example of a nonrelativistic theory it would be useful to have a formulation where the two terms in Eq.  are treated in different ways. While the term in the first line (the contribution from bound state exchange processes) is treated most efficient in terms of the Hubbard-Stratonovich field $\Phi$, this is different for the second term. For a relativistic field theory this corresponds to the contribution from photon or other exchange processes and it is therefore most efficient to write it in terms of this exchange field. In other words, for the nonrelativistic theory, the Hubbard-Stratonovich transformation should be constructed such that the effective action $\Gamma_{k=0}$ contains as an explicit contribution only the classical term (the second term in Eq. ). All additional terms should be described by bound state exchange processes. For the flowing action one can use the same prescription. To that end it is necessary to work with a scale-dependent Hubbard-Stratonovich transformation, however. An exact flow equation that can be used for this purpose was derived in [@FW09] and will be discussed in the next section.

Scale dependent Hubbard-Stratonovich transformation {#sec:scaledeppartialbos}
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In this section we investigate the bound state problem using a $k$-dependent version of the 