Large-Nc equivalence and the sign problem at finite baryon density

QCD with a finite baryon chemical potential, despite its importance, is not well understood because the standard lattice QCD simulation is not applicable due to the sign problem. Although QCD-like theories which do not suffer from the sign problem have been studied intensively, relation to QCD with a finite baryon chemical potential was not clear. This paper introduces large-$N_c$ equivalences between QCD and various QCD-like theories. These equivalences lead us to a unified viewpoint for QCD with baryon and isospin chemical potentials, $SO(2N_c)$ and $Sp(2N_c)$ gauge theories, QCD with adjoint matters and two-color QCD. In particular QCD with the baryon chemical potential is large-$N_c$ equivalent to its phase quenched version in a certain parameter region, which is relevant for heavy ion collision experiments. All previous simulation results which study the effect of the phase confirm the phase quench approximation is quantitatively good already at Nc=3; it is so good that often two theories give the same value within error. Therefore the phase quenched simulation is the best strategy for the QCD critical point search. At small volume one can study a tiny 1/Nc effect by the phase reweighting; the large-Nc equivalence guarantees that the phase reweighing method works without suffering from the overlapping problem.