Euclidean 4d Exact Solitons in a Skyrme Type Model

We introduce a Skyrme type, four dimensional Euclidean field theory made of a triplet of scalar fields ~n, taking values on the sphere S2, and an additional real scalar field φ, which is dynamical only on a three dimensional surface embedded in IR. Using a special ansatz we reduce the 4d non-linear equations of motion into linear ordinary differential equations, which lead to the construction of an infinite number of exact soliton solutions with vanishing Euclidean action. The theory possesses a mass scale which fixes the size of the solitons in way which differs from Derrick’s scaling arguments. The model may be relevant to the study of the low energy limit of pure SU(2) Yang-Mills theory. The aim of this Letter is to present a new four dimensional Euclidean field theory and to construct an infinite number of exact soliton solutions for it. The physical motivation of our work is twofold. First, it contributes to the development of exact methods, in the framework of integrable field theories, for the study of non-perturbative aspects of physical theories [1]. Second, it may connect to recent attempts to understand the role of solitons in the low energy limit of pure Yang-Mills theories [2]. The Skyrme [3] and Skyrme-Faddeev [4] models are effective field theories possessing topological solitons, with several applications in various areas of physics. Such theories can not be solved exactly, and those solutions are constructed through numerical methods. The lack of an exact closed form for the solitons prevents the understanding of several properties of the phenomena involved. It is therefore of physical interest to construct similar theories possessing exact solutions. In Ref. [5] we have introduced a model with the same field content and topology as the Skyrme-Faddeev theory, and have constructed an infinite number of exact soliton solutions with arbitrary topological Hopf charges. That model is integrable in the sense of [1] and possesses an infinite number of local conservation laws. The exact solution was obtained through an ansatz that, as explained in [6], originates from the conformal symmetry of the static equations of motion. In this Letter we propose a model with a Skyrme-Faddeev scalar field taking values on the sphere S, together with an extra real scalar field. An important aspect of the model is that, although defined on a four dimensional Euclidean space IR, the extra field lives only on a three dimensional surface embedded on IR. That fact introduces a mass parameter which fixes the size of the solitons, but in a way which differs from Derrick’s scaling arguments. The soliton solutions are constructed through an ansatz, introduced in [6, 5], which reduces the four dimensional non-linear equations of motion into linear ordinary differential equations. The physical boundary conditions are such that the Euclidean action vanishes when evaluated on all soliton solutions. The theory can perhaps be viewed as an effective theory for the pure SU(2) Yang-Mills theory in a way explained at the end of the Letter. The model considered in this Letter is defined by the action