yields the crucial contribution to the gap equation discussed in this paper is highlighted in gray.[]{data-label="fig:triangles"}](triangle-diagram-DW){width="\columnwidth"}

We can straightforwardly adopt the formalism developed in the context of superconductivity [@KosMillisLarkin-PRB2004; @FischerEtAl2017] to calculate the leading-order corrections to the zero-temperature mean-field gap equation due to Gaussian fluctuations of the order parameter for density-wave order. We consider the regimes of small and large momenta/frequencies separately, which is possible [@VaksGalitskiiLarkin-JETP1962] since the integrals only depend on the combination $r=\sqrt{\omega^2+(v_\mathrm{F}|{\boldsymbol{q}}|\cos\theta)^2}/(2\Phi_0)$, where $\theta$ denotes the angle between fermionic momentum ${\boldsymbol{k}}$ and bosonic momentum ${\boldsymbol{q}}$. The crucial contribution to the fluctuation corrections, as given by Eq.  and diagrammatically represented in Fig. \[fig:gap-equation\](b), stems from the regime where $r\gg1$ and $v_\mathrm{F}|{\boldsymbol{q}}|>\omega$, whereas long-wavelength fluctuations lead to corrections that are negligible compared to the mean-field terms. This is due to the fact that the fermionic triangle (cf. Fig. \[fig:triangles\]) associated with the derivative of the polarization function becomes structureless in the limit $r\ll1$ and hence fluctuation corrections reduce to the simpler form shown in Fig. \[fig:long-wavelength\].

In the regime of interest, the fluctuation propagator is dominated by the normal part of the polarization function, while its derivative is largely determined by the anomalous part. Therefore, the leading contribution to 