The Chiral Condensate in a Finite Volume

TL;DR

This paper develops a new approach in chiral perturbation theory at finite volume, allowing for a unified treatment across different regimes and enabling the calculation of the quark condensate and Dirac operator spectral density.

Contribution

It introduces a method that handles zero momentum modes separately, maintaining infrared finiteness and connecting p-regime and epsilon-regime results in finite-volume chiral perturbation theory.

Findings

Computed the quark condensate across different regimes.

Derived the spectral density of the Dirac operator.

Established a smooth connection between regimes.

Abstract

Chiral perturbation theory at finite four-volume V (=L^3T) is reconsidered with a view towards finding a computational scheme that can deal with any value of M_\pi L, where M_\pi is a generic Nambu-Goldstone mass. The momentum zero modes that cause the usual p-expansion to fail in the chiral limit are treated separately, and partly integrated out to all orders. In this way the theory remains infrared finite in the perturbative expansion, and the chiral limit can be considered at finite volume. We illustrate the technique by computing the quark condensate in a finite volume, smoothly connecting standard results in the p-regime for larger masses with those of the epsilon-regime for smaller masses. From the partially quenched theory we also obtain the spectral density of the Dirac operator, a smooth function from the microscopic region to the bulk region of the p-regime.