Bhabha scattering
$e^+e^- \to e^+e^-$
Elastic scattering of an electron and positron. Two tree-level diagrams contribute: the s-channel (annihilation into a virtual photon) and the t-channel (photon exchange). Their interference produces the characteristic Bhabha angular distribution.
Bhabha scattering is used as a luminosity monitor at electron-positron colliders. The forward peak from the t-channel provides a large, calculable cross section.
Order $\alpha$ diagrams
2 diagrams.
Order $\alpha^{2}$ diagrams
3 diagrams.
From diagrams to cross section
How the individual diagram contributions combine into the physical result.
Sum the two channels
$$i\mathcal{M} = i\mathcal{M}_s + i\mathcal{M}_t$$
The total amplitude is the sum of the s-channel (annihilation) and t-channel (exchange) diagrams. Both must be included — neither is gauge invariant alone.
Square and average
$$\overline{|\mathcal{M}|^2} = \overline{|\mathcal{M}_s|^2} + \overline{|\mathcal{M}_t|^2} + 2\mathrm{Re}\,\overline{\mathcal{M}_s\mathcal{M}_t^*}$$
Three terms: the s-channel squared, the t-channel squared, and the s-t interference. The interference term is physically essential — it determines the angular distribution.
Full Bhabha amplitude (massless limit)
$$\overline{|\mathcal{M}|^2} = 2e^4\left[\frac{t^2+u^2}{s^2} + \frac{s^2+u^2}{t^2} - \frac{2u^2}{st}\right]$$
The interference term $-2u^2/(st)$ is negative — destructive interference between annihilation and exchange.
Differential cross section
$$\frac{d\sigma}{d\Omega} = \frac{\alpha^2}{2s}\left[\frac{t^2+u^2}{s^2} + \frac{s^2+u^2}{t^2} - \frac{2u^2}{st}\right]$$
The forward peak ($\theta \to 0$) is dominated by the t-channel $1/t^2$ pole. At $\theta = 90°$ the s-channel dominates. Bhabha scattering is used as a luminosity monitor at $e^+e^-$ colliders because the forward cross section is large and calculable.