a minus sign for fermions. In addition we use in Eq.  the matrix of second functional derivatives $$\begin{aligned}
&&\Gamma^{(2)}_k(p,q)=\\
\nonumber
&&\begin{pmatrix} \overset{\rightharpoonup}{\delta}_{\psi_1^*(p)} \\ \overset{\rightharpoonup}{\delta}_{\psi_2^*(p)}\\
\overset{\rightharpoonup}{\delta}_{\psi_1(-p)}\\
\overset{\rightharpoonup}{\delta}_{\psi_2(-p)} \end{pmatrix} \Gamma_k[\psi]  \begin{pmatrix} 
\overset{\leftharpoonup}{\delta}_{\psi_1(q)} && \overset{\leftharpoonup}{\delta}_{\psi_2(q)} && \overset{\leftharpoonup}{\delta}_{\psi_1^*(-q)} &&
\overset{\leftharpoonup}{\delta}_{\psi_2^*(-q)} \end{pmatrix}\end{aligned}$$ and the cutoff matrix $${\cal R}_k(p,q) = \begin{pmatrix} \frac{-R_k(\vec p^2)}{2M_1}  && 0 && 0 && 0 \\ 0 && \frac{-R_k(\vec p^2)}{2M_2}  && 0 && 0 \\ 0 && 0 && \frac{R_k(\vec p^2)}{2M_1}  && 0 \\ 0 && 0 && 0 && \frac{R_k(\vec p^2)}{2M_2} \end{pmatrix} \delta(p-q),
\label{eq:secondfunctder}$$ with $$\delta(p) = (2\pi)^4 \delta(p_0) \delta^{(3)}(\vec p).
\label{eq:}$$

For large values of $k$ the flowing action approaches the microscopic action $$\lim_{k\to\infty} \Gamma_k[\psi] = S[\psi],$$ while for $k\to0$ the flowing action equals the quantum effective action $\Gamma$ which is the generating functional of one-particle irreducible Feynman diagrams $$\lim_{k\to0}\Gamma_k[\psi]=\Gamma[\psi].$$ We note that the flow equation can be written as $$\partial_k \Gamma_k[\psi] = \tilde \partial_k \frac{1}{2} \text{STr} \ln (\Gamma^{(2)}[\psi]+{\cal R}_k)$$ with the formal derivative $\tilde \partial_k$ that applies to the cutoff term ${\cal R}_k$, only. For reviews of the flow equation method see [@ReviewRG; @Pawlowski; @Metzner; @SalmhoferHonerkamp].

Fixed Hubbard-Stratonovich transformation {#sec:Partialbosonization}
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In this section we use a Hubbard-Stratonovich transformation [@HS] to calculate the effective bound state propagator. We modify 