Bhabha scattering: Differential cross section

To leading order, the spin-averaged differential cross section for this process is

$\frac{\mathrm{d} \sigma}{\mathrm{d} (\cos\theta)} = \frac{\pi \alpha^2}{s} \left( u^2 \left( \frac{1}{s} + \frac{1}{t} \right)^2 + \left( \frac{t}{s} \right)^2 + \left( \frac{s}{t} \right)^2 \right) \,$

where s,t, and u are the Mandelstam variables,

α

is the fine-structure constant, and

θ

is the scattering angle.

This cross section is calculated neglecting the electron mass relative to the collision energy and including only the contribution from photon exchange. This is a valid approximation at collision energies small compared to the mass scale of the Z boson, about 91 GeV; for energies not too small compared to this mass, the contribution from Z boson exchange also becomes important.

Mandelstam variables

In this article, the Mandelstam variables are defined by

$s= \,$	$(k+p)^2= \,$	$(k'+p')^2 \approx \,$	$2 k \cdot p \approx\,$	$2 k' \cdot p' \,$
$t= \,$	$(k-k')^2= \,$	$(p-p')^2\approx \,$	$-2 k \cdot k' \approx \,$	$-2 p \cdot p' \,$
$u= \,$	$(k-p')^2= \,$	$(p-k')^2\approx \,$	$-2 k \cdot p' \approx \,$	$-2 k' \cdot p \,$