ELSEVIER Nuclear Physics B (Proc. Suppl.) 63A-C (1998) 575-577 PROCEEDINGS SUPPLEMENTS Sphaleron transition rate: recent numerical results J. Ambj0rn and A. Krasnitz* Niels Bohr Institute, Blegdamsvej 17, DK-2100 Copenhagen, Denmark We report results of a recent numerical simulation wherein the sphaleron transition rate was determined using a novel measuring technique for the topological charge in real time. Also reported is a related study of static and real-time correlation functions of gauge fields at high temperature. The high-temperature baryon-number viola- tion rate in electroweak theory is a key parameter for understanding the baryon asymmetry of the universe. At present, this nonperturbative quan- tity can only be determined in the classical ap- proximation, by means of a real-time lattice sim- ulation. This line of research was initiated in [1], and the last two years have seen much numerical effort to improve the simulations, as well as the- oretical examination of the classical approxima- tion, as reviewed by Jan Smit in this volume[2]. In particular, it became apparent recently that the method previously used to measure topologi- cal charge in real time might give rise to system- atic errors in the transition rate. In our current series of simulations we measured the topological charge using a new systematic procedure for elim- ination of these errors. We now give a brief de- scription of the measurement technique (the cool- ing method in the following), followed by results for the rate in the SU(2) Yang-Mills theory with and without the scalar field. We also discuss the closely related issue of the gauge field correlation functions at high temperature. Consider the standard lattice approximation for the topological charge per unit time i Ea j,n × Z (Go.."), (1) I::]j,~ where *Presenter at the conference. 0920-5632/98/$19.00 © 1998 Elsevier Science B.V. All rights reserved. PII S0920-5632(97)00838-4 Using the standard notation, we assign to every lattice site j link matrices Uj,n in the fundamen- tal representation of SU(2) and lattice analogs of color electric fields Ej,n, three of each, cor- responding to the three positive lattice directions n. The usual plaquette variable is denoted by Up, and a s are the three Pauli matrices. In fact, it is misleading to write the left-hand side of (1) as a time derivative of Chern-Simons number Ncs or any other fixed-time functional of the fields: the object on the right-hand side of (1) is not a total time derivative (TTD). This shortcoming of the lattice topological charge density (TCD) is re- lated to the difficulty in defining a lattice analog of the continuum Ncs with the correct properties under gauge transformations [3]. For smooth field configurations (1) tends to the continuum/~/cs. However, thermal field configurations on the lat- tice are not smooth, hence (1) does not approach a TTD in the continuum limit. The idea behind the cooling method is to re- place the Hamiltonian real-time trajectory in the phase space by another path between the same endpoints, the one along which phase-space con- figurations are predominantly smooth. Lattice topological charge (1), computed along such a path, would suffer much less from systematic er- rors than the one computed along the real-time trajectory. In general terms, the method works as follows. Let the real-time evolution of a dy- namical system be described by a Hamiltonian depending on coordinates qi and momenta Pi (i = 1,...N): H = ~,ip~/2 + V(q). Let these variables depend, along with the real time t, on a cooling time r. r, of which all the dynamical