Finding the effective Polyakov line action for SU(3) gauge theories at finite chemical potential

Motivated by the sign problem, we calculate the effective Polyakov line action corresponding to certain SU(3) lattice gauge theories on a ${16^3 \times 6}$ lattice via the "relative weights" method introduced in our previous articles. The calculation is carried out at $\beta=5.6,5.7$ for the pure gauge theory, and at $\beta=5.6$ for the gauge field coupled to a relatively light scalar particle. In the latter example we determine the effective theory also at finite chemical potential, and show how observables relevant to phase structure can be computed in the effective theory via mean field methods. In all cases a comparison of Polyakov line correlators in the effective theory and the underlying lattice gauge theory, computed numerically at zero chemical potential, shows accurate agreement down to correlator magnitudes of order $10^{-5}$. We also derive the effective Polyakov line action corresponding to a gauge theory with heavy quarks and large chemical potential, and apply mean field methods to extract observables.