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Finite density lattice QCD without extrapolations

Alexander Adam, Szabolcs Borsányi, Zoltán Fodor, Jana N. Guenther, Paolo Parotto, Attila Pásztor, Ludovica Pirelli, Chik Him Wong

TL;DR

The paper addresses the sign problem in finite-density lattice QCD by advocating a canonical-ensemble formulation that fixes baryon number and, for the first time here, strangeness, enabling direct access to finite-density thermodynamics with physical quark masses. It reconstructs the canonical partition function $Z_C(T,V,N)$ from the grand-canonical function $Z_{GC}(T,V,\mu_B)$ via a Fourier transform and recovers $\mu_B$ as a discrete derivative of the canonical free energy, with an extension to joint $B$ and $S$ sectors through a two-dimensional Fourier transform. Using 2+1 flavors on a $16^3\times8$ lattice with a $4$HEX-staggered action, the study shows that CE results converge toward GC expectations in the thermodynamic limit and provides comparisons to Hadron Resonance Gas predictions across temperatures. The approach yields purely statistical uncertainties at present, avoids $\mu_B$ extrapolation systematics, and offers a new ab initio route to finite-density QCD relevant for heavy-ion phenomenology.

Abstract

Finite density lattice QCD usually relies on extrapolations in baryon chemical potential ($μ_B$), be it Taylor expansion, T' expansion (\cite{Borsanyi:2021sxv}) or analytical continuation. However, their range of validity is difficult to control. In the canonical formulation, the baryon density is the parameter of the system, not $μ_B$. Here we demonstrate that we can access finite density QCD in the canonical formulation with physical quark masses. We present first results with both the strangeness ($n_S$) and baryon ($n_B$) densities as parameters. Specifically, we compute the QCD pressure and chemical potentials as functions of $n_B$ and $n_S$. Our computations rely on high-statistics simulations with 2+1 4HEX-staggered fermions.

Finite density lattice QCD without extrapolations

TL;DR

The paper addresses the sign problem in finite-density lattice QCD by advocating a canonical-ensemble formulation that fixes baryon number and, for the first time here, strangeness, enabling direct access to finite-density thermodynamics with physical quark masses. It reconstructs the canonical partition function from the grand-canonical function via a Fourier transform and recovers as a discrete derivative of the canonical free energy, with an extension to joint and sectors through a two-dimensional Fourier transform. Using 2+1 flavors on a lattice with a HEX-staggered action, the study shows that CE results converge toward GC expectations in the thermodynamic limit and provides comparisons to Hadron Resonance Gas predictions across temperatures. The approach yields purely statistical uncertainties at present, avoids extrapolation systematics, and offers a new ab initio route to finite-density QCD relevant for heavy-ion phenomenology.

Abstract

Finite density lattice QCD usually relies on extrapolations in baryon chemical potential (), be it Taylor expansion, T' expansion (\cite{Borsanyi:2021sxv}) or analytical continuation. However, their range of validity is difficult to control. In the canonical formulation, the baryon density is the parameter of the system, not . Here we demonstrate that we can access finite density QCD in the canonical formulation with physical quark masses. We present first results with both the strangeness () and baryon () densities as parameters. Specifically, we compute the QCD pressure and chemical potentials as functions of and . Our computations rely on high-statistics simulations with 2+1 4HEX-staggered fermions.

Paper Structure

This paper contains 6 sections, 4 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Results
  3. Volume effects in the CE
  4. Comparison with HRG
  5. Conclusions
  6. Acknowledgments

Figures (3)

  • Figure 1: Left plot: baryon particle density $n=N/V$ in terms of the baryochemical potential $\mu_B$ in the grand canonical ensemble and in the canonical ensemble for $T=140,\,170$ MeV, corresponding respectively to the simulation volumes $V\sim22, \ 13 \, \rm{fm}^3$. $n$ is computed in the GCE with a Taylor expansion. $\mu_B$ is computed in the CE as a finite difference. Analogously we compute $N_S$ as a function of $\mu_S$ (right plot). For one flavour only it is easier to compute a higher number of Fourier coefficients.
  • Figure 2: $\Delta p/T^4$ as a function of $\mu_B$ for several volumes $V$, see eq.(\ref{['pressure']}). The simulation volume $V_0$ is $\sim 13 \rm{fm}^3$ for the left plot ($T=170$ MeV) and $\sim 25 \rm{fm}^3$ for the right plot ($T=135$ MeV). GCE lines and CE datapoints tend to agree at higher volumes.
  • Figure 3: Comparison with HRG. Left plot: $\Delta p/T^4$ at fixed $B=0,...,4$. Right plot: $\Delta p/T^4$ at $B=1$, $S=0,1,-1$. Also the case $B=0,\, S=0$ is shown.