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High temperature color conductivity at next-to-leading log order

Peter Arnold, Laurence G. Yaffe

TL;DR

The paper derives the next-to-leading log order correction to the non-Abelian color conductivity in hot, weakly coupled plasmas by sequentially matching three effective theories across distance scales: Theory 1 (kinetic, LT dynamics), Theory 2 (collisional, stochastic kinetic theory), and Theory 3 (Langevin, conductivity-dominated regime). Using Wilson loops as a gauge-invariant observable, the LO result fixes σ ≈ m^2/(d γ1) and the NLLO correction is extracted from the zero-frequency, small-momentum self-energy Π00(0,k) through a careful DR-regulated matching, yielding σ−1 proportional to ln(m/γ) plus a finite constant. The analysis hinges on the structure of the linearized collision operator δĈ, its l-decomposition, and the exact evaluation of γ1, with numerical constants determined from Σm(ρ) functions. The final NLLO result provides an improved, gauge-consistent prediction for color conductivity and informs non-perturbative color dynamics and hot electroweak baryon-number violation rates.

Abstract

The non-Abelian analog of electrical conductivity at high temperature has previously been known only at leading logarithmic order: that is, neglecting effects suppressed only by an inverse logarithm of the gauge coupling. We calculate the first sub-leading correction. This has immediate application to improving, to next-to-leading log order, both effective theories of non-perturbative color dynamics, and calculations of the hot electroweak baryon number violation rate.

High temperature color conductivity at next-to-leading log order

TL;DR

The paper derives the next-to-leading log order correction to the non-Abelian color conductivity in hot, weakly coupled plasmas by sequentially matching three effective theories across distance scales: Theory 1 (kinetic, LT dynamics), Theory 2 (collisional, stochastic kinetic theory), and Theory 3 (Langevin, conductivity-dominated regime). Using Wilson loops as a gauge-invariant observable, the LO result fixes σ ≈ m^2/(d γ1) and the NLLO correction is extracted from the zero-frequency, small-momentum self-energy Π00(0,k) through a careful DR-regulated matching, yielding σ−1 proportional to ln(m/γ) plus a finite constant. The analysis hinges on the structure of the linearized collision operator δĈ, its l-decomposition, and the exact evaluation of γ1, with numerical constants determined from Σm(ρ) functions. The final NLLO result provides an improved, gauge-consistent prediction for color conductivity and informs non-perturbative color dynamics and hot electroweak baryon-number violation rates.

Abstract

The non-Abelian analog of electrical conductivity at high temperature has previously been known only at leading logarithmic order: that is, neglecting effects suppressed only by an inverse logarithm of the gauge coupling. We calculate the first sub-leading correction. This has immediate application to improving, to next-to-leading log order, both effective theories of non-perturbative color dynamics, and calculations of the hot electroweak baryon number violation rate.

Paper Structure

This paper contains 30 sections, 229 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Review of Path Integral Formulation
  3. General properties of $\delta\hat{C}$
  4. Matching Theory 2 to Theory 3
  5. Wilson loops
  6. Relation to $\sigma$ in Theory 3
  7. Relation to $\sigma$ in Theory 2
  8. Next order in the loop expansion
  9. Calculating $\Pi_{00}(0,k{\to}0)$ in Theory 2
  10. Perturbative expansion of the action
  11. Propagators
  12. Transposition
  13. Analysis of diagrams
  14. The ${\bf k}\to0$ limit
  15. Extracting UV and IR divergences
  16. ...and 15 more sections

Figures (6)

  • Figure 1: A time-like Wilson loop rectangle.
  • Figure 2: Examples of first-order contribution to the expectation of a large-time Wilson rectangle.
  • Figure 3: A next-order correction to Fig. \ref{['fig:Wilson1']}c that does not appear in Coulomb gauge.
  • Figure 4: The Coulomb-gauge interaction vertices of action (\ref{['eq:S2']}) of Theory 2. Dashed lines represent $A_0$, wavy lines ${\bf A}$, and solid lines the gauge-fixing ghost $\eta$. $n$ is any positive integer.
  • Figure 5: The first-order and next-order contributions to the Wilson loop in Coulomb gauge.
  • ...and 1 more figures