propagator of $\Phi$ can be solved exactly. More concrete, we write the term of $\Gamma_k$ that is quadratic in $\Phi$ as $$\Gamma_k^{(\Phi,2)} = \int_{p,\vec q,\vec q^\prime} \Phi^*(p,\vec q) P_\Phi(p,\vec q,\vec q^\prime) \Phi(p,\vec q^\prime)$$ with a $k$-dependent function $P_\Phi(p,\vec q,\vec q^\prime)$ denoting the inverse propagator. Its flow equation reads $$\begin{split}
& \partial_k P_\Phi(p,\vec q^\prime,\vec q^{\prime\prime}) =  \tilde \partial_k (-i) \int_{q} V(\vec q^{\prime},\vec q) V(\vec q, \vec q^{\prime\prime}) \\
& \times \frac{1}{(\eta_1 p+q)_0-\frac{1}{2M_1}[(\eta_1 \vec p+\vec q)^2+R_k((\eta_1 \vec p+\vec q)^2)]+i\epsilon}\\
& \times \frac{1}{(\eta_2 p-q)_0-\frac{1}{2M_2}[(\eta_1 \vec p-\vec q)^2+R_k((\eta_1 \vec p-\vec q)^2)]+i\epsilon}.
\end{split}$$ We can write this in a symbolic notation as $$\partial_k P_\Phi(p) = - \tilde \partial_k (V A_k^{-1}(p) V)$$ where the “matrix indices” $\vec q$ etc. and the corresponding integrations are implicit. We use the abbreviation $$\begin{split}
& A_k^{-1}(p,\vec q,\vec q^\prime) = i \int_{q_0}   \delta^{(3)}(\vec q-\vec q^\prime) \\
& \times \frac{1}{(\eta_1 p+q)_0-\frac{1}{2M_1}[(\eta_1 \vec p+\vec q)^2+R_k((\eta_1 \vec p+\vec q)^2)]+i\epsilon}\\
& \times \frac{1}{(\eta_2 p-q)_0-\frac{1}{2M_2}[(\eta_1 \vec p-\vec q)^2+R_k((\eta_1 \vec p-\vec q)^2)]+i\epsilon}\\
& = \delta^{(3)}(\vec q-\vec q^\prime) {\bigg ( }p_0-\frac{1}{2M_1}[(\eta_1 \vec p+\vec q)^2+R_k((\eta_1\vec p+\vec q)^2)]\\
& \quad \quad -\frac{1}{2M_2}[(\eta_1 \vec p-\vec q)^2+R_k((\eta_1\vec p-\vec q)^2)]+i\epsilon {\bigg )}^{-1}.
\end{split}
\label{eq:akinverse}$$ Since the only $k$-dependence in this expression comes from $R_k$, the flow equation for $P_\Phi$ can be integrated directly. 