[@SimilarityRG] has been employed to investigate for example a two-dimensional particle with contact potential [@GlazekWilson]. Note that the flow equation of Wegner and the similarity renormalization group break space-time symmetry explicitly. This is in contrast to Wetterich’s formulation of renormalization in terms of the effective average action or flowing action $\Gamma_k$ [@CWFloweq]. Ellwanger demonstrated that this formulation can be used to investigate bound state formation for relativistic theories [@Ellwanger]. By tracing the flow of the four-point vertex, Ellwanger calculates for example binding energies for the Wick-Cutkosky model and finds good agreement with known results in various limits. As a drawback of his approach one might see that properties of the bound states are somewhat hidden in the analytic structure of the four-point vertex. This makes is it hard to refine approximations or to investigate interactions between bound states or spontaneous symmetry breaking.

These difficulties can be overcome by introducing auxiliary fields for composite operators as will be discussed in more detail below. It is important to adapt the composite fields while solving the flow equation. The reason is that one wants to absorb the essential parts of the four-point vertex by means of an Hubbard-Stratonovich transformation at each step in 