On the spectrum of the Faddeev-Popov operator in topological background fields

TL;DR

This paper analytically investigates the spectrum of the Faddeev-Popov operator in topological backgrounds within SU(2) Yang-Mills theory to explore potential links between topological excitations and confinement mechanisms.

Contribution

It provides an analytical study of the Faddeev-Popov spectrum in topological backgrounds, highlighting differences between instantons and center vortices in relation to confinement.

Findings

A single instanton yields few zero-modes, likely irrelevant to confinement.

A center vortex with sufficient flux produces many zero-modes, potentially related to confinement.

Eigenstates in vortex backgrounds meet conditions necessary for quark confinement.

Abstract

In the Gribov-Zwanziger scenario the confinement of gluons is attributed to an enhancement of the spectrum of the Faddeev-Popov operator near eigenvalue zero. This has been observed in functional and also in lattice calculations. The linear rise of the quark-anti-quark potential and thus quark confinement on the other hand seems to be connected to topological excitations. To investigate whether a connection exists between both aspects of confinement, the spectrum of the Faddeev-Popov operator in two topological background fields is determined analytically in SU(2) Yang-Mills theory. It is found that a single instanton, which is likely irrelevant to quark confinement, also sustains only few additional zero-modes. A center vortex, which is likely important to quark confinement, is found to contribute much more zero-modes, provided the vortex is of sufficient flux. Furthermore, the corresponding eigenstates in the vortex case satisfy one necessary condition for the confinement of quarks.