Chiral Symmetry Restoration from a Boundary

TL;DR

This paper investigates how boundary conditions, specifically homogeneous Dirichlet conditions, can induce a phase transition leading to the restoration of chiral symmetry in QCD, with implications for lattice computations.

Contribution

It demonstrates that boundary conditions can non-perturbatively restore chiral symmetry in QCD, using the sigma model to analyze the effects of boundary-induced phase transitions.

Findings

Chiral symmetry is restored if the compact dimension is less than 2.0 fm.

For lengths greater than 4 fm, a uniform chiral condensate forms.

Finite-size corrections decay as a power law, not exponentially.

Abstract

The boundary of a manifold can alter the phase of a theory in the bulk. We explore the possibility of a boundary-induced phase transition for the chiral symmetry of QCD. In particular, we investigate the consequences of imposing homogeneous Dirichlet boundary conditions on the quark fields. Such boundary conditions are sometimes employed in lattice gauge theory computations, for example, when including external electromagnetic fields, or when computing quark propagators with a reduced temporal extent. Homogeneous Dirichlet boundary conditions force the chiral condensate to vanish at the boundary, and thereby obstruct the spontaneous breaking of chiral symmetry in the bulk. We show the restoration of chiral symmetry due to a boundary is a non-perturbative phenomenon depending upon the mechanism of spontaneous symmetry breaking, and utilize the sigma model to exemplify the issues. Within this model, we find that chiral symmetry is completely restored if the length of the compact direction is less than 2.0 fm. For lengths greater than about 4 fm, an approximately uniform chiral condensate forms centered about the midpoint of the compact direction. While the volume-averaged condensate approaches the infinite volume value as the compact direction becomes very long, the finite-size corrections are shown to be power law rather than exponential.