Study of the maximal Abelian gauge inSU(2)Euclidean Yang-Mills theory in the presence of the Gribov horizon

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 *

Study of the zero modes of the Faddeev–Popov operator in the maximal Abelian gauge

“…Although being out of the aim of the present work, it is worth mentioning that a treatment of the Gribov issue similar to that of the Landau gauge [3,16] can be implemented in the maximal Abelian gauge, see for instance refs. [6,7,8,9,10]. Let us turn thus to the characterization of the zero modes of the Faddeev-Popov operator M ab , which clearly affect expression (19).…”

“…On the other hand, the Hilbert norm A 2 is known to be deeply related to the Gribov issue. In fact, several properties of the Gribov region in both Landau and maximal Abelian gauge can be obtained by looking at all relative minima of A 2 along the gauge orbits, see [3,4,6,7,8,9,10].…”

Gribov ambiguities at the Landau-maximal Abelian interpolating gauge

“…Of course, they play a fundamental role for a consistent quantization, but their elimination provides a profound change in the gluon and ghost propagators, specially at the infrared regime. This is a well-known feature for Landau and maximal Abelian gauges; see for instance [1,16]. In fact, the inclusion of dimension 2 condensates makes the analytic result of the propagators to stay in harmony with lattice results [12,13,32,33].…”

“…This redundancy is the reason why only the first equation of (26) is considered as the copies equation for the MAG. A final comment is that the Faddeev-Popov operator is hermitian in this case; see [16,17] and references therein for more details.…”

“…This is easy to verify. On the other hand, the usual MAG [16,17] has only off-diagonal components for the Faddeev-Popov operator. In this sense, we have to be careful with the trivial term because, when we choose η = 0 for the interpolating parameter, the mixed and purely diagonal components of the Faddeev-Popov operator will not vanish and this will provide a different trivial term with respect to the usual MAG case.…”