Why is Thomson scattering the low energy limit of Compton scattering? I understand why the classical Thomson scattering computation should not match the QED one, since a classical field is composed of many photons, while Compton scattering involves a single photon. This paper did the relativistic classical computation, and indeed it does not match.
Therefore, the fact that they both match at low frequencies is somewhat puzzling to me. How can we argue for that?
One possible explanation I can think of is that one cannot exactly determine the number of photons at low energies. If I have a detector of photons, and I send a beam of photons with very low energy, Heisenberg uncertainty principle gives me large time uncertainty which forbids from pinpointing exactly when I detected a photon, therefore I can't say for sure how many photons the beam has. Hence, in the low energy limit it does not matter how many photons I have, one or many, the result should be the same, which is Thomson scattering. However, I tried computing the QED tree-level process with 2 incoming identical photons + 1 incoming electron and 1 outgoing photon + 1 outgoing electron , and this process obviously occurs at $O(e^3)$ so there's no possible way it can match the $O(e^2)$ computation.
 A: The limit you have to take to go from the Klein-Nishina computation to the Thomson case is the non-relativistic limit.
The KN computation "is "quantum" in the sense that you are treating the photon as a point-like object having a well defined momentum and having an hard scattering with an electron with an arbitrarily large momentum transfer.
In the Thomson case the computation is "classical" in the sense that you are considering an incoming EM wave with a given frequency. The Thomson computation is however also "non-relativistic", since the frequency of the outgoing wave is computed using non-relativistic equations of motion. This is basically neglecting any electron recoil and forces the outgoing frequency to be the same as the ingoing frequency. This approximation is good when $\hbar \omega \ll m_e c^2$. This is indeed the limit in which the Klein-Nishina and Thomson computation match.
If you take the formula for the ougtoing frequency of the photon
$$\omega' = \frac{\omega}{1+\frac{\omega}{m}(1-\cos\theta)}$$
you see that if $\omega \ll m$ then $\omega \sim \omega'$. Taking the same limit in the Klein-Nishina formula gives back the Thomson cross section.
Additional complications might arise, and this could be what it is been computed in the paper you linked (not sure though), when the classical amplitude of the electric field is large (for example for strong lasers). In that case you might have additional photons in your scattering process and you obtain a non-linear Compton scattering, but that's a whole other thing and it's unrelated to both Thomson and KN computations.
