Equation of State of QCD with $N_f=3$ flavours up to the electroweak scale

TL;DR

This work presents a non-perturbative determination of the QCD Equation of State for $N_f=3$ in the temperature range $T=3$ GeV to $165$ GeV using shifted boundary conditions and a Schrödinger functional renormalization scheme. The primary observable is the entropy density $s(T)$, computed on the lattice at multiple spacings and extrapolated to the continuum with subpercent precision, from which the pressure $p(T)$ and energy density $e(T)$ are derived. The results approach the Stefan–Boltzmann limit at high temperature and are described by perturbative coefficients plus non-perturbative terms, indicating sizable non-perturbative contributions up to the electroweak scale. The methodology avoids explicit power-divergence subtractions, enabling a controlled access to the EoS up to high temperatures and is extensible to $N_f=5$.

Abstract

The Equation of State of Quantum Chromodynamics with $N_f=3$ flavours is determined non-perturbatively with a precision of about $0.5\%-1.0\%$ in the range of temperatures between 3 GeV and 165 GeV. The computation is carried out by numerical simulations of the gauge theory discretized on the lattice. At each given temperature the entropy density is computed at several lattice spacings in order to extrapolate the results to the continuum limit. The pressure and energy density are then determined by integrating the entropy density with respect to the temperature. The numerical data show a linear behaviour in the strong coupling constant squared, which points to the Stefan-Boltzmann limit at infinite temperature. They are also compatible with the known perturbative formula supplemented by higher order terms in the coupling constant, containing non-perturbative contributions. This parametrization describes well our data together with those present in the literature down to 500 MeV.

Figures (2)

• Figure 1: Left: continuum limit extrapolation of $s/T^3$. The black points are the one-loop improved non-perturbative results at each given temperature and $a/L_0$, plotted as function of $(a/L_0)^2$. Data at each temperature $T_n$, $n=0,1,...,8$ have been shifted downward by $n$ for better readability. Red continuum lines and bands represent the continuum limit extrapolation, while red dashed lines and shadowed bands are continuations of the fit function for comparison with $L_0/a=4$ data (not included in the fit). The continuum limit results are represented by red crosses. Right: Continuum results of $s/T^3$ as a function of the coupling $\hat{g}^2$. The orange and green bands are fits to the fit ansatz Eq. \ref{['eq:fit_ansatz_1']}, see main text for details.
• Figure 2: Left: the continuum results of $s/T^3$ (black points) are fitted (blue band) to the fit ansatz Eq. \ref{['eq:fit_ansatz_2']}. The comparison with non-perturbative results from the literature (Refs. Borsanyi:2013biaHotQCD:2014kolBazavov:2017dsy) is also shown. Right: entropy density, energy density and pressure for $T\geq 500$ MeV.