= e^{-M_k^\dagger} {\cal R}_k e^{-M_k}, \\ 
& \widehat{\partial_k {\cal R}_k} = e^{-M_k^\dagger} \partial_k {\cal R}_k e^{-M_k}
\end{split}
\label{eq:abbreviations}$$ and functional derivatives are now taken with respect to $\hat \Phi$. It is important to note that the matrices $\partial_k Q$, $\widehat{\partial_k Q}$ and $M_k$ have entries only in the $\Phi$-$\Phi$-block. For example, Eq.  implies a transformation of the cutoff term for the composite bosons but not for the fundamental fields $\psi_1, \psi_2$. For simplicity we drop the hats at most places below.

To investigate the implications of the flow equation we use again a truncation of the flowing action in terms of a vertex expansion. Due to the nonrelativistic dispersion relation and the instantaneous interaction, this leads to exact flow equations for the considered $n$-point functions since their flow equations decouple from higher vertex functions. More concrete, we choose as our truncation $$\begin{aligned}
\nonumber
\Gamma_k &=& \int_p \left\{\psi_1^*\left(p_0-\frac{1}{2M_1}\vec p^2\right)\psi_1+\psi_2^*\left(p_0-\frac{1}{2M_2}\vec p^2\right)\psi_2\right\}\\
\nonumber
&& - \int_{p,\vec q,\vec q^\prime} (\psi_1\psi_2)^*(p,\vec q) V(\vec q- \vec q^\prime) (\psi_1\psi_2)(p,\vec q^\prime)\\
\nonumber
&& - \int_{p,\vec q,\vec q^\prime} (\psi_1\psi_2)^*(p,\vec q) \lambda_\psi(p,\vec q, \vec q^\prime) (\psi_1\psi_2)(p,\vec q^\prime)\\
\nonumber
&& - \int_{p,\vec q,\vec q^\prime} \left\{ \Phi^*(p,\vec q) h_\Phi(p,\vec q, \vec q^\prime) (\psi_1\psi_2)(p,\vec q^\prime)+c.c. \right\}\\
&& + \int_{p,\vec q, \vec q^\prime} \Phi^*(p,\vec 