Distribution law of the Dirac eigenmodes in QCD

TL;DR

This paper investigates the distribution of Dirac eigenmodes in QCD, revealing that both near-zero and higher-lying modes follow the Gaussian Unitary Ensemble distribution, indicating a broader origin of randomness beyond chiral symmetry breaking.

Contribution

The study demonstrates that eigenmode distributions in QCD follow RMT GUE predictions regardless of their relation to chiral symmetry breaking, challenging previous assumptions.

Findings

Both near-zero and higher-lying modes follow GUE distribution.

Randomness in eigenmodes is not solely due to SBCS.

Eigenmode distributions are linked to confinement physics.

Abstract

The near-zero modes of the Dirac operator are connected to spontaneous breaking of chiral symmetry in QCD (SBCS) via the Banks-Casher relation. At the same time the distribution of the near-zero modes is well described by the Random Matrix Theory (RMT) with the Gaussian Unitary Ensemble (GUE). Then it has become a standard lore that a randomness, as observed through distributions of the near-zero modes of the Dirac operator, is a consequence of SBCS. The higher-lying modes of the Dirac operator are not affected by SBCS and are sensitive to confinement physics and related $SU(2)_{CS}$ and $SU(2N_F)$ symmetries. We study the distribution of the near-zero and higher-lying eigenmodes of the overlap Dirac operator within $N_F=2$ dynamical simulations. We find that both the distributions of the near-zero and higher-lying modes are perfectly described by GUE of RMT. This means that randomness, while consistent with SBCS, is not a consequence of SBCS and is related to some more general property of QCD in confinement regime.