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

TL;DR

The paper introduces the SO(3) real algebra method to tackle the sign problem in SU(3) lattice QCD at finite baryon density by decomposing gauge variables into an SO(3) part and an SU(3)/SO(3) coset. It defines a maximal SO(3) gauge to render the SU(3)/SO(3) component small and uses the real, nonnegative determinant of the SO(3) sector to drive Monte Carlo sampling, with the remaining SU(3) determinant ratio treated via reweighting. The framework combines continuum and lattice formulations, detailing the factorization, gauge fixing, and practical Monte Carlo steps, while highlighting the conditions under which phase fluctuations of the determinant ratio remain manageable. If successful, this approach could enable finite-density QCD simulations in a parameter window and offer insights into the QCD phase diagram, potentially extendable to finite temperature and SU(N) generalizations.

Abstract

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.