QCD at Zero Baryon Density and the Polyakov Loop Paradox

TL;DR

This paper compares grand canonical and canonical partition functions in QCD at zero baryon density, clarifying the Polyakov loop paradox and demonstrating their equivalence in the thermodynamic limit.

Contribution

It provides a detailed comparison of the two ensembles in QCD, explaining the Polyakov loop discrepancy and confirming their equivalence for thermodynamic observables.

Findings

Free energy densities are equal in the thermodynamic limit.

Polyakov loop expectation value is zero in the canonical ensemble.

Differences between ensembles vanish with increasing volume.

Abstract

We compare the grand canonical partition function at fixed chemical potential mu with the canonical partition function at fixed baryon number B, formally and by numerical simulations at mu=0 and B=0 with four flavours of staggered quarks. We verify that the free energy densities are equal in the thermodynamic limit, and show that they can be well described by the hadron resonance gas at T < T_c and by the free fermion gas at T>T_c. Small differences between the two ensembles, for thermodynamic observables characterising the deconfinement phase transition, vanish with increasing lattice size. These differences are solely caused by contributions of non-zero baryon density sectors, which are exponentially suppressed with increasing volume. The Polyakov loop shows a different behaviour: for all temperatures and volumes, its expectation value is exactly zero in the canonical formulation, whereas it is always non-zero in the commonly used grand-canonical formulation. We clarify this paradoxical difference, and show that the non-vanishing Polyakov loop expectation value is due to contributions of non-zero triality states, which are not physical, because they give zero contribution to the partition function.