Summary: We propose a framework for quantitative evaluation of dynamical tendency for polarization in an arbitrary random variable that can be decomposed into a pair of orthogonal subspaces. The method uses measures based on comparisons of given dynamics to its counterpart with statistically independent components. The formalism of previously considered \(X\)-distributions is used to express the aforementioned comparisons, in effect putting the former approach on solid footing. Our analysis leads to the definition of a suitable correlation coefficient with clear statistical meaning. We apply the method to the dynamics induced by pure-glue lattice QCD in local left-right components of overlap Dirac eigenmodes. It is found that, in finite physical volume, there exists a non-zero physical scale in the spectrum of eigenvalues such that eigenmodes at smaller (fixed) eigenvalues exhibit convex \(X\)-distribution (positive correlation), while at larger eigenvalues the distribution is concave (negative correlation). This chiral polarization scale thus separates a regime where dynamics enhances chirality relative to statistical independence from a regime where it suppresses it, and gives an objective definition to the notion of “low” and “high” Dirac eigenmode. We propose to investigate whether the polarization scale remains non-zero in the infinite volume limit, in which case it would represent a new kind of low energy scale in QCD.
For Part I see [I. Horváth, Nucl. Phys. B 710, No. 1–2, 464–484 (2005; Zbl 1115.81402); erratum ibid. 714, No. 1–2, 175–176 (2005; Zbl 1207.81170)].