Counterterms for Linear Divergences in Real-Time Classical Gauge Theories at High Temperature

Real-time classical SU($N$) gauge theories at non-zero temperature contain linear divergences. We introduce counterterms for these divergences in the equations of motion in the continuum and on the lattice. These counterterms can be given in terms of auxiliary fields that satisfy local equations of motion. We present a lattice model with 6+1D auxiliary fields that for IR-sensitive quantities yields cut-off independent results to leading order in the coupling. Also an approximation with 5+1D auxiliary fields is discussed.Comment: 10 pages, major change

Non-equilibrium field theory

I discuss various topics in relativistic non-equilibrium field theory related to high energy physics and cosmology. I focus on non-perturbative problems and how they can be treated on the lattice.Comment: Plenary talk given at the 18th International Symposium on Lattice Field Theory (Lattice 2000), Bangalore, India, 17-22 Aug 200

Abelian hard thermal loops on a lattice

In Abelian theories, one can write for the hard thermal loop equations of motions a local formulation that is more economical than the traditional Vlasov formulation and is in explicitly canonical form. I show how this formulation can be used for simulating non-equilibrium dynamics in the Abelian Higgs model.Comment: 3 pages. Talk given at LATTICE99 (electroweak), Pisa, Ital

The finite temperature real time \hbar^2 corrections in quantum mechanics

We study non-perturbative real time correlation functions at finite temperature. In order to see whether the classical term gives a good approximation in the high temperature limit T >> \hbar\omega, we consider the first \hbar^2 quantum corrections. We find that for the simplest non-trivial case, the quantum mechanical anharmonic oscillator, the classical result is reliable only for moderately large times: after some time t_* the classical approximation breaks down even at high temperatures. Moreover, the result for the first quantum corrections cannot, in general, be reproduced by modifying the parameters of the classical theory.Comment: 28 pages, 7 figure

Effective theory for real-time dynamics in hot gauge theories

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.Comment: 11 pages, LaTeX, major rewriting, new title, new reference

Ultrasoft Amplitudes in Hot QCD

By using the Boltzmann equation describing the relaxation of colour excitations in the QCD plasma, we obtain effective amplitudes for the ultrasoft colour fields carrying momenta of order $g^2 T$. These amplitudes are of the same order in $g$ as the hard thermal loops (HTL), which they generalize by including the effects of the collisions among the hard particles. The ultrasoft amplitudes share many of the remarkable properties of the HTL's: they are gauge invariant, obey simple Ward identities, and, in the static limit, reduce to the usual Debye mass for the electric fields. However, unlike the HTL's, which correspond effectively to one-loop diagrams, the ultrasoft amplitudes resum an infinite number of diagrams of the bare perturbation theory. By solving the linearized Boltzmann equation, we obtain a formula for the colour conductivity which accounts for the contributions of the hard and soft modes beyond the leading logarithmic approximation.Comment: 38 pages, 10 figures, LaTeX. An extension of the previous results is included (in Sec. 4.2); also, some minor modifications here and there. Final version, to appear in Nucl. Phys.

Isotropization by QCD Plasma Instabilities

Numerical solutions of the Wong-Yang-Mills equations with anisotropic particle momentum distributions are presented. Their isotropization by collective effects due to the classical Yang-Mills field is shown.Comment: 4 pages, 4 figures, contribution to the Quark Matter 2005 proceeding

Chern-Simons Number Diffusion and Hard Thermal Loops on the Lattice

We develop a discrete lattice implementation of the hard thermal loop effective action by the method of added auxiliary fields. We use the resulting model to measure the sphaleron rate (topological susceptibility) of Yang-Mills theory at weak coupling. Our results give parametric behavior in accord with the arguments of Arnold, Son, and Yaffe, and are in quantitative agreement with the results of Moore, Hu, and Muller.Comment: 43 pages, 6 figure

From hard thermal loops to Langevin dynamics

In hot non-Abelian gauge theories, processes characterized by the momentum scale $g^2 T$ (such as electroweak baryon number violation in the very early universe) are non-perturbative. An effective theory for the soft ($|\vec{p}|\sim g^2 T$) field modes is obtained by integrating out momenta larger than $g^2 T$. Starting from the hard thermal loop effective theory, which is the result of integrating out the scale $T$, it is shown how to integrate out the scale $gT$ in an expansion in the gauge coupling $g$. At leading order in $g$, one obtains Vlasov-Boltzmann equations for the soft field modes, which contain a Gaussian noise and a collision term. The 2-point function of the noise and the collision term are explicitly calculated in a leading logarithmic approximation. In this approximation the Boltzmann equation is solved. The resulting effective theory for the soft field modes is described by a Langevin equation. It determines the parametric form of the hot baryon number violation rate as $\Gamma = \kappa g^{10} \log(1/g) T^4$, and it allows for a calculation of $\kappa$ on the lattice.Comment: 42 pages, 2 figures, uses elsart.sty; explanatory paragraph added to the introduction, 4 references added, a few quotations added; to appear in Nucl. Phys.

Effective Classical Theory for Real-Time SU(N) Gauge Theories at High Temperature

We derive an effective classical theory for real-time SU($N$) gauge theories at high temperature. By separating off and integrating out quantum fluctuations we obtain a 3D classical path integral over the initial fields and conjugate momenta. The leading hard mode contribution is incorporated in the equations of motion for the classical fields. This yields the gauge invariant hard thermal loop (HTL) effective equation of motion. No gauge-variant terms are generated as in treatments with an intermediate momentum cut-off. Quantum corrections to classical correlation functions can be calculated perturbatively. The 4D renormalizability of the theory ensures that the 4D counterterms are sufficient to render the theory finite. The HTL contributions of the quantum fluctuations provide the counterterms for the linear divergences in the classical theory.Comment: 13 pages, 1 figure, title changed, discussion on the classical approximation include