Chern-Simons number diffusion with hard thermal loops

We construct an extension of the standard Kogut-Susskind lattice model for classical $\mathrm{}(3+1)\mathrm{}$-dimensional Yang-Mills theory, in which “classical particle” degrees of freedom are added. We argue that this will correctly reproduce the “hard thermal loop” effects of hard degrees of freedom, while giving a local implementation which is numerically tractable. We prove that the extended system is Hamiltonian and has the same thermodynamics as dimensionally reduced hot Yang-Mills theory put on a lattice. We present a numerical update algorithm and study the Abelian theory to verify that the classical gauge theory self-energy is correctly modified. Then we use the extended system to study the diffusion constant for the Chern-Simons number. We verify the Arnold-Son-Yaffe picture that the diffusion constant is inversely proportional to the hard thermal loop strength. Our numbers correspond to a diffusion constant of $\Gamma =29\mathrm{}\pm 6{\alpha }_{\mathrm{w}}^5{T}^4$ for $m_{\mathrm{D}}^2{=11g}^2{T}^2/6.\mathrm{}$