of the fields $\psi$. This can be done by solving the field equation $$\frac{\delta \Gamma_k}{\delta \Phi}=0$$ for the field $\Phi$ as a functional of $\psi$. Plugging this solution $\Phi[\psi]$ into the action leads to $$\Gamma_k^{(\psi)} = \Gamma_k[\psi,\Phi[\psi]].$$ In general, the dependence of $\Gamma_k$ on $\Phi$ is complicated and undoing the Hubbard-Stratonovich transformation leads to complicated interaction vertices for the field $\psi$. However, it is quite easy to calculate the effective vertex with two incoming and two outgoing $\psi$-particles. We note that the only contribution to this interaction is given by a tree level diagram involving the propagator $G_k$. The corresponding term in $\Gamma_{k=0}^{(\psi)}$ is $$\begin{split}
& \Gamma_{k=0}^{(\psi,4)} = -\int_{p,\vec y_1,\vec y_2} (\psi_1\psi_2)(p,\vec y_1)^* g_n(\vec y_1) V(\vec y_1) \\
& \times \frac{1}{p_0-\frac{1}{2(M_1+M_2)}\vec p^2-E_n} V(\vec y_2) g^*_n(\vec y_2) (\psi_1\psi_2)(p,\vec y_2)\\
& - \int_{p,\vec y} (\psi_1\psi_2)^*(p,\vec y) V(\vec y) (\psi_1\psi_2)(p,\vec y).
\end{split}
\label{eq:efffourfermioninteraction}$$ This expression has a simple interpretation. The first term is the contribution from bound state exchange processes, while the second term is just the classical term. For an electromagnetic Coulomb interaction, the second term can be seen as a contribution from photon exchange processes.

Let us summarize what we have done in this section. Starting from the microscopic model in 