Physics of Strongly coupled Quark-Gluon Plasma

Physics of Strongly coupled Quark-Gluon Plasma

Edward Shuryak
Department of Physics and Astronomy
University at Stony Brook,NY 11794 USA
shuryak@tonic.physics.sunysb.edu
Abstract

This review cover our current understanding of strongly coupled Quark-Gluon Plasma (sQGP), especially theoretical progress in (i) explaining the RHIC data by hydrodynamics, (ii) describing lattice data using electric-magnetic duality; (iii) understanding of gauge-string duality known as AdS/CFT and its application for “conformal” plasma. In view of interdisciplinary nature of the subject, we include brief introduction into several topics “for pedestrians”. Some fundamental questions addressed are: Why is sQGP such a good liquid? What is the nature of (de)confinement and what do we know about “magnetic” objects creating it? Do they play any important role in sQGP physics? Can we understand the AdS/CFT predictions, from the gauge theory side? Can they be tested experimentally? Can AdS/CFT duality help us understand rapid equilibration/entropy production? Can we work out a complete dynamical “gravity dual” to heavy ion collisions?

1 Introduction

Soon after discovery of QCD, people used asymptotic freedom to argue that very hot/dense matter must be weakly coupled [1] and thus deconfined. My own entrance to this field started with the question: while fluctuations of gauge field lead to anti-screening, the famous negative coefficient leading to asymptotic freedom, what their fluctuations would do? Explicit calculation [2], using the Coulomb gauge, have produced sign of the Debye mass, opposite to that of virtual gluons and the same as in QED! Thus I called this phase of matter “plasma”, putting it even in the paper title111A small anecdote is related to that: in those days plasma research in Russia was semi-classified and thus the paper returned to me with a note saying that it lacks proper permissions. I found a quick fix to this problem: a letter from renown plasma physicist saying this plasma has nothing to do with “real plasma physics”. And perhaps it was true: I got an invitation to speak about it at some plasma physics meeting only 30 years later. . One more important finding of that calculation was that static magnetic screening is absent, also like in QED: we will see that this conclusion would be seriously modified non-perturbatively.

The next 25 years theory of QGP was based on pQCD. The program to do so, with a number of “signals” was proposed in my other paper [3] : those were based on high energy hadronic/nuclear collisions and included several perturbative process, from gluonic production of new quark flavors to charmonium dissociation by gluonic “photoeffect”. A lot of efforts have been made to derive perturbative series for finite- thermodynamics, and eventually all calculable terms have been calculated, see e.g. [4]. Convergence was bad, but we thought that some clever re-summation can still make it work.

But whether QGP can or cannot be created experimentally was not at all clear. In fact all the way to the year 2000, when RHIC started, most theorists argued that nothing else but a “firework of mini-jets” can possibly be seen at RHIC.

And yet, in my talk at QM99 I predicted based on hydro calculations, that elliptic flow at RHIC would be twice than at SPS. It did not take long after the start of RHIC operation to see that this is indeed what happens: in fact a rather perfect case for hydrodynamical explosion was made both from radial and elliptic flows. After lots of debates, this period culminated with the “discovery” workshop of 2004 and subsequent “white papers” from 4 experimental collaborations which documented it.

Theory of QGP is still profoundly affected by this “paradigm shift” to the so called strong-coupling regime. We are still in so-to-say non-equilibrium transition, as huge amount of physics issues required to be learned. Some came from other fields, including physics of strongly coupled QED plasmas and trapped ultracold gases with large scattering length. String theory provided a remarkable tool – the AdS/CFT correspondence – which related heavy ions to the the fascinating physics of strong gravity and black holes. Another important trend is that transport properties of QGP and non-equilibrium dynamics came to the forefront: and for those the Euclidean approaches (lattice, instantons) we used before is much less suited than for thermodynamics. All of it made the last 5 years the time of unprecedented challenges.

Because the issues we discuss incorporate several fields of physics, some introductory parts of this review are marked “for pedestrians”. Indeed, heavy ion physics did not have much in common with string theory and black holes, or dilute quantum gases, so some basic definitions and main physics statements (made at an “intuitive” level) may be helpful to some readers. We start with two such introductory subsections, about classical strongly coupled plasmas and quantum ultracold gases, which we will not discuss in this review in depth.

1.1 Strongly coupled plasmas for pedestrians

By definition, plasmas are states of matter in which particles are “charged” and thus interact via long range (massless) gauge222 Why only gauge and not scalar fields? Indeed, supersymmetric models have massless scalars which in many cases create the so called BPS situation, in which gauge repulsion is canceled by scalar attraction: and we will call them plasmas as well. However this is as far as it goes: a generic massless scalar, attractive in all channels and not restricted by supersymmetry, is just a recipe for instability and should not be considered at all. fields. This separate it from “neutral” gases, liquids or solids in which the interparticle interaction is short range. Sometimes plasmas were called “the 4-th state of matter”, but this does not comply with standard terminology: in fact plasmas can themselves be gases, liquids or solids.

Classical plasmas are of course those which does not involve quantum mechanics or . Let me start with counting the parameters of the problem. There are 4 variables 333Recall that the problem is not only classical but it is also nonrelativistic: thus no or . – the particle mass and density, the temperature and the Coulomb charge. Three of them can be used as units of mass, length and time: thus only one combination remains. The standard choice is the so called plasma parameter, which can be loosely defined as the ratio of interaction energy to kinetic energy, and is more technically defined as

(1)

where are respectively the ion charge, the Wigner-Seitz radius and the temperature: this form is convenient to use because it only involves the input parameters, such as the temperature and density, while average potential energy is not so easily available. The meaning of it is the same, and the values of all observables - like transport coefficients we will discuss below – are usually expressed as a function of .

Depending on magnitude of this parameter classical plasmas have the following regimes:
i. a weakly coupled or gas regime, for ;
ii. a liquid regime for ;
iii. a glassy liquid regime for ;
iv. a solid regime for .

Existence of permanent correlation between the particles is seen in the simplest way via density-density correlation functions

(2)

with is the number of particles, is the position of the i-th-particle at time . characterizes the likelihood to find 2 particles a distance away from each other at time . Here are some examples, from our own (non-Abelian) MD simulations [5], which show that liquid regime demonstrate nearest-neighbor peaks, and crystals have peaks corresponding to longer range order. They also show “healing” of correlations with time in gases, but much less so in liquids and solids.

Figure 1: (Color online) correlation function for , respectively. Red circles correspond to , and blue squares correspond to .

The case of small is widely discussed in statistical mechanics courses: let me just remind the reader that it is in this case when one can use Boltzmann eqn, cascades and other simple tools appropriate for a gas. Unlike gases and solids, the interplay of local order and randomness at large distances makes liquids difficult to treat theoretically444I heard an opinion, ascribed to a lecture of V.Weisskopf, that if theorists would invent the Universe from scratch, without any experiment, they would never think about liquids.. Thus, in spite of their crucial importance for a lot of chemistry in general and our life in particular, most physics and statistical mechanics courses tend to either omit them completely or tell as little as possible about them. It is possibly worth reminding heavy ion practitioners, that for liquids neither Boltzmann equation nor cascades can be used because particle are strongly correlated with several neighbors at all times. The very idea of “scattering” and cross section involves particles coming from and going to infinity: it is appropriate for dilute gases but not condensed matter where interparticle distances do not exceed the range of the forces at any time.

Strongly coupled classical electromagnetic plasmas can be studied experimentally: they are not at all exotic objects. For example, table solt can be considered a crystalline plasma made of permanently charged ions and . At (still too small to ionize non-valence electrons) one gets a molten solt, which is liquid plasma with . A more famous object of recent experimentation is a charged dust: in space (e.g. at International Space Station) it has been put into a nice crystal and studied in depth. For example, one can get a particle piercing it and creating Mach cones, several if there are excitations other than the sound. One can find more introductory information and references on the subject in Mrowczynski and Thoma review [6].

We will not discuss any theory of it, but just note that starting from about 1980’s availability of computational resources get sufficient to use “Molecular Dynamics” (MD). This means that one can write equations of motion (EOM) for the interparticle forces and directly solve them for say particles. It is this simplest but powerful theoretical tool we will use to access properties of strongly coupled QGP by classical simulations.

1.2 Strongly coupled ultracold gases

Back in 1999 G.Bertsch formulated a “many-body challenge problem,” asking: what are the ground state properties of a two-species fermion system in the limit that the scattering length of its interaction approaches infinity? (This limit is usually referred to as the “unitary limit” because the scattering cross section reaches its unitarity limit per s-wave.) Such problem was originally set up as a parameter-free model for a fictitious dilute neutron matter: recall that nn scattering length is indeed huge because of near-zero-energy isoscalar virtual state.

The answer was provided experimentally by atomic experimentalists, who found a way to modify the interaction strength in ultracold trapped systems by the magnetic field, which can shift level positions till they cross and form the so called “Feshbach resonances”. As a result, the interaction measure – the scattering length in units of interparticle spacing – can be changed in a wide interval, practically from (very strong attraction) to (very strong repulsion). Systems of ultracold atoms became a very hot topic a decade earlier, when laser cooling techniques were developed so that temperatures got low enough (so that atomic thermal de Brogle wavelength get comparable to interparticle distances) to observe Bose-Einstein condensation (BEC) for bosonic atoms. Last years was the time of strongly coupled quantum gases made of fermions, possessing superfluidity and huge pairing gaps. Never before one had manybody quantum systems with widely tunable interparticle interaction amenable to experimentation, and clearly it created a kind of revolution in quantum manybody physics.

For some technical reasons, BEC in strongly systems (of interest to us in respect to monopole condensation/confinement) is not yet studied, while strongly coupled fermions have been investigated quite extensively. Let me briefly review the questions which were of the main interest of the atomic community. One of them is whether two known weakly coupled phases – BCS superconductor for small and negative and BEC of bound atom pairs at small positive – join smoothly or there exist a discontinuity at the Feshbach resonance. The answer seems to be the former, namely two phases do join smoothly. Another issue is whether low- strongly coupled fermions is in a gaped superfluid state: it has been answered positively, and phenomena as complex as Abrikosov’s lattice of vortexes has been observed.

Clearly, there are many fascinating phenomena in this field, but let us focus on just two issues most relevant for this review and sQGP:
(i) whether transition from weakly to strongly coupling regime is reflected in an unusually small viscosity, seen in onset of hydrodynamical behavior in unexpectedly small systems;
(ii) possible universal limits in the infinite interaction limit for pairing, in connection to color superconductivity of quark matter.

Rather early it has been observed that the answer to the former question is clearly affirmative. When the trap is switched off, the atoms in a weakly coupled gas simply fly away with their thermal/quantum velocities, displaying angular distribution of velocities of the trap shape. However in strongly coupled regime, with the particle mean free path smaller than the system size, hydrodynamical flow develops. Deformed traps thus develop “elliptic flows” in the direction of maximal pressure gradient, in a way analogous to what is happening in heavy ion collisions: see Fig.2(a). This is by no means trivial: the interparticle distances are about 1000 times the atomic size, and the total number of atoms is only (only few times more than in central heavy ion collisions at RHIC): similar number of water molecules would show any hydro!

Figure 2: (a) False color absorption images of a strongly interacting degenerate Fermi gas of ultracold Li atoms as a function of time after release from a laser trap. From O’Hara et al.[7] (b)“Quantum viscosity” in strongly-interacting Fermi gas (trap-averaged). (c) Same data for the shear viscosity as in units of the entropy density as a function of energy . The lower green dotted line shows the string theory prediction . The light blue bar shows the estimate for a quark-gluon plasma (QGP) while the blue solid bar shows the estimate for He, near the - point.

The next questions was whether one can use hydrodynamics quantitatively, find out its accuracy and quantify the viscosity. From “released traps” the experiments switched to quadrupole vibrational modes: as the trapped system has a cigar-shape with much weaker focusing along axes compared to axial ones, there is softer -vibrations and higher frequency “axial” mode. I will not go into vast literature and simply say that the value of the vibrational frequencies are indeed given by hydro, reaching near-percent accuracy at the Feshbach resonance point (where ).

The first study of viscosity has been made by Gelman, myself and Zahed [8]: from available data on different vibrational modes – z-mode and axially symmetric radial mode of a cigar-shaped atomic cloud – we tried to deduce viscosity and found that the values are roughly consistent with each other. Instead of going into details of that work, let us discuss what one would expect based on “universality” arguments. The main point is that if certain observable are finite in the limit, the value can only depend on few parameters – e.g. the particle mass , the density and Plank constant when the temperature is zero. By dimensional analysis, at , the energy density of infinitely strongly coupled gas can only be a number times the energy density of the ideal Fermi gas with the same density,

(3)

with some constant .

Assuming that there can be nonzero “quantum viscosity” at zero for large amplitude oscillations, we [8] proposed to measure viscosity in units of , the only combination with the right dimension. Experimental data by Turlapov et al. [9] for viscosity in such units are shown in Fig.2(b). It is plotted as a function of energy per particle in Fermi energy units, , which basically characterizes the excitation temperature: (corresponding to the left side of the figure) is close to the ground state . From the measured points it seems that is actually vanishing in the ground state, about linearly in . Another way to express viscosity – familiar from Black Hole physics– is to express it as with being the entropy density. Such ratio is shown in Fig.2(c): and it looks like this one reaches nonzero value as . Its magnitude makes strongly coupled fermionic atoms to be the “second best liquid” known, in between the sQGP (light blue band below) and the “bronze winner” (former champion) , liquid , shown by dark blue.

Few people (myself included) may find it fascinating, but most of atomic quantum gases community do not care and even unaware that some qualitatively new regime happens close to the Feshbach resonance: global parameters (like in total energy which was subject of many theoretical and numerical works) are smooth there. Small viscosity of quantum gases does not yet have microscopic explanations: certainly not by “unitary” cross section and kinetic theory. Linear behavior of and in reminds that of electrons in solids: but strong coupling actually destroys the Fermi surface (as experiments measuring momentum distribution show quite clearly) and makes a very strong superconductor. The critical point in units is at : a look at the discussed figure reveals no changes in viscosity (nor in oscillation frequencies themselves) visible by an eye.

Apart of being in general related to the issue of sQGP as “the most perfect liquid”, the strongly coupled atomic superconductor provides some valuable information on how large the pairing gaps can possibly become in cold quark matter with color superconductivity. Assuming that near deconfinement cold quark matter is also strongly coupled, with weakly bound diquarks playing the role of Feshbach resonance, in [10] I have used of the ratio (the pairing critical temperature to the Fermi energy) and data on strongly coupled fermionic atoms to get upper limit on the corresponding transition to the color superconductivity. Fig.3 explains the argument. “Universality” tells that the critical temperature must be simply proportional to the Fermi energy

(4)

with the universal constant . In fact two values of show some change: the phase transition to superfluidity at and which experimentalists (Kinast et al) interpret as a transition to a regime where not only there is no of atomic pairs, but even the pairs themselves are melted. Using these two values as slopes of solid and dashed lines in Fig.3(b), one can see that all chemical freezeouts in heavy ion collisions (points) are above possible domain of pairing, even at infinite coupling. At finite coupling the gaps and is even smaller.

Figure 3: (a) Schematic phase diagram for QCD, in the plane baryon chemical potential - temperature. M (multifragmentation) point is the endpoint of nuclear gas-liquid transition. E is a similar endpoint separating the first order transition to the right from a crossover to the left of it. (Black) solid lines show phase boundaries, dashed lines are curves of marginal stability of indicated states. Two dash-dotted straight lines are related with bounds from atomic experiments we discuss in the text, they intersect with unbinding of diquark Cooper pairs (D) and most strongly coupled point (S), which is at the maximum of the transition line and is also a divider between BCS-like and BEC-like color superconductor. Other lines are zero binding for singlet , singlet , octet and finally triplet states. (right) Compilation of experimental data on the chemical freezeout from different experiments: squares (circles) are for fits at mid-rapidity (all particles), respectively. Two solid lines are the phase transition lines with the quark effective mass and , two dashed lines show pair unbinding lines for the same masses.

Let me finish with one recent development, providing quite fascinating prospects if true. The idea is that infinitely strongly interacting atoms may have “gravity dual” description similar to AdS/CFT. Son [11] and also Balasubramanian and McGreevy [12] have argued that since at the interaction can be treated as a kind of boundary condition when two atoms meet, and this problem has what he called “Schreodinger symmetry”, a subgroup of conformal symmetry. They found examples of the metric possessing this very symmetry and suggested it be the desired “gravity dual” . (The reader should however be reminded that there is no proof of its existence or in fact any empirical confirmation of this idea so far.)

2 Heavy Ion Collisions

2.1 Heavy ion collisions and flows for pedestrians

Let me start with brief history. At one hand, high energy physics was for all of its history been interested in “high energy asymptotic” . In practice this mostly was reduced to global features like total and elastic cross sections, as it was thought that multibody final state produced is too complicated to get some sufficiently simple theoretical treatment. Nuclear physicists started from nuclear collisions at nonrelativistic domain: their cross section was obvious and the main objective was understanding of “excited matter”. Experimental program at Relativistic Heavy Ion Collider (RHIC) at Brookhaven and Large Hadronic Collider (LHC) at CERN is so to say a brainchild of both communities.

What do we mean saying that collisions produced “excited matter”? Should one expect simplification if it is the case? The answer is that we are not interested in excited system produced (e.g. in “elementary” pp collisions), but mostly in a macroscopically large fireball whose size (macro scale) greatly exceed the micro scale of correlations inside it.

(5)

In weakly coupled systems (gases) is the mean free path length, or relaxation time. In strongly coupled setting such as AdS/CFT the temperature would be the only dimensional parameter describing the microscopic physics: most dissipative phenomena have the scale as a relaxation scale.

Have we reached this regime? Perhaps good illustration that the question is nontrivial are my own two attempts to apply macroscopic physics for high energy processes. In 1971 I proposed “spherical explosion” in annihilation into many pions [13], only to be killed by discovery of asymptotic freedom in 1973 and jets in evens in 1976. In 1979 Zhirov and myself [14] looked at fresh results from pp collisions at then-new ISR collider at CERN555 L. van Hove and few others, including myself, put forward a proposal to accelerate heavy ions in ISR, to see what happens. CERN leaders basically rejected it; the experiments never went beyond the alpha-particles, and then this first hadronic collider was physically destroyed. As we know now, QGP could have been discovered and studied at ISR 20 years prior to RHIC. . The general idea of the paper was to look for collective transverse flows. We argued that since secondaries have different masses, the kinematic effect of a collective motion (flow velocity) can be separated from their thermal spectra. More specifically, if the “matter flow” is only longitudinal, along the beam direction, the transverse momenta spectra of different secondaries would be just thermal

(6)

where the mass and transverse momentum combine into a “transverse mass” . The ISR data indeed showed this “ scaling”: but there was no sign of the deviations from it due to transverse flow. (Lacking the effect, we argued that it is the “vacuum pressure” which stops it.)

Heavy ion collisions are used to make the system larger. The macro scale – nuclear sizes – for Au (or Pb) used in Brookhaven (or CERN) have typical and radius . At RHIC the temperature T changes from the initial value of about , thus can be viewed as a large parameter. In fact experiments with beam, , also showed good macroscopic behavior, as are rather peripheral collisions. The boundary between macroscopic and microscopic systems should be somewhere, as collisions are not hydro-like: but the exact location of it is not yet known: small systems of several nucleons are subject of large fluctuations, and it is not easy666On top of that, there is experimental background, as very peripheral collisions of two beams is hard to tell from beam-residual gas interaction. to study them.

Once produced, the fireball is expanding hydrodynamically up to the so called freezeout conditions. People familiar with Big Bang can recognize existence of multiple freezeout, for each reaction at its own time: and in the Little Bang it is similar. The so called chemical freezeout (at which particle composition gets frozen) is at at RHIC, while re-scattering continue to (for central collisions). In units of QCD critical temperature, it is variation from about to , conveniently bracketing the QCD deconfinement/chiral restoration transition.

Flow velocity is decomposed into longitudinal (along the beam) and transverse components. The latter is further split into radial flow (present even for axially symmetric central collisions) and elliptic flow which exists only for non-central collisions. The reason elliptic flow to be very important was pointed out by H.Sorge: it is developed than the radial one, and thus most sensitive to the QGP era than other flows.

The radial flow has Hubble-like profile, with transverse rapidity growing roughly linear till the edge of the fireball. The maximal radial flow velocity at RHIC is about . This radial flow is firmly established from a combined analysis of particle spectra, HBT correlations, a deuteron coalescence and other observables: so we will not show

At non-zero impact parameter the original excited system has a deformed almond-like shape: thus its expansion in the transverse plane can be described by the (Fourier) series in azimuthal angle

(7)

where average is over all particles777This is written as if the direction of impact parameter of the event is known. I would not go into how it is determined in detail but just say that it usually comes from counters sitting at rapidity far from the region where most observations are done, at mid-rapidity . . At mid-rapidity only even harmonics are allowed, the second one characterizes the so called elliptic flow. Multiple arguments of the parameter stand for the collision energy , transverse momentum , particle mass/type , rapidity , centrality888In real experiments the measure of impact parameter is the so called the number of participants , which changes from zero at most peripheral collisions to at central collisions. This number is the total number of nucleons minus the so called spectators, which fly forward and are directly observed by forward-backward calorimeters. and the atomic number characterizing the colliding system.

If in high energy collisions of hadrons and/or nuclei a macroscopically large excited system is produced, its expansion and decay can be described by relativistic hydrodynamics. Its history starts with the pioneer paper by Fermi of 1951 [16] who proposed a statistical model to Lorentz-contracted initial state. Pomeranchuk [17] then pointed out that initial Fermi stage cannot be the final stage of the collisions since strong interaction in the system persists: he proposed a freezeout temperature . L.D.Landau [15] then explain that one should use relativistic hydrodynamics in between those two stages, saving Fermi’s prediction of the multiplicity by entropy conservation.

(Before we go into details, a comment: often hydrodynamics is considered as some consequence of kinetic equations, but in fact applicability conditions of both approaches are far from being coincident. In particular, for the former approach the stronger the interaction in the system is, the better. Kinetic approach, on the contrary, was never formulated but for weakly interacting systems: and as we repeatedly emphasize in this review, it is so for QGP.)

The conceptual basis of the ideal hydrodynamics is very simple: it is just a set of local conservation laws for the stress tensor () and for the conserved currents (),

(8)

The is in form for and related to the bulk properties of the fluid and its 4-velocity by the relations,

(9)

Here is the energy density, is the pressure, is the number density of the corresponding current, and is the proper velocity of the fluid. In strong interactions, the conserved currents are isospin (), strangeness (), and baryon number (). For the hydrodynamic evolution, isospin symmetry is assumed and the net strangeness is set to zero; therefore only the baryon current is considered below.

Let me add a simple heuristic argument why the first term in the stress tensor has and not any other combination. The point is and themselves are defined up to a constant (which depending on context we call the bag constant, or cosmological term or dark energy). The combination does not have it, and it is also proportional to entropy which is defined uniquely. How this argument complies with the last term in (9)? Well it has B but without velocity, so in hydro eqns there is only pressure gradient and this B term disappears as well.

In order to close up this set of equations, one needs also the equation of state (EoS) . One should also be aware of two thermodynamical differentials

(10)

and the definition of the sound velocity

(11)

and that . Using these equations and the thermodynamical relations in the form

(12)

one may show that these equations imply another nontrivial conservation law, namely, the conservation of the entropy

(13)

Therefore in the ideal hydro all the entropy is produced only in the discontinuities – shock waves – which are not actually there is application we discuss. Thus the “initial entropy” is simply passed to the solution as an initial parameter, determined in the earlier (violent) stage of the collision: (this is similar to Big Bang cosmology, in which “entropy production” stage is also very different from stages of cosmological evolution we can observe by e.g. Nucleosynthesis.)

Next order effects in micro-to-macro expansion is the domain of “viscous hydrodynamics”: we will discuss their applications to data description as well as their derivation from AdS/CFT settings.

Figure 4: The left and right sides show the hydrodynamic solution at the SPS and RHIC. The thin lines show contours of constant transverse fluid rapidity () with values 0.1,0.2,…,0.7 . The thick lines show contours of constant energy density. denotes the energy density where . and denote the energy density where the matter shifts from hadronic to mixed and mixed to a QGP, respectively. The shift to hadronic cascade is made at . denotes the mean transverse rapidity weighted with the total entropy flowing through the energy density contours. Walking along these contours, the line is broken into segments by dashed and then solid lines. 20% of the total entropy passing through the entire arc passes through each segment.

The simplest Bjorken 1+1 dim solution is a good example “for pedestrians”, reminding how to write hydro in arbitrary coordinates.

(14)

where the semicolon indicates a covariant derivative. For tensors of rank 1 and 2 it reads explicitly

(15)

where the comma denotes a simple partial derivative and the Christoffel symbols are given by derivatives of the metric tensor :

(16)

As an example, let us do the following transformation from Cartesian to light cone coordinates:

In the new coordinate system the velocity field (after inserting ) is given by

(17)

with , , and .

Now we turn to the metric of the new system. We have

(18)

and therefore

(19)

The only non-vanishing Christoffel symbols are

(20)

The 1+1d equations for boost-invariant solution can be written in the following way

(21)
(22)

The so called Bjorken [18] solution999 We call it following established tradition, although the existence of such simple solution was first noticed by Landau and it was included in his classic paper as some intermediate step. The space-time picture connected with such scaling regime was discussed in refs [20, 21] before Bjorken, and some estimates for the energy above which the transition to the scaling regime were expected to happen were also discussed in my paper [3] as well. is obtained if the velocity is given by the velocity where is the proper time. In this 1-d-Hubble regime there is no longitudinal acceleration at all: all volume elements are expanded linearly with time and move along straight lines from the collision point. The spatial and the energy-momentum rapidity are just equal to each other. Exactly as in the Big Bang, for each ”observer” ( the volume element ) the picture is just the same, with the pressure from the left compensated by that from the right. The history is also the same for all volume elements, if it is expressed in its own proper time .

Thus the entropy conservation becomes the following (ordinary) differential equation in proper time

(23)

which has the obvious solution

(24)

Let us compare three simple cases: (i) hadronic matter, (ii) quark-gluon plasma and (iii) the mixed phase (existing if there are first order transitions in the system). In the first case we adopt the equation of state suggested in [22] . If so, the decrease of the energy density with time is given by

(25)

In the QGP case the same law holds, but with

In the mixed phase the pressure remains constant , therefore

(26)

So far all dissipative phenomena were ignored. Including first dissipative terms into our equations one has

(27)

Note that ignoring one finds in the r.h.s. exactly the combination which also appears in the sound attenuation, so the correction to ideal case is . Thus the length directly tells us the magnitude of the dissipative corrections. At time one has to abandon the hydrodynamics altogether, as the dissipative corrections cannot be ignored. Since the correction is negative, it reduces the rate of the entropy decrease with time. Another way to say that, is that the total positive sign shows that some amount of entropy is generated by the dissipative term. We will discuss “gravity dual” to this solution in the last chapter.

2.2 Collective flows and hydrodynamics

Any treatment of the explosion by hydro in general includes (i) the initiation; (ii) hydro evolution and (iii) freezeout. Geometrically, (i) and (iii) form two 3-hypersurfaces in 4 dimensions, together constituting a boundary of 4-volume region in which hydro eqns are solved. Only the stage (ii) deals with the Equation of State and transport properties of new form of matter, the QGP – while (i) and (iii) have to deal with stages at which hydrodynamics is not applicable. The stage (i) is least understood theoretically: yet the uncertainties it produces are rather minor. Correct treatment of the freezeout stage (iii) is much more important for calculations of the final spectra which are compared to data: and that is where most intensive debates were. Fortunately the latest hadronic stage is not that mysterious, as some theorists think: it happens in a dilute gas-like medium made of mostly pions and their resonances. We understand their low energy scattering cross sections quite well, the corresponding cascade codes have been extensively tested using the low energy heavy ion collisions at AGS (BNL) and SPS(CERN). It just needs a bit of extra work.

Solving hydro eqns is not simple – they are nonlinear and prone to instabilities – but solutions are easily controllable by the energy, momentum and entropy (for ideal hydro) conservation: thus I presume it was done correctly by all groups. At years 2000-2004 most groups used an approximation in which one out of 3+1 (coordinates+time) – the longitudinal spatial rapidity – was taken as irrelevant, thus switching to 2+1. Example of output from such hydro calculations [19], with properly chosen EoS101010LH8 mentioned in the figure means the“latent heat” of magnitude at the QCD transition temperature. is shown in Fig.4, for two collision energies, per nucleon, marked SPS and RHIC and shown at left and right figure. The only thing I would like to mention is that the fraction of time spent in the QGP phase (below the curve) is not that large: but its existence crucially change flow pattern at RHIC. As one can see from the right plot, all lines of constant rapidity are nearly vertical and nearly equidistant: it means after QGP the flow gets a simple Hubble-like flow with and time-independent constant .

Location of the freezeout surface is also a nontrivial task. In our first application of hydrodynamics for radial transverse flow at SPS [23] we developed rather tedious “differential freezeout”, for each geometry and each secondary particle was applied. We calculated individual reaction rates for different secondaries and matched them with hydro expansion rates locally. In spite of relative success of the work, there were no followers to this approach at RHIC. Selecting hydrodynamics as a Ph.D. topic for my graduate student, Derek Teaney [19], we had in mind a different procedure for freezout, namely switching at the onset of hadronic phase ( in the figure above) to a hadronic cascade (RQMD) which automatically leads to earlier freezeout for smaller systems or particles with smaller hadronic cross sections111111Note that e.g. meson and the nucleon have about the same mass of 1 GeV, but cross section of re-scattering on pions ranging from few mb for the former to 200 mb for the latter, at the resonance peak dominating re-scattering.. The same approach was later used by Hirano et al, who generalized our rapidity-independent hydro to the full 3+1 dimensional case, successfully reproducing also the dependence on rapidity [24]. They confirmed that linearly decreases from its maximum at mid-rapidity linearly, for the same reason as it decreases toward smaller energies: there is less matter there, shorter QGP phase and also shorter hadronic phase due to earlier freezeout. Independently developed code by Nonaka and Bass [25], with a different cascade code UrQMD, later also confirmed these calculations and well reproduce data for all dependences. These three groups have basically covered all outstanding issues related to applications of the ideal hydrodynamics to RHIC data.

One might think that, after a couple of groups checked the calculations themselves, the rest would be history if all heavy ion community would recognize/accept it. Unfortunately, it is not the case. Many hydro groups have not implemented hadronic freezeout, ending hydro at some (arbitrarily selected) isoterms . There is absolutely no reason to think it is the right choice. Even thinking about freezout as a local concept, one finds that it is determined not by local density but by local expansion rate of matter. It is well known for more than a decade [23] that the isoterms do resemble the lines of fixed expansion rate. Furthermore, while hydro solutions simply scale with the size/time of the system121212Hydro eqns have only first derivative over coordinates, which can thus be rescaled., the freezout conditions (involving the reaction rates) do not. It is thus not at all surprising, that many results based on unrealistic freezeout show qualitatively wrong dependencies.

Figure 5: Top two panels: On the left, proton vs. for minimum-bias collisions at RHIC [27, 28] are compared with hydro calculations [19, 29, 30, 31], and on the right is the same comparison for pions. Bottom two panels: . On the left, for protons, for 0–5% centrality bin collisions at RHIC [32] are compared with the same hydro calculations. On the the right, the same comparison for pions.

Let me start with few plots from PHENIX “white paper” [26] in Fig.5 for protons and pions, to illustrate some important physics points explaining different level of success of different authors. The lower part of Fig.5 shows measured spectra of protons and pions, in comparison with different hydro calculations. The shape of those is very different mostly because heavy protons and light pions have different thermal motion at the time of freezout, in spite of the same collective radial flow. Note that nearly every group has a correct shape of the spectra (and thus correct radial flow velocity), but they don’t aways have the normalization (for the nucleons) correctly: it is because of “chemical freezout” not implemented by some groups. There is no problem for hydro+cascade model [19] (the red curve).

Now we switch to elliptic flow, shown as the ratio of observed momentum anisotropy divided by calculated spatial elliptic anisotropy , shown in the upper part of Fig.5. Although some calculations are not too close to the data, the overall magnitude of the effect and its dependence is clearly reproduced. However, that is only true for the “good” dependence . (Actually there is another one, the dependence of on the particle mass, which everybody get right. See one nice example of that from Hirano in Fig. 6(d).)

Unfortunately other dependences of are not so forgiving as and they show qualitative differences between models which do and do not include hadronic freezeout properly. Those include the dependences of the elliptic flow on (i) collision energy , (ii) centrality or number of participants , (iii) and rapidity . Let me start with the collision energy: Heinz and Kolb in their large hydro review [30] give their excitation curve for elliptic flows shown in Fig.6(a). All variants of their prediction for the elliptic flow has rapid rise on the left side of the plot (at low collision energies) with about constant saturating values at higher energies (one variant even reaches a peak followed by a ). As one sees from see Fig.6(b) from the same review, this is the trend observed in the RHIC domain: the data show a slow rise without peaks or saturation. Kolb and Heinz thus concluded that hydro is supported by the data, at all collision energies below RHIC. This lead to a myth about a “hydro limit” which was “never reached before RHIC” which was (and still is) repeated from one conference to another. Finally Fig.6(c) from that review display rapidity dependence of from a calculation by Hirano (before he switched to hydro+cascade): the conclusion was that hydro only works at mid-rapidity.

Figure 6: (a) Excitation function of the elliptic (solid) and radial (dashed) flow for Pb+Pb or Au+Au collisions at  fm from a hydrodynamic calculation.[33] The collision energy is parametrized on the horizontal axis in terms of total particle multiplicity density at this impact parameter. (b) A compilation of data vs. collision energy from mid-central (12–34% of the total cross section) Pb+Pb and Au+Au collisions. (c) -integrated elliptic flow for minimum bias Au+Au collisions at  GeV as a function of pseudorapidity, compared with data from PHOBOS and STAR. (d)Transverse momentum dependence of for pions, kaons and protons. Filled plots are the results from the hybrid model. The impact parameter in the model simulation is 7.2 fm which corresponds to 20-30% centrality.

All those results are for fixed-T freezeout, which is not based on anything and thus is simply wrong. Here what hydro+cascades approach finds for all of these observables. The energy excitation curves of from [19] and [24] are shown in Fig. 7, left and right. When correct freezout is implemented, the elliptic flow is rising steadily all the way from SPS to RHIC, as the data do. The reason Heinz et al (as well as curved marked 120 Mev and 100 MeV in left and right plots) strongly overshoot the data is simply because the freezout does occur at the same at different collision energy. In fact, it is independently measured (from radial flow for central collisions) that while the freezeout temperature at SPS is about 140 MeV, it is as low as 90 MeV at RHIC. The trend is well understood: the larger is the system, the it is at the beginning, and the it gets at the end of the explosion!

Figure 7: ( Two examples of the excitation function of in Au+Au collisions:(a) from [19] and (b) from [24]. Points obtained with fixed- (120 Mev in (a) and 100 MeV in (b)) freezeout strongly overpredict the data at low energies, but with cascade freezeout (red curves in both) those describe data quite well. (Two triangular points in (a) correspond to SPS and RHIC collision energies.)

Another “hydro problem” discussed in Kolb-Heinz review, the rapidity dependence of , also went away as soon as correct freezeout was used by Hirano et al. The results (circles in Fig.8 ) are right on top of the data (triangles), without any change of any parameters. The reason is exactly the same as for the energy dependence: in fact one can check that show good “limiting fragmentation properties, depending basically on , the distance to beam rapidity . One can see the difference in centrality dependence as well, in the left side of Fig.8 .

Intermediate summary: these “problems” (and associated myths) were caused by wrong freezeout. Matter at this time is a dilute pion gas, which is not a good liquid, neither at SPS, not at RHIC and will not be at LHC, and cascade is the best131313 This does not imply that we have complete confidence in many details of those cascades. To name one outstanding issue: the precise in-matter modification of hadronic resonances like etc, dominating the cross sections, is being addressed but still far from been solved. approach we have to describe it. As far as we can now test, two other evolution eras – sQGP and “mixed” or near- one – can be surprisingly well described by the hydrodynamics.

Figure 8: (From [24]) (Left) Centrality dependence of . The solid (dashed) line results from a full ideal fluid dynamic approach (a hybrid model). For reference, a result from a hadronic cascade model is also shown (dash-dotted line). (Right) Pseudorapidity dependence of . The solid (dashed) line is the result from a full ideal hydrodynamic approach with MeV ( MeV). Filled circles are the result from the hybrid model. All data are from the PHOBOS Collaboration. [34]
Figure 9: (From [36]) Elliptic flow dependence on (Left) centrality and (Right) , compared to viscous hydro with variable viscosity.

The last statement should not be understood that the agreement with all details of the data is perfect. Theoretically, one should always ask about accuracy and applicability limits of this hydrodynamical description. As emphasized by Teaney [35] the answer should be obtained by calculation the role of viscous effects. Since we will have extensive derivation of “derivative expansion” in AdS/CFT language at the end of this paper, I will not go into details here, going directly into new developments.

In order to get more accurate account of viscosity effect on flow, a new round of studies has been performed during the last year. Relativistic Navier-Stokes has some problem with causality, thus ‘higher order” methods has been used. Apart from viscosity those methods have another parameter, the relaxation time , which is can either be used as a regulator – and its value put to zero at the end – or as a real representation of two-gradient terms. There are 4 groups who have reported solving 2+1 dim higher order hydrodynamics. P. and U. Romatschke [36], Dusling and Teaney [37] , Heinz and Song141414This group originally found viscosity effects about twice larger than others: but it was found in their later work that this happened because of different account for some higher order term. In the limit, in which Navier-Stokes limits is supposed to be recovered, all results are now consistent with others. [38]. Molnar [39] have compared viscous hydro with some version of his parton cascade and found good agreement when the parameters are tuned appropriately: but cascade describe even at larger momenta. the solution is supposed to converge to that of the Navier-Stokes eqn, avoiding the causality problems.

In Fig.9 we show Romatschke’s results: literally taken they favor very small viscosity, even less than the famous lower bound. Now, is the accuracy level really allows us to extract ? The uncertainties in the initial state deformation [40, 41] are at the 10% level, comparable to the viscosity effect itself. EoS can probably be constrained better, but I think uncertainties related to freezeout – not yet discussed at all – are also at 10 percent level, although they can also be reduced down to few percent level provided more efforts to understand hadronic resonances/interactions at the hadronic stage will be made. All of it leaves us with a statement that while literally fits require or less, we can only conclude that it is definitely . 151515Unfortunately I am skeptical about magnitude of systematic errors of any lattice results for (such as [42]): while the Euclidean correlation functions themselves are quite accurate, the spectral density is obtained by rather arbitrary choice between many excellent possible fits. . Even so, sQGP is still the most perfect liquid known.

In summary: hydrodynamics+hadronic cascades reproduces all RHIC data on radial and elliptic flows of various secondaries, as a function of centrality,rapidity or energy are reproduced till , which is 99% of particles. Contrary to predictions of some, CuCu data match AuAu well, so Cu is large enough to be treated hydrodynamically. New round of studies last year included viscosity and relaxation time parameters on top of ideal hydro: viscosity values is limited to very small value.

2.3 Jets quenching and correlations

Pairs of partons can collide at small impact parameter: in pp collisions this produces a pair of large hadronic jets, which are (nearly) back-to-back in transverse plane (because total transverse momentum due to “intrinsic’ parton is small). We can use therefore those high- partons as a kind of x-rays, penetrating through the medium on its way outward and in principle providing its “tomography”.

We will not go into this subject in depth (see e.g. PHENIX “white paper” [26] but just note that accurate calibration of structure functions have been made in pp and dAu collisions, as well as with hard photon measurements (which are not interacting with the QGP). Thus we know quite well how many jets are being produced, for any impact parameter. The number of hard hadrons at transverse momentum relative to those expected to be produced as calculated from the parton model is called . If this quantity is 1, it means the jets are all accounted for and none is lost in the medium. This is what indeed is observed with direct photons, not interacting with the matter, see Fig.10(a). It was quite unexpected that for mesons this ratio was found to be rapidly decreasing and then in a wide range of momenta its value is only for central AuAu collisions, which means that of jets are absorbed.

Figure 10: (left) from PHENIX for and STAR for protons and charged pions. (right top) Two particle correlation function from STAR; (right bottom) from PHENIX for single leptons (closed points) and (open points) compared to theoretical calculations

Theory of quenching mechanisms included gluon radiation on uncorrelated centers (with Landau-Pomeranchuk-Migdal effect) [43], synchrotron-like radiation on coherent fields [44, 45], as well as losses due to elastic scattering. Comparing two radiation mechanisms in general let me just remind that the synchrotron-like radiation gives the energy loss growing quadratically with energy, it is stronger at high energies than radiation from uncorrelated kicks which gives only the first power : but then correlation length in “GLASMA” fields and their lifetime is limited. As for elastic scattering losses, it depends on what are the couplings and especially masses of quasiparticles (quarks, gluons or maybe monopoles near ) on which scattering occurs. The rate of energy loss itself is an order of magnitude larger than pQCD predictions. Multiple phenomenological fits for the were made: but they depend on models.

Furthermore, as noted in my paper [46], any such model has predictions for ellipticity in the range below its “geometric limit” for infinite quenching, while the data showed exceeding such limits for all models used. In simpler words, those models could fit but not a double plot . The reason of such large is not yet found, to my knowledge.

Further crucial test of this theory came from experimental observation of “single lepton” quenching and : those leptons come from semileptonic decays of quarks. At the same heavy quarks have smaller velocity, and if the main quenching mechanism be radiative, it should reduce quenching accordingly. The data however do show any serious reduction, with the same value for single leptons as for pions (coming mostly from gluon jets). This fact cast doubts at any perturbative mechanism of energy loss, since re-scattering of a gluon should be larger than that of a quark by the Casimir (color charge) ratio 9/4.

Moore and Teaney [47] developed a general framework of dealing with heavy quark dynamics in QGP, by invoking Focker-Plank or Langevin eqn. They have provided a general argument that if quark mass relative to temperature is large, relaxation of heavy quark is happening slowly and thus justify the Langevin’s uncorrelated kicks assumption.

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Here is a momentum drag coefficient and delivers random momentum kicks which are uncorrelated in time. is the mean squared momentum transfer per unit time. The usual diffusion constant in space is related to those parameters by

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In Fig.11 we show the calculated dependence of quenching and elliptic flow for leptons, resulting from the Langevin process (calculated on top of hydro evolution). As one can see stronger coupling leads to smaller and larger : comparison with data (value about .2 for and yellow band for ) clearly favor the smallest diffusion constant, about . Further work on heavy quark diffusion by Rapp and collaborators [48] have tried to specify the diffusion constant from data better, and also suggested its explanation using heavy-light resonances.

Figure 11: (from Moore and Teaney) (a) The charm quark nuclear modification factor and (b) elliptic flow for representative values of the diffusion coefficient given in the legend. In this model, the drag is proportional to the velocity, .

The next RHIC discovery was associated with “jet correlations”, which means that in events triggered by one hard particle with large one look for a second “companion particle” correlated161616It means that thousands of particles correlated with the trigger are statistically subtracted. with it. While in and peripheral collisions one sees “back-to-back jets”, with two peaks and relative azimuthal angle values near zero or , nuclear collisions typically show (or strongly reduced) peak at . Further subtraction of flow in the correlation functions revealed new peaks shifted from the direction of the companion jet by a large angle , see Fig.12(b).

Where the energy of the quenched jets go? Thinking about this question at the time we came with the answer: provided energy is deposited locally, hydrodynamics should provide a detailed prediction. Thus new hydrodynamical phenomenon171717In the field of heavy ion collisions Mach cone emission was actively discussed in 1970’s by Greiner et al: but it turned out not to work because nuclear matter – unlike sQGP – is not a particularly good liquid. The nucleon m.f.p. in nuclear matter is about 1.5 is not much smaller than the nuclear size. suggested in [49, 50], – the so called conical flow – is induced by jets quenched in sQGP. The kinematics is explained in Fig.12(a) which show a plane transverse to the beam. Two oppositely moving jets originate from the hard collision point B. Due to strong quenching, the survival of the trigger jet biases it to be produced close to the surface and to move outward. This forces its companion to move inward through matter and to be maximally quenched. The energy deposition starts at point B, thus a spherical sound wave appears (the dashed circle in Fig.12left ). Further energy deposition is along the jet line, and is propagating with a speed of light, till the leading parton is found at point A at the moment of the snapshot. The expected Mach cone angle is given by

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Here angular bracket means not only the ensemble average but also the average over the time from appearance of the wave to its observation181818Recall that the speed of sound changes significantly during the evolution, becoming small near .

Experimental correlation functions include the usual elliptic flow and the serious experimental issue was whether the peaks I just described are not the artifact of elliptic flow subtraction. By Quark Matter 05 this was shown to be the case, see Fig.12( top right), which shows PHENIX data selected in bins with specific angle between trigger jet and the reaction plane: the shape and position of the maximum (shown by blue lines) are the same while elliptic flow has a very different phase at all these bins. The position of the cone is on angle relative to reaction plane Fig.12(right top), centrality (not shown here) and Fig.12(right down): so one may think it is an universal property of the medium. The angle values themselves are a bit different, with 1.2 rad preferred by Phenix and 1.36 rad by Star data: those correspond to amazingly small velocity of the sound wave191919Note, it is the velocity not squared. The speed of sound can be seen as a dash curve in Fig.15(right). , indicating perhaps that what we see was related to the region. Another evidence for that is observation of conical structure at low collision (SPS) energies, reported at QM08 by CERES collaboration: at such energies near- region dominates.

At the last Quark Matter 08 large set of 3-particle correlation data ( a hard trigger plus companions) have been presented both by STAR and PHENIX collaborations. Although those are too technical to be shown here, the overall conclusion is that Mach cone structure is more likely explanation of the data than other possibilities such as “deflected jets”.

Figure 12: (left) A schematic picture of flow created by a jet going through the fireball. The trigger jet is going to the right from the origination point B. The companion quenched jet is moving to the left, heating the matter (in shadowed area) and producing a shock cone with a flow normal to it, at the Mach angle , where are jet and sound velocities.
(right top)The background subtracted correlation functions from PHENIX experiments, a distribution in azimuthal angle between the trigger jet and associated particle. Unlike in pp and collisions where the decay of the companion jet create a peak at (STAR plot), central collisions show a minimum at that angle and a maximum corresponding to the Mach angle

A number of authors have by now reproduced the very existence of conical flow in hydro, see e.g. Baeuchle et al[51]: but really quantitative study of its excitation is still to be done. Casalderrey and myself[52] have shown, using conservation of adiabatic invariants, that fireball expansion should in fact greatly enhance the sonic boom: the reason is similar to enhancement of a sea wave (such as tsunami) as it goes onshore. They also showed that data exclude 1-st order phase transition, because in this case conical flow would stop and split into two, which is not observed.

Antinori and myself[53] suggested a decisive test by b-quark jets. Those can be tagged experimentally even when semi-relativistic: the Mach cone should then shrink, till it goes to zero at the critical velocity . (Gluon radiation behaves oppositely, expanding with .)

The experiments with tagged -jets seem to be even more important in view of recent studies by Guylassy et al [193, 192] who found (using AdS/CFT results from [157] and specific model relating it to Cooper-Fry formula) that when the velocity of the jet approach the angle of the peak does not accurately follow the Mach angle but remains always larger. They have further found that the main part of the peak comes not from cones but from the non-hydrodynamical near-jet zone (they call the “neck”): what is the nature of those large angle emission remains unknown. This group have further studied weak-coupling (pQCD-based) version of the near zone [192], finding that in this case most of the flow remains at small angles, with very small but visible peaks at Mach angle, but no trace of large-angle emission predicted by AdS/CFT.

Let me finally mention the main open question, which is the absolute and relative amplitudes of excitation of two hydrodynamical modes, the (responsible for the Mach cone structure) and the mode, which show matter co-moving forward behind the jet. As emphasized in our works, this question cannot be answered from hydro itself, as close to the jet it looses its applicability. As we will see later, this ratio was recently found from AdS/CFT: but we don’t know yet of it does or does not agree with the data.

2.4 Charmonium suppression

Charmonium suppression is one of the classic probes: since charm quark pairs originate during early hard processes, they go through all stages of the evolution of the system. A small fraction of such pairs produce bound states. By comparing the yield of these states in heavy ion collisions to that in pp collisions (where matter is absent) one can observe their survival probability, giving us important information about the properties of the medium.

Many mechanisms of suppression in matter were proposed over the years. The first was suggested by myself in the original “QGP paper” [3], it is a gluonic analog to “photo-effect” . Perturbative calculations of its rate (see e.g. Kharzeev et al [54]) leads to a large excitation rates. Indeed, since charmonia are surrounded by many gluons in QGP, and nearly each has energy sufficient for excitation, one may think would have hard time surviving. That was the first preliminary conclusion: nearly all charmonium states at RHIC should be rapidly destroyed. If so, the observed may only come from charm quarks at chemical freezout, as advocated e.g. by Andronic et al [55].

However the argument given above is valid only if QGP is a weakly coupled gas, so that charm quarks would fly away from each other as soon as enough energy is available. As was recently shown by Young and myself [56], in strongly coupled QGP the fate charmonium is very different. Multiple momentum exchanges with matter will lead to equilibration in momentum space , while equilibration in position space is very slow and diffusive in nature. Persistent attractions between and makes the possibility of returning back to the ground state for the quite substantial, leading to a substantially higher survival probability. For the sake of argument, imagine the matter so dense that any diffusion of and is completely stopped: then, after this situation changes by hadronization, one would still find them close to each other and thus – with its by far the largest density at the origin – will be obtained again. Thus, strongly coupled – sticky - plasma may actually the .

Matsui and Satz [57] have proposed another idea and asked a different question: up to which does charmonium survive as a bound state? They argued that because of the deconfinement and the Debye screening, the effective attraction in QGP is simply too small to hold them together. Satz and others in 1980’s have used the free energy potential, obtained from the lattice, as an effective potential in Schreodinger eqn.

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They have shown that as the Debye screening radius decreases with and becomes smaller than the r.m.s. radii of corresponding states , those states should subsequently melt. Furthermore, it was found that for the melting point is nearly exactly , making it a famous “QGP signal”.

Figure 13: (From [56]). (a)Distribution over quark pair separation at fixed after 9 fm/c of Langevin evolution, (red squares) and (green triangles) the potential. Strong enhancement at small distances due to potential is revealed. (b)The points are PHENIX data for anomalous suppression of in AuAu min bias collisions . Two curves are Langevin model, with (solid, upper) and without (dashed, lower) feed-down.

Dedicated lattice studies [58, 59] extracted quarkonia spectral densities using the so called maximal entropy method (MEM) to analyze the temporal (Euclidean time) correlators. Contrary to the above-mentioned predictions, the peaks corresponding to states remains basically unchanged with in this region, indicating the dissolution temperature is as high as . Mocsy et al [60] have used the Schrödinger equation for the Green function in order to find an effective potential which would describe best not only the lowest s-wave states, but the whole spectral density. Recently [61] they have argued that a near-threshold enhancement is hard to distinguish from a true bound state: according to these authors, the above mentioned MEM dissolution temperature is perhaps too high. My view is that the collisional width of states in sQGP is probably large and thus the discussion of MEM data is rather academic: in any case we dont observe in plasma but after it, and a survival is a real-time issue which cannot be answered by the lattice anyway.

Let us now briefly review the experimental situation. For a long time it was dominated by the SPS experiments NA38/50/60, who have observed both “normal” nuclear absorption and an “anomalous” suppression, maximal in central PbPb collisions. Since at RHIC QGP has a longer lifetime and reaches a higher energy density, straightforward extrapolations of the naive melting scenarios predicted near-total suppression. And yet, the first RHIC data apparently indicate a survival probability very similar to that observed at the SPS.

One possible explanation [62, 63] is that the suppression is (nearly exactly) by a recombination process from unrelated (or non-diagonal) pairs floating in the medium. However this scenario needs quite accurate fine-tuning of two mechanisms. It also would require rapidity and momentum distributions of the at RHIC be completely different from those in a single hard production.

Another logical possibility advocated by Karsch, Kharzeev and Satz [64] is that actually does both at SPS and RHIC: all the (so called anomalous, or nonnuclear) suppression observed is simply due to suppression of feed-down from higher charmonium states, and . (Those are feeding down about 40% of in pp collisions.) These authors however have not explained survival probability can be close to one.

Young and myself [56] did exactly that, followed Langevin dynamics of charm quark pairs, propagating on top of (hydro) expanding fireball. The treatment basically is the same as that discussed in the preceding section, where heavy quark diffusion constant has been derived. One new important element though is the effective potential, which we found is slowing down dissolution of the pair quite substantially, l see Fig.13(a) leading to “quasiequilibrium” situation in which ratios of different charmonium states are close to equilibrium ones at corresponding , while the probability is continue to leak into unbound pairs which occupy slowly growing volume. The main finding of this work is that the lifetime of sQGP is not sufficient to reach the equilibrium distribution of the pairs in space, allowing for a significant fraction of to survive through of the sQGP era. This probability for charmonium dissociation in sQGP is much large than in perturbative estimates, or for Langevin diffusion which would not include strong mutual interaction. We have not yet answered many other questions: e.g. what happens during of the “mixed phase”. (In view of it seem to be a magnetic plasma, as we will argue below, the mutual attraction of charmed quarks gets only stronger there,)

3 From lattice QCD to sQGP

This section has been more difficult to write than others, because a connection between lattice results and the “strongly coupled” regime of QGP at remains indirect. Perhaps by itself it is rather unconvincing for a critical reader, as it was for many lattice practitioners, who are still quite reluctant to accept the “paradigm shift” of 2004. There are quite serious reasons for that. One (which we will discuss in the AdS/CFT section) is that the difference between weakly and strongly coupled regimes is deceivingly small in thermodynamical quantities. The second reason is that by performing Euclidean rotation of the formalism and correlators, one indeed gets rid of the unwanted phase factors, but a heavy price for that is extremely limited ability to understand real-time transport properties – diffusion constants, viscosity and so on – which turned out to be at the heart of this debate. As usual, we start with introduction for pedestrians which experts should jump over.

3.1 The QCD phase diagram for pedestrians

QCD phase diagram is quite multidimensional: apart from the temperature one can introduce chemical potentials for each quark flavors . One however only consider 2 combinations of those,the and chemical potentials. Then one can vary parameters of the theory itself, such as the number of colors or quark masses : in many cases however we will only discuss certain limits, for example in our discussion below a shorthand notation “2 flavors” would imply massless quarks and infinitely heavy (or just absent) quarks.

Three main phenomena will be under discussion: (i) confinement, (ii) chiral symmetry breaking and (iii) color superconductivity. The minimalistic phase diagram may have only three main phases: (a) hadronic, at low and , which is both confined and has broken chiral symmetry; (b) Quark-Gluon plasma (QGP) at high and , where all kind of condensates are absent; and (c) color superconductor (CS) at high and low . There can of course be many more phases, as these features are not really exclusive– e.g. there can be coexisting chiral and CS condensates.

Note that two last phenomena are due to different kinds of , chiral breaking (ii) is due to quark-antiquark pairing, its nonzero order parameter is the “quark condensate” ; while color superconductivity (iii) is due to quark-quark pairing and its nonzero order parameters are a set of diquark condensates , with color indices a,b and the flavor ones i,j arranged in various ways. (The spinor indices will be always suppressed because it is believed that whatever color-flavor structure of the condensates can be, the diquark spin remains , in most cases except very exotic ones.)

By confinement we mean more specifically confinement, which means that electric color field is expelled into flux tubes, making quark-antiquark potentials linear. component of the gauge field is expelled from the low temperature phase: we will in fact see that it actually dominates it202020We will not discuss AdS/QCD in this review: let me still remind its practitioners that popular models of confinement – hard or soft “walls” ending the holographic space and forcing strings to generate linear potential – are over-simplistic, because the same thing happens with electric and magnetic strings.. Linear potential is synonymous to the “Wilson area law” of a large Wilson loop

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with the nonzero string tension. From pioneering lattice calculation by Creutz in 1980’s we know that pure gauge theories indeed have this feature, although mathematical “proof” of that remains famously elusive. Polyakov introduced in 1978 his famous loop and argued that it provides a parameter, with nonzero value at . We will discuss issues related to this below in some detail.

The area law and Polyakov loop average are of course true order parameters only in pure gauge theory without quarks. Since quark pair can be produced, the lattice static potentials show linear behavior up to certain distances. Many people thus think that in QCD with quarks the confining phase cannot be strictly defined and thus there should be no real phase transition separating it from the deconfined QGP.

This is however still disputed. Another possible order parameter (albeit nonlocal) is based on the idea of the “dual superconductor” by tHooft and Mandelstamm [103] which suggested long ago an existence of a nonzero magnetic condensate of some bosonic objects. Di Giacomo and collaborators have implemented the corresponding observable – called “Pisa order parameter” - which is an operator adding one explicit monopole to the vacuum. If so, the deconfinement should be a true phase transition: but numerical lattice data for generic nonzero quark masses show only a rapid crossover. This contradiction is not yet resolved.

To show one example in which all used order parameters are studied together, let me discuss the work by Pisa group on QCD [65] shown in Fig.14. In this theory a quark has the same color as gauge particles, and – unlike the usual quarks – it makes chiral restoration temperature distinctly different212121Most lattice works agree on coinciding deconfinement and chiral symmetry in fundamental – real-world – QCD, but it is still debated and I refrained from taking sides and show one but not the other plot. from deconfinement. (Larger color representation (charge) of quarks is believed to lead to stronger pairing and thus higher melting temperature for the chiral condensate.) The Polyakov loop shown in figure (a) behaves as disorder parameter at deconfinement indeed, the same point as indicated by Pisa parameter (c), while the chiral condensate and its susceptibility (b) indicate much smoother higher- chiral restoration transition. Thus in adjoint QCD we see a presence of one more phase (on top of the minimal list of three given above), a deconfined but chirally broken phase which is usually called “a plasma of constituent quarks”. It has all the requisites of a chirally broken phase, such as massless Goldstone bosons – pions, etc.

Figure 14: (a) The Polyakov loop, with and lattice. (b) Susceptibility of the chiral condensate, with and two different lattices: , . The data are compatible with a crossover at the chiral transition. In the inset, the chiral condensate within the same range of values. A clear jump is visible at the deconfinement phase transition. (c) The parameter, with , , for two different spatial volumes.

In general it is believed that with growing all transition become sharper. Arguments about likely shift in real-world QCD from near-crossover at zero and to real second-order point (the so called QCD critical point) and then first order line has been put forward by Rajagopal,Stephanov and myself [66], as well as few proposals how one can experimentally search for it. RHIC specialized run with greatly reduced beam energy is planned for 2009: perhaps it will shed light on this issue.

What happens if one changes another famous parameter, the number of colors by increasing ? Because gluons are adjoint and their effects are , they dominate all quark-induced effects. It has been recently argued by McLerran and Pisarski [67] that this implies that at large enough . The phase in between –chirally restored and confined – they called a “quarkionic” phase, thinking about quark-filled Fermi sphere but with baryonic excitations at the surface. Glozman [68] further suggested even more exotic possibility: particle excitations in form of (chirally symmetric) baryons, baryonic holes. The issue is not yet settled and such phases have not been seen on the lattice222222Chirally symmetric baryons can certainly exist: in fact Liao and myself [69] have argued that those are needed to explain lattice susceptibilities in the usual QGP near deconfinement at .. My view is that both these exotic ideas are perhaps excluded in a confined phase, so it should be just baryonic.

In the discussion above we have ignored short 1-st order transition line between the “vacuum-like” and “nuclear matter-like” regions, at . Since the usual nuclear matter is Fermi-liquid, it is not qualitatively different from the bulk of hadronic phase: thus one can go around this phase transition. This is not the case for larger , as in this case baryons are becoming heavier while the nuclear forces are believed to have a smooth limit

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The obvious consequence of this is crystallization, as kinetic energy gets subleading to the potential one. Examples of specific calculations with the skyrmions are well known232323These works also found another phase at higher density, in which skyrmions “fragment” into objects of fractional topology: they interpreted those as a ”chirally restored” baryonic phase. I however don’t know if one can trust the model all the way to this phase. [70]. Is there a possibility that solidification happens even for physical QCD? Old Migdal’s pion condensation was of this nature, and for more recent study of crystalline matter see Rapp, myself and Zahed [71]. There can also be a crystalline color superconductor, known also as LOFF phase, see more in the review [72]. Instead of going any further into the zoo of possibilities, let me stop with a joke: perhaps QCD would not have less phases than water does, and this is quite a lot.

Finally, for completeness, let me mention the opposite direction, increasing the number of light quarks . At some point asymptotic freedom will disappear, but before that there should be a critical line (or more) at which Banks-Zaks infrared fixed point will make the theory conformal in IR. As the corresponding fixed point coupling gets less and less, chiral symmetry breaking and confinement must go: again either together or separately. All this territory is amenable to lattice analysis but remains largely unexplored.

Figure 15: The trace anomaly, (left) calculated on lattices with temporal extent and the ratio of pressure and energy density as well as the velocity of sound obtained in calculations with the p4fat3 action on (short dashes) and (long dashes).

3.2 Main QGP properties from the lattice

The thermodynamical observables – pressure and energy density – from the lattice is the simplest global observable, thus they were calculated more than a decade ago (and used in hydro calculations). Let me show two recent plots from Karsch [73] which depict them in a combinations which reveal somewhat more than standard plots of .

The first combination is related to the famous scale anomaly: the fact that its value relative to themselves goes to zero means that QGP gets more and more conformal. However the way it goes down is not – as simple bag model predicts: this phenomenon is not yet explained.

As shown in [73] and earlier papers, quark and gluon quasiparticles dominate at . But what happens below that? Gluon/quark masses are too high to explain the peak of . Chernodub et al [74] have shown that lattice magnetic objects – monopoles and vortices – reproduce the shape242424Unfortunately not the absolute normalization: this issue deserves further studies as the lattice used in this work is rather small. of this curve well.

The second combination252525Its special role in hydrodynamics was emphasized [23] and its minimum got a spatial name - “the softest point”. Indeed, the gradient of pressure provides a force and energy density a mass to be moved, so it is proportional to hydrodynamical acceleration of matter. is - now conformity is seen as the place where this ratio reaches 1/3. One can clearly see that it is not yet reached at RHIC, but it is the case at LHC. This is one of the reasons why LHC experiments will be decisive in proving (or disproving) whether the AdS/CFT duality can (or cannot) be used in the conformal window of finite- QCD.

Let me now turn to the screening lengths. As I already mentioned, QGP got its name after it was found [2] that thermal gluons – unlike virtual ones – lead to electric screening of the charge in weakly coupled regime (high ). The corresponding electric (or a Debye) mass is . Static screening does not appear via perturbative diagrams; but it has been soon conjectured by Polyakov [75] that magnetic screening should appear non-perturbatively, at the smaller “magnetic scale” .

To illustrate whether lattice results on the screening masses are or are not in agreement with that, we show their -dependence calculated by Nakamura et al [76], see Fig.16(a). Note that at high the electric mass is indeed significantly larger than the magnetic one, but it vanishes at – here electric objects gets too heavy and “electric part” of QGP effectively disappears. However magnetic screening mass grows continuously toward : thus the two cross each other, around .

These observation were in fact the starting point for Liao and myself in thinking about “magnetic scenario” for the near- region. We had used the screening masses to get an idea about density of electric and magnetic objects, one should conclude that QGP switched from electric to magnetic plasma somewhere around of

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.


Figure 16: (upper left) Temperature dependence of electric and magnetic screening masses according to Nakamura et al [76]. The dotted line is fitted by the assumption, . For the electric mass, the dashed and solid lines represent LOP and HTL re-summation results, respectively. (upper right) Temperature dependence of the effective string tensions of the free and potential energies . (down) The energy and entropy (as ) of two static quarks separated by large distance, in -flavor QCD according to [77].

The static potential between quark and anti-quark is another traditional observable, by means of which quark confinement in Non-Abelian gauge theories was established. It was originally inferred from heavy meson spectrum and Regge trajectories, and then studied in great detail numerically, through lattice gauge theories, for review see e.g. [78]. It is usually represented as a sum of a Coulomb part , dominant at small distances, and a linear part dominant at large distances. The latter, related with existence of electric flux tubes, is a manifestation of quark confinement. The string tension in the vacuum () has been consistently determined by different methods to be about

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Studies of the static potential have been extended to finite . In particular, deconfinement temperature is defined as a disappearance of the linear behavior as a signal of deconfinement at in the corresponding free energy . Bielefeld-BNL group has published lattice results for static free energy, as well as internal energy and entropy

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at both below and above , see [79, 80].

Remarkable features of these results include:
1. The linear (in r) part of the potentials. Their effective tensions are shown in Fig.16(top right). While that for free energy vanishes at (by definition), that for potential energy extends till at least about 1.3, with a peak values about 5 times (!) the . Similar behavior is seen in entropy,while canceling in free energy. The widths of these peaks provide a natural definition of “near-” region as
2.Although potentials at large distances are finite , near their values reach very large magnitudes, see Fig.16(down). The corresponding large entropy means that really huge number of states is involved ;
The origin of this large energy and entropy associated with static pairs near , remains mysterious: many attempts (e.g. [81]) failed to explain it. Below we will return to this phenomenon in connection with “magnetic plasma” scenario.

Before looking for explanations, however, let us focus on physical difference between F and U, based on papers by Zahed, Liao and myself [82, 83], in which they are related to what happens for and motion of the charges. To be specific, let’s consider a pair of static charges held by external hands. Suppose they are close initially , and then are separated to some finite . This can be done in two possible ways, adiabatically slowly or very fast. The difference between them in thermodynamical and quantum-mechanical contexts are known in many fields of physics. Perhaps the oldest is the so called Landau-Zener problem [84] of electron motion, following the motion of two nucleus in a diatomic molecule. While nuclei change their relative distance with velocity , the electrons are in a specific quantum state with the energy depending in . The issue is probability of the level crossings, which appear when there are two quantum levels crossing each other, at some separation . When the two nucleus approach the crossing at very slowly, then the electrons may jump from one energy state to another, always selecting the lowest energy state. However if the two nucleus move fast, there is large probability for the electron to remain in the original state. More quantitatively, Landau-Zener showed that this probability is given by

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where is the non-diagonal transition matrix element of the Hamiltonian; but we will only need the limits of large and small . The adiabatic limit obviously corresponds to free energy measured on the lattice. The “potential energy” means that no entropy is generated: this implies that there was no transition from the original pure state at T=0 into multiple states as level crossing occur: thus it corresponds to fast motion limit. The positivity of entropy means that always.

This discussion is very relevant for the problem of effective potential to be used in Schreodinger eqn for the bound states, e.g. in charmonium problem. Zahed and myself [82] argued that in this case one should use the energy: provide much more stable bound states, delaying melting to higher . Several authors (e.g. [85]) have used effective potentials in between those two limiting cases. With such potentials not only charmonium but also light quark mesons get bound, as also are baryons and “gluonic chains” [86] and also colored binary states. However, in a liquid with the parameters we expect from such interaction the number of nearest neighbors associated with one charge is expected to be , and thus it is not clear what is the role of the binary states. Quantum manybody studies of these systems are not attempted yet, and we don’t know if there is any sense to identify them, and if so how wide those states may be.

3.3 Polyakov loop, “Higgsing” and deformations of QCD

Physics of monopoles, to which we will turn shortly below, has been originally developed in the Georgi-Glashow model or =2 SUSY which has adjoint scalars. Those may have some nonzero expectation values – this phenomenon would be colloquially referred to as “Higgsing” below. If so, the color group is broken, generically to diagonal . QCD-like theories are much more difficult precisely because they do have elementary scalars, making Higgsing much more subtle, with its role at finite presumably played by the zeroth component of the gauge field .

The definition of the Polyakov loop [75] is a holonomy of the gauge field across the periodic direction

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where is the Matsubara time. If it has VEV one can think of as a diagonal color matrix, with some eigenvalues : Polyakov’s view on confinement is that the eigenvalues widely fluctuate and : in the deconfined (plasma phase) these eigenvalues fluctuate little near minima of the effective action, at . At high – that is in weak coupling – one can calculate this effective potential perturbatively. In zeroth order, costs no energy, but its coupling to gluons in the heat bath leads to shifting the gluon states262626Here we mean gluodynamics only: we turn to quarks a bit later. and as a result one gets the following effective action

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This pushes away from zero, to the minima mentioned. One set of lattice data on has been already shown in Fig.14(a): as one can see the breaking indeed happens and thus it is indeed a disorder parameter.

Opinions on the role of the breaking of the symmetry vastly duffer: while some think it the very essence of confinement others think it has no dynamical meaning at all. To exemplify this polemics the reader may e.g. see Ref. [87] in which Smilga pointed out that it is highly suspicious that the pure gauge theory – which has no such symmetry and can be formulated as gauge theory rather than , with explicitly eliminated. Smilga further argued using simpler examples that effective potential is gauge-dependent concept and should be treated with care, he emphasized that although at finite lattices/coupling one may apparently see domains, only one minimum is physical and the so called bubbles with nonzero surface tension are not really there. For comparison of these two lattice formulation see deForcrand [88]. The question is no longer debatable when there are fundamental fermions in the theory, as they see different phases and thus no longer respect the symmetry and one vacuum with real Polyakov line VEV is preferred.

Figure 17: Schematic phase diagram for adjoint QCD with Shifman-Unsal deformations, on a plane deformation parameter - temperature . At zero deformation (left side of the plot) one has the usual adjoint QCD in which deconfinement happens below chiral symmetry restoration. However, as grows and we deform into a theory without deconfinement, the order must be reversed, giving place to a new phase in the right upper corner, which is confined but chirally symmetric.

Before we discuss phenomenology of the Polyakov line and its effective potential in strong coupling (on the lattice) in real-world QCD, let us turn to interesting “deformations” of QCD recently discussed by Unsal [89] and Shifman and Unsal [90]. The central idea is to deform some QCD-like theory into something else to which the answer is either known or can be obtained perturbatively. Although they consider several different fermion representations, for simplicity we will only consider in this section the case of fermions, for which the interplay of gluons and fermions is simpler. Before going into details, here is the list of deformations:
(i) rotating fermionic boundary conditions along the time from anti-periodic to periodic
(ii) putting the theory into a spatial box with variable size
(iii) introducing additional potential for the Polyakov loop.

The first deformation is obtained via the introduction of the phase into fermion boundary condition, allowing to interpolate smoothly between the periodic and anti-periodic boundary condition

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into fermionic contribution into a line, with changing from (anti-periodic) to (periodic) continuously. The effective potential becomes then

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When ( boundary conditions) and the number of flavors two terms in the effective action simply cancel: this happens because this theory is the =1 SUSY gluodynamics with the supersymmetry remaining unbroken272727We remind the reader that normal thermal boundary conditions obviously break the symmetry between fermions and bosons.. The famous argument based on Witten index applies in this case, telling that the number of vacua cannot be changed with any deformation. When the fermionic terms dominate and the sign of the potential is reversed. It means in this case one has the theory in which is not pushed to large values and there is no breaking and thus no deconfinement even at weak coupling.

Let us now think about deconfinement and chiral symmetry of the adjoint QCD for (the upper limit from asymptotic freedom) in the plane, shown in Fig.17. When we have the usual (undeformed) adjoint QCD in which (see Fig.14) . But when there should be no deconfinement phase at all, which means that grows indefinitely before crossing the vertical line. There should however still be chiral symmetry breaking at any : in fact the theory with periodic fermions has well known dyons with fermionic zero modes, which generate NJL-type interaction and chiral symmetry breaking provided is small enough to get the coupling large enough282828The physics of chiral transition is the same transition between “instanton liquid” and “instanton-antiinstanton molecules described by Ilgenfritz and myself [91], see also references for later studies in the review [92].. As a result, there should be an intersection between the deconfinement line and the chiral restoration line at some , as shown in Fig.17: after the deconfinement and chiral symmetry restoration lines have crossed one finds a qualitatively new phase, confinement but with chiral symmetry .

Now let us turn to the second Shifman-Unsal deformation of the QCD-like theories: if one formulates the theory on space with (periodic for fermions) compact direction of the variable length , one can gradually interpolate between 4-d and 3-d gauge theories. The major difference between those, as explained by Polyakov [93] many years ago, is that while 4-d instanton-antiinstanton interaction is short range , the 3d instantons (that is, monopoles) interact by a long range magnetic Coulomb . The result is that the 3d theory is confined by monopoles-instantons, while the 4d theory is not confined by its instantons. Moreover, it happens even in the weak coupling regime, in which the instanton-monopole density is exponentially small.

Specific mechanism for QCD on was discussed by Unsal in a separate paper [94] for periodic fermions, it is condensation of magnetic charge 2 “bions” – pairs of certain dyons – bound by fermion-induced forces in spite of mutual magnetic repulsion. The binding is analogous to instanton-antiinstanton molecule formation, the confinement at high is like Polyakov mechanism in 3dim [93].

The third Shifman-Unsal deformation is done by an to the QCD action of artificial potential for the , e.g.

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with some coefficients chosen at will. The authors themselves argued that if the coefficients are chosen in order to the generated by quantum fluctuations naturally, one should be able to delay deconfinement (increase ), to the extent that it will occur in the weakly coupled domain and make it tractable in a (semiclassical) controlled approximation.

Perhaps it is also interesting to go to another direction as well, , reaching for the regime in which electric theory is even stronger coupled than usual but its dual –magnetic theory of monopoles – will gets perturbative instead. Both deformations can easily be done on the lattice: a possibility to check continuity of the underlying physics of both deconfinement and chiral restoration all the way from strong to weak coupling will surely contribute a lot into our understand of both.

3.4 Phenomenology of in QCD

The early history of perturbative derivation of the effective potential for (or ) can be found in the classic review [95]. Instead of repeating here well known results, let me refer to more recent attempts to combine known perturbative results with the lattice data include a paper by Pisarski [96] where one can find the details of the recommended effective Lagrangian.

One more motivation to study the QCD deformation via adding extra potential for Polyakov loop on the lattice comes from heavy ion phenomenology. Dumitru and collaborators [97] have used this form of the effective Lagrangian to study real-time evolution of . The main conclusion from their work is that that belongs to the class of so called slow variables, and its evolution in heavy ion collisions has to be treated separately from the overall equilibration. They have performed numerical solution of the EOM for it, starting from “suddenly quenched” value corresponding to its vacuum form, moving toward its minimum at the deconfined phase at . The main finding of this work is that the relaxation of this variable is very slow, taking about 40 fm/c or so. This time significantly exceeds the QGP lifetime at RHIC which is only about 5 fm/c or so, which suggests that in real collisions we should treat as essentially random variable frozen at some value and color direction during hydro evolution. It means there is a chaotic out-of-equilibrium Higgsing, slowly rolling down, like in cosmological inflationary models: thus one would like to know as much as possible about phase transitions and EoS for values of .

Another active direction using effective potential is the so called PNJL model [98], which combines the Polyakov loop with well known Nambu-Iona-Lasinio model for chiral symmetry breaking, see also [99]. Quite impressive results for QCD thermodynamics were obtained along this path by the group of Weise [100]. Let me give their notations and the parameterization of the potential. A background color gauge field , where with the gauge fields and the generators . The matrix valued, constant field relates to the (traced) Polyakov loop as follows:

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In the so-called Polyakov gauge one chooses a diagonal representation for the matrix , which leaves only two independent variables, and . The potential involves the logarithm of , the Jacobi determinant which results from integrating out six non-diagonal generators while keeping the two diagonal ones, , to represent :

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with

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The logarithmic divergence of as automatically limits the Polyakov loop to be always smaller than 1, reaching this value asymptotically only as . The parameters and are determined to reproduce lattice data for the thermodynamics of pure gauge lattice QCD up to about twice the critical temperature292929At much higher temperatures, where transverse gluons begin to dominate, the PNJL model is not supposed to be applicable.. The values of these parameters are