Mesoscopic QCD and the Theta Vacua

TL;DR

This paper analyzes the QCD partition function across all topological sectors, revealing how the vacuum structure and chiral condensate depend on the vacuum angle heta, and identifying a first-order phase transition at heta=\pi.

Contribution

It provides a detailed analytical study of the heta dependence in QCD with multiple flavors, including the derivation of the partition function and the characterization of Dashen's phenomena.

Findings

Chiral condensate decreases monotonically with heta.

A cusp in the condensate at heta=\pi for degenerate quark masses.

Discontinuity in the energy density derivative at heta=\pi indicating a phase transition.

Abstract

The partition function of QCD is analyzed for an arbitrary number of flavors, N_f, and arbitrary quark masses including the contributions from all topological sectors in the Leutwyler--Smilga regime. For given N_f and arbitrary vacuum angle, \theta, the partition function can be reduced to N_f-2 angular integrations of single Bessel functions. For two and three flavors, the \theta dependence of the QCD vacuum is studied in detail. For N_f= 2 and 3, the chiral condensate decreases monotonically as \theta increases from zero to \pi and the chiral condensate develops a cusp at \theta=\pi for degenerate quark masses in the macroscopic limit. We find a discontinuity at \theta=\pi in the first derivative of the energy density with respect to \theta for degenerate quark masses. This corresponds to the first--order phase transition in which CP is spontaneously broken, known as Dashen's phenomena.