which the system becomes unstable with respect to pairing fluctuations describing a homogeneous, static field:$$\label{instability condition}
\beta \sum _{l,m,l',m'}\Delta ^{*}_{l,m,0}\left( 0\right) \Gamma^{-1}_{l,m,l',m'}\left( 0,0\right) \Delta _{l',m',0}\left( 0\right) =0.$$ We obtain $$\label{Tc eq}
1=-\frac{V_{l}}{\left( 2\pi \right) ^{3}}\int _{0}^{\infty }d\left| \mathbf{k}\right| \left| \mathbf{k}\right| ^{2}\left| R_{l}\left( \left| \mathbf{k}\right| \right) \right| ^{2}\frac{1-2f\left( \beta _{c}\varepsilon _{\mathbf{k}}\right) }{2\varepsilon _{\mathbf{k}}}$$ which is diagonal in $ l $ and degenerate in $ m $. In general (\[Tc eq\]) has several solutions $ \beta _{c}\equiv 1/k_{B}T_{c} $, corresponding to the formation of a superconducting state with different values of the angular momentum quantum number $ l=0,2,4,\ldots  $ Evidently the highest $ T_{c} $ corresponds to the true superconducting instability, and it gives the angular momentum quantum number of the Cooper pairs in the superconducting state, just below $ T_{c} $.

As expected [@Randeria-95] Eq. (\[Tc eq\]) has the same form as in the mean-field theory and so the weak-coupling limit the Gaussian theory reduces to it. In particular the argument for exotic pairing in the BCS limit that we recalled above applies also to the critical temperature: thus at intermediate densities we expect pairs to form and condense simultaneously, at $ T_{c} $, with angular momentum quantum number $ l_{max}=2 