ar X iv : h ep - t h / 05 09 01 4 v 2 1 1 O ct 2 00 5 Spinning U ( 1 ) gauged Skyrmions

Introduction.– Many nonlinear classical field theories on flat spacetime backgrounds admit soliton solutions. These nonsingular solutions describe particle-like, localised configurations with finite energy. There has been some interest in recent years in the issue of globally regular spinning soliton solutions. However, to the best of our knowledge, to date no stationary and spinning solitons were found. (We describe single lumps with angular momentum as spinning, and reserve rotating for more general (gravitating-) solutions, including multilumps.) Notably, it is known that that finite energy solutions of the Yang-Mills-Higgs (YMH) system with a nonvanishing magnetic charge have zero angular momentum [1, 2] . Moreover, as found in [5], none of the known gauge field solitons with gauge group SU(2) (e.g. dyons, sphalerons, vortices) admit spinning generalizations within the stationary, axially symmetric, one-soliton sector. To date two types of spinning solitons have been found in the literature, a) Q-balls solitons in a complex scalar field theory with a non-renormalizable self-interaction [6], which are nontopological solitons so their stability is not guaranteed by a topological charge, and b) the electrically charged dipole monopole– antimonopole pair [7] of the YMH system with vanishing topological charge, which is not topologically stable even in the limit of vanishing angular momentum. It is our purpose here to construct a soliton which has intrinsic angular momentum and presents a topologically stable limit . Our definition for a ’soliton presenting a topologically stable limit’ is, a finite energy spinning lump which is topologically stable in the limit of vanishing angular momentum. This configuration corresponds to axially symmetric, electrically charged solutions of the U(1) gauged Skyrme model. Concerning the question of the existence of any given topologically stable solution, this is quite an intricate matter that deserves a brief description. To start with, there must be a valid topological lower bound on the energy, which may or may not be saturated, and for the Skyrmion it is not. Then there is the question whether any given field configuration (the solution) does minimise the energy? For the Skyrme model, this is a difficult problem for two reasons: a) because the sigma model fields are constrained, and b) because in addition to the quadratic kinetic term there is also a quartic kinetic term. Thus for the 1-Skyrmion, the existence proof is given by [9] and, [10], while for axially symmetric case, to the best of our knowledge, there is no rigorous existence proof. So axially symmetric Skyrmions and their magnetically gauged counterparts are supported only numerically. In addition, when a nonvanishing electric field is present, as it is in the present work, the functional misnimised is not the positive definite energy but the indefinite action. The proof of existence of such solutions, namely that for YMH dyons, is given by [11], but again it is too hard to adapt this proof for the gauged (and ungauged) Skyrme model. Thus the existence of the U(1) gauged axially symmetric solutions of the present paper, and those of [8], are supported only numerically.