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

TL;DR

This paper derives an analytical formula for the distribution of the k-th smallest Dirac eigenvalue in QCD with imaginary isospin chemical potential, enabling lattice determination of the pion decay constant F.

Contribution

It introduces a new analytical expression for the k-th eigenvalue distribution using chiral Random-Two Matrix Theory, linking it to the pion decay constant in the epsilon-regime.

Findings

Exact eigenvalue distribution formulas for finite and infinite N

Generalizations of Dyson's integration Theorem and Sonine's identity

Potential for lattice QCD to determine F from eigenvalue data

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.