I need to evaluate or work out the asymptotic scaling of the following integral:
\begin{equation} I~=~\int_{\mathbb{R}^3} dq d^2p \frac{e^{i\vec{p}\cdot \vec{r}}e^{iq z}}{p^2 + \frac{1}{g^2}q^4} \end{equation}
where $g$ is some parameter, $\vec{p} = (p_x,p_y)$, $r=(x,y)$ and we have explicitly distinguished the $z$-momenta: $q= p_z$. I am interested in the large distance limit. So if there is some cutoff scale $l$ then I want the asymptotic form when $r,z>>l$ and also when $z>>r>>l$.
It's a bit like the calculation to derive the Coulomb interaction between two-point sources in 3+1D Maxwell theory except here, in the 'propagator', the $z$-component of the momenta is $q^4$ rather than $q^2$. I have tried various tricks like the Schwinger method to no avail. I would be grateful for any guidance.
