The diagram itself looks like that things would happen in some specific order, but that is probably just an illusion.
q p k
~~~~~ --------------- ~~~~~~
| | virtual
| | electron
| | p - k
~~~~~ --------------- ~~~~~~
-q -p -k
Classically, in Thomson or Compton scattering, the incoming electromagnetic wave shakes the electron, and the shaking electron simultaneously produces the scattered wave.
In the diagram above, we have photon-photon scattering. Each internal line describes an instance of Compton scattering.
In an earlier blog posting, we noted that "off-shell" electrons can classically exist if they are interacting. A free electron is classically on-shell.
In the diagram, all the electron lines are off-shell. We suggest that all the electrons and positrons are interacting with other particles in the diagram.
A photon on the right side is being born at the same time that the virtual electron is still absorbing the photon on the left side.
The electron and the positron are interacting with the photons, but not with each other? If the mutual interaction of the pair is negligible, then the 1 / r^2 potential which governs pair production / annihilation has to be produced by the incoming photons.
How would the incoming photons interact with a dipole?
They might start to pull the dipole apart, but at the same time, the dipole would transmit an electromagnetic wave because the charges in the dipole would be accelerating.
Rayleigh scattering
If the dipole is much shorter (< 1 / 10) than the incoming wave, we speak about Rayleigh scattering.
Rayleigh scattering is much larger at short wavelengths. That is why the sky appears blue.
The virtual pair in photon-photon scattering can be understood as a tiny dipole which the much longer electromagnetic waves are disturbing.
The cross section of Rayleigh scattering from a droplet is
σ = 2/3 π^5 d^6 / λ^4
* (n^2 - 1)^2 / (n^2 + 2)^2,
where d is the droplet diameter, λ is the wavelength of light, and n is the refractive index of the droplet.
Liang and Czarnecki (2011) quote the classic Euler and Kockel (1935) cross section for low-energy photon-photon scattering:
dσ / dΩ = 139 α^4 / (180 π)^2
* E^6 / m_e^8
* (3 + cos^2 θ)^2,
where α = 1 / 137 and E is the photon energy.
Let us think about a photon coming from, e.g., the left. It meets another photon that comes from the opposite direction. Let us imagine that the other photon is a Rayleigh type "droplet". What is the size of the droplet?
Let us use green light, λ = 532 nm. The cross section for photon-photon scattering is assumed to be 10^-67 m^2, and it has been measured to be smaller than 10^-59 m^2.
The corresponding Rayleigh droplet would be 0.3 * 10^-15 m in diameter. We assume that the coefficient at the end (n^2 ...) has a value ~ 1.
If the wavelength of the photons is λ = 2 * 10^-12 m, then photon-photon scattering has a roughly microbarn cross section, that is, 10^-34 m^2.
The corresponding Rayleigh droplet would have a diameter of 30 * 10^-15 m.
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