Gauge Invariant Monopoles in Su(2) Gluodynamics

We introduce a gauge invariant topological definition of monopole charge in pure SU(2) gluodynamics. The non-trivial topology is provided by hedgehog configurations of the non-Abelian field strength tensor on the two-sphere surrounding the monopole. It is shown that this definition can be formulated entirely in terms of Wilson loops which makes the gauge invariance manifest. Moreover, it counts correctly the monopole charge in case of spontaneously broken gauge symmetry and of pure Abelian gauge fields. Introduction The issue of magnetic monopoles in gauge models has a long history. The most famous result remains the Dirac quantization condition [1]. Although the monopoles themselves appeared to be purely mathematical constructions there were many exciting theoretical issues elucidated in the next 50 years or so following the Dirac’s paper, for a review see [2]. The next breakthrough was the realization [3] that magnetic monopoles naturally appear as classical configurations in the context of grand unified theories. Since then the main development is the observation and accumulation of data on the monopoles in lattice gauge theories (see Ref. [4] for a review). The monopoles are intrinsically a U(1) object and in the non-Abelian case, with no spontaneous symmetry breaking, the choice of a particular U(1) turns out to be a matter of gauge fixing. As a result, the definition of the monopoles is not unique and it is a separate issue which monopoles are relevant physically. Despite of the gauge dependence of monopoles in non-Abelian models the crucial point is that it is only the compactness of the gauge group that makes the very existence of monopoles possible [5]. Therefore the lattice regularization is singled out since it explicitly preserves the global structure of the gauge group. In particular, any non-Abelian lattice action is a periodic functional which permits for this reason existence of chromo-magnetic singular fluxes (Dirac strings) [6, 7]. In this paper we show that the monopoles defined as the end-points of non-Abelian Dirac strings are SU(2) gauge invariant. It is important that the singularity of the non-Abelian flux does not necessarily imply singular gauge potentials, contrary to the Abelian case. Whether the potentials are singular or not depends now on a particular gauge. We also formulate the monopole charge in terms of physical fluxes (Wilson loops) alone thus demonstrating explicitly its gauge invariance. To substantiate our definition of the magnetic charge we first consider a particular case of spontaneously broken gauge symmetry, namely, the Georgi-Glashow model and show that at the level of classical field configurations our formulation is identical to the well known definition of the ’t Hooft-Polyakov monopole charge. Then we rederive it in terms of Wilson loops which allows us to check that in the limit of pure Abelian gauge fields our construction is the same as for the Abelian monopole. At the level of classical field configurations it is also possible to estimate the self-energy of the monopole, which turns out to be linearly divergent in the ultraviolet. Thus, our definition of the monopole charge if applied to classical field configurations would select the singular Wu-Yang monopole [8]. However, it is well known that the Wu-Yang solution is in fact unstable [9]. Physically the instability arises because the interaction of spins with the magnetic field is so strong that the massless gluons fall onto the monopole center. Thus, it is an open dynamical question, what kind of the field configurations would be primarily identified as having the magnetic charge. Most probably, one should rely on the numerical simulations to answer this question. Thus, it is worth mentioning that our, pure topological definition of the monopole charge can be implemented in the quantum context [10] of the lattice gauge theories.