Dimensionally reduced expression for the QCD fermion determinant at finite temperature and chemical potential

Abstract

A dimensionally reduced expression for the QCD fermion determinant at finite temperature and chemical potential is derived which sheds light on the determinant’s dependence on these quantities. This is done via a partial zeta regularization, formally applying a general formula for the zeta determinant of a differential operator in one variable with operator-valued coefficients. The resulting expression generalizes the known one for the free fermion determinant, obtained via Matsubara frequency summation, to the case of a general background gauge field; moreover there is no undetermined overall factor. Rigorous versions of the result are obtained in a continuous time–lattice space setting. The determinant expression reduces to a remarkably simple form in the low temperature limit. A program for using this to obtain insight into the QCD phase transition at zero temperature and nonzero density is outlined.