A study of the maximal Abelian gauge in SU(2) Euclidean Yang-Mills theory in the presence of the Gribov horizon

TL;DR

This paper investigates the maximal Abelian gauge in SU(2) Euclidean Yang-Mills theory considering the Gribov horizon, demonstrating renormalizability and compatibility with residual gauge symmetries, and introducing a BRST invariant gluon operator.

Contribution

It introduces a local, renormalizable horizon function for the maximal Abelian gauge, extending Zwanziger's approach and proving all-order renormalizability while maintaining residual gauge symmetries.

Findings

The horizon term is compatible with the residual U(1) Ward identity.

The nonrenormalization theorem Z_g Z_A^{1/2}=1 holds with the Gribov horizon.

A BRST invariant dimension two gluon operator is introduced and shown to be multiplicatively renormalizable.

Abstract

We pursue the study of SU(2) Euclidean Yang-Mills theory in the maximal Abelian gauge by taking into account the effects of the Gribov horizon. The Gribov approximation, previously introduced in [1], is improved through the introduction of the horizon function, which is constructed under the requirements of localizability and renormalizability. By following Zwanziger's treatment of the horizon function in the Landau gauge, we prove that, when cast in local form, the horizon term of the maximal Abelian gauge leads to a quantized theory which enjoys multiplicative renormalizability, a feature which is established to all orders by means of the algebraic renormalization. Furthermore, it turns out that the horizon term is compatible with the local residual U(1) Ward identity, typical of the maximal Abelian gauge, which is easily derived. As a consequence, the nonrenormalization theorem, Z_{g}Z_{A}^{1/2}=1, relating the renormalization factors of the gauge coupling constant Z_{g} and of the diagonal gluon field Z_{A}, still holds in the presence of the Gribov horizon. Finally, we notice that a generalized dimension two gluon operator can be also introduced. It is BRST invariant on-shell, a property which ensures its multiplicative renormalizability. Its anomalous dimension is not an independent parameter of the theory, being obtained from the renormalization factors of the gauge coupling constant and of the diagonal antighost field.