Why don't photons break up into multiple lower energy photons when they scatter?

Question

This question was asked and is similar to what I am asking. Why don't photons split up into multiple lower energy versions of themselves?

However, that question got answered on the basis of considering an isolated photon decaying into low energy photons. I am not asking that.

Consider a photon undergoing Compton scattering. Why would it not produce multiple low energy photons rather than one photon at the Compton-scattered energy and momentum? Or consider a photon scattering off a nucleus. Why can it not be scattered into multiple low energy photons?

An isolated photon has difficulty decaying because of conserved quantum numbers such as angular momentum, linear momentum, and energy. However, when scattering off a charged particle, this is not the case.

Answer 1

In principle it can. If you look at the fundamental process, such as Compton scattering then you see that the incoming photon is first absorbed and then reradiated by the electron. The electron can also radiate the photon multiple times, provided that energy and momentum is conserved, but for each extra radiations, it requires an extra interaction vertex. Each such vertex comes with a suppression factor. Therefore, the probability for the radiation of multiple photons is reduced relative to the radiation of a single photon. The constraints due to the conservation of momentum and energy may reduce the probability for multiple photons even further.