The Analysis of Space-Time Structure in QCD Vacuum II: Dynamics of Polarization and Absolute X-Distribution

TL;DR

This paper introduces a framework to quantify polarization tendencies in complex systems, applies it to lattice QCD eigenmodes, and identifies a scale separating different polarization regimes, potentially revealing a new low-energy scale in QCD.

Contribution

It develops a formalism for measuring polarization dynamics and applies it to QCD, defining a polarization scale that distinguishes regimes of chirality enhancement and suppression.

Findings

Existence of a non-zero polarization scale in finite volume QCD.

Eigenmodes below the scale show positive correlation (chirality enhancement).

Eigenmodes above the scale show negative correlation (chirality suppression).

Abstract

We propose a framework for quantitative evaluation of dynamical tendency for polarization in 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 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.