
G^{-1}\equiv G_0^{-1}+\Delta\end{aligned}$$ where $\Delta$ is given by: $$\begin{aligned}
&&\Delta(k,\tilde{\omega};k',\tilde{\omega}) \equiv \nonumber\\
&-&g\left(\begin{array}{cc}
0&\psi_B(k+k',\tilde{\omega}+\tilde{\omega}')\\
\bar{\psi}_B(k+k',\tilde{\omega}+\tilde{\omega}')&0\\
\end{array}\right).\end{aligned}$$ The last term in equation (\[sbeff\]) can be expanded into a sum: $$\begin{aligned}
-\ln \det(G_0G^{-1})
=\sum_{l=1}^{\infty}\frac{\text{Tr}\left[(G_0\Delta)^{2l}\right]}{2l}.
\label{suml}\end{aligned}$$ Below, we will evaluate explicitely the $l=1$ and $l=2$ terms in the sum and, following [@GT; @PS], we will give an order of magnitude estimate for the higher order terms showing that they are negligible in the broad resonance limit.

**$l=1$ term**: $$\begin{aligned}
\frac{1}{2}\text{Tr}\left[(G_0\Delta)^{2}\right]=
\int_{k,\tilde{\omega}}\bar{\psi}_B(k,\tilde{\omega})\psi_B(k,\tilde{\omega})
\Pi(k,\tilde{\omega})\end{aligned}$$ After resonance ($\nu<0$), we have $2|\mu|\simeq |\epsilon_b|>
|\epsilon_{\star}|\gg \epsilon_F$, as discussed in Appendix A. As an order of magnitude $|i\tilde{\omega}|$ gives a contribution of the order of the kinetic energy $\sim|k^2/4m|\sim \epsilon_F$. As a result, we obtain $$\begin{aligned}
\Pi(k,\tilde{\omega})\simeq \Pi(0,0)\simeq \epsilon_b-\nu\end{aligned}$$ where we used $\mu\simeq \epsilon_b/2$ and the bound state equation (\[bse\]). Thus the $l=1$ term just gives rise to an effective chemical potential for the dimers equal to $2\mu -\epsilon_b$.

**$l=2$ term**: $$\begin{aligned}
\frac{1}{4}\text{Tr}\left[(G_0\Delta)^{4}\right]&=& \frac{1}{2}
\int_{k_1,\tilde{\omega}_1} \int_{k_2,\tilde{\omega}_2}
\int_{k_3,\tilde{\omega}_3} \bar{\psi}_{B}(1+2-3)\nonumber \\ &\times&
\bar{\psi}_{B}(3)\psi_{B}(2)\psi_{B}(1)g_B(1,2,3)\end{aligned}$$ in an obvious short hand notation for the associated wave vectors and frequencies, and $$\begin{aligned}
g_B(1,2,3)&\equiv& g^4\int_{k,\tilde{\omega}}\Big[
(i\tilde{\omega}-\xi_k) \nonumber \\ &\times&
(i\tilde{\omega}_2-i\tilde{\ome