It is known that one could recover electrostatic Coulomb potential from QED as a nonrelativistic Born approximation. The 3D case could be found in the standard textbook P&S Intro to QFT. This method indeed could (or should can) be generalized to arbitrary spatial dimension $d$. Theories in different dimension share free propagator in common form $$ G_{00}(q)=\frac{i}{q^2+i\epsilon} $$ due to the fundamental quadratic principle in physics. In the nonrelativistic limit, $$ G_{00}(q)\sim\frac{i}{-|\mathbf q|^2+i\epsilon}+\mathcal O(|\mathbf q|^4). $$ The Coulomb potential is its spaitial Fourier transformation $$ \int\mathrm d^d\mathbf q\ \frac{ie^{i\mathbf q\cdot\mathbf r}}{-|\mathbf q|^2+i\epsilon}=\int\mathrm dq\mathrm d\Omega\ \frac{iq^{d-1}e^{iqr\cos\theta}}{-q^2+i\epsilon}=\frac{1}{r^{d-2}}\int\mathrm du\mathrm d\Omega\ \frac{iu^{d-1}e^{iu\cos\theta}}{-u^2+i\epsilon'} $$ Assuming the integral has been properly regularized, it is a constant independent of $r$ so that Coulomb potential $$V(r)=\frac C{r^{d-2}}.$$
The problem arises in the case $d=2$. In this case $V(r)$ is a constant, which violates the result from applying Gauss's law in 2D $$V(r)=C\ln(\frac rL).$$
So what is wrong here?
----(update)----
Inspired by the post mentioned in @Chiral Anormal's comment, I found the integral could be expressed with help of Bessel functions $$ \int\mathrm dq\mathrm d\theta\frac{qe^{iqr\cos\theta}}{-q^2+i\epsilon}=2\pi\int\mathrm d q\ \frac{qJ_0(qr)}{-q^2+i\epsilon}=-2\pi K_0(-i^{3/2}r\sqrt{\epsilon}) $$
As is commonly done in QFT the divergence was arranged by $\epsilon$ and asymptotically $$ -2\pi K_0(-i^{3/2}r\sqrt{\epsilon})\sim 2\pi \ln(r\sqrt{\epsilon})+\mathcal O(1),\quad \epsilon\rightarrow 0 $$ as expected. The "characteristic length" $L$ in logarithmic plays a role as the regulator was argued by @Luboš Motl in Coulomb potential in 2D. It partially solves my question. However, still, I am troubled about the consistency of the QED propagator in 1+2d spacetime and the gap in my argument, i.e., how the last integral depends on $r$.
