Spectral sum rules and Selberg's integral

TL;DR

This paper derives spectral sum rules for different chiral random matrix ensembles using Selberg's integral, connecting spectral correlations with low-energy effective theories.

Contribution

It provides a unified derivation of Leutwyler-Smilga sum rules for one and two flavors across three chiral ensembles using Selberg's integral.

Findings

Sum rules for N_f=1 are identical across ensembles

Spectral correlations relate to low-energy effective partition functions

Derived all Leutwyler-Smilga sum rules for specified ensembles

Abstract

Using Selberg's integral formula we derive all Leutwyler-Smilga type sum rules for one and two flavors, and for each of the three chiral random matrix ensembles. In agreement with arguments from effective field theory, all sum rules for $N_f = 1$ coincide for the three ensembles. The connection between spectral correlations and the low-energy effective partition function is discussed.