QCD at zero baryon density and the Polyakov loop paradox

We compare the grand-canonical partition function at fixed chemical potential μ with the canonical partition function at fixed baryon number B, formally and by numerical simulations at μ=0 and $B=0$ with four flavors 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<Tc and by the free fermion gas at T>Tc. Small differences between the two ensembles, for thermodynamic observables characterizing the deconfinement phase transition, vanish with increasing lattice size. These differences are solely caused by contributions of nonzero baryon density sectors, which are exponentially suppressed with increasing volume. The Polyakov loop shows a different behavior: for all temperatures and volumes, its expectation value is exactly zero in the canonical formulation, whereas it is always nonzero in the commonly used grand-canonical formulation. We clarify this paradoxical difference, and show that the nonvanishing Polyakov loop expectation value is due to contributions of nonzero triality states, which are not physical, because they give zero contribution to the partition function.