The kth smallest Dirac operator eigenvalue and the pion decay constant

Abstract

We derive an analytical expression for the distribution of the kth smallest Dirac eigenvalue in QCD with an imaginary isospin chemical potential in the Dirac operator for arbitrary gauge field topology ν. Because of its dependence on the pion decay constant Fπ 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 kth 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 generalizations of Dyson’s integration theorem and Sonine’s identity.