Motion of Chern-Simons number at high temperatures under a chemical potential

I investigate the evolution of finite temperature, classical Yang-Mills field equations under the influence of a chemical potential for Chern-Simons number N_cs. The rate of N_cs diffusion,, Γ_d, and the linear response of N_cs to a chemical potential, Γ_μ, are both computed; the relation Γ_d = 2Γ_μ is satisfied numerically and the results agree with the recent measurement of Γ_d by Ambjørn and Krasnitz. The response of N_cs under chemical potential remains linear at least to μ = 6T, which is impossible if there is a free energy barrier to the motion of N_cs. The possibility that the result depends on lattice artefacts via hard thermal loops is investigated by changing the lattice action and by examining elongated rectangular lattices; provided that the lattice is fine enough, the result is weakly if at all dependent on the specifics of the cutoff. I also compare SU(2) with SU(3) and find Γ_SU(3) 7(_{s/w)⁴ΓSU(2)}.     

1＋1维U（1）Higgs模型的Non－contractable loop和Sphaleron

讨论圆周上的Ｕ（１）Ｈｉｇｇｓ模型，构造出了Ｎｏｎ－ｃｏｎｔｒａｃｔａｂｌｅｌｏｏｐｓ，并给出了Ｓｐｈａｌｅｒｏｎ和多Ｓｐｈａｌｅｒｏｎ解．     

Bödeker’s effective theory: From Langevin dynamics to Dyson–Schwinger equations

The dynamics of weakly coupled, non-abelian gauge fields at high temperature is non-perturbative if the characteristic momentum scale is of order |k|g²T. Such a situation is typical for the processes of electroweak baryon number violation in the early Universe. Bödeker has derived an effective theory that describes the dynamics of the soft field modes by means of a Langevin equation. This effective theory has been used for lattice calculations so far [G.D. Moore, Nucl. Phys. B568 (2000) 367. Available from: <hep-ph/9810313>; G.D. Moore, Phys. Rev. D62 (2000) 085011. Available from: <hep-ph/0001216>]. In this work we provide a complementary, more analytic approach based on Dyson–Schwinger equations. Using methods known from stochastic quantitation, we recast Bödeker’s Langevin equation in the form of a field theoretic path integral. We introduce gauge ghosts in order to help control possible gauge artefacts that might appear after truncation, and which leads to a BRST symmetric formulation and to corresponding Ward identities. A second set of Ward identities, reflecting the origin of the theory in a stochastic differential equation, is also obtained. Finally, Dyson–Schwinger equations are derived.     

Improved Hamiltonian for Minkowski Yang-Mills theory

I develop an improved Hamiltonian for classical, Minkowski Yang-Mills theory, which evolves infrared fields with tree level corrections from lattice spacing a beginning at O(a⁴). I use it to investigate the response of Chern-Simons number to a chemical potential, and to compute the maximal Lyapunov exponent. The Lyapunov exponent has a small a limit, and the Chern-Simons number response appears to be approaching one at the finest lattices considered. In both cases the limit is within 10% of the limit found using the unimproved (Kogut-Susskind) Hamiltonian. For the maximal Lyapunov exponent the limits differ between Hamiltonians by about 5%, significant at about 5σ, indicating that while a small a limit exists, its value depends on the specifics of the lattice cutoff. For Chern-Simons number the difference between Hamiltonians is within statistical errors of about 10%, which constitutes an upper bound on the lattice regulation dependence.