The k-th Smallest Dirac Operator Eigenvalue and the Pion Decay Constant

Abstract

We derive an analytical expression for the distribution of the k-th smallest Dirac eigenvalue in QCD with imaginary isospin chemical potential in the Dirac operator. Because of its dependence on the pion decay constant F through the chemical potential in the epsilon-regime of chiral perturbation theory this can be used for lattice determinations of that low-energy constant. On the technical side we use a chiral Random-Two Matrix Theory, where we express the k-th eigenvalue distribution through the joint probability of the ordered k smallest eigenvalues. The latter can be computed exactly for finite and infinite N, for which we derive generalisations of Dyson's integration Theorem and Sonine's identity.

27 pages, 5 figures; v2: typos corrected, published version

References in corpus (10)

• Unquenched QCD Dirac Operator Spectra at Nonzero Baryon Chemical Potential
• Chiral Symmetry Breaking and the Dirac Spectrum at Nonzero Chemical Potential
• A New Method for Determining $F_{\pi }$ on the Lattice
• A new Chiral Two-Matrix Theory for Dirac Spectra with Imaginary Chemical Potential
• Parameters of the lowest order chiral Lagrangian from fermion eigenvalues
• Microscopic eigenvalue correlations in QCD with imaginary isospin chemical potential
• Extracting $F_{\pi }$ from small lattices: unquenched results
• Quarks with Twisted Boundary Conditions in the Epsilon Regime
• Individual complex Dirac eigenvalue distributions from random matrix theory and comparison to quenched lattice QCD with a quark chemical potential
• Determining F_pi from spectral sum rules

Cited by in corpus (2)

• New Ways to Determine Low-Energy Constants with Wilson Fermions
• Universality crossover between chiral random matrix ensembles and twisted SU(2) lattice Dirac spectra