Complex saddle points in QCD at finite temperature and density

The sign problem in QCD at finite temperature and density leads naturally to the consideration of complex saddle points of the action or effective action. The global symmetry $\mathcal{CK}$ of the finite-density action, where $\mathcal{C}$ is charge conjugation and $\mathcal{K}$ is complex conjugation, constrains the eigenvalues of the Polyakov loop operator $P$ at a saddle point in such a way that the action is real at a saddle point, and net color charge is zero. The values of $Tr_{F}P$ and $Tr_{F}P^{\dagger}$ at the saddle point, are real but not identical, indicating the different free energy cost associated with inserting a heavy quark versus an antiquark into the system. At such complex saddle points, the mass matrix associated with Polyakov loops may have complex eigenvalues, reflecting oscillatory behavior in color-charge densities. We illustrate these properties with a simple model which includes the one-loop contribution of gluons and massless quarks moving in a constant Polyakov loop background. Confinement-deconfinement effects are modeled phenomenologically via an added potential term depending on the Polyakov loop eigenvalues. For sufficiently large $T$ and $\mu$, the results obtained reduce to those of perturbation theory at the complex saddle point. These results may be experimentally relevant for the CBM experiment at FAIR.