Partition function zeros in QCD

The spectral correlations of the QCD Dirac operator in the 4-dimensional Euclidean box, 1/ΛQCD ≪ L ≪ 1/mπ, usually denoted as Leutwyler-Smilga regime [3], have been the object of intense study in recent years. They contain valuable information about the chiral properties of the QCD vacuum. Provided that chiral symmetry is spontaneously broken, the above regime can be described either by an effective Lagrangian method [3] or by means of χRMT (for a recent review see [1]). The universal limit in which QCD and χRMT coincide is the limit N → ∞ , λ → 0 , m → 0, in which the microscopic variables, ζ ≡ 2N λ and μ ≡ 2N m, are kept fixed and N is identified as the dimensionless volume. (Here, λ denotes an eigenvalue of the Dirac operator and m is the dimensionless quarkmass parameter.) A way to get information about phase transitions is to study the scaling behaviour of the zeros of the partition function [4]. The authors of [3] noticed a qualitative agreement of the QCD partition function zeros and the averaged Dirac operator eigenvalues for one flavour. We show [2] with a “trapping argument” in the context of χRMT that the zeros and the average eigenvalue positions are indeed intimately connected. It is also known [1] that the spectral correlations of QCD follow the predictions of χRMT only up to a certain energy scale, the Thouless energy Ec. Beyond this scale correlations die out. By introducing the concept of normal modes we find an alternative way of studying χRMT. It enables us to remove certain restrictions imposed by Ec and to use χRMT to describe QCD beyond the Leutwyler-Smilga regime. The χRMT partition function corresponding toNf flavours, topological sector ν, quarks in the fundamental and gluons in the adjoint representation is given by [1]