287. Heavy quark thermalization in classical lattice gauge theory

This is a guest post by Marcus Tassler from the U. of Munster. Dmitry.

Dmitry has kindly invited me to present a recent article in which we use classical lattice gauge theory as a simple and numerically accessible model to study the dynamics of gauge theories at high temperature. We apply this technique to investigate the momentum diffusion of heavy quarks, which is observed to be surprisingly rapid in heavy-ion experiments, and point out the possibility to measure the diffusion constant in ordinary lattice simulations. The work I will present was done in collaboration with Mikko Laine, Guy D. Moore and Owe Philipsen and is available as arXiv:0902.2856.

Typical momentum scales of the quark-gluon plasma
Before going into some detail I will briefly introduce the typical momentum scales of the quark-gluon plasma at high temperature:

1. Hard momentum scale: p~T
   The corresponding length scale is the average inter particle spacing 1/T (with T being the temperature of the plasma).
2. Electric scale: p~gT
   Due to interactions with the plasma the gluon acquires a thermal mass, the so called Debye mass m_D~gT, which suppresses electric interactions over distances larger than 1/m_D.
3. Magnetic scale: p~g²T
   At this momentum scale magnetic interactions experience a dynamical screening which suppresses them over distances larger than 1/g²T. Chromo-magnetic forces are the most far reaching interactions in the plasma.

The long ranged thermal physics of the soft electric and magnetic scales is responsible for collective phenomena and has long been argued to be essentially classical at high temperatures. At length scales smaller than the average inter particle spacing 1/T thermal effects necessarily become irrelevant and quantum physics regularizes the high energy behaviour of the system.

How to study the dynamics of the plasma on the lattice ?
To study the dynamics of the plasma we now introduce a lattice cutoff to separate the thermal scales from the quantum scale. Choosing a lattice spacing much larger than the typical inter particle spacing a>>1/T the hard scale is replaced by the borders of the Brillouin zone. Having isolated the thermal momentum scales the system becomes amenable to a treatment by classical statistics. We choose a temporal gauge and replace the classical partition function by the corresponding partition function for gauge fields discretized on a spatial (i.e. 3-dimensional) lattice. To study the time evolution of the statistical ensemble we solve the nonabelian Maxwell equations of the lattice system which are shown in the following:

	Faradays law
	Amperes law
	Gauss constraints

Table 1: Maxwell equations for Yang-Mills theory on the lattice.

Links and plaquettes are denoted as U_i and U_ij while the generators of SU(N) are represented by T^a. Faradays law of induction defines the color electric field E_i=E_i^aT^a, which evolves according to Amperes law. The equation of motion is obtained directly from a variation of the lattice action. Gauss’s law finally serves as a constraint for the field configurations of the statistical ensemble. The time coordinate is continuous and we are working in Minkowski space. At high temperatures a staggering agreement with the predictions from perturbation theory is observed for a wide range of quantities (i.e. any quantity which is insensitive to the hard scale), while at intermediate temperatures the classical system captures the otherwise inaccessible non-perturbative dynamics of the plasma. We therefore argue for the use of this classical lattice system as a simple numerically amenable model to study the dynamics of the plasma and point to the successfull application of similar techniques in areas ranging from electroweak physics to inflationary preheting.

The puzzle of the rapid diffusion of heavy quarks in a quark-gluon plasma
We use this approach to investigate the random walk of heavy quarks in a quark-gluon plasma which is described by the following Langevin equation:

The second equation expresses that the random force, i.e. the small kicks a heavy quark experiences when interacting with gluons in the plasma, is uncorrelated in time and direction. There are two coefficients, the relaxation rate and the diffusion constant, which are related by a fluctuation-dissipation relation (M is the mass of the quark):

To describe the diffusion of heavy quarks it is therefore sufficient to determine the diffusion constant which can be obtained by calculating or measuring the correlator of two chromoelectric fields seperated in time and connected by a Wilson line W:

The heavy quark diffusion constant, as obtained from a measurement of this correlator on a classical lattice, is shown in figure 1.

Fig. 1: The diffusion constant as obtained from the classical lattice simulations plotted against the inverse lattice coupling.

The solid line is the prediction from perturbation theory which is reproduced exactly by the lattice data at high temperatures. This agreement is expected since the perturbative expansion is known to agree with a classical approximation at asymptotically high temperatures. In the intermediate temperature range, relevant for the physics of heavy-ion collisions, the heavy quark diffusion constant is found to be much larger than expected in perturbation theory. This indicates that the rapid thermalization of heavy quarks observed in heavy-ion experiments can indeed be understood by studying the non-perturbative dynamics of the quark-gluon plasma. We test the reliability of the result by comparing different discretization schemes and find that the results are consistent at week couplings. In the strong coupling regime, where it is natural to expect any classical approximation to be invalid, the results start to depend on the discretization scheme which is analyzed in detail in the article.

Fig. 2: Spectrum of the correlator as obtained from the simulations.

A new quantity to be measured on the lattice ?
Any attempt to measure the diffusion constant in an ordinary lattice simulation is faced with the serious problem that the corresponding correlator is defined in Minkowski space and needs to be obtained by analytical continuation. To illustrate that this is possible in principle the shape of the correlator is shown in figure 2. The spectrum, as obtained from the classical lattice simulations, turns out to be very flat around the origin for physical couplings. Since the frequency spectrum in euclidean space is a discrete spectrum of Matsubara modes this is just what is needed for a reliable measurement of the diffusion constant in ordinary lattice simulations. We therefore point out that this is a promising new quantity which can be measured in the context of the full quantum theory.