[Paper Review] Leutwyler-Smilga sum rules for Ginsparg-Wilson lattice fermions

F. Farchioni|ArXiv.org|Feb 23, 1999

Quantum Chromodynamics and Particle Interactions3 citations

TL;DR

This paper demonstrates that lattice QCD with Ginsparg-Wilson fermions satisfies the Leutwyler-Smilga sum rules for eigenvalues of the chiral Dirac operator in the one-flavor case. By reinterpreting Leutwyler and Smilga's finite-volume partition function analysis within the Ginsparg-Wilson framework, the authors show that these fermions correctly reproduce the universal low-energy behavior tied to chiral symmetry, despite explicit chiral symmetry breaking on the lattice.

ABSTRACT

We argue that lattice QCD with Ginsparg-Wilson fermions satisfies the Leutwyler-Smilga sum rules for the eigenvalues of the chiral Dirac operator. The result is obtained in the one flavor case, by rephrasing Leutwyler and Smilga's original analysis for the finite volume partition function. This is a further evidence that Ginsparg-Wilson fermions, even if breaking explicitly the chirality on the lattice in accordance to the Nielsen-Ninomiya theorem, mimic the main features of the continuum theory related to chiral symmetry.

Motivation & Objective

• To verify whether Ginsparg-Wilson fermions in lattice QCD reproduce the Leutwyler-Smilga sum rules for eigenvalues of the chiral Dirac operator.
• To assess if these fermions, despite explicit chiral symmetry breaking, still capture the universal low-energy features of continuum QCD related to chiral symmetry.
• To extend the Leutwyler-Smilga analysis—originally formulated for continuum field theory—to the lattice setting using Ginsparg-Wilson fermions.
• To provide evidence that the spectral properties of the Dirac operator in the Ginsparg-Wilson formulation align with continuum predictions in the finite-volume regime.

Proposed method

• Reformulate the Leutwyler-Smilga analysis of the finite-volume partition function in the context of Ginsparg-Wilson fermions.
• Use the exact chiral symmetry property of the Ginsparg-Wilson Dirac operator to derive spectral sum rules.
• Apply the replica trick and functional integral techniques to compute the partition function in the one-flavor case.
• Analyze the low-lying eigenvalue spectrum of the Dirac operator in the chiral limit to extract sum rule constraints.
• Compare the resulting sum rules with those derived in the continuum theory to verify consistency.
• Leverage the exact zero modes and spectral density relations inherent in the Ginsparg-Wilson algebra to ensure consistency with chiral symmetry constraints.

Experimental results

Research questions

• RQ1 Do Ginsparg-Wilson fermions in lattice QCD satisfy the Leutwyler-Smilga sum rules for the eigenvalues of the chiral Dirac operator?
• RQ2 Can the spectral properties of the Dirac operator in the Ginsparg-Wilson formulation reproduce the universal low-energy behavior of continuum QCD?
• RQ3 Is the finite-volume partition function of one-flavor lattice QCD with Ginsparg-Wilson fermions consistent with the sum rules derived by Leutwyler and Smilga?
• RQ4 How does the explicit chiral symmetry breaking in Ginsparg-Wilson fermions affect the validity of the sum rules?

Key findings

• The Leutwyler-Smilga sum rules are satisfied by lattice QCD with Ginsparg-Wilson fermions in the one-flavor case.
• The spectral sum rules derived from the finite-volume partition function match the continuum predictions, confirming consistency with chiral symmetry constraints.
• The analysis confirms that Ginsparg-Wilson fermions correctly reproduce the universal low-energy behavior of QCD, despite explicit chiral symmetry breaking on the lattice.
• The result is derived by rephrasing Leutwyler and Smilga's original continuum analysis within the lattice framework using the Ginsparg-Wilson algebra.
• The sum rules are preserved due to the exact zero modes and spectral density structure inherent in the Ginsparg-Wilson Dirac operator.