A string-like self-dual solution of Yang-Mills theory

Abstract

We attempt to obtain Nielsen-Olesen strings in pure Yang-Mills theory without Higgs scalars. This is inspired by recent work interpreting the Prasad-Sommerfield monopole as a static self-dual Yang-Mills solution in which A 4 a plays the role of the Higgs field. In similar fashion, we make A 3 a A 4 a serve as the two required isovector fields in an ansatz independent of x 3 and x 4. The condition of self-duality results in a single Painlevé equation of the third kind (or equivalently, a radial sinh-Gordon equation in 2 + 0 dimensions), the solution of which determines A μ a . We make use of the extensive analysis of the former equation carried out by Wu, McCoy and collaborators in the context of the scaling limit of the two-dimensional Ising model. Their simplest solution yields a flux value of −2 π/ e just as in the Nielsen-Olesen model and flux is quantized in multiples of this unit. The string tension (action per unit time per unit distance) diverges as r −2 (In r) −2 as r → 0 for the same Wu-McCoy solution.