i.e., taking into account contributions up to $\mathcal{O}[(\delta{\boldsymbol{\Phi}})^2]$.

![Diagrammatic representation of contributions to the gap equation. Straight lines represent the fermionic propagators while wiggly lines stand for the fluctuation propagator. The order parameter (indicated by a dashed line) is added for the sake of clarity here, however, it does not contribute to the derivative ${\mathrm{d}}E/{\mathrm{d}}\Phi_0^2$. The logarithmic contribution to the mean-field gap equation is shown in (a), while the structure of Gaussian fluctuation corrections to the gap equation is presented in (b).[]{data-label="fig:gap-equation"}](gap-equation){width="\columnwidth"}

Including fluctuations around the saddle-point configuration results in additional contributions to the ground-state energy, $E(\Phi_0)=E_\mathrm{MF}(\Phi_0)+E_\mathrm{fluct}(\Phi_0)$, which are also reflected in the gap equation as $$\frac{1}{{\rho_{\mathrm{F}}}L^d}\frac{{\mathrm{d}}E(\Phi_0)}{{\mathrm{d}}\Phi_0^2}=\frac{1}{\lambda}-\ln\Big(\frac{2E_\mathrm{F}}{\Phi_0}\Big)+\frac{1}{{\rho_{\mathrm{F}}}L^d}\frac{{\mathrm{d}}E_\mathrm{fluct}(\Phi_0)}{{\mathrm{d}}\Phi_0^2}=0{~ ,}\label{eq:gap-equation-fluct}$$ where we introduced the dimensionless interaction $\lambda={\rho_{\mathrm{F}}}V$. Unless the additional contribution stemming from fluctuations is smaller than the first two terms already appearing at mean-field level \[cf. Eq. \], the mean-field theory is not justified. The logarithmic contribution already appearing at mean-field level corresponds to the fermionic loop diagram depicted in Fig. \[fig:gap-equation\](a). The Gaussian fluctuation corrections studied here take the form presented in Fig. \[fig:gap-equation\](b). In the remainder, we show that such terms indeed give rise to a logarithmic contribution to the gap equation. The usual contributions due to critical fluctuations, 