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Debye screening mass of hot Yang-Mills theory to three-loop order

Ioan Ghisoiu, Jan Moller, York Schroder

TL;DR

This work computes the Debye screening mass of hot Yang–Mills theory to three-loop order within a three-scale EFT framework, enabling a gauge-invariant determination of a ${g^7}$ contribution to the QCD pressure via matching QCD to EQCD (and MQCD). The calculation combines a basis transformation of three-loop sum-integrals, Tarasov tensor reduction, and IBP reductions to a small set of master integrals, followed by careful renormalization and numerical evaluation of the running coupling. The main result is a finite, gauge-parameter–independent expression for the Debye mass with well-behaved convergence and a closed-form soft-scale contribution to the pressure at ${\mathcal{O}}(g^7)$, providing a nontrivial test of the EFT approach and a path toward a full high-order comparison with lattice results. This advances the perturbative understanding of hot QCD and reinforces the EFT framework used for multi-scale thermal systems, with implications for precision thermodynamics and transport parameter studies.

Abstract

Building upon our earlier work, we compute a Debye mass of finite-temperature Yang-Mills theory to three-loop order. As an application, we determine a $g^7$ contribution to the thermodynamic pressure of hot QCD.

Debye screening mass of hot Yang-Mills theory to three-loop order

TL;DR

This work computes the Debye screening mass of hot Yang–Mills theory to three-loop order within a three-scale EFT framework, enabling a gauge-invariant determination of a contribution to the QCD pressure via matching QCD to EQCD (and MQCD). The calculation combines a basis transformation of three-loop sum-integrals, Tarasov tensor reduction, and IBP reductions to a small set of master integrals, followed by careful renormalization and numerical evaluation of the running coupling. The main result is a finite, gauge-parameter–independent expression for the Debye mass with well-behaved convergence and a closed-form soft-scale contribution to the pressure at , providing a nontrivial test of the EFT approach and a path toward a full high-order comparison with lattice results. This advances the perturbative understanding of hot QCD and reinforces the EFT framework used for multi-scale thermal systems, with implications for precision thermodynamics and transport parameter studies.

Abstract

Building upon our earlier work, we compute a Debye mass of finite-temperature Yang-Mills theory to three-loop order. As an application, we determine a contribution to the thermodynamic pressure of hot QCD.

Paper Structure

This paper contains 14 sections, 35 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Setup: effective theory and matching
  3. Screening mass in QCD
  4. Screening mass in EQCD
  5. Completing the calculation
  6. Basis transformation
  7. Renormalization
  8. Numerical evaluation
  9. A $g^7$ contribution to the QCD pressure
  10. Conclusions
  11. Lower-order ingredients
  12. Master sum-integrals
  13. Coefficients of Eq. (\ref{['eq:PiOld']})
  14. IBP relations for basis transformation

Figures (2)

  • Figure 1: The three non-trivial three-loop sum-integrals $J_{11}$, $J_{12}$ and $J_{13}$ of Eq. (\ref{['eq:newBasis']}) that are needed for the Debye mass. A dot on a line stands for an extra power of the corresponding propagator, a cross denotes an extra factor of $P_0^2$ in the numerator, and a slash means the line appears in the numerator.
  • Figure 2: Left: dimensionless Debye mass $m_{\mathrm{ER}}^2/T^2$ as a function of the temperature $T$, in units of ${\Lambda_{\overline{\hbox{\tiny\rm{MS}}}}}$ and with the variation of $\bar{\mu} = (0.5 ... 2.0) \times \bar{\mu}_{\text{opt}}$. Right: variation of the mass with respect to $\bar{\mu}$ at the scale $T/{\Lambda_{\overline{\hbox{\tiny\rm{MS}}}}} = 2$ and with $\mu_3 = \mu_{3,\text{opt}}$.