Chiral symmetry and spectrum of Euclidean Dirac operator in QCD

TL;DR

This paper derives exact relations for the spectral density of the Euclidean Dirac operator in QCD, providing theoretical benchmarks for lattice simulations and insights into the QCD vacuum structure.

Contribution

It presents new exact relations for the spectral density in QCD based on chiral symmetry, applicable in both infinite and finite volume regimes.

Findings

Derived relations for spectral density $ ho(\lambda)$ in QCD.

Results applicable to both thermodynamic and finite volume limits.

Provides benchmarks for numerical lattice QCD simulations.

Abstract

Some exact relations for the spectral density $\rho(\lambda)$ of the Euclidean Dirac operator in $QCD$ are derived. They follow directly from the chiral symmetry of the $QCD$ lagrangian with massless quarks. New results are obtained both in thermodynamic limit when the Euclidean volume $V$ is sent to infinity and also in the theory defined in finite volume where the spectrum is discrete and a nontrivial information on $\rho(\lambda)$ in the region $\lambda \sim 1/(|<\bar{q}q>_0|V)$ (the characteristic level spacing) can be obtained. These exact results should be confronted with "experimental" numerical simulations on the lattices and in some particular models for $QCD$ vacuum structure and may serve as a nontrivial test of the validity of these simulations.