On the dipole swing and the search for frame independence in the dipole model

Small-x evolution in QCD is conveniently described by Mueller's dipole model which, however, does not include saturation effects in a way consistent with boost invariance. In this paper we first show that the recently studied zero and one dimensional toy models exhibiting saturation and explicit boost invariance can be interpreted in terms positive definite k -> k+1 dipole vertices. Such k -> k+1 vertices can in the full model be generated by combining the usual dipole splitting with k - 1 simultaneous dipole swings. We show that, for a system consisting of N dipoles, one needs to combine the dipole splitting with at most N - 1 simultaneous swings in order to generate all colour correlations induced by the multiple dipole interactions.