feynman-diagramsparticle-physicsphotonsquantum-electrodynamics
I understand the basic concepts of a Feynman diagram, but I don't get what is going on here. I have named the photons ABCD and the fermions 1234 for clarity.
I'm going to assume that the vertical axis is time (although I suspect this doesn't matter).
I can accept that photon A turns into an electron(2)-positron(1) pair, but I don't quite know what happens next. Positron 2 absorbs photon B to become positron 4? And what's going on at the top?
Not sure where and how you got your magnificent misimpression that your diagram vanishes.
Of course it does not: it is one of the 8 tree diagrams 
of hard bremsstrahlung, e.g. from S M Swanson, Phys Rev 154 (1967) 1601, all related to each other by suitable crossings and permutations of external momenta. They yield the standard O(α3) cross section for the process.
The amp and its line-perm brethren are physical indeed. An awful amount of actual experimental physics relies on it, and it has an elegant form, cf. section 3 of Berends & Kleiss. There is no good reason it should vanish.
There are more diagrams that contribute to the LHS of your equation. In particular, there are vertex correction diagrams which correspond to the "cross terms" of the RHS.
Best Answer
It is not a good idea to see a Feynman diagram as some sort of collision process really happening.
The diagram is just a term in the perturbative expansion of a quantum mechanical transition amplitude (in other words, a nice "graphical" way to represent a bunch of integrals).
The only actual observed objects are two incoming photons with a certain energy, and two outgoing photons with different energies. The probability of such a process (given the income and outcome) to happen is given by the amplitude associated to this particular Feynman graph. It is also worth noting that such probability is not completely accurate, since a single diagram is just one term of the infinite formal perturbative expansion giving the transition amplitude.