Two-loop QCD gauge coupling at high temperatures

TL;DR

The authors perform a two-loop matching calculation to obtain the high-temperature EQCD gauge coupling $g_E^2$ in terms of the 4d QCD coupling, using background-field gauge and a Taylor expansion of the gluon self-energy. This yields explicit 2-loop corrections and a new $O(g^6T^4)$ contribution to the hot-QCD pressure, improving the perturbative description of thermal QCD. Numerically, the 2-loop result reduces the renormalization-scale dependence and lies about 20% below the 1-loop value, indicating better perturbative stability near the deconfinement transition. The improved coupling allows a parameter-free 3d/4d comparison of the spatial string tension, with the 2-loop predictions aligning well with lattice data down to temperatures close to $T_c$, supporting the view that hard scales are perturbative while soft scales require non-perturbative treatment.

Abstract

We determine the 2-loop effective gauge coupling of QCD at high temperatures, defined as a matching coefficient appearing in the dimensionally reduced effective field theory. The result allows to improve on one of the classic non-perturbative probes for the convergence of the weak-coupling expansion at high temperatures, the comparison of full and effective theory determinations of an observable called the spatial string tension. We find surprisingly good agreement almost down to the critical temperature of the deconfinement phase transition. We also determine one new contribution of order O(g^6T^4) to the pressure of hot QCD.

Figures (4)

• Figure 1: The 1-loop and 2-loop self-energy diagrams in the background field gauge. Wavy lines represent gauge fields, dotted lines ghosts, and solid lines fermions. The 2-loop graphs have been divided into two-particle-irreducible and two-particle-reducible contributions.
• Figure 2: A comparison of 1-loop and 2-loop values for $g_{\hbox{\scriptsize E}}^2/T$, as a function of $\bar{\mu}/T$, for a fixed $T/{\Lambda_{\overline{\hbox{\tiny\rm{MS}}}}} =2.0$ and $N_{\rm f} = 0,2,3$.
• Figure 3: The 1-loop and 2-loop values for $g_{\hbox{\scriptsize E}}^2/T$, as a function of $T/{\Lambda_{\overline{\hbox{\tiny\rm{MS}}}}}$ (solid lines). For each $T$ the scale $\bar{\mu}$ has been fixed to the "principal of minimal sensitivity" point $\bar{\mu}_{\hbox{\scriptsize opt}}$ following from the 1-loop expression, and varied then in the range $\bar{\mu} = (0.5 ... 2.0) \times \bar{\mu}_{\hbox{\scriptsize opt}}$ (the grey bands).
• Figure 4: We compare 4d lattice data for the spatial string tension, taken from Ref. boyd, with expressions obtained by combining 1-loop and 2-loop results for $g_{\hbox{\scriptsize E}}^2$ together with Eq. (\ref{['gMgE']}) and the non-perturbative value of the string tension of 3d SU(3) gauge theory, Eq. (\ref{['sigma_3d']}). The upper edges of the bands correspond to $T_{\rm c}/{\Lambda_{\overline{\hbox{\tiny\rm{MS}}}}} = 1.35$, the lower edges to $T_{\rm c}/{\Lambda_{\overline{\hbox{\tiny\rm{MS}}}}} = 1.10$.