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Chern-Simons number diffusion with hard thermal loops

Guy D. Moore, Chaoran Hu, Berndt Muller

TL;DR

Problem: accurately capturing hard thermal loop effects in real-time, nonperturbative dynamics of hot non-Abelian plasmas, including N_CS diffusion, within classical lattice simulations. Approach: extend the Kogut–Susskind lattice model by coupling to a bath of classical particles obeying Wong equations, creating a local Hamiltonian system that reproduces HTL physics. Contributions: demonstration of energy and symplectic measure conservation, thermodynamic equivalence to dimensionally reduced hot Yang–Mills theory, abelian retarded propagator tests confirming correct self-energy, and a measurement of N_CS diffusion that exhibits Arnold–Son–Yaffe scaling; quantitative result Gamma = 29 ± 6 α_w^5 T^4 for m_D^2 = 11 g^2 T^2/6. Significance: provides a practical, scalable framework for simulating HTL-influenced infrared dynamics and informs electroweak baryogenesis studies.

Abstract

We construct an extension of the standard Kogut-Susskind lattice model for classical 3+1 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 Chern-Simons number. We verify the Arnold-Son-Yaffe picture that the diffusion constant is inversely proportional to hard thermal loop strength. Our numbers correspond to a diffusion constant of Gamma = 29 +- 6 alpha_w^5 T^4 for m_D^2 = 11 g^2 T^2/6.

Chern-Simons number diffusion with hard thermal loops

TL;DR

Problem: accurately capturing hard thermal loop effects in real-time, nonperturbative dynamics of hot non-Abelian plasmas, including N_CS diffusion, within classical lattice simulations. Approach: extend the Kogut–Susskind lattice model by coupling to a bath of classical particles obeying Wong equations, creating a local Hamiltonian system that reproduces HTL physics. Contributions: demonstration of energy and symplectic measure conservation, thermodynamic equivalence to dimensionally reduced hot Yang–Mills theory, abelian retarded propagator tests confirming correct self-energy, and a measurement of N_CS diffusion that exhibits Arnold–Son–Yaffe scaling; quantitative result Gamma = 29 ± 6 α_w^5 T^4 for m_D^2 = 11 g^2 T^2/6. Significance: provides a practical, scalable framework for simulating HTL-influenced infrared dynamics and informs electroweak baryogenesis studies.

Abstract

We construct an extension of the standard Kogut-Susskind lattice model for classical 3+1 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 Chern-Simons number. We verify the Arnold-Son-Yaffe picture that the diffusion constant is inversely proportional to hard thermal loop strength. Our numbers correspond to a diffusion constant of Gamma = 29 +- 6 alpha_w^5 T^4 for m_D^2 = 11 g^2 T^2/6.

Paper Structure

This paper contains 3 sections, 14 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Hard thermal loops and particles
  3. The lattice system

Figures (2)

  • Figure 1: An illustration of the degrees of freedom of the proposed system. Lattice sites are large dots, and the solid lines joining them are links. Classical fields take values at sites (Higgs fields) or on links (connections $U$ and electric fields $E$). Particles (the small dots) take on real valued coordinates and momenta (illustrated with arrows). A particle's charge affects the classical fields as if it resided at the nearest lattice site. The dotted lines (really planes, extending out of the page) are barriers between the region nearest one site and that nearest another, that is, faces of the dual lattice. When a particle crosses a barrier, the charge is parallel transported to the new box, the $E$ field on the link orthogonal to the barrier receives a kick, and the particle momentum orthogonal to the barrier is changed to conserve energy.
  • Figure :