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

TL;DR

The paper develops a lattice implementation of hard thermal loop effects for SU(2) Yang–Mills theory using auxiliary fields expanded in spherical harmonics and truncated at l_max. This local, discretized HTL theory reproduces correct thermodynamics for l_max≥1 and yields a controllable spectrum for the gauge-field propagator, enabling efficient real-time simulations. Measuring the Chern–Simons diffusion rate Γ via calibrated cooling, the authors confirm the Arnold–Son–Yaffe scaling Γ ∝ α^5 T^4 with a suppressed dependence on the HTL strength, and obtain a MSM value Γ ≈ 25.4 α^5 T^4, in agreement with previous particle-based HTL results and pure YM lattice studies. The approach demonstrates robust convergence with l_max and provides a practical, accurate method to quantify baryon-number violation in the hot early universe.

Abstract

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.

Figures (8)

• Figure 1: Left: The inverse propagator Eq. (\ref{['invprop']}) with $l_{\rm max}=10$, plotted against $\omega/m_{\rm D}$ with fixed $k=0.4 m_{\rm D}$. Right: The positive frequency poles of the propagator at $l_{\rm max} = 10$. In these figures, one can clearly see the development of the cut in the interval $-k \le \omega \le k$, and the two plasmon poles at $\omega^2 \approx m^2_D/3 + 6k^2/5$.
• Figure 2: The spectral density $\rho/\omega$ at $k=0.4 m_{\rm D}$ for propagators without the $l$-cutoff and with various values of $l_{\rm max}$. The spectral density for finite $l_{\rm max}$ is a sum of form $\sum_{\alpha} C_\alpha \delta(\omega-\omega_\alpha)$. The plot symbols are plotted at coordinates $(\omega_\alpha, 2C_\alpha/(\omega_{\alpha+1}-\omega_{\alpha-1}))$, which makes it possible to compare the different $l_{\rm max}$-values.
• Figure 3: How the $N_{\rm CS}$ evolution is measured (after broken_nonpert). Top horizontal line shows the configurations (solid circles) generated by the lattice equations of motion. Every few timesteps, the configurations are cooled a fixed cooling length, giving a parallel cooled trajectory (open circles). Now the fields are smooth enough so that ${\bf E\cdot B}$ can be reliably integrated, giving $\delta N_{\rm CS}(t)$ along the cooled trajectory. In longer intervals, the $N_{\rm CS}$ measurement is "grounded" by cooling all the way to a vacuum configuration. If $\delta N_{\rm CS}$ along paths like $V_1\rightarrow A \rightarrow B \rightarrow V_2$ is always close to an integer, we know that the integration errors are small. The residual deviation from an integer value is subtracted from $\delta N_{\rm CS}(A\rightarrow B)$, cancelling the accumulation of errors.
• Figure 4: The dependence of $\Gamma$ on $l_{\rm max}$ on a lattice of size $24^3$,$\beta_{\rm L} = 8.7$.
• Figure 5: The Chern-Simons number diffusion rate $\Gamma$ in physical units. Dashed line: fit to linear + second order term, Eq. (\ref{['fit1']}); continuous line: fit to linear + a log-term, Eq. (\ref{['fit2']}).