important in the long-wavelength limit and for small frequencies, correspond to the limit where the fermionic triangle becomes structureless, cf. Fig. \[fig:long-wavelength\], but our analysis shows that the internal structure of the triangle diagram is indeed important.

![Fluctuation corrections to the gap equation in the long-wavelength regime ($|{\boldsymbol{q}}|\ll2\xi^{-1}$) and for small frequencies ($\omega\ll2\Phi_0$). The fermionic triangle part of the fluctuation corrections to the gap equation \[also shown in Fig. \[fig:gap-equation\](b)\] becomes structureless. This limit corresponds to the usual contributions known from the consideration of critical fluctuations. Note that in the opposite regime of short-wavelength fluctuations ($|{\boldsymbol{q}}|\gg2\xi^{-1}$), the inner structure of the fermionic triangle becomes crucial, cf. Fig. \[fig:triangles\].[]{data-label="fig:long-wavelength"}](long-wavelength){width="\columnwidth"}

If the Gaussian fluctuation corrections are of the same order as the mean-field contribution to the zero-temperature gap equation, they can be effectively restated as a modification of the prefactor of the logarithmic contribution. Then the solution of Eq.  can be written in the familiar form  by introducing the effective interaction $$\lambda_\mathrm{eff}=\lambda\bigg[1-\frac{1}{{\rho_{\mathrm{F}}}L^{d}\ln\big(\frac{2 E_\mathrm{F}}{\Phi_0}\big)}\frac{{\mathrm{d}}E_\mathrm{fluct}(\Phi_0)}{{\mathrm{d}}\Phi_0^2}\bigg]{~ .}$$ Hence the sign of the fluctuation contribution plays an interesting role as negative fluctuation corrections further enhance the gap as compared to its mean-field value whereas positive fluctuation contributions result in a reduction, i.e., weaken the ordered state.

