Chiral Symmetry and the Low-Energy Spectrum of the QCD Dirac Operator

TL;DR

This paper links the low-energy spectrum of the QCD Dirac operator to chiral symmetry breaking, showing how it relates to Goldstone modes, the pion decay constant, and random matrix theory predictions, with implications for lattice QCD.

Contribution

It demonstrates the connection between the Dirac spectrum's slope, chiral symmetry, and Goldstone modes, providing a theoretical framework for lattice QCD observations.

Findings

The Dirac spectrum slope is determined by the pion decay constant.

Eigenvalue correlations match random matrix theory predictions.

Continuum results are potentially observable in lattice QCD simulations.

Abstract

The order parameter of the chiral phase transition is directly related to the infrared part of the spectrum of the QCD Dirac operator. This part of the spectrum follows from the low energy limit of QCD which is given by a partition function of weakly interacting Goldstone modes. We find that the slope of the Dirac spectrum is determined by the pion decay constant whereas for $\lambda \ll 1/L^2 \Lambda_{\rm QCD}$ the correlations of the Dirac eigenvalues are given by a random matrix theory with the global symmetries of the QCD partition function. A possible observation of these continuum results in lattice QCD with staggered fermions is discussed.