(\psi_1\psi_2)^*(x,\vec y)\, V(\vec y)\, (\psi_1\psi_2)(x,\vec y)\\
&=& -\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).\end{aligned}$$ In the last line we used $$V(\vec q-\vec q^\prime) = \int_{\vec y} e^{i(\vec q-\vec q^\prime)\vec y} \,V(\vec y).$$

In this work we investigate the microscopic action in Eq.  using the functional integral formalism. The partition function as a functional of the source field $j$ is given by $$Z[j] = \int D \psi \; e^{iS_\psi[\psi]-i\int_p\sum_{i=1,2} \{j_i^*(p) \psi_i(p)+c.c.\}}.
\label{eq:partitionfunctnoRk}$$ In order to derive an exact flow equation we modify this expression by introducing an infrared cutoff term $$Z_k[j] = e^{-i W_k[j]} = \int D \psi \, e^{i S_\psi[\psi]+i\Delta S_k[\psi]-i\int \{j^*\psi+c.c.\}},$$ where we choose $$\begin{aligned}
\nonumber
\Delta S_k & = \int_p & \psi_1^*(p)\left[-\frac{1}{2M_1}R_k(\vec p^2)\right]\psi_1(p) \\
&& + \psi_2^*(p)\left[-\frac{1}{2M_2}R_k(\vec p^2)\right]\psi_2(p),
\label{eq:cutoffaction}\end{aligned}$$ with $R_k\to\infty$ for $k\to0$ and $R_k\to0$ for $k\to0$. The flowing action is now defined as the modified Legendre transform $$\Gamma_k[\psi] = \int_p \sum_{i=1,2}\{j_i^*(p)\psi_i(p)+c.c.\} -W_k[j]-\Delta S_k[\psi],$$ where $\psi(p)=\frac{\delta W_k}{\delta j^*(p)}$ is now the expectation value. The flowing action satisfies the exact flow equation [@CWFloweq] $$\partial_k \Gamma_k[\psi] = \frac{1}{2} \text{STr} (\Gamma^{(2)}[\psi]+{\cal R}_k)^{-1} \partial_k {\cal R}_k.
\label{eq:exactfloweqCW}$$ The operation $STr$ is a trace over both continuous and discrete degrees of freedom such as momentum or spin. It also includes 