For a charged Dirac particle, incoming momenta and outgoing momenta obey . The two QED diagrams have internal fermion momenta and , giving and channels. The amplitude is proportional to , where . It is related to particle-antiparticle two-photon annihilation by crossing symmetry, with appropriate changes to external wavefunctions and conjugations.
Double Compton scattering is , together with its inverse. The forward reaction has one incoming and two outgoing photons, so it changes their number and can source the divergence of the photon number current. Ordinary Compton scattering has one incoming and one outgoing photon and conserves photon number. Forward and inverse rates can balance without either reaction individually conserving it.
Articles by others on the same topic
Compton scattering is a quantum mechanical phenomenon that describes the elastic scattering of X-rays or gamma rays off charged particles, most commonly electrons. This effect is significant because it demonstrates the particle-like behavior of photons, the quantum particles of light. The process occurs when a photon collides with a free or loosely bound electron. During the collision, energy and momentum are conserved, leading to an increase in the wavelength of the scattered photon (which corresponds to a decrease in its energy).
Classic theory predicts that the output frequency must be the same as the input one since the electromagnetic wave makes the electron vibrate with same frequency as itself, which then irradiates further waves.
But the output waves are longer because photons are discrete and energy is proportional to frequency:
The formula is exactly that of two relativistic billiard balls colliding.
Compton Scattering by Compton Scattering (2017)
Source. Experiment with a caesium-137 source.