can be stated in terms of the polarization matrix $\boldsymbol{\Pi}_q$ as $$\boldsymbol{\mathcal{D}}^{-1}_q=\frac{1}{V}\boldsymbol{\mathds{1}}-\boldsymbol{\Pi}_q{~ .}$$ For density-wave order, transverse and longitudinal fluctuations are not coupled and hence the polarization matrix is diagonal. The Gaussian fluctuation corrections can thus be expressed in terms of longitudinal and transverse contributions using $$E_\mathrm{fluct}(\Phi_0)=\frac{L^d}{2}\int_q\ln\Big[4^N\Big(\frac{1}{V}-\Pi^\perp_q\Big)^{N-1}\Big(\frac{1}{V}-\Pi^\parallel_q\Big)\Big]{~ .}$$ Both the longitudinal and the transverse contribution consist of a normal part $\Pi^\mathrm{n}_q=\frac{1}{2}\int_k(G_{k+q}G_{-k}+G_{-k}G_{k-q})$ which survives the limit $\Phi_0\rightarrow0$, and an anomalous contribution $\Pi^\mathrm{a}_q=-\int_kF_kF_{k+q}$ which vanishes in the high-temperature normal state, $$\begin{aligned}
 \Pi^\perp_q&=\Pi^\mathrm{n}_q-\Pi^\mathrm{a}_q{~ ,}\label{eq:Pi-perp} \\
 \Pi^\parallel_q&=\Pi^\mathrm{n}_q+\Pi^\mathrm{a}_q{~ .}\label{eq:Pi-parallel}\end{aligned}$$ These integrals (see Appendix \[app:polarization-function\] for details) are of the same structure as those arising in the context of superconductivity, and hence we can build on the results obtained by previous studies [@KosMillisLarkin-PRB2004; @FischerEtAl2017] in the remainder.

Results and Discussion {#sec:results}
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![Diagrammatic representation of the terms contributing to the derivative of the polarization function. For the definition of the diagrammatic elements see Fig. \[fig:diagrammatic-elements\]. (a) and (b) Derivatives of the anomalous part, while (c) represents the derivative of the normal contribution. While in the long-wavelength regime, the fermionic triangle part depicted here becomes structureless (cf. Fig. \[fig:long-wavelength\]), it is of particular importance for the contributions stemming from the regime of short wavelengths. The diagram which 