Spectral Sum Rules of the Dirac operator and Partially Quenched Chiral Condensates

TL;DR

This paper derives high-order spectral sum rules for the Dirac operator using Virasoro constraints and computes expansions of the partially quenched chiral condensate, confirming results with Random Matrix Theory and supersymmetry methods.

Contribution

It introduces a novel derivation of spectral sum rules and expansions for the chiral condensate using Virasoro constraints and the replica method, extending previous results.

Findings

Derived generalized Leutwyler-Smilga spectral sum rules to high order.

Computed low-mass and large-mass expansions of the partially quenched chiral condensate.

Confirmed that results match earlier findings from Random Matrix Theory and supersymmetric approaches.

Abstract

Exploiting Virasoro constraints on the effective finite-volume partition function, we derive generalized Leutwyler-Smilga spectral sum rules of the Dirac operator to high order. By introducing $N_v$ fermion species of equal masses, we next use the Virasoro constraints to compute two (low-mass and large-mass) expansions of the partially quenched chiral condensate through the replica method of letting $N_v \to 0$. The low-mass expansion can only be pushed to a certain finite order due to de Wit-'t Hooft poles, but the large-mass expansion can be carried through to arbitrarily high order. Results agree exactly with earlier results obtained through both Random Matrix Theory and the supersymmetric method.