How to interpret the Feymann diagram of two photons scattering off each other? I've attached a Feynmann diagram of two photons scattering with the nuclear reaction: 
γ + γ --> γ + γ

May I ask how would one interpret this diagram?
My attempt in interpreting is that the photon in the bottom left interacts with the photon in the top left by some form of virtual particle (shown by the black arrow line, which I can't seem to wrap my head around what virtual particle can exist between two photons scattering; in fact can photons scatter?). However, for some reason the two photons disappear into some virtual particles, and then reappear back in the bottom right and top right corners.
May I also ask is it correct to interpret any particles that begin and end in a basic interaction vertex shown in a Feynmann diagram to be virtual particle (as is the case in my interpretation)?
 A: As far as my understanding of QED it goes like this:
All we can know are the initial and final states: 2 initial and 2 final states of definite momentum.
What happens in between is: everything. However, that is too much, so we do a perturbative expansion of the amplitude. This diagram is the leading order, where the photons scatter off of a virtual electron-position pair (or any other charged particle, but let's stick with $e^+e^-$).
An important feature of Feynman diagrams is that four-momentum is conserved at all vertices: hence the electrons is off-shell:
$$ p^{\mu}p_{\mu} \ne m_e^2 $$
That is, it's a virtual particle.
So what is the four momentum? Based on the earlier statement that "everything" can happen, it can have any four momentum, as long as it's conserved at the vertices. So, you have to integrate over $d^4p$.
Regarding the interpretation the 2 photons are absorbed by a particle, and then later readmitted: that may be misleading.
In the t-channel (scattering), the diagram is not time-ordered.
The exchange particle has a space-like four-momentum and the Feynman diagram represents 2 old-school time-ordered diagrams.
("A" emits a photon that "B" then absorbs and "B" emits a photon that "A" then absorbs).
In the s-channel (annihilation) the diagram represents both cases:
1) The initial state particles annihilate into a virtual particle which then decays into the final state particles
2) the final state particles are emitted while creating a virtual particle which is then is destroyed by absorbing the initial state particles.
So I would assign an order to the operations in a Feynman diagram with some trepidation. The key is to include all crossing symmetries (the u-channel) because the final state photons are identical particles. (I think you have to permute all the indices to compute the leading order amplitude--so if you can time order the emission and absorption, why bother, it's only half, or a quarter, of the story anyway.)
