All of QED is built from a single vertex: an electron line, a positron line, and a photon line meeting at a point, with factor -ie gamma^mu. Why can't there be a vertex with two photon lines and no fermion lines?
ABecause photons are electrically neutral and do not couple directly to each other — the photon-photon interaction is absent from the QED Lagrangian because it is not gauge invariant
BBecause two photons would violate energy conservation
CBecause the Dirac equation does not allow it
DBecause photon-photon scattering has never been observed experimentally
The QED Lagrangian is L = psi-bar(i gamma^mu D_mu - m)psi - (1/4)F^2, where D_mu = partial_mu + ieA_mu is the covariant derivative. The interaction term is -e psi-bar gamma^mu psi A_mu, which involves exactly one photon field and two fermion fields. There is no A^3 or A^4 term because the electromagnetic field is abelian (U(1) gauge theory). Photon-photon scattering does occur in QED, but only through a fermion loop (a box diagram with four vertices), making it a higher-order process suppressed by alpha^2. Direct photon self-coupling is a feature of non-abelian gauge theories like QCD.
Question 2 Multiple Choice
Compton scattering (photon + electron -> photon + electron) has two tree-level Feynman diagrams. These are the s-channel (electron absorbs the photon, propagates, then emits the final photon) and the u-channel (crossed diagram). Why is there no t-channel diagram?
AA t-channel diagram would have a photon exchanged between two electrons, but there is only one electron in Compton scattering
BA t-channel diagram would require a photon-photon-electron vertex, which does not exist in QED — the exchanged particle would need to couple to two photons on one side, but the QED vertex always has exactly one photon
CThe t-channel is suppressed by an additional power of alpha
DThe t-channel diagram is included in the s-channel after crossing symmetry
In the t-channel, the exchanged particle would connect an incoming electron-outgoing electron vertex on one side and an incoming photon-outgoing photon vertex on the other. The first vertex exists (-ie gamma^mu), but the second would require a photon-photon coupling, which is absent from QED at tree level. The only tree-level diagrams for Compton scattering are those where the electron propagates between the two photon-electron vertices: one where absorption precedes emission (s-channel) and one with the opposite order (u-channel). Both diagrams must be included and their amplitudes added before squaring.
Question 3 True / False
Electron-positron annihilation into two photons (e+e- -> gamma gamma) is closely related to Compton scattering by crossing symmetry.
TTrue
FFalse
Answer: True
Crossing symmetry relates processes obtained by moving particles between the initial and final states (and replacing particles with antiparticles). Compton scattering (e + gamma -> e + gamma) becomes pair annihilation (e + e+ -> gamma + gamma) when you cross the final-state electron into an initial-state positron and the initial-state photon into a final-state photon. The Feynman diagrams have the same topology, and the amplitudes are related by analytic continuation of the Mandelstam variables (s <-> t or s <-> u). This means you can often compute one process and obtain the others by relabeling momenta.
Question 4 Short Answer
Calculate the tree-level amplitude for electron-muon scattering (e- mu- -> e- mu-) and explain why this is the simplest non-trivial QED process.
Think about your answer, then reveal below.
Model answer: There is only one tree-level diagram: a single virtual photon exchanged in the t-channel. The amplitude is M = (-ie)^2 [u-bar(p3) gamma^mu u(p1)] (-ig_{mu nu}/q^2) [u-bar(p4) gamma^nu u(p2)] = ie^2 [u-bar(p3) gamma^mu u(p1)] [u-bar(p4) gamma_mu u(p2)] / q^2, where q = p1 - p3 is the momentum transfer. This is the simplest non-trivial QED process because the electron and muon are distinguishable, so there is no exchange (u-channel) diagram and no identical-particle complications. The single diagram has two vertices (factor e^2), one photon propagator (factor 1/q^2), and two fermion currents.
This amplitude is the prototype for all QED calculations. The structure — two fermion currents connected by a photon propagator — appears in every electromagnetic scattering process. More complex processes (Bhabha scattering, Moller scattering) add exchange diagrams or additional channels but are built from the same ingredients.