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

The Lorentz force is $d\mathbf{p}/dt=q\mathbf{v}\times \mathbf{B}$. Because it is perpendicular to the velocity, it changes the direction of momentum without changing its magnitude. For a charged particle in a uniform magnetic field, the radius of curvature is $R=p/(|q|B)$. Over the short track, the bending angle is $\varphi \simeq L/R$. Equivalently, integrate the force for a time $L/v$ while neglecting the small rotation in its direction. The magnitude of the acquired momentum component perpendicular to the incident direction isIts direction follows $q\mathbf{v}\times \mathbf{B}$; a signed component can therefore be written $qBL$ after fixing an oriented bending axis. Chord length and arc length agree at this order in $L/R$.

The sagitta is the maximum distance between the circular track and its chord. The perpendicular bisector of a chord of length $L$ gives a right triangle with hypotenuse $R$ and sides $L/2$ and $R-s$. HenceAt fixed electric charge, magnetic field and lever arm, $p=|q|B{L}^2/(8s)$, so differentiation and propagation of uncertainty giveThus a fixed spatial measurement uncertainty produces a relative momentum uncertainty growing linearly with $p$. If momentum is expressed in $\mathrm{GeV}/c$, $B$ in tesla, $L,s$ in metres and $z=q/e$, restore the usual conversion $p=0.2998|z|BR$.

For azimuthally symmetric Rutherford scattering, integrate over the solid angle, $d\Omega =2\pi \sin \theta d\theta$. Substituting $u=\sin (\theta /2)$ gives $\sin \theta d\theta =4udu$, soThe small lower cutoff dominates when ${\theta }_{\max }\gg {\theta }_{\min }$ and ${\theta }_{\min }\ll 1$. Using $\sin ({\theta }_{\min }/2)\simeq {\theta }_{\min }/2$, the leading total cross section isA nonzero cutoff is essential: the unscreened forward cross section otherwise diverges.

Let $\ell =\ln ({\theta }_{\max }/{\theta }_{\min })$. The supplied second moment gives the single-scatter root mean square angleFor independent scatters with uniform azimuth, the small angular-deflection vectors have zero mean. Their cross terms average to zero, so their variances add. After $N$ scatters,This is the root mean square magnitude of the two-dimensional angular deflection. Its projection into one specified plane has a variance smaller by a factor of two, hence a root mean square smaller by $\sqrt{2}$.

For a thickness $x$ with number density ${\rho }_n$ of scattering centres, the mean count is $N={\rho }_{n}x\sigma$. Using the total cross section in i and the variance addition in ii,ThusThe stated assumptions make the prefactor independent of $x$ and approximately independent of $p$. The lower-cutoff factors cancel except inside the slowly varying logarithm. Treating $N$ as a Poisson random variable rather than a fixed count gives the same result after replacing $N$ by its mean.

The magnetic bending angle is $\varphi \simeq |q|BL/p$. For a momentum estimate based on this angle, propagation of uncertainty gives $\sigma (p)/p=\sigma (\varphi )/\varphi$. Comparing the total Multiple Coulomb scattering angle with the bend therefore gives the commonly intended characteristic estimateIts independence of $p$ is the key result for relativistic particles: the scattering angle and magnetic bend both scale as $1/p$.

A precise standard deviation requires specifying how tracks are measured. If $\varphi$ is obtained from independently measured entrance and exit directions and all the material lies between them, the bending-plane projected root mean square is $\mathrm{RMS}(\theta )/\sqrt{2}$. For that estimator the equality isA three-point sagitta fit weights scatter positions differently again. The paper does not specify these measurement details, so it fixes the scaling and characteristic estimate, but not one universal geometric coefficient for all track fits.

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.