On the zero modes of the Faddev-Popov operator in the Landau gauge

TL;DR

This paper constructs explicit examples of zero modes of the Faddeev-Popov operator in the Landau gauge for SU(2) and SU(3) in Euclidean space, including the first non-abelian SU(3) configuration in 4D.

Contribution

It provides the first explicit non-abelian gauge field configurations leading to zero modes of the Faddeev-Popov operator in SU(3) in 4D.

Findings

Explicit zero modes constructed for SU(2) and SU(3) in Euclidean space.

First non-abelian SU(3) configuration in 4D with non-vanishing nonlinear term.

Demonstrates the existence of non-trivial gauge field configurations with zero modes.

Abstract

Following Henyey procedure, we construct examples of zero modes of the Faddev-Popov operator in the Landau gauge in Euclidean space in D dimensions, for both SU(2) and SU(3 groups. We consider gauge field configurations $A^a_\mu$ which give rise to a field strength, $F^a_{\mu\nu} =\partial_\mu A^a_\nu -\partial_\nu A^a_\mu + f^{abc}A^b_\mu A^c_\nu$, whose nonlinear term, $ f^{abc}A^b_\mu A^c_\nu$, turns out to be nonvanishing. To our knowledge, this is the first time where such a non-abelian configuration is explicitly obtained in the case of SU(3) in 4D.