Random matrix theory and spectral sum rules for the Dirac operator in QCD

TL;DR

This paper develops a random matrix model that captures the spectral properties of the Dirac operator in QCD, demonstrating the universality of certain spectral sum rules across different physical scenarios.

Contribution

It introduces a random matrix model equivalent to the low energy QCD partition function, establishing universal sum rules for Dirac eigenvalues based on symmetry considerations.

Findings

Sum rules are universal and depend only on symmetries.

Sum rules hold for an interacting instanton liquid.

The spectral density near zero exhibits universal approach to the thermodynamic limit.

Abstract

We construct a random matrix model that, in the large $N$ limit, reduces to the low energy limit of the QCD partition function put forward by Leutwyler and Smilga. This equivalence holds for an arbitrary number of flavors and any value of the QCD vacuum angle. In this model, moments of the inverse squares of the eigenvalues of the Dirac operator obey sum rules, which we conjecture to be universal. In other words, the validity of the sum rules depends only on the symmetries of the theory but not on its details. To illustrate this point we show that the sum rules hold for an interacting liquid of instantons. The physical interpretation is that the way the thermodynamic limit of the spectral density near zero is approached is universal. However, its value, $i.e.$ the chiral condensate, is not.