Past exam of the particle physics course of the University of Oxford / 2016 / 8 / 1

Collider luminosity is the proportionality factor between a process cross section and its interaction rate: $dN/dt=\mathcal{L}\sigma$ before efficiency corrections. It has units of inverse area per time. Integrated collider luminosity is ${\mathcal{L}}_{\mathrm{int}}=\int \mathcal{L}(t)dt$, with inverse-area units, and gives $N=\sigma {\mathcal{L}}_{\mathrm{int}}$ for a constant cross section and unit efficiency.

At an $e^{+}{e}^{-}$ collider, count small-angle Bhabha scattering in a dedicated forward detector. Its calculable quantum electrodynamics cross section in a precisely defined angular acceptance provides $\mathcal{L}=N/(ϵ\sigma \Delta t)$, after background and radiative corrections. The inner angular boundary is especially important because the rate is sharply forward-peaked.

At a collider with two proton beams, a van der Meer scan calibrates a visible interaction monitor by separating the beams horizontally and vertically while recording the overlap-dependent rate. Known bunch populations and the measured overlap give an absolute collider luminosity and hence the monitor's visible cross section. Its rate then tracks collider luminosity during normal running. Correlations between transverse coordinates require a nonfactorizing scan analysis rather than blindly multiplying two one-dimensional profiles.