Past exam of the particle physics course of the University of Oxford / 2022 / 4 / c / Solution

At high momentum, the fixed spatial measurement uncertainty dominates because ii grows as $p$. At lower, still relativistic momentum, Multiple Coulomb scattering dominates the relative uncertainty, which approaches the approximately constant term in iv. Below the relativistic regime the additional velocity dependence of scattering must also be included.

Three complementary improvements are: increase $B$ and the tracking lever arm $L$ to enlarge the magnetic bend; improve hit-position resolution and alignment to reduce $\sigma (s)$; and reduce passive material and detector-layer thickness to reduce $x$ and hence Multiple Coulomb scattering. For independent contributions one commonly writeswhere $C$ is the scattering term for the specified material distribution and track estimator.