fluctuations of the amplitude and the phase of the order parameter. Therefore, a natural question in the context of density-wave instabilities is whether an analogous mechanism of cancellation of longitudinal and transverse fluctuation corrections ensures the validity of the respective mean-field theories – or whether quantum fluctuations become sizable such that the mean-field description is not justified. This question will be addressed in the remainder of this paper.

For density-wave instabilities, the order parameter governing the effective action is a real $N$-component vector, $\boldsymbol{\Phi}(\boldsymbol{r},\tau)\in\mathds{R}^N$, where $\tau$ refers to the imaginary time in Matsubara formalism. This order parameter can be split into the static and homogeneous mean-field value and spatial and temporal fluctuations around this mean-field value as $$\boldsymbol{\Phi}(\boldsymbol{r},\tau)=\Phi_0\hat{\boldsymbol{e}}+\delta\boldsymbol{\Phi}(\boldsymbol{r},\tau){~ .}$$ Here, we introduced $\hat{\boldsymbol{e}}$ as the unit vector along the direction of the mean-field order parameter and the magnitude of the mean-field order parameter $\Phi_0$ can be determined self-consistently from the gap equation. The fluctuations $\delta\boldsymbol{\Phi}$ of the order parameter around the mean-field configuration can be further split into one longitudinal mode $\parallel \hat{\boldsymbol{e}}$ and $N-1$ transverse modes $\perp \hat{\boldsymbol{e}}$. In the remainder of this paper, we restrict ourselves to the leading contribution and evaluate the fluctuation corrections for Gaussian fluctuations, 