$.

Let us now focus on the BE limit. For sufficiently strong coupling and low densities the inverse propagator $ \Gamma ^{-1}_{l,m,l',m'}\left( \mathbf{q},i\omega _{\nu }\right)  $ can be expanded to the lowest non-trivial order in the pair’s total momentum $ \hbar \mathbf{q} $ and frequency $ \omega _{\nu } $ [@Nozieres-SchmittRink-85]. Such expansion was carried out in Ref. [@Quintanilla-Gyorffy-Annett-Wallington-02] for the DSM, following closely the procedure of [@Zwerger-92; @Zwerger-97]. The derivation is entirely analogous in the present slightly more general case. In short we find that, after appropriate rescaling of the fields, $$\label{Gamma INV bosons}
\Gamma ^{-1}_{l,m,l',m'}\left( \mathbf{q},i\omega _{\nu }\right) =\left[ -i\omega _{\nu }-\mu ^{b}_{l}\left( \beta ,\mu \right) \right] \delta _{l,l'}\delta _{m,m'}+\sum _{i,j=x,y,z}\frac{\hbar ^{2}\mathbf{q}_{i}\mathbf{q}_{j}}{2m_{l,m,l',m'}^{b,ij}\left( \beta ,\mu \right) }.$$ Thus the action $ S_{eff}^{(2)}\left[ \Delta ^{*},\Delta \right]  $ describes an ideal Bose gas made out of bosons that propagate with some effective mass $ m_{l,m,l',m'}^{b,ij}\left( \beta ,\mu \right)  $ and chemical potential $ \mu ^{b}_{l}\left( \beta ,\mu \right)  $ (these quantities are given by explicit formulae which we shall omit, for brevity). In general the effective masses $ m_{l,m,l',m'}^{b,ij}\left( \beta ,\mu \right)  $ represent anisotropic dispersion relations that are different for PP in distinct internal states. Moreover, their off-diagonal values ($ 