Study of Yang–Mills–Chern–Simons theory in presence of the Gribov horizon
Introduction
Three-dimensional Yang–Mills theory is one of the most important models in which it is possible to analyse unsolved non-perturbative problems such as colour confinement. The theory is simpler than QCD, but it is still highly non-trivial. It has local degrees of freedom and the coupling constant is dimensionful. Moreover, it can be viewed as an approximation to the high temperature phase of QCD with the mass gap serving as the magnetic mass.
A very interesting term which can be added to the three-dimensional Yang–Mills theory is the Chern–Simons term [1], [2]1: this term provides mass for the gauge field which is of topological origin. Therefore, while pure Yang–Mills is known to be a confining theory, the addition of the topological Chern–Simons term has the effect of generating a de-confined massive excitation. Said otherwise, the theory undergoes a change of regime, passing from a confined to a de-confined regime.
The purpose of this paper is that of discussing, within a quantum field theory framework, how this change of regime is driven by the presence of the Chern–Simons term. To that aim, we shall take into account the non-perturbative effects arising from the Gribov horizon [5].2 This will enable us to encode non perturbative effects into the two-point gluon correlation function whose analytic structure can be employed to analyse how the theory moves from one regime to another when varying the coupling constant and the Chern–Simons mass parameter . We remind here that the presence of the Gribov phenomenon is a general feature of the quantization procedure of nonabelian gauge theories, the existence of Gribov copies being in fact a well known property of any local covariant renormalizable gauge fixing [8] (see also [9]). The presence of gauge copies gives rise to zero modes of the Faddeev–Popov operator which invalidate the usual Faddeev–Popov construction.
A successful method to deal with the issue of the Gribov copies is that of restricting the domain of integration in the functional integral to the so-called Gribov region [5], [6], [7], which is the set of all transverse field configurations for which the Faddeev–Popov operator is strictly positive, namely . The region has been proven to be bounded in all directions in field space [10], its boundary being the first Gribov horizon. Moreover, all gauge orbits pass through at least once [11], a property which strongly supports the restriction to . Remarkably, the whole procedure results in a local and renormalizable action known as the Gribov–Zwanziger action [12], [13]. More recently, a refinement of the Gribov–Zwanziger action has been worked out in [14], [15] by taking into account the effects of dimension of two condensates. The resulting two-point gluon correlation function turns out to be in excellent agreement with the most recent lattice data [16], allowing for nontrivial analytic estimates of the first glueball states [17], [18]. Let us also mention that the Refined Gribov–Zwanziger framework has been employed in the study of the Casimir energy [19], producing the correct sign for the Casimir force within the MIT bag model, clarifying a long-standing problem. Also, in a series of papers [20], [21], [22], [23], the Gribov–Zwanziger setup has been employed in order to study, in the continuum, the transition between the confining and non-confining regimes when Higgs fields are present. Also in this case, the non-perturbative gluon two-point correlation function obtained by taking into account the effects of the Gribov horizon turns out to be a useful quantity in order to obtain information about the transition from the confining to the Higgs regime. As discussed in detail in [20], [21], [22], [23], the gluon correlation function undergoes a continuous change from a confining expression of the Gribov type, characterized by the presence of unphysical complex conjugate poles, to a Yukawa type propagator with a real pole, indicating that the theory is in the Higgs regime. The emerging picture is in full agreement with the renewed Fradkin–Shenker work [24].
In the present paper, we shall implement the restriction to the Gribov region in Yang–Mills–Chern–Simons theory by working out the non-perturbative expression of the two point gauge correlation function. Further, we shall vary the gauge coupling constant and the Chern–Simons mass and discuss how the poles of this correlation function get modified, thus obtaining information on how the theory passes form the confining to the non-confining regimes.
The paper is organized as follows: in Section 2, the gluon propagator and the Gribov gap equation for Yang–Mills–Chern–Simons theory are obtained. In Section 3, the behaviour of the poles of the gauge propagator as functions of the two parameters is discussed. In Section 4 we present our conclusions.
Section snippets
Gauge propagator for Yang–Mills–Chern–Simons action in presence of the Gribov horizon
We start by considering the Yang–Mills–Chern–Simons action in Euclidean flat space quantized in the Landau gauge, namely Here, stands for the Chern–Simons mass, is the Lagrange multiplier enforcing the Landau gauge, , and are the Faddeev–Popov ghosts. This theory is known as the topologically massive non-Abelian gauge theory, because of the massive gluon propagator [1], [2], given by
Analytic structure of the gauge propagator and the different regimes of the theory
The propagator in expression (25) depends on the coupling constant and on the Chern–Simons mass , and exhibits a rather complex pole structure. The poles of the propagator are functions of the parameters . As such, the study of their behaviour when varying is of great help in understanding the different regimes in which the theory may be found, as recently discussed in the case of Yang–Mills theories in the presence of Higgs fields [20], [21], [22], [23] as well as of gauge
Conclusion
In this paper the Gribov semi-classical approach to eliminate gauge copies has been applied to Yang–Mills Chern–Simons theory in three dimensions. Unlike what happens in pure Yang–Mills theory, whose propagator is always confining at zero temperature within the Gribov semi-classical approach, the presence of the Chern–Simons topological term gives rise to a new regime in which a physical massive mode can propagate. In particular, the present analysis shows that there is a range of parameters,
Acknowledgements
This work of F.C. and A.G is partially supported by the FONDECYT grants N 1120352 andN 3130679. A.G. also wants to thank the Max Planck Institute for its hospitality, where this manuscript was partially written. The Centro de Estudios Cientificos (CECs) is funded by the Chilean Government through the Centers of Excellence Base Financing Program of CONICYT.
The third author’s work is supported by FAPERJ, Fundação de Amparo à Pesquisa do Estado do Rio de Janeiro, under the program Cientista do
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