Z=\int D\left[ \Delta ^{*},\Delta \right] \, e^{-S_{eff}\left[ \Delta ^{*},\Delta \right] }.$$ We then expand it to the lowest non-trivial (Gaussian) order: $$\label{Seff expansion}
S_{eff}\left[ \Delta ^{*},\Delta \right] \approx S_{eff}\left[ \Delta ^{*(0)},\Delta ^{(0)}\right] +S^{(2)}_{eff}\left[ \Delta ^{*},\Delta \right] .$$ Here $ \Delta _{l,m,\mathbf{q}}^{*(0)}\left( \tau \right) ,\Delta _{l,m,\mathbf{q}}^{(0)}\left( \tau \right)  $ is the configuration of the pairing fields at the saddle point and the Gaussian contribution, $ S_{eff}^{(2)}\left[ \Delta ^{*},\Delta \right]  $, takes into account fluctuations of the fields about that saddle point.

Exotic pairs in the ground state \[SEC-gs\]
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In the ground state, the pairing fields become “frozen” in their saddle-point configurations [@Randeria-95]. $ S_{eff}^{(2)}\left[ \Delta ^{*},\Delta \right]  $ acquires a trivial form such that $ \int D\left[ \Delta ^{*},\Delta \right] \exp \left\{ -S_{eff}^{(2)}\left[ \Delta ^{*},\Delta \right] \right\} =1 $, Eq. (\[Z with Seff\]) thus becoming $ Z=\exp \left\{ -S_{eff}\left[ \Delta ^{*(0)},\Delta ^{(0)}\right] \right\}  $. As usual we look for a stationary and homogeneous saddle point:$$\begin{aligned}
\Delta _{l,m,\mathbf{q}}^{(0)}\left( \tau \right)  & = & \Delta _{l,m}^{(0)}\delta _{\mathbf{q},0},\\
\Delta _{l,m,\mathbf{q}}^{*(0)}\left( \tau \right)  & = & \Delta _{l,m}^{(0)*}\delta _{\mathbf{q},0},\end{aligned}$$ which yields the BCS ground state [@DeGennes-66]. As is well-known this state can describe, at least variationally, the BCS to Bose crossover at $ T=0 