When an electron and a positron meet, they annihilate each other, converting into two gamma rays. Unless all of them hit in a straight line (very unlikely), they should emit Bremsstrahlung before the collision, but I think this is not observed. Can someone explain why the Bremsstrahlung is missing?
Thinking of the interaction in terms of Bremsstralung is equivalent to treating each particle as acting independently in an external field, but as the particle interact under the field of their partner a better point of view is to consider the system as (excited or even unbound) state of positronium.
That is the dynamics of the system is better understood in terms of the relaxation of excited atomic states. And indeed the atomic spectrum of positronium can be measured.