equations [@Randeria-95] take in the case of our fairly general model. In terms of the bosonic Matsubara frequencies $ \omega _{\nu }=2\nu \pi /\beta  $ the quadratic contribution to the effective action has the form $$\label{Sb}
S_{eff}^{(2)}\left[ \Delta ^{*},\Delta \right] =\beta \sum _{\mathbf{q}\nu }\sum _{l,m,l',m'}\Delta ^{*}_{l,m,\mathbf{q}}\left( \omega _{\nu }\right) \Gamma ^{-1}_{l,m,l',m'}\left( \mathbf{q},i\omega _{\nu }\right) \Delta _{l',m',\mathbf{q}}\left( \omega _{\nu }\right)$$ where the sums on $ l $ and $ l' $ extend only over values of the angular momentum quantum number having the same parity (both even or both odd). As usual, we start by writing out the inverse bosonic propagator explicitly. It is$$\begin{aligned}
\Gamma ^{-1}_{l,m,l',m'}\left( \mathbf{q},i\omega _{\nu }\right)  & = & -\frac{L^{3}}{V_{l}}\delta _{l,l'}\delta _{m,m'}-\sum _{\mathbf{k}}i^{l'-l}\phi _{l,m,\mathbf{k}}^{*}\phi _{l',m',\mathbf{k}}\times \nonumber \\
 &  & \hspace {-2cm}\times \frac{1}{\beta }\sum _{n}G_{0}\left( \frac{\mathbf{q}}{2}+\mathbf{k},i\omega _{n}\right) G_{0}\left( \frac{\mathbf{q}}{2}-\mathbf{k},i\omega _{\nu }-i\omega _{n}\right) \label{inv Gamma} \end{aligned}$$ where $ G_{0}\left( \mathbf{k},i\omega _{n}\right) \equiv \left( \varepsilon _{\mathbf{k}}-i\omega _{n}\right) ^{-1} $ is a free-electron Green’s function and $ \omega _{n}=(2n+1)\pi /\beta  $ a fermion Matsubara frequency. This is a slightly more general form of the similar expression found in the literature [@Nozieres-SchmittRink-85; @Zwerger-92; @SaDeMelo-Randeria-Engelbrecht-93; @Zwerger-97; @Quintanilla-Gyorffy-Annett-Wallington-02]. Such fairly general bosonic propagator, like the one obtained in Ref. [@Quintanilla-Gyorffy-Annett-Wallington-02] for the DSM, describes not 