QED Feynman rules

The QED Feynman rules combine the Dirac propagator, photon propagator, and vertex $-ie{\gamma }^{\mu }$ with external Dirac spinors and photon polarization vectors. In a covariant gauge they generate perturbative electron-photon scattering amplitudes.

Each elementary QED vertex has one photon leg, so counting these legs gives $V=2I_{\gamma }+E_{\gamma }$. An amplitude with two external photons has even $V$. In particular Compton scattering has no cubic-order elementary-vertex contribution. With four external legs and $V=4$, a connected diagram has four internal lines and one loop.

For massless quarks at photon-exchange tree level, the initial-spin-averaged, final-color-summed amplitude is $\overline{|\mathcal{M}{|}^2}=8N_c{e}^4{Q}_q^2[(p_1\cdot q_1)(p_2\cdot q_2)+(p_1\cdot q_2)(p_2\cdot q_1)]/s^2$. In the centre-of-mass frame this is $N_c{e}^4{Q}_q^2(1+{\cos }^2\theta )$. With $N_c=3$ and the standard two-body flux, its integrated cross-section is the displayed result.