the functional integral over the field $\psi$ in Eq.  by including another integral over the (composite) field $\Phi$ $$\begin{aligned}
\nonumber
Z[j,J] &=& \int D\psi \; e^{i S_\psi[\psi]+i\int_p \sum_i \{j_i^*(p) \psi_i(p)+c.c.\}}\\
& \times & \int D \Phi \; e^{i S_\text{pb}[\psi,\Phi]+i \int_{p,\vec q}\{J^*(p,\vec q)\Phi(p,\vec q)+c.c.\}}\end{aligned}$$ with $$\begin{aligned}
\nonumber
S_\text{pb}[\psi,\Phi] &=& \int_{p,\vec q,\vec q^\prime} \left[\Phi^*(p,\vec q) -(\psi_1\psi_2)^*(p,\vec q)\right]\\
&& V(\vec q-\vec q^\prime) \left[\Phi(p,\vec q^{\prime})-(\psi_1\psi_2)(p,\vec q^\prime)\right].
\label{eq:spb1}\end{aligned}$$ For $J=0$ the functional integral over $\Phi$ contributes only a multiplicative constant to $Z$ which is irrelevant for most purposes. In the combined action $S=S_\psi+S_\text{pb}$ the term quadratic in $\psi$ cancels and we arrive at $$\begin{aligned}
\nonumber
S &=& \int_p \psi_1^*(p_0-\frac{1}{2M_1}\vec p^2) \psi_1 + \psi_2^*(p_0-\frac{1}{2M_2}\vec p^2)\psi_2\\
\nonumber
&&- \int_{p,\vec q, \vec q^\prime} \left\{\Phi^*(p,\vec q)V(\vec q-\vec q^\prime)(\psi_1\psi_2)(p,\vec q)+c.c.\right\}\\
&&+ \int_{p,\vec q,\vec q^\prime} \Phi^*(p,\vec q) V(\vec q-\vec q^\prime) \Phi(p,\vec q^\prime).
\label{eq:HStransformedaction}\end{aligned}$$ Note that there is a certain ambiguity in the precise from of the Hubbard-Stratonovich transformation. Linear transformations of the field $\Phi$ are always possible. For the Hubbard-Stratonovich transformed action in Eq.  it is now possible to solve some parts of the functional integral.

Let us consider the functional renormalization group equations for this theory when an infrared cutoff of the form in Eq.  is added for the field 