q) P_\Phi(p,\vec q,\vec q^\prime) \Phi(p,\vec q^\prime).
\label{eq:truncationrebosonization}\end{aligned}$$ Once again, the propagators for the fields $\psi_1, \psi_2$ remain unmodified by the renormalization flow, while the functions $\lambda_\psi$, $h_\Phi$ and $P_\Phi$ are $k$-dependent objects. For $k=\Lambda$ they have the initial values $$\begin{aligned}
\nonumber
\lambda_{\psi,\Lambda}(p,\vec q,\vec q^\prime) &=& 0,\\
\nonumber
h_{\Phi,\Lambda} (p,\vec q,\vec q^\prime) &=& V(\vec q-\vec q^\prime),\\
P_{\Phi,\Lambda} (p,\vec q,\vec q^\prime) &=& Q_\Lambda(\vec q,\vec q^\prime) = \infty\; \delta^{(3)}(\vec q-\vec q^\prime).\end{aligned}$$ Projecting the flow equation onto the truncation in Eq.  we find for $\lambda_\psi=0$ the flow equations (again in matrix notation and suppressing the argument $p$ at several places) $$\begin{aligned}
\nonumber
\partial_k \lambda_\psi &=& \tilde \partial_k (V A_k^{-1} V) + h_\Phi (\partial_k Q^{-1}) h_\Phi,\\
\nonumber
\partial_k h_\Phi &=& \tilde \partial_k (h_\Phi A_k^{-1} V) - P_\Phi (\partial_k Q^{-1}) h_\Phi\\
\nonumber
&&- (\partial_k M_k^\dagger) h_\Phi,\\
\nonumber
\partial_k P_\Phi &=& -\tilde \partial_k (h_\Phi A_k^{-1} h_\Phi) - P_\Phi (\partial_k Q^{-1}) P_\Phi\\
&& - (\partial_k M_k^\dagger) P_\Phi - P_\Phi (\partial_k M_k).
\label{eq:floweq234}\end{aligned}$$ As discussed in ref. [@FW09] we can now use our freedom in the choice of $\partial_k Q^{-1}$ to enforce $\partial_k \lambda_\psi=0$, i.e. $$\lambda_\psi(p,\vec q, \vec q^\prime)=0 \quad \text{for all } k.$$ This fixes $\partial_k Q^{-1}=-h_\Phi^{-1} V (\partial_k A_k^{-1}) V h_\Phi^{-1}$. In addition, the $k$-dependent field rescaling 