Jean-Paul Blaizot

Bose–Einstein condensation and thermalization of the quark–gluon plasma

Abstract In ultra-relativistic heavy ion collisions, the matter formed shortly after the collision is a dense, out of equilibrium, system of gluons characterized by a semi-hard momentum scale Q s . Simple power counting arguments indicate that this system is over-occupied: the gluon occupation number is parametrically large when compared to a system in thermal equilibrium with the same energy density. On short time scales, soft elastic scattering tends to drive the system toward the formation of a Bose–Einstein condensate that contains a large fraction of the gluons while contributing little to the energy density. The lifetime and existence of this condensate depends on whether inelastic processes, that occur on the same time scale as elastic processes, preferably increase or decrease the number of gluons. During this overpopulated stage, and all the way to thermalization, the system behaves as a strongly interacting fluid, even though the elementary coupling constant is small. Finally, we argue that while complete isotropization may never be reached, the system may yet evolve for a long time with a fixed anisotropy between average longitudinal and transverse momenta.

Non linear gluon evolution in path-integral form

We explore and clarify the connections between two different forms of the renormalization 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.

Gluon Transport Equation in the Small Angle Approximation and the Onset of Bose-Einstein Condensation

Abstract In this paper, we study the evolution of a dense system of gluons, such as those produced in the early stages of ultra-relativistic heavy ion collisions. We describe the approach to thermal equilibrium using the small angle approximation for gluon scattering in a Boltzmann equation that includes the effects of Bose statistics. In the present study we ignore the effect of the longitudinal expansion, i.e., we restrict ourselves to spatially uniform systems, with spherically symmetric momentum distributions. Furthermore we take into account only elastic scattering, i.e., we neglect inelastic, number changing, processes. We solve the transport equation for various initial conditions that correspond to small or large initial gluon phase-space densities. For a small initial phase-space density, the system evolves towards thermal equilibrium, as expected. For a large enough initial phase-space density the equilibrium state contains a Bose condensate. We present numerical evidence that such over-populated systems reach the onset of Bose–Einstein condensation in a finite time. The approach to condensation is characterized by a scaling behavior that we briefly analyze.

EMMI rapid reaction task force on ‘Thermalization in non-Abelian plasmas’

Recently, different proposals have been put forward on how thermalization proceeds in heavy-ion collisions in the idealized limit of very large nuclei at sufficiently high energy. Important aspects of the parametric estimates at weak coupling may be tested using well-established classical-statistical lattice simulations of the far-from-equilibrium gluon dynamics. This has to be confronted with strong coupling scenarios in related theories based on gauge-string dualities. Furthermore, closely related questions about far-from-equilibrium dynamics arise in early-universe cosmology and in non-relativistic systems of ultracold atoms. These were central topics of the EMMI Rapid Reaction Task Force meeting held on 12–14 December 2011, at the University of Heidelberg, which we report on. Communicated by Professor Achim Schwenk

A path integral for heavy-quarks in a hot plasma

Abstract 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.

Heavy quark bound states in a quark–gluon plasma: Dissociation and recombination

Abstract We present a comprehensive approach to the dynamics of heavy quarks in a quark–gluon plasma, including the possibility of bound state formation and dissociation. In this exploratory paper, we restrict ourselves to the case of an Abelian plasma, but the extension of the techniques used to the non-Abelian case is doable. A chain of well defined approximations leads eventually to a generalized Langevin equation, where the force and the noise terms are determined from a correlation function of the equilibrium plasma, and depend explicitly on the configuration of the heavy quarks. We solve the Langevin equation for various initial conditions, numbers of heavy quark–antiquark pairs and temperatures of the plasma. Results of simulations illustrate several expected phenomena: dissociation of bound states as a result of combined effects of screening of the potential and collisions with the plasma constituent, formation of bound pairs (recombination) that occurs when enough heavy quarks are present in the system.

Energy flow along the medium-induced parton cascade

We discuss the dynamics of parton cascades that develop in dense QCD matter, and contrast their properties with those of similar cascades of gluon radiation in vacuum. We argue that such cascades belong to two distinct classes that are characterized respectively by an increasing or a constant (or decreasing) branching rate along the cascade. In the former class, of which the BDMPS, medium-induced, cascade constitutes a typical example, it takes a finite time to transport a finite amount of energy to very soft quanta, while this time is essentially infinite in the latter case, to which the DGLAP cascade belongs. The medium induced cascade is accompanied by a constant flow of energy towards arbitrary soft modes, leading eventually to the accumulation of the initial energy of the leading particle at zero energy. It also exhibits scaling properties akin to wave turbulence. These properties do not show up in the cascade that develops in vacuum. There, the energy accumulates in the spectrum at smaller and smaller energy as the cascade develops, but the energy never flows all the way down to zero energy. Our analysis suggests that the way the energy is shared among the offsprings of a splitting gluon has little impact on the qualitative properties of the cascades, provided the kernel that governs the splittings is not too singular.

Quark production, Bose–Einstein condensates and thermalization of the quark–gluon plasma

Abstract In this paper, we study the thermalization of gluons and N f flavors of massless quarks and antiquarks in a spatially homogeneous system. First, two coupled transport equations for gluons and quarks (and antiquarks) are derived within the diffusion approximation of the Boltzmann equation, with only 2 ↔ 2 processes included in the collision term. Then, these transport equations are solved numerically in order to study the thermalization of the quark–gluon plasma. At initial time, we assume that only gluons are present and we choose the gluon distribution of a form inspired by the color glass picture, namely f = f 0 θ ( 1 − p Q s ) with Q s the saturation momentum and f 0 a constant. The subsequent evolution of the system may, or may not, lead to the formation of a (transient) Bose condensate (BEC) of gluons, depending on the value of f 0 . In fact, we observe, depending on the value of f 0 , three different patterns: (a) thermalization without BEC for f 0 ≤ f 0 t , (b) thermalization with transient BEC for f 0 t f 0 ≤ f 0 c , and (c) thermalization with BEC for f 0 c f 0 . The values of f 0 t and f 0 c depend on N f . When f 0 ≳ 1 > f 0 c , the onset of BEC occurs at a finite time t c ∼ 1 ( α s f 0 ) 2 1 Q s . We also find that quark production slows down the thermalization process: the equilibration time for N f = 3 is typically about 5 to 6 times longer than that for N f = 0 at the same Q s and f 0 .

Thermalization and Bose-Einstein Condensation in Overpopulated Glasma

Abstract We report recent progress on understanding the thermalization of the quark-gluon plasma during the early stage in a heavy ion collision. The initially high overpopulation in the far-from-equilibrium gluonic matter (“Glasma”) is shown to play a crucial role. The strongly interacting nature (and thus fast evolution) naturally arises as an emergent property of this pre-equilibrium matter where the intrinsic coupling is weak but the highly occupied gluon states coherently amplify the scattering. A possible transient Bose-Einstein Condensate is argued to form dynamically on a rather general ground. We develop a kinetic approach for describing its evolution toward thermalization as well as the onset of condensation.

Searching evidence for the color glass condensate at RHIC

Abstract This contribution discusses the phenomenon of parton saturation, the color glass picture of hadronic wavefunctions, and their relevance in the early stages of nucleus–nucleus collisions. Evidence for the color glass condensate in the presently available RHIC data is critically reviewed.