Past exam of the advanced quantum field theory for particle physics course of the University of Oxford / 2021 / 2 / b

There are exactly four tree-level Feynman diagrams: one four-photon contact graph and one exchange graph for each of the three partitions $12|34$, $13|24$ and $14|23$ of the external legs. The exchange graphs contain two cubic Feynman vertices. A fermion loop or a closed Faddeev-Popov ghost line has positive loop order, so neither enters here.

Write $p_1+p_2+p_3+p_4=0$ and take physical photon polarization vectors ${\epsilon }_i$ with $p_i\cdot {\epsilon }_i=0$ and $p_i^2=0$. At the cubic Feynman vertex attached to legs $i,j$, the two terms containing $p_i\cdot {\epsilon }_i$ or $p_j\cdot {\epsilon }_j$ vanish. If $q=p_i+p_j$ flows into the internal line, the remaining factor at that end isAt the other end the internal incoming four-momentum is $q$ rather than $-q$, giving $+2\lambda q^{\nu }({\epsilon }_k\cdot {\epsilon }_l)$. Multiplication by the photon propagator givesThe cancellation holds first at generic nonzero channel Mandelstam invariants and then by continuation at degenerate kinematics. Summing the three exchange graphs produces $4i{\lambda }^{2}P$, whereThe quartic Feynman vertex contributes $-4i{\lambda }^{2}P$. ThereforeThis is a cancellation for physical on shell photon polarization vectors. The off shell four-photon correlation function need not vanish. Physical light-by-light scattering first occurs through charged-particle loop Feynman diagrams.