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The pressure of the SU(N) lattice gauge theory at large-N

Barak Bringoltz, Michael Teper

TL;DR

This work probes whether the well-known high-temperature deficit in pressure and entropy seen in SU(3) persists in the large-$N$ limit by computing bulk thermodynamics for SU(4) and SU(8) on finite lattices and comparing to SU(3). Using the integral method with Wilson action and careful finite-volume checks, the authors find that the normalized pressure $p/T^4$, energy density $\epsilon/T^4$, entropy $S/T^3$, and the interaction measure $\Delta/ T^4$ exhibit little $N$-dependence in the range $T_c \le T \le 1.6\,T_c$ (and up to $\sim 2.5\,T_c$ for $\Delta$), indicating that the deficit persists in the planar limit. This constrains explanations based on perturbative expansions, resonance models, or topological fluctuations, and strengthens the case that gravity-dual (AdS/CFT) approaches may capture the relevant strong-coupling features of deconfined gauge matter at large $N$. The results thus provide a bridge between large-$N$ holographic ideas and real-world QCD thermodynamics, while highlighting the need for a continuum extrapolation in future work.

Abstract

We calculate bulk thermodynamic properties, such as the pressure, energy density, and entropy, in SU(4) and SU(8) lattice gauge theories, for the range of temperatures T <= 2.0Tc and T <= 1.6Tc respectively. We find that the N=4,8 results are very close to each other, and to what one finds in SU(3), and are far from the asymptotic free-gas value. We conclude that any explanation of the high-T pressure (or entropy) deficit must be such as to survive the N-->oo limit. We give some examples of this constraint in action and comment on what this implies for the relevance of gravity duals.

The pressure of the SU(N) lattice gauge theory at large-N

TL;DR

This work probes whether the well-known high-temperature deficit in pressure and entropy seen in SU(3) persists in the large- limit by computing bulk thermodynamics for SU(4) and SU(8) on finite lattices and comparing to SU(3). Using the integral method with Wilson action and careful finite-volume checks, the authors find that the normalized pressure , energy density , entropy , and the interaction measure exhibit little -dependence in the range (and up to for ), indicating that the deficit persists in the planar limit. This constrains explanations based on perturbative expansions, resonance models, or topological fluctuations, and strengthens the case that gravity-dual (AdS/CFT) approaches may capture the relevant strong-coupling features of deconfined gauge matter at large . The results thus provide a bridge between large- holographic ideas and real-world QCD thermodynamics, while highlighting the need for a continuum extrapolation in future work.

Abstract

We calculate bulk thermodynamic properties, such as the pressure, energy density, and entropy, in SU(4) and SU(8) lattice gauge theories, for the range of temperatures T <= 2.0Tc and T <= 1.6Tc respectively. We find that the N=4,8 results are very close to each other, and to what one finds in SU(3), and are far from the asymptotic free-gas value. We conclude that any explanation of the high-T pressure (or entropy) deficit must be such as to survive the N-->oo limit. We give some examples of this constraint in action and comment on what this implies for the relevance of gravity duals.

Paper Structure

This paper contains 9 sections, 13 equations, 5 figures, 9 tables.

Table of Contents

  1. Introduction
  2. Lattice set up and methodology
  3. The method used
  4. Average plaquette
  5. Finite volume effects
  6. The deconfined phase
  7. The confined phase
  8. Results
  9. Summary and discussion

Figures (5)

  • Figure 1: The pressure, normalized to the lattice Stephan-Boltzmann pressure, including the full discretization errors. The symbol's vertical sizes are representing the largest error bars (which are received for the highest temperature). The solid line is for $SU(3)$ and $L_t=6$ from Boyd:1996bx.
  • Figure 2: Results for $\Delta(T)/T^4=T\frac{\partial p/T^4}{\partial T}$, normalized by the same coefficient as we normalize the pressure. The solid line is for $SU(3)$ and $L_t=6$ from Boyd:1996bx.
  • Figure 3: Results for $\Delta(T)/T^4=T\frac{\partial p/T^4}{\partial T}$, normalized to the free-gas result. The lines are for $SU(3)$ and $L_t=4,6$ from Boyd:1996bx. Red triangles correspond to $L_t=5$, and changing $\beta$, while blue circles correspond to changing $L_t$ and keeping a fixed $\beta=5.800$.
  • Figure 4: Results for $\Delta(T)/T^4=T\frac{\partial p/T^4}{\partial T}$ for $N=3,4,8$, by fixing $\beta=\beta_c(L_t=5)$, while changing $L_t=2,3,4,5$.
  • Figure 5: Results for energy density and entropy, normalized to the lattice Stephan-Boltzmann result, including the full discretization errors. The solid line is for $SU(3)$ and $L_t=6$ from Boyd:1996bx.