Hamiltonian for the situation at hand. At chemical potential $ \mu  $, it is [@Nozieres-SchmittRink-85 Eq. (1) (for example)]$$\begin{aligned}
\hat{H}-\mu \hat{N} & = & \sum _{\mathbf{k},\sigma }\varepsilon _{\mathbf{k}}\hat{c}_{\mathbf{k},\sigma }^{+}\hat{c}_{\mathbf{k},\sigma }+\frac{1}{L^{3}}\sum _{\mathbf{q},\mathbf{k},\mathbf{k}'}V\left( \mathbf{k}-\mathbf{k}'\right) \times \nonumber \\
 &  & \hspace {2cm}\times \hat{c}^{+}_{\mathbf{q}/2+\mathbf{k},\uparrow }\hat{c}^{+}_{\mathbf{q}/2-\mathbf{k},\downarrow }\hat{c}_{\mathbf{q}/2-\mathbf{k}',\downarrow }\hat{c}_{\mathbf{q}/2+\mathbf{k}',\uparrow },\label{our H in k-space} \end{aligned}$$ It is particularly illustrative to re-write it in the following way:$$\begin{aligned}
\hat{H}-\mu \hat{N} & = & \sum _{\mathbf{k},\sigma }\varepsilon _{\mathbf{k}}\hat{c}_{\mathbf{k},\sigma }^{+}\hat{c}_{\mathbf{k},\sigma }+\frac{1}{L^{3}}\sum _{\kappa ,l,m}\sum _{\mathbf{q}}V_{\kappa ,l}\hat{b}^{+}_{\kappa ,l,m,\mathbf{q}}\hat{b}_{\kappa ,l,m,\mathbf{q}}.\label{our H in k-space separated} \end{aligned}$$ Here, $ \hat{c}^{+}_{\mathbf{k},\sigma },\hat{c}_{\mathbf{k},\sigma } $ create and annihilate, respectively, and electron with momentum $ \hbar \mathbf{k} $ and spin $ \sigma =\uparrow ,\downarrow  $. $ \varepsilon _{\mathbf{k}}\equiv \hbar ^{2}\left| \mathbf{k}\right| ^{2}/2m^{*}-\mu  $ is the single-particle dispersion relation ($ m^{*} $ is the effective mass of an electron) and $ L^{3} $ is the (very large) sample volume. The additional operators$$\begin{aligned}
\hat{b}^{+}_{\kappa ,l,m,\mathbf{q}} & \equiv  & \sum _{\mathbf{k}}\phi _{\kappa ,l,m,\mathbf{k}}\hat{c}^{+}_{\mathbf{q}/2+\mathbf{k}\uparrow }\hat{c}^{+}_{\mathbf{q}/2-\mathbf{k}\downarrow },\label{b+ operator} \\
\hat{b}_{\kappa ,l,m,\mathbf{q}} & \equiv  & \sum _{\mathbf{k}}\phi _{\kappa ,l,m,\mathbf{k}}^{*}\hat{c}_{\mathbf{q}/2-\mathbf{k}\downarrow }\hat{c}_{\mathbf{q}/2+\mathbf{k}\uparrow }.\label{b operator} \end{aligned}$$ create and annihilate, respectively, a pair with opposite spins, total momentum **$ \hbar \mathbf{q} $** and internal wave function $ \phi _{\kappa ,l,m,\mathbf{k}}\equiv R_{\kappa ,l}\left( \left| \mathbf{k}\right| \right) Y_{l,m}\left( \hat{\mathbf{k}}\right)  $, where 