We just covered Feynman diagrams for the first time today, so please bear with me as a student who is not yet educated well in this subject. One question we did asked: "Draw the Feynman diagram for the interaction between an electron and a positron, producing two photons". The correct answer allegedly is this:
I understand that $\beta^-$, $\beta^+$ and $\gamma$ refer to the electron, positron and photons respectively, but we learnt Feynman diagrams as representative of boson interactions, or exchange particle interactions (is there a difference?), and in this diagram there is an unlabelled line running from the electron to the positron – all the diagrams we covered in class had labels with the relevant boson. I understand that this is the annihilation process of matter-antimatter, but what is the boson that triggers it? I've read around the subject (erratically and informally) and have heard of bosons for the EM force and strong/weak forces, but… an annihilation boson? I've no idea what this unlabelled line is supposed to represent, and I didn't have a chance to ask my teacher today.
If someone could please suggest what the unlabelled line is representing, and whether or not it even necessarily represents anything at all, I would be grateful. Also, are bosons synonymous with exchange particles in this context?
P.S. I am familiar with StackExchange guidelines (but never really use the physics site) and if this question doesn't make the cut as is stands I'll happily edit it.
Best Answer
There’s an error in your diagram which is abetting your confusion: the arrow on the positron line should point into the past, not into the future. This is a way of encoding that the electron and positron are, fundamentally, excitations of the same four-component quantum field. The mathematical transformation $\mathit{CP}$ which changes the particle into the antiparticle winds up being the same as the time-reversal transformation $T$.
Feynman and Wheeler were very excited by the idea that a positron is an electron moving backwards in time; Wheeler proposed, half-seriously, that perhaps there’s only one electron, zipping back and forth from the beginning of the universe to the end.
Every Feynman vertex must conserve electrical charge, lepton number, and various other quantum numbers that you’ll learn about later. So the line between your photon vertices must also be an excitation of the electron field. If it goes off into the future, it’s an electron; if it goes off into the past, it’s a positron; if it goes off in a spacelike direction, you’re in the middle of doing an integral.
“But then, why can’t an electron and a positron annihilate to make a single photon?” you ask. Because once you’re done, the initial state and the final state must also conserve energy and momentum; an electron-positron pair has a zero-momentum rest frame, but a photon doesn’t. Two is the minimum number of photons. Ortho-positronium, which has odd parity, decays to three photons.