k}\cdot {\bm p}) \bigg\{ \frac{\lambda}{4\pi} f_1 I^{(0)}({\bm k},s) \nonumber \\
& \qquad\qquad\qquad + \frac{15 \lambda}{8\pi} \Delta A(k\epsilon) (f_2 - f_1) I^{(2)}({\bm k},s)
\bigg\} \Bigg],
\label{I2sol}\end{aligned}$$ where $$\begin{aligned}
f_n = 2\pi \int_{-1}^{1} dx \frac{x^{2n}}{s+\lambda + i v_s k x}.\end{aligned}$$

Having found the explicit expression for $\hat{h}({\bm k}, {\bm p},s)$, we proceed to calculate the combined correlation function, Eq.. Below, we show that only a small number of terms from Eqs. and - contribute to $C(R,T)$.

$C(R,T)$ in terms of $\hat{h}({\bm k}, {\bm p},s)$ {#subsection:hhat}
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In what follows, it will be convenient to re-write $C(R,T)$ in terms of the Fourier and Laplace transforms of all quantities. Substituting Eq. into Eq., and using the Fourier representation of the regularised dipolar field, Eq., we obtain $$\begin{aligned}
&C(R,T) = \lim_{t\rightarrow\infty} {\mathcal L}^{-1}_{s_1,t} {\mathcal L}^{-1}_{s_2,t+T} \frac{\kappa^2}{(2\pi)^3 V} \nonumber \\
&\qquad \times \int d{\bm k} e^{-i {\bm k}\cdot{\bm R}} \frac{A^2(k\epsilon)}{k^2} 
\int d{\bm p}_1 d{\bm p}_2 
(\hat{\bm k} \cdot {\bm p}_1) (\hat{\bm k} \cdot {\bm p}_2) \nonumber \\
& \qquad\qquad\qquad \times \big( \delta^{\alpha\beta} - \hat{k}^\alpha \hat{k}^\beta \big) p_1^{\beta} 
\big( \delta^{\alpha\gamma} - \hat{k}^\alpha \hat{k}^\gamma \big) p_2^{\gamma} \nonumber \\
&  \qquad\qquad\qquad \times \langle \hat{h}({\bm k}, {\bm p}_1,s_1)\hat{h}(-{\bm k}, {\bm p}_2,s_2) \rangle_{\hat{\xi}},
\label{CRTlaplace}\end{aligned}$$ where ${\mathcal L}^{-1}_{s,t}$ formally denotes the inverse 