Hubbard Stratonovich transformation. Instead of $S_\text{pb}$ in Eq.  we employ $$\begin{split}
& S_\text{pb}=\int_p\left[\Phi^*(p)-(\psi_1\psi_2)^*(p) V Q_\Lambda^{-1}(p)\right]\; Q(p)\\
& \left[\Phi(p)-Q_\Lambda^{-1}(p) V (\psi_1\psi_2)(p)\right].
\end{split}
\label{eq:Spb2}$$ Again we use a matrix notation where the summation over the index $\vec q$ etc. is left implicit. In Eq.  the matrix $Q(p)$ is $k$-dependent, while $Q_\Lambda(p)$ equals $Q(p)$ for $k=\Lambda$ but is independent of $k$. We choose $$\begin{split}
& \lim_{\Lambda\to\infty} Q_\Lambda(p,\vec q,\vec q^\prime) = \infty \;\delta^{(3)}(\vec q-\vec q^\prime),\\ 
& \lim_{\Lambda\to\infty} Q_\Lambda^{-1}(p,\vec q,\vec q^\prime)=0.
\end{split}$$ In this limit the combined action $S=S_\psi+S_\text{pb}$ becomes $$\begin{aligned}
\nonumber
S &=& \int_p \, \psi_1^*\left(p_0-\frac{1}{2M_1}\vec p^2\right)\psi_1 + \psi_2^*\left(p_0-\frac{1}{2M_2}\vec p^2\right)\psi_2\\
\nonumber
&& - \int_{p,\vec q,\vec q^\prime} (\psi_1\psi_2)^*(p,\vec q) V(\vec q-\vec q^\prime) (\psi_1\psi_2)(p,\vec q^\prime)\\
\nonumber
&& - \int_{p,\vec q,\vec q^\prime} \left\{ \Phi^*(p,\vec q) V(\vec q-\vec q^\prime) (\psi_1\psi_2)(p,\vec q^\prime)+c.c.\right\}\\
&& + \int_{p,\vec q, \vec q^\prime} \Phi^*(p,\vec q) Q(p,\vec q,\vec q^\prime) \Phi(p,\vec q^\prime).
\label{eq:combinedaction2}\end{aligned}$$ The field $\Phi$ is very “massive” for large $\Lambda$. The action in Eq.  is the starting point for the flow of the functional $\Gamma_k[\psi,\Phi]$ for $k=\Lambda$. In ref. [@FW09] we showed that the flowing action for a scale dependent Hubbard-Stratonovich transformation of the form in Eq. satisfies the exact flow equation $$\begin{aligned}
\nonumber
\partial_k \Gamma_k &=& \frac{1}{2} \text{STr} (\Gamma_k^{(2)}+{\cal R}_k)^{-1} (\partial_k {\cal R}_k 