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We show that the CJT thermodynamic potential of dynamical quark models with aquark propagator represented by complex conjugate mass poles inevitablyexhibits thermodynamic instabilities. We find that the minimal coupling of thequark sector to a Polyakov loop potential can strongly suppress but notcompletely remove such instabilities. This general effect is explicitlydemonstrated in the framework of a covariant, chirally symmetric, effectivequark model.

Thermodynamic instabilities in dynamical quark models with complex conjugate mass poles