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This document summarizes the efforts of the EMMI Rapid Reaction Task Force on "Suppression and (re)generation of quarkonium in heavy-ion collisions at the LHC", centered around their 2019 and 2022 meetings. It provides a review of existing experimental results and theoretical approaches, including lattice QCD calculations and semiclassical and quantum approaches for the dynamical evolution of quarkonia in the quark-gluon plasma as probed in high-energy heavy-ion collisions. The key ingredients of the transport models are itemized to facilitate comparisons of calculated quantities such as reaction rates, binding energies, and nuclear modification factors. A diagnostic assessment of the various results is attempted and coupled with an outlook for the future., Comment: 54 pages, 19 figures

We propose a model for the propagation of a heavy-quark in a hot plasma, to be viewed as a first step towards a full description of the dynamics of heavy quark systems in a quark-gluon plasma, including bound state formation. The heavy quark is treated as a non relativistic particle interacting with a fluctuating field, whose correlator is determined by a hard thermal loop approximation. This approximation, which concerns only the medium in which the heavy quark propagates, is the only one that is made, and it can be improved. The dynamics of the heavy quark is given exactly by a quantum mechanical path integral that is calculated in this paper in the Euclidean space-time using numerical Monte Carlo techniques. The spectral function of the heavy quark in the medium is then reconstructed using a Maximum Entropy Method. The path integral is also evaluated exactly in the case where the mass of the heavy quark is infinite; one then recovers known results concerning the complex optical potential that controls the long time behavior of the heavy quark. The heavy quark correlator and its spectral function is also calculated semi-analytically at the one-loop order, which allows for a detailed description of the coupling between the heavy quark and the plasma collective modes.

We study the gluon distribution in nucleus-nucleus collisions in the framework of the Color-Glass-Condensate. Approximate analytical solutions are compared to numerical solutions of the non-linear Yang-Mills equations. We find that the full numerical solution can be well approximated by taking the full initial condition of the fields in Coulomb gauge and using a linearized solution for the time evolution. We also compare kt-factorized approximations to the full solution., Comment: 13 pages, 12 figures, RevTeX

The heavy-quark spectral function in a hot plasma is reconstructed from the corresponding euclidean propagator. The latter is evaluated through a path-integral simulation. A weak-coupling calculation is also performed, allowing to interpret the qualitative behavior of the spectral function in terms of quite general physical processes., Comment: 4 pages, 3 figures - To appear in the conference proceedings for Quark Matter 2009, March 30 - April 4, Knoxville, Tennessee

We demonstrate the power of a recently-proposed approximation scheme for the non-perturbative renormalization group that gives access to correlation functions over their full momentum range. We solve numerically the leading-order flow equations obtained within this scheme, and compute the two-point functions of the O(N) theories at criticality, in two and three dimensions. Excellent results are obtained for both universal and non-universal quantities at modest numerical cost., Comment: 4 pages, 1 figure

We apply the previously proposed scheme of approximately self-consistent hard-thermal-loop resummations in the entropy of high-temperature QCD to N=4 supersymmetric Yang-Mills (SYM) theories and compare with a (uniquely determined) R[4,4] Pad\'e approximant that interpolates accurately between the known perturbative result and the next-to-leading order strong-coupling result obtained from AdS/CFT correspondence. We find good agreement up to couplings where the entropy has dropped to about 85% of the Stefan-Boltzmann value. This is precisely the regime which in purely gluonic QCD corresponds to temperatures above 2.5 times the deconfinement temperature and for which this method of hard-thermal-loop resummation has given similar good agreement with lattice QCD results. This suggests that in this regime the entropy of both QCD and N=4 SYM is dominated by effectively weakly coupled hard-thermal-loop quasiparticle degrees of freedom. In N=4 SYM, strong-coupling contributions to the thermodynamic potential take over when the entropy drops below 85% of the Stefan-Boltzmann value., Comment: 14 pages, 2 figures, JHEP3. v2: revised and expanded, with unchanged HTL results but corrected NLO strong-coupling result from AdS/CFT (which is incorrectly reproduced in almost all previous papers comparing weak and strong coupling results of N=4 SYM) and novel (unique) Pade approximant interpolating between weak and strong coupling results

We investigate a zero-dimensional toy model originally introduced by Mueller and Salam which mimics high-energy scattering in QCD in the presence of both gluon saturation and gluon number fluctuations, and hence of Pomeron loops. Unlike other toy models of the reaction-diffusion type, the model studied in this paper is consistent with boost invariance and, related to that, it exhibits a mechanism for particle saturation close to that of the JIMWLK equation in QCD, namely the saturation of the emission rate due to high-density effects. Within this model, we establish the dominant high-energy behaviour of the S-matrix element for the scattering between a target obtained by evolving one particle and a projectile made with exactly n particles. Remarkably, we find that all such matrix elements approach the black disk limit S=0 at high rapidity Y, with the same exponential law: ~ exp(-Y) for all values of n. This is so because the S-matrix is dominated by rare target configurations which involve only few particles. We also find that the bulk distribution for a saturated system is of the Poisson type., Comment: 34 pages, 9 figures. Some explanations added on the frame-dependence of the relevant configurations (new section 3.3)

We show how the Landau-Pomeranchuk-Migdal effect on photon production rates in a quark-gluon plasma can be derived via the usual Boltzmann equation. To do this, we first derive the electromagnetic polarization tensor using linear response theory, and then formulate the Boltzmann equation including the collisions mediated by soft gluon exchanges. We then identify the resulting expression for the production rate with that obtained by the field-theoretic formalism recently proposed by Arnold, Moore and Yaffe. To illustrate the LPM effect we solve the Boltzmann equation in the diffusion approximation., Comment: 17 pages, 4 figures Minor rewriting. In particular, the specific small angle behavior of the collision term involved in the LPM effect is better explained. Version accepted for publication in Physics Letters B

We propose a method to solve the Non Perturbative Renormalization Group equations for the $n$-point functions. In leading order, it consists in solving the equations obtained by closing the infinite hierarchy of equations for the $n$-point functions. This is achieved: i) by exploiting the decoupling of modes and the analyticity of the $n$-point functions at small momenta: this allows us to neglect some momentum dependence of the vertices entering the flow equations; ii) by relating vertices at zero momenta to derivatives of lower order vertices with respect to a constant background field. Although the approximation is not controlled by a small parameter, its accuracy can be systematically improved. When it is applied to the O(N) model, its leading order is exact in the large $N$ limit; in this case, one recovers known results in a simple and direct way, i.e., without introducing an auxiliary field., Comment: Minor changes. Version to be published

We propose an effective theory which governs Pomeron dynamics in QCD at high energy, in the leading logarithmic approximation, and in the limit where N_c, the number of colors, is large. In spite of its remarkably simple structure, this effective theory generates precisely the evolution equations for scattering amplitudes that have been recently deduced from a more complete microscopic analysis. It accounts for the BFKL evolution of the Pomerons together with their interactions: dissociation (one Pomeron splitting into two) and recombination (two Pomerons merging into one). It is constructed by exploiting a duality principle relating the evolutions in the target and the projectile, more precisely, splitting and merging processes, or fluctuations in the dilute regime and saturation effects in the dense regime. The simplest Pomeron loop calculated with the effective theory is free of both ultraviolet or infrared singularities., Comment: 13 pages, 1 figure

We discuss proton-nucleus collisions in the framework of the color glass condensate. By assuming that the proton can be described as a low density color source, we solve exactly the Yang-Mills equations corresponding to this type of collision, and then use this solution in order to calculate inclusive gluon production or quark-antiquark production. Our result shows that k_T-factorization, while valid for gluon production, is violated for quark pair production in proton-nucleus collisions., Comment: Talk given at SEWM2004, Helsinki, June 2004

We compute the production of quark-antiquark pairs in high energy collisions between a small and a large projectile, as in proton-nucleus collisions, in the framework of the Color Glass Condensate. We derive a general expression for quark pair-production, which is not k_t-factorizable. However, k_t-factorization is recovered in the limit of large mass pairs or large quark--anti-quark momenta. Our results are amenable to a simple interpretation and suggest how multi-parton correlations at small x can be quantified in high-energy proton/deuteron-nucleus collisions., Comment: 37 pages, 20 postscript figures

We study gluon production in high energy proton-nucleus collisions in the semi-classical framework of the Color Glass Condensate. We develop a general formalism to compute gluon fields in covariant gauge to lowest order in the classical field of the proton and to all orders in the classical field of the nucleus. The use of the covariant gauge makes the diagrammatic interpretation of the solution more transparnt. k_t-factorization holds to this order for gluon production -- Our results for the gluon distribution are equivalent to the prior diagrammatic analysis of Kovchegov and Mueller. We also show that these results are equivalent to the computation of gluon production by Dumitru and McLerran in the Fock-Schwinger gauge. We demonstrate how the Cronin effect arises in this approach, and examine its behavior in the two extreme limits of a) no small-x quantum evolution, and b) fully saturated quantum evolution. In both cases, the formalism reduces to Glauber's formalism of multiple scatterings. We comment on the possible implications of this study for the interpretation of the recent results on Deuteron-Gold collisions at the Relativistic Heavy Ion Collider (RHIC)., Comment: 46 pages, 12 postscript figures

This paper has been withdrawn by the authors. We have discovered an error in our estimate of c, leading to a larger discrepancy with the exact result. This prevents us to judge the quality of the approximation from its leading order prediction, and we withdraw the paper until we have completed the next to leading order calculation, Comment: This paper has been withdrawn

We review the various methods which have been employed recently to describe the thermodynamics of the high temperature quark-gluon plasma using weak coupling techniques, and we compare their results with those of most recent lattice gauge calculations. Many of the difficulties encountered with perturbation theory at finite temperature are in fact not specific to QCD but are present in any field theory at finite temperature and will be discussed first in the simple example of the scalar field theory. We discuss the merits and limitations of various techniques which have been used to go beyond perturbation theory in the soft sector, such as dimensional reduction, screened perturbation theory or hard-thermal-loop perturbation theory, and Phi-derivable approximations. In the last part of the review, we focus on the later, which lead to a remarkably simple expression for the entropy of the quark-gluon plasma. When complemented with further, physically motivated, approximations, this approach reproduces accurately the entropy obtained from lattice gauge calculations at temperatures above 2.5 T_c, where T_c is the deconfinement temperature. This calculation thus provides also support to the physical picture of the quark-gluon plasma as a gas of weakly interacting quasiparticles., Comment: Review for "Quark-Gluon Plasma 3", eds. R.C. Hwa and X.-N. Wang, World Scientific, Singapore. 63 pages, 21 figures. v2: minor corrections and 2 references added

The successive perturbative estimates of the pressure of QCD at high temperature T show no sign of convergence, unless the coupling constant g is unrealistically small. Exploiting known results of an effective field theory which separates hard (order 2 pi T) and soft (order gT) contributions, we explore the accuracy of simple resummations which at a given loop order systematically treat hard contributions strictly perturbatively, but soft contributions without truncations. This turns out to improve significantly the two-loop and the three-loop results in that both remain below the ideal-gas value, and the degree of renormalization scale dependence decreases as one goes from two to three loop order, whereas it increases in the conventional perturbative results. Including the four-loop logarithms recently obtained by Kajantie et al., we find that this trend continues and that with a particular sublogarithmic constant the untruncated four-loop result is close to the three-loop result, which itself agrees well with available lattice results down to temperatures of about 2.5 T_c. We also investigate the possibility of optimization by using a variational (``screened'') perturbation theory in the effective theory. At two loops, this gives a result below the ideal gas value, and also closer to lattice results than the recent two-loop hard-thermal-loop-screened result of Andersen et al. While at three-loop order the gap equation of dimensionally reduced screened perturbation theory does not have a solution in QCD, this is remedied upon inclusion of the four-loop logarithms., Comment: v2: 18 pages, 8 figures, REVTEX4, 2 additional subsections discussing the effect of 4-loop logarithms; v3: references updated, minor additions and a new paragraph in sect. IIIC

We present new results on the Gross-Neveu model at finite temperature and at next-to-leading order in the 1/N expansion. In particular, a new expression is obtained for the effective potential which is explicitly invariant under renormalization group transformations. The model is used as a playground to investigate various features of field theory at finite temperature. For example we verify that, as expected from general arguments, the cancellation of ultraviolet divergences takes place at finite temperature without the need for introducing counterterms beyond those of zero-temperature. As well known, the discrete chiral symmetry of the 1+1 dimensional model is spontaneously broken at zero temperature and restored, in leading order, at some temperature T_c; we find that the 1/N approximation breaks down for temperatures below T_c: As the temperature increases, the fluctuations become eventually too large to be treated as corrections, and a Landau pole invalidates the calculation of the effective potential in the vicinity of its minimum. Beyond T_c, the 1/N expansion becomes again regular: it predicts that in leading order the system behaves as a free gas of massless fermions and that, at the next-to-leading order, it remains weakly interacting. In the limit of large temperature, the pressure coincides with that given by perturbation theory with a coupling constant defined at a scale of the order of the temperature, as expected from asymptotic freedom., Comment: 77 pages, 19 figures (some of them bitmaped, for original figures contact authors)

We compare our previously proposed hard-thermal-loop (HTL) resummed calculation of quark number susceptibilities using a self-consistent two-loop approximation to the quark density with a recent calculation of the same quantity at the one-loop level in a variant of HTL-screened perturbation theory. Besides pointing out conceptual problems with the latter approach, we show that it severely over-includes the leading-order interaction effects while including none of the plasmon term which after all is the reason to construct improved resummation schemes., Comment: 6 pages, 6 figures. Revised version to appear in Eur. J. Phys. C

We explore and clarify the connections between two different forms of the renormalisation group equations describing the quantum evolution of hadronic structure functions at small $x$. This connection is established via a Langevin formulation and associated path integral solutions that highlight the statistical nature of the quantum evolution, pictured here as a random walk in the space of Wilson lines. The results confirm known approximations, form the basis for numerical simulations and widen the scope for further analytical studies., Comment: 32 pages

We compute analytically the diagonal quark number susceptibilities for a quark-gluon plasma at finite temperature and zero chemical potential, and compare with recent lattice results. The calculation uses the approximately self-consistent resummation of hard thermal and dense loops that we have developed previously. For temperatures between 1.5 to 5 $T_c$, our results follow the same trend as the lattice data, but exceed them in magnitude by about 5-10%. We also compute the lowest order contribution, of order $\alpha_s^3\log(1/\alpha_s)$, to the off-diagonal susceptibility. This contribution, which is not a part of our self-consistent calculation, is numerically small, but not small enough to be compatible with a recent lattice simulation., Comment: 13 pages, 5 figures, uses elsart.cls; v2: minor corrections; v3: sign in eq.(1) corrected

We discuss a recent approach for overcoming the poor convergence of the perturbative expansion for the thermodynamic potential of QCD. This approach is based on self-consistent approximations which allow for a gauge-invariant and manifestly ultraviolet-finite resummation of the essential physics of the hard thermal/dense loops. The results thus obtained are in good agreement with available lattice data down to temperatures of about twice the critical temperature. Calculations for a plasma with finite quark density (i.e., with a non-zero chemical potential $\mu$) are no more difficult than at $\mu=0$., Comment: 4 pages LaTeX2e, contribution to the proceedings of the 15th International Conference on Ultra-Relativistic Nucleus-Nucleus Collisions (QM 2001), Long Island, New York, January 15 - 20, 2001

We propose a gauge-invariant and manifestly UV finite resummation of the physics of hard thermal/dense loops (HTL/HDL) in the thermodynamics of the quark-gluon plasma. The starting point is a simple, effectively one-loop expression for the entropy or the quark density which is derived from the fully self-consistent two-loop skeleton approximation to the free energy, but subject to further approximations, whose quality is tested in a scalar toy model. In contrast to the direct HTL/HDL-resummation of the one-loop free energy, in our approach both the leading-order (LO) and the next-to-leading order (NLO) effects of interactions are correctly reproduced and arise from kinematical regimes where the HTL/HDL are justifiable approximations. The LO effects are entirely due to the (asymptotic) thermal masses of the hard particles. The NLO ones receive contributions both from soft excitations, as described by the HTL/HDL propagators, and from corrections to the dispersion relation of the hard excitations, as given by HTL/HDL perturbation theory. The numerical evaluations of our final expressions show very good agreement with lattice data for zero-density QCD, for temperatures above twice the transition temperature., Comment: 62 pages REVTEX, 14 figures; v2: numerous clarifications, sect. 2C shortened, new material in sect. 3C; v3: more clarifications, one appendix removed, alternative implementation of the NLO effects, corrected eq. (5.16)

Self-consistent approximations allowing the calculation of the entropy and the baryon density of a quark-gluon plasma are presented. These approximations incorporate the essential physics of the hard thermal loops, involve only ultraviolet-finite quantities, and are free from overcounting ambiguities. While being nonperturbative in the strong coupling constant $g$, agreement with ordinary perturbation theory is achieved up to and including order $g^3$. It is shown how the pressure can be reconstructed from the entropy and the baryon density taking into account the scale anomaly. The results obtained are in good agreement with available lattice data down to temperatures of about twice the critical temperature., Comment: 13 pages, 4 figures, LATEX file using elsart.sty; v2: minor rewording in introduction, obvious sign error corrected in sect. 4

This talk is a brief overview of the present status of our understanding of nucleus-nucleus collisions at high energy and the search for signals of the quark-gluon plasma., Comment: flatex qm99.tex, 2 files XIV International Conference on `Ultra-Relativistic Nucleon-Nucleon Collisions' Torino, Italie 1999-05-10 1999-05-15 May 10-15, 1999 [SPhT-T99/103]

Self-consistent approximations in terms of fully dressed propagators provide a simple expression for the entropy of an ultrarelativistic plasma, which isolates the contribution of the elementary excitations as a leading contribution. Further approximations, whose validity is checked on a soluble model involving a scalar field, allow us to calculate the entropy of the QCD plasma. We obtain an accurate description of lattice data for purely gluonic QCD, down to temperatures of about twice the transition temperature., Comment: 12 pages, 3 figures, REVTEX (minor modifications)

The theoretical ideas relevant for the physics of the disoriented chiral condensate (DCC) are reviewed., Comment: 18 pages LaTex, based on invited lecture given by A.Krzywicki at the workshop "Quark, plasma and beyond", Zif, Bielefeld, May 1996 ; a reference is corrected

When computed to one-loop order in resummed perturbation theory, the non-abelian Debye mass appears to be logarithmically sensitive to the magnetic scale $g^2T$. More generally, we show that in higher orders power-like infrared divergences forbid the use of perturbation theory to calculate the corrections to Debye screening. A similar infrared problem occurs in the determination of the mass-shell for the scalar propagator in 2+1-dimensional scalar electrodynamics. In this context, we provide a non-perturbative approach which solves the infrared problems and allows for an accurate calculation of the scalar propagator in the vicinity of the mass-shell., Comment: 29 pages, LaTex, 7 figures (not included, available upon request)

We use the linear $\sigma$ model to analyse the dynamics of a disoriented chiral condensate. For idealized boundary conditions appropriate to high energy collisions, the problem can be reduced to a one dimensional one. The evolution of the chiral state is then that of a simple dynamical system and can be studied analytically., Comment: 14 pages Latex, LPTHE Orsay 94/18 , SPhT T94/019