The dual quark condensate in local and nonlocal NJL models: An order parameter for deconfinement?

We study the behavior of the dual quark condensate Σ1Σ1 in the Nambu–Jona-Lasinio (NJL) model and its nonlocal variant. In quantum chromodynamics Σ1Σ1 can be related to the breaking of the center symmetry and is therefore an (approximate) order parameter of confinement. The deconfinement transition is then signaled by a strong rise of Σ1Σ1 as a function of temperature. However, a similar behavior is also seen in the NJL model, which is known to have no confinement. Indeed, it was shown that in this model the rise of Σ1Σ1 is triggered by the chiral   phase transition. In order to shed more light on this issue, we calculate Σ1Σ1 for several variants of the NJL model, some of which have been suggested to be confining. Switching between “confining” and “non-confining” models and parametrizations we find no qualitative difference in the behavior of Σ1Σ1, namely, it always rises in the region of the chiral phase transition. We conclude that without having established a relation to the center symmetry in a given model, Σ1Σ1 should not blindly be regarded as an order parameter of confinement.