Rotating regular solutions in Einstein-Yang-Mills-Higgs theory

We construct new axially symmetric rotating solutions of Einstein-Yang-Mills-Higgs theory. These globally regular configurations possess a nonvanishing electric charge which equals the total angular momentum, and zero topological charge, representing a monopole-antimonopole system rotating around the symmetry axis through their common center of mass. Introduction.– Rotation is an universal phenomenon, which seems to be shared by all objects, at all possible scales. For a gravitating Maxwell field, the Kerr-Newman black hole solutions represent the only asymptotically flat configurations with nonzero angular momentum. However, no regular rotating solution is found in the limit of zero event horizon radius. The inclusion of a larger (non Abelian) gauge group in the theory leads to the possibility of regularising these configurations, as evidenced by the Bartnick-McKinnon (BM) solution of the Einstein-Yang-Mills (EYM) equations [1]. However, to date no explicit example of an asymptotically flat regular rotating solution with non Abelian matter fields is known [2]. Although predicted perturbatively [3], no rotating generalisations of the BM solution seem to exist [4, 5] . The situation is more complicated in a spontaneously broken gauge theory. As discussed in [6, 4, 2] for a Higgs field in the adjoint representation (the case considered in this letter), the Julia-Zee dyons do no present generalisations with a nonvanishing angular momentum. In fact the general result presented in [9] proves that the angular momentum of any regular solution with a nonvanishing magnetic charge is zero . This however leaves open the possibility of the existence of rotating Einstein-Yang-Mills-Higgs (EYMH) solutions in the topologically trivial sector of the theory. We have in mind solutions described by an equal number of monopoles and antimonopoles situated on the z-axis with zero net magnetic charge like those in [11] and [12], gravitating and in flat space respectively; but unlike the latter [11, 12], with nonzero electric charge. Although the density of the magnetic field is locally nonzero, the magnetic charge of these configurations measured at infinity would vanish. This, in the presence of an electric charge, results in nonzero angular momentum. Despite the presence of some comments in the literature on the possible existence of such solutions, no explicit construction has been attempted. Here we construct numerically the simplest example of a regular rotating solution in a spontaneously broken gauge theory. It represents an asymptotically flat, electrically charged monopole-antimonopole (MA) system rotating around their common center of mass. For a vanishing electric field, the solution reduces to the static axially symmetric MA configurations discussed in [11]. Axially symmetric ansatz and general relations.– Our study of the SU(2)-EYMH system is based upon the action