SO(3) real algebra method for SU(3) QCD at finite baryon-number densities

For SU(3) lattice QCD calculations at finite baryon-number densities, we propose the ``SO(3) real algebra method'', in which the SU(3) gauge variable is divided into the SO(3) and SU(3)/SO(3) parts. In this method, we introduce the ``maximal SO(3) gauge'' by minimizing the SU(3)/SO(3) part of the SU(3) gauge variable. In the Monte Carlo calculation, the SO(3) real algebra method employs the SO(3) fermionic determinant, i.e., the fermionic determinant of the SO(3) part of the SU(3) gauge variable, in the maximal SO(3) gauge, as well as the positive SU(3) gauge action factor $e^{-S_G}$. Here, the SO(3) fermionic determinant is real, and it is non-negative for the even-number flavor case ($N_f=2n$) of the same quark mass, e.g., $m_u=m_d$. The SO(3) real algebra method alternates between the maximal SO(3) gauge fixing and Monte Carlo updates on the SO(3) determinant and $e^{-S_G}$. After the most importance sampling, the ratio of the SU(3) and SO(3) fermionic determinants is treated as a weight factor. If the phase factor of the ratio does not fluctuate significantly among the sampled gauge configurations for a set of parameters (e.g., volume, chemical potential, and quark mass), then SU(3) lattice QCD calculations at finite densities would be feasible.