Non-perturbative aspects of hot QCD

In high temperature QCD, the perturbation theory is plagued with infrared divergences which reflect long-range non-perturbative phenomena. I argue that it is possible to study such phenomena within a classical thermal field theory which can be put on a three-dimensional lattice. The classical theory is an effective theory for the soft, nonperturbative modes, as obtained after integrating out the hard modes in perturbation theory. It is well suited for numerical studies of the nonperturbative real-time dynamics, which cannot be studied within the standard, imaginary-time formulations of lattice QCD.

I present an effective classical theory which describes the non-perturbative real-time dynamics in hot gauge theories and has the potential for numerical implementation. * Based on a talk given at TFT98, Regensburg, Germany, August 10-14, 1998 1 For definitness, I shall mostly use a QCD-inspired terminology. But the present considerations apply to any hot Yang-Mills theory, in particular, to the electroweak plasma at high temperature (T ≫ Tc).

2016

In high temperature QCD, the perturbation theory is plagued with infrared divergences which reflect long-range non-perturbative phenomena. I argue that it is possible to study such phenomena within a classical thermal field theory which can be put on a three-dimensional lattice. The classical theory is an effective theory for the soft, nonperturbative modes, as obtained after integrating out the hard modes in perturbation theory. It is well suited for numerical studies of the nonperturbative real-time dynamics, which cannot be studied within the standard, imaginary-time formulations of lattice QCD. Invited talk at the Third Workshop on "Continuous Advances in QCD",

Journal of High Energy Physics, 2003

Quasiparticle dynamics in relativistic plasmas associated with hot, weakly-coupled gauge theories (such as QCD at asymptotically high temperature T ) can be described by an effective kinetic theory, valid on sufficiently large time and distance scales. The appropriate Boltzmann equations depend on effective scattering rates for various types of collisions that can occur in the plasma. The resulting effective kinetic theory may be used to evaluate observables which are dominantly sensitive to the dynamics of typical ultrarelativistic excitations. This includes transport coefficients (viscosities and diffusion constants) and energy loss rates. In this paper, we show how to formulate effective Boltzmann equations which will be adequate to compute such observables to leading order in the running coupling g(T ) of high-temperature gauge theories [and all orders in 1/ log g(T ) −1 ]. As previously proposed in the literature, a leading-order treatment requires including both 2 ↔ 2 particle scattering processes as well as effective "1 ↔ 2" collinear splitting processes in the Boltzmann equations. The latter account for nearly collinear bremsstrahlung and pair production/annihilation processes which take place in the presence of fluctuations in the background gauge field. Our effective kinetic theory is applicable not only to near-equilibrium systems (relevant for the calculation of transport coefficients), but also to highly non-equilibrium situations, provided some simple conditions on distribution functions are satisfied.

Physics Letters B, 1998

For a high temperature non-Abelian plasma, we reformulate the hard thermal loop approximation as an effective classical thermal field theory for the soft modes. The effective theory is written in local Hamiltonian form, and the thermal partition function is explicitly constructed. It involves an ultraviolet cutoff which separates between hard and soft degrees of freedom in a gauge-invariant way, together with counterterms which cancel the cutoff dependence in the soft correlation functions. The effective theory is well suited for numerical studies of the non-perturbative dynamics in real time, in particular, for the computation of the baryon number violation rate at high temperature.

Nuclear Physics B, 1994

The longwavelength excitations of a quark-gluon plasma at high temperature can be described as collective oscillations of gauge and fermionic average fields. We show that, at leading order in the coupling strength, the Dyson-Schwinger equations for the N -point functions reduce to a set of coupled equations for these average fields and their induced sources which involve only 2-point functions. The equations for the 2-point functions describe the dynamics of the hard plasma particles in the presence of soft background fields. They may be given the form of simple kinetic equations. Both the wavelength and the amplitude of the collective modes are controlled by the coupling strength, and we show that it is important to take this properly into account in order to obtain consistent equations of motion which are covariant under gauge transformations. By solving the kinetic equations for the 2point functions with well defined (i.e. retarded or advanced) boundary conditions, we obtain the induced currents in closed forms. These act as generating functionals for the one-particle irreducible amplitudes with soft external lines, and yield in particular all the so called "hard thermal loops" identified in the diagrammatic analysis.

Physical Review D, 1994

In this work the determination of low-energy bound states in Quantum Chromodynamics is recast so that it is linked to a weak-coupling problem. This allows one to approach the solution with the same techniques which solve Quantum Electrodynamics: namely, a combination of weak-coupling diagrams and many-body quantum mechanics. The key to eliminating necessarily nonperturbative effects is the use of a bare Hamiltonian in which quarks and gluons have nonzero constituent masses rather than the zero masses of the current picture. The use of constituent masses cuts off the growth of the running coupling constant and makes it possible that the running coupling never leaves the perturbative domain. For stabilization purposes an artificial potential is added to the Hamiltonian, but with a coefficient that vanishes at the physical value of the coupling constant. The weak-coupling approach potentially reconciles the simplicity of the Constituent Quark Model with the complexities of Quantum Chromodynamics. The penalty for achieving this perturbative picture is the necessity of formulating the dynamics of QCD in light-front coordinates and of dealing with the complexities of renormalization which such a formulation entails. We describe the renormalization process first using a qualitative phase space cell analysis, and we then set up a precise similarity renormalization scheme with cutoffs on constituent momenta and exhibit calculations to second order. We outline further computations that remain to be carried out. There is an initial nonperturbative but nonrelativistic calculation of the hadronic masses that determines the artificial potential, with binding energies required to be fourth order in the coupling as in QED. Next there is a calculation of the leading radiative corrections to these masses, which requires our renormalization program. Then the real struggle of finding the right extensions to perturbation theory to study the strong-coupling behavior of bound states can begin.

Physical Review D, 2003

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 2πT) 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 2.5T 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.

EPJ Web of Conferences

The lecture is divided in two parts. The first one deals with an introduction to the physics of hot, dense many-particle systems in quantum field theory [1,2]. The basics of the path integral approach to the partition function are explained for the example of chiral quark models. The QCD phase diagram is discussed in the meanfield approximation while QCD bound states in the medium are treated in the rainbow-ladder approximation (Gaussian fluctuations). Special emphasis is devoted to the discussion of the Mott effect, i.e. the transition of bound states to unbound, but resonant scattering states in the continnum under the influence of compression and heating of the system. Three examples are given: (1) the QCD model phase diagram with chiral symmetry restoration and color superconductivity [3], (2) the Schrödinger equation for heavy-quarkonia [4], and (2) Pions [5] as well as Kaons and D-mesons in the finite-temperature Bethe-Salpeter equation [6]. We discuss recent applications of this quantum field theoretical approach to hot and dense quark matter for a description of anomalous J/ψ supression in heavy-ion collisions [7] and for the structure and cooling of compact stars with quark matter interiors [8]. The second part provides a detailed introduction to the Polyakov-loop Nambu-Jona-Lasinio model [9] for thermodynamics and mesonic correlations [10] in the phase diagram of quark matter. Important relationships of low-energy QCD like the Gell-Mann-Oakes-Renner relation are generalized to finite temperatures. The effect of including the coupling to the Polyakov-loop potential on the phase diagram and mesonic correlations is discussed. An outlook is given to effects of nonlocality of the interactions [11] and of mesonic correlations in the medium [12] which go beyond the meanfield description.

Eprint Arxiv Hep Ph 9509298, 1995

Classical transport theory for colored particles is reviewed and used to derive the hard thermal loops of QCD. A perturbative study of the non-Abelian transport equations that preserves their gauge symmetry is used to compute the induced color current in a hot quark-gluon plasma. From this approach the effective action of hard thermal loops can be derived. This derivation is more direct than alternative ones based on perturbative quantum field theory, and shows that hard thermal effects in hot QCD are essentially classical.