Individual Eigenvalue Distributions of Chiral Random Two-Matrix Theory and the Determination of F_pi

TL;DR

This paper derives analytical formulas for individual eigenvalue distributions of Dirac operators in a chiral Random Two-Matrix framework, providing a potential new method to determine F_pi from lattice simulations.

Contribution

It introduces explicit calculations of eigenvalue distributions and gap probabilities in chiral Random Two-Matrix Theory, linking them to the effective chiral Lagrangian in finite volume.

Findings

Derived explicit formulas for eigenvalue distributions and correlations.

Defined and computed gap probabilities at finite N and in the scaling limit.

Presented a new kernel for the gap probability related to external sources.

Abstract

Dirac operator eigenvalues split into two when subjected to two different external vector sources. In a specific finite-volume scaling regime of gauge theories with fermions, this problem can be mapped to a chiral Random Two-Matrix Theory. We derive analytical expressions to leading order in the associated finite-volume expansion, showing how individual Dirac eigenvalue distributions and their correlations equivalently can be computed directly from the effective chiral Lagrangian in the epsilon-regime. Because of its equivalence to chiral Random Two-Matrix Theory, we use the latter for all explicit computations. On the mathematical side, we define and determine gap probabilities and individual eigenvalue distributions in that theory at finite N, and also derive the relevant scaling limit as N is taken to infinity. In particular, the gap probability for one Dirac eigenvalue is given in terms of a new kernel that depends on the external vector source. This expression may give a new and simple way of determining the pion decay constant F_pi from lattice gauge theory simulations.