course, our analysis does not imply that the effect of fluctuations is merely to replace $\lambda$ by $\lambda_\mathrm{eff}$ of Eq. , as there is no guarantee that even higher-order processes will not give rise to corrections of the same order. Moreover, we find that the additional logarithmic contribution to the gap equation stemming from fluctuations originates from the derivative of the anomalous polarization function, and the crucial process is the interaction of quasiparticles through the exchange of fluctuations depicted in Fig. \[fig:triangles\](a).

Interestingly, we find that the impact of longitudinal and transverse modes is quite different: Longitudinal fluctuations always yield $E_\mathrm{fluct}>0$ and are thus detrimental to the formation of long-range order. Transverse fluctuations, on the other hand, only yield $E_\mathrm{fluct}>0$ in the long-wavelength regime. In the opposite regime of transverse fluctuations on lengthscales smaller than the coherence length, we find that the respective fluctuation correction surprisingly lowers the energy. Furthermore, this contribution is the dominant one since it yields the additional logarithm to the gap equation in the case of perfect nesting that ultimately leads to an increase of the gap as compared to its mean-field value. It is due to the twist of the “phase” induced by the excitation 