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The Pressure in 2, 2+1 and 3 Flavour QCD

F. Karsch, E. Laermann, A. Peikert

TL;DR

This study computes the high-temperature QCD pressure for two, two-plus-one, and three light flavors using a $16^3\times4$ lattice with improved gauge and staggered fermion actions and the integral method. It shows that flavour dependence is largely governed by the Stefan-Boltzmann limit, while a heavier strange quark reduces the pressure more than an ideal gas would suggest, highlighting nonperturbative effects. By analyzing cut-off effects and providing a continuum-limit estimate for $T\gtrsim 2T_c$, it offers a first quantitative picture of the massless QCD equation of state in this regime and outlines the path toward more precise determinations with larger $N_\tau$ and cross-checks using alternative discretizations.

Abstract

We calculate the pressure in QCD with two and three light quarks on a lattice of size 16^3x4 using tree level improved gauge and fermion actions. We argue that for temperatures T > 2T_c systematic effects due to the finite lattice cut-off and non-vanishing quark masses are below 15% in this calculation and give an estimate for the continuum extrapolated pressure in QCD with massless quarks. We find that the flavour dependence of the pressure is dominated by that of the Stefan-Boltzmann constant. Furthermore we perform a calculation of the pressure using 2 light (m_u,d/T=0.4) and one heavier quark (m_s/T = 1). In this case the pressure is reduced relative to that of three flavour QCD. This effect is stronger than expected from the mass dependence of an ideal Fermi gas.

The Pressure in 2, 2+1 and 3 Flavour QCD

TL;DR

This study computes the high-temperature QCD pressure for two, two-plus-one, and three light flavors using a lattice with improved gauge and staggered fermion actions and the integral method. It shows that flavour dependence is largely governed by the Stefan-Boltzmann limit, while a heavier strange quark reduces the pressure more than an ideal gas would suggest, highlighting nonperturbative effects. By analyzing cut-off effects and providing a continuum-limit estimate for , it offers a first quantitative picture of the massless QCD equation of state in this regime and outlines the path toward more precise determinations with larger and cross-checks using alternative discretizations.

Abstract

We calculate the pressure in QCD with two and three light quarks on a lattice of size 16^3x4 using tree level improved gauge and fermion actions. We argue that for temperatures T > 2T_c systematic effects due to the finite lattice cut-off and non-vanishing quark masses are below 15% in this calculation and give an estimate for the continuum extrapolated pressure in QCD with massless quarks. We find that the flavour dependence of the pressure is dominated by that of the Stefan-Boltzmann constant. Furthermore we perform a calculation of the pressure using 2 light (m_u,d/T=0.4) and one heavier quark (m_s/T = 1). In this case the pressure is reduced relative to that of three flavour QCD. This effect is stronger than expected from the mass dependence of an ideal Fermi gas.

Paper Structure

This paper contains 6 sections, 9 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Improved action, Cut-off dependence
  3. The numerical calculation
  4. The pressure
  5. Continuum Limit: An Estimate
  6. Conclusions

Figures (3)

  • Figure 1: String tension calculated on $16^4$ lattices (a) and action differences from $16^4$ and $16^3\times 4$ lattices (b) for $n_f=2$, 3 and (2+1).
  • Figure 2: The pressure for $n_f=2$, 2+1 and 3 calculated with the p4-action (a) and the normalized values $p/p_{\rm SB}$ (b).The arrows indicate the continuum ideal gas limits for two and three flavour QCD with quarks of mass $m/T=0.4$ as well as the case of two flavour QCD with $m/T=0.4$ and an additional heavier quark of mass $m_s/T=1$.
  • Figure 3: The pressure for $n_f=2$. Shown are results obtained with the p4-action on lattices with temporal extent $N_\tau=4$ (line) as well as with the standard staggered fermion action on $N_\tau=4$ (squares) and 6 (circles, triangles) lattices. Also shown is an estimate of the continuum equation of state for massless QCD (dashed band), based on the assumption that the systematic error of the current analysis is $(15\pm 5)\%$.