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 ${\mathrm{Tr}}_{F}P$ and ${\mathrm{Tr}}_F{P}^{†}$ 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 two flavors of 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 temperature $T$ and quark chemical potential $\mu$, the results obtained reduce to those of perturbation theory at the complex saddle point. These results may be experimentally relevant for the compressed baryonic matter experiment at FAIR.

Figure 1

$\frac{1}{3}{\mathrm{Tr}}_{F}P$ (solid line) and $\frac{1}{3}{\mathrm{Tr}}_F{P}^{†}$ (dotted line) as functions of $T$ at constant $\mu$ for values up to $\mu =450\mathrm{MeV}$ with $N_f=2$.

Figure 2

$\frac{1}{3}{\mathrm{Tr}}_{F}P$ (solid line) and $\frac{1}{3}{\mathrm{Tr}}_F{P}^{†}$ (dotted line) as a function of $\mu$ at constant $T$ for values up to $T=250\mathrm{MeV}$ with $N_f=2$.

Figure 3

The shaded region of the $\mu -T$ plane indicates where the mass matrix is complex. High-$T$ and low-$T$ approximations to the boundary are also shown.