The derivation of Jacobian matrices for the propagation of track parameter uncertainties in the presence of magnetic fields and detector material

Abstract: In high-energy physics experiments, the trajectories of charged particles are reconstructed using track reconstruction algorithms. Such algorithms need to both identify the set of measurements from a single charged particle and to fit the parameters by propagating tracks along the measurements. The propagation of the track parameter uncertainties is an important component in the track fitting to get the optimal precision in the fitted parameters. The error propagation is performed at intersections between the track and local coordinate frames defined on detector components by calculating a Jacobian matrix corresponding to the local-to-local frame transport. This paper derives the Jacobian matrix in a general manner to harmonize with numerical integration methods developed for inhomogeneous magnetic fields and materials. The Jacobian and transported covariance matrices are validated by simulating the propagation of charged particles between two frames and comparing with the results of numerical methods.