- {\cal R}_k (\partial_k Q^{-1}){\cal R}_k)\\
&&-\frac{1}{2} \Gamma_k^{(1)} (\partial_k Q^{-1}) \Gamma_k^{(1)}+\gamma_k
\label{eq:exactflow}\end{aligned}$$ where $\gamma_k$ is a field independent constant that is irrelevant for most purposes and will be dropped from here on. The flow equation holds for fixed ($k$-independent) field $\Phi$. For our purpose it is useful to perform a $k$-dependent linear transformation on the field $$\begin{aligned}
\nonumber
\hat \Phi(p,\vec q) &=& \int_{\vec q^\prime} \left(e^{M_k(p)}\right)(\vec q,\vec q^\prime) \Phi(p,\vec q^\prime),\\
\hat \Phi^*(p,\vec q) &=& \int_{\vec q^\prime} \Phi^*(p,\vec q^\prime) \left(e^{M_k^\dagger(p)}\right)(\vec q^\prime,\vec q),\end{aligned}$$ with an exponentiated matrix $e^{M_k(p)}$ with “indices” $\vec q,\vec q^\prime$. This gives the flow equation $$\begin{aligned}
\nonumber
\partial_k \Gamma_k {\big |}_{\hat\Phi} &=& \partial_k \Gamma_k {\big |}_\Phi - \int_{p,\vec q}\frac{\delta \Gamma_k}{\delta \hat \Phi(p,\vec q)} \partial_k \hat \Phi(p,\vec q){\big |}_\Phi\\
\nonumber
&& -\int_{p,\vec q}(\partial_k \hat \Phi^*(p,\vec q)){\big |}_\Phi\frac{\delta \Gamma_k}{\delta \hat \Phi^*(p,\vec q)}\\
\nonumber
&=& \frac{1}{2} \text{STr} (\Gamma_k^{(2)}+\hat {\cal R}_k)^{-1} (\widehat{\partial_k {\cal R}_k} - \hat {\cal R}_k (\widehat{\partial_k Q^{-1}}) \hat {\cal R}_k)\\
\nonumber
&& -\frac{1}{2} \Gamma_k^{(1)} (\widehat{\partial_k Q^{-1}}) \Gamma_k^{(1)} \\
&& -  \frac{\delta\Gamma_k}{\delta \hat \Phi}(\partial_k M_k) \hat \Phi- \hat \Phi^* (\partial_k M_k^\dagger) \frac{\delta\Gamma_k}{\delta \hat \Phi^*} .
\label{eq:floweqwithvariablechange}\end{aligned}$$ In the last equation we used an obvious matrix notation and the abbreviations $$\begin{split}
& \widehat{\partial_k Q^{-1}} = e^{M_k^\dagger} (\partial_k Q^{-1}) e^{M_k}, \quad \hat {\cal R}_k 