in general provide information about multiple sus-ceptibilities, or higher derivatives of the free energy over T or various chemical potentials.This idea further was applied toward location of thecritical point in [71], suggesting the low energy scan pro- gram. The results of the RHIC scan are however still toofluent to be reviewed, even after QM14.Let me instead comment on another development, onthe interface of the lattice and heavy ion communities.Calculation of susceptibilities on the lattice, includingthe most difficult near- T c region, reached significant ma-turity. Karsch stressed at many meetings, that highersusceptibilities now can be compared to the data and thefreezeout curve be reconstructed, even without using theparticle ratios. I will follow here [40] from which Fig.57is taken: Fig(a) compares new set of freezeout param-eters (points) compared to earlier one from the parti-cle ratios. Fig.(b) shows that as µ grows a consistencybetween different ratios is getting worse. It is howevergenerally believed that at chemical freezeout the HadronResonance Gas model is generally describing the QCDthermodynamics, expect perhaps in the vicinity of theQCD critical point. 40 60 80 100 120 140 160 0 100 200 300 400 500 600 700 800 T ( M e V ) µ B (MeV) This workCleymans et al., PRC (2006) Figure 3: (Color online) Freeze-out parameters in the ( T − µ B ) plane:comparison between the curve obtained in Ref. [28] (red band) andthe values obtained in the present analysis from a combined fit of σ /M for net-electric charge and net protons (blue symbols). freeze-out temperature and baryo-chemical potential forthe different collision energies are given in Table 1. Conclusions In conclusion, our study shows that we can simulta-neously describe the net-electric charge fluctuations andthe lower-order cumulants of the net-proton multiplic-ity distributions measured at RHIC for collision energiesspanning over more than an order of magnitude ( √ s =(11 . − σ /M for net-electric chargeand net-proton number, we obtain the freeze-out condi-tions summarized in Table 1. The resulting freeze-outtemperatures are constrained to better than 5 MeV for √ s > . T ch and µ B,ch [17].We note that a useful cross-check of our extracted che-mical freeze-out parameters can be provided through theindependent determination of the same parameters via acommon fit of standard SHMs to experimental particleyields or ratios [27, 28, 40]. At first glance, our param-eters are below those extracted from SHM fits as is alsoevident from Fig. 3. We note, however, that the latestLHC data [41] seem to suggest a separation of chemicalfreeze-out parameters according to particle flavor, which is χ Q / χ Q √ s (GeV)HRG modelSTAR data 0.2 0.4 0.6 0.8 1 10 100 χ B / χ B √ s (GeV)HRG modelSTAR data-6-4-2 0 2 4 6 8 10 10 100 χ Q / χ Q √ s (GeV)HRG modelSTAR data 0 0.2 0.4 0.6 0.8 1 10 100 χ B / χ B √ s (GeV)HRG modelSTAR data Figure 4: (Color online) Comparison between HRG model resultsfor χ X /χ X and χ X /χ X , with X = Q (left) and X = B (right)as functions of √ s (blue crosses), and experimental data for themost central collisions (0 − √ s (GeV) HRG model, χ / χ STAR data, χ / χ HRG model, χ / χ STAR data, χ / χ χ B / χ B √ s (GeV)HRG modelSTAR data Figure 5: (Color online) Left: Comparison between HRG model re-sults and experimental data for the most central collisions (0 − σ /M of net-electric charge (blue, upper sym-bols) and Sσ of net protons (red, lower symbols). The experimentaldata have been fitted in the HRG model in order to extract thefreeze-out parameters for each collision energy. Right: Net-proton σ /M calculated with the freeze-out conditions obtained from thesimultaneous fit shown in the left panel. also supported by recent lattice QCD simulations [42] andsequential SHMs [43]. Therefore, special emphasis shouldbe given to a fit to the light-quark particles only, i.e. pionsand (anti-)protons, which dominate the net-electric chargeand net-proton measurements, respectively. At the LHC,the proton data [44] indicate a rather low freeze-out tem-perature (smaller than 150 MeV), which is in line withour results. Preliminary results from the RHIC beam en-ergy scan [45] show freeze-out temperatures ranging from(140 − √ s = (7 . − 100 120 140 160 180 200 0 50 100 150 200 250 300 350 T ( M e V ) µ B (MeV) Fit1: χ / χ and χ / χ Fit2: χ / χ and χ / χ Figure 6: (Color online) Freeze-out parameters in the ( T − µ B ) plane:comparison between the values obtained by a combined fit of σ /M for net-electric charge and net protons (blue circles), and the val-ues obtained by fitting σ /M for net-electric charge and Sσ for netprotons (red squares). √ s [GeV] µ B,ch [MeV] T ch [MeV]11.5 326.7 ± ± ± ± ± ± ± ± ± ± ± ± Table 1: In this table we list the values of µ B,ch and T ch at chemicalfreeze-out, corresponding to the relative collision energies. Thesevalues are based on our combined fit to the data in Fig. 1. the higher-order fluctuations. Acknowledgements We gratefully acknowledge useful discussions with BillLlope. This work is supported by the Italian Ministryof Education, Universities and Research under the FirbResearch Grant RBFR0814TT, the Hessian LOEWE ini-tiative Helmholtz International Center for FAIR, and theUS Department of Energy grants DE-FG02-03ER41260,DE-FG02-05ER41367 and DE-FG02-07ER41521. References [1] Y. Aoki, G. Endrodi, Z. Fodor, S. D. Katz and K. K. Szabo,Nature , 675 (2006).[2] Y. Aoki et al. , Phys. Lett. B , 46 (2006); Y. Aoki etal. , JHEP , 088 (2009); S. Borsanyi et al. [Wuppertal-Budapest Coll.], JHEP , 073 (2010).[3] J. Berges and K. Rajagopal, Nucl. Phys. B , 215 (1999)[4] A. M. Halasz, A. D. Jackson, R. E. Shrock, M. A. Stephanovand J. J. M. Verbaarschot, Phys. Rev. D , 096007 (1998)[5] M. A. Stephanov, K. Rajagopal and E. V. Shuryak, Phys. Rev.Lett. , 4816 (1998). [6] M. A. Stephanov, K. Rajagopal and E. V. Shuryak, Phys. Rev.D , 114028 (1999).[7] R. V. Gavai and S. Gupta, Phys. Rev. D , 114503 (2008).[8] G. Endrodi, Z. Fodor, S. D. Katz and K. K. Szabo, JHEP , 001 (2011)[9] L. Adamczyk et al. [STAR Collaboration], Phys. Rev. Lett. , 032302 (2014).[10] L. Adamczyk et al. [STAR Collaboration], arXiv:1402.1558[nucl-ex].[11] D. McDonald [STAR Collaboration], Nucl. Phys. A904-905 ,907c (2013).[12] N. R. Sahoo [STAR Collaboration], Acta Phys. Polon. Suppl. , 437 (2013).[13] F. Karsch, Central Eur. J. Phys. , 1234 (2012).[14] A. Bazavov et al. , Phys. Rev. Lett. , 192302 (2012).[15] S. Mukherjee and M. Wagner, PoS CPOD 2013 , 039 (2013).[16] S. Borsanyi, Z. Fodor, S. D. Katz, S. Krieg, C. Ratti andK. K. Szabo, Phys. Rev. Lett. , 062005 (2013).[17] S. Borsanyi, Z. Fodor, S. D. Katz, S. Krieg, C. Ratti andK. K. Szabo, arXiv:1403.4576 [hep-lat].[18] M. Nahrgang, T. Schuster, M. Mitrovski, R. Stock and M. Ble-icher, Eur. Phys. J. C , 2143 (2012).[19] N. R. Sahoo, S. De and T. K. Nayak, Phys. Rev. C , 044906(2013).[20] V. V. Begun, M. I. Gorenstein, M. Hauer, V. P. Konchakovskiand O. S. Zozulya, Phys. Rev. C , 044903 (2006).[21] F. Karsch and K. Redlich, Phys. Lett. B , 136 (2011).[22] J. Fu, Phys. Lett. B , 144 (2013).[23] P. Garg, D. K. Mishra, P. K. Netrakanti, B. Mohanty,A. K. Mohanty, B. K. Singh and N. Xu, Phys. Lett. B ,691 (2013).[24] M. Nahrgang, M. Bluhm, P. Alba, R. Bellwied and C. Ratti,arXiv:1402.1238 [hep-ph].[25] S. Borsanyi, Z. Fodor, S. D. Katz, S. Krieg, C. Ratti and K. Sz-abo, JHEP , 138 (2012).[26] P. Braun-Munzinger, K. Redlich and J. Stachel, In *Hwa, R.C.(ed.) et al.: Quark gluon plasma* 491-599.[27] F. Becattini, J. Manninen and M. Gazdzicki, Phys. Rev. C ,044905 (2006)[28] J. Cleymans, H. Oeschler, K. Redlich and S. Wheaton, Phys.Rev. C , 034905 (2006).[29] J. Manninen and F. Becattini, Phys. Rev. C , 054901 (2008)[30] A. Andronic, P. Braun-Munzinger, K. Redlich and J. Stachel,J. Phys. G , 124081 (2011).[31] F. Karsch, K. Redlich, and A. Tawfik, Phys. Lett. B , 67(2003).[32] A. Tawfik, Phys. Rev. D , 054502 (2005).[33] D. Teaney, arXiv:nucl-th/0204023.[34] H. Bebie, P. Gerber, J.L. Goity, and H. Leutwyler, Nucl. Phys.B , 95 (1992).[35] M. Bluhm, P. Alba, W. Alberico, A. Beraudo and C. Ratti,arXiv:1306.6188 [hep-ph].[36] J. Beringer et al. [Particle Data Group], Phys. Rev. D ,010001 (2012).[37] S. Jeon and V. Koch, Phys. Rev. Lett. , 5435 (1999).[38] M. Kitazawa and M. Asakawa, Phys. Rev. C , 021901 (2012).[39] M. Kitazawa and M. Asakawa, Phys. Rev. C , 024904 (2012)[Erratum-ibid. C , 069902 (2012)].[40] A. Andronic, P. Braun-Munzinger and J. Stachel, Nucl. Phys.A , 167 (2006).[41] B. Abelev et al. [ALICE Collaboration], Phys. Rev. Lett. ,252301 (2012).[42] R. Bellwied, S. Borsanyi, Z. Fodor, S. D. Katz and C. Ratti,Phys. Rev. Lett. , 202302 (2013).[43] K. A. Bugaev, D. R. Oliinychenko, V. V. Sagun, A. I. Ivanyt-skyi, J. Cleymans, E. G. Nikonov and G. M. Zinovjev,arXiv:1312.5149 [hep-ph].[44] R. Preghenella [ALICE Collaboration], Acta Phys. Polon. B , 555 (2012).[45] S. Das [STAR Collaboration], arXiv:1402.0255 [nucl-ex]. FIG. 57: (Color online) (a) Freeze-out parameters in the( T − µ B ) plane: comparison between the curve obtained inCleymans et al (red band) and the values obtained in thepresent analysis from a combined fit for net-electric chargeand net protons (blue symbols). (b) The freezeout parametersfitted from two different set of rations, as shown in the plot FIG. 58: (color online) The temperature of the kinetic freeze-out versus mean velocity of the radial flow, fitted to ALICEspectra of the identified secondaries ( π, K, p, Λ , Ξ , Ω). B. From chemical to kinetic freezeouts Separation in magnitude of the elastic and inelastic(low energy) hadronic reactions is the basis of the “twofreezeouts” paradigm, with separate chemical T ch andkinetic T kin temperatures.Its effectiveness became even more clear at LHC, whichwe illustrate by Fig.58 containing the “blast wave” fittedparameters to the ALICE spectra of . Unlike T ch , thekinetic one T kin strongly depends on centrality of PbPbcollisions, decreasing to values below 0.1 GeV for themost central bins. Cooling from 0.16 to 0.1 GeV maynot look so dramatic: but thermodynamical quantitiesin this region are proportional to high power of T andthey do change strongly.First, does on understand the dependences displayedin Fig.58 at all? The most central AA collisions producethe largest systems, which have the highest T at the earlystages, and also the lowest T at the end. One popular wayto explain it is to mention, that the energy conservationnegatively correlates the flow velocity < β T > and T kin ,so more “explosive” systems cool more.While correct, this simple idea seems to be in contra-diction to the pp, pA data also shown in Fig.58, and forwhich T ch and T kin are close. An explanation to that,discussed in the middle of this review, is that high mul-tiplicity bins, displaying strong radial flow, achieve it viamore extreme conditions at their early stage, by increas-ing the initial T i . In other words, higher multiplicity inthis case means large energy/particle, which reveals it-self in the flow without cooling. (In AA collisions T i alsodepends on centrality, but much weaker.)The kinetics of the freezeout – that the rescatteringprobability on the way out should be about 1/2 – is ofcourse central to defining T kin . Cold fireballs created incentral PbPb have thousands of particles and freezeoutlate, at times reaching 15 fm. The highest multiplicitiesobserved in pp, pA have 10-20 times less particles, andfreezeout time/size of 3 fm or so.In summary, contrary to belief of many, in PbPb LHC collisions one does have an extended phase of the colli-sion, in which matter cools deep into the hadronic phase .This opens some interesting questions related to hadronicphase, which can now be addressed experimentally.Let me mention only one issue here. Since between T ch and T kin the particle numbers are conserved, one shouldintroduce new nonzero chemical potentials, not associ-ated with conserved quantum numbers like charge andbaryon number. In particular, there should be nonzerochemical potentials for pions.Whether there are nontrivial fugacity factors at thekinetic freezeout can be directly observed in the pionspectra, because in this case Bose enhancement can bemeasured. This idea is at least 20 years old [58]: itsexperimental manifestation was displayed in the paperby me and Hung [59] is shown in Fig.59(a). (The ref-erence SS collisions is much smaller system than PbPb,and thus its chemical and kinetic freeze outs should beclose.) Fig.59(b) shows that the same effect shows up,now at LHC. The fit without chemical equilibrium, withnonzero pion µ on top, provides a better description tothe spectra at small p t < M eV .Interesting that the parameter of the fit in this lastwork gives µ π ≈ m π , so the authors speculated if theconditions for pion Bose-Einstein Condensation (BEC)were actually reached. If this indeed becomes true, ithas been many times suggested previously that the fem-toscopy parameter λ should show it, as it is sensitive to“degree of coherence” of the pion source. Recent fem-toscopy data on 2 and 3 identical pions from ALICE,discussed in [155], can be indeed fitted with a coherentsource. The fraction of coherent pions coming from thisfit is as large as 23% ± T , and related to the inverse spatial size ofthe BEC cloud.This indeed sounds like what is observed in heavy ioncollisions: as one goes to most central collisions and thekinetic freezeout T kin get below 100 MeV, the p t spectrado become enhanced at small momenta. The differencehowever is in the shape: we don’t see a new Gaussian, asin the atomic experiments, but thermal spectrum modi-fied by µ .The condensate should be a separate component, with µ being exactly m π and independent on T . If BEC cloudcontains about 1/4 of all pions, its diameter should belarge, at least of the order of 2-3 fm. The correspond-ing width of momentum distribution, from uncertaintyrelation, should be as small as say < p t >< . GeV .Looking back to Fig.59(b) one however finds, that suchsoft secondaries seem to be outside of the acceptance. So,3even if BEC component is there, we so far cannot see it,neither with ALICE nor with any other LHC detectors!How then can we get their influence in the femtoscopy?This issue can perhaps be clarified by a short dedicatedrun, in which the ALICE detector switches to smaller(say 1/2 of the current value) magnetic field, to improvethe low p t acceptance. (Yes, recalculating all the effi-ciencies is a lot of extra work, but perhaps it is worthclarifying this interesting issue.) C. The search for the critical point and RHIC lowenergy scan The main idea [71] is 15 years old and well known:critical point – if exists – should lead to large correlation PbPb/SS positive pions, NA44 mu=60,80,100 MeV FIG. 6. The ratio of π + p t spectra for PbPb to SS collisions. Points are experimental datafrom NA44 experiment, three curves correspond to pion chemical potential µ π = 60,80 and 100MeV (from bottom up). ν ( G e V ) Pion Collision Rate Fit to PPVW ν π in π KPPVW ν π in π KAGS ν π N SPS ν π N FIG. 7. Pion collision rates ν = τ − coll [GeV] in a pion-kaon-nucleon gas versus temperature T[GeV]. FIG. 59: (Color online) (a) Points show the ratio of PbPb toSS spectra, from NA44. Three curves are for pion chemicalpotential µ π = 60 , , MeV , from [59]. (b) (From [60])Dots are ALICE transverse momentum spectra of pions inthe low- p t region, compared to the model with (upper plot)and without (lower plot) pion chemical potential. ! ¼ h " i ¼ TV ; ! ¼ h " i ¼ $ TV ; ! ¼ h " i c " h " i h " i ¼ TV ½ ð $ Þ $ ’ : (3)The critical point is characterized by ! 1 . The centralobservation in this Letter is that the higher moments (cu-mulants) ! and ! diverge with much faster than thequadratic moment ! .To be precise, the correlators scale slightly differentlythan Eqs. (3) suggest, e.g., h " i ( % . Since the anoma-lous dimension % ) : is very small, the differencebetween the actual asymptotic scaling and Eqs. (3) isdiscernible only for very large values of , irrelevant inthe context of this study. More importantly, the parameters $ and $ also scale with [see Eq. (12)].Of course, the fluctuations of the critical mode are notmeasured directly in heavy-ion collision experiments.These fluctuations do, however, influence fluctuations ofmultiplicities, momentum distributions, ratios, etc., of ob-served particles, such as pions or protons, to which thecritical mode couples [4]. The purpose of this Letter is todetermine the magnitude of these effects. Critical contribution to experimental observables.— Weshall now estimate the effect of the critical point fluctua-tions on the observables such as the pion multiplicityfluctuations. Using a similar approach, it should bestraightforward to construct corresponding estimates forsuch observables as charge, proton number, transversemomentum fluctuations, etc., as well as to take into ac-count acceptance cuts.We shall focus on the most singular contribution, pro-portional to a power of the correlation length . Thiscontribution can be found using an intuitive picture de-scribed in Ref. [4]: One considers a joint probability dis-tribution for the occupation numbers n p of observedparticles (e.g., pions) together with the value of the criticalmode field " (more precisely, its zero-momentum mode " ), the latter treated as classical. Because of coupling ofthe critical mode of the type "&& , the fluctuations of theoccupation numbers receive an additional contribution,proportional to the corresponding correlation functions(moments) of the fluctuations of " given by Eq. (3). Inthis Letter, however, it will be more convenient to useinstead the more formal diagrammatic method of Ref. [9]. Cubic cumulant.— The 3-particle correlator receives thefollowing most singular contribution from the " fluctua-tions, given by the diagram in Fig. 1: h ’ n p ’ n p ’ n p i " ¼ $ V T ! Gm " " v p ! p v p ! p v p ! p : (4)Subscript " indicates that only the critical mode contribu-tion is considered. As in Refs. [4,9], we denoted "&& coupling by G and introduced a shorthand notation forthe variance of the occupation number distribution: v p ¼ ! n p ð * ! n p Þ , where the ‘‘ þ ’’ is for the Bose particles. Since the total multiplicity is just the sum of all occu-pation numbers and thus ’ N ¼ X p ’ n p ; (5)the cubic moment of the pion multiplicity distribution isgiven by h ð ’ N Þ i ¼ V Z p Z p Z p h ’ n p ’ n p ’ n p i ; (6)where R p " R d p = ð & Þ . Since h ð ’ N Þ i scales as V , it isconvenient to normalize it by the mean total multiplicity ! N ,which scales similarly. Thus we define ! ð N Þ " h ð ’ N Þ i ! N (7)and find ! ð N Þ " ¼ $ T G m " ! Z p v p ! p " ! Z p ! n p " : (8) Quartic cumulant.— The leading contribution to the con-nected 4-particle correlator is given by the sum of twotypes of diagrams in Fig. 2: h ’ n p ’ n p ’ n p ’ n p i c; " ¼ V T ! $ m " " $ $! Gm " " , v p ! p v p ! p v p ! p v p ! p : (9)The quartic cumulant of multiplicity fluctuations isgiven by FIG. 1. Diagrammatic representation of the contribution to thethree-particle correlator from the critical mode " . Wavy linesrepresent propagators of the " field, each contributing factor =m " , and crossed circles represent insertions of ’ n p into thecorrelator Eq. (4)—see Ref. [9] for details.FIG. 2. Diagrammatic representation of the critical mode con-tribution to the connected four-particle correlator. Same notationas in Fig. 1. PRL week ending23 JANUARY 2009 FIG. 60: The enhanced contribution to 4 particle correlator,from [152]. lengths and enhanced e-by-e fluctuations, similar to crit-ical opalescence known in many cases. Mathematicallyspeaking one may go for effects given by diagrams whichhave as many critical propagators as possible. Stephanov[152] pointed out that since the n -particle correlators maycontain up to n such propagators, they are more sensi-tive to large correlation length: 3 particle correlators are ∼ ξ , 4-particle ones ∼ ξ , see Fig.60, and so on. Thewavy line at zero 4-momentum is ∼ /m σ ∼ ξ : butthe prediction is not just the power of the propagatorsbecause the coupling of critical modes by itself vanishesas certain power of ξ given by the critical indices. Thequartic one in the diagram considered is ∼ /ξ so thetotal power is 7, not 8.Let me first retell the same story in a simpler language.The critical field we here call σ should be viewed as somestochastic/fluctuating background field coupled to fluc-tuations in particle number (circles with crosses in thediagram above). One can view it being proportional tosome stochastic potential ∆ V ( x ) which enter the proba-bility in the usual way P ∼ exp ( − ∆ V ( x ) /T ), so that inits minima the probability is larger and more particles –e.g. 4 mentioned above – all gather there. The criticalpoint is special in that the scale of the correlation length ξ of this potential increases, and thus more particles havea chance to get into the same fluctuations.Which particles we speak about? In principle sigma isscalar-isoscalar, so any one of them. In our paper [71]and in [152] the simplest coupling was considered as anexample, the σππ one, and so the particles were pions.Let me now argue that using the nucleons should workeven better. First, the powers of the baryon density n B = N N − N ¯ N correlated together are the susceptibili-ties calculated on the lattice as derivatives over µ B . Sec-ond, we know from the nucleon forces – e.g. the simplestversion of the Walecka model – that σ is the main compo-nent of the attractive nuclear potential which binds thenuclei. ∆ V = g σNN πr exp ( − m σ r ) (86)In vacuum the typical mass m σ ∼ M eV and theinter-nucleon distance r ∼ . f m are combined into smallsuppression factor ∼ exp ( − (cid:28) V ∼ − M eV is much smaller4than the nucleon mass, in spite of strong coupling. (Atsmaller distance r the repulsive omega contribution dom-inates the attractive sigma one.).Can it be so, that at the QCD critical point m σ → V , perhaps even larger than thefreezeout temperature T . Furthermore, if say ξ = 2 f m ,the volume 4 πξ / ∼ f m is large enough to collectmany nucleons, not just 3 or 4, as Stephanov suggested.So, such clustering of the nucleons should produce largenuclear fragments, a new cute signal of the critical point!The argument, unfortunately, is rather naive. The crit-ical mode which gets long-range is not just the σ field but– because we are at nonzero density – a certain combi-nation with ω . Therefore the repulsive forces betweennucleons should be getting longer range as well. To tellwhat happens we need a reliable theory or some dedi-cated experiments. Fortunately, we can do it in the com-ing low energy scan. Isoscalar sigma interacts with scalar– net baryon – density n s = N N + N ¯ N , while omega in-teracts with n B = N N − N ¯ N . The powers of these differby the non-diagonal terms such as nucleon-antinucleoncorrelators C m,n = < N mN N n ¯ N > which can and shouldbe measured. Perhaps restricting kinematics of all parti-cles involved – rapidity and momenta differences – wouldfurther enhance the signal.Let me now jump to the latest STAR data, presented atCPOD 2014 by Luo, and shown in Fig.61. The proton 4-point correlator has an interesting structure: a minimumat √ s = 20 − GeV and perhaps a maximum at lowenergy (?). Antiprotons have a similar shape but withmuch smaller amplitude. Theory in fact predicts someoscillatory behavior of the kurtosis near the critical point:it can be that.(However before getting excited by the new large signal– with large error bar – let me remind that it appearedas a result of particle ID improvement, from 0 . < p t < . GeV /c to now reaching p t = 2 GeV /c . The newlyopen kinematic window should be sensitive to hydro flow,and potentially to its fluctuations.) FIG. 61: The kurtosis – 4 particle correlator – in units ofthe width, as a function of the collision energy √ s, GeV . Also near critical point one expect significant modifi-cation of attractive (sigma-related) nuclear forces: canthose affect the 4-proton (antiprotons) correlations inquestion?Clearly more data at the low energies are needed to un-derstand what is going on, and whether one does indeeddiscovered the QCD critical point or not. XIII. SUMMARY AND DISCUSSIONA. Progress on the big questions Before we go to summaries of the particular subjects,let me remind my list of “big questions” mentioned inthe Introduction:I. Can one locate the “soft-to-hard” boundary ,where the transition from strong to weak coupling (per-turbative) regimes take place?II. Can one locate the ‘micro-to- macro” bound-ary , where the transition from single-participant to col-lective regime takes place?III. Can we experimentally identify signals ofthe QCD phase transition , in particularly locate theQCD critical point?Somewhat surprisingly, the sharpest observed transi-tion which we discussed above is in the profile of the pp elastic amplitude shown in Fig.47(a). Although in-directly, rather sharp transition from black to gray isclaimed to be related to the phase transition from thedeconfined (gluonic) to confined (stringy) regimes of thePomeron. At one hand, its sharpness is surprising be-cause it is associated not with macroscopically large withquite small system – the Pomeron or a pair of strings.At the other, the analogy basically originates from thephase transition in gluodynamics (strings at early stageare considered excitable but not breakable, so no quarksyet), which is in fact a very strong first order transition.Micro-to-macro transition in pA and pp collisions, as afunction of multiplcity, is a debatable case. Data on themean p ⊥ and slopes shown in Fig.19 does indicate set-ting of the radial flow: but we know the radial flow canbe “faked”. At the same time, the v { } as a functionof multiplicity are rather flat. Its more-particle version v { n } , n > T, µ points on the phase diagram, but notyet good enough to discover/disprove the critical point. B. Sounds The first triumph of hydrodynamics, at the onset ofRHIC program, was description of the “Little Bang” ingreat details: the radial and elliptic flows were obtainedas a function of p ⊥ , centrality, rapidity, particle type andcollision energy. The second one, discussed above in de-tail, is even more spectacular description of higher az-imuthal harmonics of the flow, with m = 3 − 6. As it hasbeen repeatedly emphasized, those are basically soundharmonics, which should become even more obvious as m grows. The viscous damping of these modes actuallyagree with acoustic-inspired formulae very well.Another phenomenon, well known for the Big Bangperturbations, is the “phase factor”. Common freezeouttime for all flows does not imply the same phases, as theirfrequencies are m -dependent: and as the phases rotateone should see maxima/minima. The only experimentalindication for that is enhancement of the triangular flow m = 3 over the damping curve, and even the elliptic one m = 2 for the ultra-central bin.We emphasized that we only observed harmonics with m < m = 20 or so, and models like IP glasmapredict harmonics by the hundreds. Perhaps one caninvent other observable manifestations of those modes,such as Magneto-Sono-Luminescence process in whichthey are converted into electromagnetic signals.The damping of harmonics with m > C. The conflicting views of the initial state Perhaps the most important conceptual controversy inthe field remains the conflicting conclusions coming fromweakly coupled and strongly coupled scenarios of the ini-tial state and equilibration.Significant progress in the theory of weakly coupledinitial state is the adoption of the concept of turbulentcascade , with stationary and time-dependent self-similarsolutions. Both classical glue simulation and gluon cas-cades came up with out-of-equilibrium attractors possess- ing power spectra with certain indices, which are qual-itatively different from the equilibrium. From practicalperspective, these studies suggest that the stress tensorremains anisotropic for a long time. However more recentworks indicate that nontrivial attarctor solution is onlyapproached if the coupling is unrealistically small.Strongly coupled approaches, especially based onAdS/CFT and related models, view equilibration as aprocess dual to the gravitational collapse resulting inblack hole production in the bulk. As soon as sometrapped surface (a black hole) is there, the equilibra-tion is very rapid: any kind of “debries” simply falls intoit. Mathematically, the non-hydro modes have imaginaryparts comparable to the real one, which numerically arequite large (50). So, in this scenario, equilibration is ex-tremely rapid: there are no cascades or even gluon quasi-particles themselves, the only light propagating modesare sounds.Whether the stress tensor remains anisotropic beyondthe short initial period or not is still an open question.Theoreticall efforts to combine hydrodynamics with out-of-equilibrium parameterization of the stress tensor werediscussed above, and they will sure allow to model thesituation at any realistic anisotropy. In order to decidewhich picture is correct one needs to think about ex-perimental observables sensitive to the early stage. (Myspecific proposal – the dilepton polarization – has beendiscussed in section IX C.) D. The smallest drops of sQGP The major experimental discovery which came fromthe first years of LHC operation has been collectiveanisotropies in high multiplicity pA and pp collisions.One point of view – admittedly advocated above – isthat in those cases there are explosive QGP fireballs.While small then those produced in AA collisions, theyare still “macroscopically large” and can be describedhydrodynamically. Strong arguments for this are strongradial and elliptic flows in those systems.The opposite point of view is that from the smallestto the highest multiplicity binds the pA and pp collisionsproduce microscopic systems which can be discussed dy-namically, by the same models as used for minimally bi-ased pp . The issue is reduced to “the shape of Pomeron”problem, and models based on pQCD (BFKL or colorglass) or confining one (stringy Pomeron) need to be de-veloped much further to see if such hopes are correct.High collectivity of angular anisotropies may still be aresult of certain shape of the process. Experiments atRHIC with d and He beams however disfavor such sce-nario, in my opinion.Groups working on both scenarios now try to figure outthe limits of their approaches, which is always a goodthing. Inside hydrodynamics, for example, one studyhigher gradients and their effect. Inside the string-basedpicture we discussed a string-string interaction – ignored6for long time by event generators – leading to “spaghetticollapse” at certain density.Meanwhile, phenomenologists describe the data. Hy-drodynamical treatment of high multiplicity pA, pp events seem to be rather successful: but those requirereally small initial sizes and high temperatures of thefireball produced. But we do not really understand howsuch systems can be produced. In particular, the case ofcentral pA collisions is contested between the IP-glasmamodel and a string-based initial state picture. So far onehas very little theoretical control over the initial state ofthe high multiplicity pp : in anywhere glasma should bethere. It is difficult to study it for statistical reasons, butsince this is the highest density system we have by now,it should be pursued. E. Heavy quarks and quarkonia LHC data confirmed what has been already hinted bythe RHIC data: significant fraction of the observed char-monia comes from recombination at the chemical freeze-out of charm quarks. The “surviving charmonia” fractioncontinue to be reduced. Such major change in charmquark behavior, from “heavy-like” to ”light-quark-like”is clearly an important discovery.Let me clarify this last statement. It remains true that c, b quarks are produced differently from the light ones,namely in the initial partonic processes. Yet their inter-action with the ambient matter is strong. At large p t we observe quenching R c,bAA comparable to that of glu-ons/light quarks. At small p t we observe an elliptic flowof open charm and changes in spectra.Langevin/Fokker-Planck studies however suggest that c quarks are not moving with the flow. At early time c, b quarks are produced with large p t and start decelerating,due to drag, while the matter is slowly accelerating dueto pressure gradients: their velocities move toward eachother, yet they do not match even by the end. As aresult, charm radial/elliptic flows are not given by theCooper-Fry expression. The recombinant charmonia areperhaps an exception: whether those actually co-movewith the flow still needs to be established.On the theory front, Langevin/Fokker-Planck studieshas induced new conceptual developments. In particular,we discussed new set of solutions of those for charmonia,the quasi-equilibrium attractors with constant particleflux. Those states, not the original bound states like J/ψ, ψ (cid:48) etc, provide a convenient basis for evaluation ofthe speed of relaxation and out-of-equilibrium correctionsto current charm hadronization models. F. Jets The theory of hard processes – jets, charm/bottomproduction – were based on factorization theorems and a concept of structure functions. It is a solid founda-tion, but a very restrictive one. When one asks questionsabout say jets in high multiplicity bins of pp collisions,one soon realizes the “corresponding structure functions”do not exist: that concept has only been defined forthe “untouched proton” in inclusive setting. Univer-sal structure functions, measured rather than calculated,had served us since 1970’s, but now they cannot be usedanymore. If certain fluctuation of a nucleon is selected,new models and measurements are needed.Of course, there are practical limits: hard processescosts several orders of magnitude, and central pA costsalso about factor 20-100 down the probability. Yethigh LHC luminosity plus specialized triggers should beenough to get to some of those issues in the near future.Jet quenching in central pA remains to be understood.Scaling arguments, like the ones we used for hydro insmaller-but-hotter systems, can and should be developedand confronted with data.We argued above that in AA collisions jet quenchingparameter ˆ q seem to be strongly enhanced at the near- T c region. Small systems evolution and freezeout is verydifferent: this should play a role in jet quenching. Acknowledgements. This paper is a summaryfrom multiple conversations with colleagues, at seminars,workshops and conferences. They are too many to at-tempt to name them here (see long list of references be-low), but still I need to thank them for patiently teachingme about this or that idea or experimental findings. Thiswork was supported in part by the U.S. Department ofEnergy, Office of Science, under Contract No. DE-FG-88ER40388. Appendix A: Bjorken flow The idea of rapidity-independent “scaling” distribu-tion of secondaries originates from Feynman’s early dis-cussion of the parton model, around 1970. The existenceof rapidity-independent hydro solution was perhaps firstnoticed by Landau, who used rapidity variable in his clas-sic paper, as a somewhat trivial case. The space-timepicture connected with such scaling regime was discussedin refs [158, 159] before Bjorken’s famous paper [160] inwhich the solution was spelled out explicitly.It is instructive first to describe it in the original Carte-sian coordinates. There is no dependence on transversecoordinates x, y , only on time t and longitudinal coordi-nate z . The 1+1d equations ∂ µ T µν = 0 can be re-writtenin the following way ∂∂t ( s cosh y ) + ∂∂z ( s sinh y ) = 0 (A1) ∂∂t ( T sinh y ) + ∂∂z ( T cosh y ) = 0 (A2)7where u µ = ( cosh ( y ) , sinh ( y ), and T, s are the tempera-ture and the energy density. The first equation manifeststhe entropy conservation.The central point is the 1-d-Hubble ansatz for the 4-velocity u µ = ( t, , , z ) /τ (A3)where τ = t − z is the proper time. Note that allvolume elements are expanded linearly with time andmove along straight lines from the collision point. Thespatial η = tanh − ( z/t ) and the momentum rapidities y = tanh − v are just equal to each other. Exactly as inthe Big Bang, for each ”observer” ( the volume element) the picture is just the same, with the pressure from theleft compensated by that from the right. The history isalso the same for all volume elements, if it is expressedin its own proper time τ . Thus one has s ( τ ) , T ( τ ). Usingthis ansatz, the entropy conservation becomes an ordi-nary differential equation in proper time τds ( τ ) dτ + sτ = 0 (A4)with an obvious solution s = constτ (A5)So far all dissipative phenomena were ignored. Includingfirst dissipative terms into our equations one finds thefollowing source for the entropy current1 (cid:15) + p d(cid:15)dτ = 1 s dsdτ = − τ (cid:18) − (4 / η + ξ ( (cid:15) + p ) τ (cid:19) (A6)with shear and bulk viscosities η, ξ , which tells us thatone has to abandon ideal hydrodynamics at sufficientlyearly time.Alternatively, one can start with curved coordinates τ, η from the beginning, and look for η -independentsolution. Those are co-moving coordinates, in those u µ = (1 , , , 0) but the equations obtain extra term fromChristoffel symbols. Appendix B: Gubser flow The Gubser flow [21, 68] is a solution which keeps theboost-invariance and the axial symmetry in the trans-verse plane of the Bjorken flow, but replaces the trans-lational invariance in the transverse plane by symmetryunder special conformal transformation. Therefore, onerestriction is that the matter is required to be conformal,with the EOS (cid:15) = 3 p . Another is that the colliding sys-tems has to be of a particular shape, corresponding toconformal map of the sphere onto the transverse plane.The ideal hydro solution has three parameters: Oneis dimensional q , it defines the size of the system (and is roughly corresponding to the radii of the colliding nuclei).The other two are dimensionless, f ∗ characterizes thenumber of degrees of freedom in the matter, and ˆ (cid:15) theamount of entropy in the system.The original setting uses the coordinates we usedabove, the proper time -spatial rapidity - transverse ra-dius - azimuthal angle (¯ τ , η, ¯ r, φ ) with the metric ds = − d ¯ τ + ¯ τ dη + d ¯ r + ¯ r dφ , (B1)The dimensionless coordinates ¯ τ = qτ, ¯ r = qr are rescaledversions of the actual coordinates.Looking for solutions independent on both “angles” η, φ and using transverse rapidity u µ = ( − cosh κ ( τ, r ) , , sinh κ ( τ, r ) , 0) (B2)Gubser obtained the following solution v ⊥ = tanh κ ( τ, r ) = (cid:18) q τ r q τ + q r (cid:19) (B3) (cid:15) = ˆ (cid:15) (2 q ) / τ / (1 + 2 q ( τ + r ) + q ( τ − r ) ) / (B4)where ˆ (cid:15) is the second parameter. In [68] Gubser andYarom re-derived the same solution by going into the co-moving frame. In order to do so they rescaled the metric ds = τ d ˆ s (B5)and performed a coordinate transformation from the τ, r to a new set ρ, θ given by:sinh ρ = − − q τ + q r qτ (B6)tan θ = 2 qr q τ − q r (B7)In the new coordinates the rescaled metric reads: d ˆ s = − dρ + cosh ρ (cid:0) dθ + sin θdφ (cid:1) + dη (B8)and we will use ρ as the “new time” coordinate and θ as a new “space” coordinate. In the new coordinates thefluid is at rest.The relation between the velocity in Minkowski spacein the ( τ, r, φ, η ) coordinates and the one in the rescaledmetric in ( ρ, θ, φ, η ) coordinates corresponds to: u µ = τ ∂ ˆ x ν ∂ ˆ x µ ˆ u ν , (B9)while the energy density transforms as: (cid:15) = τ − ˆ (cid:15) .The temperature (in the rescaled frame, ˆ T = τ f / ∗ T ,with f ∗ = (cid:15)/T = 11 as in [21]) is now dependent onlyon the new time ρ , in the case with nonzero viscosity thesolution is ˆ T = ˆ T (cosh ρ ) / + H sinh ρ ρ ) / F (cid:18) , 76 ; 52 , − sinh ρ (cid:19) (B10) H is a dimensionless constant made out of theshear viscosity and the temperature, η = H T and F is the hypergeometric function. In the inviscid case thesolution is just the first term of expression (B10), andof course it also conserves the entropy in this case. Thepicture of the explosion is obtained by transformationfrom this expression back to τ, r coordinates.Small perturbations to the Gubser flow obey linearizedequations which have also been derived in [68]. We startwith the zero viscosity case, so that the background tem-perature (now to be called T ) will be given by just thefirst term in (B10). The perturbations over the previoussolution are defined byˆ T = ˆ T (1 + δ ) (B11) u µ = u µ + u µ (B12)with ˆ u µ = ( − , , , 0) (B13)ˆ u µ = (0 , u θ ( ρ, θ, φ ) , u φ ( ρ, θ, φ ) , 0) (B14) δ = δ ( ρ, θ, φ ) (B15)Plugging expressions (B11),(B12) into the hydrody-namic equations and only keeping linear terms in theperturbation, one can get a system of coupled 1-st orderdifferential equations. Furthermore, if one ignores theviscosity terms, one may exclude velocity and get thefollowing (second order) closed equation for the temper-ature perturbation. ∂ δ∂ρ − 13 cosh ρ (cid:18) ∂ δ∂θ + 1tan θ ∂δ∂θ + 1sin θ ∂ δ∂φ (cid:19) + 43 tanh ρ ∂δ∂ρ = 0 (B16)(Since the initial perturbations are assumed to berapidity-independent, we also ignored this coordinatehere.) It has a number of remarkable properties: all 4 coor-dinates can be separated δ ( ρ, θ, φ ) = R ( ρ )Θ( θ )Φ( θ ) anda general solution is given by R ( ρ ) = C (cosh ρ ) / P / − + √ λ +1 (tanh ρ )+ C (cosh ρ ) / Q / − + √ λ +1 (tanh ρ Θ( θ ) = C P ml (cos θ ) + C Q ml (cos θ )Φ( φ ) = C e imφ + C e − imφ (B17)where λ = l ( l + 1) and P and Q are associated Legendrepolynomials. The part of the solution depending on θ and φ can be combined in order to form spherical har-monics Y lm ( θ, φ ), such that δ ( ρ, θ, φ ) ∝ R l ( ρ ) Y lm ( θ, φ ).This property should have been anticipated, as one ofthe main ideas of Gubser has been to introduce a coordi-nate which together with φ make a map on a 2-d sphere.Gubser setting was used as a theoretical laboratoryever since. A complete Green function has been con-structed [36], leading to pictures of sound circles we dis-cussed at the beginning of this review. Generalizationto perturbations by the quenching jets, with the soundspropagating in the rapidity direction, was done in [49].For the second order (the Israel-Stuart version) of the hy-drodynamics in it has been done in [156, 157]. Recentlythe Boltzmann equation in tau-approximation has alsobeen solved in it, see [53] discussed in section IV C.There are also a number of phenomenological applica-tions. Without going into those, let me just commentthat those are limited by the fact that at large r thepower tail of the solution is completely inadequate forheavy ion collisions. So to say, Gubser solution is likean explosion in atmosphere, while the real ones are invacuum. As a result, in applications one basically has toamputate the unphysical regions. [1] E. V. Shuryak, Sov. Phys. JETP , 212 (1978) [Zh.Eksp. Teor. Fiz. , 408 (1978)].[2] E. V. Shuryak, Phys. Rept. , 71 (1980).[3] D. J. Gross, R. D. Pisarski and L. G. Yaffe, Rev. Mod.Phys. , 43 (1981).[4] E. V. Shuryak, The QCD vacuum, hadrons and super-dense matter (second edition, World scientific, 2004).[5] T. Schafer and E. V. Shuryak, Rev. Mod. Phys. , 323(1998) [hep-ph/9610451].[6] For recent extensive discussion of the topological soli-tons in gauge theory see talks at the workshop’s web-page, http://scgp.stonybrook.edu/archives/13417[7] For recent extensive discussion of the resurgencetheory, transseries and solitons/saddle solutions seehttp://scgp.stonybrook.edu/archives/10828[8] J. Liao and E. Shuryak, Phys. Rev. C , 054907 (2007) [hep-ph/0611131].[9] Y. Hidaka and R. D. Pisarski, Phys. Rev. D , 071501(2008) [arXiv:0803.0453 [hep-ph]].[10] A. D’Alessandro, M. D’Elia and E. V. Shuryak, Phys.Rev. D , 094501 (2010) [arXiv:1002.4161 [hep-lat]].[11] J. Liao and E. Shuryak, Phys. Rev. Lett. , 162302(2008) [arXiv:0804.0255 [hep-ph]].[12] C. Ratti and E. Shuryak, Phys. Rev. D , 034004(2009) [arXiv:0811.4174 [hep-ph]].[13] J. Liao and E. Shuryak, Phys. Rev. D , 094007 (2010)[arXiv:0804.4890 [hep-ph]].[14] J. M. Maldacena, Int. J. Theor. Phys. , 1113(1999) [Adv. Theor. Math. Phys. , 231 (1998)] [hep-th/9711200].[15] M.Tannebaum, Nucl.Phys.A 931 (2014) c877, PHENIXCollab. S. S. Adler, et al., PRC 89, 044905 (2014) [16] D. Teaney, J. Lauret and E. V. Shuryak, Phys. Rev.Lett. , 4783 (2001) [nucl-th/0011058].[17] D. Teaney, J. Lauret and E. V. Shuryak, nucl-th/0110037.[18] T. Hirano, U. W. Heinz, D. Kharzeev, R. Laceyand Y. Nara, Phys. Lett. B , 299 (2006) [nucl-th/0511046].[19] J. P. Blaizot and J. Y. Ollitrault, Adv. Ser. Direct. HighEnergy Phys. , 393 (1990).[20] D. Teaney, Phys. Rev. C , 034913 (2003) [nucl-th/0301099].[21] S. S. Gubser, arXiv:1006.0006 [hep-th].[22] S. S. Gubser and A. Yarom, arXiv:1012.1314 [hep-th].[23] B. Schenke and R. Venugopalan, arXiv:1405.3605 [nucl-th].[24] Rajeev S. Bhalerao and Jean-Yves Ollitrault, Phys.Lett.B641, 260264 (2006), arXiv:nucl-th/0607009 [nucl-th].[25] J. B. Rose, J. F. Paquet, G. S. Denicol, M. Luzum,B. Schenke, S. Jeon and C. Gale, arXiv:1408.0024 [nucl-th].[26] Jan Fiete Grosse-Oetringhaus (ALICE), QM2014,Nucl.Phys. A931(2014) 22.[27] R.Granier de Cassagnac (CMS) Nucl.Phys.A 931 (2014)c22[28] B. A. Gelman, E. V. Shuryak and I. Zahed, Phys. Rev.C , 044908 (2006) [nucl-th/0601029].[29] B. A. Gelman, E. V. Shuryak and I. Zahed, Phys. Rev.C , 044909 (2006) [nucl-th/0605046].[30] G. Basar and D. Teaney, arXiv:1312.6770 [nucl-th].[31] D. Teaney and L. Yan, Phys. Rev. C , 044908 (2012)[arXiv:1206.1905 [nucl-th]].[32] J. Jia (for the ATLAS Collaboration), arXiv:1107.1468.[33] K. Aamodt et al (ALICE Collaboration)arXiv:1105.3865, Phys. Rev. Lett. , 032301(2011)[34] U. Heinz and R. Snellings, Ann. Rev. Nucl. Part. Sci. , 123 (2013) [arXiv:1301.2826 [nucl-th]].[35] R. A. Lacey, Y. Gu, X. Gong, D. Reynolds, N. N. Aji-tanand, J. M. Alexander, A. Mwai and A. Taranenko,arXiv:1301.0165 [nucl-ex].[36] P. Staig and E. Shuryak, Phys. Rev. C , 044912 (2011)[arXiv:1105.0676 [nucl-th]].[37] A. Andronic, P. Braun-Munzinger, K. Redlich andJ. Stachel, PoS CPOD , 044 (2007) [arXiv:0710.1851[nucl-th]].[38] V. Khachatryan et al. [CMS Collaboration],arXiv:1410.1804 [nucl-ex].[39] T. Iritani, G. Cossu and S. Hashimoto, PoS LATTICE , 376 (2014) [arXiv:1311.0218 [hep-lat]].[40] P. Alba, W. Alberico, R. Bellwied, M. Bluhm,V. M. Sarti, M. Nahrgang and C. Ratti, arXiv:1403.4903[hep-ph].[41] S. Chatrchyan et al. [CMS Collaboration], EPJC 74(2014) 2847 [arXiv:1307.3442 [hep-ex]].[42] P.Bozek, Phys. Rev. C 85, 014911[43] P.Bozek and W.Broniowski, Phys.Lett.B 739 (2014) 308[44] J.L. Nagle, A. Adare, S. Beckman, T. Koblesky, J. Or-juela Koop, D. McGlinchey, P. Romatschke, J. Carlson,J.E. Lynn, and M. McCumber, PRL 113, 112301[45] V.E. Zakharov, V.S. Lvov, G. Falkovich, “Kolmogorovspectra of turbulence I. Wave turbulence.”, SpringerVerlag. ISBN 3-540-54533-6.[46] C. Young, J. I. Kapusta, C. Gale, S. Jeon andB. Schenke, arXiv:1407.1077 [nucl-th]. [47] P. Staig and E. Shuryak, Phys. Rev. C , 034908 (2011)[arXiv:1008.3139 [nucl-th]].[48] E. Shuryak and P. Staig, Phys. Rev. C , no. 6, 064905(2013) [arXiv:1306.2938 [nucl-th]].[49] E. Shuryak and P. Staig, Phys. Rev. C , no. 5, 054903(2013) [arXiv:1307.2568].[50] A. Mazeliauskas and D. Teaney, Phys. Rev. C , no.4, 044902 (2015) [arXiv:1501.03138 [nucl-th]].[51] M. Martinez and M. Strickland, Phys. Rev. C ,024906 (2010) [arXiv:0909.0264 [hep-ph]].[52] W. Florkowski and R. Ryblewski, Phys. Rev. C 83,034907 (2011), 1007.0130.[53] G. S. Denicol, U. W. Heinz, M. Martinez, J. Noronhaand M. Strickland, Phys. Rev. D , no. 12, 125026(2014) [arXiv:1408.7048 [hep-ph]].[54] M. Lublinsky and E. Shuryak, Phys. Rev. D , 065026(2009) [arXiv:0905.4069 [hep-ph]].[55] W.Florkowski, R.Ryblewski, NPA, 931 (2014) 343347[56] R. A. Janik and R. B. Peschanski, Phys. Rev. D ,045013 (2006) [hep-th/0512162].[57] M. P. Heller, R. A. Janik and P. Witaszczyk, Phys. Rev.D , 126002 (2012) [arXiv:1203.0755 [hep-th]].[58] H. Bebie,P.Gerber , J.L. Goity, H. Leutwyler.Nucl.Phys.B378:95-130,1992[59] C. M. Hung and E. V. Shuryak, Phys. Rev. C , 1891(1998) [hep-ph/9709264].[60] V. Begun, W. Florkowski and M. Rybczynski,arXiv:1312.1487 [nucl-th].[61] D. Kharzeev, Y. V. Kovchegov and K. Tuchin, Phys.Lett. B , 23 (2004) [hep-ph/0405045].[62] [ATLAS Collaboration], Measurement of the centralitydependence of the charged particle pseudorapidity dis-tribution in proton-lead collisions ATLAS-CONF-2013-096 https://cds.cern.ch/record/1599773[63] J. D. Bjorken, S. J. Brodsky and A. Scharff Goldhaber,Phys. Lett. B , 344 (2013) [arXiv:1308.1435 [hep-ph]].[64] G. R. Farrar and J. D. Allen, EPJ Web Conf. , 07007(2013) [arXiv:1307.2322 [hep-ph]].[65] T. Kalaydzhyan and E. Shuryak, arXiv:1407.3270 [hep-ph].[66] P. K. Kovtun and A. O. Starinets, Phys. Rev. D ,086009 (2005) [hep-th/0506184].[67] P. M. Chesler and L. G. Yaffe, JHEP , 086 (2014)[arXiv:1309.1439 [hep-th]].[68] S. S. Gubser and A. Yarom, Nucl. Phys. B , 469(2011) [arXiv:1012.1314 [hep-th]].[69] P. Braun-Munzinger, J. Stachel and C. Wetterich, Phys.Lett. B , 61 (2004) [nucl-th/0311005].[70] E. V. Shuryak, Phys. Lett. B , 9 (1998) [hep-ph/9704456].[71] M. A. Stephanov, K. Rajagopal and E. V. Shuryak,Phys. Rev. D , 114028 (1999) [hep-ph/9903292].[72] M. Lublinsky and E. Shuryak, Phys. Rev. C , 061901(2011) [arXiv:1108.3972 [hep-ph]].[73] T. Kalaydzhyan and E. Shuryak, arXiv:1402.7363 [hep-ph].[74] M. He, R. J. Fries and R. Rapp, Phys. Rev. C , 014903(2012) [arXiv:1106.6006 [nucl-th]].[75] E. Witten, Phys. Rev. D , 272 (1984).[76] M. Hindmarsh, S. J. Huber, K. Rummukainen andD. J. Weir, Phys. Rev. Lett. , 041301 (2014)[arXiv:1304.2433 [hep-ph]].[77] T. Kalaydzhyan and E. Shuryak, arXiv:1412.5147 [hep- ph].[78] L. D. McLerran and R. Venugopalan, Phys. Rev. D ,3352 (1994) [hep-ph/9311205].[79] R. Baier, A. H. Mueller, D. Schiff and D. T. Son, Phys.Lett. B , 51 (2001) [hep-ph/0009237].[80] T. Epelbaum, Nuclear Physics A 931 (2014) 337342[81] , J.Berges, B.Schenke, S.Schlichting , R.Venugopalan,Nuclear Physics A 931 (2014) 348353[82] T.Lappi, A.Dumitru and Y.Nara, Nuclear Physics A 931(2014) c354[83] A. Kurkela and Y. Zhu, arXiv:1506.06647 [hep-ph].[84] E. V. Shuryak, Phys. Lett. B , 150 (1978) [Sov. J.Nucl. Phys. , 408 (1978)] [Yad. Fiz. , 796 (1978)].[85] M. Chiu, T. K. Hemmick, V. Khachatryan, A. Leonidov,J. Liao and L. McLerran, Nucl. Phys. A , 16 (2013)[arXiv:1202.3679 [nucl-th]].[86] E. V. Shuryak and L. Xiong, Phys. Rev. Lett. , 2241(1993) [hep-ph/9301218].[87] G. Basar, D. E. Kharzeev and E. V. Shuryak,arXiv:1402.2286 [hep-ph].[88] K. Tuchin, Phys. Rev. C , 024911 (2013)[arXiv:1305.5806 [hep-ph]].[89] E. Shuryak, arXiv:1203.1012 [nucl-th].[90] E. Shuryak and I. Zahed, Phys. Rev. C , no. 4, 044915(2013) [arXiv:1301.4470 [hep-ph]].[91] T. Kalaydzhyan and E. Shuryak, arXiv:1404.1888 [hep-ph].[92] Collective string interactions in AdS/QCD models andhigh multiplicity pA collisions, I.Iatrakis, A.Ramamurtiand E.Shuryak, in progress[93] P. A. R. Ade et al. [Planck Collaboration],arXiv:1303.5075 [astro-ph.CO].[94] S. Floerchinger and U. A. Wiedemann, JHEP , 100(2011) [arXiv:1108.5535 [nucl-th]].[95] L. P. Csernai, F. Becattini and D. J. Wang, J. Phys.Conf. Ser. , 012054 (2014).[96] P. Kovtun, J. Phys. A , 473001 (2012)[arXiv:1205.5040 [hep-th]].[97] P. Kovtun, G. D. Moore and P. Romatschke, Phys. Rev.D , 025006 (2011) [arXiv:1104.1586 [hep-ph]].[98] L. D. Landau, Izv. Akad. Nauk Ser. Fiz. , 51 (1953).[99] E. V. Shuryak and O. V. Zhirov, Phys. Lett. B , 253(1979).[100] T. C. Brooks et al. [MiniMax Collaboration], (Privatecommunication via J.D.Bjorken)[101] V. Khachatryan et al. [CMS Collaboration], JHEP , 091 (2010) [arXiv:1009.4122 [hep-ex]].[102] S. Chatrchyan et al. [CMS Collaboration], Phys. Lett.B , 795 (2013) [arXiv:1210.5482 [nucl-ex]].[103] S. Chatrchyan et al. [CMS Collaboration], Eur. Phys. J.C , 2164 (2012) [arXiv:1207.4724 [hep-ex]].[104] B. Abelev et al. [ALICE Collaboration], Phys. Lett. B , 29 (2013) [arXiv:1212.2001 [nucl-ex]].[105] A. Adare et al. [PHENIX Collaboration], Phys. Rev.Lett. , 212301 (2013) [arXiv:1303.1794 [nucl-ex]].[106] G. Goldhaber, S. Goldhaber, W. Y. Lee and A. Pais,Phys. Rev. , 300 (1960).[107] G. I. Kopylov and M. I. Podgoretsky, Sov. J. Nucl. Phys. , 336 (1974) [Yad. Fiz. , 656 (1973)].[108] E. V. Shuryak, Yad. Fiz. , 1302 (1973).[109] S. Pratt, Phys. Rev. Lett. , 232301 (2009)[arXiv:0811.3363 [nucl-th]].[110] A. N. Makhlin and Y. M. Sinyukov, Z. Phys. C , 69(1988). [111] P. Bozek, arXiv:1408.1264 [nucl-th].[112] K. Aamodt et al. [ALICE Collaboration], Phys. Rev. D , 112004 (2011) [arXiv:1101.3665 [hep-ex]].[113] Y.Hirono and E.Shuryak, Femtoscopic signiture ofstrong radial flow in high-multiplicity pp collisions, inprogress[114] E. Shuryak, S. J. Sin and I. Zahed, J. Korean Phys. Soc. , 384 (2007) [hep-th/0511199].[115] T. Matsui and H. Satz, Phys. Lett. B , 416 (1986).[116] N. Brambilla, M. A. Escobedo, J. Ghiglieri and A. Vairo,JHEP , 130 (2013) [arXiv:1303.6097 [hep-ph]].[117] A. Mocsy, P. Petreczky and M. Strickland, Int. J. Mod.Phys. A , 1340012 (2013) [arXiv:1302.2180 [hep-ph]].[118] O. Kaczmarek and F. Zantow, PoS LAT , 192(2006) [hep-lat/0510094].[119] D. E. Kharzeev, Phys. Rev. D , 074007 (2014)[arXiv:1409.2496 [hep-ph]].[120] C. Young and E. Shuryak, Phys. Rev. C , 034907(2009) [arXiv:0803.2866 [nucl-th]].[121] P. Romatschke, arXiv:1504.02529 [nucl-th].[122] J. Liao and E. Shuryak, Phys. Rev. D , 094007 (2010)[arXiv:0804.4890 [hep-ph]].[123] J. -P. Blaizot, Y. Mehtar-Tani and M. A. C. Torres,arXiv:1407.0326 [hep-ph].[124] E. V. Shuryak, Phys. Rev. C , 027902 (2002) [nucl-th/0112042].[125] J. Liao and E. Shuryak, Phys. Rev. Lett. , 202302(2009) [arXiv:0810.4116 [nucl-th]].[126] X. Zhang and J. Liao, arXiv:1311.5463 [nucl-th].[127] J. Xu, J. Liao and M. Gyulassy, arXiv:1508.00552 [hep-ph].[128] E. Shuryak, Phys. Rev. C , 024907 (2012)[arXiv:1203.6614 [hep-ph]].[129] S. Lin and E. Shuryak, Phys. Rev. D , 125018 (2008)[arXiv:0808.0910 [hep-th]].[130] V. Balasubramanian, A. Bernamonti, J. de Boer,N. Copland, B. Craps, E. Keski-Vakkuri, B. Mullerand A. Schafer et al. , Phys. Rev. D , 026010 (2011)[arXiv:1103.2683 [hep-th]].[131] S. Lin and E. Shuryak, Phys. Rev. D , 085013 (2008)[hep-ph/0610168].[132] S. Lin and E. Shuryak, Phys. Rev. D , 085014 (2008)[arXiv:0711.0736 [hep-th]].[133] S. S. Gubser, S. S. Pufu and A. Yarom, Phys. Rev. D , 066014 (2008) [arXiv:0805.1551 [hep-th]].[134] S. Lin and E. Shuryak, Phys. Rev. D , 124015 (2009)[arXiv:0902.1508 [hep-th]].[135] S. S. Gubser, S. S. Pufu and A. Yarom, JHEP ,050 (2009) [arXiv:0902.4062 [hep-th]].[136] S. Lin and E. Shuryak, Phys. Rev. D , 045025 (2011)[arXiv:1011.1918 [hep-th]].[137] U. Gursoy and E. Kiritsis, JHEP , 032 (2008)[arXiv:0707.1324 [hep-th]].[138] U. Gursoy, E. Kiritsis and F. Nitti, JHEP , 019(2008) [arXiv:0707.1349 [hep-th]].[139] D. Aren, I. Iatrakis, M. Jrvinen and E. Kiritsis, JHEP , 068 (2013) [arXiv:1309.2286 [hep-ph]].[140] Ioannis Iatrakis, Adith Ramamurti and EdwardShuryak, QCD strings and their interaction in theAdS/QCD, in progress[141] E. A. Kuraev, L. N. Lipatov and V. S. Fadin, Sov. Phys.JETP , 199 (1978) Ya. Ya. Balitsky and L. N. Lipa-tov, Sov. J. Nucl. Phys. , 22 (1978)[142] A. Stoffers and I. Zahed, Phys. Rev. D , 075023 (2013) [arXiv:1205.3223 [hep-ph]].[143] G. Basar, D. E. Kharzeev, H. U. Yee and I. Zahed, Phys.Rev. D , 105005 (2012) [arXiv:1202.0831 [hep-th]].[144] E. Shuryak and I. Zahed, Phys. Rev. D , 094001(2014) [arXiv:1311.0836 [hep-ph]].[145] B. Svetitsky, Phys. Rev. D 37, 2484 (1988)[146] G. D. Moore and D. Teaney, Phys. Rev. C , 064904(2005) [hep-ph/0412346].[147] M. Strickland, arXiv:1410.5786 [nucl-th].[148] J. Xu, J. Liao and M. Gyulassy, arXiv:1508.00552 [hep-ph].[149] A.Andronic, Nucl.Phys.A 931 (2014) c135.[150] A.Beraudo,Nucl.Phys.A 931 (2014) c145.[151] R. J. Fries, B. Muller, C. Nonaka and S. A. Bass, Phys.Rev. C , 044902 (2003) [nucl-th/0306027].[152] M. A. Stephanov, Phys. Rev. Lett. , 032301 (2009)[arXiv:0809.3450 [hep-ph]].[153] R. Rapp and E. V. Shuryak, nucl-th/0202059.[154] Y. Qian and I. Zahed, arXiv:1411.3653 [hep-ph].[155] C.Loizides [ALICE collaboration], talk at The 2nd In-ternational Conference on the initial stages..., Napa,Dec.2014[156] H. Marrochio, J. Noronha, G. S. Denicol, M. Luzum,S. Jeon and C. Gale, Phys. Rev. C , no. 1, 014903(2015) [arXiv:1307.6130 [nucl-th]].[157] L. G. Pang, Y. Hatta, X. N. Wang and B. W. Xiao,Phys. Rev. D , no. 7, 074027 (2015) [arXiv:1411.7767[hep-ph]].[158] C. B. Chin, E. C. G. Sudarshan and K. H. Wang, Phys.Rev 012(1975)902[159] M. I. Gorenshtein, V. A. Zhdanov and Yu. M. Sinjukov, ZhETF Phys. Rev. D27(1983)140[161] P. Danielewicz, M. Gyulassy, Phys.Rev.