Feynman rules for QED
The complete set of rules for computing amplitudes
External lines
| Particle | Incoming | Outgoing |
|---|---|---|
| Electron | $u^s(p)$ | $\bar{u}^{s'}(p')$ |
| Positron | $\bar{v}^r(q)$ | $v^{r'}(q')$ |
| Photon | $\epsilon^\mu_\lambda(k)$ | $\epsilon^{*\mu}_\lambda(k)$ |
Propagators (internal lines)
| Particle | Propagator |
|---|---|
| Electron/positron | $\dfrac{i(\not{p}+m)}{p^2-m^2+i\epsilon}$ |
| Photon (Feynman gauge) | $\dfrac{-i\eta_{\mu\nu}}{k^2+i\epsilon}$ |
Vertex
QED has exactly one interaction vertex:
$$-ie\gamma^\mu$$
Two fermion lines and one photon line meet at each vertex. The $\gamma^\mu$ index contracts with the photon propagator or polarization vector.
Integration and combinatorics
| Rule | Factor |
|---|---|
| Loop momentum integral | $\displaystyle\int\frac{d^4k}{(2\pi)^4}$ for each undetermined loop momentum |
| Closed fermion loop | $(-1)$ and a trace over spinor indices |
| Momentum conservation | $(2\pi)^4\delta^4(\sum p_{\text{in}} - \sum p_{\text{out}})$ at each vertex |
| Symmetry factor | $1/S$ for diagrams with $S$ identical configurations |
From amplitude to cross section
For unpolarized $2 \to 2$ scattering:
$$d\sigma = \frac{1}{2E_1\,2E_2\,|v_1 - v_2|}\,\overline{|\mathcal{M}|^2}\,d\Phi_2$$
where the spin-averaged squared amplitude is:
$$\overline{|\mathcal{M}|^2} = \frac{1}{(2s_1+1)(2s_2+1)}\sum_{\text{spins}}|\mathcal{M}|^2$$
and the two-body phase space in the center-of-mass frame is:
$$d\Phi_2 = \frac{|\mathbf{p}_f|}{16\pi^2\sqrt{s}}\,d\Omega$$
Useful identities
Contraction of gamma matrices ($d = 4$):
$$\gamma^\mu\gamma_\mu = 4, \quad \gamma^\mu\gamma^\nu\gamma_\mu = -2\gamma^\nu, \quad \gamma^\mu\gamma^\nu\gamma^\rho\gamma_\mu = 4\eta^{\nu\rho}$$
Feynman parametrization:
$$\frac{1}{AB} = \int_0^1 dx\,\frac{1}{[xA+(1-x)B]^2}$$
$$\frac{1}{ABC} = 2\int_0^1 dx\,dy\,dz\,\delta(x+y+z-1)\,\frac{1}{[xA+yB+zC]^3}$$
Standard $d$-dimensional loop integral:
$$\int\frac{d^dk}{(2\pi)^d}\frac{1}{(k^2-\Delta)^n} = \frac{i(-1)^n}{(4\pi)^{d/2}}\frac{\Gamma(n-d/2)}{\Gamma(n)}\frac{1}{\Delta^{n-d/2}}$$