exchange processes. As a second step we choose a $k$-dependent linear field redefinition (encoded by the matrix $M_k$) such that also the Yukawa interaction $h_\Phi$ remains $k$-independent. While the first step (to enforce vanishing $\lambda_\psi$) is rather natural, the second is more arbitrary. In our case the choice of $M_k$ was guided by the insight we obtained in section \[sec:Partialbosonization\]. More general it is useful to redefine the composite fields such that the Yukawa-like interaction $h_\Phi$ remains independent of the center of mass momentum $p$, which is always possible. For some problems and for suitable choices of the cutoff function $R_k$ it may also be possible to choose the basis for the composite fields $\Phi$ such that the inverse propagator $P_\Phi$ is a diagonal matrix. As discussed in section \[sec:Partialbosonization\] this is possible for the problem considered here for very simple cutoff functions such as $R_k=k^2$. For more complicated $R_k$ one can still diagonalize $P_\Phi$ but the drawback is then that $h_\Phi$ might depend on $p$.

Approximative solutions {#sec:approximationschemes}
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Up to this point we discussed the application of the functional RG to the nonrelativistic two-body problem in a rather formal way. No approximations were needed, but we had to 