Exact knot solutions in a generalized Skyrme-Faddeev model

Abstract

We propose a generalized Skyrme-Faddeev type theory with an additional scalar field. In a special case of model parameters one has a theory which admits exact knot solutions given by a class of exact toroidal solitons from Aratyn-Ferreira-Zimerman (AFZ) integrable CP 1 model. In a general case the theory admits an exact knot solution for a unit Hopf charge. For higher Hopf charges we perform numeric analysis of the solutions and obtain estimates for the knot energies using energy minimization procedure based on ansatz with AFZ field configurations and with rational functions. We show that AFZ configurations provide a better approximate solutions. The corresponding knot energies are in a good agreement with a standard law for the low energy bound, EH Q [6,7]. The Skyrme-Faddeev theory [8,9] represents another kind of effective theories which supposed to be induced from quantum chromodynamics (QCD) in low energy regime. The theory possesses topological knot solitons classified by the homotopy group π 3 (CP 1 ) = Z, i.e., by the topological Hopf charge. It is proposed to interpret such knot solutions as color electric and color magnetic glueball states [8,[10][11][12][13]. One should notice, that derivation of a strict expression for a low energy effective action from the basics of QCD is an extremely difficult problem [14][15][16]. So far there exists a number of various extended Skyrme-Faddeev models where some exact and numeric knot and vortex solutions have been found [17][18][19][20].In the present paper we consider an extended SkyrmeFaddeev model with an additional scalar field. In general the low energy effective action of QCD contains scalar fields [10] which originate from unknown coefficient correlation functions in the effective action and play roles of order parameters in the effective theory [14]. We consider a minimal extension of the Skyrme-Faddeev theory to find out a model which is similar to the Skyrme theory, and, at the same time, inherits properties of integrable CP 1 models.So far, all known topological solutions in (3+1) Skyrme type models are obtained numerically except the original Skyrmion soliton [1]. Some exact vortex solutions have been obtained in a generalized Skyrme-Faddeev theory for a special set of model parameters which imply the existence of integrable submodel in the theory [17][18][19]. A family of exact analytic knot solutions with toroidal configuration have been constructed in special integrable CP 1 models [21,22]. Unfortunately, physical applications of such solutions in QCD remain unclear. We propose a Skyrme-Faddeev-type model which admits exact knot solutions and which can give some hints towards constructing a more realistic low energy QCD effective theory. Let us consider an extended Skyrme-Faddeev model defined by the following Lagrangianwheren is a unit triplet field in adjoint representation of SU (2) color group, i.e., CP 1 field, φ is a real scalar field, and H µν is a magnetic field defined as followsµ, β, ν, ξ are model parameters. The first two terms in (1) ...